diff --git a/code/drasil-printers/lib/Language/Drasil/Markdown/Print.hs b/code/drasil-printers/lib/Language/Drasil/Markdown/Print.hs index 49012ab94e..4284bb394b 100644 --- a/code/drasil-printers/lib/Language/Drasil/Markdown/Print.hs +++ b/code/drasil-printers/lib/Language/Drasil/Markdown/Print.hs @@ -14,7 +14,7 @@ import Language.Drasil.Printing.AST (ItemType(Flat, Nested), ListType(Ordered, Unordered, Definitions, Desc, Simple), Expr, Expr(..), Spec(Quote, EmptyS, Ref, HARDNL, E, (:+:)), Label, LinkType(Internal, Cite2, External), OverSymb(Hat), Fonts(Emph, Bold), - Spacing(Thin), Fence(Abs), Ops(Perc)) + Spacing(Thin), Fence(Abs), Ops(Perc, Mul)) import Language.Drasil.Printing.Citation (BibRef) import Language.Drasil.Printing.LayoutObj (Project(Project), LayoutObj(..), Filename, RefMap, File(File)) @@ -139,7 +139,8 @@ pExpr (Label s) = printMath $ TeX.pExpr (Label s') pExpr (Sub e) = bslash <> unders <> braces (pExpr e) pExpr (Sup e) = hat <> braces (pExpr e) pExpr (Over Hat s) = printMath $ commandD "hat" (pExpr' s) -pExpr (MO Perc) = bslash <> printMath (TeX.pExpr (MO Perc)) +pExpr (MO o) + | o == Perc || o == Mul = bslash <> printMath (TeX.pExpr (MO o)) pExpr (Fenced l r m) = fence TeX.Open l <> pExpr m <> fence TeX.Close r where fence _ Abs = text "\\|" diff --git a/code/drasil-printers/lib/Language/Drasil/Printing/AST.hs b/code/drasil-printers/lib/Language/Drasil/Printing/AST.hs index 7df5be8785..b863d3294d 100644 --- a/code/drasil-printers/lib/Language/Drasil/Printing/AST.hs +++ b/code/drasil-printers/lib/Language/Drasil/Printing/AST.hs @@ -12,7 +12,7 @@ data Ops = IsIn | Integer | Real | Rational | Natural | Boolean | Comma | Prime | Ln | Sin | Cos | Tan | Sec | Csc | Cot | Arcsin | Arccos | Arctan | Not | Dim | Exp | Neg | Cross | Dot | Scale | Eq | NEq | Lt | Gt | LEq | GEq | Impl | Iff | Subt | And | Or | Add | Mul | Summ | Inte | Prod | Point | Perc | LArrow | RArrow | ForAll - | VAdd | VSub | Partial + | VAdd | VSub | Partial deriving Eq -- | Holds the type of "text fencing" ("(), {}, |, ||"). data Fence = Paren | Curly | Norm | Abs diff --git a/code/drasil-printers/lib/Language/Drasil/TeX/Print.hs b/code/drasil-printers/lib/Language/Drasil/TeX/Print.hs index e764e47b46..cb01da7114 100644 --- a/code/drasil-printers/lib/Language/Drasil/TeX/Print.hs +++ b/code/drasil-printers/lib/Language/Drasil/TeX/Print.hs @@ -166,7 +166,7 @@ pOps Subt = pure hyph pOps And = commandD "land" empty pOps Or = commandD "lor" empty pOps Add = pure pls -pOps Mul = pure $ text " " +pOps Mul = pure $ text "\\," pOps Summ = command0 "displaystyle" <> command0 "sum" pOps Prod = command0 "displaystyle" <> command0 "prod" pOps Inte = texSym "int" diff --git a/code/stable/dblpend/SRS/HTML/DblPend_SRS.html b/code/stable/dblpend/SRS/HTML/DblPend_SRS.html index e605baa470..a73c717cc0 100644 --- a/code/stable/dblpend/SRS/HTML/DblPend_SRS.html +++ b/code/stable/dblpend/SRS/HTML/DblPend_SRS.html @@ -761,7 +761,7 @@

Theoretical Models

Equation - \[\symbf{F}=m \symbf{a}\text{(}t\text{)}\] + \[\symbf{F}=m\,\symbf{a}\text{(}t\text{)}\] Description @@ -824,7 +824,7 @@

General Definitions

Equation - \[{v_{\text{x}1}}={w_{1}} {L_{1}} \cos\left({θ_{1}}\right)\] + \[{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\] @@ -868,15 +868,15 @@

We also know the horizontal position that is defined in DD:positionXDD1

- \[{p_{\text{x}1}}={L_{1}} \sin\left({θ_{1}}\right)\] + \[{p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right)\]

Applying this,

- \[{v_{\text{x}1}}=\frac{\,d{L_{1}} \sin\left({θ_{1}}\right)}{\,dt}\] + \[{v_{\text{x}1}}=\frac{\,d{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt}\]

L1 is constant with respect to time, so

- \[{v_{\text{x}1}}={L_{1}} \frac{\,d\sin\left({θ_{1}}\right)}{\,dt}\] + \[{v_{\text{x}1}}={L_{1}}\,\frac{\,d\sin\left({θ_{1}}\right)}{\,dt}\]

Therefore, using the chain rule,

- \[{v_{\text{x}1}}={w_{1}} {L_{1}} \cos\left({θ_{1}}\right)\] + \[{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]
@@ -902,7 +902,7 @@

Equation - \[{v_{\text{y}1}}={w_{1}} {L_{1}} \sin\left({θ_{1}}\right)\] + \[{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\] @@ -946,15 +946,15 @@

We also know the vertical position that is defined in DD:positionYDD1

- \[{p_{\text{y}1}}=-{L_{1}} \cos\left({θ_{1}}\right)\] + \[{p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right)\]

Applying this,

- \[{v_{\text{y}1}}=-\left(\frac{\,d{L_{1}} \cos\left({θ_{1}}\right)}{\,dt}\right)\] + \[{v_{\text{y}1}}=-\left(\frac{\,d{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}\right)\]

L1 is constant with respect to time, so

- \[{v_{\text{y}1}}=-{L_{1}} \frac{\,d\cos\left({θ_{1}}\right)}{\,dt}\] + \[{v_{\text{y}1}}=-{L_{1}}\,\frac{\,d\cos\left({θ_{1}}\right)}{\,dt}\]

Therefore, using the chain rule,

- \[{v_{\text{y}1}}={w_{1}} {L_{1}} \sin\left({θ_{1}}\right)\] + \[{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]

@@ -980,7 +980,7 @@

Equation - \[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right)\] + \[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\] @@ -1027,13 +1027,13 @@

We also know the horizontal position that is defined in DD:positionXDD2

- \[{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right)\] + \[{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)\]

Applying this,

- \[{v_{\text{x}2}}=\frac{\,d{p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right)}{\,dt}\] + \[{v_{\text{x}2}}=\frac{\,d{p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt}\]

L1 is constant with respect to time, so

- \[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right)\] + \[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]

@@ -1059,7 +1059,7 @@

Equation - \[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right)\] + \[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\] @@ -1106,11 +1106,11 @@

We also know the vertical position that is defined in DD:positionYDD2

- \[{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right)\] + \[{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)\]

Applying this,

- \[{v_{\text{y}2}}=-\left(\frac{\,d{p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right)}{\,dt}\right)\] + \[{v_{\text{y}2}}=-\left(\frac{\,d{p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}\right)\]

Therefore, using the chain rule,

- \[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right)\] + \[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]

@@ -1136,7 +1136,7 @@

Equation - \[{a_{\text{x}1}}=-{w_{1}}^{2} {L_{1}} \sin\left({θ_{1}}\right)+{α_{1}} {L_{1}} \cos\left({θ_{1}}\right)\] + \[{a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\] @@ -1183,15 +1183,15 @@

Earlier, we found the horizontal velocity to be

- \[{v_{\text{x}1}}={w_{1}} {L_{1}} \cos\left({θ_{1}}\right)\] + \[{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]

Applying this to our equation for acceleration

- \[{a_{\text{x}1}}=\frac{\,d{w_{1}} {L_{1}} \cos\left({θ_{1}}\right)}{\,dt}\] + \[{a_{\text{x}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}\]

By the product and chain rules, we find

- \[{a_{\text{x}1}}=\frac{\,d{w_{1}}}{\,dt} {L_{1}} \cos\left({θ_{1}}\right)-{w_{1}} {L_{1}} \sin\left({θ_{1}}\right) \frac{\,d{θ_{1}}}{\,dt}\] + \[{a_{\text{x}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\cos\left({θ_{1}}\right)-{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt}\]

Simplifying,

- \[{a_{\text{x}1}}=-{w_{1}}^{2} {L_{1}} \sin\left({θ_{1}}\right)+{α_{1}} {L_{1}} \cos\left({θ_{1}}\right)\] + \[{a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\]

@@ -1217,7 +1217,7 @@

Equation - \[{a_{\text{y}1}}={w_{1}}^{2} {L_{1}} \cos\left({θ_{1}}\right)+{α_{1}} {L_{1}} \sin\left({θ_{1}}\right)\] + \[{a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\] @@ -1264,15 +1264,15 @@

Earlier, we found the vertical velocity to be

- \[{v_{\text{y}1}}={w_{1}} {L_{1}} \sin\left({θ_{1}}\right)\] + \[{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]

Applying this to our equation for acceleration

- \[{a_{\text{y}1}}=\frac{\,d{w_{1}} {L_{1}} \sin\left({θ_{1}}\right)}{\,dt}\] + \[{a_{\text{y}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt}\]

By the product and chain rules, we find

- \[{a_{\text{y}1}}=\frac{\,d{w_{1}}}{\,dt} {L_{1}} \sin\left({θ_{1}}\right)+{w_{1}} {L_{1}} \cos\left({θ_{1}}\right) \frac{\,d{θ_{1}}}{\,dt}\] + \[{a_{\text{y}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt}\]

Simplifying,

- \[{a_{\text{y}1}}={w_{1}}^{2} {L_{1}} \cos\left({θ_{1}}\right)+{α_{1}} {L_{1}} \sin\left({θ_{1}}\right)\] + \[{a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\]

@@ -1298,7 +1298,7 @@

Equation - \[{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2} {L_{2}} \sin\left({θ_{2}}\right)+{α_{2}} {L_{2}} \cos\left({θ_{2}}\right)\] + \[{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\] @@ -1348,13 +1348,13 @@

Earlier, we found the horizontal velocity to be

- \[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right)\] + \[{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]

Applying this to our equation for acceleration

- \[{a_{\text{x}2}}=\frac{\,d{v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right)}{\,dt}\] + \[{a_{\text{x}2}}=\frac{\,d{v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}\]

By the product and chain rules, we find

- \[{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2} {L_{2}} \sin\left({θ_{2}}\right)+{α_{2}} {L_{2}} \cos\left({θ_{2}}\right)\] + \[{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)\]

@@ -1380,7 +1380,7 @@

Equation - \[{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2} {L_{2}} \cos\left({θ_{2}}\right)+{α_{2}} {L_{2}} \sin\left({θ_{2}}\right)\] + \[{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\] @@ -1430,13 +1430,13 @@

Earlier, we found the horizontal velocity to be

- \[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right)\] + \[{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]

Applying this to our equation for acceleration

- \[{a_{\text{y}2}}=\frac{\,d{v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right)}{\,dt}\] + \[{a_{\text{y}2}}=\frac{\,d{v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt}\]

By the product and chain rules, we find

- \[{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2} {L_{2}} \cos\left({θ_{2}}\right)+{α_{2}} {L_{2}} \sin\left({θ_{2}}\right)\] + \[{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)\]

@@ -1458,7 +1458,7 @@

Equation - \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)+{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)+{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)\] @@ -1500,7 +1500,7 @@

Detailed derivation of force on the first object:

- \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)+{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)+{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)\]
@@ -1522,7 +1522,7 @@

Detailed derivation of force on the first object:

Equation - \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{1}} \symbf{g}\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}}\,\cos\left({θ_{1}}\right)-{\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\symbf{g}\] @@ -1570,7 +1570,7 @@

Detailed derivation of force on the first object:

Detailed derivation of force on the first object:

- \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{1}} \symbf{g}\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}}\,\cos\left({θ_{1}}\right)-{\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\symbf{g}\]
@@ -1592,7 +1592,7 @@

Detailed derivation of force on the first object:

Equation - \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)\] @@ -1628,7 +1628,7 @@

Detailed derivation of force on the first object:

Detailed derivation of force on the second object:

- \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}} \sin\left({θ_{2}}\right)\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)\]
@@ -1650,7 +1650,7 @@

Detailed derivation of force on the second object:

Equation - \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{2}} \symbf{g}\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\symbf{g}\] @@ -1692,7 +1692,7 @@

Detailed derivation of force on the second object:

Detailed derivation of force on the second object:

- \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{2}} \symbf{g}\] + \[\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\symbf{g}\]
@@ -1777,7 +1777,7 @@

Data Definitions

Equation - \[{p_{\text{x}1}}={L_{1}} \sin\left({θ_{1}}\right)\] + \[{p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right)\] Description @@ -1840,7 +1840,7 @@

Data Definitions

Equation - \[{p_{\text{y}1}}=-{L_{1}} \cos\left({θ_{1}}\right)\] + \[{p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right)\] Description @@ -1904,7 +1904,7 @@

Data Definitions

Equation - \[{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right)\] + \[{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)\] @@ -1972,7 +1972,7 @@

Data Definitions

Equation - \[{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right)\] + \[{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)\] @@ -2157,7 +2157,7 @@

Instance Models

Equation - \[{α_{1}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{-g \left(2 {m_{1}}+{m_{2}}\right) \sin\left({θ_{1}}\right)-{m_{2}} g \sin\left({θ_{1}}-2 {θ_{2}}\right)-2 \sin\left({θ_{1}}-{θ_{2}}\right) {m_{2}} \left({w_{2}}^{2} {L_{2}}+{w_{1}}^{2} {L_{1}} \cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{1}} \left(2 {m_{1}}+{m_{2}}-{m_{2}} \cos\left(2 {θ_{1}}-2 {θ_{2}}\right)\right)}\] + \[{α_{1}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{-g\,\left(2\,{m_{1}}+{m_{2}}\right)\,\sin\left({θ_{1}}\right)-{m_{2}}\,g\,\sin\left({θ_{1}}-2\,{θ_{2}}\right)-2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,{m_{2}}\,\left({w_{2}}^{2}\,{L_{2}}+{w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{1}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)}\] @@ -2259,7 +2259,7 @@

Instance Models

Equation - \[{α_{2}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{2 \sin\left({θ_{1}}-{θ_{2}}\right) \left({w_{1}}^{2} {L_{1}} \left({m_{1}}+{m_{2}}\right)+g \left({m_{1}}+{m_{2}}\right) \cos\left({θ_{1}}\right)+{w_{2}}^{2} {L_{2}} {m_{2}} \cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{2}} \left(2 {m_{1}}+{m_{2}}-{m_{2}} \cos\left(2 {θ_{1}}-2 {θ_{2}}\right)\right)}\] + \[{α_{2}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,\left({w_{1}}^{2}\,{L_{1}}\,\left({m_{1}}+{m_{2}}\right)+g\,\left({m_{1}}+{m_{2}}\right)\,\cos\left({θ_{1}}\right)+{w_{2}}^{2}\,{L_{2}}\,{m_{2}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{2}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)}\] @@ -2327,25 +2327,25 @@

Detailed derivation of angle of the second rod:

By solving equations GD:xForce2 and GD:yForce2 for T2 sin(θ2) and T2 cos(θ2) and then substituting into equation GD:xForce1 and GD:yForce1 , We can get equations 1 and 2:

- \[{m_{1}} {a_{\text{x}1}}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)-{m_{2}} {a_{\text{x}2}}\] + \[{m_{1}}\,{a_{\text{x}1}}=-{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{x}2}}\]

- \[{m_{1}} {a_{\text{y}1}}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{m_{2}} {a_{\text{y}2}}-{m_{2}} g-{m_{1}} g\] + \[{m_{1}}\,{a_{\text{y}1}}={\symbf{T}_{1}}\,\cos\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{y}2}}-{m_{2}}\,g-{m_{1}}\,g\]

Multiply the equation 1 by cos(θ1) and the equation 2 by sin(θ1) and rearrange to get:

- \[{\symbf{T}_{1}} \sin\left({θ_{1}}\right) \cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right) \left({m_{1}} {a_{\text{x}1}}+{m_{2}} {a_{\text{x}2}}\right)\] + \[{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right)\]

- \[{\symbf{T}_{1}} \sin\left({θ_{1}}\right) \cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right) \left({m_{1}} {a_{\text{y}1}}+{m_{2}} {a_{\text{y}2}}+{m_{2}} g+{m_{1}} g\right)\] + \[{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)\]

This leads to the equation 3

- \[\sin\left({θ_{1}}\right) \left({m_{1}} {a_{\text{y}1}}+{m_{2}} {a_{\text{y}2}}+{m_{2}} g+{m_{1}} g\right)=-\cos\left({θ_{1}}\right) \left({m_{1}} {a_{\text{x}1}}+{m_{2}} {a_{\text{x}2}}\right)\] + \[\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right)\]

Next, multiply equation GD:xForce2 by cos(θ2) and equation GD:yForce2 by sin(θ2) and rearrange to get:

- \[{\symbf{T}_{2}} \sin\left({θ_{2}}\right) \cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right) {m_{2}} {a_{\text{x}2}}\] + \[{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}}\]

- \[{\symbf{T}_{1}} \sin\left({θ_{2}}\right) \cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right) \left({m_{2}} {a_{\text{y}2}}+{m_{2}} g\right)\] + \[{\symbf{T}_{1}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)\]

which leads to equation 4

- \[\sin\left({θ_{2}}\right) \left({m_{2}} {a_{\text{y}2}}+{m_{2}} g\right)=-\cos\left({θ_{2}}\right) {m_{2}} {a_{\text{x}2}}\] + \[\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}}\]

By giving equations GD:accelerationX1 and GD:accelerationX2 and GD:accelerationY1 and GD:accelerationY2 plus additional two equations, 3 and 4, we can get IM:calOfAngle1 and IM:calOfAngle2 via a computer algebra program:

diff --git a/code/stable/dblpend/SRS/Jupyter/DblPend_SRS.ipynb b/code/stable/dblpend/SRS/Jupyter/DblPend_SRS.ipynb index 70e01be4ba..631977b995 100644 --- a/code/stable/dblpend/SRS/Jupyter/DblPend_SRS.ipynb +++ b/code/stable/dblpend/SRS/Jupyter/DblPend_SRS.ipynb @@ -413,7 +413,7 @@ "\n", "Equation\n", "\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}$$\n", + "$$\\symbf{F}=m\\,\\symbf{a}\\text{(}t\\text{)}$$\n", "\n", "\n", "\n", @@ -478,7 +478,7 @@ "\n", "Equation\n", "\n", - "$${v_{\\text{x}1}}={w_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${v_{\\text{x}1}}={w_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "\n", "\n", "\n", @@ -514,16 +514,16 @@ "$$\\symbf{v}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{p}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "We also know the horizontal position that is defined in [DD:positionXDD1](#DD:positionXDD1)\n", - "$${p_{\\text{x}1}}={L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${p_{\\text{x}1}}={L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "\n", "Applying this,\n", - "$${v_{\\text{x}1}}=\\frac{\\,d{L_{1}} \\sin\\left({θ_{1}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{x}1}}=\\frac{\\,d{L_{1}}\\,\\sin\\left({θ_{1}}\\right)}{\\,dt}$$\n", "\n", "$L_1$ is constant with respect to time, so\n", - "$${v_{\\text{x}1}}={L_{1}} \\frac{\\,d\\sin\\left({θ_{1}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{x}1}}={L_{1}}\\,\\frac{\\,d\\sin\\left({θ_{1}}\\right)}{\\,dt}$$\n", "\n", "Therefore, using the chain rule,\n", - "$${v_{\\text{x}1}}={w_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${v_{\\text{x}1}}={w_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "
\n", "\n", "\n", @@ -550,7 +550,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -586,16 +586,16 @@ "$$\\symbf{v}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{p}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "We also know the vertical position that is defined in [DD:positionYDD1](#DD:positionYDD1)\n", - "$${p_{\\text{y}1}}=-{L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${p_{\\text{y}1}}=-{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "\n", "Applying this,\n", - "$${v_{\\text{y}1}}=-\\left(\\frac{\\,d{L_{1}} \\cos\\left({θ_{1}}\\right)}{\\,dt}\\right)$$\n", + "$${v_{\\text{y}1}}=-\\left(\\frac{\\,d{L_{1}}\\,\\cos\\left({θ_{1}}\\right)}{\\,dt}\\right)$$\n", "\n", "$L_1$ is constant with respect to time, so\n", - "$${v_{\\text{y}1}}=-{L_{1}} \\frac{\\,d\\cos\\left({θ_{1}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{y}1}}=-{L_{1}}\\,\\frac{\\,d\\cos\\left({θ_{1}}\\right)}{\\,dt}$$\n", "\n", "Therefore, using the chain rule,\n", - "$${v_{\\text{y}1}}={w_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${v_{\\text{y}1}}={w_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${v_{\\text{y}1}}={w_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${v_{\\text{y}1}}={w_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "
\n", @@ -622,7 +622,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -659,13 +659,13 @@ "$$\\symbf{v}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{p}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "We also know the horizontal position that is defined in [DD:positionXDD2](#DD:positionXDD2)\n", - "$${p_{\\text{x}2}}={p_{\\text{x}1}}+{L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${p_{\\text{x}2}}={p_{\\text{x}1}}+{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "\n", "Applying this,\n", - "$${v_{\\text{x}2}}=\\frac{\\,d{p_{\\text{x}1}}+{L_{2}} \\sin\\left({θ_{2}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{x}2}}=\\frac{\\,d{p_{\\text{x}1}}+{L_{2}}\\,\\sin\\left({θ_{2}}\\right)}{\\,dt}$$\n", "\n", "$L_1$ is constant with respect to time, so\n", - "$${v_{\\text{x}2}}={v_{\\text{x}1}}+{w_{2}} {L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${v_{\\text{x}2}}={v_{\\text{x}1}}+{w_{2}}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${v_{\\text{x}2}}={v_{\\text{x}1}}+{w_{2}} {L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${v_{\\text{x}2}}={v_{\\text{x}1}}+{w_{2}}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "
\n", @@ -692,7 +692,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -729,13 +729,13 @@ "$$\\symbf{v}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{p}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "We also know the vertical position that is defined in [DD:positionYDD2](#DD:positionYDD2)\n", - "$${p_{\\text{y}2}}={p_{\\text{y}1}}-{L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${p_{\\text{y}2}}={p_{\\text{y}1}}-{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "\n", "Applying this,\n", - "$${v_{\\text{y}2}}=-\\left(\\frac{\\,d{p_{\\text{y}1}}-{L_{2}} \\cos\\left({θ_{2}}\\right)}{\\,dt}\\right)$$\n", + "$${v_{\\text{y}2}}=-\\left(\\frac{\\,d{p_{\\text{y}1}}-{L_{2}}\\,\\cos\\left({θ_{2}}\\right)}{\\,dt}\\right)$$\n", "\n", "Therefore, using the chain rule,\n", - "$${v_{\\text{y}2}}={v_{\\text{y}1}}+{w_{2}} {L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${v_{\\text{y}2}}={v_{\\text{y}1}}+{w_{2}}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${v_{\\text{y}2}}={v_{\\text{y}1}}+{w_{2}} {L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${v_{\\text{y}2}}={v_{\\text{y}1}}+{w_{2}}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", @@ -762,7 +762,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -799,16 +799,16 @@ "$$\\symbf{a}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Earlier, we found the horizontal velocity to be\n", - "$${v_{\\text{x}1}}={w_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${v_{\\text{x}1}}={w_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "\n", "Applying this to our equation for acceleration\n", - "$${a_{\\text{x}1}}=\\frac{\\,d{w_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right)}{\\,dt}$$\n", + "$${a_{\\text{x}1}}=\\frac{\\,d{w_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)}{\\,dt}$$\n", "\n", "By the product and chain rules, we find\n", - "$${a_{\\text{x}1}}=\\frac{\\,d{w_{1}}}{\\,dt} {L_{1}} \\cos\\left({θ_{1}}\\right)-{w_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right) \\frac{\\,d{θ_{1}}}{\\,dt}$$\n", + "$${a_{\\text{x}1}}=\\frac{\\,d{w_{1}}}{\\,dt}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)-{w_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)\\,\\frac{\\,d{θ_{1}}}{\\,dt}$$\n", "\n", "Simplifying,\n", - "$${a_{\\text{x}1}}=-{w_{1}}^{2} {L_{1}} \\sin\\left({θ_{1}}\\right)+{α_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${a_{\\text{x}1}}=-{w_{1}}^{2}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)+{α_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${a_{\\text{x}1}}=-{w_{1}}^{2} {L_{1}} \\sin\\left({θ_{1}}\\right)+{α_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${a_{\\text{x}1}}=-{w_{1}}^{2}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)+{α_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "
\n", @@ -835,7 +835,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -872,16 +872,16 @@ "$$\\symbf{a}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Earlier, we found the vertical velocity to be\n", - "$${v_{\\text{y}1}}={w_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${v_{\\text{y}1}}={w_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "\n", "Applying this to our equation for acceleration\n", - "$${a_{\\text{y}1}}=\\frac{\\,d{w_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right)}{\\,dt}$$\n", + "$${a_{\\text{y}1}}=\\frac{\\,d{w_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)}{\\,dt}$$\n", "\n", "By the product and chain rules, we find\n", - "$${a_{\\text{y}1}}=\\frac{\\,d{w_{1}}}{\\,dt} {L_{1}} \\sin\\left({θ_{1}}\\right)+{w_{1}} {L_{1}} \\cos\\left({θ_{1}}\\right) \\frac{\\,d{θ_{1}}}{\\,dt}$$\n", + "$${a_{\\text{y}1}}=\\frac{\\,d{w_{1}}}{\\,dt}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)+{w_{1}}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)\\,\\frac{\\,d{θ_{1}}}{\\,dt}$$\n", "\n", "Simplifying,\n", - "$${a_{\\text{y}1}}={w_{1}}^{2} {L_{1}} \\cos\\left({θ_{1}}\\right)+{α_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${a_{\\text{y}1}}={w_{1}}^{2}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)+{α_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${a_{\\text{y}1}}={w_{1}}^{2} {L_{1}} \\cos\\left({θ_{1}}\\right)+{α_{1}} {L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${a_{\\text{y}1}}={w_{1}}^{2}\\,{L_{1}}\\,\\cos\\left({θ_{1}}\\right)+{α_{1}}\\,{L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "
\n", @@ -908,7 +908,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -946,13 +946,13 @@ "$$\\symbf{a}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Earlier, we found the horizontal velocity to be\n", - "$${v_{\\text{x}2}}={v_{\\text{x}1}}+{w_{2}} {L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${v_{\\text{x}2}}={v_{\\text{x}1}}+{w_{2}}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "\n", "Applying this to our equation for acceleration\n", - "$${a_{\\text{x}2}}=\\frac{\\,d{v_{\\text{x}1}}+{w_{2}} {L_{2}} \\cos\\left({θ_{2}}\\right)}{\\,dt}$$\n", + "$${a_{\\text{x}2}}=\\frac{\\,d{v_{\\text{x}1}}+{w_{2}}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)}{\\,dt}$$\n", "\n", "By the product and chain rules, we find\n", - "$${a_{\\text{x}2}}={a_{\\text{x}1}}-{w_{2}}^{2} {L_{2}} \\sin\\left({θ_{2}}\\right)+{α_{2}} {L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${a_{\\text{x}2}}={a_{\\text{x}1}}-{w_{2}}^{2}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)+{α_{2}}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${a_{\\text{x}2}}={a_{\\text{x}1}}-{w_{2}}^{2} {L_{2}} \\sin\\left({θ_{2}}\\right)+{α_{2}} {L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${a_{\\text{x}2}}={a_{\\text{x}1}}-{w_{2}}^{2}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)+{α_{2}}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "
\n", @@ -979,7 +979,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1017,13 +1017,13 @@ "$$\\symbf{a}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Earlier, we found the horizontal velocity to be\n", - "$${v_{\\text{y}2}}={v_{\\text{y}1}}+{w_{2}} {L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${v_{\\text{y}2}}={v_{\\text{y}1}}+{w_{2}}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "\n", "Applying this to our equation for acceleration\n", - "$${a_{\\text{y}2}}=\\frac{\\,d{v_{\\text{y}1}}+{w_{2}} {L_{2}} \\sin\\left({θ_{2}}\\right)}{\\,dt}$$\n", + "$${a_{\\text{y}2}}=\\frac{\\,d{v_{\\text{y}1}}+{w_{2}}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)}{\\,dt}$$\n", "\n", "By the product and chain rules, we find\n", - "$${a_{\\text{y}2}}={a_{\\text{y}1}}+{w_{2}}^{2} {L_{2}} \\cos\\left({θ_{2}}\\right)+{α_{2}} {L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${a_{\\text{y}2}}={a_{\\text{y}1}}+{w_{2}}^{2}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)+{α_{2}}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${a_{\\text{y}2}}={a_{\\text{y}1}}+{w_{2}}^{2} {L_{2}} \\cos\\left({θ_{2}}\\right)+{α_{2}} {L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${a_{\\text{y}2}}={a_{\\text{y}1}}+{w_{2}}^{2}\\,{L_{2}}\\,\\cos\\left({θ_{2}}\\right)+{α_{2}}\\,{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", @@ -1050,7 +1050,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1084,7 +1084,7 @@ "\n", "#### Detailed derivation of force on the first object:\n", "\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{1}} \\sin\\left({θ_{1}}\\right)+{\\symbf{T}_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{1}}\\,\\sin\\left({θ_{1}}\\right)+{\\symbf{T}_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{1}} \\sin\\left({θ_{1}}\\right)+{\\symbf{T}_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{1}}\\,\\sin\\left({θ_{1}}\\right)+{\\symbf{T}_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", @@ -1111,7 +1111,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1147,7 +1147,7 @@ "\n", "#### Detailed derivation of force on the first object:\n", "\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{1}} \\cos\\left({θ_{1}}\\right)-{\\symbf{T}_{2}} \\cos\\left({θ_{2}}\\right)-{m_{1}} \\symbf{g}$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{1}}\\,\\cos\\left({θ_{1}}\\right)-{\\symbf{T}_{2}}\\,\\cos\\left({θ_{2}}\\right)-{m_{1}}\\,\\symbf{g}$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{1}} \\cos\\left({θ_{1}}\\right)-{\\symbf{T}_{2}} \\cos\\left({θ_{2}}\\right)-{m_{1}} \\symbf{g}$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{1}}\\,\\cos\\left({θ_{1}}\\right)-{\\symbf{T}_{2}}\\,\\cos\\left({θ_{2}}\\right)-{m_{1}}\\,\\symbf{g}$$\n", "
\n", @@ -1174,7 +1174,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1206,7 +1206,7 @@ "\n", "#### Detailed derivation of force on the second object:\n", "\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=-{\\symbf{T}_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
\n", @@ -1233,7 +1233,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1267,7 +1267,7 @@ "\n", "#### Detailed derivation of force on the second object:\n", "\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{2}} \\cos\\left({θ_{2}}\\right)-{m_{2}} \\symbf{g}$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{2}}\\,\\cos\\left({θ_{2}}\\right)-{m_{2}}\\,\\symbf{g}$$\n", "\n", "### Data Definitions\n", "\n", @@ -1366,7 +1366,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1435,7 +1435,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1504,7 +1504,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1574,7 +1574,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1783,7 +1783,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1871,7 +1871,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1917,28 +1917,28 @@ "\n", "\n", "By solving equations [GD:xForce2](#GD:xForce2) and [GD:yForce2](#GD:yForce2) for $T_2sin(θ_2)$ and $T_2cos(θ_2)$ and then substituting into equation [GD:xForce1](#GD:xForce1) and [GD:yForce1](#GD:yForce1) , We can get equations 1 and 2:\n", - "$${m_{1}} {a_{\\text{x}1}}=-{\\symbf{T}_{1}} \\sin\\left({θ_{1}}\\right)-{m_{2}} {a_{\\text{x}2}}$$\n", + "$${m_{1}}\\,{a_{\\text{x}1}}=-{\\symbf{T}_{1}}\\,\\sin\\left({θ_{1}}\\right)-{m_{2}}\\,{a_{\\text{x}2}}$$\n", "\n", "\n", - "$${m_{1}} {a_{\\text{y}1}}={\\symbf{T}_{1}} \\cos\\left({θ_{1}}\\right)-{m_{2}} {a_{\\text{y}2}}-{m_{2}} g-{m_{1}} g$$\n", + "$${m_{1}}\\,{a_{\\text{y}1}}={\\symbf{T}_{1}}\\,\\cos\\left({θ_{1}}\\right)-{m_{2}}\\,{a_{\\text{y}2}}-{m_{2}}\\,g-{m_{1}}\\,g$$\n", "\n", "Multiply the equation 1 by $cos(θ_1)$ and the equation 2 by $sin(θ_1)$ and rearrange to get:\n", - "$${\\symbf{T}_{1}} \\sin\\left({θ_{1}}\\right) \\cos\\left({θ_{1}}\\right)=-\\cos\\left({θ_{1}}\\right) \\left({m_{1}} {a_{\\text{x}1}}+{m_{2}} {a_{\\text{x}2}}\\right)$$\n", + "$${\\symbf{T}_{1}}\\,\\sin\\left({θ_{1}}\\right)\\,\\cos\\left({θ_{1}}\\right)=-\\cos\\left({θ_{1}}\\right)\\,\\left({m_{1}}\\,{a_{\\text{x}1}}+{m_{2}}\\,{a_{\\text{x}2}}\\right)$$\n", "\n", "\n", - "$${\\symbf{T}_{1}} \\sin\\left({θ_{1}}\\right) \\cos\\left({θ_{1}}\\right)=\\sin\\left({θ_{1}}\\right) \\left({m_{1}} {a_{\\text{y}1}}+{m_{2}} {a_{\\text{y}2}}+{m_{2}} g+{m_{1}} g\\right)$$\n", + "$${\\symbf{T}_{1}}\\,\\sin\\left({θ_{1}}\\right)\\,\\cos\\left({θ_{1}}\\right)=\\sin\\left({θ_{1}}\\right)\\,\\left({m_{1}}\\,{a_{\\text{y}1}}+{m_{2}}\\,{a_{\\text{y}2}}+{m_{2}}\\,g+{m_{1}}\\,g\\right)$$\n", "\n", "This leads to the equation 3\n", - "$$\\sin\\left({θ_{1}}\\right) \\left({m_{1}} {a_{\\text{y}1}}+{m_{2}} {a_{\\text{y}2}}+{m_{2}} g+{m_{1}} g\\right)=-\\cos\\left({θ_{1}}\\right) \\left({m_{1}} {a_{\\text{x}1}}+{m_{2}} {a_{\\text{x}2}}\\right)$$\n", + "$$\\sin\\left({θ_{1}}\\right)\\,\\left({m_{1}}\\,{a_{\\text{y}1}}+{m_{2}}\\,{a_{\\text{y}2}}+{m_{2}}\\,g+{m_{1}}\\,g\\right)=-\\cos\\left({θ_{1}}\\right)\\,\\left({m_{1}}\\,{a_{\\text{x}1}}+{m_{2}}\\,{a_{\\text{x}2}}\\right)$$\n", "\n", "Next, multiply equation [GD:xForce2](#GD:xForce2) by $cos(θ_2)$ and equation [GD:yForce2](#GD:yForce2) by $sin(θ_2)$ and rearrange to get:\n", - "$${\\symbf{T}_{2}} \\sin\\left({θ_{2}}\\right) \\cos\\left({θ_{2}}\\right)=-\\cos\\left({θ_{2}}\\right) {m_{2}} {a_{\\text{x}2}}$$\n", + "$${\\symbf{T}_{2}}\\,\\sin\\left({θ_{2}}\\right)\\,\\cos\\left({θ_{2}}\\right)=-\\cos\\left({θ_{2}}\\right)\\,{m_{2}}\\,{a_{\\text{x}2}}$$\n", "\n", "\n", - "$${\\symbf{T}_{1}} \\sin\\left({θ_{2}}\\right) \\cos\\left({θ_{2}}\\right)=\\sin\\left({θ_{2}}\\right) \\left({m_{2}} {a_{\\text{y}2}}+{m_{2}} g\\right)$$\n", + "$${\\symbf{T}_{1}}\\,\\sin\\left({θ_{2}}\\right)\\,\\cos\\left({θ_{2}}\\right)=\\sin\\left({θ_{2}}\\right)\\,\\left({m_{2}}\\,{a_{\\text{y}2}}+{m_{2}}\\,g\\right)$$\n", "\n", "which leads to equation 4\n", - "$$\\sin\\left({θ_{2}}\\right) \\left({m_{2}} {a_{\\text{y}2}}+{m_{2}} g\\right)=-\\cos\\left({θ_{2}}\\right) {m_{2}} {a_{\\text{x}2}}$$\n", + "$$\\sin\\left({θ_{2}}\\right)\\,\\left({m_{2}}\\,{a_{\\text{y}2}}+{m_{2}}\\,g\\right)=-\\cos\\left({θ_{2}}\\right)\\,{m_{2}}\\,{a_{\\text{x}2}}$$\n", "\n", "By giving equations [GD:accelerationX1](#GD:accelerationX1) and [GD:accelerationX2](#GD:accelerationX2) and [GD:accelerationY1](#GD:accelerationY1) and [GD:accelerationY2](#GD:accelerationY2) plus additional two equations, 3 and 4, we can get [IM:calOfAngle1](#IM:calOfAngle1) and [IM:calOfAngle2](#IM:calOfAngle2) via a computer algebra program:\n", "\n", diff --git a/code/stable/dblpend/SRS/PDF/DblPend_SRS.tex b/code/stable/dblpend/SRS/PDF/DblPend_SRS.tex index 6d4593e9ff..48dcfea599 100644 --- a/code/stable/dblpend/SRS/PDF/DblPend_SRS.tex +++ b/code/stable/dblpend/SRS/PDF/DblPend_SRS.tex @@ -368,7 +368,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)} + \symbf{F}=m\,\symbf{a}\text{(}t\text{)} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -407,7 +407,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {v_{\text{x}1}}={w_{1}} {L_{1}} \cos\left({θ_{1}}\right) + {v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -435,22 +435,22 @@ \subsubsection{General Definitions} We also know the horizontal position that is defined in \hyperref[DD:positionXDD1]{DD:positionXDD1} \begin{displaymath} -{p_{\text{x}1}}={L_{1}} \sin\left({θ_{1}}\right) +{p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} Applying this, \begin{displaymath} -{v_{\text{x}1}}=\frac{\,d{L_{1}} \sin\left({θ_{1}}\right)}{\,dt} +{v_{\text{x}1}}=\frac{\,d{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt} \end{displaymath} ${L_{1}}$ is constant with respect to time, so \begin{displaymath} -{v_{\text{x}1}}={L_{1}} \frac{\,d\sin\left({θ_{1}}\right)}{\,dt} +{v_{\text{x}1}}={L_{1}}\,\frac{\,d\sin\left({θ_{1}}\right)}{\,dt} \end{displaymath} Therefore, using the chain rule, \begin{displaymath} -{v_{\text{x}1}}={w_{1}} {L_{1}} \cos\left({θ_{1}}\right) +{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} \medskip \noindent @@ -467,7 +467,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {v_{\text{y}1}}={w_{1}} {L_{1}} \sin\left({θ_{1}}\right) + {v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -495,22 +495,22 @@ \subsubsection{General Definitions} We also know the vertical position that is defined in \hyperref[DD:positionYDD1]{DD:positionYDD1} \begin{displaymath} -{p_{\text{y}1}}=-{L_{1}} \cos\left({θ_{1}}\right) +{p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} Applying this, \begin{displaymath} -{v_{\text{y}1}}=-\left(\frac{\,d{L_{1}} \cos\left({θ_{1}}\right)}{\,dt}\right) +{v_{\text{y}1}}=-\left(\frac{\,d{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt}\right) \end{displaymath} ${L_{1}}$ is constant with respect to time, so \begin{displaymath} -{v_{\text{y}1}}=-{L_{1}} \frac{\,d\cos\left({θ_{1}}\right)}{\,dt} +{v_{\text{y}1}}=-{L_{1}}\,\frac{\,d\cos\left({θ_{1}}\right)}{\,dt} \end{displaymath} Therefore, using the chain rule, \begin{displaymath} -{v_{\text{y}1}}={w_{1}} {L_{1}} \sin\left({θ_{1}}\right) +{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} \medskip \noindent @@ -527,7 +527,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right) + {v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -556,17 +556,17 @@ \subsubsection{General Definitions} We also know the horizontal position that is defined in \hyperref[DD:positionXDD2]{DD:positionXDD2} \begin{displaymath} -{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right) +{p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} Applying this, \begin{displaymath} -{v_{\text{x}2}}=\frac{\,d{p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right)}{\,dt} +{v_{\text{x}2}}=\frac{\,d{p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt} \end{displaymath} ${L_{1}}$ is constant with respect to time, so \begin{displaymath} -{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right) +{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} \medskip \noindent @@ -583,7 +583,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right) + {v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -612,17 +612,17 @@ \subsubsection{General Definitions} We also know the vertical position that is defined in \hyperref[DD:positionYDD2]{DD:positionYDD2} \begin{displaymath} -{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right) +{p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} Applying this, \begin{displaymath} -{v_{\text{y}2}}=-\left(\frac{\,d{p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right)}{\,dt}\right) +{v_{\text{y}2}}=-\left(\frac{\,d{p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt}\right) \end{displaymath} Therefore, using the chain rule, \begin{displaymath} -{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right) +{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \medskip \noindent @@ -639,7 +639,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {a_{\text{x}1}}=-{w_{1}}^{2} {L_{1}} \sin\left({θ_{1}}\right)+{α_{1}} {L_{1}} \cos\left({θ_{1}}\right) + {a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -669,22 +669,22 @@ \subsubsection{General Definitions} Earlier, we found the horizontal velocity to be \begin{displaymath} -{v_{\text{x}1}}={w_{1}} {L_{1}} \cos\left({θ_{1}}\right) +{v_{\text{x}1}}={w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} Applying this to our equation for acceleration \begin{displaymath} -{a_{\text{x}1}}=\frac{\,d{w_{1}} {L_{1}} \cos\left({θ_{1}}\right)}{\,dt} +{a_{\text{x}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)}{\,dt} \end{displaymath} By the product and chain rules, we find \begin{displaymath} -{a_{\text{x}1}}=\frac{\,d{w_{1}}}{\,dt} {L_{1}} \cos\left({θ_{1}}\right)-{w_{1}} {L_{1}} \sin\left({θ_{1}}\right) \frac{\,d{θ_{1}}}{\,dt} +{a_{\text{x}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\cos\left({θ_{1}}\right)-{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt} \end{displaymath} Simplifying, \begin{displaymath} -{a_{\text{x}1}}=-{w_{1}}^{2} {L_{1}} \sin\left({θ_{1}}\right)+{α_{1}} {L_{1}} \cos\left({θ_{1}}\right) +{a_{\text{x}1}}=-{w_{1}}^{2}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} \medskip \noindent @@ -701,7 +701,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {a_{\text{y}1}}={w_{1}}^{2} {L_{1}} \cos\left({θ_{1}}\right)+{α_{1}} {L_{1}} \sin\left({θ_{1}}\right) + {a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -731,22 +731,22 @@ \subsubsection{General Definitions} Earlier, we found the vertical velocity to be \begin{displaymath} -{v_{\text{y}1}}={w_{1}} {L_{1}} \sin\left({θ_{1}}\right) +{v_{\text{y}1}}={w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} Applying this to our equation for acceleration \begin{displaymath} -{a_{\text{y}1}}=\frac{\,d{w_{1}} {L_{1}} \sin\left({θ_{1}}\right)}{\,dt} +{a_{\text{y}1}}=\frac{\,d{w_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right)}{\,dt} \end{displaymath} By the product and chain rules, we find \begin{displaymath} -{a_{\text{y}1}}=\frac{\,d{w_{1}}}{\,dt} {L_{1}} \sin\left({θ_{1}}\right)+{w_{1}} {L_{1}} \cos\left({θ_{1}}\right) \frac{\,d{θ_{1}}}{\,dt} +{a_{\text{y}1}}=\frac{\,d{w_{1}}}{\,dt}\,{L_{1}}\,\sin\left({θ_{1}}\right)+{w_{1}}\,{L_{1}}\,\cos\left({θ_{1}}\right)\,\frac{\,d{θ_{1}}}{\,dt} \end{displaymath} Simplifying, \begin{displaymath} -{a_{\text{y}1}}={w_{1}}^{2} {L_{1}} \cos\left({θ_{1}}\right)+{α_{1}} {L_{1}} \sin\left({θ_{1}}\right) +{a_{\text{y}1}}={w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}\right)+{α_{1}}\,{L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} \medskip \noindent @@ -763,7 +763,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2} {L_{2}} \sin\left({θ_{2}}\right)+{α_{2}} {L_{2}} \cos\left({θ_{2}}\right) + {a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -794,17 +794,17 @@ \subsubsection{General Definitions} Earlier, we found the horizontal velocity to be \begin{displaymath} -{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right) +{v_{\text{x}2}}={v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} Applying this to our equation for acceleration \begin{displaymath} -{a_{\text{x}2}}=\frac{\,d{v_{\text{x}1}}+{w_{2}} {L_{2}} \cos\left({θ_{2}}\right)}{\,dt} +{a_{\text{x}2}}=\frac{\,d{v_{\text{x}1}}+{w_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right)}{\,dt} \end{displaymath} By the product and chain rules, we find \begin{displaymath} -{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2} {L_{2}} \sin\left({θ_{2}}\right)+{α_{2}} {L_{2}} \cos\left({θ_{2}}\right) +{a_{\text{x}2}}={a_{\text{x}1}}-{w_{2}}^{2}\,{L_{2}}\,\sin\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} \medskip \noindent @@ -821,7 +821,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2} {L_{2}} \cos\left({θ_{2}}\right)+{α_{2}} {L_{2}} \sin\left({θ_{2}}\right) + {a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -852,17 +852,17 @@ \subsubsection{General Definitions} Earlier, we found the horizontal velocity to be \begin{displaymath} -{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right) +{v_{\text{y}2}}={v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} Applying this to our equation for acceleration \begin{displaymath} -{a_{\text{y}2}}=\frac{\,d{v_{\text{y}1}}+{w_{2}} {L_{2}} \sin\left({θ_{2}}\right)}{\,dt} +{a_{\text{y}2}}=\frac{\,d{v_{\text{y}1}}+{w_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right)}{\,dt} \end{displaymath} By the product and chain rules, we find \begin{displaymath} -{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2} {L_{2}} \cos\left({θ_{2}}\right)+{α_{2}} {L_{2}} \sin\left({θ_{2}}\right) +{a_{\text{y}2}}={a_{\text{y}1}}+{w_{2}}^{2}\,{L_{2}}\,\cos\left({θ_{2}}\right)+{α_{2}}\,{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \medskip \noindent @@ -879,7 +879,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)+{\symbf{T}_{2}} \sin\left({θ_{2}}\right) + \symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)+{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -904,7 +904,7 @@ \subsubsection{General Definitions} \paragraph{Detailed derivation of force on the first object:} \label{GD:xForce1Deriv} \begin{displaymath} -\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)+{\symbf{T}_{2}} \sin\left({θ_{2}}\right) +\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)+{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \medskip \noindent @@ -921,7 +921,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{1}} \symbf{g} + \symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}}\,\cos\left({θ_{1}}\right)-{\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\symbf{g} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -948,7 +948,7 @@ \subsubsection{General Definitions} \paragraph{Detailed derivation of force on the first object:} \label{GD:yForce1Deriv} \begin{displaymath} -\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{1}} \symbf{g} +\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{1}}\,\cos\left({θ_{1}}\right)-{\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{1}}\,\symbf{g} \end{displaymath} \medskip \noindent @@ -965,7 +965,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}} \sin\left({θ_{2}}\right) + \symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -988,7 +988,7 @@ \subsubsection{General Definitions} \paragraph{Detailed derivation of force on the second object:} \label{GD:xForce2Deriv} \begin{displaymath} -\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}} \sin\left({θ_{2}}\right) +\symbf{F}=m \symbf{a}\text{(}t\text{)}=-{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \medskip \noindent @@ -1005,7 +1005,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{2}} \symbf{g} + \symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\symbf{g} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1030,7 +1030,7 @@ \subsubsection{General Definitions} \paragraph{Detailed derivation of force on the second object:} \label{GD:yForce2Deriv} \begin{displaymath} -\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}} \cos\left({θ_{2}}\right)-{m_{2}} \symbf{g} +\symbf{F}=m \symbf{a}\text{(}t\text{)}={\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)-{m_{2}}\,\symbf{g} \end{displaymath} \subsubsection{Data Definitions} \label{Sec:DDs} @@ -1090,7 +1090,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {p_{\text{x}1}}={L_{1}} \sin\left({θ_{1}}\right) + {p_{\text{x}1}}={L_{1}}\,\sin\left({θ_{1}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1131,7 +1131,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {p_{\text{y}1}}=-{L_{1}} \cos\left({θ_{1}}\right) + {p_{\text{y}1}}=-{L_{1}}\,\cos\left({θ_{1}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1172,7 +1172,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}} \sin\left({θ_{2}}\right) + {p_{\text{x}2}}={p_{\text{x}1}}+{L_{2}}\,\sin\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1214,7 +1214,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}} \cos\left({θ_{2}}\right) + {p_{\text{y}2}}={p_{\text{y}1}}-{L_{2}}\,\cos\left({θ_{2}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1345,7 +1345,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - {α_{1}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{-g \left(2 {m_{1}}+{m_{2}}\right) \sin\left({θ_{1}}\right)-{m_{2}} g \sin\left({θ_{1}}-2 {θ_{2}}\right)-2 \sin\left({θ_{1}}-{θ_{2}}\right) {m_{2}} \left({w_{2}}^{2} {L_{2}}+{w_{1}}^{2} {L_{1}} \cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{1}} \left(2 {m_{1}}+{m_{2}}-{m_{2}} \cos\left(2 {θ_{1}}-2 {θ_{2}}\right)\right)} + {α_{1}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{-g\,\left(2\,{m_{1}}+{m_{2}}\right)\,\sin\left({θ_{1}}\right)-{m_{2}}\,g\,\sin\left({θ_{1}}-2\,{θ_{2}}\right)-2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,{m_{2}}\,\left({w_{2}}^{2}\,{L_{2}}+{w_{1}}^{2}\,{L_{1}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{1}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1406,7 +1406,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - {α_{2}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{2 \sin\left({θ_{1}}-{θ_{2}}\right) \left({w_{1}}^{2} {L_{1}} \left({m_{1}}+{m_{2}}\right)+g \left({m_{1}}+{m_{2}}\right) \cos\left({θ_{1}}\right)+{w_{2}}^{2} {L_{2}} {m_{2}} \cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{2}} \left(2 {m_{1}}+{m_{2}}-{m_{2}} \cos\left(2 {θ_{1}}-2 {θ_{2}}\right)\right)} + {α_{2}}\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\right)=\frac{2\,\sin\left({θ_{1}}-{θ_{2}}\right)\,\left({w_{1}}^{2}\,{L_{1}}\,\left({m_{1}}+{m_{2}}\right)+g\,\left({m_{1}}+{m_{2}}\right)\,\cos\left({θ_{1}}\right)+{w_{2}}^{2}\,{L_{2}}\,{m_{2}}\,\cos\left({θ_{1}}-{θ_{2}}\right)\right)}{{L_{2}}\,\left(2\,{m_{1}}+{m_{2}}-{m_{2}}\,\cos\left(2\,{θ_{1}}-2\,{θ_{2}}\right)\right)} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1436,42 +1436,42 @@ \subsubsection{Instance Models} \paragraph{Detailed derivation of angle of the second rod:} \label{IM:calOfAngle2Deriv} -By solving equations \hyperref[GD:xForce2]{GD:xForce2} and \hyperref[GD:yForce2]{GD:yForce2} for ${\symbf{T}_{2}} \sin\left({θ_{2}}\right)$ and ${\symbf{T}_{2}} \cos\left({θ_{2}}\right)$ and then substituting into equation \hyperref[GD:xForce1]{GD:xForce1} and \hyperref[GD:yForce1]{GD:yForce1} , We can get equations 1 and 2: +By solving equations \hyperref[GD:xForce2]{GD:xForce2} and \hyperref[GD:yForce2]{GD:yForce2} for ${\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)$ and ${\symbf{T}_{2}}\,\cos\left({θ_{2}}\right)$ and then substituting into equation \hyperref[GD:xForce1]{GD:xForce1} and \hyperref[GD:yForce1]{GD:yForce1} , We can get equations 1 and 2: \begin{displaymath} -{m_{1}} {a_{\text{x}1}}=-{\symbf{T}_{1}} \sin\left({θ_{1}}\right)-{m_{2}} {a_{\text{x}2}} +{m_{1}}\,{a_{\text{x}1}}=-{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{x}2}} \end{displaymath} \begin{displaymath} -{m_{1}} {a_{\text{y}1}}={\symbf{T}_{1}} \cos\left({θ_{1}}\right)-{m_{2}} {a_{\text{y}2}}-{m_{2}} g-{m_{1}} g +{m_{1}}\,{a_{\text{y}1}}={\symbf{T}_{1}}\,\cos\left({θ_{1}}\right)-{m_{2}}\,{a_{\text{y}2}}-{m_{2}}\,g-{m_{1}}\,g \end{displaymath} Multiply the equation 1 by $\cos\left({θ_{1}}\right)$ and the equation 2 by $\sin\left({θ_{1}}\right)$ and rearrange to get: \begin{displaymath} -{\symbf{T}_{1}} \sin\left({θ_{1}}\right) \cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right) \left({m_{1}} {a_{\text{x}1}}+{m_{2}} {a_{\text{x}2}}\right) +{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right) \end{displaymath} \begin{displaymath} -{\symbf{T}_{1}} \sin\left({θ_{1}}\right) \cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right) \left({m_{1}} {a_{\text{y}1}}+{m_{2}} {a_{\text{y}2}}+{m_{2}} g+{m_{1}} g\right) +{\symbf{T}_{1}}\,\sin\left({θ_{1}}\right)\,\cos\left({θ_{1}}\right)=\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right) \end{displaymath} This leads to the equation 3 \begin{displaymath} -\sin\left({θ_{1}}\right) \left({m_{1}} {a_{\text{y}1}}+{m_{2}} {a_{\text{y}2}}+{m_{2}} g+{m_{1}} g\right)=-\cos\left({θ_{1}}\right) \left({m_{1}} {a_{\text{x}1}}+{m_{2}} {a_{\text{x}2}}\right) +\sin\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{y}1}}+{m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g+{m_{1}}\,g\right)=-\cos\left({θ_{1}}\right)\,\left({m_{1}}\,{a_{\text{x}1}}+{m_{2}}\,{a_{\text{x}2}}\right) \end{displaymath} Next, multiply equation \hyperref[GD:xForce2]{GD:xForce2} by $\cos\left({θ_{2}}\right)$ and equation \hyperref[GD:yForce2]{GD:yForce2} by $\sin\left({θ_{2}}\right)$ and rearrange to get: \begin{displaymath} -{\symbf{T}_{2}} \sin\left({θ_{2}}\right) \cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right) {m_{2}} {a_{\text{x}2}} +{\symbf{T}_{2}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}} \end{displaymath} \begin{displaymath} -{\symbf{T}_{1}} \sin\left({θ_{2}}\right) \cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right) \left({m_{2}} {a_{\text{y}2}}+{m_{2}} g\right) +{\symbf{T}_{1}}\,\sin\left({θ_{2}}\right)\,\cos\left({θ_{2}}\right)=\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right) \end{displaymath} which leads to equation 4 \begin{displaymath} -\sin\left({θ_{2}}\right) \left({m_{2}} {a_{\text{y}2}}+{m_{2}} g\right)=-\cos\left({θ_{2}}\right) {m_{2}} {a_{\text{x}2}} +\sin\left({θ_{2}}\right)\,\left({m_{2}}\,{a_{\text{y}2}}+{m_{2}}\,g\right)=-\cos\left({θ_{2}}\right)\,{m_{2}}\,{a_{\text{x}2}} \end{displaymath} By giving equations \hyperref[GD:accelerationX1]{GD:accelerationX1} and \hyperref[GD:accelerationX2]{GD:accelerationX2} and \hyperref[GD:accelerationY1]{GD:accelerationY1} and \hyperref[GD:accelerationY2]{GD:accelerationY2} plus additional two equations, 3 and 4, we can get \hyperref[IM:calOfAngle1]{IM:calOfAngle1} and \hyperref[IM:calOfAngle2]{IM:calOfAngle2} via a computer algebra program: diff --git a/code/stable/dblpend/SRS/mdBook/src/SecDDs.md b/code/stable/dblpend/SRS/mdBook/src/SecDDs.md index 043e989b16..fd3d0134b1 100644 --- a/code/stable/dblpend/SRS/mdBook/src/SecDDs.md +++ b/code/stable/dblpend/SRS/mdBook/src/SecDDs.md @@ -29,7 +29,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Horizontal position of the first object | |Symbol |\\({p\_{\text{x}1}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{p\_{\text{x}1}}={L\_{1}} \sin\left({θ\_{1}}\right)\\] | +|Equation |\\[{p\_{\text{x}1}}={L\_{1}}\\,\sin\left({θ\_{1}}\right)\\] | |Description|
  • \\({p\_{\text{x}1}}\\) is the horizontal position of the first object (\\({\text{m}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
| |Notes |
  • \\({p\_{\text{x}1}}\\) is the horizontal position
  • \\({p\_{\text{x}1}}\\) is shown in [Fig:dblpend](./SecPhysSyst.md#Figure:dblpend).
| |Source |-- | @@ -46,7 +46,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Vertical position of the first object | |Symbol |\\({p\_{\text{y}1}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{p\_{\text{y}1}}=-{L\_{1}} \cos\left({θ\_{1}}\right)\\] | +|Equation |\\[{p\_{\text{y}1}}=-{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\] | |Description|
  • \\({p\_{\text{y}1}}\\) is the vertical position of the first object (\\({\text{m}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
| |Notes |
  • \\({p\_{\text{y}1}}\\) is the vertical position
  • \\({p\_{\text{y}1}}\\) is shown in [Fig:dblpend](./SecPhysSyst.md#Figure:dblpend).
| |Source |-- | @@ -63,7 +63,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Horizontal position of the second object | |Symbol |\\({p\_{\text{x}2}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{p\_{\text{x}2}}={p\_{\text{x}1}}+{L\_{2}} \sin\left({θ\_{2}}\right)\\] | +|Equation |\\[{p\_{\text{x}2}}={p\_{\text{x}1}}+{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\] | |Description|
  • \\({p\_{\text{x}2}}\\) is the horizontal position of the second object (\\({\text{m}}\\))
  • \\({p\_{\text{x}1}}\\) is the horizontal position of the first object (\\({\text{m}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
| |Notes |
  • \\({p\_{\text{x}2}}\\) is the horizontal position
  • \\({p\_{\text{x}2}}\\) is shown in [Fig:dblpend](./SecPhysSyst.md#Figure:dblpend).
| |Source |-- | @@ -80,7 +80,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Vertical position of the second object | |Symbol |\\({p\_{\text{y}2}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{p\_{\text{y}2}}={p\_{\text{y}1}}-{L\_{2}} \cos\left({θ\_{2}}\right)\\] | +|Equation |\\[{p\_{\text{y}2}}={p\_{\text{y}1}}-{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\] | |Description|
  • \\({p\_{\text{y}2}}\\) is the vertical position of the second object (\\({\text{m}}\\))
  • \\({p\_{\text{y}1}}\\) is the vertical position of the first object (\\({\text{m}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
| |Notes |
  • \\({p\_{\text{y}2}}\\) is the vertical position
  • \\({p\_{\text{y}2}}\\) is shown in [Fig:dblpend](./SecPhysSyst.md#Figure:dblpend).
| |Source |-- | diff --git a/code/stable/dblpend/SRS/mdBook/src/SecGDs.md b/code/stable/dblpend/SRS/mdBook/src/SecGDs.md index f46be1c57e..04e652b3a5 100644 --- a/code/stable/dblpend/SRS/mdBook/src/SecGDs.md +++ b/code/stable/dblpend/SRS/mdBook/src/SecGDs.md @@ -12,7 +12,7 @@ This section collects the laws and equations that will be used to build the inst |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(x\\)-component of velocity of the first object | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{v\_{\text{x}1}}={w\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right)\\] | +|Equation |\\[{v\_{\text{x}1}}={w\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\] | |Description|
  • \\({v\_{\text{x}1}}\\) is the horizontal velocity of the first object (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({w\_{1}}\\) is the angular velocity of the first object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | @@ -25,19 +25,19 @@ At a given point in time, velocity is defined in [DD:positionGDD](./SecDDs.md#DD We also know the horizontal position that is defined in [DD:positionXDD1](./SecDDs.md#DD:positionXDD1) -\\[{p\_{\text{x}1}}={L\_{1}} \sin\left({θ\_{1}}\right)\\] +\\[{p\_{\text{x}1}}={L\_{1}}\\,\sin\left({θ\_{1}}\right)\\] Applying this, -\\[{v\_{\text{x}1}}=\frac{\\,d{L\_{1}} \sin\left({θ\_{1}}\right)}{\\,dt}\\] +\\[{v\_{\text{x}1}}=\frac{\\,d{L\_{1}}\\,\sin\left({θ\_{1}}\right)}{\\,dt}\\] \\({L\_{1}}\\) is constant with respect to time, so -\\[{v\_{\text{x}1}}={L\_{1}} \frac{\\,d\sin\left({θ\_{1}}\right)}{\\,dt}\\] +\\[{v\_{\text{x}1}}={L\_{1}}\\,\frac{\\,d\sin\left({θ\_{1}}\right)}{\\,dt}\\] Therefore, using the chain rule, -\\[{v\_{\text{x}1}}={w\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right)\\] +\\[{v\_{\text{x}1}}={w\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\]
@@ -49,7 +49,7 @@ Therefore, using the chain rule, |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(y\\)-component of velocity of the first object | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{v\_{\text{y}1}}={w\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right)\\] | +|Equation |\\[{v\_{\text{y}1}}={w\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)\\] | |Description|
  • \\({v\_{\text{y}1}}\\) is the vertical velocity of the first object (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({w\_{1}}\\) is the angular velocity of the first object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | @@ -62,19 +62,19 @@ At a given point in time, velocity is defined in [DD:positionGDD](./SecDDs.md#DD We also know the vertical position that is defined in [DD:positionYDD1](./SecDDs.md#DD:positionYDD1) -\\[{p\_{\text{y}1}}=-{L\_{1}} \cos\left({θ\_{1}}\right)\\] +\\[{p\_{\text{y}1}}=-{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\] Applying this, -\\[{v\_{\text{y}1}}=-\left(\frac{\\,d{L\_{1}} \cos\left({θ\_{1}}\right)}{\\,dt}\right)\\] +\\[{v\_{\text{y}1}}=-\left(\frac{\\,d{L\_{1}}\\,\cos\left({θ\_{1}}\right)}{\\,dt}\right)\\] \\({L\_{1}}\\) is constant with respect to time, so -\\[{v\_{\text{y}1}}=-{L\_{1}} \frac{\\,d\cos\left({θ\_{1}}\right)}{\\,dt}\\] +\\[{v\_{\text{y}1}}=-{L\_{1}}\\,\frac{\\,d\cos\left({θ\_{1}}\right)}{\\,dt}\\] Therefore, using the chain rule, -\\[{v\_{\text{y}1}}={w\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right)\\] +\\[{v\_{\text{y}1}}={w\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)\\]
@@ -86,7 +86,7 @@ Therefore, using the chain rule, |:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(x\\)-component of velocity of the second object | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{v\_{\text{x}2}}={v\_{\text{x}1}}+{w\_{2}} {L\_{2}} \cos\left({θ\_{2}}\right)\\] | +|Equation |\\[{v\_{\text{x}2}}={v\_{\text{x}1}}+{w\_{2}}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\] | |Description|
  • \\({v\_{\text{x}2}}\\) is the horizontal velocity of the second object (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({v\_{\text{x}1}}\\) is the horizontal velocity of the first object (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({w\_{2}}\\) is the angular velocity of the second object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | @@ -99,15 +99,15 @@ At a given point in time, velocity is defined in [DD:positionGDD](./SecDDs.md#DD We also know the horizontal position that is defined in [DD:positionXDD2](./SecDDs.md#DD:positionXDD2) -\\[{p\_{\text{x}2}}={p\_{\text{x}1}}+{L\_{2}} \sin\left({θ\_{2}}\right)\\] +\\[{p\_{\text{x}2}}={p\_{\text{x}1}}+{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\] Applying this, -\\[{v\_{\text{x}2}}=\frac{\\,d{p\_{\text{x}1}}+{L\_{2}} \sin\left({θ\_{2}}\right)}{\\,dt}\\] +\\[{v\_{\text{x}2}}=\frac{\\,d{p\_{\text{x}1}}+{L\_{2}}\\,\sin\left({θ\_{2}}\right)}{\\,dt}\\] \\({L\_{1}}\\) is constant with respect to time, so -\\[{v\_{\text{x}2}}={v\_{\text{x}1}}+{w\_{2}} {L\_{2}} \cos\left({θ\_{2}}\right)\\] +\\[{v\_{\text{x}2}}={v\_{\text{x}1}}+{w\_{2}}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\]
@@ -119,7 +119,7 @@ Applying this, |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(y\\)-component of velocity of the second object | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{v\_{\text{y}2}}={v\_{\text{y}1}}+{w\_{2}} {L\_{2}} \sin\left({θ\_{2}}\right)\\] | +|Equation |\\[{v\_{\text{y}2}}={v\_{\text{y}1}}+{w\_{2}}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\] | |Description|
  • \\({v\_{\text{y}2}}\\) is the vertical velocity of the second object (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({v\_{\text{y}1}}\\) is the vertical velocity of the first object (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({w\_{2}}\\) is the angular velocity of the second object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | @@ -132,15 +132,15 @@ At a given point in time, velocity is defined in [DD:positionGDD](./SecDDs.md#DD We also know the vertical position that is defined in [DD:positionYDD2](./SecDDs.md#DD:positionYDD2) -\\[{p\_{\text{y}2}}={p\_{\text{y}1}}-{L\_{2}} \cos\left({θ\_{2}}\right)\\] +\\[{p\_{\text{y}2}}={p\_{\text{y}1}}-{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\] Applying this, -\\[{v\_{\text{y}2}}=-\left(\frac{\\,d{p\_{\text{y}1}}-{L\_{2}} \cos\left({θ\_{2}}\right)}{\\,dt}\right)\\] +\\[{v\_{\text{y}2}}=-\left(\frac{\\,d{p\_{\text{y}1}}-{L\_{2}}\\,\cos\left({θ\_{2}}\right)}{\\,dt}\right)\\] Therefore, using the chain rule, -\\[{v\_{\text{y}2}}={v\_{\text{y}1}}+{w\_{2}} {L\_{2}} \sin\left({θ\_{2}}\right)\\] +\\[{v\_{\text{y}2}}={v\_{\text{y}1}}+{w\_{2}}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\]
@@ -152,7 +152,7 @@ Therefore, using the chain rule, |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(x\\)-component of acceleration of the first object | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[{a\_{\text{x}1}}=-{w\_{1}}^{2} {L\_{1}} \sin\left({θ\_{1}}\right)+{α\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right)\\] | +|Equation |\\[{a\_{\text{x}1}}=-{w\_{1}}^{2}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)+{α\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\] | |Description|
  • \\({a\_{\text{x}1}}\\) is the horizontal acceleration of the first object (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({w\_{1}}\\) is the angular velocity of the first object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
  • \\({α\_{1}}\\) is the angular acceleration of the first object (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | @@ -165,19 +165,19 @@ Our acceleration is: Earlier, we found the horizontal velocity to be -\\[{v\_{\text{x}1}}={w\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right)\\] +\\[{v\_{\text{x}1}}={w\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\] Applying this to our equation for acceleration -\\[{a\_{\text{x}1}}=\frac{\\,d{w\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right)}{\\,dt}\\] +\\[{a\_{\text{x}1}}=\frac{\\,d{w\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)}{\\,dt}\\] By the product and chain rules, we find -\\[{a\_{\text{x}1}}=\frac{\\,d{w\_{1}}}{\\,dt} {L\_{1}} \cos\left({θ\_{1}}\right)-{w\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right) \frac{\\,d{θ\_{1}}}{\\,dt}\\] +\\[{a\_{\text{x}1}}=\frac{\\,d{w\_{1}}}{\\,dt}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)-{w\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)\\,\frac{\\,d{θ\_{1}}}{\\,dt}\\] Simplifying, -\\[{a\_{\text{x}1}}=-{w\_{1}}^{2} {L\_{1}} \sin\left({θ\_{1}}\right)+{α\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right)\\] +\\[{a\_{\text{x}1}}=-{w\_{1}}^{2}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)+{α\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\]
@@ -189,7 +189,7 @@ Simplifying, |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(y\\)-component of acceleration of the first object | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[{a\_{\text{y}1}}={w\_{1}}^{2} {L\_{1}} \cos\left({θ\_{1}}\right)+{α\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right)\\] | +|Equation |\\[{a\_{\text{y}1}}={w\_{1}}^{2}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)+{α\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)\\] | |Description|
  • \\({a\_{\text{y}1}}\\) is the vertical acceleration of the first object (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({w\_{1}}\\) is the angular velocity of the first object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
  • \\({α\_{1}}\\) is the angular acceleration of the first object (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | @@ -202,19 +202,19 @@ Our acceleration is: Earlier, we found the vertical velocity to be -\\[{v\_{\text{y}1}}={w\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right)\\] +\\[{v\_{\text{y}1}}={w\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)\\] Applying this to our equation for acceleration -\\[{a\_{\text{y}1}}=\frac{\\,d{w\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right)}{\\,dt}\\] +\\[{a\_{\text{y}1}}=\frac{\\,d{w\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)}{\\,dt}\\] By the product and chain rules, we find -\\[{a\_{\text{y}1}}=\frac{\\,d{w\_{1}}}{\\,dt} {L\_{1}} \sin\left({θ\_{1}}\right)+{w\_{1}} {L\_{1}} \cos\left({θ\_{1}}\right) \frac{\\,d{θ\_{1}}}{\\,dt}\\] +\\[{a\_{\text{y}1}}=\frac{\\,d{w\_{1}}}{\\,dt}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)+{w\_{1}}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)\\,\frac{\\,d{θ\_{1}}}{\\,dt}\\] Simplifying, -\\[{a\_{\text{y}1}}={w\_{1}}^{2} {L\_{1}} \cos\left({θ\_{1}}\right)+{α\_{1}} {L\_{1}} \sin\left({θ\_{1}}\right)\\] +\\[{a\_{\text{y}1}}={w\_{1}}^{2}\\,{L\_{1}}\\,\cos\left({θ\_{1}}\right)+{α\_{1}}\\,{L\_{1}}\\,\sin\left({θ\_{1}}\right)\\]
@@ -226,7 +226,7 @@ Simplifying, |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(x\\)-component of acceleration of the second object | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[{a\_{\text{x}2}}={a\_{\text{x}1}}-{w\_{2}}^{2} {L\_{2}} \sin\left({θ\_{2}}\right)+{α\_{2}} {L\_{2}} \cos\left({θ\_{2}}\right)\\] | +|Equation |\\[{a\_{\text{x}2}}={a\_{\text{x}1}}-{w\_{2}}^{2}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)+{α\_{2}}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\] | |Description|
  • \\({a\_{\text{x}2}}\\) is the horizontal acceleration of the second object (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({a\_{\text{x}1}}\\) is the horizontal acceleration of the first object (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({w\_{2}}\\) is the angular velocity of the second object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
  • \\({α\_{2}}\\) is the angular acceleration of the second object (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | @@ -239,15 +239,15 @@ Our acceleration is: Earlier, we found the horizontal velocity to be -\\[{v\_{\text{x}2}}={v\_{\text{x}1}}+{w\_{2}} {L\_{2}} \cos\left({θ\_{2}}\right)\\] +\\[{v\_{\text{x}2}}={v\_{\text{x}1}}+{w\_{2}}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\] Applying this to our equation for acceleration -\\[{a\_{\text{x}2}}=\frac{\\,d{v\_{\text{x}1}}+{w\_{2}} {L\_{2}} \cos\left({θ\_{2}}\right)}{\\,dt}\\] +\\[{a\_{\text{x}2}}=\frac{\\,d{v\_{\text{x}1}}+{w\_{2}}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)}{\\,dt}\\] By the product and chain rules, we find -\\[{a\_{\text{x}2}}={a\_{\text{x}1}}-{w\_{2}}^{2} {L\_{2}} \sin\left({θ\_{2}}\right)+{α\_{2}} {L\_{2}} \cos\left({θ\_{2}}\right)\\] +\\[{a\_{\text{x}2}}={a\_{\text{x}1}}-{w\_{2}}^{2}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)+{α\_{2}}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)\\]
@@ -259,7 +259,7 @@ By the product and chain rules, we find |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(y\\)-component of acceleration of the second object | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[{a\_{\text{y}2}}={a\_{\text{y}1}}+{w\_{2}}^{2} {L\_{2}} \cos\left({θ\_{2}}\right)+{α\_{2}} {L\_{2}} \sin\left({θ\_{2}}\right)\\] | +|Equation |\\[{a\_{\text{y}2}}={a\_{\text{y}1}}+{w\_{2}}^{2}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)+{α\_{2}}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\] | |Description|
  • \\({a\_{\text{y}2}}\\) is the vertical acceleration of the second object (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({a\_{\text{y}1}}\\) is the vertical acceleration of the first object (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({w\_{2}}\\) is the angular velocity of the second object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
  • \\({α\_{2}}\\) is the angular acceleration of the second object (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | @@ -272,15 +272,15 @@ Our acceleration is: Earlier, we found the horizontal velocity to be -\\[{v\_{\text{y}2}}={v\_{\text{y}1}}+{w\_{2}} {L\_{2}} \sin\left({θ\_{2}}\right)\\] +\\[{v\_{\text{y}2}}={v\_{\text{y}1}}+{w\_{2}}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\] Applying this to our equation for acceleration -\\[{a\_{\text{y}2}}=\frac{\\,d{v\_{\text{y}1}}+{w\_{2}} {L\_{2}} \sin\left({θ\_{2}}\right)}{\\,dt}\\] +\\[{a\_{\text{y}2}}=\frac{\\,d{v\_{\text{y}1}}+{w\_{2}}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)}{\\,dt}\\] By the product and chain rules, we find -\\[{a\_{\text{y}2}}={a\_{\text{y}1}}+{w\_{2}}^{2} {L\_{2}} \cos\left({θ\_{2}}\right)+{α\_{2}} {L\_{2}} \sin\left({θ\_{2}}\right)\\] +\\[{a\_{\text{y}2}}={a\_{\text{y}1}}+{w\_{2}}^{2}\\,{L\_{2}}\\,\cos\left({θ\_{2}}\right)+{α\_{2}}\\,{L\_{2}}\\,\sin\left({θ\_{2}}\right)\\]
@@ -292,14 +292,14 @@ By the product and chain rules, we find |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Horizontal force on the first object | |Units |\\({\text{N}}\\) | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{1}} \sin\left({θ\_{1}}\right)+{\boldsymbol{T}\_{2}} \sin\left({θ\_{2}}\right)\\] | +|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{1}}\\,\sin\left({θ\_{1}}\right)+{\boldsymbol{T}\_{2}}\\,\sin\left({θ\_{2}}\right)\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({\boldsymbol{T}\_{1}}\\) is the tension of the first object (\\({\text{N}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
  • \\({\boldsymbol{T}\_{2}}\\) is the tension of the second object (\\({\text{N}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | #### Detailed derivation of force on the first object: {#GD:xForce1Deriv} -\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{1}} \sin\left({θ\_{1}}\right)+{\boldsymbol{T}\_{2}} \sin\left({θ\_{2}}\right)\\] +\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{1}}\\,\sin\left({θ\_{1}}\right)+{\boldsymbol{T}\_{2}}\\,\sin\left({θ\_{2}}\right)\\]
@@ -311,14 +311,14 @@ By the product and chain rules, we find |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Vertical force on the first object | |Units |\\({\text{N}}\\) | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{1}} \cos\left({θ\_{1}}\right)-{\boldsymbol{T}\_{2}} \cos\left({θ\_{2}}\right)-{m\_{1}} \boldsymbol{g}\\] | +|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{1}}\\,\cos\left({θ\_{1}}\right)-{\boldsymbol{T}\_{2}}\\,\cos\left({θ\_{2}}\right)-{m\_{1}}\\,\boldsymbol{g}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({\boldsymbol{T}\_{1}}\\) is the tension of the first object (\\({\text{N}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
  • \\({\boldsymbol{T}\_{2}}\\) is the tension of the second object (\\({\text{N}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
  • \\({m\_{1}}\\) is the mass of the first object (\\({\text{kg}}\\))
  • \\(\boldsymbol{g}\\) is the gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | #### Detailed derivation of force on the first object: {#GD:yForce1Deriv} -\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{1}} \cos\left({θ\_{1}}\right)-{\boldsymbol{T}\_{2}} \cos\left({θ\_{2}}\right)-{m\_{1}} \boldsymbol{g}\\] +\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{1}}\\,\cos\left({θ\_{1}}\right)-{\boldsymbol{T}\_{2}}\\,\cos\left({θ\_{2}}\right)-{m\_{1}}\\,\boldsymbol{g}\\]
@@ -330,14 +330,14 @@ By the product and chain rules, we find |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Horizontal force on the second object | |Units |\\({\text{N}}\\) | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{2}} \sin\left({θ\_{2}}\right)\\] | +|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{2}}\\,\sin\left({θ\_{2}}\right)\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({\boldsymbol{T}\_{2}}\\) is the tension of the second object (\\({\text{N}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | #### Detailed derivation of force on the second object: {#GD:xForce2Deriv} -\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{2}} \sin\left({θ\_{2}}\right)\\] +\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=-{\boldsymbol{T}\_{2}}\\,\sin\left({θ\_{2}}\right)\\]
@@ -349,11 +349,11 @@ By the product and chain rules, we find |:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Vertical force on the second object | |Units |\\({\text{N}}\\) | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{2}} \cos\left({θ\_{2}}\right)-{m\_{2}} \boldsymbol{g}\\] | +|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{2}}\\,\cos\left({θ\_{2}}\right)-{m\_{2}}\\,\boldsymbol{g}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({\boldsymbol{T}\_{2}}\\) is the tension of the second object (\\({\text{N}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
  • \\({m\_{2}}\\) is the mass of the second object (\\({\text{kg}}\\))
  • \\(\boldsymbol{g}\\) is the gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) | #### Detailed derivation of force on the second object: {#GD:yForce2Deriv} -\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{2}} \cos\left({θ\_{2}}\right)-{m\_{2}} \boldsymbol{g}\\] +\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}={\boldsymbol{T}\_{2}}\\,\cos\left({θ\_{2}}\right)-{m\_{2}}\\,\boldsymbol{g}\\] diff --git a/code/stable/dblpend/SRS/mdBook/src/SecIMs.md b/code/stable/dblpend/SRS/mdBook/src/SecIMs.md index 37638fc9eb..7bb357e157 100644 --- a/code/stable/dblpend/SRS/mdBook/src/SecIMs.md +++ b/code/stable/dblpend/SRS/mdBook/src/SecIMs.md @@ -15,7 +15,7 @@ This section transforms the problem defined in the [problem description](./SecPr |Output |\\({θ\_{1}}\\) | |Input Constraints |\\[{L\_{1}}\gt{}0\\]\\[{L\_{2}}\gt{}0\\]\\[{m\_{1}}\gt{}0\\]\\[{m\_{2}}\gt{}0\\] | |Output Constraints| | -|Equation |\\[{α\_{1}}\left({θ\_{1}},{θ\_{2}},{w\_{1}},{w\_{2}}\right)=\frac{-g \left(2 {m\_{1}}+{m\_{2}}\right) \sin\left({θ\_{1}}\right)-{m\_{2}} g \sin\left({θ\_{1}}-2 {θ\_{2}}\right)-2 \sin\left({θ\_{1}}-{θ\_{2}}\right) {m\_{2}} \left({w\_{2}}^{2} {L\_{2}}+{w\_{1}}^{2} {L\_{1}} \cos\left({θ\_{1}}-{θ\_{2}}\right)\right)}{{L\_{1}} \left(2 {m\_{1}}+{m\_{2}}-{m\_{2}} \cos\left(2 {θ\_{1}}-2 {θ\_{2}}\right)\right)}\\] | +|Equation |\\[{α\_{1}}\left({θ\_{1}},{θ\_{2}},{w\_{1}},{w\_{2}}\right)=\frac{-g\\,\left(2\\,{m\_{1}}+{m\_{2}}\right)\\,\sin\left({θ\_{1}}\right)-{m\_{2}}\\,g\\,\sin\left({θ\_{1}}-2\\,{θ\_{2}}\right)-2\\,\sin\left({θ\_{1}}-{θ\_{2}}\right)\\,{m\_{2}}\\,\left({w\_{2}}^{2}\\,{L\_{2}}+{w\_{1}}^{2}\\,{L\_{1}}\\,\cos\left({θ\_{1}}-{θ\_{2}}\right)\right)}{{L\_{1}}\\,\left(2\\,{m\_{1}}+{m\_{2}}-{m\_{2}}\\,\cos\left(2\\,{θ\_{1}}-2\\,{θ\_{2}}\right)\right)}\\] | |Description |
  • \\({α\_{1}}\\) is the angular acceleration of the first object (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
  • \\({w\_{1}}\\) is the angular velocity of the first object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({w\_{2}}\\) is the angular velocity of the second object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\(g\\) is the magnitude of gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({m\_{1}}\\) is the mass of the first object (\\({\text{kg}}\\))
  • \\({m\_{2}}\\) is the mass of the second object (\\({\text{kg}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
| |Notes |
  • \\({θ\_{1}}\\) is calculated by solving the ODE here together with the initial conditions and [IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2).
| |Source |-- | @@ -34,7 +34,7 @@ This section transforms the problem defined in the [problem description](./SecPr |Output |\\({θ\_{2}}\\) | |Input Constraints |\\[{L\_{1}}\gt{}0\\]\\[{L\_{2}}\gt{}0\\]\\[{m\_{1}}\gt{}0\\]\\[{m\_{2}}\gt{}0\\] | |Output Constraints| | -|Equation |\\[{α\_{2}}\left({θ\_{1}},{θ\_{2}},{w\_{1}},{w\_{2}}\right)=\frac{2 \sin\left({θ\_{1}}-{θ\_{2}}\right) \left({w\_{1}}^{2} {L\_{1}} \left({m\_{1}}+{m\_{2}}\right)+g \left({m\_{1}}+{m\_{2}}\right) \cos\left({θ\_{1}}\right)+{w\_{2}}^{2} {L\_{2}} {m\_{2}} \cos\left({θ\_{1}}-{θ\_{2}}\right)\right)}{{L\_{2}} \left(2 {m\_{1}}+{m\_{2}}-{m\_{2}} \cos\left(2 {θ\_{1}}-2 {θ\_{2}}\right)\right)}\\] | +|Equation |\\[{α\_{2}}\left({θ\_{1}},{θ\_{2}},{w\_{1}},{w\_{2}}\right)=\frac{2\\,\sin\left({θ\_{1}}-{θ\_{2}}\right)\\,\left({w\_{1}}^{2}\\,{L\_{1}}\\,\left({m\_{1}}+{m\_{2}}\right)+g\\,\left({m\_{1}}+{m\_{2}}\right)\\,\cos\left({θ\_{1}}\right)+{w\_{2}}^{2}\\,{L\_{2}}\\,{m\_{2}}\\,\cos\left({θ\_{1}}-{θ\_{2}}\right)\right)}{{L\_{2}}\\,\left(2\\,{m\_{1}}+{m\_{2}}-{m\_{2}}\\,\cos\left(2\\,{θ\_{1}}-2\\,{θ\_{2}}\right)\right)}\\] | |Description |
  • \\({α\_{2}}\\) is the angular acceleration of the second object (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
  • \\({θ\_{1}}\\) is the angle of the first rod (\\({\text{rad}}\\))
  • \\({θ\_{2}}\\) is the angle of the second rod (\\({\text{rad}}\\))
  • \\({w\_{1}}\\) is the angular velocity of the first object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({w\_{2}}\\) is the angular velocity of the second object (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{1}}\\) is the length of the first rod (\\({\text{m}}\\))
  • \\({m\_{1}}\\) is the mass of the first object (\\({\text{kg}}\\))
  • \\({m\_{2}}\\) is the mass of the second object (\\({\text{kg}}\\))
  • \\(g\\) is the magnitude of gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({L\_{2}}\\) is the length of the second rod (\\({\text{m}}\\))
| |Notes |
  • \\({θ\_{2}}\\) is calculated by solving the ODE here together with the initial conditions and [IM:calOfAngle1](./SecIMs.md#IM:calOfAngle1).
| |Source |-- | @@ -42,36 +42,36 @@ This section transforms the problem defined in the [problem description](./SecPr #### Detailed derivation of angle of the second rod: {#IM:calOfAngle2Deriv} -By solving equations [GD:xForce2](./SecGDs.md#GD:xForce2) and [GD:yForce2](./SecGDs.md#GD:yForce2) for \\({\boldsymbol{T}\_{2}} \sin\left({θ\_{2}}\right)\\) and \\({\boldsymbol{T}\_{2}} \cos\left({θ\_{2}}\right)\\) and then substituting into equation [GD:xForce1](./SecGDs.md#GD:xForce1) and [GD:yForce1](./SecGDs.md#GD:yForce1) , We can get equations 1 and 2: +By solving equations [GD:xForce2](./SecGDs.md#GD:xForce2) and [GD:yForce2](./SecGDs.md#GD:yForce2) for \\({\boldsymbol{T}\_{2}}\\,\sin\left({θ\_{2}}\right)\\) and \\({\boldsymbol{T}\_{2}}\\,\cos\left({θ\_{2}}\right)\\) and then substituting into equation [GD:xForce1](./SecGDs.md#GD:xForce1) and [GD:yForce1](./SecGDs.md#GD:yForce1) , We can get equations 1 and 2: -\\[{m\_{1}} {a\_{\text{x}1}}=-{\boldsymbol{T}\_{1}} \sin\left({θ\_{1}}\right)-{m\_{2}} {a\_{\text{x}2}}\\] +\\[{m\_{1}}\\,{a\_{\text{x}1}}=-{\boldsymbol{T}\_{1}}\\,\sin\left({θ\_{1}}\right)-{m\_{2}}\\,{a\_{\text{x}2}}\\] -\\[{m\_{1}} {a\_{\text{y}1}}={\boldsymbol{T}\_{1}} \cos\left({θ\_{1}}\right)-{m\_{2}} {a\_{\text{y}2}}-{m\_{2}} g-{m\_{1}} g\\] +\\[{m\_{1}}\\,{a\_{\text{y}1}}={\boldsymbol{T}\_{1}}\\,\cos\left({θ\_{1}}\right)-{m\_{2}}\\,{a\_{\text{y}2}}-{m\_{2}}\\,g-{m\_{1}}\\,g\\] Multiply the equation 1 by \\(\cos\left({θ\_{1}}\right)\\) and the equation 2 by \\(\sin\left({θ\_{1}}\right)\\) and rearrange to get: -\\[{\boldsymbol{T}\_{1}} \sin\left({θ\_{1}}\right) \cos\left({θ\_{1}}\right)=-\cos\left({θ\_{1}}\right) \left({m\_{1}} {a\_{\text{x}1}}+{m\_{2}} {a\_{\text{x}2}}\right)\\] +\\[{\boldsymbol{T}\_{1}}\\,\sin\left({θ\_{1}}\right)\\,\cos\left({θ\_{1}}\right)=-\cos\left({θ\_{1}}\right)\\,\left({m\_{1}}\\,{a\_{\text{x}1}}+{m\_{2}}\\,{a\_{\text{x}2}}\right)\\] -\\[{\boldsymbol{T}\_{1}} \sin\left({θ\_{1}}\right) \cos\left({θ\_{1}}\right)=\sin\left({θ\_{1}}\right) \left({m\_{1}} {a\_{\text{y}1}}+{m\_{2}} {a\_{\text{y}2}}+{m\_{2}} g+{m\_{1}} g\right)\\] +\\[{\boldsymbol{T}\_{1}}\\,\sin\left({θ\_{1}}\right)\\,\cos\left({θ\_{1}}\right)=\sin\left({θ\_{1}}\right)\\,\left({m\_{1}}\\,{a\_{\text{y}1}}+{m\_{2}}\\,{a\_{\text{y}2}}+{m\_{2}}\\,g+{m\_{1}}\\,g\right)\\] This leads to the equation 3 -\\[\sin\left({θ\_{1}}\right) \left({m\_{1}} {a\_{\text{y}1}}+{m\_{2}} {a\_{\text{y}2}}+{m\_{2}} g+{m\_{1}} g\right)=-\cos\left({θ\_{1}}\right) \left({m\_{1}} {a\_{\text{x}1}}+{m\_{2}} {a\_{\text{x}2}}\right)\\] +\\[\sin\left({θ\_{1}}\right)\\,\left({m\_{1}}\\,{a\_{\text{y}1}}+{m\_{2}}\\,{a\_{\text{y}2}}+{m\_{2}}\\,g+{m\_{1}}\\,g\right)=-\cos\left({θ\_{1}}\right)\\,\left({m\_{1}}\\,{a\_{\text{x}1}}+{m\_{2}}\\,{a\_{\text{x}2}}\right)\\] Next, multiply equation [GD:xForce2](./SecGDs.md#GD:xForce2) by \\(\cos\left({θ\_{2}}\right)\\) and equation [GD:yForce2](./SecGDs.md#GD:yForce2) by \\(\sin\left({θ\_{2}}\right)\\) and rearrange to get: -\\[{\boldsymbol{T}\_{2}} \sin\left({θ\_{2}}\right) \cos\left({θ\_{2}}\right)=-\cos\left({θ\_{2}}\right) {m\_{2}} {a\_{\text{x}2}}\\] +\\[{\boldsymbol{T}\_{2}}\\,\sin\left({θ\_{2}}\right)\\,\cos\left({θ\_{2}}\right)=-\cos\left({θ\_{2}}\right)\\,{m\_{2}}\\,{a\_{\text{x}2}}\\] -\\[{\boldsymbol{T}\_{1}} \sin\left({θ\_{2}}\right) \cos\left({θ\_{2}}\right)=\sin\left({θ\_{2}}\right) \left({m\_{2}} {a\_{\text{y}2}}+{m\_{2}} g\right)\\] +\\[{\boldsymbol{T}\_{1}}\\,\sin\left({θ\_{2}}\right)\\,\cos\left({θ\_{2}}\right)=\sin\left({θ\_{2}}\right)\\,\left({m\_{2}}\\,{a\_{\text{y}2}}+{m\_{2}}\\,g\right)\\] which leads to equation 4 -\\[\sin\left({θ\_{2}}\right) \left({m\_{2}} {a\_{\text{y}2}}+{m\_{2}} g\right)=-\cos\left({θ\_{2}}\right) {m\_{2}} {a\_{\text{x}2}}\\] +\\[\sin\left({θ\_{2}}\right)\\,\left({m\_{2}}\\,{a\_{\text{y}2}}+{m\_{2}}\\,g\right)=-\cos\left({θ\_{2}}\right)\\,{m\_{2}}\\,{a\_{\text{x}2}}\\] By giving equations [GD:accelerationX1](./SecGDs.md#GD:accelerationX1) and [GD:accelerationX2](./SecGDs.md#GD:accelerationX2) and [GD:accelerationY1](./SecGDs.md#GD:accelerationY1) and [GD:accelerationY2](./SecGDs.md#GD:accelerationY2) plus additional two equations, 3 and 4, we can get [IM:calOfAngle1](./SecIMs.md#IM:calOfAngle1) and [IM:calOfAngle2](./SecIMs.md#IM:calOfAngle2) via a computer algebra program: diff --git a/code/stable/dblpend/SRS/mdBook/src/SecTMs.md b/code/stable/dblpend/SRS/mdBook/src/SecTMs.md index d75cf5f7fe..431ad945c9 100644 --- a/code/stable/dblpend/SRS/mdBook/src/SecTMs.md +++ b/code/stable/dblpend/SRS/mdBook/src/SecTMs.md @@ -39,7 +39,7 @@ This section focuses on the general equations and laws that DblPend is based on. |Refname |TM:NewtonSecLawMot | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's second law of motion | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}\\] | +|Equation |\\[\boldsymbol{F}=m\\,\boldsymbol{a}\text{(}t\text{)}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The net force \\(\boldsymbol{F}\\) on a body is proportional to the acceleration \\(\boldsymbol{a}\text{(}t\text{)}\\) of the body, where \\(m\\) denotes the mass of the body as the constant of proportionality.
| |Source |-- | diff --git a/code/stable/gamephysics/SRS/HTML/GamePhysics_SRS.html b/code/stable/gamephysics/SRS/HTML/GamePhysics_SRS.html index 165afcc379..f755547c49 100644 --- a/code/stable/gamephysics/SRS/HTML/GamePhysics_SRS.html +++ b/code/stable/gamephysics/SRS/HTML/GamePhysics_SRS.html @@ -858,7 +858,7 @@

Theoretical Models

- + @@ -952,7 +952,7 @@

Theoretical Models

@@ -1017,7 +1017,7 @@

Theoretical Models

- + @@ -1081,7 +1081,7 @@

General Definitions

@@ -1136,7 +1136,7 @@

Detailed derivation of gravitational acceleration:

From Newton's law of universal gravitation, we have:

- \[\symbf{F}=\frac{G {m_{1}} {m_{2}}}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}}\] + \[\symbf{F}=\frac{G\,{m_{1}}\,{m_{2}}}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}}\]

The above equation governs the gravitational attraction between two bodies. Suppose that one of the bodies is significantly more massive than the other, so that we concern ourselves with the force the massive body exerts on the lighter body. Further, suppose that the Cartesian coordinate system is chosen such that this force acts on a line which lies along one of the principal axes. Then our unit vector directed from the center of the large mass to the center of the smaller mass d̂ for the x or y axes is:

@@ -1144,15 +1144,15 @@

Detailed derivation of gravitational acceleration:

Given the above assumptions, let M and m be the mass of the massive and light body respectively. Equating F above with Newton's second law for the force experienced by the light body, we get:

- \[{\symbf{F}_{\symbf{g}}}=G \frac{M m}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}}=m \symbf{g}\] + \[{\symbf{F}_{\symbf{g}}}=G\,\frac{M\,m}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}}=m\,\symbf{g}\]

where g is the gravitational acceleration. Dividing the above equation by m, we have:

- \[G \frac{M}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}}=\symbf{g}\] + \[G\,\frac{M}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}}=\symbf{g}\]

and thus the negative sign indicates that the force is an attractive force:

- \[\symbf{g}=-G \frac{M}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}}\] + \[\symbf{g}=-G\,\frac{M}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}}\]
@@ -1172,7 +1172,7 @@

Detailed derivation of gravitational acceleration:

@@ -1270,7 +1270,7 @@

Data Definitions

@@ -1786,7 +1786,7 @@

Data Definitions

- + @@ -1976,7 +1976,7 @@

Data Definitions

- + @@ -2015,11 +2015,11 @@

Data Definitions

Detailed derivation of impulse (vector):

Newton's second law of motion states:

- \[\symbf{F}=m \symbf{a}\text{(}t\text{)}=m \frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt}\] + \[\symbf{F}=m\,\symbf{a}\text{(}t\text{)}=m\,\frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt}\]

Rearranging:

- \[\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m \left(\int_{{\symbf{v}\text{(}t\text{)}_{1}}}^{{\symbf{v}\text{(}t\text{)}_{2}}}{1}\,d\symbf{v}\text{(}t\text{)}\right)\] + \[\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m\,\left(\int_{{\symbf{v}\text{(}t\text{)}_{1}}}^{{\symbf{v}\text{(}t\text{)}_{2}}}{1}\,d\symbf{v}\text{(}t\text{)}\right)\]

Integrating the right hand side:

- \[\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m {\symbf{v}\text{(}t\text{)}_{2}}-m {\symbf{v}\text{(}t\text{)}_{1}}=m Δ\symbf{v}\] + \[\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m\,{\symbf{v}\text{(}t\text{)}_{2}}-m\,{\symbf{v}\text{(}t\text{)}_{1}}=m\,Δ\symbf{v}\]
@@ -2042,7 +2042,7 @@

Detailed derivation of impulse (vector):

- + @@ -2101,7 +2101,7 @@

Detailed derivation of impulse (vector):

- + @@ -2382,7 +2382,7 @@

Detailed derivation of j-th body's angular acceleration:

diff --git a/code/stable/gamephysics/SRS/Jupyter/GamePhysics_SRS.ipynb b/code/stable/gamephysics/SRS/Jupyter/GamePhysics_SRS.ipynb index 083438d8da..fd3a499006 100644 --- a/code/stable/gamephysics/SRS/Jupyter/GamePhysics_SRS.ipynb +++ b/code/stable/gamephysics/SRS/Jupyter/GamePhysics_SRS.ipynb @@ -348,7 +348,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -453,7 +453,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -510,7 +510,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -577,7 +577,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -618,19 +618,19 @@ "\n", "\n", "From [Newton's law of universal gravitation](#TM:UniversalGravLaw), we have:\n", - "$$\\symbf{F}=\\frac{G {m_{1}} {m_{2}}}{{\\|\\symbf{d}\\|^{2}}} \\symbf{\\hat{d}}$$\n", + "$$\\symbf{F}=\\frac{G\\,{m_{1}}\\,{m_{2}}}{{\\|\\symbf{d}\\|^{2}}}\\,\\symbf{\\hat{d}}$$\n", "\n", "The above equation governs the gravitational attraction between two bodies. Suppose that one of the bodies is significantly more massive than the other, so that we concern ourselves with the force the massive body exerts on the lighter body. Further, suppose that the Cartesian coordinate system is chosen such that this force acts on a line which lies along one of the principal axes. Then our unit vector directed from the center of the large mass to the center of the smaller mass $d̂$ for the x or y axes is:\n", "$$\\symbf{\\hat{d}}=\\frac{\\symbf{d}}{\\|\\symbf{d}\\|}$$\n", "\n", "Given the above assumptions, let $M$ and $m$ be the mass of the massive and light body respectively. Equating $F$ above with Newton's second law for the force experienced by the light body, we get:\n", - "$${\\symbf{F}_{\\symbf{g}}}=G \\frac{M m}{{\\|\\symbf{d}\\|^{2}}} \\symbf{\\hat{d}}=m \\symbf{g}$$\n", + "$${\\symbf{F}_{\\symbf{g}}}=G\\,\\frac{M\\,m}{{\\|\\symbf{d}\\|^{2}}}\\,\\symbf{\\hat{d}}=m\\,\\symbf{g}$$\n", "\n", "where $g$ is the gravitational acceleration. Dividing the above equation by $m$, we have:\n", - "$$G \\frac{M}{{\\|\\symbf{d}\\|^{2}}} \\symbf{\\hat{d}}=\\symbf{g}$$\n", + "$$G\\,\\frac{M}{{\\|\\symbf{d}\\|^{2}}}\\,\\symbf{\\hat{d}}=\\symbf{g}$$\n", "\n", "and thus the negative sign indicates that the force is an attractive force:\n", - "$$\\symbf{g}=-G \\frac{M}{{\\|\\symbf{d}\\|^{2}}} \\symbf{\\hat{d}}$$\n", + "$$\\symbf{g}=-G\\,\\frac{M}{{\\|\\symbf{d}\\|^{2}}}\\,\\symbf{\\hat{d}}$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{2}} \\cos\\left({θ_{2}}\\right)-{m_{2}} \\symbf{g}$$\n", + "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}={\\symbf{T}_{2}}\\,\\cos\\left({θ_{2}}\\right)-{m_{2}}\\,\\symbf{g}$$\n", "
Equation\n", - "$${p_{\\text{x}1}}={L_{1}} \\sin\\left({θ_{1}}\\right)$$\n", + "$${p_{\\text{x}1}}={L_{1}}\\,\\sin\\left({θ_{1}}\\right)$$\n", "
Equation\n", - "$${p_{\\text{y}1}}=-{L_{1}} \\cos\\left({θ_{1}}\\right)$$\n", + "$${p_{\\text{y}1}}=-{L_{1}}\\,\\cos\\left({θ_{1}}\\right)$$\n", "
Equation\n", - "$${p_{\\text{x}2}}={p_{\\text{x}1}}+{L_{2}} \\sin\\left({θ_{2}}\\right)$$\n", + "$${p_{\\text{x}2}}={p_{\\text{x}1}}+{L_{2}}\\,\\sin\\left({θ_{2}}\\right)$$\n", "
Equation\n", - "$${p_{\\text{y}2}}={p_{\\text{y}1}}-{L_{2}} \\cos\\left({θ_{2}}\\right)$$\n", + "$${p_{\\text{y}2}}={p_{\\text{y}1}}-{L_{2}}\\,\\cos\\left({θ_{2}}\\right)$$\n", "
Equation\n", - "$${α_{1}}\\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\\right)=\\frac{-g \\left(2 {m_{1}}+{m_{2}}\\right) \\sin\\left({θ_{1}}\\right)-{m_{2}} g \\sin\\left({θ_{1}}-2 {θ_{2}}\\right)-2 \\sin\\left({θ_{1}}-{θ_{2}}\\right) {m_{2}} \\left({w_{2}}^{2} {L_{2}}+{w_{1}}^{2} {L_{1}} \\cos\\left({θ_{1}}-{θ_{2}}\\right)\\right)}{{L_{1}} \\left(2 {m_{1}}+{m_{2}}-{m_{2}} \\cos\\left(2 {θ_{1}}-2 {θ_{2}}\\right)\\right)}$$\n", + "$${α_{1}}\\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\\right)=\\frac{-g\\,\\left(2\\,{m_{1}}+{m_{2}}\\right)\\,\\sin\\left({θ_{1}}\\right)-{m_{2}}\\,g\\,\\sin\\left({θ_{1}}-2\\,{θ_{2}}\\right)-2\\,\\sin\\left({θ_{1}}-{θ_{2}}\\right)\\,{m_{2}}\\,\\left({w_{2}}^{2}\\,{L_{2}}+{w_{1}}^{2}\\,{L_{1}}\\,\\cos\\left({θ_{1}}-{θ_{2}}\\right)\\right)}{{L_{1}}\\,\\left(2\\,{m_{1}}+{m_{2}}-{m_{2}}\\,\\cos\\left(2\\,{θ_{1}}-2\\,{θ_{2}}\\right)\\right)}$$\n", "
Equation\n", - "$${α_{2}}\\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\\right)=\\frac{2 \\sin\\left({θ_{1}}-{θ_{2}}\\right) \\left({w_{1}}^{2} {L_{1}} \\left({m_{1}}+{m_{2}}\\right)+g \\left({m_{1}}+{m_{2}}\\right) \\cos\\left({θ_{1}}\\right)+{w_{2}}^{2} {L_{2}} {m_{2}} \\cos\\left({θ_{1}}-{θ_{2}}\\right)\\right)}{{L_{2}} \\left(2 {m_{1}}+{m_{2}}-{m_{2}} \\cos\\left(2 {θ_{1}}-2 {θ_{2}}\\right)\\right)}$$\n", + "$${α_{2}}\\left({θ_{1}},{θ_{2}},{w_{1}},{w_{2}}\\right)=\\frac{2\\,\\sin\\left({θ_{1}}-{θ_{2}}\\right)\\,\\left({w_{1}}^{2}\\,{L_{1}}\\,\\left({m_{1}}+{m_{2}}\\right)+g\\,\\left({m_{1}}+{m_{2}}\\right)\\,\\cos\\left({θ_{1}}\\right)+{w_{2}}^{2}\\,{L_{2}}\\,{m_{2}}\\,\\cos\\left({θ_{1}}-{θ_{2}}\\right)\\right)}{{L_{2}}\\,\\left(2\\,{m_{1}}+{m_{2}}-{m_{2}}\\,\\cos\\left(2\\,{θ_{1}}-2\\,{θ_{2}}\\right)\\right)}$$\n", "
Equation\[\symbf{F}=m \symbf{a}\text{(}t\text{)}\]\[\symbf{F}=m\,\symbf{a}\text{(}t\text{)}\]
Description
Equation - \[\symbf{F}=G \frac{{m_{1}} {m_{2}}}{\|\symbf{d}\|^{2}} \symbf{\hat{d}}=G \frac{{m_{1}} {m_{2}}}{\|\symbf{d}\|^{2}} \frac{\symbf{d}}{\|\symbf{d}\|}\] + \[\symbf{F}=G\,\frac{{m_{1}}\,{m_{2}}}{\|\symbf{d}\|^{2}}\,\symbf{\hat{d}}=G\,\frac{{m_{1}}\,{m_{2}}}{\|\symbf{d}\|^{2}}\,\frac{\symbf{d}}{\|\symbf{d}\|}\]
Equation\[\symbf{τ}=\symbf{I} α\]\[\symbf{τ}=\symbf{I}\,α\]
Description
Equation - \[\symbf{g}=-\frac{G M}{\|\symbf{d}\|^{2}} \symbf{\hat{d}}\] + \[\symbf{g}=-\frac{G\,M}{\|\symbf{d}\|^{2}}\,\symbf{\hat{d}}\]
Equation - \[j=\frac{-\left(1+{C_{\text{R}}}\right) {{\symbf{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\symbf{n}}{\left(\frac{1}{{m_{\text{A}}}}+\frac{1}{{m_{\text{B}}}}\right) \|\symbf{n}\|^{2}+\frac{\|{\symbf{u}_{\text{A}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{A}}}}+\frac{\|{\symbf{u}_{\text{B}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{B}}}}}\] + \[j=\frac{-\left(1+{C_{\text{R}}}\right)\,{{\symbf{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\symbf{n}}{\left(\frac{1}{{m_{\text{A}}}}+\frac{1}{{m_{\text{B}}}}\right)\,\|\symbf{n}\|^{2}+\frac{\|{\symbf{u}_{\text{A}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{A}}}}+\frac{\|{\symbf{u}_{\text{B}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{B}}}}}\]
Equation - \[{\symbf{p}\text{(}t\text{)}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}} {\symbf{p}\text{(}t\text{)}_{j}}}}{{m_{T}}}\] + \[{\symbf{p}\text{(}t\text{)}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}}\,{\symbf{p}\text{(}t\text{)}_{j}}}}{{m_{T}}}\]
Equation\[KE=m \frac{\|\symbf{v}\text{(}t\text{)}\|^{2}}{2}\]\[KE=m\,\frac{\|\symbf{v}\text{(}t\text{)}\|^{2}}{2}\]
Description
Equation\[\symbf{J}=m Δ\symbf{v}\]\[\symbf{J}=m\,Δ\symbf{v}\]
Description
Equation\[PE=m \symbf{g} h\]\[PE=m\,\symbf{g}\,h\]
Description
Equation\[\symbf{I}=\displaystyle\sum{{m_{j}} {d_{j}}^{2}}\]\[\symbf{I}=\displaystyle\sum{{m_{j}}\,{d_{j}}^{2}}\]
Description
Equation - \[{\symbf{v}\text{(}t\text{)}_{\text{A}}}\left({t_{\text{c}}}\right)={\symbf{v}\text{(}t\text{)}_{\text{A}}}\left(t\right)+\frac{j}{{m_{\text{A}}}} \symbf{n}\] + \[{\symbf{v}\text{(}t\text{)}_{\text{A}}}\left({t_{\text{c}}}\right)={\symbf{v}\text{(}t\text{)}_{\text{A}}}\left(t\right)+\frac{j}{{m_{\text{A}}}}\,\symbf{n}\]
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}$$\n", + "$$\\symbf{F}=m\\,\\symbf{a}\\text{(}t\\text{)}$$\n", "
Equation\n", - "$$\\symbf{F}=G \\frac{{m_{1}} {m_{2}}}{\\|\\symbf{d}\\|^{2}} \\symbf{\\hat{d}}=G \\frac{{m_{1}} {m_{2}}}{\\|\\symbf{d}\\|^{2}} \\frac{\\symbf{d}}{\\|\\symbf{d}\\|}$$\n", + "$$\\symbf{F}=G\\,\\frac{{m_{1}}\\,{m_{2}}}{\\|\\symbf{d}\\|^{2}}\\,\\symbf{\\hat{d}}=G\\,\\frac{{m_{1}}\\,{m_{2}}}{\\|\\symbf{d}\\|^{2}}\\,\\frac{\\symbf{d}}{\\|\\symbf{d}\\|}$$\n", "
Equation\n", - "$$\\symbf{τ}=\\symbf{I} α$$\n", + "$$\\symbf{τ}=\\symbf{I}\\,α$$\n", "
Equation\n", - "$$\\symbf{g}=-\\frac{G M}{\\|\\symbf{d}\\|^{2}} \\symbf{\\hat{d}}$$\n", + "$$\\symbf{g}=-\\frac{G\\,M}{\\|\\symbf{d}\\|^{2}}\\,\\symbf{\\hat{d}}$$\n", "
\n", @@ -657,7 +657,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -741,7 +741,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1348,7 +1348,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1556,7 +1556,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1597,13 +1597,13 @@ "\n", "\n", "Newton's second law of motion states:\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}=m \\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", + "$$\\symbf{F}=m\\,\\symbf{a}\\text{(}t\\text{)}=m\\,\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Rearranging:\n", - "$$\\int_{{t_{1}}}^{{t_{2}}}{\\symbf{F}}\\,dt=m \\left(\\int_{{\\symbf{v}\\text{(}t\\text{)}_{1}}}^{{\\symbf{v}\\text{(}t\\text{)}_{2}}}{1}\\,d\\symbf{v}\\text{(}t\\text{)}\\right)$$\n", + "$$\\int_{{t_{1}}}^{{t_{2}}}{\\symbf{F}}\\,dt=m\\,\\left(\\int_{{\\symbf{v}\\text{(}t\\text{)}_{1}}}^{{\\symbf{v}\\text{(}t\\text{)}_{2}}}{1}\\,d\\symbf{v}\\text{(}t\\text{)}\\right)$$\n", "\n", "Integrating the right hand side:\n", - "$$\\int_{{t_{1}}}^{{t_{2}}}{\\symbf{F}}\\,dt=m {\\symbf{v}\\text{(}t\\text{)}_{2}}-m {\\symbf{v}\\text{(}t\\text{)}_{1}}=m Δ\\symbf{v}$$\n", + "$$\\int_{{t_{1}}}^{{t_{2}}}{\\symbf{F}}\\,dt=m\\,{\\symbf{v}\\text{(}t\\text{)}_{2}}-m\\,{\\symbf{v}\\text{(}t\\text{)}_{1}}=m\\,Δ\\symbf{v}$$\n", "
\n", "\n", "
Equation\n", - "$$j=\\frac{-\\left(1+{C_{\\text{R}}}\\right) {{\\symbf{v}\\text{(}t\\text{)}_{\\text{i}}}^{\\text{A}\\text{B}}}\\cdot{}\\symbf{n}}{\\left(\\frac{1}{{m_{\\text{A}}}}+\\frac{1}{{m_{\\text{B}}}}\\right) \\|\\symbf{n}\\|^{2}+\\frac{\\|{\\symbf{u}_{\\text{A}\\text{P}}}\\text{*}\\symbf{n}\\|^{2}}{{\\symbf{I}_{\\text{A}}}}+\\frac{\\|{\\symbf{u}_{\\text{B}\\text{P}}}\\text{*}\\symbf{n}\\|^{2}}{{\\symbf{I}_{\\text{B}}}}}$$\n", + "$$j=\\frac{-\\left(1+{C_{\\text{R}}}\\right)\\,{{\\symbf{v}\\text{(}t\\text{)}_{\\text{i}}}^{\\text{A}\\text{B}}}\\cdot{}\\symbf{n}}{\\left(\\frac{1}{{m_{\\text{A}}}}+\\frac{1}{{m_{\\text{B}}}}\\right)\\,\\|\\symbf{n}\\|^{2}+\\frac{\\|{\\symbf{u}_{\\text{A}\\text{P}}}\\text{*}\\symbf{n}\\|^{2}}{{\\symbf{I}_{\\text{A}}}}+\\frac{\\|{\\symbf{u}_{\\text{B}\\text{P}}}\\text{*}\\symbf{n}\\|^{2}}{{\\symbf{I}_{\\text{B}}}}}$$\n", "
Equation\n", - "$${\\symbf{p}\\text{(}t\\text{)}_{\\text{CM}}}=\\frac{\\displaystyle\\sum{{m_{j}} {\\symbf{p}\\text{(}t\\text{)}_{j}}}}{{m_{T}}}$$\n", + "$${\\symbf{p}\\text{(}t\\text{)}_{\\text{CM}}}=\\frac{\\displaystyle\\sum{{m_{j}}\\,{\\symbf{p}\\text{(}t\\text{)}_{j}}}}{{m_{T}}}$$\n", "
Equation\n", - "$$KE=m \\frac{\\|\\symbf{v}\\text{(}t\\text{)}\\|^{2}}{2}$$\n", + "$$KE=m\\,\\frac{\\|\\symbf{v}\\text{(}t\\text{)}\\|^{2}}{2}$$\n", "
Equation\n", - "$$\\symbf{J}=m Δ\\symbf{v}$$\n", + "$$\\symbf{J}=m\\,Δ\\symbf{v}$$\n", "
\n", @@ -1637,7 +1637,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1709,7 +1709,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1995,7 +1995,7 @@ "\n", "\n", "\n", "\n", "\n", diff --git a/code/stable/gamephysics/SRS/PDF/GamePhysics_SRS.tex b/code/stable/gamephysics/SRS/PDF/GamePhysics_SRS.tex index 0a7c7bff97..f892cb91a2 100644 --- a/code/stable/gamephysics/SRS/PDF/GamePhysics_SRS.tex +++ b/code/stable/gamephysics/SRS/PDF/GamePhysics_SRS.tex @@ -367,7 +367,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)} + \symbf{F}=m\,\symbf{a}\text{(}t\text{)} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -431,7 +431,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=G \frac{{m_{1}} {m_{2}}}{\|\symbf{d}\|^{2}} \symbf{\hat{d}}=G \frac{{m_{1}} {m_{2}}}{\|\symbf{d}\|^{2}} \frac{\symbf{d}}{\|\symbf{d}\|} + \symbf{F}=G\,\frac{{m_{1}}\,{m_{2}}}{\|\symbf{d}\|^{2}}\,\symbf{\hat{d}}=G\,\frac{{m_{1}}\,{m_{2}}}{\|\symbf{d}\|^{2}}\,\frac{\symbf{d}}{\|\symbf{d}\|} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -468,7 +468,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{τ}=\symbf{I} α + \symbf{τ}=\symbf{I}\,α \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -510,7 +510,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{g}=-\frac{G M}{\|\symbf{d}\|^{2}} \symbf{\hat{d}} + \symbf{g}=-\frac{G\,M}{\|\symbf{d}\|^{2}}\,\symbf{\hat{d}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -538,7 +538,7 @@ \subsubsection{General Definitions} From \hyperref[TM:UniversalGravLaw]{Newton's law of universal gravitation}, we have: \begin{displaymath} -\symbf{F}=\frac{G {m_{1}} {m_{2}}}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}} +\symbf{F}=\frac{G\,{m_{1}}\,{m_{2}}}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}} \end{displaymath} The above equation governs the gravitational attraction between two bodies. Suppose that one of the bodies is significantly more massive than the other, so that we concern ourselves with the force the massive body exerts on the lighter body. Further, suppose that the Cartesian coordinate system is chosen such that this force acts on a line which lies along one of the principal axes. Then our unit vector directed from the center of the large mass to the center of the smaller mass $\symbf{\hat{d}}$ for the x or y axes is: @@ -548,17 +548,17 @@ \subsubsection{General Definitions} Given the above assumptions, let $M$ and $m$ be the mass of the massive and light body respectively. Equating $\symbf{F}$ above with Newton's second law for the force experienced by the light body, we get: \begin{displaymath} -{\symbf{F}_{\symbf{g}}}=G \frac{M m}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}}=m \symbf{g} +{\symbf{F}_{\symbf{g}}}=G\,\frac{M\,m}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}}=m\,\symbf{g} \end{displaymath} where $\symbf{g}$ is the gravitational acceleration. Dividing the above equation by $m$, we have: \begin{displaymath} -G \frac{M}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}}=\symbf{g} +G\,\frac{M}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}}=\symbf{g} \end{displaymath} and thus the negative sign indicates that the force is an attractive force: \begin{displaymath} -\symbf{g}=-G \frac{M}{{\|\symbf{d}\|^{2}}} \symbf{\hat{d}} +\symbf{g}=-G\,\frac{M}{{\|\symbf{d}\|^{2}}}\,\symbf{\hat{d}} \end{displaymath} \medskip \noindent @@ -575,7 +575,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - j=\frac{-\left(1+{C_{\text{R}}}\right) {{\symbf{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\symbf{n}}{\left(\frac{1}{{m_{\text{A}}}}+\frac{1}{{m_{\text{B}}}}\right) \|\symbf{n}\|^{2}+\frac{\|{\symbf{u}_{\text{A}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{A}}}}+\frac{\|{\symbf{u}_{\text{B}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{B}}}}} + j=\frac{-\left(1+{C_{\text{R}}}\right)\,{{\symbf{v}\text{(}t\text{)}_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\symbf{n}}{\left(\frac{1}{{m_{\text{A}}}}+\frac{1}{{m_{\text{B}}}}\right)\,\|\symbf{n}\|^{2}+\frac{\|{\symbf{u}_{\text{A}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{A}}}}+\frac{\|{\symbf{u}_{\text{B}\text{P}}}\text{*}\symbf{n}\|^{2}}{{\symbf{I}_{\text{B}}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -630,7 +630,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{p}\text{(}t\text{)}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}} {\symbf{p}\text{(}t\text{)}_{j}}}}{{m_{T}}} + {\symbf{p}\text{(}t\text{)}_{\text{CM}}}=\frac{\displaystyle\sum{{m_{j}}\,{\symbf{p}\text{(}t\text{)}_{j}}}}{{m_{T}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -983,7 +983,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - KE=m \frac{\|\symbf{v}\text{(}t\text{)}\|^{2}}{2} + KE=m\,\frac{\|\symbf{v}\text{(}t\text{)}\|^{2}}{2} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1104,7 +1104,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{J}=m Δ\symbf{v} + \symbf{J}=m\,Δ\symbf{v} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1131,17 +1131,17 @@ \subsubsection{Data Definitions} Newton's second law of motion states: \begin{displaymath} -\symbf{F}=m \symbf{a}\text{(}t\text{)}=m \frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt} +\symbf{F}=m\,\symbf{a}\text{(}t\text{)}=m\,\frac{\,d\symbf{v}\text{(}t\text{)}}{\,dt} \end{displaymath} Rearranging: \begin{displaymath} -\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m \left(\int_{{\symbf{v}\text{(}t\text{)}_{1}}}^{{\symbf{v}\text{(}t\text{)}_{2}}}{1}\,d\symbf{v}\text{(}t\text{)}\right) +\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m\,\left(\int_{{\symbf{v}\text{(}t\text{)}_{1}}}^{{\symbf{v}\text{(}t\text{)}_{2}}}{1}\,d\symbf{v}\text{(}t\text{)}\right) \end{displaymath} Integrating the right hand side: \begin{displaymath} -\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m {\symbf{v}\text{(}t\text{)}_{2}}-m {\symbf{v}\text{(}t\text{)}_{1}}=m Δ\symbf{v} +\int_{{t_{1}}}^{{t_{2}}}{\symbf{F}}\,dt=m\,{\symbf{v}\text{(}t\text{)}_{2}}-m\,{\symbf{v}\text{(}t\text{)}_{1}}=m\,Δ\symbf{v} \end{displaymath} \medskip \noindent @@ -1161,7 +1161,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - PE=m \symbf{g} h + PE=m\,\symbf{g}\,h \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1204,7 +1204,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{I}=\displaystyle\sum{{m_{j}} {d_{j}}^{2}} + \symbf{I}=\displaystyle\sum{{m_{j}}\,{d_{j}}^{2}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1415,7 +1415,7 @@ \subsubsection{Instance Models} \end{displaymath} \\ \midrule Equation & \begin{displaymath} - {\symbf{v}\text{(}t\text{)}_{\text{A}}}\left({t_{\text{c}}}\right)={\symbf{v}\text{(}t\text{)}_{\text{A}}}\left(t\right)+\frac{j}{{m_{\text{A}}}} \symbf{n} + {\symbf{v}\text{(}t\text{)}_{\text{A}}}\left({t_{\text{c}}}\right)={\symbf{v}\text{(}t\text{)}_{\text{A}}}\left(t\right)+\frac{j}{{m_{\text{A}}}}\,\symbf{n} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1477,7 +1477,7 @@ \subsubsection{Data Constraints} \\ $ω$ & -- & -- & $2.1$ $\frac{\text{rad}}{\text{s}}$ & 10$\%$ \\ -$ϕ$ & -- & $0\leq{}ϕ\leq{}2 π$ & $\frac{π}{2}$ ${\text{rad}}$ & 10$\%$ +$ϕ$ & -- & $0\leq{}ϕ\leq{}2\,π$ & $\frac{π}{2}$ ${\text{rad}}$ & 10$\%$ \label{Table:InDataConstraints} \end{longtblr} \subsubsection{Properties of a Correct Solution} diff --git a/code/stable/gamephysics/SRS/mdBook/src/SecDDs.md b/code/stable/gamephysics/SRS/mdBook/src/SecDDs.md index d66042e951..a1e8d9c666 100644 --- a/code/stable/gamephysics/SRS/mdBook/src/SecDDs.md +++ b/code/stable/gamephysics/SRS/mdBook/src/SecDDs.md @@ -13,7 +13,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Center of Mass | |Symbol |\\({\boldsymbol{p}\text{(}t\text{)}\_{\text{CM}}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{\boldsymbol{p}\text{(}t\text{)}\_{\text{CM}}}=\frac{\displaystyle\sum{{m\_{j}} {\boldsymbol{p}\text{(}t\text{)}\_{j}}}}{{m\_{T}}}\\] | +|Equation |\\[{\boldsymbol{p}\text{(}t\text{)}\_{\text{CM}}}=\frac{\displaystyle\sum{{m\_{j}}\\,{\boldsymbol{p}\text{(}t\text{)}\_{j}}}}{{m\_{T}}}\\] | |Description|
  • \\({\boldsymbol{p}\text{(}t\text{)}\_{\text{CM}}}\\) is the Center of Mass (\\({\text{m}}\\))
  • \\({m\_{j}}\\) is the mass of the j-th particle (\\({\text{kg}}\\))
  • \\({\boldsymbol{p}\text{(}t\text{)}\_{j}}\\) is the position vector of the j-th particle (\\({\text{m}}\\))
  • \\({m\_{T}}\\) is the total mass of the rigid body (\\({\text{kg}}\\))
| |Notes |
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)).
| |Source |-- | @@ -166,7 +166,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Kinetic energy | |Symbol |\\(KE\\) | |Units |\\({\text{J}}\\) | -|Equation |\\[KE=m \frac{\|\boldsymbol{v}\text{(}t\text{)}\|^{2}}{2}\\] | +|Equation |\\[KE=m\\,\frac{\|\boldsymbol{v}\text{(}t\text{)}\|^{2}}{2}\\] | |Description|
  • \\(KE\\) is the kinetic energy (\\({\text{J}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{v}\text{(}t\text{)}\\) is the velocity (\\(\frac{\text{m}}{\text{s}}\\))
| |Notes |
  • Kinetic energy is the measure of the energy a body possesses due to its motion.
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)) and two-dimensional (from [A:objectDimension](./SecAssumps.md#assumpOD)).
  • No damping occurs during the simulation (from [A:dampingInvolvement](./SecAssumps.md#assumpDI)).
| |Source |-- | @@ -217,7 +217,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Impulse (vector) | |Symbol |\\(\boldsymbol{J}\\) | |Units |\\(\text{N}\text{s}\\) | -|Equation |\\[\boldsymbol{J}=m Δ\boldsymbol{v}\\] | +|Equation |\\[\boldsymbol{J}=m\\,Δ\boldsymbol{v}\\] | |Description|
  • \\(\boldsymbol{J}\\) is the impulse (vector) (\\(\text{N}\text{s}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(Δ\boldsymbol{v}\\) is the change in velocity (\\(\frac{\text{m}}{\text{s}}\\))
| |Notes |
  • An impulse (vector) \\(\boldsymbol{J}\\) occurs when a force \\(\boldsymbol{F}\\) acts over a body over an interval of time.
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)).
| |Source |-- | @@ -227,15 +227,15 @@ This section collects and defines all the data needed to build the instance mode Newton's second law of motion states: -\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}=m \frac{\\,d\boldsymbol{v}\text{(}t\text{)}}{\\,dt}\\] +\\[\boldsymbol{F}=m\\,\boldsymbol{a}\text{(}t\text{)}=m\\,\frac{\\,d\boldsymbol{v}\text{(}t\text{)}}{\\,dt}\\] Rearranging: -\\[\int\_{{t\_{1}}}^{{t\_{2}}}{\boldsymbol{F}}\\,dt=m \left(\int\_{{\boldsymbol{v}\text{(}t\text{)}\_{1}}}^{{\boldsymbol{v}\text{(}t\text{)}\_{2}}}{1}\\,d\boldsymbol{v}\text{(}t\text{)}\right)\\] +\\[\int\_{{t\_{1}}}^{{t\_{2}}}{\boldsymbol{F}}\\,dt=m\\,\left(\int\_{{\boldsymbol{v}\text{(}t\text{)}\_{1}}}^{{\boldsymbol{v}\text{(}t\text{)}\_{2}}}{1}\\,d\boldsymbol{v}\text{(}t\text{)}\right)\\] Integrating the right hand side: -\\[\int\_{{t\_{1}}}^{{t\_{2}}}{\boldsymbol{F}}\\,dt=m {\boldsymbol{v}\text{(}t\text{)}\_{2}}-m {\boldsymbol{v}\text{(}t\text{)}\_{1}}=m Δ\boldsymbol{v}\\] +\\[\int\_{{t\_{1}}}^{{t\_{2}}}{\boldsymbol{F}}\\,dt=m\\,{\boldsymbol{v}\text{(}t\text{)}\_{2}}-m\\,{\boldsymbol{v}\text{(}t\text{)}\_{1}}=m\\,Δ\boldsymbol{v}\\]
@@ -248,7 +248,7 @@ Integrating the right hand side: |Label |Potential energy | |Symbol |\\(PE\\) | |Units |\\({\text{J}}\\) | -|Equation |\\[PE=m \boldsymbol{g} h\\] | +|Equation |\\[PE=m\\,\boldsymbol{g}\\,h\\] | |Description|
  • \\(PE\\) is the potential energy (\\({\text{J}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{g}\\) is the gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(h\\) is the height (\\({\text{m}}\\))
| |Notes |
  • The potential energy of an object is the energy held by an object because of its position to other objects.
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)) and two-dimensional (from [A:objectDimension](./SecAssumps.md#assumpOD)).
  • No damping occurs during the simulation (from [A:dampingInvolvement](./SecAssumps.md#assumpDI)).
| |Source |-- | @@ -265,7 +265,7 @@ Integrating the right hand side: |Label |Moment of inertia | |Symbol |\\(\boldsymbol{I}\\) | |Units |\\(\text{kg}\text{m}^{2}\\) | -|Equation |\\[\boldsymbol{I}=\displaystyle\sum{{m\_{j}} {d\_{j}}^{2}}\\] | +|Equation |\\[\boldsymbol{I}=\displaystyle\sum{{m\_{j}}\\,{d\_{j}}^{2}}\\] | |Description|
  • \\(\boldsymbol{I}\\) is the moment of inertia (\\(\text{kg}\text{m}^{2}\\))
  • \\({m\_{j}}\\) is the mass of the j-th particle (\\({\text{kg}}\\))
  • \\({d\_{j}}\\) is the distance between the j-th particle and the axis of rotation (\\({\text{m}}\\))
| |Notes |
  • The moment of inertia \\(\boldsymbol{I}\\) of a body measures how much torque is needed for the body to achieve angular acceleration about the axis of rotation.
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)).
| |Source |-- | diff --git a/code/stable/gamephysics/SRS/mdBook/src/SecDataConstraints.md b/code/stable/gamephysics/SRS/mdBook/src/SecDataConstraints.md index 3c98deb138..fd9bb30ea8 100644 --- a/code/stable/gamephysics/SRS/mdBook/src/SecDataConstraints.md +++ b/code/stable/gamephysics/SRS/mdBook/src/SecDataConstraints.md @@ -4,18 +4,18 @@ The [Data Constraints Table](./SecDataConstraints.md#Table:InDataConstraints) sh
-|Var |Physical Constraints |Software Constraints |Typical Value |Uncert. | -|:------------------------------------|:----------------------------------|:----------------------|:---------------------------------------------------------------------------|:----------| -|\\({C\_{\text{R}}}\\) |\\(0\leq{}{C\_{\text{R}}}\leq{}1\\)|-- |\\(0.8\\) |10\\(\\%\\)| -|\\(\boldsymbol{F}\\) |-- |-- |\\(98.1\\) \\({\text{N}}\\) |10\\(\\%\\)| -|\\(G\\) |-- |-- |\\(66.743\cdot{}10^{-12}\\) \\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\\)|10\\(\\%\\)| -|\\(\boldsymbol{I}\\) |\\(\boldsymbol{I}\gt{}0\\) |-- |\\(74.5\\) \\(\text{kg}\text{m}^{2}\\) |10\\(\\%\\)| -|\\(L\\) |\\(L\gt{}0\\) |-- |\\(44.2\\) \\({\text{m}}\\) |10\\(\\%\\)| -|\\(m\\) |\\(m\gt{}0\\) |-- |\\(56.2\\) \\({\text{kg}}\\) |10\\(\\%\\)| -|\\(\boldsymbol{p}\text{(}t\text{)}\\)|-- |-- |\\(0.412\\) \\({\text{m}}\\) |10\\(\\%\\)| -|\\(\boldsymbol{v}\text{(}t\text{)}\\)|-- |-- |\\(2.51\\) \\(\frac{\text{m}}{\text{s}}\\) |10\\(\\%\\)| -|\\(\boldsymbol{τ}\\) |-- |-- |\\(200\\) \\(\text{N}\text{m}\\) |10\\(\\%\\)| -|\\(ω\\) |-- |-- |\\(2.1\\) \\(\frac{\text{rad}}{\text{s}}\\) |10\\(\\%\\)| -|\\(ϕ\\) |-- |\\(0\leq{}ϕ\leq{}2 π\\)|\\(\frac{π}{2}\\) \\({\text{rad}}\\) |10\\(\\%\\)| +|Var |Physical Constraints |Software Constraints |Typical Value |Uncert. | +|:------------------------------------|:----------------------------------|:------------------------|:---------------------------------------------------------------------------|:----------| +|\\({C\_{\text{R}}}\\) |\\(0\leq{}{C\_{\text{R}}}\leq{}1\\)|-- |\\(0.8\\) |10\\(\\%\\)| +|\\(\boldsymbol{F}\\) |-- |-- |\\(98.1\\) \\({\text{N}}\\) |10\\(\\%\\)| +|\\(G\\) |-- |-- |\\(66.743\cdot{}10^{-12}\\) \\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\\)|10\\(\\%\\)| +|\\(\boldsymbol{I}\\) |\\(\boldsymbol{I}\gt{}0\\) |-- |\\(74.5\\) \\(\text{kg}\text{m}^{2}\\) |10\\(\\%\\)| +|\\(L\\) |\\(L\gt{}0\\) |-- |\\(44.2\\) \\({\text{m}}\\) |10\\(\\%\\)| +|\\(m\\) |\\(m\gt{}0\\) |-- |\\(56.2\\) \\({\text{kg}}\\) |10\\(\\%\\)| +|\\(\boldsymbol{p}\text{(}t\text{)}\\)|-- |-- |\\(0.412\\) \\({\text{m}}\\) |10\\(\\%\\)| +|\\(\boldsymbol{v}\text{(}t\text{)}\\)|-- |-- |\\(2.51\\) \\(\frac{\text{m}}{\text{s}}\\) |10\\(\\%\\)| +|\\(\boldsymbol{τ}\\) |-- |-- |\\(200\\) \\(\text{N}\text{m}\\) |10\\(\\%\\)| +|\\(ω\\) |-- |-- |\\(2.1\\) \\(\frac{\text{rad}}{\text{s}}\\) |10\\(\\%\\)| +|\\(ϕ\\) |-- |\\(0\leq{}ϕ\leq{}2\\,π\\)|\\(\frac{π}{2}\\) \\({\text{rad}}\\) |10\\(\\%\\)| **

Input Data Constraints

** diff --git a/code/stable/gamephysics/SRS/mdBook/src/SecGDs.md b/code/stable/gamephysics/SRS/mdBook/src/SecGDs.md index c977a04160..3e649f70cb 100644 --- a/code/stable/gamephysics/SRS/mdBook/src/SecGDs.md +++ b/code/stable/gamephysics/SRS/mdBook/src/SecGDs.md @@ -12,7 +12,7 @@ This section collects the laws and equations that will be used to build the inst |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Acceleration due to gravity | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[\boldsymbol{g}=-\frac{G M}{\|\boldsymbol{d}\|^{2}} \boldsymbol{\hat{d}}\\] | +|Equation |\\[\boldsymbol{g}=-\frac{G\\,M}{\|\boldsymbol{d}\|^{2}}\\,\boldsymbol{\hat{d}}\\] | |Description|
  • \\(\boldsymbol{g}\\) is the gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(G\\) is the gravitational constant (\\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\\))
  • \\(M\\) is the mass of the larger rigid body (\\({\text{kg}}\\))
  • \\(\|\boldsymbol{d}\|\\) is the Euclidean norm of the distance between the center of mass of two bodies (\\({\text{m}}\\))
  • \\(\boldsymbol{\hat{d}}\\) is the unit vector directed from the center of the large mass to the center of the smaller mass (\\({\text{m}}\\))
| |Notes |
  • If one of the masses is much larger than the other, it is convenient to define a gravitational field around the larger mass as shown above. The negative sign in the equation indicates that the force is an attractive force.
| |Source |[Definition of Gravitational Acceleration](https://en.wikipedia.org/wiki/Gravitational_acceleration) | @@ -22,7 +22,7 @@ This section collects the laws and equations that will be used to build the inst From [Newton's law of universal gravitation](./SecTMs.md#TM:UniversalGravLaw), we have: -\\[\boldsymbol{F}=\frac{G {m\_{1}} {m\_{2}}}{{\|\boldsymbol{d}\|^{2}}} \boldsymbol{\hat{d}}\\] +\\[\boldsymbol{F}=\frac{G\\,{m\_{1}}\\,{m\_{2}}}{{\|\boldsymbol{d}\|^{2}}}\\,\boldsymbol{\hat{d}}\\] The above equation governs the gravitational attraction between two bodies. Suppose that one of the bodies is significantly more massive than the other, so that we concern ourselves with the force the massive body exerts on the lighter body. Further, suppose that the Cartesian coordinate system is chosen such that this force acts on a line which lies along one of the principal axes. Then our unit vector directed from the center of the large mass to the center of the smaller mass \\(\boldsymbol{\hat{d}}\\) for the x or y axes is: @@ -30,15 +30,15 @@ The above equation governs the gravitational attraction between two bodies. Supp Given the above assumptions, let \\(M\\) and \\(m\\) be the mass of the massive and light body respectively. Equating \\(\boldsymbol{F}\\) above with Newton's second law for the force experienced by the light body, we get: -\\[{\boldsymbol{F}\_{\boldsymbol{g}}}=G \frac{M m}{{\|\boldsymbol{d}\|^{2}}} \boldsymbol{\hat{d}}=m \boldsymbol{g}\\] +\\[{\boldsymbol{F}\_{\boldsymbol{g}}}=G\\,\frac{M\\,m}{{\|\boldsymbol{d}\|^{2}}}\\,\boldsymbol{\hat{d}}=m\\,\boldsymbol{g}\\] where \\(\boldsymbol{g}\\) is the gravitational acceleration. Dividing the above equation by \\(m\\), we have: -\\[G \frac{M}{{\|\boldsymbol{d}\|^{2}}} \boldsymbol{\hat{d}}=\boldsymbol{g}\\] +\\[G\\,\frac{M}{{\|\boldsymbol{d}\|^{2}}}\\,\boldsymbol{\hat{d}}=\boldsymbol{g}\\] and thus the negative sign indicates that the force is an attractive force: -\\[\boldsymbol{g}=-G \frac{M}{{\|\boldsymbol{d}\|^{2}}} \boldsymbol{\hat{d}}\\] +\\[\boldsymbol{g}=-G\\,\frac{M}{{\|\boldsymbol{d}\|^{2}}}\\,\boldsymbol{\hat{d}}\\]
@@ -50,7 +50,7 @@ and thus the negative sign indicates that the force is an attractive force: |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Impulse for Collision | |Units |\\(\text{N}\text{s}\\) | -|Equation |\\[j=\frac{-\left(1+{C\_{\text{R}}}\right) {{\boldsymbol{v}\text{(}t\text{)}\_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\boldsymbol{n}}{\left(\frac{1}{{m\_{\text{A}}}}+\frac{1}{{m\_{\text{B}}}}\right) \|\boldsymbol{n}\|^{2}+\frac{\|{\boldsymbol{u}\_{\text{A}\text{P}}}\text{\*}\boldsymbol{n}\|^{2}}{{\boldsymbol{I}\_{\text{A}}}}+\frac{\|{\boldsymbol{u}\_{\text{B}\text{P}}}\text{\*}\boldsymbol{n}\|^{2}}{{\boldsymbol{I}\_{\text{B}}}}}\\] | +|Equation |\\[j=\frac{-\left(1+{C\_{\text{R}}}\right)\\,{{\boldsymbol{v}\text{(}t\text{)}\_{\text{i}}}^{\text{A}\text{B}}}\cdot{}\boldsymbol{n}}{\left(\frac{1}{{m\_{\text{A}}}}+\frac{1}{{m\_{\text{B}}}}\right)\\,\|\boldsymbol{n}\|^{2}+\frac{\|{\boldsymbol{u}\_{\text{A}\text{P}}}\text{\*}\boldsymbol{n}\|^{2}}{{\boldsymbol{I}\_{\text{A}}}}+\frac{\|{\boldsymbol{u}\_{\text{B}\text{P}}}\text{\*}\boldsymbol{n}\|^{2}}{{\boldsymbol{I}\_{\text{B}}}}}\\] | |Description|
  • \\(j\\) is the impulse (scalar) (\\(\text{N}\text{s}\\))
  • \\({C\_{\text{R}}}\\) is the coefficient of restitution (Unitless)
  • \\({{\boldsymbol{v}\text{(}t\text{)}\_{\text{i}}}^{\text{A}\text{B}}}\\) is the initial relative velocity between rigid bodies of A and B (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(\boldsymbol{n}\\) is the collision normal vector (\\({\text{m}}\\))
  • \\({m\_{\text{A}}}\\) is the mass of rigid body A (\\({\text{kg}}\\))
  • \\({m\_{\text{B}}}\\) is the mass of rigid body B (\\({\text{kg}}\\))
  • \\(\|\boldsymbol{n}\|\\) is the length of the normal vector (\\({\text{m}}\\))
  • \\(\|{\boldsymbol{u}\_{\text{A}\text{P}}}\text{\*}\boldsymbol{n}\|\\) is the length of the perpendicular vector to the contact displacement vector of rigid body A (\\({\text{m}}\\))
  • \\({\boldsymbol{I}\_{\text{A}}}\\) is the moment of inertia of rigid body A (\\(\text{kg}\text{m}^{2}\\))
  • \\(\|{\boldsymbol{u}\_{\text{B}\text{P}}}\text{\*}\boldsymbol{n}\|\\) is the length of the perpendicular vector to the contact displacement vector of rigid body B (\\({\text{m}}\\))
  • \\({\boldsymbol{I}\_{\text{B}}}\\) is the moment of inertia of rigid body B (\\(\text{kg}\text{m}^{2}\\))
| |Notes |
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)) and two-dimensional (from [A:objectDimension](./SecAssumps.md#assumpOD)).
  • A right-handed coordinate system is used (from [A:axesDefined](./SecAssumps.md#assumpAD)).
  • All collisions are vertex-to-edge (from [A:collisionType](./SecAssumps.md#assumpCT)).
| |Source |[Impulse for Collision Ref](http://www.chrishecker.com/images/e/e7/Gdmphys3.pdf) | diff --git a/code/stable/gamephysics/SRS/mdBook/src/SecIMs.md b/code/stable/gamephysics/SRS/mdBook/src/SecIMs.md index 0726e28871..10642e0227 100644 --- a/code/stable/gamephysics/SRS/mdBook/src/SecIMs.md +++ b/code/stable/gamephysics/SRS/mdBook/src/SecIMs.md @@ -75,7 +75,7 @@ Performing the derivative, we obtain: |Output |\\({t\_{\text{c}}}\\) | |Input Constraints |\\[t\gt{}0\\]\\[j\gt{}0\\]\\[{m\_{\text{A}}}\gt{}0\\]\\[\boldsymbol{n}\gt{}0\\] | |Output Constraints|\\[{t\_{\text{c}}}\gt{}0\\] | -|Equation |\\[{\boldsymbol{v}\text{(}t\text{)}\_{\text{A}}}\left({t\_{\text{c}}}\right)={\boldsymbol{v}\text{(}t\text{)}\_{\text{A}}}\left(t\right)+\frac{j}{{m\_{\text{A}}}} \boldsymbol{n}\\] | +|Equation |\\[{\boldsymbol{v}\text{(}t\text{)}\_{\text{A}}}\left({t\_{\text{c}}}\right)={\boldsymbol{v}\text{(}t\text{)}\_{\text{A}}}\left(t\right)+\frac{j}{{m\_{\text{A}}}}\\,\boldsymbol{n}\\] | |Description |
  • \\({\boldsymbol{v}\text{(}t\text{)}\_{\text{A}}}\\) is the velocity at point A (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({t\_{\text{c}}}\\) is the denotes the time at collision (\\({\text{s}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(j\\) is the impulse (scalar) (\\(\text{N}\text{s}\\))
  • \\({m\_{\text{A}}}\\) is the mass of rigid body A (\\({\text{kg}}\\))
  • \\(\boldsymbol{n}\\) is the collision normal vector (\\({\text{m}}\\))
| |Notes |
  • The output of the instance model will be the functions of position, velocity, orientation, and angular acceleration over time that satisfy the equations for the velocity and angular acceleration, with the given initial conditions for position, velocity, orientation, and angular acceleration. The motion is translational, so the position, velocity, orientation, and angular acceleration functions are for the centre of mass (from [DD:ctrOfMass](./SecDDs.md#DD:ctrOfMass)).
  • All bodies are assumed to be rigid (from [A:objectTy](./SecAssumps.md#assumpOT)) and two-dimensional (from [A:objectDimension](./SecAssumps.md#assumpOD)).
  • A right-handed coordinate system is used (from [A:axesDefined](./SecAssumps.md#assumpAD)).
  • All collisions are vertex-to-edge (from [A:collisionType](./SecAssumps.md#assumpCT)).
  • It is currently assumed that no damping occurs during the simulation (from [A:dampingInvolvement](./SecAssumps.md#assumpDI)) and that no constraints are involved (from [A:constraintsAndJointsInvolvement](./SecAssumps.md#assumpCAJI)).
  • \\(j\\) is defined in [GD:impulse](./SecGDs.md#GD:impulse)
| |Source |-- | diff --git a/code/stable/gamephysics/SRS/mdBook/src/SecTMs.md b/code/stable/gamephysics/SRS/mdBook/src/SecTMs.md index f8912d9620..7831dc482d 100644 --- a/code/stable/gamephysics/SRS/mdBook/src/SecTMs.md +++ b/code/stable/gamephysics/SRS/mdBook/src/SecTMs.md @@ -11,7 +11,7 @@ This section focuses on the general equations and laws that GamePhysics is based |Refname |TM:NewtonSecLawMot | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's second law of motion | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}\\] | +|Equation |\\[\boldsymbol{F}=m\\,\boldsymbol{a}\text{(}t\text{)}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The net force \\(\boldsymbol{F}\\) on a body is proportional to the acceleration \\(\boldsymbol{a}\text{(}t\text{)}\\) of the body, where \\(m\\) denotes the mass of the body as the constant of proportionality.
| |Source |-- | @@ -41,7 +41,7 @@ This section focuses on the general equations and laws that GamePhysics is based |Refname |TM:UniversalGravLaw | |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's law of universal gravitation | -|Equation |\\[\boldsymbol{F}=G \frac{{m\_{1}} {m\_{2}}}{\|\boldsymbol{d}\|^{2}} \boldsymbol{\hat{d}}=G \frac{{m\_{1}} {m\_{2}}}{\|\boldsymbol{d}\|^{2}} \frac{\boldsymbol{d}}{\|\boldsymbol{d}\|}\\] | +|Equation |\\[\boldsymbol{F}=G\\,\frac{{m\_{1}}\\,{m\_{2}}}{\|\boldsymbol{d}\|^{2}}\\,\boldsymbol{\hat{d}}=G\\,\frac{{m\_{1}}\\,{m\_{2}}}{\|\boldsymbol{d}\|^{2}}\\,\frac{\boldsymbol{d}}{\|\boldsymbol{d}\|}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(G\\) is the gravitational constant (\\(\frac{\text{m}^{3}}{\text{kg}\text{s}^{2}}\\))
  • \\({m\_{1}}\\) is the mass of the first body (\\({\text{kg}}\\))
  • \\({m\_{2}}\\) is the mass of the second body (\\({\text{kg}}\\))
  • \\(\|\boldsymbol{d}\|\\) is the Euclidean norm of the distance between the center of mass of two bodies (\\({\text{m}}\\))
  • \\(\boldsymbol{\hat{d}}\\) is the unit vector directed from the center of the large mass to the center of the smaller mass (\\({\text{m}}\\))
  • \\(\boldsymbol{d}\\) is the distance between the center of mass of the rigid bodies (\\({\text{m}}\\))
| |Notes |
  • Two rigid bodies in the universe attract each other with a force \\(\boldsymbol{F}\\) that is directly proportional to the product of their masses, \\({m\_{1}}\\) and \\({m\_{2}}\\), and inversely proportional to the squared distance \\({\|\boldsymbol{d}\|^{2}}\\) between them.
| |Source |-- | @@ -56,7 +56,7 @@ This section focuses on the general equations and laws that GamePhysics is based |Refname |TM:NewtonSecLawRotMot | |:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's second law for rotational motion | -|Equation |\\[\boldsymbol{τ}=\boldsymbol{I} α\\] | +|Equation |\\[\boldsymbol{τ}=\boldsymbol{I}\\,α\\] | |Description|
  • \\(\boldsymbol{τ}\\) is the torque (\\(\text{N}\text{m}\\))
  • \\(\boldsymbol{I}\\) is the moment of inertia (\\(\text{kg}\text{m}^{2}\\))
  • \\(α\\) is the angular acceleration (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Notes |
  • The net torque \\(\boldsymbol{τ}\\) on a rigid body is proportional to its angular acceleration \\(α\\), where \\(\boldsymbol{I}\\) denotes the moment of inertia of the rigid body as the constant of proportionality.
  • We also assume that all rigid bodies involved are two-dimensional (from [A:objectDimension](./SecAssumps.md#assumpOD)).
| |Source |-- | diff --git a/code/stable/glassbr/SRS/HTML/GlassBR_SRS.html b/code/stable/glassbr/SRS/HTML/GlassBR_SRS.html index f927ef8e71..e2751ddee5 100644 --- a/code/stable/glassbr/SRS/HTML/GlassBR_SRS.html +++ b/code/stable/glassbr/SRS/HTML/GlassBR_SRS.html @@ -1014,20 +1014,20 @@

Data Definitions

@@ -1316,7 +1316,7 @@

Data Definitions

- + @@ -1448,7 +1448,7 @@

Instance Models

@@ -1621,7 +1621,7 @@

Instance Models

@@ -1712,7 +1712,7 @@

Instance Models

@@ -1885,7 +1885,7 @@

Instance Models

@@ -2045,7 +2045,7 @@

Instance Models

- + diff --git a/code/stable/glassbr/SRS/Jupyter/GlassBR_SRS.ipynb b/code/stable/glassbr/SRS/Jupyter/GlassBR_SRS.ipynb index 1d8e7caed8..3e3987eff4 100644 --- a/code/stable/glassbr/SRS/Jupyter/GlassBR_SRS.ipynb +++ b/code/stable/glassbr/SRS/Jupyter/GlassBR_SRS.ipynb @@ -517,7 +517,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -850,7 +850,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -997,7 +997,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1172,7 +1172,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1260,7 +1260,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1435,7 +1435,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1603,7 +1603,7 @@ "\n", "\n", "\n", "\n", "\n", diff --git a/code/stable/glassbr/SRS/PDF/GlassBR_SRS.tex b/code/stable/glassbr/SRS/PDF/GlassBR_SRS.tex index eb8ca4c2c6..42c61976b7 100644 --- a/code/stable/glassbr/SRS/PDF/GlassBR_SRS.tex +++ b/code/stable/glassbr/SRS/PDF/GlassBR_SRS.tex @@ -475,20 +475,20 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - h=\frac{1}{1000} \begin{cases} - 2.16, & t=2.5\\ - 2.59, & t=2.7\\ - 2.92, & t=3.0\\ - 3.78, & t=4.0\\ - 4.57, & t=5.0\\ - 5.56, & t=6.0\\ - 7.42, & t=8.0\\ - 9.02, & t=10.0\\ - 11.91, & t=12.0\\ - 15.09, & t=16.0\\ - 18.26, & t=19.0\\ - 21.44, & t=22.0 - \end{cases} + h=\frac{1}{1000}\,\begin{cases} + 2.16, & t=2.5\\ + 2.59, & t=2.7\\ + 2.92, & t=3.0\\ + 3.78, & t=4.0\\ + 4.57, & t=5.0\\ + 5.56, & t=6.0\\ + 7.42, & t=8.0\\ + 9.02, & t=10.0\\ + 11.91, & t=12.0\\ + 15.09, & t=16.0\\ + 18.26, & t=19.0\\ + 21.44, & t=22.0 + \end{cases} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -689,7 +689,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {w_{\mathit{TNT}}}=w \mathit{TNT} + {w_{\mathit{TNT}}}=w\,\mathit{TNT} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -780,7 +780,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - B=\frac{k}{\left(a b\right)^{m-1}} \left(E h^{2}\right)^{m} \mathit{LDF} e^{J} + B=\frac{k}{\left(a\,b\right)^{m-1}}\,\left(E\,h^{2}\right)^{m}\,\mathit{LDF}\,e^{J} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -892,7 +892,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - \mathit{NFL}=\frac{{\hat{q}_{\text{tol}}} E h^{4}}{\left(a b\right)^{2}} + \mathit{NFL}=\frac{{\hat{q}_{\text{tol}}}\,E\,h^{4}}{\left(a\,b\right)^{2}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -949,7 +949,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - \hat{q}=\frac{q \left(a b\right)^{2}}{E h^{4} \mathit{GTF}} + \hat{q}=\frac{q\,\left(a\,b\right)^{2}}{E\,h^{4}\,\mathit{GTF}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1062,7 +1062,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - {J_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P_{\text{b}\text{tol}}}}\right) \frac{\left(a b\right)^{m-1}}{k \left(E h^{2}\right)^{m} \mathit{LDF}}\right) + {J_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P_{\text{b}\text{tol}}}}\right)\,\frac{\left(a\,b\right)^{m-1}}{k\,\left(E\,h^{2}\right)^{m}\,\mathit{LDF}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1163,7 +1163,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - \mathit{LR}=\mathit{NFL} \mathit{GTF} \mathit{LSF} + \mathit{LR}=\mathit{NFL}\,\mathit{GTF}\,\mathit{LSF} \end{displaymath} \\ \midrule Description & \begin{symbDescription} diff --git a/code/stable/glassbr/SRS/mdBook/src/SecDDs.md b/code/stable/glassbr/SRS/mdBook/src/SecDDs.md index a38db5468b..a531bb23da 100644 --- a/code/stable/glassbr/SRS/mdBook/src/SecDDs.md +++ b/code/stable/glassbr/SRS/mdBook/src/SecDDs.md @@ -8,16 +8,16 @@ This section collects and defines all the data needed to build the instance mode -|Refname |DD:minThick | -|:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| -|Label |Minimum thickness | -|Symbol |\\(h\\) | -|Units |\\({\text{m}}\\) | -|Equation |\\[h=\frac{1}{1000} \begin{cases}2.16, & t=2.5\\\\2.59, & t=2.7\\\\2.92, & t=3.0\\\\3.78, & t=4.0\\\\4.57, & t=5.0\\\\5.56, & t=6.0\\\\7.42, & t=8.0\\\\9.02, & t=10.0\\\\11.91, & t=12.0\\\\15.09, & t=16.0\\\\18.26, & t=19.0\\\\21.44, & t=22.0\end{cases}\\]| -|Description|
  • \\(h\\) is the minimum thickness (\\({\text{m}}\\))
  • \\(t\\) is the nominal thickness \\(t\in{}\{2.5,2.7,3.0,4.0,5.0,6.0,8.0,10.0,12.0,16.0,19.0,22.0\}\\) (\\({\text{mm}}\\))
| -|Notes |
  • \\(t\\) is a function that maps from the nominal thickness (\\(h\\)) to the minimum thickness.
| -|Source |[astm2009](./SecReferences.md#astm2009) | -|RefBy |[IM:sdfTol](./SecIMs.md#IM:sdfTol), [IM:riskFun](./SecIMs.md#IM:riskFun), [IM:nFL](./SecIMs.md#IM:nFL), and [IM:dimlessLoad](./SecIMs.md#IM:dimlessLoad) | +|Refname |DD:minThick | +|:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +|Label |Minimum thickness | +|Symbol |\\(h\\) | +|Units |\\({\text{m}}\\) | +|Equation |\\[h=\frac{1}{1000}\\,\begin{cases}2.16, & t=2.5\\\\2.59, & t=2.7\\\\2.92, & t=3.0\\\\3.78, & t=4.0\\\\4.57, & t=5.0\\\\5.56, & t=6.0\\\\7.42, & t=8.0\\\\9.02, & t=10.0\\\\11.91, & t=12.0\\\\15.09, & t=16.0\\\\18.26, & t=19.0\\\\21.44, & t=22.0\end{cases}\\]| +|Description|
  • \\(h\\) is the minimum thickness (\\({\text{m}}\\))
  • \\(t\\) is the nominal thickness \\(t\in{}\{2.5,2.7,3.0,4.0,5.0,6.0,8.0,10.0,12.0,16.0,19.0,22.0\}\\) (\\({\text{mm}}\\))
| +|Notes |
  • \\(t\\) is a function that maps from the nominal thickness (\\(h\\)) to the minimum thickness.
| +|Source |[astm2009](./SecReferences.md#astm2009) | +|RefBy |[IM:sdfTol](./SecIMs.md#IM:sdfTol), [IM:riskFun](./SecIMs.md#IM:riskFun), [IM:nFL](./SecIMs.md#IM:nFL), and [IM:dimlessLoad](./SecIMs.md#IM:dimlessLoad) |
@@ -97,7 +97,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Equivalent TNT charge mass | |Symbol |\\({w\_{\mathit{TNT}}}\\) | |Units |\\({\text{kg}}\\) | -|Equation |\\[{w\_{\mathit{TNT}}}=w \mathit{TNT}\\] | +|Equation |\\[{w\_{\mathit{TNT}}}=w\\,\mathit{TNT}\\] | |Description|
  • \\({w\_{\mathit{TNT}}}\\) is the equivalent TNT charge mass (\\({\text{kg}}\\))
  • \\(w\\) is the charge weight (\\({\text{kg}}\\))
  • \\(\mathit{TNT}\\) is the TNT equivalent factor (Unitless)
| |Source |[astm2009](./SecReferences.md#astm2009) | |RefBy |[DD:calofDemand](./SecDDs.md#DD:calofDemand) | diff --git a/code/stable/glassbr/SRS/mdBook/src/SecIMs.md b/code/stable/glassbr/SRS/mdBook/src/SecIMs.md index a7cae011f7..feb73e7712 100644 --- a/code/stable/glassbr/SRS/mdBook/src/SecIMs.md +++ b/code/stable/glassbr/SRS/mdBook/src/SecIMs.md @@ -17,7 +17,7 @@ The goal [GS:Predict-Glass-Withstands-Explosion](./SecGoalStmt.md#willBreakGS) i |Output |\\(B\\) | |Input Constraints |\\[a\gt{}0\\]\\[0\lt{}b\leq{}a\\] | |Output Constraints| | -|Equation |\\[B=\frac{k}{\left(a b\right)^{m-1}} \left(E h^{2}\right)^{m} \mathit{LDF} e^{J}\\] | +|Equation |\\[B=\frac{k}{\left(a\\,b\right)^{m-1}}\\,\left(E\\,h^{2}\right)^{m}\\,\mathit{LDF}\\,e^{J}\\] | |Description |
  • \\(B\\) is the risk of failure (Unitless)
  • \\(k\\) is the surface flaw parameter (\\(\frac{\text{m}^{12}}{\text{N}^{7}}\\))
  • \\(a\\) is the plate length (long dimension) (\\({\text{m}}\\))
  • \\(b\\) is the plate width (short dimension) (\\({\text{m}}\\))
  • \\(m\\) is the surface flaw parameter (\\(\frac{\text{m}^{12}}{\text{N}^{7}}\\))
  • \\(E\\) is the modulus of elasticity of glass (\\({\text{Pa}}\\))
  • \\(h\\) is the minimum thickness (\\({\text{m}}\\))
  • \\(\mathit{LDF}\\) is the load duration factor (Unitless)
  • \\(J\\) is the stress distribution factor (Function) (Unitless)
| |Notes |
  • \\(a\\) and \\(b\\) are the dimensions of the plate, where (\\(a\geq{}b\\)).
  • \\(h\\) is defined in [DD:minThick](./SecDDs.md#DD:minThick) and is based on the nominal thicknesses.
  • \\(\mathit{LDF}\\) is defined in [DD:loadDurFactor](./SecDDs.md#DD:loadDurFactor).
  • \\(J\\) is defined in [IM:stressDistFac](./SecIMs.md#IM:stressDistFac).
| |Source |[astm2009](./SecReferences.md#astm2009), [beasonEtAl1998](./SecReferences.md#beasonEtAl1998) (Eqs. 4-5), and [campidelli](./SecReferences.md#campidelli) (Eq. 14) | @@ -55,7 +55,7 @@ The goal [GS:Predict-Glass-Withstands-Explosion](./SecGoalStmt.md#willBreakGS) i |Output |\\(\mathit{NFL}\\) | |Input Constraints |\\[a\gt{}0\\]\\[0\lt{}b\leq{}a\\] | |Output Constraints| | -|Equation |\\[\mathit{NFL}=\frac{{\hat{q}\_{\text{tol}}} E h^{4}}{\left(a b\right)^{2}}\\] | +|Equation |\\[\mathit{NFL}=\frac{{\hat{q}\_{\text{tol}}}\\,E\\,h^{4}}{\left(a\\,b\right)^{2}}\\] | |Description |
  • \\(\mathit{NFL}\\) is the non-factored load (\\({\text{Pa}}\\))
  • \\({\hat{q}\_{\text{tol}}}\\) is the tolerable load (Unitless)
  • \\(E\\) is the modulus of elasticity of glass (\\({\text{Pa}}\\))
  • \\(h\\) is the minimum thickness (\\({\text{m}}\\))
  • \\(a\\) is the plate length (long dimension) (\\({\text{m}}\\))
  • \\(b\\) is the plate width (short dimension) (\\({\text{m}}\\))
| |Notes |
  • \\({\hat{q}\_{\text{tol}}}\\) is defined in [IM:tolLoad](./SecIMs.md#IM:tolLoad).
  • \\(E\\) comes from [A:standardValues](./SecAssumps.md#assumpSV).
  • \\(h\\) is defined in [DD:minThick](./SecDDs.md#DD:minThick) and is based on the nominal thicknesses.
  • \\(a\\) and \\(b\\) are the dimensions of the plate, where (\\(a\geq{}b\\)).
| |Source |[astm2009](./SecReferences.md#astm2009) | @@ -74,7 +74,7 @@ The goal [GS:Predict-Glass-Withstands-Explosion](./SecGoalStmt.md#willBreakGS) i |Output |\\(\hat{q}\\) | |Input Constraints |\\[a\gt{}0\\]\\[0\lt{}b\leq{}a\\] | |Output Constraints| | -|Equation |\\[\hat{q}=\frac{q \left(a b\right)^{2}}{E h^{4} \mathit{GTF}}\\] | +|Equation |\\[\hat{q}=\frac{q\\,\left(a\\,b\right)^{2}}{E\\,h^{4}\\,\mathit{GTF}}\\] | |Description |
  • \\(\hat{q}\\) is the dimensionless load (Unitless)
  • \\(q\\) is the applied load (demand) (\\({\text{Pa}}\\))
  • \\(a\\) is the plate length (long dimension) (\\({\text{m}}\\))
  • \\(b\\) is the plate width (short dimension) (\\({\text{m}}\\))
  • \\(E\\) is the modulus of elasticity of glass (\\({\text{Pa}}\\))
  • \\(h\\) is the minimum thickness (\\({\text{m}}\\))
  • \\(\mathit{GTF}\\) is the glass type factor (Unitless)
| |Notes |
  • \\(q\\) is the 3 second duration equivalent pressure, as given in [DD:calofDemand](./SecDDs.md#DD:calofDemand).
  • \\(a\\) and \\(b\\) are the dimensions of the plate, where (\\(a\geq{}b\\)).
  • \\(E\\) comes from [A:standardValues](./SecAssumps.md#assumpSV).
  • \\(h\\) is defined in [DD:minThick](./SecDDs.md#DD:minThick) and is based on the nominal thicknesses.
  • \\(\mathit{GTF}\\) is defined in [DD:gTF](./SecDDs.md#DD:gTF).
| |Source |[astm2009](./SecReferences.md#astm2009) and [campidelli](./SecReferences.md#campidelli) (Eq. 7) | @@ -112,7 +112,7 @@ The goal [GS:Predict-Glass-Withstands-Explosion](./SecGoalStmt.md#willBreakGS) i |Output |\\({J\_{\text{tol}}}\\) | |Input Constraints |\\[0\leq{}{P\_{\text{b}\text{tol}}}\leq{}1\\]\\[a\gt{}0\\]\\[0\lt{}b\leq{}a\\] | |Output Constraints| | -|Equation |\\[{J\_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P\_{\text{b}\text{tol}}}}\right) \frac{\left(a b\right)^{m-1}}{k \left(E h^{2}\right)^{m} \mathit{LDF}}\right)\\] | +|Equation |\\[{J\_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P\_{\text{b}\text{tol}}}}\right)\\,\frac{\left(a\\,b\right)^{m-1}}{k\\,\left(E\\,h^{2}\right)^{m}\\,\mathit{LDF}}\right)\\] | |Description |
  • \\({J\_{\text{tol}}}\\) is the tolerable stress distribution factor (Unitless)
  • \\({P\_{\text{b}\text{tol}}}\\) is the tolerable probability of breakage (Unitless)
  • \\(a\\) is the plate length (long dimension) (\\({\text{m}}\\))
  • \\(b\\) is the plate width (short dimension) (\\({\text{m}}\\))
  • \\(m\\) is the surface flaw parameter (\\(\frac{\text{m}^{12}}{\text{N}^{7}}\\))
  • \\(k\\) is the surface flaw parameter (\\(\frac{\text{m}^{12}}{\text{N}^{7}}\\))
  • \\(E\\) is the modulus of elasticity of glass (\\({\text{Pa}}\\))
  • \\(h\\) is the minimum thickness (\\({\text{m}}\\))
  • \\(\mathit{LDF}\\) is the load duration factor (Unitless)
| |Notes |
  • \\({P\_{\text{b}\text{tol}}}\\) is entered by the user.
  • \\(a\\) and \\(b\\) are the dimensions of the plate, where (\\(a\geq{}b\\)).
  • \\(m\\), \\(k\\), and \\(E\\) come from [A:standardValues](./SecAssumps.md#assumpSV).
  • \\(h\\) is defined in [DD:minThick](./SecDDs.md#DD:minThick) and is based on the nominal thicknesses.
  • \\(\mathit{LDF}\\) is defined in [DD:loadDurFactor](./SecDDs.md#DD:loadDurFactor).
| |Source |[astm2009](./SecReferences.md#astm2009) | @@ -150,7 +150,7 @@ The goal [GS:Predict-Glass-Withstands-Explosion](./SecGoalStmt.md#willBreakGS) i |Output |\\(\mathit{LR}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[\mathit{LR}=\mathit{NFL} \mathit{GTF} \mathit{LSF}\\] | +|Equation |\\[\mathit{LR}=\mathit{NFL}\\,\mathit{GTF}\\,\mathit{LSF}\\] | |Description |
  • \\(\mathit{LR}\\) is the load resistance (\\({\text{Pa}}\\))
  • \\(\mathit{NFL}\\) is the non-factored load (\\({\text{Pa}}\\))
  • \\(\mathit{GTF}\\) is the glass type factor (Unitless)
  • \\(\mathit{LSF}\\) is the load share factor (Unitless)
| |Notes |
  • \\(\mathit{LR}\\) is also called capacity.
  • \\(\mathit{NFL}\\) is defined in [IM:nFL](./SecIMs.md#IM:nFL).
  • \\(\mathit{GTF}\\) is defined in [DD:gTF](./SecDDs.md#DD:gTF).
| |Source |[astm2009](./SecReferences.md#astm2009) | diff --git a/code/stable/hghc/SRS/HTML/HGHC_SRS.html b/code/stable/hghc/SRS/HTML/HGHC_SRS.html index aba8ef870d..f81cc09183 100644 --- a/code/stable/hghc/SRS/HTML/HGHC_SRS.html +++ b/code/stable/hghc/SRS/HTML/HGHC_SRS.html @@ -212,7 +212,7 @@

Data Definitions

@@ -261,7 +261,7 @@

Data Definitions

diff --git a/code/stable/hghc/SRS/Jupyter/HGHC_SRS.ipynb b/code/stable/hghc/SRS/Jupyter/HGHC_SRS.ipynb index 5362089534..29013d21d0 100644 --- a/code/stable/hghc/SRS/Jupyter/HGHC_SRS.ipynb +++ b/code/stable/hghc/SRS/Jupyter/HGHC_SRS.ipynb @@ -125,7 +125,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -172,7 +172,7 @@ "\n", "\n", "\n", "\n", "\n", diff --git a/code/stable/hghc/SRS/PDF/HGHC_SRS.tex b/code/stable/hghc/SRS/PDF/HGHC_SRS.tex index 16b988e5de..598c40cc41 100644 --- a/code/stable/hghc/SRS/PDF/HGHC_SRS.tex +++ b/code/stable/hghc/SRS/PDF/HGHC_SRS.tex @@ -103,7 +103,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {h_{\text{g}}}=\frac{2 {k_{\text{c}}} {h_{\text{p}}}}{2 {k_{\text{c}}}+{τ_{\text{c}}} {h_{\text{p}}}} + {h_{\text{g}}}=\frac{2\,{k_{\text{c}}}\,{h_{\text{p}}}}{2\,{k_{\text{c}}}+{τ_{\text{c}}}\,{h_{\text{p}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -134,7 +134,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {h_{\text{c}}}=\frac{2 {k_{\text{c}}} {h_{\text{b}}}}{2 {k_{\text{c}}}+{τ_{\text{c}}} {h_{\text{b}}}} + {h_{\text{c}}}=\frac{2\,{k_{\text{c}}}\,{h_{\text{b}}}}{2\,{k_{\text{c}}}+{τ_{\text{c}}}\,{h_{\text{b}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} diff --git a/code/stable/hghc/SRS/mdBook/src/SecDDs.md b/code/stable/hghc/SRS/mdBook/src/SecDDs.md index d46c929267..ee027e30fc 100644 --- a/code/stable/hghc/SRS/mdBook/src/SecDDs.md +++ b/code/stable/hghc/SRS/mdBook/src/SecDDs.md @@ -13,7 +13,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Effective heat transfer coefficient between clad and fuel surface | |Symbol |\\({h\_{\text{g}}}\\) | |Units |\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\) | -|Equation |\\[{h\_{\text{g}}}=\frac{2 {k\_{\text{c}}} {h\_{\text{p}}}}{2 {k\_{\text{c}}}+{τ\_{\text{c}}} {h\_{\text{p}}}}\\] | +|Equation |\\[{h\_{\text{g}}}=\frac{2\\,{k\_{\text{c}}}\\,{h\_{\text{p}}}}{2\\,{k\_{\text{c}}}+{τ\_{\text{c}}}\\,{h\_{\text{p}}}}\\] | |Description|
  • \\({h\_{\text{g}}}\\) is the effective heat transfer coefficient between clad and fuel surface (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({k\_{\text{c}}}\\) is the clad conductivity (Unitless)
  • \\({h\_{\text{p}}}\\) is the initial gap film conductance (Unitless)
  • \\({τ\_{\text{c}}}\\) is the clad thickness (Unitless)
|
@@ -27,5 +27,5 @@ This section collects and defines all the data needed to build the instance mode |Label |Convective heat transfer coefficient between clad and coolant | |Symbol |\\({h\_{\text{c}}}\\) | |Units |\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\) | -|Equation |\\[{h\_{\text{c}}}=\frac{2 {k\_{\text{c}}} {h\_{\text{b}}}}{2 {k\_{\text{c}}}+{τ\_{\text{c}}} {h\_{\text{b}}}}\\] | +|Equation |\\[{h\_{\text{c}}}=\frac{2\\,{k\_{\text{c}}}\\,{h\_{\text{b}}}}{2\\,{k\_{\text{c}}}+{τ\_{\text{c}}}\\,{h\_{\text{b}}}}\\] | |Description|
  • \\({h\_{\text{c}}}\\) is the convective heat transfer coefficient between clad and coolant (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({k\_{\text{c}}}\\) is the clad conductivity (Unitless)
  • \\({h\_{\text{b}}}\\) is the initial coolant film conductance (Unitless)
  • \\({τ\_{\text{c}}}\\) is the clad thickness (Unitless)
| diff --git a/code/stable/pdcontroller/SRS/HTML/PDController_SRS.html b/code/stable/pdcontroller/SRS/HTML/PDController_SRS.html index cf4e59837c..7adc2e21f0 100644 --- a/code/stable/pdcontroller/SRS/HTML/PDController_SRS.html +++ b/code/stable/pdcontroller/SRS/HTML/PDController_SRS.html @@ -668,7 +668,7 @@

Theoretical Models

@@ -777,7 +777,7 @@

Theoretical Models

- + @@ -976,7 +976,7 @@

Data Definitions

- + @@ -1036,7 +1036,7 @@

Data Definitions

- + @@ -1100,7 +1100,7 @@

Data Definitions

@@ -1194,7 +1194,7 @@

Instance Models

@@ -1233,19 +1233,19 @@

Detailed derivation of Process Variable:

The Process Variable Ys in a PD Control Loop is the product of the Process Error (from DD:ddProcessError), Control Variable (from DD:ddCtrlVar), and the Power Plant (from GD:gdPowerPlant).

- \[{Y_{\text{s}}}=\left({R_{\text{s}}}-{Y_{\text{s}}}\right) \left({K_{\text{p}}}+{K_{\text{d}}} s\right) \frac{1}{s^{2}+s+20}\] + \[{Y_{\text{s}}}=\left({R_{\text{s}}}-{Y_{\text{s}}}\right)\,\left({K_{\text{p}}}+{K_{\text{d}}}\,s\right)\,\frac{1}{s^{2}+s+20}\]

Substituting the values and rearranging the equation.

- \[s^{2} {Y_{\text{s}}}+\left(1+{K_{\text{d}}}\right) {Y_{\text{s}}} s+\left(20+{K_{\text{p}}}\right) {Y_{\text{s}}}-{R_{\text{s}}} s {K_{\text{d}}}-{R_{\text{s}}} {K_{\text{p}}}=0\] + \[s^{2}\,{Y_{\text{s}}}+\left(1+{K_{\text{d}}}\right)\,{Y_{\text{s}}}\,s+\left(20+{K_{\text{p}}}\right)\,{Y_{\text{s}}}-{R_{\text{s}}}\,s\,{K_{\text{d}}}-{R_{\text{s}}}\,{K_{\text{p}}}=0\]

Computing the Inverse Laplace Transform of a function (from TM:invLaplaceTransform) of the equation.

- \[\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right) \frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right) {y_{\text{t}}}-{K_{\text{d}}} \frac{\,d{r_{\text{t}}}}{\,dt}-{r_{\text{t}}} {K_{\text{p}}}=0\] + \[\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{y_{\text{t}}}-{K_{\text{d}}}\,\frac{\,d{r_{\text{t}}}}{\,dt}-{r_{\text{t}}}\,{K_{\text{p}}}=0\]

The Set-Point rt is a step function and a constant (from A:Set-Point). Therefore the differential of the set point is zero. Hence the equation reduces to

- \[\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right) \frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right) {y_{\text{t}}}-{r_{\text{t}}} {K_{\text{p}}}=0\] + \[\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{y_{\text{t}}}-{r_{\text{t}}}\,{K_{\text{p}}}=0\] diff --git a/code/stable/pdcontroller/SRS/Jupyter/PDController_SRS.ipynb b/code/stable/pdcontroller/SRS/Jupyter/PDController_SRS.ipynb index 3530a52fa8..d30d682e9a 100644 --- a/code/stable/pdcontroller/SRS/Jupyter/PDController_SRS.ipynb +++ b/code/stable/pdcontroller/SRS/Jupyter/PDController_SRS.ipynb @@ -340,7 +340,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -446,7 +446,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -642,7 +642,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -709,7 +709,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -777,7 +777,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -865,7 +865,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -899,16 +899,16 @@ "\n", "\n", "The Process Variable $Y_s$ in a PD Control Loop is the product of the Process Error (from [DD:ddProcessError](#DD:ddProcessError)), Control Variable (from [DD:ddCtrlVar](#DD:ddCtrlVar)), and the Power Plant (from [GD:gdPowerPlant](#GD:gdPowerPlant)).\n", - "$${Y_{\\text{s}}}=\\left({R_{\\text{s}}}-{Y_{\\text{s}}}\\right) \\left({K_{\\text{p}}}+{K_{\\text{d}}} s\\right) \\frac{1}{s^{2}+s+20}$$\n", + "$${Y_{\\text{s}}}=\\left({R_{\\text{s}}}-{Y_{\\text{s}}}\\right)\\,\\left({K_{\\text{p}}}+{K_{\\text{d}}}\\,s\\right)\\,\\frac{1}{s^{2}+s+20}$$\n", "\n", "Substituting the values and rearranging the equation.\n", - "$$s^{2} {Y_{\\text{s}}}+\\left(1+{K_{\\text{d}}}\\right) {Y_{\\text{s}}} s+\\left(20+{K_{\\text{p}}}\\right) {Y_{\\text{s}}}-{R_{\\text{s}}} s {K_{\\text{d}}}-{R_{\\text{s}}} {K_{\\text{p}}}=0$$\n", + "$$s^{2}\\,{Y_{\\text{s}}}+\\left(1+{K_{\\text{d}}}\\right)\\,{Y_{\\text{s}}}\\,s+\\left(20+{K_{\\text{p}}}\\right)\\,{Y_{\\text{s}}}-{R_{\\text{s}}}\\,s\\,{K_{\\text{d}}}-{R_{\\text{s}}}\\,{K_{\\text{p}}}=0$$\n", "\n", "Computing the Inverse Laplace Transform of a function (from [TM:invLaplaceTransform](#TM:invLaplaceTransform)) of the equation.\n", - "$$\\frac{\\,d\\frac{\\,d{y_{\\text{t}}}}{\\,dt}}{\\,dt}+\\left(1+{K_{\\text{d}}}\\right) \\frac{\\,d{y_{\\text{t}}}}{\\,dt}+\\left(20+{K_{\\text{p}}}\\right) {y_{\\text{t}}}-{K_{\\text{d}}} \\frac{\\,d{r_{\\text{t}}}}{\\,dt}-{r_{\\text{t}}} {K_{\\text{p}}}=0$$\n", + "$$\\frac{\\,d\\frac{\\,d{y_{\\text{t}}}}{\\,dt}}{\\,dt}+\\left(1+{K_{\\text{d}}}\\right)\\,\\frac{\\,d{y_{\\text{t}}}}{\\,dt}+\\left(20+{K_{\\text{p}}}\\right)\\,{y_{\\text{t}}}-{K_{\\text{d}}}\\,\\frac{\\,d{r_{\\text{t}}}}{\\,dt}-{r_{\\text{t}}}\\,{K_{\\text{p}}}=0$$\n", "\n", "The Set-Point $r_t$ is a step function and a constant (from [A:Set-Point](#setPoint)). Therefore the differential of the set point is zero. Hence the equation reduces to\n", - "$$\\frac{\\,d\\frac{\\,d{y_{\\text{t}}}}{\\,dt}}{\\,dt}+\\left(1+{K_{\\text{d}}}\\right) \\frac{\\,d{y_{\\text{t}}}}{\\,dt}+\\left(20+{K_{\\text{p}}}\\right) {y_{\\text{t}}}-{r_{\\text{t}}} {K_{\\text{p}}}=0$$\n", + "$$\\frac{\\,d\\frac{\\,d{y_{\\text{t}}}}{\\,dt}}{\\,dt}+\\left(1+{K_{\\text{d}}}\\right)\\,\\frac{\\,d{y_{\\text{t}}}}{\\,dt}+\\left(20+{K_{\\text{p}}}\\right)\\,{y_{\\text{t}}}-{r_{\\text{t}}}\\,{K_{\\text{p}}}=0$$\n", "\n", "### Data Constraints\n", "\n", diff --git a/code/stable/pdcontroller/SRS/PDF/PDController_SRS.tex b/code/stable/pdcontroller/SRS/PDF/PDController_SRS.tex index 60e611c1f0..cedde3c71e 100644 --- a/code/stable/pdcontroller/SRS/PDF/PDController_SRS.tex +++ b/code/stable/pdcontroller/SRS/PDF/PDController_SRS.tex @@ -308,7 +308,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - {F_{\text{s}}}=\int_{\mathit{-∞}}^{∞}{{f_{\text{t}}} e^{-s t}}\,dt + {F_{\text{s}}}=\int_{\mathit{-∞}}^{∞}{{f_{\text{t}}}\,e^{-s\,t}}\,dt \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -374,7 +374,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \frac{1}{m s^{2}+c s+k} + \frac{1}{m\,s^{2}+c\,s+k} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -492,7 +492,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {P_{\text{s}}}={K_{\text{p}}} {E_{\text{s}}} + {P_{\text{s}}}={K_{\text{p}}}\,{E_{\text{s}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -531,7 +531,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {D_{\text{s}}}={K_{\text{d}}} {E_{\text{s}}} s + {D_{\text{s}}}={K_{\text{d}}}\,{E_{\text{s}}}\,s \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -571,7 +571,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {C_{\text{s}}}={E_{\text{s}}} \left({K_{\text{p}}}+{K_{\text{d}}} s\right) + {C_{\text{s}}}={E_{\text{s}}}\,\left({K_{\text{p}}}+{K_{\text{d}}}\,s\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -630,7 +630,7 @@ \subsubsection{Instance Models} \end{displaymath} \\ \midrule Equation & \begin{displaymath} - \frac{\,d^{2}{y_{\text{t}}}}{\,dt^{2}}+\left(1+{K_{\text{d}}}\right) \frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right) {{y_{\text{t}}}}={r_{\text{t}}} {K_{\text{p}}} + \frac{\,d^{2}{y_{\text{t}}}}{\,dt^{2}}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{{y_{\text{t}}}}={r_{\text{t}}}\,{K_{\text{p}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -655,22 +655,22 @@ \subsubsection{Instance Models} The Process Variable ${Y_{\text{s}}}$ in a PD Control Loop is the product of the Process Error (from \hyperref[DD:ddProcessError]{DD:ddProcessError}), Control Variable (from \hyperref[DD:ddCtrlVar]{DD:ddCtrlVar}), and the Power Plant (from \hyperref[GD:gdPowerPlant]{GD:gdPowerPlant}). \begin{displaymath} -{Y_{\text{s}}}=\left({R_{\text{s}}}-{Y_{\text{s}}}\right) \left({K_{\text{p}}}+{K_{\text{d}}} s\right) \frac{1}{s^{2}+s+20} +{Y_{\text{s}}}=\left({R_{\text{s}}}-{Y_{\text{s}}}\right)\,\left({K_{\text{p}}}+{K_{\text{d}}}\,s\right)\,\frac{1}{s^{2}+s+20} \end{displaymath} Substituting the values and rearranging the equation. \begin{displaymath} -s^{2} {Y_{\text{s}}}+\left(1+{K_{\text{d}}}\right) {Y_{\text{s}}} s+\left(20+{K_{\text{p}}}\right) {Y_{\text{s}}}-{R_{\text{s}}} s {K_{\text{d}}}-{R_{\text{s}}} {K_{\text{p}}}=0 +s^{2}\,{Y_{\text{s}}}+\left(1+{K_{\text{d}}}\right)\,{Y_{\text{s}}}\,s+\left(20+{K_{\text{p}}}\right)\,{Y_{\text{s}}}-{R_{\text{s}}}\,s\,{K_{\text{d}}}-{R_{\text{s}}}\,{K_{\text{p}}}=0 \end{displaymath} Computing the Inverse Laplace Transform of a function (from \hyperref[TM:invLaplaceTransform]{TM:invLaplaceTransform}) of the equation. \begin{displaymath} -\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right) \frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right) {y_{\text{t}}}-{K_{\text{d}}} \frac{\,d{r_{\text{t}}}}{\,dt}-{r_{\text{t}}} {K_{\text{p}}}=0 +\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{y_{\text{t}}}-{K_{\text{d}}}\,\frac{\,d{r_{\text{t}}}}{\,dt}-{r_{\text{t}}}\,{K_{\text{p}}}=0 \end{displaymath} The Set-Point ${r_{\text{t}}}$ is a step function and a constant (from \hyperref[setPoint]{A:Set-Point}). Therefore the differential of the set point is zero. Hence the equation reduces to \begin{displaymath} -\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right) \frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right) {y_{\text{t}}}-{r_{\text{t}}} {K_{\text{p}}}=0 +\frac{\,d\frac{\,d{y_{\text{t}}}}{\,dt}}{\,dt}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{y_{\text{t}}}-{r_{\text{t}}}\,{K_{\text{p}}}=0 \end{displaymath} \subsubsection{Data Constraints} \label{Sec:DataConstraints} diff --git a/code/stable/pdcontroller/SRS/mdBook/src/SecDDs.md b/code/stable/pdcontroller/SRS/mdBook/src/SecDDs.md index 660ff0d56f..dc1fe98872 100644 --- a/code/stable/pdcontroller/SRS/mdBook/src/SecDDs.md +++ b/code/stable/pdcontroller/SRS/mdBook/src/SecDDs.md @@ -30,7 +30,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Proportional control in the frequency domain | |Symbol |\\({P\_{\text{s}}}\\) | |Units |Unitless | -|Equation |\\[{P\_{\text{s}}}={K\_{\text{p}}} {E\_{\text{s}}}\\] | +|Equation |\\[{P\_{\text{s}}}={K\_{\text{p}}}\\,{E\_{\text{s}}}\\] | |Description|
  • \\({P\_{\text{s}}}\\) is the Proportional control in the frequency domain (Unitless)
  • \\({K\_{\text{p}}}\\) is the Proportional Gain (Unitless)
  • \\({E\_{\text{s}}}\\) is the Process Error in the frequency domain (Unitless)
| |Notes |
  • The Proportional Controller is the product of the Proportional Gain and the Process Error (from [DD:ddProcessError](./SecDDs.md#DD:ddProcessError)). The equation is converted to the frequency domain by applying the Laplace transform (from [TM:laplaceTransform](./SecTMs.md#TM:laplaceTransform)).
| |Source |[johnson2008](./SecReferences.md#johnson2008) | @@ -47,7 +47,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Derivative control in the frequency domain | |Symbol |\\({D\_{\text{s}}}\\) | |Units |Unitless | -|Equation |\\[{D\_{\text{s}}}={K\_{\text{d}}} {E\_{\text{s}}} s\\] | +|Equation |\\[{D\_{\text{s}}}={K\_{\text{d}}}\\,{E\_{\text{s}}}\\,s\\] | |Description|
  • \\({D\_{\text{s}}}\\) is the Derivative control in the frequency domain (Unitless)
  • \\({K\_{\text{d}}}\\) is the Derivative Gain (Unitless)
  • \\({E\_{\text{s}}}\\) is the Process Error in the frequency domain (Unitless)
  • \\(s\\) is the Complex frequency-domain parameter (Unitless)
| |Notes |
  • The Derivative Controller is the product of the Derivative Gain and the differential of the Process Error (from [DD:ddProcessError](./SecDDs.md#DD:ddProcessError)). The equation is converted to the frequency domain by applying the Laplace transform (from [TM:laplaceTransform](./SecTMs.md#TM:laplaceTransform)). A pure form of the Derivative controller is used in this application (from [A:Unfiltered Derivative](./SecAssumps.md#unfilteredDerivative)).
| |Source |[johnson2008](./SecReferences.md#johnson2008) | @@ -64,7 +64,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Control Variable in the frequency domain | |Symbol |\\({C\_{\text{s}}}\\) | |Units |Unitless | -|Equation |\\[{C\_{\text{s}}}={E\_{\text{s}}} \left({K\_{\text{p}}}+{K\_{\text{d}}} s\right)\\] | +|Equation |\\[{C\_{\text{s}}}={E\_{\text{s}}}\\,\left({K\_{\text{p}}}+{K\_{\text{d}}}\\,s\right)\\] | |Description|
  • \\({C\_{\text{s}}}\\) is the Control Variable in the frequency domain (Unitless)
  • \\({E\_{\text{s}}}\\) is the Process Error in the frequency domain (Unitless)
  • \\({K\_{\text{p}}}\\) is the Proportional Gain (Unitless)
  • \\({K\_{\text{d}}}\\) is the Derivative Gain (Unitless)
  • \\(s\\) is the Complex frequency-domain parameter (Unitless)
| |Notes |
  • The Control Variable is the output of the controller. In this case, it is the sum of the Proportional (from [DD:ddPropCtrl](./SecDDs.md#DD:ddPropCtrl)) and Derivative (from [DD:ddDerivCtrl](./SecDDs.md#DD:ddDerivCtrl)) controllers. The parallel (from [A:Parallel Equation](./SecAssumps.md#parallelEq)) and de-coupled (from [A:Decoupled equation](./SecAssumps.md#decoupled)) form of the PD equation is used in this document.
| |Source |[johnson2008](./SecReferences.md#johnson2008) | diff --git a/code/stable/pdcontroller/SRS/mdBook/src/SecIMs.md b/code/stable/pdcontroller/SRS/mdBook/src/SecIMs.md index 7e0194af39..5fef25925f 100644 --- a/code/stable/pdcontroller/SRS/mdBook/src/SecIMs.md +++ b/code/stable/pdcontroller/SRS/mdBook/src/SecIMs.md @@ -15,7 +15,7 @@ This section transforms the problem defined in the [problem description](./SecPr |Output |\\({y\_{\text{t}}}\\) | |Input Constraints |\\[{r\_{\text{t}}}\gt{}0\\]\\[{K\_{\text{p}}}\gt{}0\\]\\[{K\_{\text{d}}}\gt{}0\\] | |Output Constraints|\\[{y\_{\text{t}}}\gt{}0\\] | -|Equation |\\[\frac{\\,d^{2}{y\_{\text{t}}}}{\\,dt^{2}}+\left(1+{K\_{\text{d}}}\right) \frac{\\,d{y\_{\text{t}}}}{\\,dt}+\left(20+{K\_{\text{p}}}\right) {{y\_{\text{t}}}}={r\_{\text{t}}} {K\_{\text{p}}}\\] | +|Equation |\\[\frac{\\,d^{2}{y\_{\text{t}}}}{\\,dt^{2}}+\left(1+{K\_{\text{d}}}\right)\\,\frac{\\,d{y\_{\text{t}}}}{\\,dt}+\left(20+{K\_{\text{p}}}\right)\\,{{y\_{\text{t}}}}={r\_{\text{t}}}\\,{K\_{\text{p}}}\\] | |Description |
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({y\_{\text{t}}}\\) is the Process Variable (Unitless)
  • \\({K\_{\text{d}}}\\) is the Derivative Gain (Unitless)
  • \\({K\_{\text{p}}}\\) is the Proportional Gain (Unitless)
  • \\({r\_{\text{t}}}\\) is the Set-Point (Unitless)
| |Source |[abbasi2015](./SecReferences.md#abbasi2015) and [johnson2008](./SecReferences.md#johnson2008) | |RefBy |[FR:Output-Values](./SecFRs.md#outputValues) and [FR:Calculate-Values](./SecFRs.md#calculateValues) | @@ -24,16 +24,16 @@ This section transforms the problem defined in the [problem description](./SecPr The Process Variable \\({Y\_{\text{s}}}\\) in a PD Control Loop is the product of the Process Error (from [DD:ddProcessError](./SecDDs.md#DD:ddProcessError)), Control Variable (from [DD:ddCtrlVar](./SecDDs.md#DD:ddCtrlVar)), and the Power Plant (from [GD:gdPowerPlant](./SecGDs.md#GD:gdPowerPlant)). -\\[{Y\_{\text{s}}}=\left({R\_{\text{s}}}-{Y\_{\text{s}}}\right) \left({K\_{\text{p}}}+{K\_{\text{d}}} s\right) \frac{1}{s^{2}+s+20}\\] +\\[{Y\_{\text{s}}}=\left({R\_{\text{s}}}-{Y\_{\text{s}}}\right)\\,\left({K\_{\text{p}}}+{K\_{\text{d}}}\\,s\right)\\,\frac{1}{s^{2}+s+20}\\] Substituting the values and rearranging the equation. -\\[s^{2} {Y\_{\text{s}}}+\left(1+{K\_{\text{d}}}\right) {Y\_{\text{s}}} s+\left(20+{K\_{\text{p}}}\right) {Y\_{\text{s}}}-{R\_{\text{s}}} s {K\_{\text{d}}}-{R\_{\text{s}}} {K\_{\text{p}}}=0\\] +\\[s^{2}\\,{Y\_{\text{s}}}+\left(1+{K\_{\text{d}}}\right)\\,{Y\_{\text{s}}}\\,s+\left(20+{K\_{\text{p}}}\right)\\,{Y\_{\text{s}}}-{R\_{\text{s}}}\\,s\\,{K\_{\text{d}}}-{R\_{\text{s}}}\\,{K\_{\text{p}}}=0\\] Computing the Inverse Laplace Transform of a function (from [TM:invLaplaceTransform](./SecTMs.md#TM:invLaplaceTransform)) of the equation. -\\[\frac{\\,d\frac{\\,d{y\_{\text{t}}}}{\\,dt}}{\\,dt}+\left(1+{K\_{\text{d}}}\right) \frac{\\,d{y\_{\text{t}}}}{\\,dt}+\left(20+{K\_{\text{p}}}\right) {y\_{\text{t}}}-{K\_{\text{d}}} \frac{\\,d{r\_{\text{t}}}}{\\,dt}-{r\_{\text{t}}} {K\_{\text{p}}}=0\\] +\\[\frac{\\,d\frac{\\,d{y\_{\text{t}}}}{\\,dt}}{\\,dt}+\left(1+{K\_{\text{d}}}\right)\\,\frac{\\,d{y\_{\text{t}}}}{\\,dt}+\left(20+{K\_{\text{p}}}\right)\\,{y\_{\text{t}}}-{K\_{\text{d}}}\\,\frac{\\,d{r\_{\text{t}}}}{\\,dt}-{r\_{\text{t}}}\\,{K\_{\text{p}}}=0\\] The Set-Point \\({r\_{\text{t}}}\\) is a step function and a constant (from [A:Set-Point](./SecAssumps.md#setPoint)). Therefore the differential of the set point is zero. Hence the equation reduces to -\\[\frac{\\,d\frac{\\,d{y\_{\text{t}}}}{\\,dt}}{\\,dt}+\left(1+{K\_{\text{d}}}\right) \frac{\\,d{y\_{\text{t}}}}{\\,dt}+\left(20+{K\_{\text{p}}}\right) {y\_{\text{t}}}-{r\_{\text{t}}} {K\_{\text{p}}}=0\\] +\\[\frac{\\,d\frac{\\,d{y\_{\text{t}}}}{\\,dt}}{\\,dt}+\left(1+{K\_{\text{d}}}\right)\\,\frac{\\,d{y\_{\text{t}}}}{\\,dt}+\left(20+{K\_{\text{p}}}\right)\\,{y\_{\text{t}}}-{r\_{\text{t}}}\\,{K\_{\text{p}}}=0\\] diff --git a/code/stable/pdcontroller/SRS/mdBook/src/SecTMs.md b/code/stable/pdcontroller/SRS/mdBook/src/SecTMs.md index 54072c8113..744de06e5d 100644 --- a/code/stable/pdcontroller/SRS/mdBook/src/SecTMs.md +++ b/code/stable/pdcontroller/SRS/mdBook/src/SecTMs.md @@ -11,7 +11,7 @@ This section focuses on the general equations and laws that PD Controller is bas |Refname |TM:laplaceTransform | |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Laplace Transform | -|Equation |\\[{F\_{\text{s}}}=\int\_{\mathit{-∞}}^{∞}{{f\_{\text{t}}} e^{-s t}}\\,dt\\] | +|Equation |\\[{F\_{\text{s}}}=\int\_{\mathit{-∞}}^{∞}{{f\_{\text{t}}}\\,e^{-s\\,t}}\\,dt\\] | |Description|
  • \\({F\_{\text{s}}}\\) is the Laplace Transform of a function (Unitless)
  • \\({f\_{\text{t}}}\\) is the Function in the time domain (Unitless)
  • \\(s\\) is the Complex frequency-domain parameter (Unitless)
  • \\(t\\) is the time (\\({\text{s}}\\))
| |Notes |
  • Bilateral Laplace Transform. The Laplace transforms are typically inferred from a pre-computed table of Laplace Transforms ([laplaceWiki](./SecReferences.md#laplaceWiki)).
| |Source |[laplaceWiki](./SecReferences.md#laplaceWiki) | @@ -41,7 +41,7 @@ This section focuses on the general equations and laws that PD Controller is bas |Refname |TM:tmSOSystem | |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Second Order Mass-Spring-Damper System | -|Equation |\\[\frac{1}{m s^{2}+c s+k}\\] | +|Equation |\\[\frac{1}{m\\,s^{2}+c\\,s+k}\\] | |Description|
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(s\\) is the Complex frequency-domain parameter (Unitless)
  • \\(c\\) is the Damping coefficient of the spring (Unitless)
  • \\(k\\) is the Stiffness coefficient of the spring (\\({\text{s}}\\))
| |Notes |
  • The Transfer Function (from [A:Transfer Function](./SecAssumps.md#pwrPlantTxFnx)) of a Second Order System (mass-spring-damper) is characterized by this equation.
| |Source |[abbasi2015](./SecReferences.md#abbasi2015) | diff --git a/code/stable/projectile/Lesson/Projectile_Lesson.ipynb b/code/stable/projectile/Lesson/Projectile_Lesson.ipynb index de69ca623b..8ee5acbe72 100644 --- a/code/stable/projectile/Lesson/Projectile_Lesson.ipynb +++ b/code/stable/projectile/Lesson/Projectile_Lesson.ipynb @@ -60,21 +60,21 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "$$v={v^{\\text{i}}}+{a^{c}} t$$\n", "\n" + "$$v={v^{\\text{i}}}+{a^{c}}\\,t$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "$$p={p^{\\text{i}}}+{v^{\\text{i}}} t+\\frac{{a^{c}} t^{2}}{2}$$\n", "\n" + "$$p={p^{\\text{i}}}+{v^{\\text{i}}}\\,t+\\frac{{a^{c}}\\,t^{2}}{2}$$\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "$$v^{2}={v^{\\text{i}}}^{2}+2 {a^{c}} \\left(p-{p^{\\text{i}}}\\right)$$\n", "\n" + "$$v^{2}={v^{\\text{i}}}^{2}+2\\,{a^{c}}\\,\\left(p-{p^{\\text{i}}}\\right)$$\n", "\n" ] }, { diff --git a/code/stable/projectile/SRS/HTML/Projectile_SRS.html b/code/stable/projectile/SRS/HTML/Projectile_SRS.html index 7e6c2e849e..cf9306f92a 100644 --- a/code/stable/projectile/SRS/HTML/Projectile_SRS.html +++ b/code/stable/projectile/SRS/HTML/Projectile_SRS.html @@ -819,7 +819,7 @@

General Definitions

- + @@ -868,7 +868,7 @@

Detailed derivation of rectilinear velocity:

Performing the integration, we have the required equation:

- \[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}} t\] + \[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\,t\]
@@ -892,7 +892,7 @@

Detailed derivation of rectilinear velocity:

@@ -939,11 +939,11 @@

Detailed derivation of rectilinear position:

From GD:rectVel, we can replace v:

- \[\int_{{p^{\text{i}}}}^{p}{1}\,dp=\int_{0}^{t}{{v^{\text{i}}}+{a^{c}} t}\,dt\] + \[\int_{{p^{\text{i}}}}^{p}{1}\,dp=\int_{0}^{t}{{v^{\text{i}}}+{a^{c}}\,t}\,dt\]

Performing the integration, we have the required equation:

- \[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}} t+\frac{{a^{c}} t^{2}}{2}\] + \[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\,t+\frac{{a^{c}}\,t^{2}}{2}\]
@@ -970,8 +970,8 @@

Detailed derivation of rectilinear position:

@@ -1027,8 +1027,8 @@

Detailed derivation of velocity vector:

\end{bmatrix}\). Since we have a Cartesian coordinate system, GD:rectVel can be applied to each coordinate of the velocity vector to yield the required equation:

\[\symbf{v}\text{(}t\text{)}=\begin{bmatrix} - {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}} t\\ - {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}} t + {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}}\,t\\ + {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}}\,t \end{bmatrix}\] @@ -1054,8 +1054,8 @@

Detailed derivation of velocity vector:

@@ -1122,8 +1122,8 @@

Detailed derivation of position vector:

\end{bmatrix}\). Since we have a Cartesian coordinate system, GD:rectPos can be applied to each coordinate of the position vector to yield the required equation:

\[\symbf{p}\text{(}t\text{)}=\begin{bmatrix} - {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}} t+\frac{{{a_{\text{x}}}^{\text{c}}} t^{2}}{2}\\ - {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}} t+\frac{{{a_{\text{y}}}^{\text{c}}} t^{2}}{2} + {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}}\,t+\frac{{{a_{\text{x}}}^{\text{c}}}\,t^{2}}{2}\\ + {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}}\,t+\frac{{{a_{\text{y}}}^{\text{c}}}\,t^{2}}{2} \end{bmatrix}\] @@ -1221,7 +1221,7 @@

Data Definitions

@@ -1290,7 +1290,7 @@

Data Definitions

@@ -1374,7 +1374,7 @@

Instance Models

@@ -1428,23 +1428,23 @@

Detailed derivation of flight duration:

We know that pyi = 0 (A:launchOrigin) and ayc = −g (A:accelYGravity). Substituting these values into the y-direction of GD:posVec gives us:

- \[{p_{\text{y}}}={{v_{\text{y}}}^{\text{i}}} t-\frac{g t^{2}}{2}\] + \[{p_{\text{y}}}={{v_{\text{y}}}^{\text{i}}}\,t-\frac{g\,t^{2}}{2}\]

To find the time that the projectile lands, we want to find the t value (tflight) where py = 0 (since the target is on the x-axis from A:targetXAxis). From the equation above we get:

- \[{{v_{\text{y}}}^{\text{i}}} {t_{\text{flight}}}-\frac{g {t_{\text{flight}}}^{2}}{2}=0\] + \[{{v_{\text{y}}}^{\text{i}}}\,{t_{\text{flight}}}-\frac{g\,{t_{\text{flight}}}^{2}}{2}=0\]

Dividing by tflight (with the constraint tflight > 0) gives us:

- \[{{v_{\text{y}}}^{\text{i}}}-\frac{g {t_{\text{flight}}}}{2}=0\] + \[{{v_{\text{y}}}^{\text{i}}}-\frac{g\,{t_{\text{flight}}}}{2}=0\]

Solving for tflight gives us:

- \[{t_{\text{flight}}}=\frac{2 {{v_{\text{y}}}^{\text{i}}}}{g}\] + \[{t_{\text{flight}}}=\frac{2\,{{v_{\text{y}}}^{\text{i}}}}{g}\]

From DD:speedIY (with vi = vlaunch) we can replace vyi:

- \[{t_{\text{flight}}}=\frac{2 {v_{\text{launch}}} \sin\left(θ\right)}{g}\] + \[{t_{\text{flight}}}=\frac{2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]
@@ -1481,7 +1481,7 @@

Detailed derivation of flight duration:

@@ -1535,19 +1535,19 @@

Detailed derivation of landing position:

We know that pxi = 0 (A:launchOrigin) and axc = 0 (A:accelXZero). Substituting these values into the x-direction of GD:posVec gives us:

- \[{p_{\text{x}}}={{v_{\text{x}}}^{\text{i}}} t\] + \[{p_{\text{x}}}={{v_{\text{x}}}^{\text{i}}}\,t\]

To find the landing position, we want to find the px value (pland) at flight duration (from IM:calOfLandingTime):

- \[{p_{\text{land}}}=\frac{{{v_{\text{x}}}^{\text{i}}}\cdot{}2 {v_{\text{launch}}} \sin\left(θ\right)}{g}\] + \[{p_{\text{land}}}=\frac{{{v_{\text{x}}}^{\text{i}}}\cdot{}2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]

From DD:speedIX (with vi = vlaunch) we can replace vxi:

- \[{p_{\text{land}}}=\frac{{v_{\text{launch}}} \cos\left(θ\right)\cdot{}2 {v_{\text{launch}}} \sin\left(θ\right)}{g}\] + \[{p_{\text{land}}}=\frac{{v_{\text{launch}}}\,\cos\left(θ\right)\cdot{}2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]

Rearranging this gives us the required equation:

- \[{p_{\text{land}}}=\frac{2 {v_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g}\] + \[{p_{\text{land}}}=\frac{2\,{v_{\text{launch}}}^{2}\,\sin\left(θ\right)\,\cos\left(θ\right)}{g}\]
diff --git a/code/stable/projectile/SRS/Jupyter/Projectile_SRS.ipynb b/code/stable/projectile/SRS/Jupyter/Projectile_SRS.ipynb index ee6b4630ae..7e5da7d379 100644 --- a/code/stable/projectile/SRS/Jupyter/Projectile_SRS.ipynb +++ b/code/stable/projectile/SRS/Jupyter/Projectile_SRS.ipynb @@ -456,7 +456,7 @@ "
\n", "\n", "\n", "\n", "\n", @@ -495,7 +495,7 @@ "$$\\int_{{v^{\\text{i}}}}^{v}{1}\\,dv=\\int_{0}^{t}{{a^{c}}}\\,dt$$\n", "\n", "Performing the integration, we have the required equation:\n", - "$$v\\text{(}t\\text{)}={v^{\\text{i}}}+{a^{c}} t$$\n", + "$$v\\text{(}t\\text{)}={v^{\\text{i}}}+{a^{c}}\\,t$$\n", "
\n", "\n", "
Equation\n", - "$$PE=m \\symbf{g} h$$\n", + "$$PE=m\\,\\symbf{g}\\,h$$\n", "
Equation\n", - "$$\\symbf{I}=\\displaystyle\\sum{{m_{j}} {d_{j}}^{2}}$$\n", + "$$\\symbf{I}=\\displaystyle\\sum{{m_{j}}\\,{d_{j}}^{2}}$$\n", "
Equation\n", - "$${\\symbf{v}\\text{(}t\\text{)}_{\\text{A}}}\\left({t_{\\text{c}}}\\right)={\\symbf{v}\\text{(}t\\text{)}_{\\text{A}}}\\left(t\\right)+\\frac{j}{{m_{\\text{A}}}} \\symbf{n}$$\n", + "$${\\symbf{v}\\text{(}t\\text{)}_{\\text{A}}}\\left({t_{\\text{c}}}\\right)={\\symbf{v}\\text{(}t\\text{)}_{\\text{A}}}\\left(t\\right)+\\frac{j}{{m_{\\text{A}}}}\\,\\symbf{n}$$\n", "
Equation - \[h=\frac{1}{1000} \begin{cases} - 2.16, & t=2.5\\ - 2.59, & t=2.7\\ - 2.92, & t=3.0\\ - 3.78, & t=4.0\\ - 4.57, & t=5.0\\ - 5.56, & t=6.0\\ - 7.42, & t=8.0\\ - 9.02, & t=10.0\\ - 11.91, & t=12.0\\ - 15.09, & t=16.0\\ - 18.26, & t=19.0\\ - 21.44, & t=22.0 - \end{cases}\] + \[h=\frac{1}{1000}\,\begin{cases} + 2.16, & t=2.5\\ + 2.59, & t=2.7\\ + 2.92, & t=3.0\\ + 3.78, & t=4.0\\ + 4.57, & t=5.0\\ + 5.56, & t=6.0\\ + 7.42, & t=8.0\\ + 9.02, & t=10.0\\ + 11.91, & t=12.0\\ + 15.09, & t=16.0\\ + 18.26, & t=19.0\\ + 21.44, & t=22.0 + \end{cases}\]
Equation\[{w_{\mathit{TNT}}}=w \mathit{TNT}\]\[{w_{\mathit{TNT}}}=w\,\mathit{TNT}\]
Description
Equation - \[B=\frac{k}{\left(a b\right)^{m-1}} \left(E h^{2}\right)^{m} \mathit{LDF} e^{J}\] + \[B=\frac{k}{\left(a\,b\right)^{m-1}}\,\left(E\,h^{2}\right)^{m}\,\mathit{LDF}\,e^{J}\]
Equation - \[\mathit{NFL}=\frac{{\hat{q}_{\text{tol}}} E h^{4}}{\left(a b\right)^{2}}\] + \[\mathit{NFL}=\frac{{\hat{q}_{\text{tol}}}\,E\,h^{4}}{\left(a\,b\right)^{2}}\]
Equation - \[\hat{q}=\frac{q \left(a b\right)^{2}}{E h^{4} \mathit{GTF}}\] + \[\hat{q}=\frac{q\,\left(a\,b\right)^{2}}{E\,h^{4}\,\mathit{GTF}}\]
Equation - \[{J_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P_{\text{b}\text{tol}}}}\right) \frac{\left(a b\right)^{m-1}}{k \left(E h^{2}\right)^{m} \mathit{LDF}}\right)\] + \[{J_{\text{tol}}}=\ln\left(\ln\left(\frac{1}{1-{P_{\text{b}\text{tol}}}}\right)\,\frac{\left(a\,b\right)^{m-1}}{k\,\left(E\,h^{2}\right)^{m}\,\mathit{LDF}}\right)\]
Equation\[\mathit{LR}=\mathit{NFL} \mathit{GTF} \mathit{LSF}\]\[\mathit{LR}=\mathit{NFL}\,\mathit{GTF}\,\mathit{LSF}\]
Description
Equation\n", - "$$h=\\frac{1}{1000} \\begin{cases} 2.16, & t=2.5\\\\ 2.59, & t=2.7\\\\ 2.92, & t=3.0\\\\ 3.78, & t=4.0\\\\ 4.57, & t=5.0\\\\ 5.56, & t=6.0\\\\ 7.42, & t=8.0\\\\ 9.02, & t=10.0\\\\ 11.91, & t=12.0\\\\ 15.09, & t=16.0\\\\ 18.26, & t=19.0\\\\ 21.44, & t=22.0 \\end{cases}$$\n", + "$$h=\\frac{1}{1000}\\,\\begin{cases} 2.16, & t=2.5\\\\ 2.59, & t=2.7\\\\ 2.92, & t=3.0\\\\ 3.78, & t=4.0\\\\ 4.57, & t=5.0\\\\ 5.56, & t=6.0\\\\ 7.42, & t=8.0\\\\ 9.02, & t=10.0\\\\ 11.91, & t=12.0\\\\ 15.09, & t=16.0\\\\ 18.26, & t=19.0\\\\ 21.44, & t=22.0 \\end{cases}$$\n", "
Equation\n", - "$${w_{\\mathit{TNT}}}=w \\mathit{TNT}$$\n", + "$${w_{\\mathit{TNT}}}=w\\,\\mathit{TNT}$$\n", "
Equation\n", - "$$B=\\frac{k}{\\left(a b\\right)^{m-1}} \\left(E h^{2}\\right)^{m} \\mathit{LDF} e^{J}$$\n", + "$$B=\\frac{k}{\\left(a\\,b\\right)^{m-1}}\\,\\left(E\\,h^{2}\\right)^{m}\\,\\mathit{LDF}\\,e^{J}$$\n", "
Equation\n", - "$$\\mathit{NFL}=\\frac{{\\hat{q}_{\\text{tol}}} E h^{4}}{\\left(a b\\right)^{2}}$$\n", + "$$\\mathit{NFL}=\\frac{{\\hat{q}_{\\text{tol}}}\\,E\\,h^{4}}{\\left(a\\,b\\right)^{2}}$$\n", "
Equation\n", - "$$\\hat{q}=\\frac{q \\left(a b\\right)^{2}}{E h^{4} \\mathit{GTF}}$$\n", + "$$\\hat{q}=\\frac{q\\,\\left(a\\,b\\right)^{2}}{E\\,h^{4}\\,\\mathit{GTF}}$$\n", "
Equation\n", - "$${J_{\\text{tol}}}=\\ln\\left(\\ln\\left(\\frac{1}{1-{P_{\\text{b}\\text{tol}}}}\\right) \\frac{\\left(a b\\right)^{m-1}}{k \\left(E h^{2}\\right)^{m} \\mathit{LDF}}\\right)$$\n", + "$${J_{\\text{tol}}}=\\ln\\left(\\ln\\left(\\frac{1}{1-{P_{\\text{b}\\text{tol}}}}\\right)\\,\\frac{\\left(a\\,b\\right)^{m-1}}{k\\,\\left(E\\,h^{2}\\right)^{m}\\,\\mathit{LDF}}\\right)$$\n", "
Equation\n", - "$$\\mathit{LR}=\\mathit{NFL} \\mathit{GTF} \\mathit{LSF}$$\n", + "$$\\mathit{LR}=\\mathit{NFL}\\,\\mathit{GTF}\\,\\mathit{LSF}$$\n", "
Equation - \[{h_{\text{g}}}=\frac{2 {k_{\text{c}}} {h_{\text{p}}}}{2 {k_{\text{c}}}+{τ_{\text{c}}} {h_{\text{p}}}}\] + \[{h_{\text{g}}}=\frac{2\,{k_{\text{c}}}\,{h_{\text{p}}}}{2\,{k_{\text{c}}}+{τ_{\text{c}}}\,{h_{\text{p}}}}\]
Equation - \[{h_{\text{c}}}=\frac{2 {k_{\text{c}}} {h_{\text{b}}}}{2 {k_{\text{c}}}+{τ_{\text{c}}} {h_{\text{b}}}}\] + \[{h_{\text{c}}}=\frac{2\,{k_{\text{c}}}\,{h_{\text{b}}}}{2\,{k_{\text{c}}}+{τ_{\text{c}}}\,{h_{\text{b}}}}\]
Equation\n", - "$${h_{\\text{g}}}=\\frac{2 {k_{\\text{c}}} {h_{\\text{p}}}}{2 {k_{\\text{c}}}+{τ_{\\text{c}}} {h_{\\text{p}}}}$$\n", + "$${h_{\\text{g}}}=\\frac{2\\,{k_{\\text{c}}}\\,{h_{\\text{p}}}}{2\\,{k_{\\text{c}}}+{τ_{\\text{c}}}\\,{h_{\\text{p}}}}$$\n", "
Equation\n", - "$${h_{\\text{c}}}=\\frac{2 {k_{\\text{c}}} {h_{\\text{b}}}}{2 {k_{\\text{c}}}+{τ_{\\text{c}}} {h_{\\text{b}}}}$$\n", + "$${h_{\\text{c}}}=\\frac{2\\,{k_{\\text{c}}}\\,{h_{\\text{b}}}}{2\\,{k_{\\text{c}}}+{τ_{\\text{c}}}\\,{h_{\\text{b}}}}$$\n", "
Equation - \[{F_{\text{s}}}=\int_{\mathit{-∞}}^{∞}{{f_{\text{t}}} e^{-s t}}\,dt\] + \[{F_{\text{s}}}=\int_{\mathit{-∞}}^{∞}{{f_{\text{t}}}\,e^{-s\,t}}\,dt\]
Equation\[\frac{1}{m s^{2}+c s+k}\]\[\frac{1}{m\,s^{2}+c\,s+k}\]
Description
Equation\[{P_{\text{s}}}={K_{\text{p}}} {E_{\text{s}}}\]\[{P_{\text{s}}}={K_{\text{p}}}\,{E_{\text{s}}}\]
Description
Equation\[{D_{\text{s}}}={K_{\text{d}}} {E_{\text{s}}} s\]\[{D_{\text{s}}}={K_{\text{d}}}\,{E_{\text{s}}}\,s\]
Description
Equation - \[{C_{\text{s}}}={E_{\text{s}}} \left({K_{\text{p}}}+{K_{\text{d}}} s\right)\] + \[{C_{\text{s}}}={E_{\text{s}}}\,\left({K_{\text{p}}}+{K_{\text{d}}}\,s\right)\]
Equation - \[\frac{\,d^{2}{y_{\text{t}}}}{\,dt^{2}}+\left(1+{K_{\text{d}}}\right) \frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right) {{y_{\text{t}}}}={r_{\text{t}}} {K_{\text{p}}}\] + \[\frac{\,d^{2}{y_{\text{t}}}}{\,dt^{2}}+\left(1+{K_{\text{d}}}\right)\,\frac{\,d{y_{\text{t}}}}{\,dt}+\left(20+{K_{\text{p}}}\right)\,{{y_{\text{t}}}}={r_{\text{t}}}\,{K_{\text{p}}}\]
Equation\n", - "$${F_{\\text{s}}}=\\int_{\\mathit{-∞}}^{∞}{{f_{\\text{t}}} e^{-s t}}\\,dt$$\n", + "$${F_{\\text{s}}}=\\int_{\\mathit{-∞}}^{∞}{{f_{\\text{t}}}\\,e^{-s\\,t}}\\,dt$$\n", "
Equation\n", - "$$\\frac{1}{m s^{2}+c s+k}$$\n", + "$$\\frac{1}{m\\,s^{2}+c\\,s+k}$$\n", "
Equation\n", - "$${P_{\\text{s}}}={K_{\\text{p}}} {E_{\\text{s}}}$$\n", + "$${P_{\\text{s}}}={K_{\\text{p}}}\\,{E_{\\text{s}}}$$\n", "
Equation\n", - "$${D_{\\text{s}}}={K_{\\text{d}}} {E_{\\text{s}}} s$$\n", + "$${D_{\\text{s}}}={K_{\\text{d}}}\\,{E_{\\text{s}}}\\,s$$\n", "
Equation\n", - "$${C_{\\text{s}}}={E_{\\text{s}}} \\left({K_{\\text{p}}}+{K_{\\text{d}}} s\\right)$$\n", + "$${C_{\\text{s}}}={E_{\\text{s}}}\\,\\left({K_{\\text{p}}}+{K_{\\text{d}}}\\,s\\right)$$\n", "
Equation\n", - "$$\\frac{\\,d^{2}{y_{\\text{t}}}}{\\,dt^{2}}+\\left(1+{K_{\\text{d}}}\\right) \\frac{\\,d{y_{\\text{t}}}}{\\,dt}+\\left(20+{K_{\\text{p}}}\\right) {{y_{\\text{t}}}}={r_{\\text{t}}} {K_{\\text{p}}}$$\n", + "$$\\frac{\\,d^{2}{y_{\\text{t}}}}{\\,dt^{2}}+\\left(1+{K_{\\text{d}}}\\right)\\,\\frac{\\,d{y_{\\text{t}}}}{\\,dt}+\\left(20+{K_{\\text{p}}}\\right)\\,{{y_{\\text{t}}}}={r_{\\text{t}}}\\,{K_{\\text{p}}}$$\n", "
Equation\[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}} t\]\[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\,t\]
Description
Equation - \[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}} t+\frac{{a^{c}} t^{2}}{2}\] + \[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\,t+\frac{{a^{c}}\,t^{2}}{2}\]
Equation \[\symbf{v}\text{(}t\text{)}=\begin{bmatrix} - {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}} t\\ - {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}} t + {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}}\,t\\ + {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}}\,t \end{bmatrix}\]
Equation \[\symbf{p}\text{(}t\text{)}=\begin{bmatrix} - {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}} t+\frac{{{a_{\text{x}}}^{\text{c}}} t^{2}}{2}\\ - {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}} t+\frac{{{a_{\text{y}}}^{\text{c}}} t^{2}}{2} + {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}}\,t+\frac{{{a_{\text{x}}}^{\text{c}}}\,t^{2}}{2}\\ + {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}}\,t+\frac{{{a_{\text{y}}}^{\text{c}}}\,t^{2}}{2} \end{bmatrix}\]
Equation - \[{{v_{\text{x}}}^{\text{i}}}={v^{\text{i}}} \cos\left(θ\right)\] + \[{{v_{\text{x}}}^{\text{i}}}={v^{\text{i}}}\,\cos\left(θ\right)\]
Equation - \[{{v_{\text{y}}}^{\text{i}}}={v^{\text{i}}} \sin\left(θ\right)\] + \[{{v_{\text{y}}}^{\text{i}}}={v^{\text{i}}}\,\sin\left(θ\right)\]
Equation - \[{t_{\text{flight}}}=\frac{2 {v_{\text{launch}}} \sin\left(θ\right)}{g}\] + \[{t_{\text{flight}}}=\frac{2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g}\]
Equation - \[{p_{\text{land}}}=\frac{2 {v_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g}\] + \[{p_{\text{land}}}=\frac{2\,{v_{\text{launch}}}^{2}\,\sin\left(θ\right)\,\cos\left(θ\right)}{g}\]
Equation\n", - "$$v\\text{(}t\\text{)}={v^{\\text{i}}}+{a^{c}} t$$\n", + "$$v\\text{(}t\\text{)}={v^{\\text{i}}}+{a^{c}}\\,t$$\n", "
\n", @@ -522,7 +522,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -562,10 +562,10 @@ "$$\\int_{{p^{\\text{i}}}}^{p}{1}\\,dp=\\int_{0}^{t}{v}\\,dt$$\n", "\n", "From [GD:rectVel](#GD:rectVel), we can replace $v$:\n", - "$$\\int_{{p^{\\text{i}}}}^{p}{1}\\,dp=\\int_{0}^{t}{{v^{\\text{i}}}+{a^{c}} t}\\,dt$$\n", + "$$\\int_{{p^{\\text{i}}}}^{p}{1}\\,dp=\\int_{0}^{t}{{v^{\\text{i}}}+{a^{c}}\\,t}\\,dt$$\n", "\n", "Performing the integration, we have the required equation:\n", - "$$p\\text{(}t\\text{)}={p^{\\text{i}}}+{v^{\\text{i}}} t+\\frac{{a^{c}} t^{2}}{2}$$\n", + "$$p\\text{(}t\\text{)}={p^{\\text{i}}}+{v^{\\text{i}}}\\,t+\\frac{{a^{c}}\\,t^{2}}{2}$$\n", "
\n", "\n", "
Equation\n", - "$$p\\text{(}t\\text{)}={p^{\\text{i}}}+{v^{\\text{i}}} t+\\frac{{a^{c}} t^{2}}{2}$$\n", + "$$p\\text{(}t\\text{)}={p^{\\text{i}}}+{v^{\\text{i}}}\\,t+\\frac{{a^{c}}\\,t^{2}}{2}$$\n", "
\n", @@ -592,7 +592,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -627,7 +627,7 @@ "\n", "\n", "For a two-dimensional Cartesian coordinate system ([A:twoDMotion](#twoDMotion) and [A:cartSyst](#cartSyst)), we can represent the velocity vector as $v(t) = \\begin{bmatrix} {v_{\\text{x}}}\\\\ {v_{\\text{y}}} \\end{bmatrix}$ and the acceleration vector as $a(t) = \\begin{bmatrix} {a_{\\text{x}}}\\\\ {a_{\\text{y}}} \\end{bmatrix}$. The acceleration is assumed to be constant ([A:constAccel](#constAccel)) and the constant acceleration vector is represented as $a^c = \\begin{bmatrix} {{a_{\\text{x}}}^{\\text{c}}}\\\\ {{a_{\\text{y}}}^{\\text{c}}} \\end{bmatrix}$. The initial velocity (at $t = 0$, from [A:timeStartZero](#timeStartZero)) is represented by $v^i = \\begin{bmatrix} {{v_{\\text{x}}}^{\\text{i}}}\\\\ {{v_{\\text{y}}}^{\\text{i}}} \\end{bmatrix}$. Since we have a Cartesian coordinate system, [GD:rectVel](#GD:rectVel) can be applied to each coordinate of the velocity vector to yield the required equation:\n", - "$$\\symbf{v}\\text{(}t\\text{)}=\\begin{bmatrix} {{v_{\\text{x}}}^{\\text{i}}}+{{a_{\\text{x}}}^{\\text{c}}} t\\\\ {{v_{\\text{y}}}^{\\text{i}}}+{{a_{\\text{y}}}^{\\text{c}}} t \\end{bmatrix}$$\n", + "$$\\symbf{v}\\text{(}t\\text{)}=\\begin{bmatrix} {{v_{\\text{x}}}^{\\text{i}}}+{{a_{\\text{x}}}^{\\text{c}}}\\,t\\\\ {{v_{\\text{y}}}^{\\text{i}}}+{{a_{\\text{y}}}^{\\text{c}}}\\,t \\end{bmatrix}$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{v}\\text{(}t\\text{)}=\\begin{bmatrix} {{v_{\\text{x}}}^{\\text{i}}}+{{a_{\\text{x}}}^{\\text{c}}} t\\\\ {{v_{\\text{y}}}^{\\text{i}}}+{{a_{\\text{y}}}^{\\text{c}}} t \\end{bmatrix}$$\n", + "$$\\symbf{v}\\text{(}t\\text{)}=\\begin{bmatrix} {{v_{\\text{x}}}^{\\text{i}}}+{{a_{\\text{x}}}^{\\text{c}}}\\,t\\\\ {{v_{\\text{y}}}^{\\text{i}}}+{{a_{\\text{y}}}^{\\text{c}}}\\,t \\end{bmatrix}$$\n", "
\n", @@ -654,7 +654,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -691,7 +691,7 @@ "\n", "\n", "For a two-dimensional Cartesian coordinate system ([A:twoDMotion](#twoDMotion) and [A:cartSyst](#cartSyst)), we can represent the position vector as $p(t) = \\begin{bmatrix} {p_{\\text{x}}}\\\\ {p_{\\text{y}}} \\end{bmatrix}$, the velocity vector as $v(t) = \\begin{bmatrix} {v_{\\text{x}}}\\\\ {v_{\\text{y}}} \\end{bmatrix}$, and the acceleration vector as $a(t) = \\begin{bmatrix} {a_{\\text{x}}}\\\\ {a_{\\text{y}}} \\end{bmatrix}$. The acceleration is assumed to be constant ([A:constAccel](#constAccel)) and the constant acceleration vector is represented as $a^c = \\begin{bmatrix} {{a_{\\text{x}}}^{\\text{c}}}\\\\ {{a_{\\text{y}}}^{\\text{c}}} \\end{bmatrix}$. The initial velocity (at $t = 0$, from [A:timeStartZero](#timeStartZero)) is represented by $v^i = \\begin{bmatrix} {{v_{\\text{x}}}^{\\text{i}}}\\\\ {{v_{\\text{y}}}^{\\text{i}}} \\end{bmatrix}$. Since we have a Cartesian coordinate system, [GD:rectPos](#GD:rectPos) can be applied to each coordinate of the position vector to yield the required equation:\n", - "$$\\symbf{p}\\text{(}t\\text{)}=\\begin{bmatrix} {{p_{\\text{x}}}^{\\text{i}}}+{{v_{\\text{x}}}^{\\text{i}}} t+\\frac{{{a_{\\text{x}}}^{\\text{c}}} t^{2}}{2}\\\\ {{p_{\\text{y}}}^{\\text{i}}}+{{v_{\\text{y}}}^{\\text{i}}} t+\\frac{{{a_{\\text{y}}}^{\\text{c}}} t^{2}}{2} \\end{bmatrix}$$\n", + "$$\\symbf{p}\\text{(}t\\text{)}=\\begin{bmatrix} {{p_{\\text{x}}}^{\\text{i}}}+{{v_{\\text{x}}}^{\\text{i}}}\\,t+\\frac{{{a_{\\text{x}}}^{\\text{c}}}\\,t^{2}}{2}\\\\ {{p_{\\text{y}}}^{\\text{i}}}+{{v_{\\text{y}}}^{\\text{i}}}\\,t+\\frac{{{a_{\\text{y}}}^{\\text{c}}}\\,t^{2}}{2} \\end{bmatrix}$$\n", "\n", "### Data Definitions\n", "\n", @@ -796,7 +796,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -865,7 +865,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -952,7 +952,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -996,19 +996,19 @@ "\n", "\n", "We know that $p_y^i = 0$ ([A:launchOrigin](#launchOrigin)) and $a_y^c = -g$ ([A:accelYGravity](#accelYGravity)). Substituting these values into the y-direction of [GD:posVec](#GD:posVec) gives us:\n", - "$${p_{\\text{y}}}={{v_{\\text{y}}}^{\\text{i}}} t-\\frac{g t^{2}}{2}$$\n", + "$${p_{\\text{y}}}={{v_{\\text{y}}}^{\\text{i}}}\\,t-\\frac{g\\,t^{2}}{2}$$\n", "\n", "To find the time that the projectile lands, we want to find the $t$ value ($t_flight$) where $p_y = 0$ (since the target is on the $x$-axis from [A:targetXAxis](#targetXAxis)). From the equation above we get:\n", - "$${{v_{\\text{y}}}^{\\text{i}}} {t_{\\text{flight}}}-\\frac{g {t_{\\text{flight}}}^{2}}{2}=0$$\n", + "$${{v_{\\text{y}}}^{\\text{i}}}\\,{t_{\\text{flight}}}-\\frac{g\\,{t_{\\text{flight}}}^{2}}{2}=0$$\n", "\n", "Dividing by $t_flight$ (with the constraint $t_flight > 0$) gives us:\n", - "$${{v_{\\text{y}}}^{\\text{i}}}-\\frac{g {t_{\\text{flight}}}}{2}=0$$\n", + "$${{v_{\\text{y}}}^{\\text{i}}}-\\frac{g\\,{t_{\\text{flight}}}}{2}=0$$\n", "\n", "Solving for $t_flight$ gives us:\n", - "$${t_{\\text{flight}}}=\\frac{2 {{v_{\\text{y}}}^{\\text{i}}}}{g}$$\n", + "$${t_{\\text{flight}}}=\\frac{2\\,{{v_{\\text{y}}}^{\\text{i}}}}{g}$$\n", "\n", "From [DD:speedIY](#DD:speedIY) (with $v^i = v_launch$) we can replace $v_y^i$:\n", - "$${t_{\\text{flight}}}=\\frac{2 {v_{\\text{launch}}} \\sin\\left(θ\\right)}{g}$$\n", + "$${t_{\\text{flight}}}=\\frac{2\\,{v_{\\text{launch}}}\\,\\sin\\left(θ\\right)}{g}$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{p}\\text{(}t\\text{)}=\\begin{bmatrix} {{p_{\\text{x}}}^{\\text{i}}}+{{v_{\\text{x}}}^{\\text{i}}} t+\\frac{{{a_{\\text{x}}}^{\\text{c}}} t^{2}}{2}\\\\ {{p_{\\text{y}}}^{\\text{i}}}+{{v_{\\text{y}}}^{\\text{i}}} t+\\frac{{{a_{\\text{y}}}^{\\text{c}}} t^{2}}{2} \\end{bmatrix}$$\n", + "$$\\symbf{p}\\text{(}t\\text{)}=\\begin{bmatrix} {{p_{\\text{x}}}^{\\text{i}}}+{{v_{\\text{x}}}^{\\text{i}}}\\,t+\\frac{{{a_{\\text{x}}}^{\\text{c}}}\\,t^{2}}{2}\\\\ {{p_{\\text{y}}}^{\\text{i}}}+{{v_{\\text{y}}}^{\\text{i}}}\\,t+\\frac{{{a_{\\text{y}}}^{\\text{c}}}\\,t^{2}}{2} \\end{bmatrix}$$\n", "
Equation\n", - "$${{v_{\\text{x}}}^{\\text{i}}}={v^{\\text{i}}} \\cos\\left(θ\\right)$$\n", + "$${{v_{\\text{x}}}^{\\text{i}}}={v^{\\text{i}}}\\,\\cos\\left(θ\\right)$$\n", "
Equation\n", - "$${{v_{\\text{y}}}^{\\text{i}}}={v^{\\text{i}}} \\sin\\left(θ\\right)$$\n", + "$${{v_{\\text{y}}}^{\\text{i}}}={v^{\\text{i}}}\\,\\sin\\left(θ\\right)$$\n", "
Equation\n", - "$${t_{\\text{flight}}}=\\frac{2 {v_{\\text{launch}}} \\sin\\left(θ\\right)}{g}$$\n", + "$${t_{\\text{flight}}}=\\frac{2\\,{v_{\\text{launch}}}\\,\\sin\\left(θ\\right)}{g}$$\n", "
\n", @@ -1055,7 +1055,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1099,16 +1099,16 @@ "\n", "\n", "We know that $p_x^i = 0$ ([A:launchOrigin](#launchOrigin)) and $a_x^c = 0$ ([A:accelXZero](#accelXZero)). Substituting these values into the x-direction of [GD:posVec](#GD:posVec) gives us:\n", - "$${p_{\\text{x}}}={{v_{\\text{x}}}^{\\text{i}}} t$$\n", + "$${p_{\\text{x}}}={{v_{\\text{x}}}^{\\text{i}}}\\,t$$\n", "\n", "To find the landing position, we want to find the $p_x$ value ($p_land$) at flight duration (from [IM:calOfLandingTime](#IM:calOfLandingTime)):\n", - "$${p_{\\text{land}}}=\\frac{{{v_{\\text{x}}}^{\\text{i}}}\\cdot{}2 {v_{\\text{launch}}} \\sin\\left(θ\\right)}{g}$$\n", + "$${p_{\\text{land}}}=\\frac{{{v_{\\text{x}}}^{\\text{i}}}\\cdot{}2\\,{v_{\\text{launch}}}\\,\\sin\\left(θ\\right)}{g}$$\n", "\n", "From [DD:speedIX](#DD:speedIX) (with $v^i = v_launch$) we can replace $v_x^i$:\n", - "$${p_{\\text{land}}}=\\frac{{v_{\\text{launch}}} \\cos\\left(θ\\right)\\cdot{}2 {v_{\\text{launch}}} \\sin\\left(θ\\right)}{g}$$\n", + "$${p_{\\text{land}}}=\\frac{{v_{\\text{launch}}}\\,\\cos\\left(θ\\right)\\cdot{}2\\,{v_{\\text{launch}}}\\,\\sin\\left(θ\\right)}{g}$$\n", "\n", "Rearranging this gives us the required equation:\n", - "$${p_{\\text{land}}}=\\frac{2 {v_{\\text{launch}}}^{2} \\sin\\left(θ\\right) \\cos\\left(θ\\right)}{g}$$\n", + "$${p_{\\text{land}}}=\\frac{2\\,{v_{\\text{launch}}}^{2}\\,\\sin\\left(θ\\right)\\,\\cos\\left(θ\\right)}{g}$$\n", "
\n", "\n", "
Equation\n", - "$${p_{\\text{land}}}=\\frac{2 {v_{\\text{launch}}}^{2} \\sin\\left(θ\\right) \\cos\\left(θ\\right)}{g}$$\n", + "$${p_{\\text{land}}}=\\frac{2\\,{v_{\\text{launch}}}^{2}\\,\\sin\\left(θ\\right)\\,\\cos\\left(θ\\right)}{g}$$\n", "
\n", diff --git a/code/stable/projectile/SRS/PDF/Projectile_SRS.tex b/code/stable/projectile/SRS/PDF/Projectile_SRS.tex index aad3b97af2..b77de65d2c 100644 --- a/code/stable/projectile/SRS/PDF/Projectile_SRS.tex +++ b/code/stable/projectile/SRS/PDF/Projectile_SRS.tex @@ -383,7 +383,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}} t + v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\,t \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -417,7 +417,7 @@ \subsubsection{General Definitions} Performing the integration, we have the required equation: \begin{displaymath} -v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}} t +v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\,t \end{displaymath} \medskip \noindent @@ -434,7 +434,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}} t+\frac{{a^{c}} t^{2}}{2} + p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\,t+\frac{{a^{c}}\,t^{2}}{2} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -469,12 +469,12 @@ \subsubsection{General Definitions} From \hyperref[GD:rectVel]{GD:rectVel}, we can replace $v$: \begin{displaymath} -\int_{{p^{\text{i}}}}^{p}{1}\,dp=\int_{0}^{t}{{v^{\text{i}}}+{a^{c}} t}\,dt +\int_{{p^{\text{i}}}}^{p}{1}\,dp=\int_{0}^{t}{{v^{\text{i}}}+{a^{c}}\,t}\,dt \end{displaymath} Performing the integration, we have the required equation: \begin{displaymath} -p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}} t+\frac{{a^{c}} t^{2}}{2} +p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\,t+\frac{{a^{c}}\,t^{2}}{2} \end{displaymath} \medskip \noindent @@ -492,8 +492,8 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} \symbf{v}\text{(}t\text{)}=\begin{bmatrix} - {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}} t\\ - {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}} t + {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}}\,t\\ + {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}}\,t \end{bmatrix} \end{displaymath} \\ \midrule @@ -532,8 +532,8 @@ \subsubsection{General Definitions} \begin{displaymath} \symbf{v}\text{(}t\text{)}=\begin{bmatrix} - {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}} t\\ - {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}} t + {{v_{\text{x}}}^{\text{i}}}+{{a_{\text{x}}}^{\text{c}}}\,t\\ + {{v_{\text{y}}}^{\text{i}}}+{{a_{\text{y}}}^{\text{c}}}\,t \end{bmatrix} \end{displaymath} \medskip @@ -552,8 +552,8 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} \symbf{p}\text{(}t\text{)}=\begin{bmatrix} - {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}} t+\frac{{{a_{\text{x}}}^{\text{c}}} t^{2}}{2}\\ - {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}} t+\frac{{{a_{\text{y}}}^{\text{c}}} t^{2}}{2} + {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}}\,t+\frac{{{a_{\text{x}}}^{\text{c}}}\,t^{2}}{2}\\ + {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}}\,t+\frac{{{a_{\text{y}}}^{\text{c}}}\,t^{2}}{2} \end{bmatrix} \end{displaymath} \\ \midrule @@ -598,8 +598,8 @@ \subsubsection{General Definitions} \begin{displaymath} \symbf{p}\text{(}t\text{)}=\begin{bmatrix} - {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}} t+\frac{{{a_{\text{x}}}^{\text{c}}} t^{2}}{2}\\ - {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}} t+\frac{{{a_{\text{y}}}^{\text{c}}} t^{2}}{2} + {{p_{\text{x}}}^{\text{i}}}+{{v_{\text{x}}}^{\text{i}}}\,t+\frac{{{a_{\text{x}}}^{\text{c}}}\,t^{2}}{2}\\ + {{p_{\text{y}}}^{\text{i}}}+{{v_{\text{y}}}^{\text{i}}}\,t+\frac{{{a_{\text{y}}}^{\text{c}}}\,t^{2}}{2} \end{bmatrix} \end{displaymath} \subsubsection{Data Definitions} @@ -662,7 +662,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {{v_{\text{x}}}^{\text{i}}}={v^{\text{i}}} \cos\left(θ\right) + {{v_{\text{x}}}^{\text{i}}}={v^{\text{i}}}\,\cos\left(θ\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -703,7 +703,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {{v_{\text{y}}}^{\text{i}}}={v^{\text{i}}} \sin\left(θ\right) + {{v_{\text{y}}}^{\text{i}}}={v^{\text{i}}}\,\sin\left(θ\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -759,7 +759,7 @@ \subsubsection{Instance Models} \end{displaymath} \\ \midrule Equation & \begin{displaymath} - {t_{\text{flight}}}=\frac{2 {v_{\text{launch}}} \sin\left(θ\right)}{g} + {t_{\text{flight}}}=\frac{2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -790,27 +790,27 @@ \subsubsection{Instance Models} We know that ${{p_{\text{y}}}^{\text{i}}}=0$ (\hyperref[launchOrigin]{A:launchOrigin}) and ${{a_{\text{y}}}^{\text{c}}}=-g$ (\hyperref[accelYGravity]{A:accelYGravity}). Substituting these values into the y-direction of \hyperref[GD:posVec]{GD:posVec} gives us: \begin{displaymath} -{p_{\text{y}}}={{v_{\text{y}}}^{\text{i}}} t-\frac{g t^{2}}{2} +{p_{\text{y}}}={{v_{\text{y}}}^{\text{i}}}\,t-\frac{g\,t^{2}}{2} \end{displaymath} To find the time that the projectile lands, we want to find the $t$ value (${t_{\text{flight}}}$) where ${p_{\text{y}}}=0$ (since the target is on the $x$-axis from \hyperref[targetXAxis]{A:targetXAxis}). From the equation above we get: \begin{displaymath} -{{v_{\text{y}}}^{\text{i}}} {t_{\text{flight}}}-\frac{g {t_{\text{flight}}}^{2}}{2}=0 +{{v_{\text{y}}}^{\text{i}}}\,{t_{\text{flight}}}-\frac{g\,{t_{\text{flight}}}^{2}}{2}=0 \end{displaymath} Dividing by ${t_{\text{flight}}}$ (with the constraint ${t_{\text{flight}}}\gt{}0$) gives us: \begin{displaymath} -{{v_{\text{y}}}^{\text{i}}}-\frac{g {t_{\text{flight}}}}{2}=0 +{{v_{\text{y}}}^{\text{i}}}-\frac{g\,{t_{\text{flight}}}}{2}=0 \end{displaymath} Solving for ${t_{\text{flight}}}$ gives us: \begin{displaymath} -{t_{\text{flight}}}=\frac{2 {{v_{\text{y}}}^{\text{i}}}}{g} +{t_{\text{flight}}}=\frac{2\,{{v_{\text{y}}}^{\text{i}}}}{g} \end{displaymath} From \hyperref[DD:speedIY]{DD:speedIY} (with ${v^{\text{i}}}={v_{\text{launch}}}$) we can replace ${{v_{\text{y}}}^{\text{i}}}$: \begin{displaymath} -{t_{\text{flight}}}=\frac{2 {v_{\text{launch}}} \sin\left(θ\right)}{g} +{t_{\text{flight}}}=\frac{2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g} \end{displaymath} \medskip \noindent @@ -841,7 +841,7 @@ \subsubsection{Instance Models} \end{displaymath} \\ \midrule Equation & \begin{displaymath} - {p_{\text{land}}}=\frac{2 {v_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g} + {p_{\text{land}}}=\frac{2\,{v_{\text{launch}}}^{2}\,\sin\left(θ\right)\,\cos\left(θ\right)}{g} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -872,22 +872,22 @@ \subsubsection{Instance Models} We know that ${{p_{\text{x}}}^{\text{i}}}=0$ (\hyperref[launchOrigin]{A:launchOrigin}) and ${{a_{\text{x}}}^{\text{c}}}=0$ (\hyperref[accelXZero]{A:accelXZero}). Substituting these values into the x-direction of \hyperref[GD:posVec]{GD:posVec} gives us: \begin{displaymath} -{p_{\text{x}}}={{v_{\text{x}}}^{\text{i}}} t +{p_{\text{x}}}={{v_{\text{x}}}^{\text{i}}}\,t \end{displaymath} To find the landing position, we want to find the ${p_{\text{x}}}$ value (${p_{\text{land}}}$) at flight duration (from \hyperref[IM:calOfLandingTime]{IM:calOfLandingTime}): \begin{displaymath} -{p_{\text{land}}}=\frac{{{v_{\text{x}}}^{\text{i}}}\cdot{}2 {v_{\text{launch}}} \sin\left(θ\right)}{g} +{p_{\text{land}}}=\frac{{{v_{\text{x}}}^{\text{i}}}\cdot{}2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g} \end{displaymath} From \hyperref[DD:speedIX]{DD:speedIX} (with ${v^{\text{i}}}={v_{\text{launch}}}$) we can replace ${{v_{\text{x}}}^{\text{i}}}$: \begin{displaymath} -{p_{\text{land}}}=\frac{{v_{\text{launch}}} \cos\left(θ\right)\cdot{}2 {v_{\text{launch}}} \sin\left(θ\right)}{g} +{p_{\text{land}}}=\frac{{v_{\text{launch}}}\,\cos\left(θ\right)\cdot{}2\,{v_{\text{launch}}}\,\sin\left(θ\right)}{g} \end{displaymath} Rearranging this gives us the required equation: \begin{displaymath} -{p_{\text{land}}}=\frac{2 {v_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g} +{p_{\text{land}}}=\frac{2\,{v_{\text{launch}}}^{2}\,\sin\left(θ\right)\,\cos\left(θ\right)}{g} \end{displaymath} \medskip \noindent diff --git a/code/stable/projectile/SRS/mdBook/src/SecDDs.md b/code/stable/projectile/SRS/mdBook/src/SecDDs.md index 0566a9f43a..2ccdfca5ce 100644 --- a/code/stable/projectile/SRS/mdBook/src/SecDDs.md +++ b/code/stable/projectile/SRS/mdBook/src/SecDDs.md @@ -30,7 +30,7 @@ This section collects and defines all the data needed to build the instance mode |Label |\\(x\\)-component of initial velocity | |Symbol |\\({{v\_{\text{x}}}^{\text{i}}}\\) | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{{v\_{\text{x}}}^{\text{i}}}={v^{\text{i}}} \cos\left(θ\right)\\] | +|Equation |\\[{{v\_{\text{x}}}^{\text{i}}}={v^{\text{i}}}\\,\cos\left(θ\right)\\] | |Description|
  • \\({{v\_{\text{x}}}^{\text{i}}}\\) is the \\(x\\)-component of initial velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({v^{\text{i}}}\\) is the initial speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(θ\\) is the launch angle (\\({\text{rad}}\\))
| |Notes |
  • \\({v^{\text{i}}}\\) is from [DD:vecMag](./SecDDs.md#DD:vecMag).
  • \\(θ\\) is shown in [Fig:Launch](./SecPhysSyst.md#Figure:Launch).
| |Source |-- | @@ -47,7 +47,7 @@ This section collects and defines all the data needed to build the instance mode |Label |\\(y\\)-component of initial velocity | |Symbol |\\({{v\_{\text{y}}}^{\text{i}}}\\) | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{{v\_{\text{y}}}^{\text{i}}}={v^{\text{i}}} \sin\left(θ\right)\\] | +|Equation |\\[{{v\_{\text{y}}}^{\text{i}}}={v^{\text{i}}}\\,\sin\left(θ\right)\\] | |Description|
  • \\({{v\_{\text{y}}}^{\text{i}}}\\) is the \\(y\\)-component of initial velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({v^{\text{i}}}\\) is the initial speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(θ\\) is the launch angle (\\({\text{rad}}\\))
| |Notes |
  • \\({v^{\text{i}}}\\) is from [DD:vecMag](./SecDDs.md#DD:vecMag).
  • \\(θ\\) is shown in [Fig:Launch](./SecPhysSyst.md#Figure:Launch).
| |Source |-- | diff --git a/code/stable/projectile/SRS/mdBook/src/SecGDs.md b/code/stable/projectile/SRS/mdBook/src/SecGDs.md index 17e8335716..9a8a60acca 100644 --- a/code/stable/projectile/SRS/mdBook/src/SecGDs.md +++ b/code/stable/projectile/SRS/mdBook/src/SecGDs.md @@ -12,7 +12,7 @@ This section collects the laws and equations that will be used to build the inst |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Rectilinear (1D) velocity as a function of time for constant acceleration | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}} t\\] | +|Equation |\\[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\\,t\\] | |Description|
  • \\(v\text{(}t\text{)}\\) is the 1D speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({v^{\text{i}}}\\) is the initial speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({a^{c}}\\) is the constant acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
| |Source |[hibbeler2004](./SecReferences.md#hibbeler2004) (pg. 8) | |RefBy |[GD:velVec](./SecGDs.md#GD:velVec) and [GD:rectPos](./SecGDs.md#GD:rectPos) | @@ -29,7 +29,7 @@ Rearranging and integrating, we have: Performing the integration, we have the required equation: -\\[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}} t\\] +\\[v\text{(}t\text{)}={v^{\text{i}}}+{a^{c}}\\,t\\]
@@ -41,7 +41,7 @@ Performing the integration, we have the required equation: |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Rectilinear (1D) position as a function of time for constant acceleration | |Units |\\({\text{m}}\\) | -|Equation |\\[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}} t+\frac{{a^{c}} t^{2}}{2}\\] | +|Equation |\\[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\\,t+\frac{{a^{c}}\\,t^{2}}{2}\\] | |Description|
  • \\(p\text{(}t\text{)}\\) is the 1D position (\\({\text{m}}\\))
  • \\({p^{\text{i}}}\\) is the initial position (\\({\text{m}}\\))
  • \\({v^{\text{i}}}\\) is the initial speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({a^{c}}\\) is the constant acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Source |[hibbeler2004](./SecReferences.md#hibbeler2004) (pg. 8) | |RefBy |[GD:posVec](./SecGDs.md#GD:posVec) | @@ -58,11 +58,11 @@ Rearranging and integrating, we have: From [GD:rectVel](./SecGDs.md#GD:rectVel), we can replace \\(v\\): -\\[\int\_{{p^{\text{i}}}}^{p}{1}\\,dp=\int\_{0}^{t}{{v^{\text{i}}}+{a^{c}} t}\\,dt\\] +\\[\int\_{{p^{\text{i}}}}^{p}{1}\\,dp=\int\_{0}^{t}{{v^{\text{i}}}+{a^{c}}\\,t}\\,dt\\] Performing the integration, we have the required equation: -\\[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}} t+\frac{{a^{c}} t^{2}}{2}\\] +\\[p\text{(}t\text{)}={p^{\text{i}}}+{v^{\text{i}}}\\,t+\frac{{a^{c}}\\,t^{2}}{2}\\]
@@ -74,7 +74,7 @@ Performing the integration, we have the required equation: |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Velocity vector as a function of time for 2D motion under constant acceleration | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{{v\_{\text{x}}}^{\text{i}}}+{{a\_{\text{x}}}^{\text{c}}} t\\\\{{v\_{\text{y}}}^{\text{i}}}+{{a\_{\text{y}}}^{\text{c}}} t\end{bmatrix}\\] | +|Equation |\\[\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{{v\_{\text{x}}}^{\text{i}}}+{{a\_{\text{x}}}^{\text{c}}}\\,t\\\\{{v\_{\text{y}}}^{\text{i}}}+{{a\_{\text{y}}}^{\text{c}}}\\,t\end{bmatrix}\\] | |Description|
  • \\(\boldsymbol{v}\text{(}t\text{)}\\) is the velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({{v\_{\text{x}}}^{\text{i}}}\\) is the \\(x\\)-component of initial velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({{a\_{\text{x}}}^{\text{c}}}\\) is the \\(x\\)-component of constant acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({{v\_{\text{y}}}^{\text{i}}}\\) is the \\(y\\)-component of initial velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({{a\_{\text{y}}}^{\text{c}}}\\) is the \\(y\\)-component of constant acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy | | @@ -83,7 +83,7 @@ Performing the integration, we have the required equation: For a two-dimensional Cartesian coordinate system ([A:twoDMotion](./SecAssumps.md#twoDMotion) and [A:cartSyst](./SecAssumps.md#cartSyst)), we can represent the velocity vector as \\(\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{v\_{\text{x}}}\\\\{v\_{\text{y}}}\end{bmatrix}\\) and the acceleration vector as \\(\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}{a\_{\text{x}}}\\\\{a\_{\text{y}}}\end{bmatrix}\\). The acceleration is assumed to be constant ([A:constAccel](./SecAssumps.md#constAccel)) and the constant acceleration vector is represented as \\({\boldsymbol{a}^{\text{c}}}=\begin{bmatrix}{{a\_{\text{x}}}^{\text{c}}}\\\\{{a\_{\text{y}}}^{\text{c}}}\end{bmatrix}\\). The initial velocity (at \\(t=0\\), from [A:timeStartZero](./SecAssumps.md#timeStartZero)) is represented by \\({\boldsymbol{v}^{\text{i}}}=\begin{bmatrix}{{v\_{\text{x}}}^{\text{i}}}\\\\{{v\_{\text{y}}}^{\text{i}}}\end{bmatrix}\\). Since we have a Cartesian coordinate system, [GD:rectVel](./SecGDs.md#GD:rectVel) can be applied to each coordinate of the velocity vector to yield the required equation: -\\[\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{{v\_{\text{x}}}^{\text{i}}}+{{a\_{\text{x}}}^{\text{c}}} t\\\\{{v\_{\text{y}}}^{\text{i}}}+{{a\_{\text{y}}}^{\text{c}}} t\end{bmatrix}\\] +\\[\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{{v\_{\text{x}}}^{\text{i}}}+{{a\_{\text{x}}}^{\text{c}}}\\,t\\\\{{v\_{\text{y}}}^{\text{i}}}+{{a\_{\text{y}}}^{\text{c}}}\\,t\end{bmatrix}\\]
@@ -95,7 +95,7 @@ For a two-dimensional Cartesian coordinate system ([A:twoDMotion](./SecAssumps.m |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Position vector as a function of time for 2D motion under constant acceleration | |Units |\\({\text{m}}\\) | -|Equation |\\[\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{{p\_{\text{x}}}^{\text{i}}}+{{v\_{\text{x}}}^{\text{i}}} t+\frac{{{a\_{\text{x}}}^{\text{c}}} t^{2}}{2}\\\\{{p\_{\text{y}}}^{\text{i}}}+{{v\_{\text{y}}}^{\text{i}}} t+\frac{{{a\_{\text{y}}}^{\text{c}}} t^{2}}{2}\end{bmatrix}\\] | +|Equation |\\[\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{{p\_{\text{x}}}^{\text{i}}}+{{v\_{\text{x}}}^{\text{i}}}\\,t+\frac{{{a\_{\text{x}}}^{\text{c}}}\\,t^{2}}{2}\\\\{{p\_{\text{y}}}^{\text{i}}}+{{v\_{\text{y}}}^{\text{i}}}\\,t+\frac{{{a\_{\text{y}}}^{\text{c}}}\\,t^{2}}{2}\end{bmatrix}\\] | |Description|
  • \\(\boldsymbol{p}\text{(}t\text{)}\\) is the position (\\({\text{m}}\\))
  • \\({{p\_{\text{x}}}^{\text{i}}}\\) is the \\(x\\)-component of initial position (\\({\text{m}}\\))
  • \\({{v\_{\text{x}}}^{\text{i}}}\\) is the \\(x\\)-component of initial velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({{a\_{\text{x}}}^{\text{c}}}\\) is the \\(x\\)-component of constant acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\({{p\_{\text{y}}}^{\text{i}}}\\) is the \\(y\\)-component of initial position (\\({\text{m}}\\))
  • \\({{v\_{\text{y}}}^{\text{i}}}\\) is the \\(y\\)-component of initial velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\({{a\_{\text{y}}}^{\text{c}}}\\) is the \\(y\\)-component of constant acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy |[IM:calOfLandingDist](./SecIMs.md#IM:calOfLandingDist) and [IM:calOfLandingTime](./SecIMs.md#IM:calOfLandingTime) | @@ -104,4 +104,4 @@ For a two-dimensional Cartesian coordinate system ([A:twoDMotion](./SecAssumps.m For a two-dimensional Cartesian coordinate system ([A:twoDMotion](./SecAssumps.md#twoDMotion) and [A:cartSyst](./SecAssumps.md#cartSyst)), we can represent the position vector as \\(\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{p\_{\text{x}}}\\\\{p\_{\text{y}}}\end{bmatrix}\\), the velocity vector as \\(\boldsymbol{v}\text{(}t\text{)}=\begin{bmatrix}{v\_{\text{x}}}\\\\{v\_{\text{y}}}\end{bmatrix}\\), and the acceleration vector as \\(\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}{a\_{\text{x}}}\\\\{a\_{\text{y}}}\end{bmatrix}\\). The acceleration is assumed to be constant ([A:constAccel](./SecAssumps.md#constAccel)) and the constant acceleration vector is represented as \\({\boldsymbol{a}^{\text{c}}}=\begin{bmatrix}{{a\_{\text{x}}}^{\text{c}}}\\\\{{a\_{\text{y}}}^{\text{c}}}\end{bmatrix}\\). The initial velocity (at \\(t=0\\), from [A:timeStartZero](./SecAssumps.md#timeStartZero)) is represented by \\({\boldsymbol{v}^{\text{i}}}=\begin{bmatrix}{{v\_{\text{x}}}^{\text{i}}}\\\\{{v\_{\text{y}}}^{\text{i}}}\end{bmatrix}\\). Since we have a Cartesian coordinate system, [GD:rectPos](./SecGDs.md#GD:rectPos) can be applied to each coordinate of the position vector to yield the required equation: -\\[\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{{p\_{\text{x}}}^{\text{i}}}+{{v\_{\text{x}}}^{\text{i}}} t+\frac{{{a\_{\text{x}}}^{\text{c}}} t^{2}}{2}\\\\{{p\_{\text{y}}}^{\text{i}}}+{{v\_{\text{y}}}^{\text{i}}} t+\frac{{{a\_{\text{y}}}^{\text{c}}} t^{2}}{2}\end{bmatrix}\\] +\\[\boldsymbol{p}\text{(}t\text{)}=\begin{bmatrix}{{p\_{\text{x}}}^{\text{i}}}+{{v\_{\text{x}}}^{\text{i}}}\\,t+\frac{{{a\_{\text{x}}}^{\text{c}}}\\,t^{2}}{2}\\\\{{p\_{\text{y}}}^{\text{i}}}+{{v\_{\text{y}}}^{\text{i}}}\\,t+\frac{{{a\_{\text{y}}}^{\text{c}}}\\,t^{2}}{2}\end{bmatrix}\\] diff --git a/code/stable/projectile/SRS/mdBook/src/SecIMs.md b/code/stable/projectile/SRS/mdBook/src/SecIMs.md index 0c9bf24ddb..7d28a185ef 100644 --- a/code/stable/projectile/SRS/mdBook/src/SecIMs.md +++ b/code/stable/projectile/SRS/mdBook/src/SecIMs.md @@ -15,7 +15,7 @@ This section transforms the problem defined in the [problem description](./SecPr |Output |\\({t\_{\text{flight}}}\\) | |Input Constraints |\\[{v\_{\text{launch}}}\gt{}0\\]\\[0\lt{}θ\lt{}\frac{π}{2}\\] | |Output Constraints|\\[{t\_{\text{flight}}}\gt{}0\\] | -|Equation |\\[{t\_{\text{flight}}}=\frac{2 {v\_{\text{launch}}} \sin\left(θ\right)}{g}\\] | +|Equation |\\[{t\_{\text{flight}}}=\frac{2\\,{v\_{\text{launch}}}\\,\sin\left(θ\right)}{g}\\] | |Description |
  • \\({t\_{\text{flight}}}\\) is the flight duration (\\({\text{s}}\\))
  • \\({v\_{\text{launch}}}\\) is the launch speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(θ\\) is the launch angle (\\({\text{rad}}\\))
  • \\(g\\) is the magnitude of gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The constraint \\(0\lt{}θ\lt{}\frac{π}{2}\\) is from [A:posXDirection](./SecAssumps.md#posXDirection) and [A:yAxisGravity](./SecAssumps.md#yAxisGravity), and is shown in [Fig:Launch](./SecPhysSyst.md#Figure:Launch).
  • \\(g\\) is defined in [A:gravAccelValue](./SecAssumps.md#gravAccelValue).
  • The constraint \\({t\_{\text{flight}}}\gt{}0\\) is from [A:timeStartZero](./SecAssumps.md#timeStartZero).
| |Source |-- | @@ -25,23 +25,23 @@ This section transforms the problem defined in the [problem description](./SecPr We know that \\({{p\_{\text{y}}}^{\text{i}}}=0\\) ([A:launchOrigin](./SecAssumps.md#launchOrigin)) and \\({{a\_{\text{y}}}^{\text{c}}}=-g\\) ([A:accelYGravity](./SecAssumps.md#accelYGravity)). Substituting these values into the y-direction of [GD:posVec](./SecGDs.md#GD:posVec) gives us: -\\[{p\_{\text{y}}}={{v\_{\text{y}}}^{\text{i}}} t-\frac{g t^{2}}{2}\\] +\\[{p\_{\text{y}}}={{v\_{\text{y}}}^{\text{i}}}\\,t-\frac{g\\,t^{2}}{2}\\] To find the time that the projectile lands, we want to find the \\(t\\) value (\\({t\_{\text{flight}}}\\)) where \\({p\_{\text{y}}}=0\\) (since the target is on the \\(x\\)-axis from [A:targetXAxis](./SecAssumps.md#targetXAxis)). From the equation above we get: -\\[{{v\_{\text{y}}}^{\text{i}}} {t\_{\text{flight}}}-\frac{g {t\_{\text{flight}}}^{2}}{2}=0\\] +\\[{{v\_{\text{y}}}^{\text{i}}}\\,{t\_{\text{flight}}}-\frac{g\\,{t\_{\text{flight}}}^{2}}{2}=0\\] Dividing by \\({t\_{\text{flight}}}\\) (with the constraint \\({t\_{\text{flight}}}\gt{}0\\)) gives us: -\\[{{v\_{\text{y}}}^{\text{i}}}-\frac{g {t\_{\text{flight}}}}{2}=0\\] +\\[{{v\_{\text{y}}}^{\text{i}}}-\frac{g\\,{t\_{\text{flight}}}}{2}=0\\] Solving for \\({t\_{\text{flight}}}\\) gives us: -\\[{t\_{\text{flight}}}=\frac{2 {{v\_{\text{y}}}^{\text{i}}}}{g}\\] +\\[{t\_{\text{flight}}}=\frac{2\\,{{v\_{\text{y}}}^{\text{i}}}}{g}\\] From [DD:speedIY](./SecDDs.md#DD:speedIY) (with \\({v^{\text{i}}}={v\_{\text{launch}}}\\)) we can replace \\({{v\_{\text{y}}}^{\text{i}}}\\): -\\[{t\_{\text{flight}}}=\frac{2 {v\_{\text{launch}}} \sin\left(θ\right)}{g}\\] +\\[{t\_{\text{flight}}}=\frac{2\\,{v\_{\text{launch}}}\\,\sin\left(θ\right)}{g}\\]
@@ -56,7 +56,7 @@ From [DD:speedIY](./SecDDs.md#DD:speedIY) (with \\({v^{\text{i}}}={v\_{\text{lau |Output |\\({p\_{\text{land}}}\\) | |Input Constraints |\\[{v\_{\text{launch}}}\gt{}0\\]\\[0\lt{}θ\lt{}\frac{π}{2}\\] | |Output Constraints|\\[{p\_{\text{land}}}\gt{}0\\] | -|Equation |\\[{p\_{\text{land}}}=\frac{2 {v\_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g}\\] | +|Equation |\\[{p\_{\text{land}}}=\frac{2\\,{v\_{\text{launch}}}^{2}\\,\sin\left(θ\right)\\,\cos\left(θ\right)}{g}\\] | |Description |
  • \\({p\_{\text{land}}}\\) is the landing position (\\({\text{m}}\\))
  • \\({v\_{\text{launch}}}\\) is the launch speed (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(θ\\) is the launch angle (\\({\text{rad}}\\))
  • \\(g\\) is the magnitude of gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The constraint \\(0\lt{}θ\lt{}\frac{π}{2}\\) is from [A:posXDirection](./SecAssumps.md#posXDirection) and [A:yAxisGravity](./SecAssumps.md#yAxisGravity), and is shown in [Fig:Launch](./SecPhysSyst.md#Figure:Launch).
  • \\(g\\) is defined in [A:gravAccelValue](./SecAssumps.md#gravAccelValue).
  • The constraint \\({p\_{\text{land}}}\gt{}0\\) is from [A:posXDirection](./SecAssumps.md#posXDirection).
| |Source |-- | @@ -66,19 +66,19 @@ From [DD:speedIY](./SecDDs.md#DD:speedIY) (with \\({v^{\text{i}}}={v\_{\text{lau We know that \\({{p\_{\text{x}}}^{\text{i}}}=0\\) ([A:launchOrigin](./SecAssumps.md#launchOrigin)) and \\({{a\_{\text{x}}}^{\text{c}}}=0\\) ([A:accelXZero](./SecAssumps.md#accelXZero)). Substituting these values into the x-direction of [GD:posVec](./SecGDs.md#GD:posVec) gives us: -\\[{p\_{\text{x}}}={{v\_{\text{x}}}^{\text{i}}} t\\] +\\[{p\_{\text{x}}}={{v\_{\text{x}}}^{\text{i}}}\\,t\\] To find the landing position, we want to find the \\({p\_{\text{x}}}\\) value (\\({p\_{\text{land}}}\\)) at flight duration (from [IM:calOfLandingTime](./SecIMs.md#IM:calOfLandingTime)): -\\[{p\_{\text{land}}}=\frac{{{v\_{\text{x}}}^{\text{i}}}\cdot{}2 {v\_{\text{launch}}} \sin\left(θ\right)}{g}\\] +\\[{p\_{\text{land}}}=\frac{{{v\_{\text{x}}}^{\text{i}}}\cdot{}2\\,{v\_{\text{launch}}}\\,\sin\left(θ\right)}{g}\\] From [DD:speedIX](./SecDDs.md#DD:speedIX) (with \\({v^{\text{i}}}={v\_{\text{launch}}}\\)) we can replace \\({{v\_{\text{x}}}^{\text{i}}}\\): -\\[{p\_{\text{land}}}=\frac{{v\_{\text{launch}}} \cos\left(θ\right)\cdot{}2 {v\_{\text{launch}}} \sin\left(θ\right)}{g}\\] +\\[{p\_{\text{land}}}=\frac{{v\_{\text{launch}}}\\,\cos\left(θ\right)\cdot{}2\\,{v\_{\text{launch}}}\\,\sin\left(θ\right)}{g}\\] Rearranging this gives us the required equation: -\\[{p\_{\text{land}}}=\frac{2 {v\_{\text{launch}}}^{2} \sin\left(θ\right) \cos\left(θ\right)}{g}\\] +\\[{p\_{\text{land}}}=\frac{2\\,{v\_{\text{launch}}}^{2}\\,\sin\left(θ\right)\\,\cos\left(θ\right)}{g}\\]
diff --git a/code/stable/sglpend/SRS/HTML/SglPend_SRS.html b/code/stable/sglpend/SRS/HTML/SglPend_SRS.html index 763b362f33..d5db24f3b3 100644 --- a/code/stable/sglpend/SRS/HTML/SglPend_SRS.html +++ b/code/stable/sglpend/SRS/HTML/SglPend_SRS.html @@ -715,7 +715,7 @@

Theoretical Models

- + @@ -761,7 +761,7 @@

Theoretical Models

- + @@ -830,7 +830,7 @@

General Definitions

@@ -872,15 +872,15 @@

\[\symbf{v}\text{(}t\text{)}=\frac{\,d\symbf{p}\text{(}t\text{)}}{\,dt}\]

We also know the horizontal position

- \[{p_{\text{x}}}={L_{\text{rod}}} \sin\left({θ_{p}}\right)\] + \[{p_{\text{x}}}={L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Applying this,

- \[{v_{\text{x}}}=\frac{\,d{L_{\text{rod}}} \sin\left({θ_{p}}\right)}{\,dt}\] + \[{v_{\text{x}}}=\frac{\,d{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt}\]

Lrod is constant with respect to time, so

- \[{v_{\text{x}}}={L_{\text{rod}}} \frac{\,d\sin\left({θ_{p}}\right)}{\,dt}\] + \[{v_{\text{x}}}={L_{\text{rod}}}\,\frac{\,d\sin\left({θ_{p}}\right)}{\,dt}\]

Therefore, using the chain rule,

- \[{v_{\text{x}}}=ω {L_{\text{rod}}} \cos\left({θ_{p}}\right)\] + \[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
@@ -906,7 +906,7 @@

@@ -948,15 +948,15 @@

\[\symbf{v}\text{(}t\text{)}=\frac{\,d\symbf{p}\text{(}t\text{)}}{\,dt}\]

We also know the vertical position

- \[{p_{\text{y}}}=-{L_{\text{rod}}} \cos\left({θ_{p}}\right)\] + \[{p_{\text{y}}}=-{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

Applying this,

- \[{v_{\text{y}}}=-\left(\frac{\,d{L_{\text{rod}}} \cos\left({θ_{p}}\right)}{\,dt}\right)\] + \[{v_{\text{y}}}=-\left(\frac{\,d{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\right)\]

Lrod is constant with respect to time, so

- \[{v_{\text{y}}}=-{L_{\text{rod}}} \frac{\,d\cos\left({θ_{p}}\right)}{\,dt}\] + \[{v_{\text{y}}}=-{L_{\text{rod}}}\,\frac{\,d\cos\left({θ_{p}}\right)}{\,dt}\]

Therefore, using the chain rule,

- \[{v_{\text{y}}}=ω {L_{\text{rod}}} \sin\left({θ_{p}}\right)\] + \[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
@@ -982,7 +982,7 @@

@@ -1027,15 +1027,15 @@

Earlier, we found the horizontal velocity to be

- \[{v_{\text{x}}}=ω {L_{\text{rod}}} \cos\left({θ_{p}}\right)\] + \[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]

Applying this to our equation for acceleration

- \[{a_{\text{x}}}=\frac{\,dω {L_{\text{rod}}} \cos\left({θ_{p}}\right)}{\,dt}\] + \[{a_{\text{x}}}=\frac{\,dω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\]

By the product and chain rules, we find

- \[{a_{\text{x}}}=\frac{\,dω}{\,dt} {L_{\text{rod}}} \cos\left({θ_{p}}\right)-ω {L_{\text{rod}}} \sin\left({θ_{p}}\right) \frac{\,d{θ_{p}}}{\,dt}\] + \[{a_{\text{x}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)-ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt}\]

Simplifying,

- \[{a_{\text{x}}}=-ω^{2} {L_{\text{rod}}} \sin\left({θ_{p}}\right)+α {L_{\text{rod}}} \cos\left({θ_{p}}\right)\] + \[{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
@@ -1061,7 +1061,7 @@

@@ -1106,15 +1106,15 @@

Earlier, we found the vertical velocity to be

- \[{v_{\text{y}}}=ω {L_{\text{rod}}} \sin\left({θ_{p}}\right)\] + \[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]

Applying this to our equation for acceleration

- \[{a_{\text{y}}}=\frac{\,dω {L_{\text{rod}}} \sin\left({θ_{p}}\right)}{\,dt}\] + \[{a_{\text{y}}}=\frac{\,dω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt}\]

By the product and chain rules, we find

- \[{a_{\text{y}}}=\frac{\,dω}{\,dt} {L_{\text{rod}}} \sin\left({θ_{p}}\right)+ω {L_{\text{rod}}} \cos\left({θ_{p}}\right) \frac{\,d{θ_{p}}}{\,dt}\] + \[{a_{\text{y}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt}\]

Simplifying,

- \[{a_{\text{y}}}=ω^{2} {L_{\text{rod}}} \cos\left({θ_{p}}\right)+α {L_{\text{rod}}} \sin\left({θ_{p}}\right)\] + \[{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
@@ -1134,7 +1134,7 @@

@@ -1166,7 +1166,7 @@

Detailed derivation of force on the pendulum:

- \[\symbf{F}=m {a_{\text{x}}}=-\symbf{T} \sin\left({θ_{p}}\right)\] + \[\symbf{F}=m\,{a_{\text{x}}}=-\symbf{T}\,\sin\left({θ_{p}}\right)\]
@@ -1186,7 +1186,7 @@

Detailed derivation of force on the pendulum:

@@ -1221,7 +1221,7 @@

Detailed derivation of force on the pendulum:

Detailed derivation of force on the pendulum:

- \[\symbf{F}=m {a_{\text{y}}}=\symbf{T} \cos\left({θ_{p}}\right)-m \symbf{g}\] + \[\symbf{F}=m\,{a_{\text{y}}}=\symbf{T}\,\cos\left({θ_{p}}\right)-m\,\symbf{g}\]
@@ -1288,21 +1288,21 @@

Consider the torque on a pendulum defined in TM:NewtonSecLawRotMot. The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string Lrod multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:

- \[\symbf{τ}=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right)\] + \[\symbf{τ}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

So then

- \[\symbf{I} α=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right)\] + \[\symbf{I}\,α=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

Therefore,

- \[\symbf{I} \frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right)\] + \[\symbf{I}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

Substituting for I

- \[m {L_{\text{rod}}}^{2} \frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right)\] + \[m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right)\]

Crossing out m and Lrod we have

- \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right) \sin\left({θ_{p}}\right)\] + \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right)\,\sin\left({θ_{p}}\right)\]

For small angles, we approximate sin θp to θp

- \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right) {θ_{p}}\] + \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right)\,{θ_{p}}\]

Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency

@@ -1325,7 +1325,7 @@

- + @@ -1372,7 +1372,7 @@

Detailed derivation of the period of the pendulum:

Therefore from the data definition of the equation for angular frequency, we have

- \[T=2 π \sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}}\] + \[T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}}\] @@ -1408,7 +1408,7 @@

Data Definitions

@@ -1473,7 +1473,7 @@

Data Definitions

@@ -1586,7 +1586,7 @@

Data Definitions

- + @@ -1719,7 +1719,7 @@

Instance Models

- + @@ -1764,19 +1764,19 @@

Detailed derivation of angular displacement:

When the pendulum is displaced to an initial angle and released, the pendulum swings back and forth with periodic motion. By applying Newton's second law for rotational motion, the equation of motion for the pendulum may be obtained:

- \[\symbf{τ}=\symbf{I} α\] + \[\symbf{τ}=\symbf{I}\,α\]

Where τ denotes the torque, I denotes the moment of inertia and α denotes the angular acceleration. This implies:

- \[-m \symbf{g} \sin\left({θ_{p}}\right) {L_{\text{rod}}}=m {L_{\text{rod}}}^{2} \frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}\] + \[-m\,\symbf{g}\,\sin\left({θ_{p}}\right)\,{L_{\text{rod}}}=m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}\]

And rearranged as:

- \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}} \sin\left({θ_{p}}\right)=0\] + \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}}\,\sin\left({θ_{p}}\right)=0\]

If the amplitude of angular displacement is small enough, we can approximate sin(θp) = θp for the purpose of a simple pendulum at very small angles. Then the equation of motion reduces to the equation of simple harmonic motion:

- \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}} {θ_{p}}=0\] + \[\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}}\,{θ_{p}}=0\]

Thus the simple harmonic motion is:

- \[{θ_{p}}\left(t\right)={θ_{i}} \cos\left(Ω t\right)\] + \[{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right)\] diff --git a/code/stable/sglpend/SRS/Jupyter/SglPend_SRS.ipynb b/code/stable/sglpend/SRS/Jupyter/SglPend_SRS.ipynb index 659a5589e4..94ba0962bb 100644 --- a/code/stable/sglpend/SRS/Jupyter/SglPend_SRS.ipynb +++ b/code/stable/sglpend/SRS/Jupyter/SglPend_SRS.ipynb @@ -395,7 +395,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -448,7 +448,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -513,7 +513,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -549,16 +549,16 @@ "$$\\symbf{v}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{p}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "We also know the horizontal position\n", - "$${p_{\\text{x}}}={L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)$$\n", + "$${p_{\\text{x}}}={L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "Applying this,\n", - "$${v_{\\text{x}}}=\\frac{\\,d{L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{x}}}=\\frac{\\,d{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)}{\\,dt}$$\n", "\n", "$L_rod$ is constant with respect to time, so\n", - "$${v_{\\text{x}}}={L_{\\text{rod}}} \\frac{\\,d\\sin\\left({θ_{p}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{x}}}={L_{\\text{rod}}}\\,\\frac{\\,d\\sin\\left({θ_{p}}\\right)}{\\,dt}$$\n", "\n", "Therefore, using the chain rule,\n", - "$${v_{\\text{x}}}=ω {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)$$\n", + "$${v_{\\text{x}}}=ω\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)$$\n", "
\n", "\n", "
Equation\[\symbf{F}=m \symbf{a}\text{(}t\text{)}\]\[\symbf{F}=m\,\symbf{a}\text{(}t\text{)}\]
Description
Equation\[\symbf{τ}=\symbf{I} α\]\[\symbf{τ}=\symbf{I}\,α\]
Description
Equation - \[{v_{\text{x}}}=ω {L_{\text{rod}}} \cos\left({θ_{p}}\right)\] + \[{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Equation - \[{v_{\text{y}}}=ω {L_{\text{rod}}} \sin\left({θ_{p}}\right)\] + \[{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Equation - \[{a_{\text{x}}}=-ω^{2} {L_{\text{rod}}} \sin\left({θ_{p}}\right)+α {L_{\text{rod}}} \cos\left({θ_{p}}\right)\] + \[{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\]
Equation - \[{a_{\text{y}}}=ω^{2} {L_{\text{rod}}} \cos\left({θ_{p}}\right)+α {L_{\text{rod}}} \sin\left({θ_{p}}\right)\] + \[{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\]
Equation - \[\symbf{F}=m {a_{\text{x}}}=-\symbf{T} \sin\left({θ_{p}}\right)\] + \[\symbf{F}=m\,{a_{\text{x}}}=-\symbf{T}\,\sin\left({θ_{p}}\right)\]
Equation - \[\symbf{F}=m {a_{\text{y}}}=\symbf{T} \cos\left({θ_{p}}\right)-m \symbf{g}\] + \[\symbf{F}=m\,{a_{\text{y}}}=\symbf{T}\,\cos\left({θ_{p}}\right)-m\,\symbf{g}\]
Equation\[T=2 π \sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}}\]\[T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}}\]
Description
Equation - \[{{p_{\text{x}}}^{\text{i}}}={L_{\text{rod}}} \sin\left({θ_{i}}\right)\] + \[{{p_{\text{x}}}^{\text{i}}}={L_{\text{rod}}}\,\sin\left({θ_{i}}\right)\]
Equation - \[{{p_{\text{y}}}^{\text{i}}}=-{L_{\text{rod}}} \cos\left({θ_{i}}\right)\] + \[{{p_{\text{y}}}^{\text{i}}}=-{L_{\text{rod}}}\,\cos\left({θ_{i}}\right)\]
Equation\[Ω=\frac{2 π}{T}\]\[Ω=\frac{2\,π}{T}\]
Description
Equation\[{θ_{p}}\left(t\right)={θ_{i}} \cos\left(Ω t\right)\]\[{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right)\]
Description
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}$$\n", + "$$\\symbf{F}=m\\,\\symbf{a}\\text{(}t\\text{)}$$\n", "
Equation\n", - "$$\\symbf{τ}=\\symbf{I} α$$\n", + "$$\\symbf{τ}=\\symbf{I}\\,α$$\n", "
Equation\n", - "$${v_{\\text{x}}}=ω {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)$$\n", + "$${v_{\\text{x}}}=ω\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)$$\n", "
\n", @@ -585,7 +585,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -621,16 +621,16 @@ "$$\\symbf{v}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{p}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "We also know the vertical position\n", - "$${p_{\\text{y}}}=-{L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)$$\n", + "$${p_{\\text{y}}}=-{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)$$\n", "\n", "Applying this,\n", - "$${v_{\\text{y}}}=-\\left(\\frac{\\,d{L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)}{\\,dt}\\right)$$\n", + "$${v_{\\text{y}}}=-\\left(\\frac{\\,d{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)}{\\,dt}\\right)$$\n", "\n", "$L_rod$ is constant with respect to time, so\n", - "$${v_{\\text{y}}}=-{L_{\\text{rod}}} \\frac{\\,d\\cos\\left({θ_{p}}\\right)}{\\,dt}$$\n", + "$${v_{\\text{y}}}=-{L_{\\text{rod}}}\\,\\frac{\\,d\\cos\\left({θ_{p}}\\right)}{\\,dt}$$\n", "\n", "Therefore, using the chain rule,\n", - "$${v_{\\text{y}}}=ω {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)$$\n", + "$${v_{\\text{y}}}=ω\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${v_{\\text{y}}}=ω {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)$$\n", + "$${v_{\\text{y}}}=ω\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)$$\n", "
\n", @@ -657,7 +657,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -694,16 +694,16 @@ "$$\\symbf{a}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Earlier, we found the horizontal velocity to be\n", - "$${v_{\\text{x}}}=ω {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)$$\n", + "$${v_{\\text{x}}}=ω\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)$$\n", "\n", "Applying this to our equation for acceleration\n", - "$${a_{\\text{x}}}=\\frac{\\,dω {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)}{\\,dt}$$\n", + "$${a_{\\text{x}}}=\\frac{\\,dω\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)}{\\,dt}$$\n", "\n", "By the product and chain rules, we find\n", - "$${a_{\\text{x}}}=\\frac{\\,dω}{\\,dt} {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)-ω {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right) \\frac{\\,d{θ_{p}}}{\\,dt}$$\n", + "$${a_{\\text{x}}}=\\frac{\\,dω}{\\,dt}\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)-ω\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)\\,\\frac{\\,d{θ_{p}}}{\\,dt}$$\n", "\n", "Simplifying,\n", - "$${a_{\\text{x}}}=-ω^{2} {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)+α {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)$$\n", + "$${a_{\\text{x}}}=-ω^{2}\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)+α\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${a_{\\text{x}}}=-ω^{2} {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)+α {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)$$\n", + "$${a_{\\text{x}}}=-ω^{2}\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)+α\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)$$\n", "
\n", @@ -730,7 +730,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -767,16 +767,16 @@ "$$\\symbf{a}\\text{(}t\\text{)}=\\frac{\\,d\\symbf{v}\\text{(}t\\text{)}}{\\,dt}$$\n", "\n", "Earlier, we found the vertical velocity to be\n", - "$${v_{\\text{y}}}=ω {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)$$\n", + "$${v_{\\text{y}}}=ω\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "Applying this to our equation for acceleration\n", - "$${a_{\\text{y}}}=\\frac{\\,dω {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)}{\\,dt}$$\n", + "$${a_{\\text{y}}}=\\frac{\\,dω\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)}{\\,dt}$$\n", "\n", "By the product and chain rules, we find\n", - "$${a_{\\text{y}}}=\\frac{\\,dω}{\\,dt} {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)+ω {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right) \\frac{\\,d{θ_{p}}}{\\,dt}$$\n", + "$${a_{\\text{y}}}=\\frac{\\,dω}{\\,dt}\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)+ω\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)\\,\\frac{\\,d{θ_{p}}}{\\,dt}$$\n", "\n", "Simplifying,\n", - "$${a_{\\text{y}}}=ω^{2} {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)+α {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)$$\n", + "$${a_{\\text{y}}}=ω^{2}\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)+α\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${a_{\\text{y}}}=ω^{2} {L_{\\text{rod}}} \\cos\\left({θ_{p}}\\right)+α {L_{\\text{rod}}} \\sin\\left({θ_{p}}\\right)$$\n", + "$${a_{\\text{y}}}=ω^{2}\\,{L_{\\text{rod}}}\\,\\cos\\left({θ_{p}}\\right)+α\\,{L_{\\text{rod}}}\\,\\sin\\left({θ_{p}}\\right)$$\n", "
\n", @@ -803,7 +803,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -835,7 +835,7 @@ "\n", "#### Detailed derivation of force on the pendulum:\n", "\n", - "$$\\symbf{F}=m {a_{\\text{x}}}=-\\symbf{T} \\sin\\left({θ_{p}}\\right)$$\n", + "$$\\symbf{F}=m\\,{a_{\\text{x}}}=-\\symbf{T}\\,\\sin\\left({θ_{p}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{F}=m {a_{\\text{x}}}=-\\symbf{T} \\sin\\left({θ_{p}}\\right)$$\n", + "$$\\symbf{F}=m\\,{a_{\\text{x}}}=-\\symbf{T}\\,\\sin\\left({θ_{p}}\\right)$$\n", "
\n", @@ -862,7 +862,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -895,7 +895,7 @@ "\n", "#### Detailed derivation of force on the pendulum:\n", "\n", - "$$\\symbf{F}=m {a_{\\text{y}}}=\\symbf{T} \\cos\\left({θ_{p}}\\right)-m \\symbf{g}$$\n", + "$$\\symbf{F}=m\\,{a_{\\text{y}}}=\\symbf{T}\\,\\cos\\left({θ_{p}}\\right)-m\\,\\symbf{g}$$\n", "
\n", "\n", "
Equation\n", - "$$\\symbf{F}=m {a_{\\text{y}}}=\\symbf{T} \\cos\\left({θ_{p}}\\right)-m \\symbf{g}$$\n", + "$$\\symbf{F}=m\\,{a_{\\text{y}}}=\\symbf{T}\\,\\cos\\left({θ_{p}}\\right)-m\\,\\symbf{g}$$\n", "
\n", @@ -961,22 +961,22 @@ "\n", "\n", "Consider the torque on a pendulum defined in [TM:NewtonSecLawRotMot](#TM:NewtonSecLawRotMot). The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string $L_rod$ multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement:\n", - "$$\\symbf{τ}=-{L_{\\text{rod}}} m \\symbf{g} \\sin\\left({θ_{p}}\\right)$$\n", + "$$\\symbf{τ}=-{L_{\\text{rod}}}\\,m\\,\\symbf{g}\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "So then\n", - "$$\\symbf{I} α=-{L_{\\text{rod}}} m \\symbf{g} \\sin\\left({θ_{p}}\\right)$$\n", + "$$\\symbf{I}\\,α=-{L_{\\text{rod}}}\\,m\\,\\symbf{g}\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "Therefore,\n", - "$$\\symbf{I} \\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-{L_{\\text{rod}}} m \\symbf{g} \\sin\\left({θ_{p}}\\right)$$\n", + "$$\\symbf{I}\\,\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-{L_{\\text{rod}}}\\,m\\,\\symbf{g}\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "Substituting for $I$\n", - "$$m {L_{\\text{rod}}}^{2} \\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-{L_{\\text{rod}}} m \\symbf{g} \\sin\\left({θ_{p}}\\right)$$\n", + "$$m\\,{L_{\\text{rod}}}^{2}\\,\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-{L_{\\text{rod}}}\\,m\\,\\symbf{g}\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "Crossing out $m$ and $L_rod$ we have\n", - "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-\\left(\\frac{\\symbf{g}}{{L_{\\text{rod}}}}\\right) \\sin\\left({θ_{p}}\\right)$$\n", + "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-\\left(\\frac{\\symbf{g}}{{L_{\\text{rod}}}}\\right)\\,\\sin\\left({θ_{p}}\\right)$$\n", "\n", "For small angles, we approximate sin $θ_p$ to $θ_p$\n", - "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-\\left(\\frac{\\symbf{g}}{{L_{\\text{rod}}}}\\right) {θ_{p}}$$\n", + "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}=-\\left(\\frac{\\symbf{g}}{{L_{\\text{rod}}}}\\right)\\,{θ_{p}}$$\n", "\n", "Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency\n", "$$Ω=\\sqrt{\\frac{\\symbf{g}}{{L_{\\text{rod}}}}}$$\n", @@ -1006,7 +1006,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1049,7 +1049,7 @@ "$$Ω=\\sqrt{\\frac{\\symbf{g}}{{L_{\\text{rod}}}}}$$\n", "\n", "Therefore from the data definition of the equation for [angular frequency](#DD:angFrequencyDD), we have\n", - "$$T=2 π \\sqrt{\\frac{{L_{\\text{rod}}}}{\\symbf{g}}}$$\n", + "$$T=2\\,π\\,\\sqrt{\\frac{{L_{\\text{rod}}}}{\\symbf{g}}}$$\n", "\n", "### Data Definitions\n", "\n", @@ -1088,7 +1088,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1157,7 +1157,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1292,7 +1292,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1444,7 +1444,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1484,19 +1484,19 @@ "\n", "\n", "When the pendulum is displaced to an initial angle and released, the pendulum swings back and forth with periodic motion. By applying [Newton's second law for rotational motion](#TM:NewtonSecLawRotMot), the equation of motion for the pendulum may be obtained:\n", - "$$\\symbf{τ}=\\symbf{I} α$$\n", + "$$\\symbf{τ}=\\symbf{I}\\,α$$\n", "\n", "Where $τ$ denotes the torque, $I$ denotes the moment of inertia and $α$ denotes the angular acceleration. This implies:\n", - "$$-m \\symbf{g} \\sin\\left({θ_{p}}\\right) {L_{\\text{rod}}}=m {L_{\\text{rod}}}^{2} \\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}$$\n", + "$$-m\\,\\symbf{g}\\,\\sin\\left({θ_{p}}\\right)\\,{L_{\\text{rod}}}=m\\,{L_{\\text{rod}}}^{2}\\,\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}$$\n", "\n", "And rearranged as:\n", - "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}+\\frac{\\symbf{g}}{{L_{\\text{rod}}}} \\sin\\left({θ_{p}}\\right)=0$$\n", + "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}+\\frac{\\symbf{g}}{{L_{\\text{rod}}}}\\,\\sin\\left({θ_{p}}\\right)=0$$\n", "\n", "If the amplitude of angular displacement is small enough, we can approximate $sin(θ_p) = θ_p$ for the purpose of a simple pendulum at very small angles. Then the equation of motion reduces to the equation of simple harmonic motion:\n", - "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}+\\frac{\\symbf{g}}{{L_{\\text{rod}}}} {θ_{p}}=0$$\n", + "$$\\frac{\\,d\\frac{\\,d{θ_{p}}}{\\,dt}}{\\,dt}+\\frac{\\symbf{g}}{{L_{\\text{rod}}}}\\,{θ_{p}}=0$$\n", "\n", "Thus the simple harmonic motion is:\n", - "$${θ_{p}}\\left(t\\right)={θ_{i}} \\cos\\left(Ω t\\right)$$\n", + "$${θ_{p}}\\left(t\\right)={θ_{i}}\\,\\cos\\left(Ω\\,t\\right)$$\n", "\n", "### Data Constraints\n", "\n", diff --git a/code/stable/sglpend/SRS/PDF/SglPend_SRS.tex b/code/stable/sglpend/SRS/PDF/SglPend_SRS.tex index c5d9a82e4c..b9fbcba0ea 100644 --- a/code/stable/sglpend/SRS/PDF/SglPend_SRS.tex +++ b/code/stable/sglpend/SRS/PDF/SglPend_SRS.tex @@ -352,7 +352,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)} + \symbf{F}=m\,\symbf{a}\text{(}t\text{)} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -384,7 +384,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{τ}=\symbf{I} α + \symbf{τ}=\symbf{I}\,α \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -424,7 +424,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {v_{\text{x}}}=ω {L_{\text{rod}}} \cos\left({θ_{p}}\right) + {v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -452,22 +452,22 @@ \subsubsection{General Definitions} We also know the horizontal position \begin{displaymath} -{p_{\text{x}}}={L_{\text{rod}}} \sin\left({θ_{p}}\right) +{p_{\text{x}}}={L_{\text{rod}}}\,\sin\left({θ_{p}}\right) \end{displaymath} Applying this, \begin{displaymath} -{v_{\text{x}}}=\frac{\,d{L_{\text{rod}}} \sin\left({θ_{p}}\right)}{\,dt} +{v_{\text{x}}}=\frac{\,d{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt} \end{displaymath} ${L_{\text{rod}}}$ is constant with respect to time, so \begin{displaymath} -{v_{\text{x}}}={L_{\text{rod}}} \frac{\,d\sin\left({θ_{p}}\right)}{\,dt} +{v_{\text{x}}}={L_{\text{rod}}}\,\frac{\,d\sin\left({θ_{p}}\right)}{\,dt} \end{displaymath} Therefore, using the chain rule, \begin{displaymath} -{v_{\text{x}}}=ω {L_{\text{rod}}} \cos\left({θ_{p}}\right) +{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right) \end{displaymath} \medskip \noindent @@ -484,7 +484,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {v_{\text{y}}}=ω {L_{\text{rod}}} \sin\left({θ_{p}}\right) + {v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -512,22 +512,22 @@ \subsubsection{General Definitions} We also know the vertical position \begin{displaymath} -{p_{\text{y}}}=-{L_{\text{rod}}} \cos\left({θ_{p}}\right) +{p_{\text{y}}}=-{L_{\text{rod}}}\,\cos\left({θ_{p}}\right) \end{displaymath} Applying this, \begin{displaymath} -{v_{\text{y}}}=-\left(\frac{\,d{L_{\text{rod}}} \cos\left({θ_{p}}\right)}{\,dt}\right) +{v_{\text{y}}}=-\left(\frac{\,d{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt}\right) \end{displaymath} ${L_{\text{rod}}}$ is constant with respect to time, so \begin{displaymath} -{v_{\text{y}}}=-{L_{\text{rod}}} \frac{\,d\cos\left({θ_{p}}\right)}{\,dt} +{v_{\text{y}}}=-{L_{\text{rod}}}\,\frac{\,d\cos\left({θ_{p}}\right)}{\,dt} \end{displaymath} Therefore, using the chain rule, \begin{displaymath} -{v_{\text{y}}}=ω {L_{\text{rod}}} \sin\left({θ_{p}}\right) +{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right) \end{displaymath} \medskip \noindent @@ -544,7 +544,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {a_{\text{x}}}=-ω^{2} {L_{\text{rod}}} \sin\left({θ_{p}}\right)+α {L_{\text{rod}}} \cos\left({θ_{p}}\right) + {a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -573,22 +573,22 @@ \subsubsection{General Definitions} Earlier, we found the horizontal velocity to be \begin{displaymath} -{v_{\text{x}}}=ω {L_{\text{rod}}} \cos\left({θ_{p}}\right) +{v_{\text{x}}}=ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right) \end{displaymath} Applying this to our equation for acceleration \begin{displaymath} -{a_{\text{x}}}=\frac{\,dω {L_{\text{rod}}} \cos\left({θ_{p}}\right)}{\,dt} +{a_{\text{x}}}=\frac{\,dω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)}{\,dt} \end{displaymath} By the product and chain rules, we find \begin{displaymath} -{a_{\text{x}}}=\frac{\,dω}{\,dt} {L_{\text{rod}}} \cos\left({θ_{p}}\right)-ω {L_{\text{rod}}} \sin\left({θ_{p}}\right) \frac{\,d{θ_{p}}}{\,dt} +{a_{\text{x}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)-ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt} \end{displaymath} Simplifying, \begin{displaymath} -{a_{\text{x}}}=-ω^{2} {L_{\text{rod}}} \sin\left({θ_{p}}\right)+α {L_{\text{rod}}} \cos\left({θ_{p}}\right) +{a_{\text{x}}}=-ω^{2}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right) \end{displaymath} \medskip \noindent @@ -605,7 +605,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {a_{\text{y}}}=ω^{2} {L_{\text{rod}}} \cos\left({θ_{p}}\right)+α {L_{\text{rod}}} \sin\left({θ_{p}}\right) + {a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -634,22 +634,22 @@ \subsubsection{General Definitions} Earlier, we found the vertical velocity to be \begin{displaymath} -{v_{\text{y}}}=ω {L_{\text{rod}}} \sin\left({θ_{p}}\right) +{v_{\text{y}}}=ω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right) \end{displaymath} Applying this to our equation for acceleration \begin{displaymath} -{a_{\text{y}}}=\frac{\,dω {L_{\text{rod}}} \sin\left({θ_{p}}\right)}{\,dt} +{a_{\text{y}}}=\frac{\,dω\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)}{\,dt} \end{displaymath} By the product and chain rules, we find \begin{displaymath} -{a_{\text{y}}}=\frac{\,dω}{\,dt} {L_{\text{rod}}} \sin\left({θ_{p}}\right)+ω {L_{\text{rod}}} \cos\left({θ_{p}}\right) \frac{\,d{θ_{p}}}{\,dt} +{a_{\text{y}}}=\frac{\,dω}{\,dt}\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right)+ω\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)\,\frac{\,d{θ_{p}}}{\,dt} \end{displaymath} Simplifying, \begin{displaymath} -{a_{\text{y}}}=ω^{2} {L_{\text{rod}}} \cos\left({θ_{p}}\right)+α {L_{\text{rod}}} \sin\left({θ_{p}}\right) +{a_{\text{y}}}=ω^{2}\,{L_{\text{rod}}}\,\cos\left({θ_{p}}\right)+α\,{L_{\text{rod}}}\,\sin\left({θ_{p}}\right) \end{displaymath} \medskip \noindent @@ -666,7 +666,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m {a_{\text{x}}}=-\symbf{T} \sin\left({θ_{p}}\right) + \symbf{F}=m\,{a_{\text{x}}}=-\symbf{T}\,\sin\left({θ_{p}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -688,7 +688,7 @@ \subsubsection{General Definitions} \paragraph{Detailed derivation of force on the pendulum:} \label{GD:hForceOnPendulumDeriv} \begin{displaymath} -\symbf{F}=m {a_{\text{x}}}=-\symbf{T} \sin\left({θ_{p}}\right) +\symbf{F}=m\,{a_{\text{x}}}=-\symbf{T}\,\sin\left({θ_{p}}\right) \end{displaymath} \medskip \noindent @@ -705,7 +705,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m {a_{\text{y}}}=\symbf{T} \cos\left({θ_{p}}\right)-m \symbf{g} + \symbf{F}=m\,{a_{\text{y}}}=\symbf{T}\,\cos\left({θ_{p}}\right)-m\,\symbf{g} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -728,7 +728,7 @@ \subsubsection{General Definitions} \paragraph{Detailed derivation of force on the pendulum:} \label{GD:vForceOnPendulumDeriv} \begin{displaymath} -\symbf{F}=m {a_{\text{y}}}=\symbf{T} \cos\left({θ_{p}}\right)-m \symbf{g} +\symbf{F}=m\,{a_{\text{y}}}=\symbf{T}\,\cos\left({θ_{p}}\right)-m\,\symbf{g} \end{displaymath} \medskip \noindent @@ -771,32 +771,32 @@ \subsubsection{General Definitions} Consider the torque on a pendulum defined in \hyperref[TM:NewtonSecLawRotMot]{TM:NewtonSecLawRotMot}. The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string ${L_{\text{rod}}}$ multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement: \begin{displaymath} -\symbf{τ}=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right) +\symbf{τ}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right) \end{displaymath} So then \begin{displaymath} -\symbf{I} α=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right) +\symbf{I}\,α=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right) \end{displaymath} Therefore, \begin{displaymath} -\symbf{I} \frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right) +\symbf{I}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right) \end{displaymath} Substituting for $\symbf{I}$ \begin{displaymath} -m {L_{\text{rod}}}^{2} \frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}} m \symbf{g} \sin\left({θ_{p}}\right) +m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-{L_{\text{rod}}}\,m\,\symbf{g}\,\sin\left({θ_{p}}\right) \end{displaymath} Crossing out $m$ and ${L_{\text{rod}}}$ we have \begin{displaymath} -\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right) \sin\left({θ_{p}}\right) +\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right)\,\sin\left({θ_{p}}\right) \end{displaymath} For small angles, we approximate sin ${θ_{p}}$ to ${θ_{p}}$ \begin{displaymath} -\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right) {θ_{p}} +\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}=-\left(\frac{\symbf{g}}{{L_{\text{rod}}}}\right)\,{θ_{p}} \end{displaymath} Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency @@ -818,7 +818,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - T=2 π \sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}} + T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -849,7 +849,7 @@ \subsubsection{General Definitions} Therefore from the data definition of the equation for \hyperref[DD:angFrequencyDD]{angular frequency}, we have \begin{displaymath} -T=2 π \sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}} +T=2\,π\,\sqrt{\frac{{L_{\text{rod}}}}{\symbf{g}}} \end{displaymath} \subsubsection{Data Definitions} \label{Sec:DDs} @@ -873,7 +873,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {{p_{\text{x}}}^{\text{i}}}={L_{\text{rod}}} \sin\left({θ_{i}}\right) + {{p_{\text{x}}}^{\text{i}}}={L_{\text{rod}}}\,\sin\left({θ_{i}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -913,7 +913,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {{p_{\text{y}}}^{\text{i}}}=-{L_{\text{rod}}} \cos\left({θ_{i}}\right) + {{p_{\text{y}}}^{\text{i}}}=-{L_{\text{rod}}}\,\cos\left({θ_{i}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -991,7 +991,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - Ω=\frac{2 π}{T} + Ω=\frac{2\,π}{T} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1086,7 +1086,7 @@ \subsubsection{Instance Models} \end{displaymath} \\ \midrule Equation & \begin{displaymath} - {θ_{p}}\left(t\right)={θ_{i}} \cos\left(Ω t\right) + {θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1113,27 +1113,27 @@ \subsubsection{Instance Models} When the pendulum is displaced to an initial angle and released, the pendulum swings back and forth with periodic motion. By applying \hyperref[TM:NewtonSecLawRotMot]{Newton's second law for rotational motion}, the equation of motion for the pendulum may be obtained: \begin{displaymath} -\symbf{τ}=\symbf{I} α +\symbf{τ}=\symbf{I}\,α \end{displaymath} Where $\symbf{τ}$ denotes the torque, $\symbf{I}$ denotes the moment of inertia and $α$ denotes the angular acceleration. This implies: \begin{displaymath} --m \symbf{g} \sin\left({θ_{p}}\right) {L_{\text{rod}}}=m {L_{\text{rod}}}^{2} \frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt} +-m\,\symbf{g}\,\sin\left({θ_{p}}\right)\,{L_{\text{rod}}}=m\,{L_{\text{rod}}}^{2}\,\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt} \end{displaymath} And rearranged as: \begin{displaymath} -\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}} \sin\left({θ_{p}}\right)=0 +\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}}\,\sin\left({θ_{p}}\right)=0 \end{displaymath} If the amplitude of angular displacement is small enough, we can approximate $\sin\left({θ_{p}}\right)={θ_{p}}$ for the purpose of a simple pendulum at very small angles. Then the equation of motion reduces to the equation of simple harmonic motion: \begin{displaymath} -\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}} {θ_{p}}=0 +\frac{\,d\frac{\,d{θ_{p}}}{\,dt}}{\,dt}+\frac{\symbf{g}}{{L_{\text{rod}}}}\,{θ_{p}}=0 \end{displaymath} Thus the simple harmonic motion is: \begin{displaymath} -{θ_{p}}\left(t\right)={θ_{i}} \cos\left(Ω t\right) +{θ_{p}}\left(t\right)={θ_{i}}\,\cos\left(Ω\,t\right) \end{displaymath} \subsubsection{Data Constraints} \label{Sec:DataConstraints} diff --git a/code/stable/sglpend/SRS/mdBook/src/SecDDs.md b/code/stable/sglpend/SRS/mdBook/src/SecDDs.md index e287cc00a8..62c6c9fdfb 100644 --- a/code/stable/sglpend/SRS/mdBook/src/SecDDs.md +++ b/code/stable/sglpend/SRS/mdBook/src/SecDDs.md @@ -13,7 +13,7 @@ This section collects and defines all the data needed to build the instance mode |Label |\\(x\\)-component of initial position | |Symbol |\\({{p\_{\text{x}}}^{\text{i}}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{{p\_{\text{x}}}^{\text{i}}}={L\_{\text{rod}}} \sin\left({θ\_{i}}\right)\\] | +|Equation |\\[{{p\_{\text{x}}}^{\text{i}}}={L\_{\text{rod}}}\\,\sin\left({θ\_{i}}\right)\\] | |Description|
  • \\({{p\_{\text{x}}}^{\text{i}}}\\) is the \\(x\\)-component of initial position (\\({\text{m}}\\))
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\({θ\_{i}}\\) is the initial pendulum angle (\\({\text{rad}}\\))
| |Notes |
  • \\({{p\_{\text{x}}}^{\text{i}}}\\) is the horizontal position
  • \\({{p\_{\text{x}}}^{\text{i}}}\\) is shown in [Fig:sglpend](./SecPhysSyst.md#Figure:sglpend).
| |Source |-- | @@ -30,7 +30,7 @@ This section collects and defines all the data needed to build the instance mode |Label |\\(y\\)-component of initial position | |Symbol |\\({{p\_{\text{y}}}^{\text{i}}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{{p\_{\text{y}}}^{\text{i}}}=-{L\_{\text{rod}}} \cos\left({θ\_{i}}\right)\\] | +|Equation |\\[{{p\_{\text{y}}}^{\text{i}}}=-{L\_{\text{rod}}}\\,\cos\left({θ\_{i}}\right)\\] | |Description|
  • \\({{p\_{\text{y}}}^{\text{i}}}\\) is the \\(y\\)-component of initial position (\\({\text{m}}\\))
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\({θ\_{i}}\\) is the initial pendulum angle (\\({\text{rad}}\\))
| |Notes |
  • \\({{p\_{\text{y}}}^{\text{i}}}\\) is the vertical position
  • \\({{p\_{\text{y}}}^{\text{i}}}\\) is shown in [Fig:sglpend](./SecPhysSyst.md#Figure:sglpend).
| |Source |-- | @@ -64,7 +64,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Angular frequency | |Symbol |\\(Ω\\) | |Units |\\({\text{s}}\\) | -|Equation |\\[Ω=\frac{2 π}{T}\\] | +|Equation |\\[Ω=\frac{2\\,π}{T}\\] | |Description|
  • \\(Ω\\) is the angular frequency (\\({\text{s}}\\))
  • \\(π\\) is the ratio of circumference to diameter for any circle (Unitless)
  • \\(T\\) is the period (\\({\text{s}}\\))
| |Notes |
  • \\(T\\) is from [DD:periodSHMDD](./SecDDs.md#DD:periodSHMDD)
| |Source |-- | diff --git a/code/stable/sglpend/SRS/mdBook/src/SecGDs.md b/code/stable/sglpend/SRS/mdBook/src/SecGDs.md index f17a634930..13db470c0b 100644 --- a/code/stable/sglpend/SRS/mdBook/src/SecGDs.md +++ b/code/stable/sglpend/SRS/mdBook/src/SecGDs.md @@ -12,7 +12,7 @@ This section collects the laws and equations that will be used to build the inst |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(x\\)-component of velocity of the pendulum | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{v\_{\text{x}}}=ω {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)\\] | +|Equation |\\[{v\_{\text{x}}}=ω\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\] | |Description|
  • \\({v\_{\text{x}}}\\) is the \\(x\\)-component of velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(ω\\) is the angular velocity (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | @@ -25,19 +25,19 @@ At a given point in time, velocity may be defined as We also know the horizontal position -\\[{p\_{\text{x}}}={L\_{\text{rod}}} \sin\left({θ\_{p}}\right)\\] +\\[{p\_{\text{x}}}={L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\] Applying this, -\\[{v\_{\text{x}}}=\frac{\\,d{L\_{\text{rod}}} \sin\left({θ\_{p}}\right)}{\\,dt}\\] +\\[{v\_{\text{x}}}=\frac{\\,d{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)}{\\,dt}\\] \\({L\_{\text{rod}}}\\) is constant with respect to time, so -\\[{v\_{\text{x}}}={L\_{\text{rod}}} \frac{\\,d\sin\left({θ\_{p}}\right)}{\\,dt}\\] +\\[{v\_{\text{x}}}={L\_{\text{rod}}}\\,\frac{\\,d\sin\left({θ\_{p}}\right)}{\\,dt}\\] Therefore, using the chain rule, -\\[{v\_{\text{x}}}=ω {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)\\] +\\[{v\_{\text{x}}}=ω\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\]
@@ -49,7 +49,7 @@ Therefore, using the chain rule, |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(y\\)-component of velocity of the pendulum | |Units |\\(\frac{\text{m}}{\text{s}}\\) | -|Equation |\\[{v\_{\text{y}}}=ω {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)\\] | +|Equation |\\[{v\_{\text{y}}}=ω\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\] | |Description|
  • \\({v\_{\text{y}}}\\) is the \\(y\\)-component of velocity (\\(\frac{\text{m}}{\text{s}}\\))
  • \\(ω\\) is the angular velocity (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | @@ -62,19 +62,19 @@ At a given point in time, velocity may be defined as We also know the vertical position -\\[{p\_{\text{y}}}=-{L\_{\text{rod}}} \cos\left({θ\_{p}}\right)\\] +\\[{p\_{\text{y}}}=-{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\] Applying this, -\\[{v\_{\text{y}}}=-\left(\frac{\\,d{L\_{\text{rod}}} \cos\left({θ\_{p}}\right)}{\\,dt}\right)\\] +\\[{v\_{\text{y}}}=-\left(\frac{\\,d{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)}{\\,dt}\right)\\] \\({L\_{\text{rod}}}\\) is constant with respect to time, so -\\[{v\_{\text{y}}}=-{L\_{\text{rod}}} \frac{\\,d\cos\left({θ\_{p}}\right)}{\\,dt}\\] +\\[{v\_{\text{y}}}=-{L\_{\text{rod}}}\\,\frac{\\,d\cos\left({θ\_{p}}\right)}{\\,dt}\\] Therefore, using the chain rule, -\\[{v\_{\text{y}}}=ω {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)\\] +\\[{v\_{\text{y}}}=ω\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\]
@@ -86,7 +86,7 @@ Therefore, using the chain rule, |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(x\\)-component of acceleration of the pendulum | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[{a\_{\text{x}}}=-ω^{2} {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)+α {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)\\] | +|Equation |\\[{a\_{\text{x}}}=-ω^{2}\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)+α\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\] | |Description|
  • \\({a\_{\text{x}}}\\) is the \\(x\\)-component of acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(ω\\) is the angular velocity (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
  • \\(α\\) is the angular acceleration (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy | | @@ -99,19 +99,19 @@ Our acceleration is: Earlier, we found the horizontal velocity to be -\\[{v\_{\text{x}}}=ω {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)\\] +\\[{v\_{\text{x}}}=ω\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\] Applying this to our equation for acceleration -\\[{a\_{\text{x}}}=\frac{\\,dω {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)}{\\,dt}\\] +\\[{a\_{\text{x}}}=\frac{\\,dω\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)}{\\,dt}\\] By the product and chain rules, we find -\\[{a\_{\text{x}}}=\frac{\\,dω}{\\,dt} {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)-ω {L\_{\text{rod}}} \sin\left({θ\_{p}}\right) \frac{\\,d{θ\_{p}}}{\\,dt}\\] +\\[{a\_{\text{x}}}=\frac{\\,dω}{\\,dt}\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)-ω\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\,\frac{\\,d{θ\_{p}}}{\\,dt}\\] Simplifying, -\\[{a\_{\text{x}}}=-ω^{2} {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)+α {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)\\] +\\[{a\_{\text{x}}}=-ω^{2}\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)+α\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\]
@@ -123,7 +123,7 @@ Simplifying, |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The \\(y\\)-component of acceleration of the pendulum | |Units |\\(\frac{\text{m}}{\text{s}^{2}}\\) | -|Equation |\\[{a\_{\text{y}}}=ω^{2} {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)+α {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)\\] | +|Equation |\\[{a\_{\text{y}}}=ω^{2}\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)+α\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\] | |Description|
  • \\({a\_{\text{y}}}\\) is the \\(y\\)-component of acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(ω\\) is the angular velocity (\\(\frac{\text{rad}}{\text{s}}\\))
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
  • \\(α\\) is the angular acceleration (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy | | @@ -136,19 +136,19 @@ Our acceleration is: Earlier, we found the vertical velocity to be -\\[{v\_{\text{y}}}=ω {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)\\] +\\[{v\_{\text{y}}}=ω\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\] Applying this to our equation for acceleration -\\[{a\_{\text{y}}}=\frac{\\,dω {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)}{\\,dt}\\] +\\[{a\_{\text{y}}}=\frac{\\,dω\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)}{\\,dt}\\] By the product and chain rules, we find -\\[{a\_{\text{y}}}=\frac{\\,dω}{\\,dt} {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)+ω {L\_{\text{rod}}} \cos\left({θ\_{p}}\right) \frac{\\,d{θ\_{p}}}{\\,dt}\\] +\\[{a\_{\text{y}}}=\frac{\\,dω}{\\,dt}\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)+ω\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)\\,\frac{\\,d{θ\_{p}}}{\\,dt}\\] Simplifying, -\\[{a\_{\text{y}}}=ω^{2} {L\_{\text{rod}}} \cos\left({θ\_{p}}\right)+α {L\_{\text{rod}}} \sin\left({θ\_{p}}\right)\\] +\\[{a\_{\text{y}}}=ω^{2}\\,{L\_{\text{rod}}}\\,\cos\left({θ\_{p}}\right)+α\\,{L\_{\text{rod}}}\\,\sin\left({θ\_{p}}\right)\\]
@@ -160,14 +160,14 @@ Simplifying, |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Horizontal force on the pendulum | |Units |\\({\text{N}}\\) | -|Equation |\\[\boldsymbol{F}=m {a\_{\text{x}}}=-\boldsymbol{T} \sin\left({θ\_{p}}\right)\\] | +|Equation |\\[\boldsymbol{F}=m\\,{a\_{\text{x}}}=-\boldsymbol{T}\\,\sin\left({θ\_{p}}\right)\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\({a\_{\text{x}}}\\) is the \\(x\\)-component of acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(\boldsymbol{T}\\) is the tension (\\({\text{N}}\\))
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
| |Source |-- | |RefBy | | #### Detailed derivation of force on the pendulum: {#GD:hForceOnPendulumDeriv} -\\[\boldsymbol{F}=m {a\_{\text{x}}}=-\boldsymbol{T} \sin\left({θ\_{p}}\right)\\] +\\[\boldsymbol{F}=m\\,{a\_{\text{x}}}=-\boldsymbol{T}\\,\sin\left({θ\_{p}}\right)\\]
@@ -179,14 +179,14 @@ Simplifying, |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Vertical force on the pendulum | |Units |\\({\text{N}}\\) | -|Equation |\\[\boldsymbol{F}=m {a\_{\text{y}}}=\boldsymbol{T} \cos\left({θ\_{p}}\right)-m \boldsymbol{g}\\] | +|Equation |\\[\boldsymbol{F}=m\\,{a\_{\text{y}}}=\boldsymbol{T}\\,\cos\left({θ\_{p}}\right)-m\\,\boldsymbol{g}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\({a\_{\text{y}}}\\) is the \\(y\\)-component of acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
  • \\(\boldsymbol{T}\\) is the tension (\\({\text{N}}\\))
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
  • \\(\boldsymbol{g}\\) is the gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Source |-- | |RefBy | | #### Detailed derivation of force on the pendulum: {#GD:vForceOnPendulumDeriv} -\\[\boldsymbol{F}=m {a\_{\text{y}}}=\boldsymbol{T} \cos\left({θ\_{p}}\right)-m \boldsymbol{g}\\] +\\[\boldsymbol{F}=m\\,{a\_{\text{y}}}=\boldsymbol{T}\\,\cos\left({θ\_{p}}\right)-m\\,\boldsymbol{g}\\]
@@ -208,27 +208,27 @@ Simplifying, Consider the torque on a pendulum defined in [TM:NewtonSecLawRotMot](./SecTMs.md#TM:NewtonSecLawRotMot). The force providing the restoring torque is the component of weight of the pendulum bob that acts along the arc length. The torque is the length of the string \\({L\_{\text{rod}}}\\) multiplied by the component of the net force that is perpendicular to the radius of the arc. The minus sign indicates the torque acts in the opposite direction of the angular displacement: -\\[\boldsymbol{τ}=-{L\_{\text{rod}}} m \boldsymbol{g} \sin\left({θ\_{p}}\right)\\] +\\[\boldsymbol{τ}=-{L\_{\text{rod}}}\\,m\\,\boldsymbol{g}\\,\sin\left({θ\_{p}}\right)\\] So then -\\[\boldsymbol{I} α=-{L\_{\text{rod}}} m \boldsymbol{g} \sin\left({θ\_{p}}\right)\\] +\\[\boldsymbol{I}\\,α=-{L\_{\text{rod}}}\\,m\\,\boldsymbol{g}\\,\sin\left({θ\_{p}}\right)\\] Therefore, -\\[\boldsymbol{I} \frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-{L\_{\text{rod}}} m \boldsymbol{g} \sin\left({θ\_{p}}\right)\\] +\\[\boldsymbol{I}\\,\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-{L\_{\text{rod}}}\\,m\\,\boldsymbol{g}\\,\sin\left({θ\_{p}}\right)\\] Substituting for \\(\boldsymbol{I}\\) -\\[m {L\_{\text{rod}}}^{2} \frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-{L\_{\text{rod}}} m \boldsymbol{g} \sin\left({θ\_{p}}\right)\\] +\\[m\\,{L\_{\text{rod}}}^{2}\\,\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-{L\_{\text{rod}}}\\,m\\,\boldsymbol{g}\\,\sin\left({θ\_{p}}\right)\\] Crossing out \\(m\\) and \\({L\_{\text{rod}}}\\) we have -\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-\left(\frac{\boldsymbol{g}}{{L\_{\text{rod}}}}\right) \sin\left({θ\_{p}}\right)\\] +\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-\left(\frac{\boldsymbol{g}}{{L\_{\text{rod}}}}\right)\\,\sin\left({θ\_{p}}\right)\\] For small angles, we approximate sin \\({θ\_{p}}\\) to \\({θ\_{p}}\\) -\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-\left(\frac{\boldsymbol{g}}{{L\_{\text{rod}}}}\right) {θ\_{p}}\\] +\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}=-\left(\frac{\boldsymbol{g}}{{L\_{\text{rod}}}}\right)\\,{θ\_{p}}\\] Because this equation, has the same form as the equation for simple harmonic motion the solution is easy to find. The angular frequency @@ -244,7 +244,7 @@ Because this equation, has the same form as the equation for simple harmonic mot |:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |The period of the pendulum | |Units |\\({\text{s}}\\) | -|Equation |\\[T=2 π \sqrt{\frac{{L\_{\text{rod}}}}{\boldsymbol{g}}}\\] | +|Equation |\\[T=2\\,π\\,\sqrt{\frac{{L\_{\text{rod}}}}{\boldsymbol{g}}}\\] | |Description|
  • \\(T\\) is the period (\\({\text{s}}\\))
  • \\(π\\) is the ratio of circumference to diameter for any circle (Unitless)
  • \\({L\_{\text{rod}}}\\) is the length of the rod (\\({\text{m}}\\))
  • \\(\boldsymbol{g}\\) is the gravitational acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The frequency and period are defined in the data definitions for [frequency](./SecDDs.md#DD:frequencyDD) and [period](./SecDDs.md#DD:periodSHMDD) respectively
| |Source |-- | @@ -258,4 +258,4 @@ The period of the pendulum can be defined from the general definition for the eq Therefore from the data definition of the equation for [angular frequency](./SecDDs.md#DD:angFrequencyDD), we have -\\[T=2 π \sqrt{\frac{{L\_{\text{rod}}}}{\boldsymbol{g}}}\\] +\\[T=2\\,π\\,\sqrt{\frac{{L\_{\text{rod}}}}{\boldsymbol{g}}}\\] diff --git a/code/stable/sglpend/SRS/mdBook/src/SecIMs.md b/code/stable/sglpend/SRS/mdBook/src/SecIMs.md index 4065f1f208..6fa2df2eff 100644 --- a/code/stable/sglpend/SRS/mdBook/src/SecIMs.md +++ b/code/stable/sglpend/SRS/mdBook/src/SecIMs.md @@ -15,7 +15,7 @@ This section transforms the problem defined in the [problem description](./SecPr |Output |\\({θ\_{p}}\\) | |Input Constraints |\\[{L\_{\text{rod}}}\gt{}0\\]\\[{θ\_{i}}\gt{}0\\]\\[\boldsymbol{g}\gt{}0\\] | |Output Constraints|\\[{θ\_{p}}\gt{}0\\] | -|Equation |\\[{θ\_{p}}\left(t\right)={θ\_{i}} \cos\left(Ω t\right)\\] | +|Equation |\\[{θ\_{p}}\left(t\right)={θ\_{i}}\\,\cos\left(Ω\\,t\right)\\] | |Description |
  • \\({θ\_{p}}\\) is the displacement angle of the pendulum (\\({\text{rad}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({θ\_{i}}\\) is the initial pendulum angle (\\({\text{rad}}\\))
  • \\(Ω\\) is the angular frequency (\\({\text{s}}\\))
| |Notes |
  • The constraint \\({θ\_{i}}\gt{}0\\) is required. The angular frequency is defined in [GD:angFrequencyGD](./SecGDs.md#GD:angFrequencyGD).
| |Source |-- | @@ -25,20 +25,20 @@ This section transforms the problem defined in the [problem description](./SecPr When the pendulum is displaced to an initial angle and released, the pendulum swings back and forth with periodic motion. By applying [Newton's second law for rotational motion](./SecTMs.md#TM:NewtonSecLawRotMot), the equation of motion for the pendulum may be obtained: -\\[\boldsymbol{τ}=\boldsymbol{I} α\\] +\\[\boldsymbol{τ}=\boldsymbol{I}\\,α\\] Where \\(\boldsymbol{τ}\\) denotes the torque, \\(\boldsymbol{I}\\) denotes the moment of inertia and \\(α\\) denotes the angular acceleration. This implies: -\\[-m \boldsymbol{g} \sin\left({θ\_{p}}\right) {L\_{\text{rod}}}=m {L\_{\text{rod}}}^{2} \frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}\\] +\\[-m\\,\boldsymbol{g}\\,\sin\left({θ\_{p}}\right)\\,{L\_{\text{rod}}}=m\\,{L\_{\text{rod}}}^{2}\\,\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}\\] And rearranged as: -\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}+\frac{\boldsymbol{g}}{{L\_{\text{rod}}}} \sin\left({θ\_{p}}\right)=0\\] +\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}+\frac{\boldsymbol{g}}{{L\_{\text{rod}}}}\\,\sin\left({θ\_{p}}\right)=0\\] If the amplitude of angular displacement is small enough, we can approximate \\(\sin\left({θ\_{p}}\right)={θ\_{p}}\\) for the purpose of a simple pendulum at very small angles. Then the equation of motion reduces to the equation of simple harmonic motion: -\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}+\frac{\boldsymbol{g}}{{L\_{\text{rod}}}} {θ\_{p}}=0\\] +\\[\frac{\\,d\frac{\\,d{θ\_{p}}}{\\,dt}}{\\,dt}+\frac{\boldsymbol{g}}{{L\_{\text{rod}}}}\\,{θ\_{p}}=0\\] Thus the simple harmonic motion is: -\\[{θ\_{p}}\left(t\right)={θ\_{i}} \cos\left(Ω t\right)\\] +\\[{θ\_{p}}\left(t\right)={θ\_{i}}\\,\cos\left(Ω\\,t\right)\\] diff --git a/code/stable/sglpend/SRS/mdBook/src/SecTMs.md b/code/stable/sglpend/SRS/mdBook/src/SecTMs.md index bda4b36da6..2663fccd3f 100644 --- a/code/stable/sglpend/SRS/mdBook/src/SecTMs.md +++ b/code/stable/sglpend/SRS/mdBook/src/SecTMs.md @@ -39,7 +39,7 @@ This section focuses on the general equations and laws that SglPend is based on. |Refname |TM:NewtonSecLawMot | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's second law of motion | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}\\] | +|Equation |\\[\boldsymbol{F}=m\\,\boldsymbol{a}\text{(}t\text{)}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The net force \\(\boldsymbol{F}\\) on a body is proportional to the acceleration \\(\boldsymbol{a}\text{(}t\text{)}\\) of the body, where \\(m\\) denotes the mass of the body as the constant of proportionality.
| |Source |-- | @@ -54,7 +54,7 @@ This section focuses on the general equations and laws that SglPend is based on. |Refname |TM:NewtonSecLawRotMot | |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's second law for rotational motion | -|Equation |\\[\boldsymbol{τ}=\boldsymbol{I} α\\] | +|Equation |\\[\boldsymbol{τ}=\boldsymbol{I}\\,α\\] | |Description|
  • \\(\boldsymbol{τ}\\) is the torque (\\(\text{N}\text{m}\\))
  • \\(\boldsymbol{I}\\) is the moment of inertia (\\(\text{kg}\text{m}^{2}\\))
  • \\(α\\) is the angular acceleration (\\(\frac{\text{rad}}{\text{s}^{2}}\\))
| |Notes |
  • The net torque \\(\boldsymbol{τ}\\) on a rigid body is proportional to its angular acceleration \\(α\\), where \\(\boldsymbol{I}\\) denotes the moment of inertia of the rigid body as the constant of proportionality.
| |Source |-- | diff --git a/code/stable/ssp/SRS/HTML/SSP_SRS.html b/code/stable/ssp/SRS/HTML/SSP_SRS.html index 90ec369838..fd20b0acc5 100644 --- a/code/stable/ssp/SRS/HTML/SSP_SRS.html +++ b/code/stable/ssp/SRS/HTML/SSP_SRS.html @@ -1354,7 +1354,7 @@

Theoretical Models

- + @@ -1449,7 +1449,7 @@

Theoretical Models

- + @@ -1508,7 +1508,7 @@

General Definitions

@@ -1595,7 +1595,7 @@

@@ -1682,7 +1682,7 @@

@@ -1751,7 +1751,7 @@

@@ -1895,7 +1895,7 @@

@@ -1979,7 +1979,7 @@

@@ -2048,7 +2048,7 @@

- + @@ -2106,7 +2106,7 @@

@@ -2183,53 +2183,53 @@

Considering one dimension, with moments in the clockwise direction as positive and moments in the counterclockwise direction as negative, and replacing the torque symbol with the moment symbol, the equation simplifies to:

- \[M={F_{\text{rot}}} r\] + \[M={F_{\text{rot}}}\,r\]

where Frot is the force causing rotation and r is the length of the moment arm, or the distance between the force and the axis about which the rotation acts. To represent the moment equilibrium, the moments from each force acting on a slice must be considered and added together. The forces acting on a slice are all shown in Fig:ForceDiagram. The midpoint of the base of a slice is considered as the axis of rotation, from which the length of the moment arm is measured. Considering first the interslice normal force acting on slice interface i, the moment is negative because the force tends to rotate the slice in a counterclockwise direction, and the length of the moment arm is the height of the force plus the difference in height between the base at slice interface i and the base at the midpoint of slice i. Thus, the moment is expressed as:

- \[-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)\] + \[-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)\]

For the i−1th slice interface, the moment is similar but in the opposite direction:

- \[{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)\] + \[{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)\]

Next, the interslice normal water force is considered. This force is zero at the height of the water table, then increases linearly towards the base of the slice due to the increasing water pressure. For such a triangular distribution, the resultant force acts at one-third of the height. Thus, for the interslice normal water force acting on slice interface i, the moment is:

- \[-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)\] + \[-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)\]

The moment for the interslice normal water force acting on slice interface i−1 is:

- \[{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)\] + \[{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)\]

The interslice shear force at slice interface i tends to rotate in the clockwise direction, and the length of the moment arm is the length from the slice edge to the slice midpoint, equivalent to half of the width of the slice, so the moment is:

- \[{\symbf{X}}_{i} \frac{{\symbf{b}}_{i}}{2}\] + \[{\symbf{X}}_{i}\,\frac{{\symbf{b}}_{i}}{2}\]

The interslice shear force at slice interface i−1 also tends to rotate in the clockwise direction, and has the same length of the moment arm, so the moment is:

- \[{\symbf{X}}_{i-1} \frac{{\symbf{b}}_{i}}{2}\] + \[{\symbf{X}}_{i-1}\,\frac{{\symbf{b}}_{i}}{2}\]

Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is Kc Wi where Wi can be expressed as γ bi y using GD:weight where y is the height of the segment under consideration. The corresponding length of the moment arm is y, the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible (A:Negligible-Effect-Surface-Slope-Seismic). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative:

- \[-\int_{0}^{{\symbf{h}}_{i}}{{K_{\text{c}}} γ {\symbf{b}}_{i} y}\,dy\] + \[-\int_{0}^{{\symbf{h}}_{i}}{{K_{\text{c}}}\,γ\,{\symbf{b}}_{i}\,y}\,dy\]

Solving the definite integral yields:

- \[-{K_{\text{c}}} γ {\symbf{b}}_{i} \frac{{\symbf{h}}_{i}^{2}}{2}\] + \[-{K_{\text{c}}}\,γ\,{\symbf{b}}_{i}\,\frac{{\symbf{h}}_{i}^{2}}{2}\]

Using GD:weight again to express γ bi hi as Wi, the moment is:

- \[-{K_{\text{c}}} {\symbf{W}}_{i} \frac{{\symbf{h}}_{i}}{2}\] + \[-{K_{\text{c}}}\,{\symbf{W}}_{i}\,\frac{{\symbf{h}}_{i}}{2}\]

The surface hydrostatic force acts into the midpoint of the surface of the slice (A:Hydrostatic-Force-Slice-Midpoint). Thus, the vertical component of the force acts directly towards the point of rotation, and has a moment of zero. The horizontal component of the force tends to rotate in a clockwise direction and the length of the moment arm is the entire height of the slice. Thus, the moment is:

- \[{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}\] + \[{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}\]

The external force again acts into the midpoint of the slice surface, so the vertical component does not contribute to the moment, and the length of the moment arm is again the entire height of the slice. The moment is:

- \[{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i}\] + \[{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i}\]

The base hydrostatic force and slice weight both act in the direction of the point of rotation (A:Hydrostatic-Force-Slice-Midpoint), therefore both have moments of zero. Thus, all of the moments have been determined. The moment equilibrium is then represented by the sum of all moments:

- \[0=-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2} \left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i}\] + \[0=-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i}\]
@@ -2248,7 +2248,7 @@

- + @@ -2288,20 +2288,20 @@

Detailed derivation of weight:

\[\symbf{a}\text{(}t\text{)}=\begin{bmatrix} 0\\ - \symbf{g} \symbf{\hat{j}} + \symbf{g}\,\symbf{\hat{j}} \end{bmatrix}\]

Since there is only one non-zero vector component, the scalar value W will be used for the weight. In this scenario, Newton's second law of motion from TM:NewtonSecLawMot can be expressed as:

- \[W=m \symbf{g}\] + \[W=m\,\symbf{g}\]

Mass can be expressed as density multiplied by volume, resulting in:

- \[W=ρ V \symbf{g}\] + \[W=ρ\,V\,\symbf{g}\]

Substituting specific weight as the product of density and gravitational acceleration yields:

- \[W=V γ\] + \[W=V\,γ\]
@@ -2323,11 +2323,11 @@

Detailed derivation of weight:

@@ -2387,27 +2387,27 @@

For the case where the water table is above the slope surface, the weights come from the weight of the saturated soil. Substituting values for saturated soil into the equation for weight from GD:weight yields:

- \[{\symbf{W}}_{i}={\symbf{V}_{\text{sat},i}} {γ_{\text{sat}}}\] + \[{\symbf{W}}_{i}={\symbf{V}_{\text{sat},i}}\,{γ_{\text{sat}}}\]

Due to A:Plane-Strain-Conditions, only two dimensions are considered, so the areas of saturated soil are considered instead of the volumes of saturated soil. Any given slice has a trapezoidal shape. The area of a trapezoid is the average of the lengths of the parallel sides multiplied by the length between the parallel sides. The parallel sides in this case are the interslice edges and the length between them is the width of the slice. Thus, the weights are defined as:

- \[{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}\] + \[{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}\]

For the case where the water table is below the slip surface, the weights come from the weight of the dry soil. Substituting values for dry soil into the equation for weight from GD:weight yields:

- \[{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}} {γ_{\text{dry}}}\] + \[{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}}\,{γ_{\text{dry}}}\]

A:Plane-Strain-Conditions again allows for two-dimensional analysis so the areas of dry soil are considered instead of the volumes of dry soil. The trapezoidal slice shape is the same as in the previous case, so the weights are defined as:

- \[{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{dry}}}\] + \[{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{dry}}}\]

For the case where the water table is between the slope surface and slip surface, the weights are the sums of the weights of the dry portions and weights of the saturated portions of the soil. Substituting values for dry and saturated soil into the equation for weight from GD:weight and adding them together yields:

- \[{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}} {γ_{\text{dry}}}+{\symbf{V}_{\text{sat},i}} {γ_{\text{sat}}}\] + \[{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}}\,{γ_{\text{dry}}}+{\symbf{V}_{\text{sat},i}}\,{γ_{\text{sat}}}\]

A:Plane-Strain-Conditions again allows for two-dimensional analysis so the areas of dry soil and areas of saturated soil are considered instead of the volumes of dry soil and volumes of saturated soil. The water table is assumed to only intersect a slice surface or base at a slice edge (A:Water-Intersects-Surface-Edge, A:Water-Intersects-Base-Edge), so the dry and saturated portions each have trapezoidal shape. For the dry portion, the parallel sides of the trapezoid are the lengths between the slope surface and water table at the slice edges. For the saturated portion, the parallel sides of the trapezoid are the lengths between the water table and slip surface at the slice edges. Thus, the weights are defined as:

- \[{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \left(\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right) {γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}\right)\] + \[{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\left(\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right)\,{γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}\right)\]
@@ -2426,7 +2426,7 @@

- + @@ -2485,10 +2485,10 @@

@@ -2544,15 +2544,15 @@

The base hydrostatic forces come from the hydrostatic pressure exerted by the water above the base of each slice. The equation for hydrostatic pressure from GD:hsPressure is:

- \[p=γ h\] + \[p=γ\,h\]

The specific weight in this case is the unit weight of water γw. The height in this case is the height from the slice base to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint (A:Hydrostatic-Force-Slice-Midpoint). The height at the midpoint is the average of the height at slice interface i and the height at slice interface i−1:

- \[\frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\] + \[\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\]

Due to A:Plane-Strain-Conditions, only two dimensions are considered, so the base hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to base hydrostatic forces by multiplying by the corresponding length of the slice base Lb,i, assuming the water table does not intersect a slice base except at a slice edge (A:Water-Intersects-Base-Edge). Thus, in the case where the height of the water table is above the height of the slip surface, the base hydrostatic forces are defined as:

- \[{\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}} {γ_{w}} \frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\] + \[{\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}}\,{γ_{w}}\,\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\]

This equation is the non-zero case of GD:baseWtrF. The zero case is when the height of the water table is below the height of the slip surface, so there is no hydrostatic force.

@@ -2577,10 +2577,10 @@

@@ -2636,15 +2636,15 @@

The surface hydrostatic forces come from the hydrostatic pressure exerted by the water above the surface of each slice. The equation for hydrostatic pressure from GD:hsPressure is:

- \[p=γ h\] + \[p=γ\,h\]

The specific weight in this case is the unit weight of water γw. The height in this case is the height from the slice surface to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint (A:Hydrostatic-Force-Slice-Midpoint). The height at the midpoint is the average of the height at slice interface i and the height at slice interface i−1:

- \[\frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right)\] + \[\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right)\]

Due to A:Plane-Strain-Conditions, only two dimensions are considered, so the surface hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to surface hydrostatic forces by multiplying by the corresponding length of the slice surface Ls,i, assuming the water table does not intersect a slice surface except at a slice edge (A:Water-Intersects-Surface-Edge). Thus, in the case where the height of the water table is above the height of the slope surface, the surface hydrostatic forces are defined as:

- \[{\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}} {γ_{w}} \frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right)\] + \[{\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}}\,{γ_{w}}\,\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right)\]

This equation is the non-zero case of GD:srfWtrF. The zero case is when the height of the water table is below the height of the slope surface, so there is no hydrostatic force.

@@ -2682,8 +2682,8 @@

Data Definitions

@@ -2926,7 +2926,7 @@

Data Definitions

@@ -2987,7 +2987,7 @@

Data Definitions

@@ -3048,7 +3048,7 @@

Data Definitions

@@ -3270,7 +3270,7 @@

Data Definitions

@@ -3334,7 +3334,7 @@

Data Definitions

@@ -3408,7 +3408,7 @@

Data Definitions

@@ -3729,7 +3729,7 @@

Instance Models

@@ -3784,83 +3784,83 @@

The mobilized shear force defined in GD:bsShrFEq can be substituted into the definition of mobilized shear force based on the factor of safety, from GD:mobShr yielding Equation (1) below:

- \[\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{N'}}_{i} \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}}\] + \[\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{N'}}_{i}\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}}\]

An expression for the effective normal forces, N′, can be derived by substituting the normal forces equilibrium from GD:normForcEq into the definition for effective normal forces from GD:resShearWO. This results in Equation (2):

- \[{\symbf{N'}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\] + \[{\symbf{N'}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\]

Substituting Equation (2) into Equation (1) gives:

- \[\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}}\] + \[\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}}\]

Since the interslice shear forces X and interslice normal forces G are unknown, they are separated from the other terms as follows:

- \[\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}}\] + \[\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}}\]

Applying assumptions A:Seismic-Force and A:Surface-Load, which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below:

- \[\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}}\] + \[\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}}\]

The definitions of GD:resShearWO and GD:mobShearWO are present in this equation, and thus can be replaced by Ri and Ti, respectively:

- \[{\symbf{T}}_{i}+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{\text{S}}}}\] + \[{\symbf{T}}_{i}+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)}{{F_{\text{S}}}}\]

The interslice shear forces X can be expressed in terms of the interslice normal forces G using A:Interslice-Norm-Shear-Forces-Linear and GD:normShrR, resulting in:

- \[{\symbf{T}}_{i}+\left(-λ {\symbf{f}}_{i-1} {\symbf{G}}_{i-1}+λ {\symbf{f}}_{i} {\symbf{G}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-λ {\symbf{f}}_{i-1} {\symbf{G}}_{i-1}+λ {\symbf{f}}_{i} {\symbf{G}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{\text{S}}}}\] + \[{\symbf{T}}_{i}+\left(-λ\,{\symbf{f}}_{i-1}\,{\symbf{G}}_{i-1}+λ\,{\symbf{f}}_{i}\,{\symbf{G}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-λ\,{\symbf{f}}_{i-1}\,{\symbf{G}}_{i-1}+λ\,{\symbf{f}}_{i}\,{\symbf{G}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)}{{F_{\text{S}}}}\]

Rearranging yields the following:

- \[{\symbf{G}}_{i} \left(\left(λ {\symbf{f}}_{i} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}\right)={\symbf{G}}_{i-1} \left(\left(λ {\symbf{f}}_{i-1} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i-1} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}\right)+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}\] + \[{\symbf{G}}_{i}\,\left(\left(λ\,{\symbf{f}}_{i}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)={\symbf{G}}_{i-1}\,\left(\left(λ\,{\symbf{f}}_{i-1}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i-1}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}\]

The definitions for Φ and Ψ from DD:convertFunc1 and DD:convertFunc2 simplify the above to Equation (3):

- \[{\symbf{G}}_{i} {\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1} {\symbf{Φ}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}\] + \[{\symbf{G}}_{i}\,{\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}\,{\symbf{Φ}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}\]

Versions of Equation (3) instantiated for slices 1 to n are shown below:

- \[{\symbf{G}}_{1} {\symbf{Φ}}_{1}={\symbf{Ψ}}_{0} {\symbf{G}}_{0} {\symbf{Φ}}_{0}+{F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\] - \[{\symbf{G}}_{2} {\symbf{Φ}}_{2}={\symbf{Ψ}}_{1} {\symbf{G}}_{1} {\symbf{Φ}}_{1}+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\] - \[{\symbf{G}}_{3} {\symbf{Φ}}_{3}={\symbf{Ψ}}_{2} {\symbf{G}}_{2} {\symbf{Φ}}_{2}+{F_{\text{S}}} {\symbf{T}}_{3}-{\symbf{R}}_{3}\] + \[{\symbf{G}}_{1}\,{\symbf{Φ}}_{1}={\symbf{Ψ}}_{0}\,{\symbf{G}}_{0}\,{\symbf{Φ}}_{0}+{F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\] + \[{\symbf{G}}_{2}\,{\symbf{Φ}}_{2}={\symbf{Ψ}}_{1}\,{\symbf{G}}_{1}\,{\symbf{Φ}}_{1}+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\] + \[{\symbf{G}}_{3}\,{\symbf{Φ}}_{3}={\symbf{Ψ}}_{2}\,{\symbf{G}}_{2}\,{\symbf{Φ}}_{2}+{F_{\text{S}}}\,{\symbf{T}}_{3}-{\symbf{R}}_{3}\]

...

- \[{\symbf{G}}_{n-2} {\symbf{Φ}}_{n-2}={\symbf{Ψ}}_{n-3} {\symbf{G}}_{n-3} {\symbf{Φ}}_{n-3}+{F_{\text{S}}} {\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\] - \[{\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2} {\symbf{G}}_{n-2} {\symbf{Φ}}_{n-2}+{F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\] - \[{\symbf{G}}_{n} {\symbf{Φ}}_{n}={\symbf{Ψ}}_{n-1} {\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}+{F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}\] + \[{\symbf{G}}_{n-2}\,{\symbf{Φ}}_{n-2}={\symbf{Ψ}}_{n-3}\,{\symbf{G}}_{n-3}\,{\symbf{Φ}}_{n-3}+{F_{\text{S}}}\,{\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\] + \[{\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2}\,{\symbf{G}}_{n-2}\,{\symbf{Φ}}_{n-2}+{F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\] + \[{\symbf{G}}_{n}\,{\symbf{Φ}}_{n}={\symbf{Ψ}}_{n-1}\,{\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}+{F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}\]

Applying A:Edge-Slices, which says that G0 and Gn are zero, results in the following special cases: Equation (8) for the first slice:

- \[{\symbf{G}}_{1} {\symbf{Φ}}_{1}={F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\] + \[{\symbf{G}}_{1}\,{\symbf{Φ}}_{1}={F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\]

and Equation (9) for the nth slice:

- \[-\left(\frac{{F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}\] + \[-\left(\frac{{F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}\]

Substituting Equation (8) into Equation (4) yields Equation (10):

- \[{\symbf{G}}_{2} {\symbf{Φ}}_{2}={\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\] + \[{\symbf{G}}_{2}\,{\symbf{Φ}}_{2}={\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\]

which can be substituted into Equation (5) to get Equation (11):

- \[{\symbf{G}}_{3} {\symbf{Φ}}_{3}={\symbf{Ψ}}_{2} \left({\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}} {\symbf{T}}_{3}-{\symbf{R}}_{3}\] + \[{\symbf{G}}_{3}\,{\symbf{Φ}}_{3}={\symbf{Ψ}}_{2}\,\left({\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{3}-{\symbf{R}}_{3}\]

and so on until Equation (12) is obtained from Equation (7):

- \[{\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2} \left({\symbf{Ψ}}_{n-3} \left({\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\] + \[{\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2}\,\left({\symbf{Ψ}}_{n-3}\,\left({\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\]

Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in:

- \[-\left(\frac{{F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{Ψ}}_{n-2} \left({\symbf{Ψ}}_{n-3} \left({\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\] + \[-\left(\frac{{F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{Ψ}}_{n-2}\,\left({\symbf{Ψ}}_{n-3}\,\left({\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\]

This can be rearranged by multiplying both sides by Ψn−1 and then distributing the multiplication of each Ψ over addition to obtain:

- \[-\left({F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}\right)={\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{2} \left({F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{\symbf{Ψ}}_{n-1} \left({F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\right)\] + \[-\left({F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}\right)={\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{2}\,\left({F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{\symbf{Ψ}}_{n-1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\right)\]

The multiplication of the Ψ terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an R or a T. The equation can then be rearranged so terms containing an R are on one side of the equality, and terms containing a T are on the other. The multiplication by the factor of safety is common to all of the T terms, and thus can be factored out, resulting in:

- \[{F_{\text{S}}} \left({\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{1} {\symbf{T}}_{1}+{\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{2} {\symbf{T}}_{2}+{\symbf{Ψ}}_{n-1} {\symbf{T}}_{n-1}+{\symbf{T}}_{n}\right)={\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{1} {\symbf{R}}_{1}+{\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{2} {\symbf{R}}_{2}+{\symbf{Ψ}}_{n-1} {\symbf{R}}_{n-1}+{\symbf{R}}_{n}\] + \[{F_{\text{S}}}\,\left({\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{1}\,{\symbf{T}}_{1}+{\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{2}\,{\symbf{T}}_{2}+{\symbf{Ψ}}_{n-1}\,{\symbf{T}}_{n-1}+{\symbf{T}}_{n}\right)={\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{1}\,{\symbf{R}}_{1}+{\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{2}\,{\symbf{R}}_{2}+{\symbf{Ψ}}_{n-1}\,{\symbf{R}}_{n-1}+{\symbf{R}}_{n}\]

Isolating the factor of safety on the left-hand side and using compact notation for the products and sums yields Equation (13), which can also be seen in IM:fctSfty:

- \[{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}}\] + \[{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}}\]

FS depends on the unknowns λ (IM:nrmShrFor) and G (IM:intsliceFs).

@@ -3953,19 +3953,19 @@

From the moment equilibrium of GD:momentEql with the primary assumption for the Morgenstern-Price method of A:Interslice-Norm-Shear-Forces-Linear and associated definition GD:normShrR, Equation (14) can be derived:

- \[0=-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+λ \frac{{\symbf{b}}_{i}}{2} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i}\] + \[0=-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+λ\,\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i}\]

Rearranging the equation in terms of λ leads to Equation (15):

- \[λ=\frac{-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)}\] + \[λ=\frac{-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)}\]

This equation can be simplified by applying assumptions A:Seismic-Force and A:Surface-Load, which state that the seismic and external forces, respectively, are zero:

- \[λ=\frac{-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)}\] + \[λ=\frac{-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)}\]

Taking the summation of all slices, and applying A:Edge-Slices to set G0, Gn, H0, and Hn equal to zero, a general equation for the proportionality constant λ is developed in Equation (16), which combines IM:nrmShrFor, IM:nrmShrForNum, and IM:nrmShrForDen:

- \[λ=\frac{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i} \left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right) \tan\left({\symbf{α}}_{i}\right)+{\symbf{h}}_{i} -2 {\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)}}\] + \[λ=\frac{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i}\,\left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right)\,\tan\left({\symbf{α}}_{i}\right)+{\symbf{h}}_{i}\,-2\,{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)}}\]

Equation (16) for λ is a function of the unknown interslice normal forces G (IM:intsliceFs) which itself depends on the unknown factor of safety FS (IM:fctSfty).

@@ -4009,9 +4009,9 @@

@@ -4118,9 +4118,9 @@

@@ -4209,8 +4209,8 @@

@@ -4268,11 +4268,11 @@

This derivation is identical to the derivation for IM:fctSfty up until Equation (3) shown again below:

- \[{\symbf{G}}_{i} {\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1} {\symbf{Φ}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}\] + \[{\symbf{G}}_{i}\,{\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}\,{\symbf{Φ}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}\]

A simple rearrangement of Equation (3) leads to Equation (17), also seen in IM:intsliceFs:

- \[{\symbf{G}}_{i}=\frac{{\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}}\] + \[{\symbf{G}}_{i}=\frac{{\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}}\]

The cases shown in IM:intsliceFs for when i = 0, i = 1, or i = n are derived by applying A:Edge-Slices, which says that G0 and Gn are zero, to Equation (17). G depends on the unknowns FS (IM:fctSfty) and λ (IM:nrmShrFor).

diff --git a/code/stable/ssp/SRS/Jupyter/SSP_SRS.ipynb b/code/stable/ssp/SRS/Jupyter/SSP_SRS.ipynb index 3d72574a6c..42ede257e1 100644 --- a/code/stable/ssp/SRS/Jupyter/SSP_SRS.ipynb +++ b/code/stable/ssp/SRS/Jupyter/SSP_SRS.ipynb @@ -551,7 +551,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -658,7 +658,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -723,7 +723,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -797,7 +797,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -871,7 +871,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -939,7 +939,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1075,7 +1075,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1143,7 +1143,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1207,7 +1207,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1268,7 +1268,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1322,43 +1322,43 @@ "$$\\symbf{τ}=\\symbf{u}\\times\\symbf{F}$$\n", "\n", "Considering one dimension, with moments in the clockwise direction as positive and moments in the counterclockwise direction as negative, and replacing the torque symbol with the moment symbol, the equation simplifies to:\n", - "$$M={F_{\\text{rot}}} r$$\n", + "$$M={F_{\\text{rot}}}\\,r$$\n", "\n", "where $F_rot$ is the force causing rotation and $r$ is the length of the moment arm, or the distance between the force and the axis about which the rotation acts. To represent the moment equilibrium, the moments from each force acting on a slice must be considered and added together. The forces acting on a slice are all shown in [Fig:ForceDiagram](#Figure:ForceDiagram). The midpoint of the base of a slice is considered as the axis of rotation, from which the length of the moment arm is measured. Considering first the interslice normal force acting on slice interface $i$, the moment is negative because the force tends to rotate the slice in a counterclockwise direction, and the length of the moment arm is the height of the force plus the difference in height between the base at slice interface $i$ and the base at the midpoint of slice $i$. Thus, the moment is expressed as:\n", - "$$-{\\symbf{G}}_{i} \\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", + "$$-{\\symbf{G}}_{i}\\,\\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", "\n", "For the $i - 1$th slice interface, the moment is similar but in the opposite direction:\n", - "$${\\symbf{G}}_{i-1} \\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", + "$${\\symbf{G}}_{i-1}\\,\\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", "\n", "Next, the interslice normal water force is considered. This force is zero at the height of the water table, then increases linearly towards the base of the slice due to the increasing water pressure. For such a triangular distribution, the resultant force acts at one-third of the height. Thus, for the interslice normal water force acting on slice interface $i$, the moment is:\n", - "$$-{\\symbf{H}}_{i} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", + "$$-{\\symbf{H}}_{i}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", "\n", "The moment for the interslice normal water force acting on slice interface $i - 1$ is:\n", - "$${\\symbf{H}}_{i-1} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i-1}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", + "$${\\symbf{H}}_{i-1}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i-1}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)$$\n", "\n", "The interslice shear force at slice interface $i$ tends to rotate in the clockwise direction, and the length of the moment arm is the length from the slice edge to the slice midpoint, equivalent to half of the width of the slice, so the moment is:\n", - "$${\\symbf{X}}_{i} \\frac{{\\symbf{b}}_{i}}{2}$$\n", + "$${\\symbf{X}}_{i}\\,\\frac{{\\symbf{b}}_{i}}{2}$$\n", "\n", "The interslice shear force at slice interface $i - 1$ also tends to rotate in the clockwise direction, and has the same length of the moment arm, so the moment is:\n", - "$${\\symbf{X}}_{i-1} \\frac{{\\symbf{b}}_{i}}{2}$$\n", + "$${\\symbf{X}}_{i-1}\\,\\frac{{\\symbf{b}}_{i}}{2}$$\n", "\n", "Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is $K_cW_i$ where $W_i$ can be expressed as $γb_iy$ using [GD:weight](#GD:weight) where $y$ is the height of the segment under consideration. The corresponding length of the moment arm is $y$, the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible ([A:Negligible-Effect-Surface-Slope-Seismic](#assumpNESSS)). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative:\n", - "$$-\\int_{0}^{{\\symbf{h}}_{i}}{{K_{\\text{c}}} γ {\\symbf{b}}_{i} y}\\,dy$$\n", + "$$-\\int_{0}^{{\\symbf{h}}_{i}}{{K_{\\text{c}}}\\,γ\\,{\\symbf{b}}_{i}\\,y}\\,dy$$\n", "\n", "Solving the definite integral yields:\n", - "$$-{K_{\\text{c}}} γ {\\symbf{b}}_{i} \\frac{{\\symbf{h}}_{i}^{2}}{2}$$\n", + "$$-{K_{\\text{c}}}\\,γ\\,{\\symbf{b}}_{i}\\,\\frac{{\\symbf{h}}_{i}^{2}}{2}$$\n", "\n", "Using [GD:weight](#GD:weight) again to express $γb_ih_i$ as $W_i$, the moment is:\n", - "$$-{K_{\\text{c}}} {\\symbf{W}}_{i} \\frac{{\\symbf{h}}_{i}}{2}$$\n", + "$$-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}\\,\\frac{{\\symbf{h}}_{i}}{2}$$\n", "\n", "The surface hydrostatic force acts into the midpoint of the surface of the slice ([A:Hydrostatic-Force-Slice-Midpoint](#assumpHFSM)). Thus, the vertical component of the force acts directly towards the point of rotation, and has a moment of zero. The horizontal component of the force tends to rotate in a clockwise direction and the length of the moment arm is the entire height of the slice. Thus, the moment is:\n", - "$${\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right) {\\symbf{h}}_{i}$$\n", + "$${\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\,{\\symbf{h}}_{i}$$\n", "\n", "The external force again acts into the midpoint of the slice surface, so the vertical component does not contribute to the moment, and the length of the moment arm is again the entire height of the slice. The moment is:\n", - "$${\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right) {\\symbf{h}}_{i}$$\n", + "$${\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\,{\\symbf{h}}_{i}$$\n", "\n", "The base hydrostatic force and slice weight both act in the direction of the point of rotation ([A:Hydrostatic-Force-Slice-Midpoint](#assumpHFSM)), therefore both have moments of zero. Thus, all of the moments have been determined. The moment equilibrium is then represented by the sum of all moments:\n", - "$$0=-{\\symbf{G}}_{i} \\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1} \\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+\\frac{{\\symbf{b}}_{i}}{2} \\left({\\symbf{X}}_{i}+{\\symbf{X}}_{i-1}\\right)+\\frac{-{K_{\\text{c}}} {\\symbf{W}}_{i} {\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right) {\\symbf{h}}_{i}+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right) {\\symbf{h}}_{i}$$\n", + "$$0=-{\\symbf{G}}_{i}\\,\\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1}\\,\\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+\\frac{{\\symbf{b}}_{i}}{2}\\,\\left({\\symbf{X}}_{i}+{\\symbf{X}}_{i-1}\\right)+\\frac{-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}\\,{\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\,{\\symbf{h}}_{i}+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\,{\\symbf{h}}_{i}$$\n", "
\n", "\n", "
Equation\n", - "$$T=2 π \\sqrt{\\frac{{L_{\\text{rod}}}}{\\symbf{g}}}$$\n", + "$$T=2\\,π\\,\\sqrt{\\frac{{L_{\\text{rod}}}}{\\symbf{g}}}$$\n", "
Equation\n", - "$${{p_{\\text{x}}}^{\\text{i}}}={L_{\\text{rod}}} \\sin\\left({θ_{i}}\\right)$$\n", + "$${{p_{\\text{x}}}^{\\text{i}}}={L_{\\text{rod}}}\\,\\sin\\left({θ_{i}}\\right)$$\n", "
Equation\n", - "$${{p_{\\text{y}}}^{\\text{i}}}=-{L_{\\text{rod}}} \\cos\\left({θ_{i}}\\right)$$\n", + "$${{p_{\\text{y}}}^{\\text{i}}}=-{L_{\\text{rod}}}\\,\\cos\\left({θ_{i}}\\right)$$\n", "
Equation\n", - "$$Ω=\\frac{2 π}{T}$$\n", + "$$Ω=\\frac{2\\,π}{T}$$\n", "
Equation\n", - "$${θ_{p}}\\left(t\\right)={θ_{i}} \\cos\\left(Ω t\\right)$$\n", + "$${θ_{p}}\\left(t\\right)={θ_{i}}\\,\\cos\\left(Ω\\,t\\right)$$\n", "
Equation\[{τ^{\text{f}}}={σ_{N}}' \tan\left(φ'\right)+c'\]\[{τ^{\text{f}}}={σ_{N}}'\,\tan\left(φ'\right)+c'\]
Description
Equation\[\symbf{F}=m \symbf{a}\text{(}t\text{)}\]\[\symbf{F}=m\,\symbf{a}\text{(}t\text{)}\]
Description
Equation - \[{\symbf{N}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)\] + \[{\symbf{N}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)\]
Equation - \[{\symbf{S}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)\] + \[{\symbf{S}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)\]
Equation - \[{\symbf{P}}_{i}={\symbf{N'}}_{i} \tan\left({φ'}_{i}\right)+{c'}_{i} {\symbf{L}_{b,i}}\] + \[{\symbf{P}}_{i}={\symbf{N'}}_{i}\,\tan\left({φ'}_{i}\right)+{c'}_{i}\,{\symbf{L}_{b,i}}\]
Equation - \[{\symbf{S}}_{i}=\frac{{\symbf{P}}_{i}}{{F_{\text{S}}}}=\frac{{\symbf{N'}}_{i} \tan\left({φ'}_{i}\right)+{c'}_{i} {\symbf{L}_{b,i}}}{{F_{\text{S}}}}\] + \[{\symbf{S}}_{i}=\frac{{\symbf{P}}_{i}}{{F_{\text{S}}}}=\frac{{\symbf{N'}}_{i}\,\tan\left({φ'}_{i}\right)+{c'}_{i}\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}}\]
Equation - \[{\symbf{R}}_{i}=\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left({φ'}_{i}\right)+{c'}_{i} {\symbf{L}_{b,i}}\] + \[{\symbf{R}}_{i}=\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left({φ'}_{i}\right)+{c'}_{i}\,{\symbf{L}_{b,i}}\]
Equation - \[{\symbf{T}}_{i}=\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)\] + \[{\symbf{T}}_{i}=\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)\]
Equation\[\symbf{X}=λ \symbf{f} \symbf{G}\]\[\symbf{X}=λ\,\symbf{f}\,\symbf{G}\]
Description
Equation - \[0=-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2} \left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i}\] + \[0=-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i}\]
Equation\[W=V γ\]\[W=V\,γ\]
Description
Equation - \[{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \begin{cases} - \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ - \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right) {γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}, & {\symbf{y}_{\text{slope},i}}\geq{}{\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{slope},i-1}}\geq{}{\symbf{y}_{\text{wt},i-1}}\geq{}{\symbf{y}_{\text{slip},i-1}}\\ - \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{dry}}}, & {\symbf{y}_{\text{wt},i}}\lt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\lt{}{\symbf{y}_{\text{slip},i-1}} - \end{cases}\] + \[{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\begin{cases} + \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ + \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right)\,{γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}, & {\symbf{y}_{\text{slope},i}}\geq{}{\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{slope},i-1}}\geq{}{\symbf{y}_{\text{wt},i-1}}\geq{}{\symbf{y}_{\text{slip},i-1}}\\ + \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{dry}}}, & {\symbf{y}_{\text{wt},i}}\lt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\lt{}{\symbf{y}_{\text{slip},i-1}} + \end{cases}\]
Equation\[p=γ h\]\[p=γ\,h\]
Description
Equation - \[{\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}} {γ_{w}} \frac{1}{2} \begin{cases} - {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slip},i-1}}\\ - 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slip},i-1}} - \end{cases}\] + \[{\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}}\,{γ_{w}}\,\frac{1}{2}\,\begin{cases} + {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slip},i-1}}\\ + 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slip},i-1}} + \end{cases}\]
Equation - \[{\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}} {γ_{w}} \frac{1}{2} \begin{cases} - {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ - 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slope},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slope},i-1}} - \end{cases}\] + \[{\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}}\,{γ_{w}}\,\frac{1}{2}\,\begin{cases} + {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ + 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slope},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slope},i-1}} + \end{cases}\]
Equation \[\symbf{H}=\begin{cases} - \frac{\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2} {γ_{w}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}\right)^{2} {γ_{w}}, & {\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slope},i}}\\ - \frac{\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2} {γ_{w}}, & {\symbf{y}_{\text{slope},i}}\gt{}{\symbf{y}_{\text{wt},i}}\land{}{\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\\ + \frac{\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2}\,{γ_{w}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}\right)^{2}\,{γ_{w}}, & {\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slope},i}}\\ + \frac{\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2}\,{γ_{w}}, & {\symbf{y}_{\text{slope},i}}\gt{}{\symbf{y}_{\text{wt},i}}\land{}{\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\\ 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slip},i}} \end{cases}\]
Equation - \[{\symbf{L}_{b}}={\symbf{b}}_{i} \sec\left({\symbf{α}}_{i}\right)\] + \[{\symbf{L}_{b}}={\symbf{b}}_{i}\,\sec\left({\symbf{α}}_{i}\right)\]
Equation - \[{\symbf{L}_{s}}={\symbf{b}}_{i} \sec\left({\symbf{β}}_{i}\right)\] + \[{\symbf{L}_{s}}={\symbf{b}}_{i}\,\sec\left({\symbf{β}}_{i}\right)\]
Equation - \[\symbf{h}=\frac{1}{2} \left({{\symbf{h}^{\text{R}}}}_{i}+{{\symbf{h}^{\text{L}}}}_{i}\right)\] + \[\symbf{h}=\frac{1}{2}\,\left({{\symbf{h}^{\text{R}}}}_{i}+{{\symbf{h}^{\text{L}}}}_{i}\right)\]
\[\symbf{f}=\begin{cases} 1, & \mathit{const\_f}\\ - \sin\left(π \frac{{\symbf{x}_{\text{slip},i}}-{\symbf{x}_{\text{slip},0}}}{{\symbf{x}_{\text{slip},n}}-{\symbf{x}_{\text{slip},0}}}\right), & \neg{}\mathit{const\_f} + \sin\left(π\,\frac{{\symbf{x}_{\text{slip},i}}-{\symbf{x}_{\text{slip},0}}}{{\symbf{x}_{\text{slip},n}}-{\symbf{x}_{\text{slip},0}}}\right), & \neg{}\mathit{const\_f} \end{cases}\]
Equation - \[\symbf{Φ}=\left(λ {\symbf{f}}_{i} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}\] + \[\symbf{Φ}=\left(λ\,{\symbf{f}}_{i}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}\]
Equation - \[\symbf{Ψ}=\frac{\left(λ {\symbf{f}}_{i} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}}{{\symbf{Φ}}_{i-1}}\] + \[\symbf{Ψ}=\frac{\left(λ\,{\symbf{f}}_{i}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}}{{\symbf{Φ}}_{i-1}}\]
Equation - \[{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}}\] + \[{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}}\]
Equation \[{\symbf{C}_{\text{num},i}}=\begin{cases} - {\symbf{b}}_{1} \left({\symbf{G}}_{1}+{\symbf{H}}_{1}\right) \tan\left({\symbf{α}}_{1}\right), & i=1\\ - {\symbf{b}}_{i} \left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right) \tan\left({\symbf{α}}_{i}\right)+\symbf{h} -2 {\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right), & 2\leq{}i\leq{}n-1\\ - {\symbf{b}}_{n} \left({\symbf{G}}_{n-1}+{\symbf{H}}_{n-1}\right) \tan\left({\symbf{α}}_{n-1}\right), & i=n + {\symbf{b}}_{1}\,\left({\symbf{G}}_{1}+{\symbf{H}}_{1}\right)\,\tan\left({\symbf{α}}_{1}\right), & i=1\\ + {\symbf{b}}_{i}\,\left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right)\,\tan\left({\symbf{α}}_{i}\right)+\symbf{h}\,-2\,{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right), & 2\leq{}i\leq{}n-1\\ + {\symbf{b}}_{n}\,\left({\symbf{G}}_{n-1}+{\symbf{H}}_{n-1}\right)\,\tan\left({\symbf{α}}_{n-1}\right), & i=n \end{cases}\]
Equation \[{\symbf{C}_{\text{den},i}}=\begin{cases} - {\symbf{b}}_{1} {\symbf{f}}_{1} {\symbf{G}}_{1}, & i=1\\ - {\symbf{b}}_{i} \left({\symbf{f}}_{i} {\symbf{G}}_{i}+{\symbf{f}}_{i-1} {\symbf{G}}_{i-1}\right), & 2\leq{}i\leq{}n-1\\ - {\symbf{b}}_{n} {\symbf{G}}_{n-1} {\symbf{f}}_{n-1}, & i=n + {\symbf{b}}_{1}\,{\symbf{f}}_{1}\,{\symbf{G}}_{1}, & i=1\\ + {\symbf{b}}_{i}\,\left({\symbf{f}}_{i}\,{\symbf{G}}_{i}+{\symbf{f}}_{i-1}\,{\symbf{G}}_{i-1}\right), & 2\leq{}i\leq{}n-1\\ + {\symbf{b}}_{n}\,{\symbf{G}}_{n-1}\,{\symbf{f}}_{n-1}, & i=n \end{cases}\]
Equation \[{\symbf{G}}_{i}=\begin{cases} - \frac{{F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}}{{\symbf{Φ}}_{1}}, & i=1\\ - \frac{{\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}}, & 2\leq{}i\leq{}n-1\\ + \frac{{F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}}{{\symbf{Φ}}_{1}}, & i=1\\ + \frac{{\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}}, & 2\leq{}i\leq{}n-1\\ 0, & i=0\lor{}i=n \end{cases}\]
Equation\n", - "$${τ^{\\text{f}}}={σ_{N}}' \\tan\\left(φ'\\right)+c'$$\n", + "$${τ^{\\text{f}}}={σ_{N}}'\\,\\tan\\left(φ'\\right)+c'$$\n", "
Equation\n", - "$$\\symbf{F}=m \\symbf{a}\\text{(}t\\text{)}$$\n", + "$$\\symbf{F}=m\\,\\symbf{a}\\text{(}t\\text{)}$$\n", "
Equation\n", - "$${\\symbf{N}}_{i}=\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)$$\n", + "$${\\symbf{N}}_{i}=\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)$$\n", "
Equation\n", - "$${\\symbf{S}}_{i}=\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)$$\n", + "$${\\symbf{S}}_{i}=\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)$$\n", "
Equation\n", - "$${\\symbf{P}}_{i}={\\symbf{N'}}_{i} \\tan\\left({φ'}_{i}\\right)+{c'}_{i} {\\symbf{L}_{b,i}}$$\n", + "$${\\symbf{P}}_{i}={\\symbf{N'}}_{i}\\,\\tan\\left({φ'}_{i}\\right)+{c'}_{i}\\,{\\symbf{L}_{b,i}}$$\n", "
Equation\n", - "$${\\symbf{S}}_{i}=\\frac{{\\symbf{P}}_{i}}{{F_{\\text{S}}}}=\\frac{{\\symbf{N'}}_{i} \\tan\\left({φ'}_{i}\\right)+{c'}_{i} {\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", + "$${\\symbf{S}}_{i}=\\frac{{\\symbf{P}}_{i}}{{F_{\\text{S}}}}=\\frac{{\\symbf{N'}}_{i}\\,\\tan\\left({φ'}_{i}\\right)+{c'}_{i}\\,{\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", "
Equation\n", - "$${\\symbf{R}}_{i}=\\left(\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right) \\tan\\left({φ'}_{i}\\right)+{c'}_{i} {\\symbf{L}_{b,i}}$$\n", + "$${\\symbf{R}}_{i}=\\left(\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right)\\,\\tan\\left({φ'}_{i}\\right)+{c'}_{i}\\,{\\symbf{L}_{b,i}}$$\n", "
Equation\n", - "$${\\symbf{T}}_{i}=\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)$$\n", + "$${\\symbf{T}}_{i}=\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)$$\n", "
Equation\n", - "$$\\symbf{X}=λ \\symbf{f} \\symbf{G}$$\n", + "$$\\symbf{X}=λ\\,\\symbf{f}\\,\\symbf{G}$$\n", "
Equation\n", - "$$0=-{\\symbf{G}}_{i} \\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1} \\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+\\frac{{\\symbf{b}}_{i}}{2} \\left({\\symbf{X}}_{i}+{\\symbf{X}}_{i-1}\\right)+\\frac{-{K_{\\text{c}}} {\\symbf{W}}_{i} {\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right) {\\symbf{h}}_{i}+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right) {\\symbf{h}}_{i}$$\n", + "$$0=-{\\symbf{G}}_{i}\\,\\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1}\\,\\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+\\frac{{\\symbf{b}}_{i}}{2}\\,\\left({\\symbf{X}}_{i}+{\\symbf{X}}_{i-1}\\right)+\\frac{-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}\\,{\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\,{\\symbf{h}}_{i}+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\,{\\symbf{h}}_{i}$$\n", "
\n", @@ -1385,7 +1385,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1417,16 +1417,16 @@ "\n", "\n", "Under the influence of gravity, and assuming a 2D Cartesian coordinate system with down as positive, an object has an acceleration vector of:\n", - "$$\\symbf{a}\\text{(}t\\text{)}=\\begin{bmatrix} 0\\\\ \\symbf{g} \\symbf{\\hat{j}} \\end{bmatrix}$$\n", + "$$\\symbf{a}\\text{(}t\\text{)}=\\begin{bmatrix} 0\\\\ \\symbf{g}\\,\\symbf{\\hat{j}} \\end{bmatrix}$$\n", "\n", "Since there is only one non-zero vector component, the scalar value $W$ will be used for the weight. In this scenario, Newton's second law of motion from [TM:NewtonSecLawMot](#TM:NewtonSecLawMot) can be expressed as:\n", - "$$W=m \\symbf{g}$$\n", + "$$W=m\\,\\symbf{g}$$\n", "\n", "Mass can be expressed as density multiplied by volume, resulting in:\n", - "$$W=ρ V \\symbf{g}$$\n", + "$$W=ρ\\,V\\,\\symbf{g}$$\n", "\n", "Substituting specific weight as the product of density and gravitational acceleration yields:\n", - "$$W=V γ$$\n", + "$$W=V\\,γ$$\n", "
\n", "\n", "
Equation\n", - "$$W=V γ$$\n", + "$$W=V\\,γ$$\n", "
\n", @@ -1453,7 +1453,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1497,22 +1497,22 @@ "\n", "\n", "For the case where the water table is above the slope surface, the weights come from the weight of the saturated soil. Substituting values for saturated soil into the equation for weight from [GD:weight](#GD:weight) yields:\n", - "$${\\symbf{W}}_{i}={\\symbf{V}_{\\text{sat},i}} {γ_{\\text{sat}}}$$\n", + "$${\\symbf{W}}_{i}={\\symbf{V}_{\\text{sat},i}}\\,{γ_{\\text{sat}}}$$\n", "\n", "Due to [A:Plane-Strain-Conditions](#assumpPSC), only two dimensions are considered, so the areas of saturated soil are considered instead of the volumes of saturated soil. Any given slice has a trapezoidal shape. The area of a trapezoid is the average of the lengths of the parallel sides multiplied by the length between the parallel sides. The parallel sides in this case are the interslice edges and the length between them is the width of the slice. Thus, the weights are defined as:\n", - "$${\\symbf{W}}_{i}={\\symbf{b}}_{i} \\frac{1}{2} \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right) {γ_{\\text{sat}}}$$\n", + "$${\\symbf{W}}_{i}={\\symbf{b}}_{i}\\,\\frac{1}{2}\\,\\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)\\,{γ_{\\text{sat}}}$$\n", "\n", "For the case where the water table is below the slip surface, the weights come from the weight of the dry soil. Substituting values for dry soil into the equation for weight from [GD:weight](#GD:weight) yields:\n", - "$${\\symbf{W}}_{i}={\\symbf{V}_{\\text{dry},i}} {γ_{\\text{dry}}}$$\n", + "$${\\symbf{W}}_{i}={\\symbf{V}_{\\text{dry},i}}\\,{γ_{\\text{dry}}}$$\n", "\n", "[A:Plane-Strain-Conditions](#assumpPSC) again allows for two-dimensional analysis so the areas of dry soil are considered instead of the volumes of dry soil. The trapezoidal slice shape is the same as in the previous case, so the weights are defined as:\n", - "$${\\symbf{W}}_{i}={\\symbf{b}}_{i} \\frac{1}{2} \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right) {γ_{\\text{dry}}}$$\n", + "$${\\symbf{W}}_{i}={\\symbf{b}}_{i}\\,\\frac{1}{2}\\,\\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)\\,{γ_{\\text{dry}}}$$\n", "\n", "For the case where the water table is between the slope surface and slip surface, the weights are the sums of the weights of the dry portions and weights of the saturated portions of the soil. Substituting values for dry and saturated soil into the equation for weight from [GD:weight](#GD:weight) and adding them together yields:\n", - "$${\\symbf{W}}_{i}={\\symbf{V}_{\\text{dry},i}} {γ_{\\text{dry}}}+{\\symbf{V}_{\\text{sat},i}} {γ_{\\text{sat}}}$$\n", + "$${\\symbf{W}}_{i}={\\symbf{V}_{\\text{dry},i}}\\,{γ_{\\text{dry}}}+{\\symbf{V}_{\\text{sat},i}}\\,{γ_{\\text{sat}}}$$\n", "\n", "[A:Plane-Strain-Conditions](#assumpPSC) again allows for two-dimensional analysis so the areas of dry soil and areas of saturated soil are considered instead of the volumes of dry soil and volumes of saturated soil. The water table is assumed to only intersect a slice surface or base at a slice edge ([A:Water-Intersects-Surface-Edge](#assumpWISE), [A:Water-Intersects-Base-Edge](#assumpWIBE)), so the dry and saturated portions each have trapezoidal shape. For the dry portion, the parallel sides of the trapezoid are the lengths between the slope surface and water table at the slice edges. For the saturated portion, the parallel sides of the trapezoid are the lengths between the water table and slip surface at the slice edges. Thus, the weights are defined as:\n", - "$${\\symbf{W}}_{i}={\\symbf{b}}_{i} \\frac{1}{2} \\left(\\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{wt},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{wt},i-1}}\\right) {γ_{\\text{dry}}}+\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right) {γ_{\\text{sat}}}\\right)$$\n", + "$${\\symbf{W}}_{i}={\\symbf{b}}_{i}\\,\\frac{1}{2}\\,\\left(\\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{wt},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{wt},i-1}}\\right)\\,{γ_{\\text{dry}}}+\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)\\,{γ_{\\text{sat}}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${\\symbf{W}}_{i}={\\symbf{b}}_{i} \\frac{1}{2} \\begin{cases} \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right) {γ_{\\text{sat}}}, & {\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slope},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\gt{}{\\symbf{y}_{\\text{slope},i-1}}\\\\ \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{wt},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{wt},i-1}}\\right) {γ_{\\text{dry}}}+\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right) {γ_{\\text{sat}}}, & {\\symbf{y}_{\\text{slope},i}}\\geq{}{\\symbf{y}_{\\text{wt},i}}\\geq{}{\\symbf{y}_{\\text{slip},i}}\\land{}{\\symbf{y}_{\\text{slope},i-1}}\\geq{}{\\symbf{y}_{\\text{wt},i-1}}\\geq{}{\\symbf{y}_{\\text{slip},i-1}}\\\\ \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right) {γ_{\\text{dry}}}, & {\\symbf{y}_{\\text{wt},i}}\\lt{}{\\symbf{y}_{\\text{slip},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\lt{}{\\symbf{y}_{\\text{slip},i-1}} \\end{cases}$$\n", + "$${\\symbf{W}}_{i}={\\symbf{b}}_{i}\\,\\frac{1}{2}\\,\\begin{cases} \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)\\,{γ_{\\text{sat}}}, & {\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slope},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\gt{}{\\symbf{y}_{\\text{slope},i-1}}\\\\ \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{wt},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{wt},i-1}}\\right)\\,{γ_{\\text{dry}}}+\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)\\,{γ_{\\text{sat}}}, & {\\symbf{y}_{\\text{slope},i}}\\geq{}{\\symbf{y}_{\\text{wt},i}}\\geq{}{\\symbf{y}_{\\text{slip},i}}\\land{}{\\symbf{y}_{\\text{slope},i-1}}\\geq{}{\\symbf{y}_{\\text{wt},i-1}}\\geq{}{\\symbf{y}_{\\text{slip},i-1}}\\\\ \\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{slope},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)\\,{γ_{\\text{dry}}}, & {\\symbf{y}_{\\text{wt},i}}\\lt{}{\\symbf{y}_{\\text{slip},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\lt{}{\\symbf{y}_{\\text{slip},i-1}} \\end{cases}$$\n", "
\n", @@ -1539,7 +1539,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1599,7 +1599,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1641,13 +1641,13 @@ "\n", "\n", "The base hydrostatic forces come from the hydrostatic pressure exerted by the water above the base of each slice. The equation for hydrostatic pressure from [GD:hsPressure](#GD:hsPressure) is:\n", - "$$p=γ h$$\n", + "$$p=γ\\,h$$\n", "\n", "The specific weight in this case is the unit weight of water $γ_w$. The height in this case is the height from the slice base to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint ([A:Hydrostatic-Force-Slice-Midpoint](#assumpHFSM)). The height at the midpoint is the average of the height at slice interface $i$ and the height at slice interface $i - 1$:\n", - "$$\\frac{1}{2} \\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)$$\n", + "$$\\frac{1}{2}\\,\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)$$\n", "\n", "Due to [A:Plane-Strain-Conditions](#assumpPSC), only two dimensions are considered, so the base hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to base hydrostatic forces by multiplying by the corresponding length of the slice base $L_b,i$, assuming the water table does not intersect a slice base except at a slice edge ([A:Water-Intersects-Base-Edge](#assumpWIBE)). Thus, in the case where the height of the water table is above the height of the slip surface, the base hydrostatic forces are defined as:\n", - "$${\\symbf{U}_{\\text{b},i}}={\\symbf{L}_{b,i}} {γ_{w}} \\frac{1}{2} \\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)$$\n", + "$${\\symbf{U}_{\\text{b},i}}={\\symbf{L}_{b,i}}\\,{γ_{w}}\\,\\frac{1}{2}\\,\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}\\right)$$\n", "\n", "This equation is the non-zero case of [GD:baseWtrF](#GD:baseWtrF). The zero case is when the height of the water table is below the height of the slip surface, so there is no hydrostatic force.\n", "
\n", @@ -1676,7 +1676,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1718,13 +1718,13 @@ "\n", "\n", "The surface hydrostatic forces come from the hydrostatic pressure exerted by the water above the surface of each slice. The equation for hydrostatic pressure from [GD:hsPressure](#GD:hsPressure) is:\n", - "$$p=γ h$$\n", + "$$p=γ\\,h$$\n", "\n", "The specific weight in this case is the unit weight of water $γ_w$. The height in this case is the height from the slice surface to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint ([A:Hydrostatic-Force-Slice-Midpoint](#assumpHFSM)). The height at the midpoint is the average of the height at slice interface $i$ and the height at slice interface $i - 1$:\n", - "$$\\frac{1}{2} \\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slope},i-1}}\\right)$$\n", + "$$\\frac{1}{2}\\,\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slope},i-1}}\\right)$$\n", "\n", "Due to [A:Plane-Strain-Conditions](#assumpPSC), only two dimensions are considered, so the surface hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to surface hydrostatic forces by multiplying by the corresponding length of the slice surface $L_s,i$, assuming the water table does not intersect a slice surface except at a slice edge ([A:Water-Intersects-Surface-Edge](#assumpWISE)). Thus, in the case where the height of the water table is above the height of the slope surface, the surface hydrostatic forces are defined as:\n", - "$${\\symbf{U}_{\\text{g},i}}={\\symbf{L}_{s,i}} {γ_{w}} \\frac{1}{2} \\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slope},i-1}}\\right)$$\n", + "$${\\symbf{U}_{\\text{g},i}}={\\symbf{L}_{s,i}}\\,{γ_{w}}\\,\\frac{1}{2}\\,\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slope},i-1}}\\right)$$\n", "\n", "This equation is the non-zero case of [GD:srfWtrF](#GD:srfWtrF). The zero case is when the height of the water table is below the height of the slope surface, so there is no hydrostatic force.\n", "\n", @@ -1765,7 +1765,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2024,7 +2024,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2092,7 +2092,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2160,7 +2160,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2417,7 +2417,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2480,7 +2480,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2551,7 +2551,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2884,7 +2884,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -2927,68 +2927,68 @@ "\n", "\n", "The mobilized shear force defined in [GD:bsShrFEq](#GD:bsShrFEq) can be substituted into the definition of mobilized shear force based on the factor of safety, from [GD:mobShr](#GD:mobShr) yielding Equation (1) below:\n", - "$$\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{{\\symbf{N'}}_{i} \\tan\\left(φ'\\right)+c' {\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", + "$$\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{{\\symbf{N'}}_{i}\\,\\tan\\left(φ'\\right)+c'\\,{\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", "\n", "An expression for the effective normal forces, $N′$, can be derived by substituting the normal forces equilibrium from [GD:normForcEq](#GD:normForcEq) into the definition for effective normal forces from [GD:resShearWO](#GD:resShearWO). This results in Equation (2):\n", - "$${\\symbf{N'}}_{i}=\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}$$\n", + "$${\\symbf{N'}}_{i}=\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}$$\n", "\n", "Substituting Equation (2) into Equation (1) gives:\n", - "$$\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{\\left(\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right) \\tan\\left(φ'\\right)+c' {\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", + "$$\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{\\left(\\left({\\symbf{W}}_{i}-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right)\\,\\tan\\left(φ'\\right)+c'\\,{\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", "\n", "Since the interslice shear forces $X$ and interslice normal forces $G$ are unknown, they are separated from the other terms as follows:\n", - "$$\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)=\\frac{\\left(\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\cos\\left({\\symbf{ω}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}} {\\symbf{W}}_{i}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right) \\tan\\left(φ'\\right)+c' {\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", + "$$\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)=\\frac{\\left(\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\cos\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right)\\,\\tan\\left(φ'\\right)+c'\\,{\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", "\n", "Applying assumptions [A:Seismic-Force](#assumpSF) and [A:Surface-Load](#assumpSL), which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below:\n", - "$$\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)=\\frac{\\left(\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}} \\cos\\left({\\symbf{β}}_{i}\\right)\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)\\right) \\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right) \\tan\\left(φ'\\right)+c' {\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", + "$$\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)=\\frac{\\left(\\left({\\symbf{W}}_{i}+{\\symbf{U}_{\\text{g},i}}\\,\\cos\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{H}}_{i}+{\\symbf{H}}_{i-1}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)-{\\symbf{U}_{\\text{b},i}}\\right)\\,\\tan\\left(φ'\\right)+c'\\,{\\symbf{L}_{b,i}}}{{F_{\\text{S}}}}$$\n", "\n", "The definitions of [GD:resShearWO](#GD:resShearWO) and [GD:mobShearWO](#GD:mobShearWO) are present in this equation, and thus can be replaced by $R_i$ and $T_i$, respectively:\n", - "$${\\symbf{T}}_{i}+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{{\\symbf{R}}_{i}+\\left(\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)\\right) \\tan\\left(φ'\\right)}{{F_{\\text{S}}}}$$\n", + "$${\\symbf{T}}_{i}+\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{{\\symbf{R}}_{i}+\\left(\\left(-{\\symbf{X}}_{i-1}+{\\symbf{X}}_{i}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)\\right)\\,\\tan\\left(φ'\\right)}{{F_{\\text{S}}}}$$\n", "\n", "The interslice shear forces $X$ can be expressed in terms of the interslice normal forces $G$ using [A:Interslice-Norm-Shear-Forces-Linear](#assumpINSFL) and [GD:normShrR](#GD:normShrR), resulting in:\n", - "$${\\symbf{T}}_{i}+\\left(-λ {\\symbf{f}}_{i-1} {\\symbf{G}}_{i-1}+λ {\\symbf{f}}_{i} {\\symbf{G}}_{i}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{{\\symbf{R}}_{i}+\\left(\\left(-λ {\\symbf{f}}_{i-1} {\\symbf{G}}_{i-1}+λ {\\symbf{f}}_{i} {\\symbf{G}}_{i}\\right) \\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right) \\sin\\left({\\symbf{α}}_{i}\\right)\\right) \\tan\\left(φ'\\right)}{{F_{\\text{S}}}}$$\n", + "$${\\symbf{T}}_{i}+\\left(-λ\\,{\\symbf{f}}_{i-1}\\,{\\symbf{G}}_{i-1}+λ\\,{\\symbf{f}}_{i}\\,{\\symbf{G}}_{i}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)-\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)=\\frac{{\\symbf{R}}_{i}+\\left(\\left(-λ\\,{\\symbf{f}}_{i-1}\\,{\\symbf{G}}_{i-1}+λ\\,{\\symbf{f}}_{i}\\,{\\symbf{G}}_{i}\\right)\\,\\cos\\left({\\symbf{α}}_{i}\\right)+\\left(-{\\symbf{G}}_{i}+{\\symbf{G}}_{i-1}\\right)\\,\\sin\\left({\\symbf{α}}_{i}\\right)\\right)\\,\\tan\\left(φ'\\right)}{{F_{\\text{S}}}}$$\n", "\n", "Rearranging yields the following:\n", - "$${\\symbf{G}}_{i} \\left(\\left(λ {\\symbf{f}}_{i} \\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right) \\tan\\left(φ'\\right)-\\left(λ {\\symbf{f}}_{i} \\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right) {F_{\\text{S}}}\\right)={\\symbf{G}}_{i-1} \\left(\\left(λ {\\symbf{f}}_{i-1} \\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right) \\tan\\left(φ'\\right)-\\left(λ {\\symbf{f}}_{i-1} \\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right) {F_{\\text{S}}}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{i}-{\\symbf{R}}_{i}$$\n", + "$${\\symbf{G}}_{i}\\,\\left(\\left(λ\\,{\\symbf{f}}_{i}\\,\\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right)\\,\\tan\\left(φ'\\right)-\\left(λ\\,{\\symbf{f}}_{i}\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right)\\,{F_{\\text{S}}}\\right)={\\symbf{G}}_{i-1}\\,\\left(\\left(λ\\,{\\symbf{f}}_{i-1}\\,\\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right)\\,\\tan\\left(φ'\\right)-\\left(λ\\,{\\symbf{f}}_{i-1}\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right)\\,{F_{\\text{S}}}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{i}-{\\symbf{R}}_{i}$$\n", "\n", "The definitions for $Φ$ and $Ψ$ from [DD:convertFunc1](#DD:convertFunc1) and [DD:convertFunc2](#DD:convertFunc2) simplify the above to Equation (3):\n", - "$${\\symbf{G}}_{i} {\\symbf{Φ}}_{i}={\\symbf{Ψ}}_{i-1} {\\symbf{G}}_{i-1} {\\symbf{Φ}}_{i-1}+{F_{\\text{S}}} {\\symbf{T}}_{i}-{\\symbf{R}}_{i}$$\n", + "$${\\symbf{G}}_{i}\\,{\\symbf{Φ}}_{i}={\\symbf{Ψ}}_{i-1}\\,{\\symbf{G}}_{i-1}\\,{\\symbf{Φ}}_{i-1}+{F_{\\text{S}}}\\,{\\symbf{T}}_{i}-{\\symbf{R}}_{i}$$\n", "\n", "Versions of Equation (3) instantiated for slices 1 to $n$ are shown below:\n", - "$${\\symbf{G}}_{1} {\\symbf{Φ}}_{1}={\\symbf{Ψ}}_{0} {\\symbf{G}}_{0} {\\symbf{Φ}}_{0}+{F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}$$\n", - "$${\\symbf{G}}_{2} {\\symbf{Φ}}_{2}={\\symbf{Ψ}}_{1} {\\symbf{G}}_{1} {\\symbf{Φ}}_{1}+{F_{\\text{S}}} {\\symbf{T}}_{2}-{\\symbf{R}}_{2}$$\n", - "$${\\symbf{G}}_{3} {\\symbf{Φ}}_{3}={\\symbf{Ψ}}_{2} {\\symbf{G}}_{2} {\\symbf{Φ}}_{2}+{F_{\\text{S}}} {\\symbf{T}}_{3}-{\\symbf{R}}_{3}$$\n", + "$${\\symbf{G}}_{1}\\,{\\symbf{Φ}}_{1}={\\symbf{Ψ}}_{0}\\,{\\symbf{G}}_{0}\\,{\\symbf{Φ}}_{0}+{F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}$$\n", + "$${\\symbf{G}}_{2}\\,{\\symbf{Φ}}_{2}={\\symbf{Ψ}}_{1}\\,{\\symbf{G}}_{1}\\,{\\symbf{Φ}}_{1}+{F_{\\text{S}}}\\,{\\symbf{T}}_{2}-{\\symbf{R}}_{2}$$\n", + "$${\\symbf{G}}_{3}\\,{\\symbf{Φ}}_{3}={\\symbf{Ψ}}_{2}\\,{\\symbf{G}}_{2}\\,{\\symbf{Φ}}_{2}+{F_{\\text{S}}}\\,{\\symbf{T}}_{3}-{\\symbf{R}}_{3}$$\n", "\n", "...\n", - "$${\\symbf{G}}_{n-2} {\\symbf{Φ}}_{n-2}={\\symbf{Ψ}}_{n-3} {\\symbf{G}}_{n-3} {\\symbf{Φ}}_{n-3}+{F_{\\text{S}}} {\\symbf{T}}_{n-2}-{\\symbf{R}}_{n-2}$$\n", - "$${\\symbf{G}}_{n-1} {\\symbf{Φ}}_{n-1}={\\symbf{Ψ}}_{n-2} {\\symbf{G}}_{n-2} {\\symbf{Φ}}_{n-2}+{F_{\\text{S}}} {\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}$$\n", - "$${\\symbf{G}}_{n} {\\symbf{Φ}}_{n}={\\symbf{Ψ}}_{n-1} {\\symbf{G}}_{n-1} {\\symbf{Φ}}_{n-1}+{F_{\\text{S}}} {\\symbf{T}}_{n}-{\\symbf{R}}_{n}$$\n", + "$${\\symbf{G}}_{n-2}\\,{\\symbf{Φ}}_{n-2}={\\symbf{Ψ}}_{n-3}\\,{\\symbf{G}}_{n-3}\\,{\\symbf{Φ}}_{n-3}+{F_{\\text{S}}}\\,{\\symbf{T}}_{n-2}-{\\symbf{R}}_{n-2}$$\n", + "$${\\symbf{G}}_{n-1}\\,{\\symbf{Φ}}_{n-1}={\\symbf{Ψ}}_{n-2}\\,{\\symbf{G}}_{n-2}\\,{\\symbf{Φ}}_{n-2}+{F_{\\text{S}}}\\,{\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}$$\n", + "$${\\symbf{G}}_{n}\\,{\\symbf{Φ}}_{n}={\\symbf{Ψ}}_{n-1}\\,{\\symbf{G}}_{n-1}\\,{\\symbf{Φ}}_{n-1}+{F_{\\text{S}}}\\,{\\symbf{T}}_{n}-{\\symbf{R}}_{n}$$\n", "\n", "Applying [A:Edge-Slices](#assumpES), which says that $G_0$ and $G_n$ are zero, results in the following special cases: Equation (8) for the first slice:\n", - "$${\\symbf{G}}_{1} {\\symbf{Φ}}_{1}={F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}$$\n", + "$${\\symbf{G}}_{1}\\,{\\symbf{Φ}}_{1}={F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}$$\n", "\n", "and Equation (9) for the $n$th slice:\n", - "$$-\\left(\\frac{{F_{\\text{S}}} {\\symbf{T}}_{n}-{\\symbf{R}}_{n}}{{\\symbf{Ψ}}_{n-1}}\\right)={\\symbf{G}}_{n-1} {\\symbf{Φ}}_{n-1}$$\n", + "$$-\\left(\\frac{{F_{\\text{S}}}\\,{\\symbf{T}}_{n}-{\\symbf{R}}_{n}}{{\\symbf{Ψ}}_{n-1}}\\right)={\\symbf{G}}_{n-1}\\,{\\symbf{Φ}}_{n-1}$$\n", "\n", "Substituting Equation (8) into Equation (4) yields Equation (10):\n", - "$${\\symbf{G}}_{2} {\\symbf{Φ}}_{2}={\\symbf{Ψ}}_{1} \\left({F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{2}-{\\symbf{R}}_{2}$$\n", + "$${\\symbf{G}}_{2}\\,{\\symbf{Φ}}_{2}={\\symbf{Ψ}}_{1}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{2}-{\\symbf{R}}_{2}$$\n", "\n", "which can be substituted into Equation (5) to get Equation (11):\n", - "$${\\symbf{G}}_{3} {\\symbf{Φ}}_{3}={\\symbf{Ψ}}_{2} \\left({\\symbf{Ψ}}_{1} \\left({F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{3}-{\\symbf{R}}_{3}$$\n", + "$${\\symbf{G}}_{3}\\,{\\symbf{Φ}}_{3}={\\symbf{Ψ}}_{2}\\,\\left({\\symbf{Ψ}}_{1}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{3}-{\\symbf{R}}_{3}$$\n", "\n", "and so on until Equation (12) is obtained from Equation (7):\n", - "$${\\symbf{G}}_{n-1} {\\symbf{Φ}}_{n-1}={\\symbf{Ψ}}_{n-2} \\left({\\symbf{Ψ}}_{n-3} \\left({\\symbf{Ψ}}_{1} \\left({F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{n-2}-{\\symbf{R}}_{n-2}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}$$\n", + "$${\\symbf{G}}_{n-1}\\,{\\symbf{Φ}}_{n-1}={\\symbf{Ψ}}_{n-2}\\,\\left({\\symbf{Ψ}}_{n-3}\\,\\left({\\symbf{Ψ}}_{1}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{n-2}-{\\symbf{R}}_{n-2}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}$$\n", "\n", "Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in:\n", - "$$-\\left(\\frac{{F_{\\text{S}}} {\\symbf{T}}_{n}-{\\symbf{R}}_{n}}{{\\symbf{Ψ}}_{n-1}}\\right)={\\symbf{Ψ}}_{n-2} \\left({\\symbf{Ψ}}_{n-3} \\left({\\symbf{Ψ}}_{1} \\left({F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{n-2}-{\\symbf{R}}_{n-2}\\right)+{F_{\\text{S}}} {\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}$$\n", + "$$-\\left(\\frac{{F_{\\text{S}}}\\,{\\symbf{T}}_{n}-{\\symbf{R}}_{n}}{{\\symbf{Ψ}}_{n-1}}\\right)={\\symbf{Ψ}}_{n-2}\\,\\left({\\symbf{Ψ}}_{n-3}\\,\\left({\\symbf{Ψ}}_{1}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{n-2}-{\\symbf{R}}_{n-2}\\right)+{F_{\\text{S}}}\\,{\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}$$\n", "\n", "This can be rearranged by multiplying both sides by $Ψ_n - 1$ and then distributing the multiplication of each $Ψ$ over addition to obtain:\n", - "$$-\\left({F_{\\text{S}}} {\\symbf{T}}_{n}-{\\symbf{R}}_{n}\\right)={\\symbf{Ψ}}_{n-1} {\\symbf{Ψ}}_{n-2} {\\symbf{Ψ}}_{1} \\left({F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{\\symbf{Ψ}}_{n-1} {\\symbf{Ψ}}_{n-2} {\\symbf{Ψ}}_{2} \\left({F_{\\text{S}}} {\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{\\symbf{Ψ}}_{n-1} \\left({F_{\\text{S}}} {\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}\\right)$$\n", + "$$-\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{n}-{\\symbf{R}}_{n}\\right)={\\symbf{Ψ}}_{n-1}\\,{\\symbf{Ψ}}_{n-2}\\,{\\symbf{Ψ}}_{1}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}\\right)+{\\symbf{Ψ}}_{n-1}\\,{\\symbf{Ψ}}_{n-2}\\,{\\symbf{Ψ}}_{2}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{2}-{\\symbf{R}}_{2}\\right)+{\\symbf{Ψ}}_{n-1}\\,\\left({F_{\\text{S}}}\\,{\\symbf{T}}_{n-1}-{\\symbf{R}}_{n-1}\\right)$$\n", "\n", "The multiplication of the $Ψ$ terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an $R$ or a $T$. The equation can then be rearranged so terms containing an $R$ are on one side of the equality, and terms containing a $T$ are on the other. The multiplication by the factor of safety is common to all of the $T$ terms, and thus can be factored out, resulting in:\n", - "$${F_{\\text{S}}} \\left({\\symbf{Ψ}}_{n-1} {\\symbf{Ψ}}_{n-2} {\\symbf{Ψ}}_{1} {\\symbf{T}}_{1}+{\\symbf{Ψ}}_{n-1} {\\symbf{Ψ}}_{n-2} {\\symbf{Ψ}}_{2} {\\symbf{T}}_{2}+{\\symbf{Ψ}}_{n-1} {\\symbf{T}}_{n-1}+{\\symbf{T}}_{n}\\right)={\\symbf{Ψ}}_{n-1} {\\symbf{Ψ}}_{n-2} {\\symbf{Ψ}}_{1} {\\symbf{R}}_{1}+{\\symbf{Ψ}}_{n-1} {\\symbf{Ψ}}_{n-2} {\\symbf{Ψ}}_{2} {\\symbf{R}}_{2}+{\\symbf{Ψ}}_{n-1} {\\symbf{R}}_{n-1}+{\\symbf{R}}_{n}$$\n", + "$${F_{\\text{S}}}\\,\\left({\\symbf{Ψ}}_{n-1}\\,{\\symbf{Ψ}}_{n-2}\\,{\\symbf{Ψ}}_{1}\\,{\\symbf{T}}_{1}+{\\symbf{Ψ}}_{n-1}\\,{\\symbf{Ψ}}_{n-2}\\,{\\symbf{Ψ}}_{2}\\,{\\symbf{T}}_{2}+{\\symbf{Ψ}}_{n-1}\\,{\\symbf{T}}_{n-1}+{\\symbf{T}}_{n}\\right)={\\symbf{Ψ}}_{n-1}\\,{\\symbf{Ψ}}_{n-2}\\,{\\symbf{Ψ}}_{1}\\,{\\symbf{R}}_{1}+{\\symbf{Ψ}}_{n-1}\\,{\\symbf{Ψ}}_{n-2}\\,{\\symbf{Ψ}}_{2}\\,{\\symbf{R}}_{2}+{\\symbf{Ψ}}_{n-1}\\,{\\symbf{R}}_{n-1}+{\\symbf{R}}_{n}$$\n", "\n", "Isolating the factor of safety on the left-hand side and using compact notation for the products and sums yields Equation (13), which can also be seen in [IM:fctSfty](#IM:fctSfty):\n", - "$${F_{\\text{S}}}=\\frac{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{R}}_{i} \\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{R}}_{n}}{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{T}}_{i} \\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{T}}_{n}}$$\n", + "$${F_{\\text{S}}}=\\frac{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{R}}_{i}\\,\\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{R}}_{n}}{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{T}}_{i}\\,\\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{T}}_{n}}$$\n", "\n", "$F_S$ depends on the unknowns $λ$ ([IM:nrmShrFor](#IM:nrmShrFor)) and $G$ ([IM:intsliceFs](#IM:intsliceFs)).\n", "
\n", @@ -3074,16 +3074,16 @@ "\n", "\n", "From the moment equilibrium of [GD:momentEql](#GD:momentEql) with the primary assumption for the Morgenstern-Price method of [A:Interslice-Norm-Shear-Forces-Linear](#assumpINSFL) and associated definition [GD:normShrR](#GD:normShrR), Equation (14) can be derived:\n", - "$$0=-{\\symbf{G}}_{i} \\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1} \\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+λ \\frac{{\\symbf{b}}_{i}}{2} \\left({\\symbf{G}}_{i} {\\symbf{f}}_{i}+{\\symbf{G}}_{i-1} {\\symbf{f}}_{i-1}\\right)+\\frac{-{K_{\\text{c}}} {\\symbf{W}}_{i} {\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right) {\\symbf{h}}_{i}+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right) {\\symbf{h}}_{i}$$\n", + "$$0=-{\\symbf{G}}_{i}\\,\\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1}\\,\\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+λ\\,\\frac{{\\symbf{b}}_{i}}{2}\\,\\left({\\symbf{G}}_{i}\\,{\\symbf{f}}_{i}+{\\symbf{G}}_{i-1}\\,{\\symbf{f}}_{i-1}\\right)+\\frac{-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}\\,{\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\,{\\symbf{h}}_{i}+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\,{\\symbf{h}}_{i}$$\n", "\n", "Rearranging the equation in terms of $λ$ leads to Equation (15):\n", - "$$λ=\\frac{-{\\symbf{G}}_{i} \\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1} \\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+\\frac{-{K_{\\text{c}}} {\\symbf{W}}_{i} {\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right) {\\symbf{h}}_{i}+{\\symbf{Q}}_{i} \\sin\\left({\\symbf{ω}}_{i}\\right) {\\symbf{h}}_{i}}{-\\frac{{\\symbf{b}}_{i}}{2} \\left({\\symbf{G}}_{i} {\\symbf{f}}_{i}+{\\symbf{G}}_{i-1} {\\symbf{f}}_{i-1}\\right)}$$\n", + "$$λ=\\frac{-{\\symbf{G}}_{i}\\,\\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1}\\,\\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+\\frac{-{K_{\\text{c}}}\\,{\\symbf{W}}_{i}\\,{\\symbf{h}}_{i}}{2}+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\,{\\symbf{h}}_{i}+{\\symbf{Q}}_{i}\\,\\sin\\left({\\symbf{ω}}_{i}\\right)\\,{\\symbf{h}}_{i}}{-\\frac{{\\symbf{b}}_{i}}{2}\\,\\left({\\symbf{G}}_{i}\\,{\\symbf{f}}_{i}+{\\symbf{G}}_{i-1}\\,{\\symbf{f}}_{i-1}\\right)}$$\n", "\n", "This equation can be simplified by applying assumptions [A:Seismic-Force](#assumpSF) and [A:Surface-Load](#assumpSL), which state that the seismic and external forces, respectively, are zero:\n", - "$$λ=\\frac{-{\\symbf{G}}_{i} \\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1} \\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1} \\left(\\frac{1}{3} {\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2} \\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right) {\\symbf{h}}_{i}}{-\\frac{{\\symbf{b}}_{i}}{2} \\left({\\symbf{G}}_{i} {\\symbf{f}}_{i}+{\\symbf{G}}_{i-1} {\\symbf{f}}_{i-1}\\right)}$$\n", + "$$λ=\\frac{-{\\symbf{G}}_{i}\\,\\left({\\symbf{h}_{\\text{z},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{G}}_{i-1}\\,\\left({\\symbf{h}_{\\text{z},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)-{\\symbf{H}}_{i}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i}}+\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{H}}_{i-1}\\,\\left(\\frac{1}{3}\\,{\\symbf{h}_{\\text{z,w},i-1}}-\\frac{{\\symbf{b}}_{i}}{2}\\,\\tan\\left({\\symbf{α}}_{i}\\right)\\right)+{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)\\,{\\symbf{h}}_{i}}{-\\frac{{\\symbf{b}}_{i}}{2}\\,\\left({\\symbf{G}}_{i}\\,{\\symbf{f}}_{i}+{\\symbf{G}}_{i-1}\\,{\\symbf{f}}_{i-1}\\right)}$$\n", "\n", "Taking the summation of all slices, and applying [A:Edge-Slices](#assumpES) to set $G_0$, $G_n$, $H_0$, and $H_n$ equal to zero, a general equation for the proportionality constant $λ$ is developed in Equation (16), which combines [IM:nrmShrFor](#IM:nrmShrFor), [IM:nrmShrForNum](#IM:nrmShrForNum), and [IM:nrmShrForDen](#IM:nrmShrForDen):\n", - "$$λ=\\frac{\\displaystyle\\sum_{i=1}^{n}{{\\symbf{b}}_{i} \\left({{\\symbf{F}_{\\text{x}}}^{\\text{G}}}+{{\\symbf{F}_{\\text{x}}}^{\\text{H}}}\\right) \\tan\\left({\\symbf{α}}_{i}\\right)+{\\symbf{h}}_{i} -2 {\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right)}}{\\displaystyle\\sum_{i=1}^{n}{{\\symbf{b}}_{i} \\left({\\symbf{G}}_{i} {\\symbf{f}}_{i}+{\\symbf{G}}_{i-1} {\\symbf{f}}_{i-1}\\right)}}$$\n", + "$$λ=\\frac{\\displaystyle\\sum_{i=1}^{n}{{\\symbf{b}}_{i}\\,\\left({{\\symbf{F}_{\\text{x}}}^{\\text{G}}}+{{\\symbf{F}_{\\text{x}}}^{\\text{H}}}\\right)\\,\\tan\\left({\\symbf{α}}_{i}\\right)+{\\symbf{h}}_{i}\\,-2\\,{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right)}}{\\displaystyle\\sum_{i=1}^{n}{{\\symbf{b}}_{i}\\,\\left({\\symbf{G}}_{i}\\,{\\symbf{f}}_{i}+{\\symbf{G}}_{i-1}\\,{\\symbf{f}}_{i-1}\\right)}}$$\n", "\n", "Equation (16) for $λ$ is a function of the unknown interslice normal forces $G$ ([IM:intsliceFs](#IM:intsliceFs)) which itself depends on the unknown factor of safety $F_S$ ([IM:fctSfty](#IM:fctSfty)).\n", "
\n", @@ -3129,7 +3129,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -3220,7 +3220,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -3305,7 +3305,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -3349,10 +3349,10 @@ "\n", "\n", "This derivation is identical to the derivation for [IM:fctSfty](#IM:fctSfty) up until Equation (3) shown again below:\n", - "$${\\symbf{G}}_{i} {\\symbf{Φ}}_{i}={\\symbf{Ψ}}_{i-1} {\\symbf{G}}_{i-1} {\\symbf{Φ}}_{i-1}+{F_{\\text{S}}} {\\symbf{T}}_{i}-{\\symbf{R}}_{i}$$\n", + "$${\\symbf{G}}_{i}\\,{\\symbf{Φ}}_{i}={\\symbf{Ψ}}_{i-1}\\,{\\symbf{G}}_{i-1}\\,{\\symbf{Φ}}_{i-1}+{F_{\\text{S}}}\\,{\\symbf{T}}_{i}-{\\symbf{R}}_{i}$$\n", "\n", "A simple rearrangement of Equation (3) leads to Equation (17), also seen in [IM:intsliceFs](#IM:intsliceFs):\n", - "$${\\symbf{G}}_{i}=\\frac{{\\symbf{Ψ}}_{i-1} {\\symbf{G}}_{i-1}+{F_{\\text{S}}} {\\symbf{T}}_{i}-{\\symbf{R}}_{i}}{{\\symbf{Φ}}_{i}}$$\n", + "$${\\symbf{G}}_{i}=\\frac{{\\symbf{Ψ}}_{i-1}\\,{\\symbf{G}}_{i-1}+{F_{\\text{S}}}\\,{\\symbf{T}}_{i}-{\\symbf{R}}_{i}}{{\\symbf{Φ}}_{i}}$$\n", "\n", "The cases shown in [IM:intsliceFs](#IM:intsliceFs) for when $i = 0$, $i = 1$, or $i = n$ are derived by applying [A:Edge-Slices](#assumpES), which says that $G_0$ and $G_n$ are zero, to Equation (17). $G$ depends on the unknowns $F_S$ ([IM:fctSfty](#IM:fctSfty)) and $λ$ ([IM:nrmShrFor](#IM:nrmShrFor)).\n", "
\n", diff --git a/code/stable/ssp/SRS/PDF/SSP_SRS.tex b/code/stable/ssp/SRS/PDF/SSP_SRS.tex index a3b6a94cf7..a609d72610 100644 --- a/code/stable/ssp/SRS/PDF/SSP_SRS.tex +++ b/code/stable/ssp/SRS/PDF/SSP_SRS.tex @@ -543,7 +543,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - {τ^{\text{f}}}={σ_{N}}' \tan\left(φ'\right)+c' + {τ^{\text{f}}}={σ_{N}}'\,\tan\left(φ'\right)+c' \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -610,7 +610,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - \symbf{F}=m \symbf{a}\text{(}t\text{)} + \symbf{F}=m\,\symbf{a}\text{(}t\text{)} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -650,7 +650,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{N}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right) + {\symbf{N}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -699,7 +699,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{S}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right) + {\symbf{S}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -748,7 +748,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{P}}_{i}={\symbf{N'}}_{i} \tan\left({φ'}_{i}\right)+{c'}_{i} {\symbf{L}_{b,i}} + {\symbf{P}}_{i}={\symbf{N'}}_{i}\,\tan\left({φ'}_{i}\right)+{c'}_{i}\,{\symbf{L}_{b,i}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -791,7 +791,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{S}}_{i}=\frac{{\symbf{P}}_{i}}{{F_{\text{S}}}}=\frac{{\symbf{N'}}_{i} \tan\left({φ'}_{i}\right)+{c'}_{i} {\symbf{L}_{b,i}}}{{F_{\text{S}}}} + {\symbf{S}}_{i}=\frac{{\symbf{P}}_{i}}{{F_{\text{S}}}}=\frac{{\symbf{N'}}_{i}\,\tan\left({φ'}_{i}\right)+{c'}_{i}\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -876,7 +876,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{R}}_{i}=\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left({φ'}_{i}\right)+{c'}_{i} {\symbf{L}_{b,i}} + {\symbf{R}}_{i}=\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left({φ'}_{i}\right)+{c'}_{i}\,{\symbf{L}_{b,i}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -920,7 +920,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{T}}_{i}=\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right) + {\symbf{T}}_{i}=\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -960,7 +960,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{X}=λ \symbf{f} \symbf{G} + \symbf{X}=λ\,\symbf{f}\,\symbf{G} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -997,7 +997,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - 0=-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2} \left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i} + 0=-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1040,67 +1040,67 @@ \subsubsection{General Definitions} Considering one dimension, with moments in the clockwise direction as positive and moments in the counterclockwise direction as negative, and replacing the torque symbol with the moment symbol, the equation simplifies to: \begin{displaymath} -M={F_{\text{rot}}} r +M={F_{\text{rot}}}\,r \end{displaymath} where ${F_{\text{rot}}}$ is the force causing rotation and $r$ is the length of the moment arm, or the distance between the force and the axis about which the rotation acts. To represent the moment equilibrium, the moments from each force acting on a slice must be considered and added together. The forces acting on a slice are all shown in \hyperref[Figure:ForceDiagram]{Fig:ForceDiagram}. The midpoint of the base of a slice is considered as the axis of rotation, from which the length of the moment arm is measured. Considering first the interslice normal force acting on slice interface $i$, the moment is negative because the force tends to rotate the slice in a counterclockwise direction, and the length of the moment arm is the height of the force plus the difference in height between the base at slice interface $i$ and the base at the midpoint of slice $i$. Thus, the moment is expressed as: \begin{displaymath} --{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right) +-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right) \end{displaymath} For the $i-1$th slice interface, the moment is similar but in the opposite direction: \begin{displaymath} -{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right) +{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right) \end{displaymath} Next, the interslice normal water force is considered. This force is zero at the height of the water table, then increases linearly towards the base of the slice due to the increasing water pressure. For such a triangular distribution, the resultant force acts at one-third of the height. Thus, for the interslice normal water force acting on slice interface $i$, the moment is: \begin{displaymath} --{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right) +-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right) \end{displaymath} The moment for the interslice normal water force acting on slice interface $i-1$ is: \begin{displaymath} -{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right) +{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right) \end{displaymath} The interslice shear force at slice interface $i$ tends to rotate in the clockwise direction, and the length of the moment arm is the length from the slice edge to the slice midpoint, equivalent to half of the width of the slice, so the moment is: \begin{displaymath} -{\symbf{X}}_{i} \frac{{\symbf{b}}_{i}}{2} +{\symbf{X}}_{i}\,\frac{{\symbf{b}}_{i}}{2} \end{displaymath} The interslice shear force at slice interface $i-1$ also tends to rotate in the clockwise direction, and has the same length of the moment arm, so the moment is: \begin{displaymath} -{\symbf{X}}_{i-1} \frac{{\symbf{b}}_{i}}{2} +{\symbf{X}}_{i-1}\,\frac{{\symbf{b}}_{i}}{2} \end{displaymath} -Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is ${K_{\text{c}}} {\symbf{W}}_{i}$ where ${\symbf{W}}_{i}$ can be expressed as $γ {\symbf{b}}_{i} y$ using \hyperref[GD:weight]{GD:weight} where $y$ is the height of the segment under consideration. The corresponding length of the moment arm is $y$, the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible (\hyperref[assumpNESSS]{A:Negligible-Effect-Surface-Slope-Seismic}). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative: +Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is ${K_{\text{c}}}\,{\symbf{W}}_{i}$ where ${\symbf{W}}_{i}$ can be expressed as $γ\,{\symbf{b}}_{i}\,y$ using \hyperref[GD:weight]{GD:weight} where $y$ is the height of the segment under consideration. The corresponding length of the moment arm is $y$, the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible (\hyperref[assumpNESSS]{A:Negligible-Effect-Surface-Slope-Seismic}). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative: \begin{displaymath} --\int_{0}^{{\symbf{h}}_{i}}{{K_{\text{c}}} γ {\symbf{b}}_{i} y}\,dy +-\int_{0}^{{\symbf{h}}_{i}}{{K_{\text{c}}}\,γ\,{\symbf{b}}_{i}\,y}\,dy \end{displaymath} Solving the definite integral yields: \begin{displaymath} --{K_{\text{c}}} γ {\symbf{b}}_{i} \frac{{\symbf{h}}_{i}^{2}}{2} +-{K_{\text{c}}}\,γ\,{\symbf{b}}_{i}\,\frac{{\symbf{h}}_{i}^{2}}{2} \end{displaymath} -Using \hyperref[GD:weight]{GD:weight} again to express $γ {\symbf{b}}_{i} {\symbf{h}}_{i}$ as ${\symbf{W}}_{i}$, the moment is: +Using \hyperref[GD:weight]{GD:weight} again to express $γ\,{\symbf{b}}_{i}\,{\symbf{h}}_{i}$ as ${\symbf{W}}_{i}$, the moment is: \begin{displaymath} --{K_{\text{c}}} {\symbf{W}}_{i} \frac{{\symbf{h}}_{i}}{2} +-{K_{\text{c}}}\,{\symbf{W}}_{i}\,\frac{{\symbf{h}}_{i}}{2} \end{displaymath} The surface hydrostatic force acts into the midpoint of the surface of the slice (\hyperref[assumpHFSM]{A:Hydrostatic-Force-Slice-Midpoint}). Thus, the vertical component of the force acts directly towards the point of rotation, and has a moment of zero. The horizontal component of the force tends to rotate in a clockwise direction and the length of the moment arm is the entire height of the slice. Thus, the moment is: \begin{displaymath} -{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i} +{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i} \end{displaymath} The external force again acts into the midpoint of the slice surface, so the vertical component does not contribute to the moment, and the length of the moment arm is again the entire height of the slice. The moment is: \begin{displaymath} -{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i} +{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i} \end{displaymath} The base hydrostatic force and slice weight both act in the direction of the point of rotation (\hyperref[assumpHFSM]{A:Hydrostatic-Force-Slice-Midpoint}), therefore both have moments of zero. Thus, all of the moments have been determined. The moment equilibrium is then represented by the sum of all moments: \begin{displaymath} -0=-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2} \left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i} +0=-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{X}}_{i}+{\symbf{X}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i} \end{displaymath} \medskip \noindent @@ -1117,7 +1117,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - W=V γ + W=V\,γ \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1142,23 +1142,23 @@ \subsubsection{General Definitions} \begin{displaymath} \symbf{a}\text{(}t\text{)}=\begin{bmatrix} 0\\ - \symbf{g} \symbf{\hat{j}} + \symbf{g}\,\symbf{\hat{j}} \end{bmatrix} \end{displaymath} Since there is only one non-zero vector component, the scalar value $W$ will be used for the weight. In this scenario, Newton's second law of motion from \hyperref[TM:NewtonSecLawMot]{TM:NewtonSecLawMot} can be expressed as: \begin{displaymath} -W=m \symbf{g} +W=m\,\symbf{g} \end{displaymath} Mass can be expressed as density multiplied by volume, resulting in: \begin{displaymath} -W=ρ V \symbf{g} +W=ρ\,V\,\symbf{g} \end{displaymath} Substituting specific weight as the product of density and gravitational acceleration yields: \begin{displaymath} -W=V γ +W=V\,γ \end{displaymath} \medskip \noindent @@ -1175,11 +1175,11 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \begin{cases} - \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ - \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right) {γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}, & {\symbf{y}_{\text{slope},i}}\geq{}{\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{slope},i-1}}\geq{}{\symbf{y}_{\text{wt},i-1}}\geq{}{\symbf{y}_{\text{slip},i-1}}\\ - \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{dry}}}, & {\symbf{y}_{\text{wt},i}}\lt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\lt{}{\symbf{y}_{\text{slip},i-1}} - \end{cases} + {\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\begin{cases} + \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ + \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right)\,{γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}, & {\symbf{y}_{\text{slope},i}}\geq{}{\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{slope},i-1}}\geq{}{\symbf{y}_{\text{wt},i-1}}\geq{}{\symbf{y}_{\text{slip},i-1}}\\ + \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{dry}}}, & {\symbf{y}_{\text{wt},i}}\lt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\lt{}{\symbf{y}_{\text{slip},i-1}} + \end{cases} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1210,32 +1210,32 @@ \subsubsection{General Definitions} For the case where the water table is above the slope surface, the weights come from the weight of the saturated soil. Substituting values for saturated soil into the equation for weight from \hyperref[GD:weight]{GD:weight} yields: \begin{displaymath} -{\symbf{W}}_{i}={\symbf{V}_{\text{sat},i}} {γ_{\text{sat}}} +{\symbf{W}}_{i}={\symbf{V}_{\text{sat},i}}\,{γ_{\text{sat}}} \end{displaymath} Due to \hyperref[assumpPSC]{A:Plane-Strain-Conditions}, only two dimensions are considered, so the areas of saturated soil are considered instead of the volumes of saturated soil. Any given slice has a trapezoidal shape. The area of a trapezoid is the average of the lengths of the parallel sides multiplied by the length between the parallel sides. The parallel sides in this case are the interslice edges and the length between them is the width of the slice. Thus, the weights are defined as: \begin{displaymath} -{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}} +{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}} \end{displaymath} For the case where the water table is below the slip surface, the weights come from the weight of the dry soil. Substituting values for dry soil into the equation for weight from \hyperref[GD:weight]{GD:weight} yields: \begin{displaymath} -{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}} {γ_{\text{dry}}} +{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}}\,{γ_{\text{dry}}} \end{displaymath} \hyperref[assumpPSC]{A:Plane-Strain-Conditions} again allows for two-dimensional analysis so the areas of dry soil are considered instead of the volumes of dry soil. The trapezoidal slice shape is the same as in the previous case, so the weights are defined as: \begin{displaymath} -{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{dry}}} +{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{dry}}} \end{displaymath} For the case where the water table is between the slope surface and slip surface, the weights are the sums of the weights of the dry portions and weights of the saturated portions of the soil. Substituting values for dry and saturated soil into the equation for weight from \hyperref[GD:weight]{GD:weight} and adding them together yields: \begin{displaymath} -{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}} {γ_{\text{dry}}}+{\symbf{V}_{\text{sat},i}} {γ_{\text{sat}}} +{\symbf{W}}_{i}={\symbf{V}_{\text{dry},i}}\,{γ_{\text{dry}}}+{\symbf{V}_{\text{sat},i}}\,{γ_{\text{sat}}} \end{displaymath} \hyperref[assumpPSC]{A:Plane-Strain-Conditions} again allows for two-dimensional analysis so the areas of dry soil and areas of saturated soil are considered instead of the volumes of dry soil and volumes of saturated soil. The water table is assumed to only intersect a slice surface or base at a slice edge (\hyperref[assumpWISE]{A:Water-Intersects-Surface-Edge}, \hyperref[assumpWIBE]{A:Water-Intersects-Base-Edge}), so the dry and saturated portions each have trapezoidal shape. For the dry portion, the parallel sides of the trapezoid are the lengths between the slope surface and water table at the slice edges. For the saturated portion, the parallel sides of the trapezoid are the lengths between the water table and slip surface at the slice edges. Thus, the weights are defined as: \begin{displaymath} -{\symbf{W}}_{i}={\symbf{b}}_{i} \frac{1}{2} \left(\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right) {γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) {γ_{\text{sat}}}\right) +{\symbf{W}}_{i}={\symbf{b}}_{i}\,\frac{1}{2}\,\left(\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{wt},i}}+{\symbf{y}_{\text{slope},i-1}}-{\symbf{y}_{\text{wt},i-1}}\right)\,{γ_{\text{dry}}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right)\,{γ_{\text{sat}}}\right) \end{displaymath} \medskip \noindent @@ -1252,7 +1252,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - p=γ h + p=γ\,h \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1288,10 +1288,10 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}} {γ_{w}} \frac{1}{2} \begin{cases} - {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slip},i-1}}\\ - 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slip},i-1}} - \end{cases} + {\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}}\,{γ_{w}}\,\frac{1}{2}\,\begin{cases} + {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slip},i-1}}\\ + 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slip},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slip},i-1}} + \end{cases} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1320,17 +1320,17 @@ \subsubsection{General Definitions} The base hydrostatic forces come from the hydrostatic pressure exerted by the water above the base of each slice. The equation for hydrostatic pressure from \hyperref[GD:hsPressure]{GD:hsPressure} is: \begin{displaymath} -p=γ h +p=γ\,h \end{displaymath} The specific weight in this case is the unit weight of water ${γ_{w}}$. The height in this case is the height from the slice base to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint (\hyperref[assumpHFSM]{A:Hydrostatic-Force-Slice-Midpoint}). The height at the midpoint is the average of the height at slice interface $i$ and the height at slice interface $i-1$: \begin{displaymath} -\frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) +\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) \end{displaymath} Due to \hyperref[assumpPSC]{A:Plane-Strain-Conditions}, only two dimensions are considered, so the base hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to base hydrostatic forces by multiplying by the corresponding length of the slice base ${\symbf{L}_{b,i}}$, assuming the water table does not intersect a slice base except at a slice edge (\hyperref[assumpWIBE]{A:Water-Intersects-Base-Edge}). Thus, in the case where the height of the water table is above the height of the slip surface, the base hydrostatic forces are defined as: \begin{displaymath} -{\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}} {γ_{w}} \frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) +{\symbf{U}_{\text{b},i}}={\symbf{L}_{b,i}}\,{γ_{w}}\,\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slip},i-1}}\right) \end{displaymath} This equation is the non-zero case of \hyperref[GD:baseWtrF]{GD:baseWtrF}. The zero case is when the height of the water table is below the height of the slip surface, so there is no hydrostatic force. @@ -1349,10 +1349,10 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}} {γ_{w}} \frac{1}{2} \begin{cases} - {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ - 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slope},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slope},i-1}} - \end{cases} + {\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}}\,{γ_{w}}\,\frac{1}{2}\,\begin{cases} + {\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}, & {\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slope},i}}\lor{}{\symbf{y}_{\text{wt},i-1}}\gt{}{\symbf{y}_{\text{slope},i-1}}\\ + 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slope},i}}\land{}{\symbf{y}_{\text{wt},i-1}}\leq{}{\symbf{y}_{\text{slope},i-1}} + \end{cases} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1381,17 +1381,17 @@ \subsubsection{General Definitions} The surface hydrostatic forces come from the hydrostatic pressure exerted by the water above the surface of each slice. The equation for hydrostatic pressure from \hyperref[GD:hsPressure]{GD:hsPressure} is: \begin{displaymath} -p=γ h +p=γ\,h \end{displaymath} The specific weight in this case is the unit weight of water ${γ_{w}}$. The height in this case is the height from the slice surface to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint (\hyperref[assumpHFSM]{A:Hydrostatic-Force-Slice-Midpoint}). The height at the midpoint is the average of the height at slice interface $i$ and the height at slice interface $i-1$: \begin{displaymath} -\frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right) +\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right) \end{displaymath} Due to \hyperref[assumpPSC]{A:Plane-Strain-Conditions}, only two dimensions are considered, so the surface hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to surface hydrostatic forces by multiplying by the corresponding length of the slice surface ${\symbf{L}_{s,i}}$, assuming the water table does not intersect a slice surface except at a slice edge (\hyperref[assumpWISE]{A:Water-Intersects-Surface-Edge}). Thus, in the case where the height of the water table is above the height of the slope surface, the surface hydrostatic forces are defined as: \begin{displaymath} -{\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}} {γ_{w}} \frac{1}{2} \left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right) +{\symbf{U}_{\text{g},i}}={\symbf{L}_{s,i}}\,{γ_{w}}\,\frac{1}{2}\,\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}+{\symbf{y}_{\text{wt},i-1}}-{\symbf{y}_{\text{slope},i-1}}\right) \end{displaymath} This equation is the non-zero case of \hyperref[GD:srfWtrF]{GD:srfWtrF}. The zero case is when the height of the water table is below the height of the slope surface, so there is no hydrostatic force. @@ -1418,8 +1418,8 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} \symbf{H}=\begin{cases} - \frac{\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2} {γ_{w}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}\right)^{2} {γ_{w}}, & {\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slope},i}}\\ - \frac{\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2} {γ_{w}}, & {\symbf{y}_{\text{slope},i}}\gt{}{\symbf{y}_{\text{wt},i}}\land{}{\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\\ + \frac{\left({\symbf{y}_{\text{slope},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2}\,{γ_{w}}+\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slope},i}}\right)^{2}\,{γ_{w}}, & {\symbf{y}_{\text{wt},i}}\geq{}{\symbf{y}_{\text{slope},i}}\\ + \frac{\left({\symbf{y}_{\text{wt},i}}-{\symbf{y}_{\text{slip},i}}\right)^{2}}{2}\,{γ_{w}}, & {\symbf{y}_{\text{slope},i}}\gt{}{\symbf{y}_{\text{wt},i}}\land{}{\symbf{y}_{\text{wt},i}}\gt{}{\symbf{y}_{\text{slip},i}}\\ 0, & {\symbf{y}_{\text{wt},i}}\leq{}{\symbf{y}_{\text{slip},i}} \end{cases} \end{displaymath} @@ -1576,7 +1576,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{L}_{b}}={\symbf{b}}_{i} \sec\left({\symbf{α}}_{i}\right) + {\symbf{L}_{b}}={\symbf{b}}_{i}\,\sec\left({\symbf{α}}_{i}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1616,7 +1616,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {\symbf{L}_{s}}={\symbf{b}}_{i} \sec\left({\symbf{β}}_{i}\right) + {\symbf{L}_{s}}={\symbf{b}}_{i}\,\sec\left({\symbf{β}}_{i}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1656,7 +1656,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{h}=\frac{1}{2} \left({{\symbf{h}^{\text{R}}}}_{i}+{{\symbf{h}^{\text{L}}}}_{i}\right) + \symbf{h}=\frac{1}{2}\,\left({{\symbf{h}^{\text{R}}}}_{i}+{{\symbf{h}^{\text{L}}}}_{i}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1811,7 +1811,7 @@ \subsubsection{Data Definitions} Equation & \begin{displaymath} \symbf{f}=\begin{cases} 1, & \mathit{const\_f}\\ - \sin\left(π \frac{{\symbf{x}_{\text{slip},i}}-{\symbf{x}_{\text{slip},0}}}{{\symbf{x}_{\text{slip},n}}-{\symbf{x}_{\text{slip},0}}}\right), & \neg{}\mathit{const\_f} + \sin\left(π\,\frac{{\symbf{x}_{\text{slip},i}}-{\symbf{x}_{\text{slip},0}}}{{\symbf{x}_{\text{slip},n}}-{\symbf{x}_{\text{slip},0}}}\right), & \neg{}\mathit{const\_f} \end{cases} \end{displaymath} \\ \midrule @@ -1851,7 +1851,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{Φ}=\left(λ {\symbf{f}}_{i} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}} + \symbf{Φ}=\left(λ\,{\symbf{f}}_{i}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1894,7 +1894,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - \symbf{Ψ}=\frac{\left(λ {\symbf{f}}_{i} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}}{{\symbf{Φ}}_{i-1}} + \symbf{Ψ}=\frac{\left(λ\,{\symbf{f}}_{i}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}}{{\symbf{Φ}}_{i-1}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -2094,7 +2094,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - {F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}} + {F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -2124,114 +2124,114 @@ \subsubsection{Instance Models} The mobilized shear force defined in \hyperref[GD:bsShrFEq]{GD:bsShrFEq} can be substituted into the definition of mobilized shear force based on the factor of safety, from \hyperref[GD:mobShr]{GD:mobShr} yielding Equation (1) below: \begin{displaymath} -\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{N'}}_{i} \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}} +\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{N'}}_{i}\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} An expression for the effective normal forces, $\symbf{N'}$, can be derived by substituting the normal forces equilibrium from \hyperref[GD:normForcEq]{GD:normForcEq} into the definition for effective normal forces from \hyperref[GD:resShearWO]{GD:resShearWO}. This results in Equation (2): \begin{displaymath} -{\symbf{N'}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}} +{\symbf{N'}}_{i}=\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}} \end{displaymath} Substituting Equation (2) into Equation (1) gives: \begin{displaymath} -\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}} +\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} Since the interslice shear forces $\symbf{X}$ and interslice normal forces $\symbf{G}$ are unknown, they are separated from the other terms as follows: \begin{displaymath} -\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \cos\left({\symbf{ω}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}} {\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}} +\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\cos\left({\symbf{ω}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{K_{\text{c}}}\,{\symbf{W}}_{i}-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} Applying assumptions \hyperref[assumpSF]{A:Seismic-Force} and \hyperref[assumpSL]{A:Surface-Load}, which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below: \begin{displaymath} -\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}} \cos\left({\symbf{β}}_{i}\right)\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\symbf{L}_{b,i}}}{{F_{\text{S}}}} +\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)=\frac{\left(\left({\symbf{W}}_{i}+{\symbf{U}_{\text{g},i}}\,\cos\left({\symbf{β}}_{i}\right)\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{H}}_{i}+{\symbf{H}}_{i-1}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)-{\symbf{U}_{\text{b},i}}\right)\,\tan\left(φ'\right)+c'\,{\symbf{L}_{b,i}}}{{F_{\text{S}}}} \end{displaymath} The definitions of \hyperref[GD:resShearWO]{GD:resShearWO} and \hyperref[GD:mobShearWO]{GD:mobShearWO} are present in this equation, and thus can be replaced by ${\symbf{R}}_{i}$ and ${\symbf{T}}_{i}$, respectively: \begin{displaymath} -{\symbf{T}}_{i}+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{\text{S}}}} +{\symbf{T}}_{i}+\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-{\symbf{X}}_{i-1}+{\symbf{X}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)}{{F_{\text{S}}}} \end{displaymath} The interslice shear forces $\symbf{X}$ can be expressed in terms of the interslice normal forces $\symbf{G}$ using \hyperref[assumpINSFL]{A:Interslice-Norm-Shear-Forces-Linear} and \hyperref[GD:normShrR]{GD:normShrR}, resulting in: \begin{displaymath} -{\symbf{T}}_{i}+\left(-λ {\symbf{f}}_{i-1} {\symbf{G}}_{i-1}+λ {\symbf{f}}_{i} {\symbf{G}}_{i}\right) \sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-λ {\symbf{f}}_{i-1} {\symbf{G}}_{i-1}+λ {\symbf{f}}_{i} {\symbf{G}}_{i}\right) \cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right) \sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)}{{F_{\text{S}}}} +{\symbf{T}}_{i}+\left(-λ\,{\symbf{f}}_{i-1}\,{\symbf{G}}_{i-1}+λ\,{\symbf{f}}_{i}\,{\symbf{G}}_{i}\right)\,\sin\left({\symbf{α}}_{i}\right)-\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\cos\left({\symbf{α}}_{i}\right)=\frac{{\symbf{R}}_{i}+\left(\left(-λ\,{\symbf{f}}_{i-1}\,{\symbf{G}}_{i-1}+λ\,{\symbf{f}}_{i}\,{\symbf{G}}_{i}\right)\,\cos\left({\symbf{α}}_{i}\right)+\left(-{\symbf{G}}_{i}+{\symbf{G}}_{i-1}\right)\,\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)}{{F_{\text{S}}}} \end{displaymath} Rearranging yields the following: \begin{displaymath} -{\symbf{G}}_{i} \left(\left(λ {\symbf{f}}_{i} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}\right)={\symbf{G}}_{i-1} \left(\left(λ {\symbf{f}}_{i-1} \cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\symbf{f}}_{i-1} \sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right) {F_{\text{S}}}\right)+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i} +{\symbf{G}}_{i}\,\left(\left(λ\,{\symbf{f}}_{i}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)={\symbf{G}}_{i-1}\,\left(\left(λ\,{\symbf{f}}_{i-1}\,\cos\left({\symbf{α}}_{i}\right)-\sin\left({\symbf{α}}_{i}\right)\right)\,\tan\left(φ'\right)-\left(λ\,{\symbf{f}}_{i-1}\,\sin\left({\symbf{α}}_{i}\right)+\cos\left({\symbf{α}}_{i}\right)\right)\,{F_{\text{S}}}\right)+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i} \end{displaymath} The definitions for $\symbf{Φ}$ and $\symbf{Ψ}$ from \hyperref[DD:convertFunc1]{DD:convertFunc1} and \hyperref[DD:convertFunc2]{DD:convertFunc2} simplify the above to Equation (3): \begin{displaymath} -{\symbf{G}}_{i} {\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1} {\symbf{Φ}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i} +{\symbf{G}}_{i}\,{\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}\,{\symbf{Φ}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i} \end{displaymath} Versions of Equation (3) instantiated for slices 1 to $n$ are shown below: \begin{displaymath} -{\symbf{G}}_{1} {\symbf{Φ}}_{1}={\symbf{Ψ}}_{0} {\symbf{G}}_{0} {\symbf{Φ}}_{0}+{F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1} +{\symbf{G}}_{1}\,{\symbf{Φ}}_{1}={\symbf{Ψ}}_{0}\,{\symbf{G}}_{0}\,{\symbf{Φ}}_{0}+{F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1} \end{displaymath} \begin{displaymath} -{\symbf{G}}_{2} {\symbf{Φ}}_{2}={\symbf{Ψ}}_{1} {\symbf{G}}_{1} {\symbf{Φ}}_{1}+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2} +{\symbf{G}}_{2}\,{\symbf{Φ}}_{2}={\symbf{Ψ}}_{1}\,{\symbf{G}}_{1}\,{\symbf{Φ}}_{1}+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2} \end{displaymath} \begin{displaymath} -{\symbf{G}}_{3} {\symbf{Φ}}_{3}={\symbf{Ψ}}_{2} {\symbf{G}}_{2} {\symbf{Φ}}_{2}+{F_{\text{S}}} {\symbf{T}}_{3}-{\symbf{R}}_{3} +{\symbf{G}}_{3}\,{\symbf{Φ}}_{3}={\symbf{Ψ}}_{2}\,{\symbf{G}}_{2}\,{\symbf{Φ}}_{2}+{F_{\text{S}}}\,{\symbf{T}}_{3}-{\symbf{R}}_{3} \end{displaymath} ... \begin{displaymath} -{\symbf{G}}_{n-2} {\symbf{Φ}}_{n-2}={\symbf{Ψ}}_{n-3} {\symbf{G}}_{n-3} {\symbf{Φ}}_{n-3}+{F_{\text{S}}} {\symbf{T}}_{n-2}-{\symbf{R}}_{n-2} +{\symbf{G}}_{n-2}\,{\symbf{Φ}}_{n-2}={\symbf{Ψ}}_{n-3}\,{\symbf{G}}_{n-3}\,{\symbf{Φ}}_{n-3}+{F_{\text{S}}}\,{\symbf{T}}_{n-2}-{\symbf{R}}_{n-2} \end{displaymath} \begin{displaymath} -{\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2} {\symbf{G}}_{n-2} {\symbf{Φ}}_{n-2}+{F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1} +{\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2}\,{\symbf{G}}_{n-2}\,{\symbf{Φ}}_{n-2}+{F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1} \end{displaymath} \begin{displaymath} -{\symbf{G}}_{n} {\symbf{Φ}}_{n}={\symbf{Ψ}}_{n-1} {\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}+{F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n} +{\symbf{G}}_{n}\,{\symbf{Φ}}_{n}={\symbf{Ψ}}_{n-1}\,{\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}+{F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n} \end{displaymath} Applying \hyperref[assumpES]{A:Edge-Slices}, which says that ${\symbf{G}}_{0}$ and ${\symbf{G}}_{n}$ are zero, results in the following special cases: Equation (8) for the first slice: \begin{displaymath} -{\symbf{G}}_{1} {\symbf{Φ}}_{1}={F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1} +{\symbf{G}}_{1}\,{\symbf{Φ}}_{1}={F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1} \end{displaymath} and Equation (9) for the $n$th slice: \begin{displaymath} --\left(\frac{{F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1} +-\left(\frac{{F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1} \end{displaymath} Substituting Equation (8) into Equation (4) yields Equation (10): \begin{displaymath} -{\symbf{G}}_{2} {\symbf{Φ}}_{2}={\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2} +{\symbf{G}}_{2}\,{\symbf{Φ}}_{2}={\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2} \end{displaymath} which can be substituted into Equation (5) to get Equation (11): \begin{displaymath} -{\symbf{G}}_{3} {\symbf{Φ}}_{3}={\symbf{Ψ}}_{2} \left({\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}} {\symbf{T}}_{3}-{\symbf{R}}_{3} +{\symbf{G}}_{3}\,{\symbf{Φ}}_{3}={\symbf{Ψ}}_{2}\,\left({\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{3}-{\symbf{R}}_{3} \end{displaymath} and so on until Equation (12) is obtained from Equation (7): \begin{displaymath} -{\symbf{G}}_{n-1} {\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2} \left({\symbf{Ψ}}_{n-3} \left({\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1} +{\symbf{G}}_{n-1}\,{\symbf{Φ}}_{n-1}={\symbf{Ψ}}_{n-2}\,\left({\symbf{Ψ}}_{n-3}\,\left({\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1} \end{displaymath} Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in: \begin{displaymath} --\left(\frac{{F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{Ψ}}_{n-2} \left({\symbf{Ψ}}_{n-3} \left({\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1} +-\left(\frac{{F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}}{{\symbf{Ψ}}_{n-1}}\right)={\symbf{Ψ}}_{n-2}\,\left({\symbf{Ψ}}_{n-3}\,\left({\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-2}-{\symbf{R}}_{n-2}\right)+{F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1} \end{displaymath} This can be rearranged by multiplying both sides by ${\symbf{Ψ}}_{n-1}$ and then distributing the multiplication of each $\symbf{Ψ}$ over addition to obtain: \begin{displaymath} --\left({F_{\text{S}}} {\symbf{T}}_{n}-{\symbf{R}}_{n}\right)={\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{1} \left({F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{2} \left({F_{\text{S}}} {\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{\symbf{Ψ}}_{n-1} \left({F_{\text{S}}} {\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\right) +-\left({F_{\text{S}}}\,{\symbf{T}}_{n}-{\symbf{R}}_{n}\right)={\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}\right)+{\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{2}\,\left({F_{\text{S}}}\,{\symbf{T}}_{2}-{\symbf{R}}_{2}\right)+{\symbf{Ψ}}_{n-1}\,\left({F_{\text{S}}}\,{\symbf{T}}_{n-1}-{\symbf{R}}_{n-1}\right) \end{displaymath} The multiplication of the $\symbf{Ψ}$ terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an $\symbf{R}$ or a $\symbf{T}$. The equation can then be rearranged so terms containing an $\symbf{R}$ are on one side of the equality, and terms containing a $\symbf{T}$ are on the other. The multiplication by the factor of safety is common to all of the $\symbf{T}$ terms, and thus can be factored out, resulting in: \begin{displaymath} -{F_{\text{S}}} \left({\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{1} {\symbf{T}}_{1}+{\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{2} {\symbf{T}}_{2}+{\symbf{Ψ}}_{n-1} {\symbf{T}}_{n-1}+{\symbf{T}}_{n}\right)={\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{1} {\symbf{R}}_{1}+{\symbf{Ψ}}_{n-1} {\symbf{Ψ}}_{n-2} {\symbf{Ψ}}_{2} {\symbf{R}}_{2}+{\symbf{Ψ}}_{n-1} {\symbf{R}}_{n-1}+{\symbf{R}}_{n} +{F_{\text{S}}}\,\left({\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{1}\,{\symbf{T}}_{1}+{\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{2}\,{\symbf{T}}_{2}+{\symbf{Ψ}}_{n-1}\,{\symbf{T}}_{n-1}+{\symbf{T}}_{n}\right)={\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{1}\,{\symbf{R}}_{1}+{\symbf{Ψ}}_{n-1}\,{\symbf{Ψ}}_{n-2}\,{\symbf{Ψ}}_{2}\,{\symbf{R}}_{2}+{\symbf{Ψ}}_{n-1}\,{\symbf{R}}_{n-1}+{\symbf{R}}_{n} \end{displaymath} Isolating the factor of safety on the left-hand side and using compact notation for the products and sums yields Equation (13), which can also be seen in \hyperref[IM:fctSfty]{IM:fctSfty}: \begin{displaymath} -{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i} \displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}} +{F_{\text{S}}}=\frac{\displaystyle\sum_{i=1}^{n-1}{{\symbf{R}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{R}}_{n}}{\displaystyle\sum_{i=1}^{n-1}{{\symbf{T}}_{i}\,\displaystyle\prod_{v=i}^{n-1}{{\symbf{Ψ}}_{v}}}+{\symbf{T}}_{n}} \end{displaymath} ${F_{\text{S}}}$ depends on the unknowns $λ$ (\hyperref[IM:nrmShrFor]{IM:nrmShrFor}) and $\symbf{G}$ (\hyperref[IM:intsliceFs]{IM:intsliceFs}). @@ -2284,22 +2284,22 @@ \subsubsection{Instance Models} From the moment equilibrium of \hyperref[GD:momentEql]{GD:momentEql} with the primary assumption for the Morgenstern-Price method of \hyperref[assumpINSFL]{A:Interslice-Norm-Shear-Forces-Linear} and associated definition \hyperref[GD:normShrR]{GD:normShrR}, Equation (14) can be derived: \begin{displaymath} -0=-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+λ \frac{{\symbf{b}}_{i}}{2} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i} +0=-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+λ\,\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i} \end{displaymath} Rearranging the equation in terms of $λ$ leads to Equation (15): \begin{displaymath} -λ=\frac{-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+\frac{-{K_{\text{c}}} {\symbf{W}}_{i} {\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}+{\symbf{Q}}_{i} \sin\left({\symbf{ω}}_{i}\right) {\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)} +λ=\frac{-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+\frac{-{K_{\text{c}}}\,{\symbf{W}}_{i}\,{\symbf{h}}_{i}}{2}+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}+{\symbf{Q}}_{i}\,\sin\left({\symbf{ω}}_{i}\right)\,{\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)} \end{displaymath} This equation can be simplified by applying assumptions \hyperref[assumpSF]{A:Seismic-Force} and \hyperref[assumpSL]{A:Surface-Load}, which state that the seismic and external forces, respectively, are zero: \begin{displaymath} -λ=\frac{-{\symbf{G}}_{i} \left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1} \left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1} \left(\frac{1}{3} {\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2} \tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right) {\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)} +λ=\frac{-{\symbf{G}}_{i}\,\left({\symbf{h}_{\text{z},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{G}}_{i-1}\,\left({\symbf{h}_{\text{z},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)-{\symbf{H}}_{i}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i}}+\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{H}}_{i-1}\,\left(\frac{1}{3}\,{\symbf{h}_{\text{z,w},i-1}}-\frac{{\symbf{b}}_{i}}{2}\,\tan\left({\symbf{α}}_{i}\right)\right)+{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)\,{\symbf{h}}_{i}}{-\frac{{\symbf{b}}_{i}}{2}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)} \end{displaymath} Taking the summation of all slices, and applying \hyperref[assumpES]{A:Edge-Slices} to set ${\symbf{G}}_{0}$, ${\symbf{G}}_{n}$, ${\symbf{H}}_{0}$, and ${\symbf{H}}_{n}$ equal to zero, a general equation for the proportionality constant $λ$ is developed in Equation (16), which combines \hyperref[IM:nrmShrFor]{IM:nrmShrFor}, \hyperref[IM:nrmShrForNum]{IM:nrmShrForNum}, and \hyperref[IM:nrmShrForDen]{IM:nrmShrForDen}: \begin{displaymath} -λ=\frac{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i} \left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right) \tan\left({\symbf{α}}_{i}\right)+{\symbf{h}}_{i} -2 {\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i} \left({\symbf{G}}_{i} {\symbf{f}}_{i}+{\symbf{G}}_{i-1} {\symbf{f}}_{i-1}\right)}} +λ=\frac{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i}\,\left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right)\,\tan\left({\symbf{α}}_{i}\right)+{\symbf{h}}_{i}\,-2\,{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right)}}{\displaystyle\sum_{i=1}^{n}{{\symbf{b}}_{i}\,\left({\symbf{G}}_{i}\,{\symbf{f}}_{i}+{\symbf{G}}_{i-1}\,{\symbf{f}}_{i-1}\right)}} \end{displaymath} Equation (16) for $λ$ is a function of the unknown interslice normal forces $\symbf{G}$ (\hyperref[IM:intsliceFs]{IM:intsliceFs}) which itself depends on the unknown factor of safety ${F_{\text{S}}}$ (\hyperref[IM:fctSfty]{IM:fctSfty}). @@ -2326,9 +2326,9 @@ \subsubsection{Instance Models} \\ \midrule Equation & \begin{displaymath} {\symbf{C}_{\text{num},i}}=\begin{cases} - {\symbf{b}}_{1} \left({\symbf{G}}_{1}+{\symbf{H}}_{1}\right) \tan\left({\symbf{α}}_{1}\right), & i=1\\ - {\symbf{b}}_{i} \left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right) \tan\left({\symbf{α}}_{i}\right)+\symbf{h} -2 {\symbf{U}_{\text{g},i}} \sin\left({\symbf{β}}_{i}\right), & 2\leq{}i\leq{}n-1\\ - {\symbf{b}}_{n} \left({\symbf{G}}_{n-1}+{\symbf{H}}_{n-1}\right) \tan\left({\symbf{α}}_{n-1}\right), & i=n + {\symbf{b}}_{1}\,\left({\symbf{G}}_{1}+{\symbf{H}}_{1}\right)\,\tan\left({\symbf{α}}_{1}\right), & i=1\\ + {\symbf{b}}_{i}\,\left({{\symbf{F}_{\text{x}}}^{\text{G}}}+{{\symbf{F}_{\text{x}}}^{\text{H}}}\right)\,\tan\left({\symbf{α}}_{i}\right)+\symbf{h}\,-2\,{\symbf{U}_{\text{g},i}}\,\sin\left({\symbf{β}}_{i}\right), & 2\leq{}i\leq{}n-1\\ + {\symbf{b}}_{n}\,\left({\symbf{G}}_{n-1}+{\symbf{H}}_{n-1}\right)\,\tan\left({\symbf{α}}_{n-1}\right), & i=n \end{cases} \end{displaymath} \\ \midrule @@ -2386,9 +2386,9 @@ \subsubsection{Instance Models} \\ \midrule Equation & \begin{displaymath} {\symbf{C}_{\text{den},i}}=\begin{cases} - {\symbf{b}}_{1} {\symbf{f}}_{1} {\symbf{G}}_{1}, & i=1\\ - {\symbf{b}}_{i} \left({\symbf{f}}_{i} {\symbf{G}}_{i}+{\symbf{f}}_{i-1} {\symbf{G}}_{i-1}\right), & 2\leq{}i\leq{}n-1\\ - {\symbf{b}}_{n} {\symbf{G}}_{n-1} {\symbf{f}}_{n-1}, & i=n + {\symbf{b}}_{1}\,{\symbf{f}}_{1}\,{\symbf{G}}_{1}, & i=1\\ + {\symbf{b}}_{i}\,\left({\symbf{f}}_{i}\,{\symbf{G}}_{i}+{\symbf{f}}_{i-1}\,{\symbf{G}}_{i-1}\right), & 2\leq{}i\leq{}n-1\\ + {\symbf{b}}_{n}\,{\symbf{G}}_{n-1}\,{\symbf{f}}_{n-1}, & i=n \end{cases} \end{displaymath} \\ \midrule @@ -2440,8 +2440,8 @@ \subsubsection{Instance Models} \\ \midrule Equation & \begin{displaymath} {\symbf{G}}_{i}=\begin{cases} - \frac{{F_{\text{S}}} {\symbf{T}}_{1}-{\symbf{R}}_{1}}{{\symbf{Φ}}_{1}}, & i=1\\ - \frac{{\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}}, & 2\leq{}i\leq{}n-1\\ + \frac{{F_{\text{S}}}\,{\symbf{T}}_{1}-{\symbf{R}}_{1}}{{\symbf{Φ}}_{1}}, & i=1\\ + \frac{{\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}}, & 2\leq{}i\leq{}n-1\\ 0, & i=0\lor{}i=n \end{cases} \end{displaymath} @@ -2474,12 +2474,12 @@ \subsubsection{Instance Models} This derivation is identical to the derivation for \hyperref[IM:fctSfty]{IM:fctSfty} up until Equation (3) shown again below: \begin{displaymath} -{\symbf{G}}_{i} {\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1} {\symbf{Φ}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i} +{\symbf{G}}_{i}\,{\symbf{Φ}}_{i}={\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}\,{\symbf{Φ}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i} \end{displaymath} A simple rearrangement of Equation (3) leads to Equation (17), also seen in \hyperref[IM:intsliceFs]{IM:intsliceFs}: \begin{displaymath} -{\symbf{G}}_{i}=\frac{{\symbf{Ψ}}_{i-1} {\symbf{G}}_{i-1}+{F_{\text{S}}} {\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}} +{\symbf{G}}_{i}=\frac{{\symbf{Ψ}}_{i-1}\,{\symbf{G}}_{i-1}+{F_{\text{S}}}\,{\symbf{T}}_{i}-{\symbf{R}}_{i}}{{\symbf{Φ}}_{i}} \end{displaymath} The cases shown in \hyperref[IM:intsliceFs]{IM:intsliceFs} for when $i=0$, $i=1$, or $i=n$ are derived by applying \hyperref[assumpES]{A:Edge-Slices}, which says that ${\symbf{G}}_{0}$ and ${\symbf{G}}_{n}$ are zero, to Equation (17). $\symbf{G}$ depends on the unknowns ${F_{\text{S}}}$ (\hyperref[IM:fctSfty]{IM:fctSfty}) and $λ$ (\hyperref[IM:nrmShrFor]{IM:nrmShrFor}). diff --git a/code/stable/ssp/SRS/mdBook/src/SecDDs.md b/code/stable/ssp/SRS/mdBook/src/SecDDs.md index 2660dd30d1..b4968242f9 100644 --- a/code/stable/ssp/SRS/mdBook/src/SecDDs.md +++ b/code/stable/ssp/SRS/mdBook/src/SecDDs.md @@ -8,15 +8,15 @@ This section collects and defines all the data needed to build the instance mode -|Refname |DD:intersliceWtrF | -|:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| -|Label |Interslice normal water forces | -|Symbol |\\(\boldsymbol{H}\\) | -|Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[\boldsymbol{H}=\begin{cases}\frac{\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}\right)^{2}}{2} {γ\_{w}}+\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}\right)^{2} {γ\_{w}}, & {\boldsymbol{y}\_{\text{wt},i}}\geq{}{\boldsymbol{y}\_{\text{slope},i}}\\\\\frac{\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}\right)^{2}}{2} {γ\_{w}}, & {\boldsymbol{y}\_{\text{slope},i}}\gt{}{\boldsymbol{y}\_{\text{wt},i}}\land{}{\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slip},i}}\\\\0, & {\boldsymbol{y}\_{\text{wt},i}}\leq{}{\boldsymbol{y}\_{\text{slip},i}}\end{cases}\\]| -|Description|
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slope}}}\\) is the \\(y\\)-coordinates of the slope (\\({\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{y}\_{\text{slip}}}\\) is the \\(y\\)-coordinates of the slip surface (\\({\text{m}}\\))
  • \\({γ\_{w}}\\) is the unit weight of water (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
| -|Source |[fredlund1977](./SecReferences.md#fredlund1977) | -|RefBy |[GD:resShearWO](./SecGDs.md#GD:resShearWO), [IM:nrmShrForNum](./SecIMs.md#IM:nrmShrForNum), and [GD:mobShearWO](./SecGDs.md#GD:mobShearWO) | +|Refname |DD:intersliceWtrF | +|:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +|Label |Interslice normal water forces | +|Symbol |\\(\boldsymbol{H}\\) | +|Units |\\(\frac{\text{N}}{\text{m}}\\) | +|Equation |\\[\boldsymbol{H}=\begin{cases}\frac{\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}\right)^{2}}{2}\\,{γ\_{w}}+\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}\right)^{2}\\,{γ\_{w}}, & {\boldsymbol{y}\_{\text{wt},i}}\geq{}{\boldsymbol{y}\_{\text{slope},i}}\\\\\frac{\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}\right)^{2}}{2}\\,{γ\_{w}}, & {\boldsymbol{y}\_{\text{slope},i}}\gt{}{\boldsymbol{y}\_{\text{wt},i}}\land{}{\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slip},i}}\\\\0, & {\boldsymbol{y}\_{\text{wt},i}}\leq{}{\boldsymbol{y}\_{\text{slip},i}}\end{cases}\\]| +|Description|
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slope}}}\\) is the \\(y\\)-coordinates of the slope (\\({\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{y}\_{\text{slip}}}\\) is the \\(y\\)-coordinates of the slip surface (\\({\text{m}}\\))
  • \\({γ\_{w}}\\) is the unit weight of water (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
| +|Source |[fredlund1977](./SecReferences.md#fredlund1977) | +|RefBy |[GD:resShearWO](./SecGDs.md#GD:resShearWO), [IM:nrmShrForNum](./SecIMs.md#IM:nrmShrForNum), and [GD:mobShearWO](./SecGDs.md#GD:mobShearWO) |
@@ -79,7 +79,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Total base lengths of slices | |Symbol |\\({\boldsymbol{L}\_{b}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{\boldsymbol{L}\_{b}}={\boldsymbol{b}}\_{i} \sec\left({\boldsymbol{α}}\_{i}\right)\\] | +|Equation |\\[{\boldsymbol{L}\_{b}}={\boldsymbol{b}}\_{i}\\,\sec\left({\boldsymbol{α}}\_{i}\right)\\] | |Description|
  • \\({\boldsymbol{L}\_{b}}\\) is the total base lengths of slices (\\({\text{m}}\\))
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
| |Notes |
  • \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB) and \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA).
| |Source |[fredlund1977](./SecReferences.md#fredlund1977) | @@ -96,7 +96,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Surface lengths of slices | |Symbol |\\({\boldsymbol{L}\_{s}}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[{\boldsymbol{L}\_{s}}={\boldsymbol{b}}\_{i} \sec\left({\boldsymbol{β}}\_{i}\right)\\] | +|Equation |\\[{\boldsymbol{L}\_{s}}={\boldsymbol{b}}\_{i}\\,\sec\left({\boldsymbol{β}}\_{i}\right)\\] | |Description|
  • \\({\boldsymbol{L}\_{s}}\\) is the surface lengths of slices (\\({\text{m}}\\))
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
| |Notes |
  • \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB) and \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB).
| |Source |[fredlund1977](./SecReferences.md#fredlund1977) | @@ -113,7 +113,7 @@ This section collects and defines all the data needed to build the instance mode |Label |\\(y\\)-direction heights of slices | |Symbol |\\(\boldsymbol{h}\\) | |Units |\\({\text{m}}\\) | -|Equation |\\[\boldsymbol{h}=\frac{1}{2} \left({{\boldsymbol{h}^{\text{R}}}}\_{i}+{{\boldsymbol{h}^{\text{L}}}}\_{i}\right)\\] | +|Equation |\\[\boldsymbol{h}=\frac{1}{2}\\,\left({{\boldsymbol{h}^{\text{R}}}}\_{i}+{{\boldsymbol{h}^{\text{L}}}}\_{i}\right)\\] | |Description|
  • \\(\boldsymbol{h}\\) is the \\(y\\)-direction heights of slices (\\({\text{m}}\\))
  • \\({\boldsymbol{h}^{\text{R}}}\\) is the heights of the right side of slices (\\({\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{h}^{\text{L}}}\\) is the heights of the left side of slices (\\({\text{m}}\\))
| |Notes |
  • This equation is based on the assumption that the surface and base of a slice are straight lines ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)).
  • \\({\boldsymbol{h}^{\text{R}}}\\) and \\({\boldsymbol{h}^{\text{L}}}\\) are defined in [DD:sliceHghtRightDD](./SecDDs.md#DD:sliceHghtRightDD) and [DD:sliceHghtLeftDD](./SecDDs.md#DD:sliceHghtLeftDD), respectively.
| |Source |[fredlund1977](./SecReferences.md#fredlund1977) | @@ -179,7 +179,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Interslice normal to shear force ratio variation function | |Symbol |\\(\boldsymbol{f}\\) | |Units |Unitless | -|Equation |\\[\boldsymbol{f}=\begin{cases}1, & \mathit{const\_f}\\\\\sin\left(π \frac{{\boldsymbol{x}\_{\text{slip},i}}-{\boldsymbol{x}\_{\text{slip},0}}}{{\boldsymbol{x}\_{\text{slip},n}}-{\boldsymbol{x}\_{\text{slip},0}}}\right), & \neg{}\mathit{const\_f}\end{cases}\\] | +|Equation |\\[\boldsymbol{f}=\begin{cases}1, & \mathit{const\_f}\\\\\sin\left(π\\,\frac{{\boldsymbol{x}\_{\text{slip},i}}-{\boldsymbol{x}\_{\text{slip},0}}}{{\boldsymbol{x}\_{\text{slip},n}}-{\boldsymbol{x}\_{\text{slip},0}}}\right), & \neg{}\mathit{const\_f}\end{cases}\\] | |Description|
  • \\(\boldsymbol{f}\\) is the interslice normal to shear force ratio variation function (Unitless)
  • \\(π\\) is the ratio of circumference to diameter for any circle (Unitless)
  • \\({\boldsymbol{x}\_{\text{slip}}}\\) is the \\(x\\)-coordinates of the slip surface (\\({\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(n\\) is the number of slices (Unitless)
  • \\(\mathit{const\_f}\\) is the decision on f (Unitless)
| |Source |[fredlund1977](./SecReferences.md#fredlund1977) | |RefBy |[IM:nrmShrForDen](./SecIMs.md#IM:nrmShrForDen), [GD:normShrR](./SecGDs.md#GD:normShrR), [DD:convertFunc2](./SecDDs.md#DD:convertFunc2), and [DD:convertFunc1](./SecDDs.md#DD:convertFunc1) | @@ -195,7 +195,7 @@ This section collects and defines all the data needed to build the instance mode |Label |First function for incorporating interslice forces into shear force | |Symbol |\\(\boldsymbol{Φ}\\) | |Units |Unitless | -|Equation |\\[\boldsymbol{Φ}=\left(λ {\boldsymbol{f}}\_{i} \cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\boldsymbol{f}}\_{i} \sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right) {F\_{\text{S}}}\\] | +|Equation |\\[\boldsymbol{Φ}=\left(λ\\,{\boldsymbol{f}}\_{i}\\,\cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right)\\,\tan\left(φ'\right)-\left(λ\\,{\boldsymbol{f}}\_{i}\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right)\\,{F\_{\text{S}}}\\] | |Description|
  • \\(\boldsymbol{Φ}\\) is the first function for incorporating interslice forces into shear force (Unitless)
  • \\(λ\\) is the proportionality constant (Unitless)
  • \\(\boldsymbol{f}\\) is the interslice normal to shear force ratio variation function (Unitless)
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\(φ'\\) is the effective angle of friction (\\({{}^{\circ}}\\))
  • \\({F\_{\text{S}}}\\) is the factor of safety (Unitless)
| |Notes |
  • \\(\boldsymbol{f}\\) is defined in [DD:ratioVariation](./SecDDs.md#DD:ratioVariation) and \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA).
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | @@ -212,7 +212,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Second function for incorporating interslice forces into shear force | |Symbol |\\(\boldsymbol{Ψ}\\) | |Units |Unitless | -|Equation |\\[\boldsymbol{Ψ}=\frac{\left(λ {\boldsymbol{f}}\_{i} \cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\boldsymbol{f}}\_{i} \sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right) {F\_{\text{S}}}}{{\boldsymbol{Φ}}\_{i-1}}\\] | +|Equation |\\[\boldsymbol{Ψ}=\frac{\left(λ\\,{\boldsymbol{f}}\_{i}\\,\cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right)\\,\tan\left(φ'\right)-\left(λ\\,{\boldsymbol{f}}\_{i}\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right)\\,{F\_{\text{S}}}}{{\boldsymbol{Φ}}\_{i-1}}\\] | |Description|
  • \\(\boldsymbol{Ψ}\\) is the second function for incorporating interslice forces into shear force (Unitless)
  • \\(λ\\) is the proportionality constant (Unitless)
  • \\(\boldsymbol{f}\\) is the interslice normal to shear force ratio variation function (Unitless)
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\(φ'\\) is the effective angle of friction (\\({{}^{\circ}}\\))
  • \\({F\_{\text{S}}}\\) is the factor of safety (Unitless)
  • \\(\boldsymbol{Φ}\\) is the first function for incorporating interslice forces into shear force (Unitless)
| |Notes |
  • \\(\boldsymbol{f}\\) is defined in [DD:ratioVariation](./SecDDs.md#DD:ratioVariation), \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA), and \\(\boldsymbol{Φ}\\) is defined in [DD:convertFunc1](./SecDDs.md#DD:convertFunc1).
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | diff --git a/code/stable/ssp/SRS/mdBook/src/SecGDs.md b/code/stable/ssp/SRS/mdBook/src/SecGDs.md index 78d221ce55..ac6aa7569e 100644 --- a/code/stable/ssp/SRS/mdBook/src/SecGDs.md +++ b/code/stable/ssp/SRS/mdBook/src/SecGDs.md @@ -12,7 +12,7 @@ This section collects the laws and equations that will be used to build the inst |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Normal force equilibrium | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{N}}\_{i}=\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)\\] | +|Equation |\\[{\boldsymbol{N}}\_{i}=\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)\\] | |Description|
  • \\(\boldsymbol{N}\\) is the normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{X}\\) is the interslice shear forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{Q}\\) is the external forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{ω}\\) is the imposed load angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\({K\_{\text{c}}}\\) is the seismic coefficient (Unitless)
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
| |Notes |
  • This equation satisfies [TM:equilibrium](./SecTMs.md#TM:equilibrium) in the normal direction. \\(\boldsymbol{W}\\) is defined in [GD:sliceWght](./SecGDs.md#GD:sliceWght), \\({\boldsymbol{U}\_{\text{g}}}\\) is defined in [GD:srfWtrF](./SecGDs.md#GD:srfWtrF), \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB), and \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -32,7 +32,7 @@ Normal force equilibrium is derived from the free body diagram of [Fig:ForceDiag |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Base shear force equilibrium | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{S}}\_{i}=\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)\\] | +|Equation |\\[{\boldsymbol{S}}\_{i}=\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)\\] | |Description|
  • \\(\boldsymbol{S}\\) is the mobilized shear force (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{X}\\) is the interslice shear forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{Q}\\) is the external forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{ω}\\) is the imposed load angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\({K\_{\text{c}}}\\) is the seismic coefficient (Unitless)
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
| |Notes |
  • This equation satisfies [TM:equilibrium](./SecTMs.md#TM:equilibrium) in the shear direction. \\(\boldsymbol{W}\\) is defined in [GD:sliceWght](./SecGDs.md#GD:sliceWght), \\({\boldsymbol{U}\_{\text{g}}}\\) is defined in [GD:srfWtrF](./SecGDs.md#GD:srfWtrF), \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB), and \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -52,7 +52,7 @@ Base shear force equilibrium is derived from the free body diagram of [Fig:Force |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Resistive shear force | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{P}}\_{i}={\boldsymbol{N'}}\_{i} \tan\left({φ'}\_{i}\right)+{c'}\_{i} {\boldsymbol{L}\_{b,i}}\\] | +|Equation |\\[{\boldsymbol{P}}\_{i}={\boldsymbol{N'}}\_{i}\\,\tan\left({φ'}\_{i}\right)+{c'}\_{i}\\,{\boldsymbol{L}\_{b,i}}\\] | |Description|
  • \\(\boldsymbol{P}\\) is the resistive shear forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{N'}\\) is the effective normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(φ'\\) is the effective angle of friction (\\({{}^{\circ}}\\))
  • \\(c'\\) is the effective cohesion (\\({\text{Pa}}\\))
  • \\({\boldsymbol{L}\_{b}}\\) is the total base lengths of slices (\\({\text{m}}\\))
| |Notes |
  • \\({\boldsymbol{L}\_{b}}\\) is defined in [DD:lengthLb](./SecDDs.md#DD:lengthLb).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -72,7 +72,7 @@ Derived by substituting [DD:normStress](./SecDDs.md#DD:normStress) and [DD:tangS |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Mobilized shear force | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{S}}\_{i}=\frac{{\boldsymbol{P}}\_{i}}{{F\_{\text{S}}}}=\frac{{\boldsymbol{N'}}\_{i} \tan\left({φ'}\_{i}\right)+{c'}\_{i} {\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] | +|Equation |\\[{\boldsymbol{S}}\_{i}=\frac{{\boldsymbol{P}}\_{i}}{{F\_{\text{S}}}}=\frac{{\boldsymbol{N'}}\_{i}\\,\tan\left({φ'}\_{i}\right)+{c'}\_{i}\\,{\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] | |Description|
  • \\(\boldsymbol{S}\\) is the mobilized shear force (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{P}\\) is the resistive shear forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({F\_{\text{S}}}\\) is the factor of safety (Unitless)
  • \\(\boldsymbol{N'}\\) is the effective normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(φ'\\) is the effective angle of friction (\\({{}^{\circ}}\\))
  • \\(c'\\) is the effective cohesion (\\({\text{Pa}}\\))
  • \\({\boldsymbol{L}\_{b}}\\) is the total base lengths of slices (\\({\text{m}}\\))
| |Notes |
  • \\({\boldsymbol{L}\_{b}}\\) is defined in [DD:lengthLb](./SecDDs.md#DD:lengthLb).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -112,7 +112,7 @@ Derived by substituting [DD:normStress](./SecDDs.md#DD:normStress) into [TM:effS |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Resistive shear force, without interslice normal and shear forces | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{R}}\_{i}=\left(\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right) \tan\left({φ'}\_{i}\right)+{c'}\_{i} {\boldsymbol{L}\_{b,i}}\\] | +|Equation |\\[{\boldsymbol{R}}\_{i}=\left(\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right)\\,\tan\left({φ'}\_{i}\right)+{c'}\_{i}\\,{\boldsymbol{L}\_{b,i}}\\] | |Description|
  • \\(\boldsymbol{R}\\) is the resistive shear forces without the influence of interslice forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{b}}}\\) is the base hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(φ'\\) is the effective angle of friction (\\({{}^{\circ}}\\))
  • \\(c'\\) is the effective cohesion (\\({\text{Pa}}\\))
  • \\({\boldsymbol{L}\_{b}}\\) is the total base lengths of slices (\\({\text{m}}\\))
| |Notes |
  • \\(\boldsymbol{W}\\) is defined in [GD:sliceWght](./SecGDs.md#GD:sliceWght), \\({\boldsymbol{U}\_{\text{g}}}\\) is defined in [GD:srfWtrF](./SecGDs.md#GD:srfWtrF), \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB), \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA), \\(\boldsymbol{H}\\) is defined in [DD:intersliceWtrF](./SecDDs.md#DD:intersliceWtrF), \\({\boldsymbol{U}\_{\text{b}}}\\) is defined in [GD:baseWtrF](./SecGDs.md#GD:baseWtrF), and \\({\boldsymbol{L}\_{b}}\\) is defined in [DD:lengthLb](./SecDDs.md#DD:lengthLb).
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | @@ -128,7 +128,7 @@ Derived by substituting [DD:normStress](./SecDDs.md#DD:normStress) into [TM:effS |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Mobilized shear force, without interslice normal and shear forces | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{T}}\_{i}=\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)\\] | +|Equation |\\[{\boldsymbol{T}}\_{i}=\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)\\] | |Description|
  • \\(\boldsymbol{T}\\) is the mobilized shear forces without the influence of interslice forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
| |Notes |
  • \\(\boldsymbol{W}\\) is defined in [GD:sliceWght](./SecGDs.md#GD:sliceWght), \\({\boldsymbol{U}\_{\text{g}}}\\) is defined in [GD:srfWtrF](./SecGDs.md#GD:srfWtrF), \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB), \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA), and \\(\boldsymbol{H}\\) is defined in [DD:intersliceWtrF](./SecDDs.md#DD:intersliceWtrF).
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | @@ -144,7 +144,7 @@ Derived by substituting [DD:normStress](./SecDDs.md#DD:normStress) into [TM:effS |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Interslice shear forces | |Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[\boldsymbol{X}=λ \boldsymbol{f} \boldsymbol{G}\\] | +|Equation |\\[\boldsymbol{X}=λ\\,\boldsymbol{f}\\,\boldsymbol{G}\\] | |Description|
  • \\(\boldsymbol{X}\\) is the interslice shear forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(λ\\) is the proportionality constant (Unitless)
  • \\(\boldsymbol{f}\\) is the interslice normal to shear force ratio variation function (Unitless)
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
| |Notes |
  • Mathematical representation of the primary assumption for the Morgenstern-Price method ([A:Interslice-Norm-Shear-Forces-Linear](./SecAssumps.md#assumpINSFL)). \\(\boldsymbol{f}\\) is defined in [DD:ratioVariation](./SecDDs.md#DD:ratioVariation).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -160,7 +160,7 @@ Derived by substituting [DD:normStress](./SecDDs.md#DD:normStress) into [TM:effS |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Moment equilibrium | |Units |\\({\text{N}}\\) | -|Equation |\\[0=-{\boldsymbol{G}}\_{i} \left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1} \left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+\frac{{\boldsymbol{b}}\_{i}}{2} \left({\boldsymbol{X}}\_{i}+{\boldsymbol{X}}\_{i-1}\right)+\frac{-{K\_{\text{c}}} {\boldsymbol{W}}\_{i} {\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right) {\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right) {\boldsymbol{h}}\_{i}\\] | +|Equation |\\[0=-{\boldsymbol{G}}\_{i}\\,\left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1}\\,\left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\left({\boldsymbol{X}}\_{i}+{\boldsymbol{X}}\_{i-1}\right)+\frac{-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}\\,{\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}\\] | |Description|
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{h}\_{\text{z}}}\\) is the heights of interslice normal forces (\\({\text{m}}\\))
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({\boldsymbol{h}\_{\text{z,w}}}\\) is the heights of the water table (\\({\text{m}}\\))
  • \\(\boldsymbol{X}\\) is the interslice shear forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\({K\_{\text{c}}}\\) is the seismic coefficient (Unitless)
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{h}\\) is the \\(y\\)-direction heights of slices (\\({\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
  • \\(\boldsymbol{Q}\\) is the external forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{ω}\\) is the imposed load angles (\\({{}^{\circ}}\\))
| |Notes |
  • This equation satisfies [TM:equilibrium](./SecTMs.md#TM:equilibrium) for the net moment. \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB), \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA), \\(\boldsymbol{W}\\) is defined in [GD:sliceWght](./SecGDs.md#GD:sliceWght), \\(\boldsymbol{h}\\) is defined in [DD:slcHeight](./SecDDs.md#DD:slcHeight), \\({\boldsymbol{U}\_{\text{g}}}\\) is defined in [GD:srfWtrF](./SecGDs.md#GD:srfWtrF), and \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -174,55 +174,55 @@ Moment is equal to torque, so the equation from [DD:torque](./SecDDs.md#DD:torqu Considering one dimension, with moments in the clockwise direction as positive and moments in the counterclockwise direction as negative, and replacing the torque symbol with the moment symbol, the equation simplifies to: -\\[M={F\_{\text{rot}}} r\\] +\\[M={F\_{\text{rot}}}\\,r\\] where \\({F\_{\text{rot}}}\\) is the force causing rotation and \\(r\\) is the length of the moment arm, or the distance between the force and the axis about which the rotation acts. To represent the moment equilibrium, the moments from each force acting on a slice must be considered and added together. The forces acting on a slice are all shown in [Fig:ForceDiagram](./SecPhysSyst.md#Figure:ForceDiagram). The midpoint of the base of a slice is considered as the axis of rotation, from which the length of the moment arm is measured. Considering first the interslice normal force acting on slice interface \\(i\\), the moment is negative because the force tends to rotate the slice in a counterclockwise direction, and the length of the moment arm is the height of the force plus the difference in height between the base at slice interface \\(i\\) and the base at the midpoint of slice \\(i\\). Thus, the moment is expressed as: -\\[-{\boldsymbol{G}}\_{i} \left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] +\\[-{\boldsymbol{G}}\_{i}\\,\left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] For the \\(i-1\\)th slice interface, the moment is similar but in the opposite direction: -\\[{\boldsymbol{G}}\_{i-1} \left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] +\\[{\boldsymbol{G}}\_{i-1}\\,\left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] Next, the interslice normal water force is considered. This force is zero at the height of the water table, then increases linearly towards the base of the slice due to the increasing water pressure. For such a triangular distribution, the resultant force acts at one-third of the height. Thus, for the interslice normal water force acting on slice interface \\(i\\), the moment is: -\\[-{\boldsymbol{H}}\_{i} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] +\\[-{\boldsymbol{H}}\_{i}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] The moment for the interslice normal water force acting on slice interface \\(i-1\\) is: -\\[{\boldsymbol{H}}\_{i-1} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i-1}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] +\\[{\boldsymbol{H}}\_{i-1}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i-1}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)\\] The interslice shear force at slice interface \\(i\\) tends to rotate in the clockwise direction, and the length of the moment arm is the length from the slice edge to the slice midpoint, equivalent to half of the width of the slice, so the moment is: -\\[{\boldsymbol{X}}\_{i} \frac{{\boldsymbol{b}}\_{i}}{2}\\] +\\[{\boldsymbol{X}}\_{i}\\,\frac{{\boldsymbol{b}}\_{i}}{2}\\] The interslice shear force at slice interface \\(i-1\\) also tends to rotate in the clockwise direction, and has the same length of the moment arm, so the moment is: -\\[{\boldsymbol{X}}\_{i-1} \frac{{\boldsymbol{b}}\_{i}}{2}\\] +\\[{\boldsymbol{X}}\_{i-1}\\,\frac{{\boldsymbol{b}}\_{i}}{2}\\] -Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is \\({K\_{\text{c}}} {\boldsymbol{W}}\_{i}\\) where \\({\boldsymbol{W}}\_{i}\\) can be expressed as \\(γ {\boldsymbol{b}}\_{i} y\\) using [GD:weight](./SecGDs.md#GD:weight) where \\(y\\) is the height of the segment under consideration. The corresponding length of the moment arm is \\(y\\), the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible ([A:Negligible-Effect-Surface-Slope-Seismic](./SecAssumps.md#assumpNESSS)). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative: +Seismic forces act over the entire height of the slice. For each horizontal segment of the slice, the seismic force is \\({K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}\\) where \\({\boldsymbol{W}}\_{i}\\) can be expressed as \\(γ\\,{\boldsymbol{b}}\_{i}\\,y\\) using [GD:weight](./SecGDs.md#GD:weight) where \\(y\\) is the height of the segment under consideration. The corresponding length of the moment arm is \\(y\\), the height from the base of the slice to the segment under consideration. In reality, the forces near the surface of the soil mass are slightly different due to the slope of the surface, but this difference is assumed to be negligible ([A:Negligible-Effect-Surface-Slope-Seismic](./SecAssumps.md#assumpNESSS)). The resultant moment from the forces on all of the segments with an equivalent resultant length of the moment arm is determined by taking the integral over the slice height. The forces tend to rotate in the counterclockwise direction, so the moment is negative: -\\[-\int\_{0}^{{\boldsymbol{h}}\_{i}}{{K\_{\text{c}}} γ {\boldsymbol{b}}\_{i} y}\\,dy\\] +\\[-\int\_{0}^{{\boldsymbol{h}}\_{i}}{{K\_{\text{c}}}\\,γ\\,{\boldsymbol{b}}\_{i}\\,y}\\,dy\\] Solving the definite integral yields: -\\[-{K\_{\text{c}}} γ {\boldsymbol{b}}\_{i} \frac{{\boldsymbol{h}}\_{i}^{2}}{2}\\] +\\[-{K\_{\text{c}}}\\,γ\\,{\boldsymbol{b}}\_{i}\\,\frac{{\boldsymbol{h}}\_{i}^{2}}{2}\\] -Using [GD:weight](./SecGDs.md#GD:weight) again to express \\(γ {\boldsymbol{b}}\_{i} {\boldsymbol{h}}\_{i}\\) as \\({\boldsymbol{W}}\_{i}\\), the moment is: +Using [GD:weight](./SecGDs.md#GD:weight) again to express \\(γ\\,{\boldsymbol{b}}\_{i}\\,{\boldsymbol{h}}\_{i}\\) as \\({\boldsymbol{W}}\_{i}\\), the moment is: -\\[-{K\_{\text{c}}} {\boldsymbol{W}}\_{i} \frac{{\boldsymbol{h}}\_{i}}{2}\\] +\\[-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}\\,\frac{{\boldsymbol{h}}\_{i}}{2}\\] The surface hydrostatic force acts into the midpoint of the surface of the slice ([A:Hydrostatic-Force-Slice-Midpoint](./SecAssumps.md#assumpHFSM)). Thus, the vertical component of the force acts directly towards the point of rotation, and has a moment of zero. The horizontal component of the force tends to rotate in a clockwise direction and the length of the moment arm is the entire height of the slice. Thus, the moment is: -\\[{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right) {\boldsymbol{h}}\_{i}\\] +\\[{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}\\] The external force again acts into the midpoint of the slice surface, so the vertical component does not contribute to the moment, and the length of the moment arm is again the entire height of the slice. The moment is: -\\[{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right) {\boldsymbol{h}}\_{i}\\] +\\[{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}\\] The base hydrostatic force and slice weight both act in the direction of the point of rotation ([A:Hydrostatic-Force-Slice-Midpoint](./SecAssumps.md#assumpHFSM)), therefore both have moments of zero. Thus, all of the moments have been determined. The moment equilibrium is then represented by the sum of all moments: -\\[0=-{\boldsymbol{G}}\_{i} \left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1} \left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+\frac{{\boldsymbol{b}}\_{i}}{2} \left({\boldsymbol{X}}\_{i}+{\boldsymbol{X}}\_{i-1}\right)+\frac{-{K\_{\text{c}}} {\boldsymbol{W}}\_{i} {\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right) {\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right) {\boldsymbol{h}}\_{i}\\] +\\[0=-{\boldsymbol{G}}\_{i}\\,\left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1}\\,\left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\left({\boldsymbol{X}}\_{i}+{\boldsymbol{X}}\_{i-1}\right)+\frac{-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}\\,{\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}\\]
@@ -234,7 +234,7 @@ The base hydrostatic force and slice weight both act in the direction of the poi |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Weight | |Units |\\({\text{N}}\\) | -|Equation |\\[W=V γ\\] | +|Equation |\\[W=V\\,γ\\] | |Description|
  • \\(W\\) is the weight (\\({\text{N}}\\))
  • \\(V\\) is the volume (\\({\text{m}^{3}}\\))
  • \\(γ\\) is the specific weight (\\(\frac{\text{N}}{\text{m}^{3}}\\))
| |Source |[Definition of Weight](https://en.wikipedia.org/wiki/Weight) | |RefBy |[GD:sliceWght](./SecGDs.md#GD:sliceWght) and [GD:momentEql](./SecGDs.md#GD:momentEql) | @@ -243,19 +243,19 @@ The base hydrostatic force and slice weight both act in the direction of the poi Under the influence of gravity, and assuming a 2D Cartesian coordinate system with down as positive, an object has an acceleration vector of: -\\[\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}0\\\\\boldsymbol{g} \boldsymbol{\hat{j}}\end{bmatrix}\\] +\\[\boldsymbol{a}\text{(}t\text{)}=\begin{bmatrix}0\\\\\boldsymbol{g}\\,\boldsymbol{\hat{j}}\end{bmatrix}\\] Since there is only one non-zero vector component, the scalar value \\(W\\) will be used for the weight. In this scenario, Newton's second law of motion from [TM:NewtonSecLawMot](./SecTMs.md#TM:NewtonSecLawMot) can be expressed as: -\\[W=m \boldsymbol{g}\\] +\\[W=m\\,\boldsymbol{g}\\] Mass can be expressed as density multiplied by volume, resulting in: -\\[W=ρ V \boldsymbol{g}\\] +\\[W=ρ\\,V\\,\boldsymbol{g}\\] Substituting specific weight as the product of density and gravitational acceleration yields: -\\[W=V γ\\] +\\[W=V\\,γ\\]
@@ -263,41 +263,41 @@ Substituting specific weight as the product of density and gravitational acceler
-|Refname |GD:sliceWght | -|:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| -|Label |Slice weight | -|Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i} \frac{1}{2} \begin{cases}\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right) {γ\_{\text{sat}}}, & {\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slope},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\gt{}{\boldsymbol{y}\_{\text{slope},i-1}}\\\\\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{wt},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{wt},i-1}}\right) {γ\_{\text{dry}}}+\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right) {γ\_{\text{sat}}}, & {\boldsymbol{y}\_{\text{slope},i}}\geq{}{\boldsymbol{y}\_{\text{wt},i}}\geq{}{\boldsymbol{y}\_{\text{slip},i}}\land{}{\boldsymbol{y}\_{\text{slope},i-1}}\geq{}{\boldsymbol{y}\_{\text{wt},i-1}}\geq{}{\boldsymbol{y}\_{\text{slip},i-1}}\\\\\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right) {γ\_{\text{dry}}}, & {\boldsymbol{y}\_{\text{wt},i}}\lt{}{\boldsymbol{y}\_{\text{slip},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\lt{}{\boldsymbol{y}\_{\text{slip},i-1}}\end{cases}\\]| -|Description|
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slope}}}\\) is the \\(y\\)-coordinates of the slope (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slip}}}\\) is the \\(y\\)-coordinates of the slip surface (\\({\text{m}}\\))
  • \\({γ\_{\text{sat}}}\\) is the soil saturated unit weight (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
  • \\({γ\_{\text{dry}}}\\) is the soil dry unit weight (\\(\frac{\text{N}}{\text{m}^{3}}\\))
| -|Notes |
  • This equation is based on the assumption that the surface and the base of a slice are straight lines ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)). The soil dry unit weight \\({γ\_{\text{dry}}}\\) and the soil saturated unit weight \\({γ\_{\text{sat}}}\\) are not indexed by \\(i\\) because the soil is assumed to be homogeneous, with constant soil properties throughout ([A:Soil-Layer-Homogeneous](./SecAssumps.md#assumpSLH)). \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB).
| -|Source |[fredlund1977](./SecReferences.md#fredlund1977) | -|RefBy |[GD:resShearWO](./SecGDs.md#GD:resShearWO), [GD:normForcEq](./SecGDs.md#GD:normForcEq), [GD:momentEql](./SecGDs.md#GD:momentEql), [GD:mobShearWO](./SecGDs.md#GD:mobShearWO), and [GD:bsShrFEq](./SecGDs.md#GD:bsShrFEq) | +|Refname |GD:sliceWght | +|:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +|Label |Slice weight | +|Units |\\(\frac{\text{N}}{\text{m}}\\) | +|Equation |\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i}\\,\frac{1}{2}\\,\begin{cases}\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\,{γ\_{\text{sat}}}, & {\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slope},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\gt{}{\boldsymbol{y}\_{\text{slope},i-1}}\\\\\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{wt},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{wt},i-1}}\right)\\,{γ\_{\text{dry}}}+\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\,{γ\_{\text{sat}}}, & {\boldsymbol{y}\_{\text{slope},i}}\geq{}{\boldsymbol{y}\_{\text{wt},i}}\geq{}{\boldsymbol{y}\_{\text{slip},i}}\land{}{\boldsymbol{y}\_{\text{slope},i-1}}\geq{}{\boldsymbol{y}\_{\text{wt},i-1}}\geq{}{\boldsymbol{y}\_{\text{slip},i-1}}\\\\\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\,{γ\_{\text{dry}}}, & {\boldsymbol{y}\_{\text{wt},i}}\lt{}{\boldsymbol{y}\_{\text{slip},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\lt{}{\boldsymbol{y}\_{\text{slip},i-1}}\end{cases}\\]| +|Description|
  • \\(\boldsymbol{W}\\) is the weights (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slope}}}\\) is the \\(y\\)-coordinates of the slope (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slip}}}\\) is the \\(y\\)-coordinates of the slip surface (\\({\text{m}}\\))
  • \\({γ\_{\text{sat}}}\\) is the soil saturated unit weight (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
  • \\({γ\_{\text{dry}}}\\) is the soil dry unit weight (\\(\frac{\text{N}}{\text{m}^{3}}\\))
| +|Notes |
  • This equation is based on the assumption that the surface and the base of a slice are straight lines ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)). The soil dry unit weight \\({γ\_{\text{dry}}}\\) and the soil saturated unit weight \\({γ\_{\text{sat}}}\\) are not indexed by \\(i\\) because the soil is assumed to be homogeneous, with constant soil properties throughout ([A:Soil-Layer-Homogeneous](./SecAssumps.md#assumpSLH)). \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB).
| +|Source |[fredlund1977](./SecReferences.md#fredlund1977) | +|RefBy |[GD:resShearWO](./SecGDs.md#GD:resShearWO), [GD:normForcEq](./SecGDs.md#GD:normForcEq), [GD:momentEql](./SecGDs.md#GD:momentEql), [GD:mobShearWO](./SecGDs.md#GD:mobShearWO), and [GD:bsShrFEq](./SecGDs.md#GD:bsShrFEq) | #### {#GD:sliceWghtDeriv} For the case where the water table is above the slope surface, the weights come from the weight of the saturated soil. Substituting values for saturated soil into the equation for weight from [GD:weight](./SecGDs.md#GD:weight) yields: -\\[{\boldsymbol{W}}\_{i}={\boldsymbol{V}\_{\text{sat},i}} {γ\_{\text{sat}}}\\] +\\[{\boldsymbol{W}}\_{i}={\boldsymbol{V}\_{\text{sat},i}}\\,{γ\_{\text{sat}}}\\] Due to [A:Plane-Strain-Conditions](./SecAssumps.md#assumpPSC), only two dimensions are considered, so the areas of saturated soil are considered instead of the volumes of saturated soil. Any given slice has a trapezoidal shape. The area of a trapezoid is the average of the lengths of the parallel sides multiplied by the length between the parallel sides. The parallel sides in this case are the interslice edges and the length between them is the width of the slice. Thus, the weights are defined as: -\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i} \frac{1}{2} \left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right) {γ\_{\text{sat}}}\\] +\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i}\\,\frac{1}{2}\\,\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\,{γ\_{\text{sat}}}\\] For the case where the water table is below the slip surface, the weights come from the weight of the dry soil. Substituting values for dry soil into the equation for weight from [GD:weight](./SecGDs.md#GD:weight) yields: -\\[{\boldsymbol{W}}\_{i}={\boldsymbol{V}\_{\text{dry},i}} {γ\_{\text{dry}}}\\] +\\[{\boldsymbol{W}}\_{i}={\boldsymbol{V}\_{\text{dry},i}}\\,{γ\_{\text{dry}}}\\] [A:Plane-Strain-Conditions](./SecAssumps.md#assumpPSC) again allows for two-dimensional analysis so the areas of dry soil are considered instead of the volumes of dry soil. The trapezoidal slice shape is the same as in the previous case, so the weights are defined as: -\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i} \frac{1}{2} \left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right) {γ\_{\text{dry}}}\\] +\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i}\\,\frac{1}{2}\\,\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\,{γ\_{\text{dry}}}\\] For the case where the water table is between the slope surface and slip surface, the weights are the sums of the weights of the dry portions and weights of the saturated portions of the soil. Substituting values for dry and saturated soil into the equation for weight from [GD:weight](./SecGDs.md#GD:weight) and adding them together yields: -\\[{\boldsymbol{W}}\_{i}={\boldsymbol{V}\_{\text{dry},i}} {γ\_{\text{dry}}}+{\boldsymbol{V}\_{\text{sat},i}} {γ\_{\text{sat}}}\\] +\\[{\boldsymbol{W}}\_{i}={\boldsymbol{V}\_{\text{dry},i}}\\,{γ\_{\text{dry}}}+{\boldsymbol{V}\_{\text{sat},i}}\\,{γ\_{\text{sat}}}\\] [A:Plane-Strain-Conditions](./SecAssumps.md#assumpPSC) again allows for two-dimensional analysis so the areas of dry soil and areas of saturated soil are considered instead of the volumes of dry soil and volumes of saturated soil. The water table is assumed to only intersect a slice surface or base at a slice edge ([A:Water-Intersects-Surface-Edge](./SecAssumps.md#assumpWISE), [A:Water-Intersects-Base-Edge](./SecAssumps.md#assumpWIBE)), so the dry and saturated portions each have trapezoidal shape. For the dry portion, the parallel sides of the trapezoid are the lengths between the slope surface and water table at the slice edges. For the saturated portion, the parallel sides of the trapezoid are the lengths between the water table and slip surface at the slice edges. Thus, the weights are defined as: -\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i} \frac{1}{2} \left(\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{wt},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{wt},i-1}}\right) {γ\_{\text{dry}}}+\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right) {γ\_{\text{sat}}}\right)\\] +\\[{\boldsymbol{W}}\_{i}={\boldsymbol{b}}\_{i}\\,\frac{1}{2}\\,\left(\left({\boldsymbol{y}\_{\text{slope},i}}-{\boldsymbol{y}\_{\text{wt},i}}+{\boldsymbol{y}\_{\text{slope},i-1}}-{\boldsymbol{y}\_{\text{wt},i-1}}\right)\\,{γ\_{\text{dry}}}+\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\,{γ\_{\text{sat}}}\right)\\]
@@ -309,7 +309,7 @@ For the case where the water table is between the slope surface and slip surface |:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Hydrostatic pressure | |Units |\\({\text{Pa}}\\) | -|Equation |\\[p=γ h\\] | +|Equation |\\[p=γ\\,h\\] | |Description|
  • \\(p\\) is the pressure (\\({\text{Pa}}\\))
  • \\(γ\\) is the specific weight (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\(h\\) is the height (\\({\text{m}}\\))
| |Notes |
  • This equation is derived from Bernoulli's equation for a slow moving fluid through a porous material.
| |Source |[Definition of Pressure](https://en.wikipedia.org/wiki/Pressure) | @@ -321,29 +321,29 @@ For the case where the water table is between the slope surface and slip surface
-|Refname |GD:baseWtrF | -|:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| -|Label |Base hydrostatic force | -|Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{U}\_{\text{b},i}}={\boldsymbol{L}\_{b,i}} {γ\_{w}} \frac{1}{2} \begin{cases}{\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}, & {\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slip},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\gt{}{\boldsymbol{y}\_{\text{slip},i-1}}\\\\0, & {\boldsymbol{y}\_{\text{wt},i}}\leq{}{\boldsymbol{y}\_{\text{slip},i}}\land{}{\boldsymbol{y}\_{\text{wt},i-1}}\leq{}{\boldsymbol{y}\_{\text{slip},i-1}}\end{cases}\\] | -|Description|
  • \\({\boldsymbol{U}\_{\text{b}}}\\) is the base hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{L}\_{b}}\\) is the total base lengths of slices (\\({\text{m}}\\))
  • \\({γ\_{w}}\\) is the unit weight of water (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slip}}}\\) is the \\(y\\)-coordinates of the slip surface (\\({\text{m}}\\))
| -|Notes |
  • This equation is based on the assumption that the base of a slice is a straight line ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)). \\({\boldsymbol{L}\_{b}}\\) is defined in [DD:lengthLb](./SecDDs.md#DD:lengthLb).
| -|Source |[fredlund1977](./SecReferences.md#fredlund1977) | -|RefBy |[GD:resShearWO](./SecGDs.md#GD:resShearWO), [GD:effNormF](./SecGDs.md#GD:effNormF), and [GD:baseWtrF](./SecGDs.md#GD:baseWtrF) | +|Refname |GD:baseWtrF | +|:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +|Label |Base hydrostatic force | +|Units |\\(\frac{\text{N}}{\text{m}}\\) | +|Equation |\\[{\boldsymbol{U}\_{\text{b},i}}={\boldsymbol{L}\_{b,i}}\\,{γ\_{w}}\\,\frac{1}{2}\\,\begin{cases}{\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}, & {\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slip},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\gt{}{\boldsymbol{y}\_{\text{slip},i-1}}\\\\0, & {\boldsymbol{y}\_{\text{wt},i}}\leq{}{\boldsymbol{y}\_{\text{slip},i}}\land{}{\boldsymbol{y}\_{\text{wt},i-1}}\leq{}{\boldsymbol{y}\_{\text{slip},i-1}}\end{cases}\\]| +|Description|
  • \\({\boldsymbol{U}\_{\text{b}}}\\) is the base hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{L}\_{b}}\\) is the total base lengths of slices (\\({\text{m}}\\))
  • \\({γ\_{w}}\\) is the unit weight of water (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slip}}}\\) is the \\(y\\)-coordinates of the slip surface (\\({\text{m}}\\))
| +|Notes |
  • This equation is based on the assumption that the base of a slice is a straight line ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)). \\({\boldsymbol{L}\_{b}}\\) is defined in [DD:lengthLb](./SecDDs.md#DD:lengthLb).
| +|Source |[fredlund1977](./SecReferences.md#fredlund1977) | +|RefBy |[GD:resShearWO](./SecGDs.md#GD:resShearWO), [GD:effNormF](./SecGDs.md#GD:effNormF), and [GD:baseWtrF](./SecGDs.md#GD:baseWtrF) | #### {#GD:baseWtrFDeriv} The base hydrostatic forces come from the hydrostatic pressure exerted by the water above the base of each slice. The equation for hydrostatic pressure from [GD:hsPressure](./SecGDs.md#GD:hsPressure) is: -\\[p=γ h\\] +\\[p=γ\\,h\\] The specific weight in this case is the unit weight of water \\({γ\_{w}}\\). The height in this case is the height from the slice base to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint ([A:Hydrostatic-Force-Slice-Midpoint](./SecAssumps.md#assumpHFSM)). The height at the midpoint is the average of the height at slice interface \\(i\\) and the height at slice interface \\(i-1\\): -\\[\frac{1}{2} \left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\] +\\[\frac{1}{2}\\,\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\] Due to [A:Plane-Strain-Conditions](./SecAssumps.md#assumpPSC), only two dimensions are considered, so the base hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to base hydrostatic forces by multiplying by the corresponding length of the slice base \\({\boldsymbol{L}\_{b,i}}\\), assuming the water table does not intersect a slice base except at a slice edge ([A:Water-Intersects-Base-Edge](./SecAssumps.md#assumpWIBE)). Thus, in the case where the height of the water table is above the height of the slip surface, the base hydrostatic forces are defined as: -\\[{\boldsymbol{U}\_{\text{b},i}}={\boldsymbol{L}\_{b,i}} {γ\_{w}} \frac{1}{2} \left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\] +\\[{\boldsymbol{U}\_{\text{b},i}}={\boldsymbol{L}\_{b,i}}\\,{γ\_{w}}\\,\frac{1}{2}\\,\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slip},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slip},i-1}}\right)\\] This equation is the non-zero case of [GD:baseWtrF](./SecGDs.md#GD:baseWtrF). The zero case is when the height of the water table is below the height of the slip surface, so there is no hydrostatic force. @@ -353,28 +353,28 @@ This equation is the non-zero case of [GD:baseWtrF](./SecGDs.md#GD:baseWtrF). Th
-|Refname |GD:srfWtrF | -|:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| -|Label |Surface hydrostatic force | -|Units |\\(\frac{\text{N}}{\text{m}}\\) | -|Equation |\\[{\boldsymbol{U}\_{\text{g},i}}={\boldsymbol{L}\_{s,i}} {γ\_{w}} \frac{1}{2} \begin{cases}{\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slope},i-1}}, & {\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slope},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\gt{}{\boldsymbol{y}\_{\text{slope},i-1}}\\\\0, & {\boldsymbol{y}\_{\text{wt},i}}\leq{}{\boldsymbol{y}\_{\text{slope},i}}\land{}{\boldsymbol{y}\_{\text{wt},i-1}}\leq{}{\boldsymbol{y}\_{\text{slope},i-1}}\end{cases}\\]| -|Description|
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{L}\_{s}}\\) is the surface lengths of slices (\\({\text{m}}\\))
  • \\({γ\_{w}}\\) is the unit weight of water (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slope}}}\\) is the \\(y\\)-coordinates of the slope (\\({\text{m}}\\))
| -|Notes |
  • This equation is based on the assumption that the surface of a slice is a straight line ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)). \\({\boldsymbol{L}\_{s}}\\) is defined in [DD:lengthLs](./SecDDs.md#DD:lengthLs).
| -|Source |[fredlund1977](./SecReferences.md#fredlund1977) | -|RefBy |[GD:srfWtrF](./SecGDs.md#GD:srfWtrF), [GD:resShearWO](./SecGDs.md#GD:resShearWO), [IM:nrmShrForNum](./SecIMs.md#IM:nrmShrForNum), [GD:normForcEq](./SecGDs.md#GD:normForcEq), [GD:momentEql](./SecGDs.md#GD:momentEql), [GD:mobShearWO](./SecGDs.md#GD:mobShearWO), and [GD:bsShrFEq](./SecGDs.md#GD:bsShrFEq) | +|Refname |GD:srfWtrF | +|:----------|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +|Label |Surface hydrostatic force | +|Units |\\(\frac{\text{N}}{\text{m}}\\) | +|Equation |\\[{\boldsymbol{U}\_{\text{g},i}}={\boldsymbol{L}\_{s,i}}\\,{γ\_{w}}\\,\frac{1}{2}\\,\begin{cases}{\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slope},i-1}}, & {\boldsymbol{y}\_{\text{wt},i}}\gt{}{\boldsymbol{y}\_{\text{slope},i}}\lor{}{\boldsymbol{y}\_{\text{wt},i-1}}\gt{}{\boldsymbol{y}\_{\text{slope},i-1}}\\\\0, & {\boldsymbol{y}\_{\text{wt},i}}\leq{}{\boldsymbol{y}\_{\text{slope},i}}\land{}{\boldsymbol{y}\_{\text{wt},i-1}}\leq{}{\boldsymbol{y}\_{\text{slope},i-1}}\end{cases}\\]| +|Description|
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({\boldsymbol{L}\_{s}}\\) is the surface lengths of slices (\\({\text{m}}\\))
  • \\({γ\_{w}}\\) is the unit weight of water (\\(\frac{\text{N}}{\text{m}^{3}}\\))
  • \\({\boldsymbol{y}\_{\text{wt}}}\\) is the \\(y\\)-coordinates of the water table (\\({\text{m}}\\))
  • \\({\boldsymbol{y}\_{\text{slope}}}\\) is the \\(y\\)-coordinates of the slope (\\({\text{m}}\\))
| +|Notes |
  • This equation is based on the assumption that the surface of a slice is a straight line ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)). \\({\boldsymbol{L}\_{s}}\\) is defined in [DD:lengthLs](./SecDDs.md#DD:lengthLs).
| +|Source |[fredlund1977](./SecReferences.md#fredlund1977) | +|RefBy |[GD:srfWtrF](./SecGDs.md#GD:srfWtrF), [GD:resShearWO](./SecGDs.md#GD:resShearWO), [IM:nrmShrForNum](./SecIMs.md#IM:nrmShrForNum), [GD:normForcEq](./SecGDs.md#GD:normForcEq), [GD:momentEql](./SecGDs.md#GD:momentEql), [GD:mobShearWO](./SecGDs.md#GD:mobShearWO), and [GD:bsShrFEq](./SecGDs.md#GD:bsShrFEq) | #### {#GD:srfWtrFDeriv} The surface hydrostatic forces come from the hydrostatic pressure exerted by the water above the surface of each slice. The equation for hydrostatic pressure from [GD:hsPressure](./SecGDs.md#GD:hsPressure) is: -\\[p=γ h\\] +\\[p=γ\\,h\\] The specific weight in this case is the unit weight of water \\({γ\_{w}}\\). The height in this case is the height from the slice surface to the water table. This height is measured from the midpoint of the slice because the resultant hydrostatic force is assumed to act at the slice midpoint ([A:Hydrostatic-Force-Slice-Midpoint](./SecAssumps.md#assumpHFSM)). The height at the midpoint is the average of the height at slice interface \\(i\\) and the height at slice interface \\(i-1\\): -\\[\frac{1}{2} \left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slope},i-1}}\right)\\] +\\[\frac{1}{2}\\,\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slope},i-1}}\right)\\] Due to [A:Plane-Strain-Conditions](./SecAssumps.md#assumpPSC), only two dimensions are considered, so the surface hydrostatic forces are expressed as forces per meter. The pressures acting on the slices can thus be converted to surface hydrostatic forces by multiplying by the corresponding length of the slice surface \\({\boldsymbol{L}\_{s,i}}\\), assuming the water table does not intersect a slice surface except at a slice edge ([A:Water-Intersects-Surface-Edge](./SecAssumps.md#assumpWISE)). Thus, in the case where the height of the water table is above the height of the slope surface, the surface hydrostatic forces are defined as: -\\[{\boldsymbol{U}\_{\text{g},i}}={\boldsymbol{L}\_{s,i}} {γ\_{w}} \frac{1}{2} \left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slope},i-1}}\right)\\] +\\[{\boldsymbol{U}\_{\text{g},i}}={\boldsymbol{L}\_{s,i}}\\,{γ\_{w}}\\,\frac{1}{2}\\,\left({\boldsymbol{y}\_{\text{wt},i}}-{\boldsymbol{y}\_{\text{slope},i}}+{\boldsymbol{y}\_{\text{wt},i-1}}-{\boldsymbol{y}\_{\text{slope},i-1}}\right)\\] This equation is the non-zero case of [GD:srfWtrF](./SecGDs.md#GD:srfWtrF). The zero case is when the height of the water table is below the height of the slope surface, so there is no hydrostatic force. diff --git a/code/stable/ssp/SRS/mdBook/src/SecIMs.md b/code/stable/ssp/SRS/mdBook/src/SecIMs.md index 1ed2ec800c..82d62778bb 100644 --- a/code/stable/ssp/SRS/mdBook/src/SecIMs.md +++ b/code/stable/ssp/SRS/mdBook/src/SecIMs.md @@ -19,7 +19,7 @@ The Morgenstern-Price method is a vertical slice, limit equilibrium slope stabil |Output |\\({F\_{\text{S}}}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[{F\_{\text{S}}}=\frac{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{R}}\_{i} \displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{R}}\_{n}}{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{T}}\_{i} \displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{T}}\_{n}}\\] | +|Equation |\\[{F\_{\text{S}}}=\frac{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{R}}\_{i}\\,\displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{R}}\_{n}}{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{T}}\_{i}\\,\displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{T}}\_{n}}\\] | |Description |
  • \\({F\_{\text{S}}}\\) is the factor of safety (Unitless)
  • \\(\boldsymbol{R}\\) is the resistive shear forces without the influence of interslice forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{Ψ}\\) is the second function for incorporating interslice forces into shear force (Unitless)
  • \\(v\\) is the local index (Unitless)
  • \\(n\\) is the number of slices (Unitless)
  • \\(\boldsymbol{T}\\) is the mobilized shear forces without the influence of interslice forces (\\(\frac{\text{N}}{\text{m}}\\))
| |Notes |
  • \\(\boldsymbol{R}\\) is defined in [GD:resShearWO](./SecGDs.md#GD:resShearWO), \\(\boldsymbol{Ψ}\\) is defined in [DD:convertFunc2](./SecDDs.md#DD:convertFunc2), and \\(\boldsymbol{T}\\) is defined in [GD:mobShearWO](./SecGDs.md#GD:mobShearWO)
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | @@ -29,91 +29,91 @@ The Morgenstern-Price method is a vertical slice, limit equilibrium slope stabil The mobilized shear force defined in [GD:bsShrFEq](./SecGDs.md#GD:bsShrFEq) can be substituted into the definition of mobilized shear force based on the factor of safety, from [GD:mobShr](./SecGDs.md#GD:mobShr) yielding Equation (1) below: -\\[\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)=\frac{{\boldsymbol{N'}}\_{i} \tan\left(φ'\right)+c' {\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] +\\[\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)=\frac{{\boldsymbol{N'}}\_{i}\\,\tan\left(φ'\right)+c'\\,{\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] An expression for the effective normal forces, \\(\boldsymbol{N'}\\), can be derived by substituting the normal forces equilibrium from [GD:normForcEq](./SecGDs.md#GD:normForcEq) into the definition for effective normal forces from [GD:resShearWO](./SecGDs.md#GD:resShearWO). This results in Equation (2): -\\[{\boldsymbol{N'}}\_{i}=\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\\] +\\[{\boldsymbol{N'}}\_{i}=\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\\] Substituting Equation (2) into Equation (1) gives: -\\[\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)=\frac{\left(\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] +\\[\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)=\frac{\left(\left({\boldsymbol{W}}\_{i}-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right)\\,\tan\left(φ'\right)+c'\\,{\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] Since the interslice shear forces \\(\boldsymbol{X}\\) and interslice normal forces \\(\boldsymbol{G}\\) are unknown, they are separated from the other terms as follows: -\\[\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right) \sin\left({\boldsymbol{α}}\_{i}\right)=\frac{\left(\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \cos\left({\boldsymbol{ω}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}} {\boldsymbol{W}}\_{i}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right) \cos\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] +\\[\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)=\frac{\left(\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\cos\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right)\\,\tan\left(φ'\right)+c'\\,{\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] Applying assumptions [A:Seismic-Force](./SecAssumps.md#assumpSF) and [A:Surface-Load](./SecAssumps.md#assumpSL), which state that the seismic coefficient and the external forces, respectively, are zero, allows for further simplification as shown below: -\\[\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right) \sin\left({\boldsymbol{α}}\_{i}\right)=\frac{\left(\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}} \cos\left({\boldsymbol{β}}\_{i}\right)\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)\right) \sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right) \cos\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right) \tan\left(φ'\right)+c' {\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] +\\[\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)=\frac{\left(\left({\boldsymbol{W}}\_{i}+{\boldsymbol{U}\_{\text{g},i}}\\,\cos\left({\boldsymbol{β}}\_{i}\right)\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{H}}\_{i}+{\boldsymbol{H}}\_{i-1}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)-{\boldsymbol{U}\_{\text{b},i}}\right)\\,\tan\left(φ'\right)+c'\\,{\boldsymbol{L}\_{b,i}}}{{F\_{\text{S}}}}\\] The definitions of [GD:resShearWO](./SecGDs.md#GD:resShearWO) and [GD:mobShearWO](./SecGDs.md#GD:mobShearWO) are present in this equation, and thus can be replaced by \\({\boldsymbol{R}}\_{i}\\) and \\({\boldsymbol{T}}\_{i}\\), respectively: -\\[{\boldsymbol{T}}\_{i}+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \cos\left({\boldsymbol{α}}\_{i}\right)=\frac{{\boldsymbol{R}}\_{i}+\left(\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \sin\left({\boldsymbol{α}}\_{i}\right)\right) \tan\left(φ'\right)}{{F\_{\text{S}}}}\\] +\\[{\boldsymbol{T}}\_{i}+\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)=\frac{{\boldsymbol{R}}\_{i}+\left(\left(-{\boldsymbol{X}}\_{i-1}+{\boldsymbol{X}}\_{i}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)\right)\\,\tan\left(φ'\right)}{{F\_{\text{S}}}}\\] The interslice shear forces \\(\boldsymbol{X}\\) can be expressed in terms of the interslice normal forces \\(\boldsymbol{G}\\) using [A:Interslice-Norm-Shear-Forces-Linear](./SecAssumps.md#assumpINSFL) and [GD:normShrR](./SecGDs.md#GD:normShrR), resulting in: -\\[{\boldsymbol{T}}\_{i}+\left(-λ {\boldsymbol{f}}\_{i-1} {\boldsymbol{G}}\_{i-1}+λ {\boldsymbol{f}}\_{i} {\boldsymbol{G}}\_{i}\right) \sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \cos\left({\boldsymbol{α}}\_{i}\right)=\frac{{\boldsymbol{R}}\_{i}+\left(\left(-λ {\boldsymbol{f}}\_{i-1} {\boldsymbol{G}}\_{i-1}+λ {\boldsymbol{f}}\_{i} {\boldsymbol{G}}\_{i}\right) \cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right) \sin\left({\boldsymbol{α}}\_{i}\right)\right) \tan\left(φ'\right)}{{F\_{\text{S}}}}\\] +\\[{\boldsymbol{T}}\_{i}+\left(-λ\\,{\boldsymbol{f}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}+λ\\,{\boldsymbol{f}}\_{i}\\,{\boldsymbol{G}}\_{i}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)-\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)=\frac{{\boldsymbol{R}}\_{i}+\left(\left(-λ\\,{\boldsymbol{f}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}+λ\\,{\boldsymbol{f}}\_{i}\\,{\boldsymbol{G}}\_{i}\right)\\,\cos\left({\boldsymbol{α}}\_{i}\right)+\left(-{\boldsymbol{G}}\_{i}+{\boldsymbol{G}}\_{i-1}\right)\\,\sin\left({\boldsymbol{α}}\_{i}\right)\right)\\,\tan\left(φ'\right)}{{F\_{\text{S}}}}\\] Rearranging yields the following: -\\[{\boldsymbol{G}}\_{i} \left(\left(λ {\boldsymbol{f}}\_{i} \cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\boldsymbol{f}}\_{i} \sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right) {F\_{\text{S}}}\right)={\boldsymbol{G}}\_{i-1} \left(\left(λ {\boldsymbol{f}}\_{i-1} \cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right) \tan\left(φ'\right)-\left(λ {\boldsymbol{f}}\_{i-1} \sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right) {F\_{\text{S}}}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}\\] +\\[{\boldsymbol{G}}\_{i}\\,\left(\left(λ\\,{\boldsymbol{f}}\_{i}\\,\cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right)\\,\tan\left(φ'\right)-\left(λ\\,{\boldsymbol{f}}\_{i}\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right)\\,{F\_{\text{S}}}\right)={\boldsymbol{G}}\_{i-1}\\,\left(\left(λ\\,{\boldsymbol{f}}\_{i-1}\\,\cos\left({\boldsymbol{α}}\_{i}\right)-\sin\left({\boldsymbol{α}}\_{i}\right)\right)\\,\tan\left(φ'\right)-\left(λ\\,{\boldsymbol{f}}\_{i-1}\\,\sin\left({\boldsymbol{α}}\_{i}\right)+\cos\left({\boldsymbol{α}}\_{i}\right)\right)\\,{F\_{\text{S}}}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}\\] The definitions for \\(\boldsymbol{Φ}\\) and \\(\boldsymbol{Ψ}\\) from [DD:convertFunc1](./SecDDs.md#DD:convertFunc1) and [DD:convertFunc2](./SecDDs.md#DD:convertFunc2) simplify the above to Equation (3): -\\[{\boldsymbol{G}}\_{i} {\boldsymbol{Φ}}\_{i}={\boldsymbol{Ψ}}\_{i-1} {\boldsymbol{G}}\_{i-1} {\boldsymbol{Φ}}\_{i-1}+{F\_{\text{S}}} {\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}\\] +\\[{\boldsymbol{G}}\_{i}\\,{\boldsymbol{Φ}}\_{i}={\boldsymbol{Ψ}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}\\,{\boldsymbol{Φ}}\_{i-1}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}\\] Versions of Equation (3) instantiated for slices 1 to \\(n\\) are shown below: -\\[{\boldsymbol{G}}\_{1} {\boldsymbol{Φ}}\_{1}={\boldsymbol{Ψ}}\_{0} {\boldsymbol{G}}\_{0} {\boldsymbol{Φ}}\_{0}+{F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\\] +\\[{\boldsymbol{G}}\_{1}\\,{\boldsymbol{Φ}}\_{1}={\boldsymbol{Ψ}}\_{0}\\,{\boldsymbol{G}}\_{0}\\,{\boldsymbol{Φ}}\_{0}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\\] -\\[{\boldsymbol{G}}\_{2} {\boldsymbol{Φ}}\_{2}={\boldsymbol{Ψ}}\_{1} {\boldsymbol{G}}\_{1} {\boldsymbol{Φ}}\_{1}+{F\_{\text{S}}} {\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\\] +\\[{\boldsymbol{G}}\_{2}\\,{\boldsymbol{Φ}}\_{2}={\boldsymbol{Ψ}}\_{1}\\,{\boldsymbol{G}}\_{1}\\,{\boldsymbol{Φ}}\_{1}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\\] -\\[{\boldsymbol{G}}\_{3} {\boldsymbol{Φ}}\_{3}={\boldsymbol{Ψ}}\_{2} {\boldsymbol{G}}\_{2} {\boldsymbol{Φ}}\_{2}+{F\_{\text{S}}} {\boldsymbol{T}}\_{3}-{\boldsymbol{R}}\_{3}\\] +\\[{\boldsymbol{G}}\_{3}\\,{\boldsymbol{Φ}}\_{3}={\boldsymbol{Ψ}}\_{2}\\,{\boldsymbol{G}}\_{2}\\,{\boldsymbol{Φ}}\_{2}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{3}-{\boldsymbol{R}}\_{3}\\] ... -\\[{\boldsymbol{G}}\_{n-2} {\boldsymbol{Φ}}\_{n-2}={\boldsymbol{Ψ}}\_{n-3} {\boldsymbol{G}}\_{n-3} {\boldsymbol{Φ}}\_{n-3}+{F\_{\text{S}}} {\boldsymbol{T}}\_{n-2}-{\boldsymbol{R}}\_{n-2}\\] +\\[{\boldsymbol{G}}\_{n-2}\\,{\boldsymbol{Φ}}\_{n-2}={\boldsymbol{Ψ}}\_{n-3}\\,{\boldsymbol{G}}\_{n-3}\\,{\boldsymbol{Φ}}\_{n-3}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-2}-{\boldsymbol{R}}\_{n-2}\\] -\\[{\boldsymbol{G}}\_{n-1} {\boldsymbol{Φ}}\_{n-1}={\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{G}}\_{n-2} {\boldsymbol{Φ}}\_{n-2}+{F\_{\text{S}}} {\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\\] +\\[{\boldsymbol{G}}\_{n-1}\\,{\boldsymbol{Φ}}\_{n-1}={\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{G}}\_{n-2}\\,{\boldsymbol{Φ}}\_{n-2}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\\] -\\[{\boldsymbol{G}}\_{n} {\boldsymbol{Φ}}\_{n}={\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{G}}\_{n-1} {\boldsymbol{Φ}}\_{n-1}+{F\_{\text{S}}} {\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}\\] +\\[{\boldsymbol{G}}\_{n}\\,{\boldsymbol{Φ}}\_{n}={\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{G}}\_{n-1}\\,{\boldsymbol{Φ}}\_{n-1}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}\\] Applying [A:Edge-Slices](./SecAssumps.md#assumpES), which says that \\({\boldsymbol{G}}\_{0}\\) and \\({\boldsymbol{G}}\_{n}\\) are zero, results in the following special cases: Equation (8) for the first slice: -\\[{\boldsymbol{G}}\_{1} {\boldsymbol{Φ}}\_{1}={F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\\] +\\[{\boldsymbol{G}}\_{1}\\,{\boldsymbol{Φ}}\_{1}={F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\\] and Equation (9) for the \\(n\\)th slice: -\\[-\left(\frac{{F\_{\text{S}}} {\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}}{{\boldsymbol{Ψ}}\_{n-1}}\right)={\boldsymbol{G}}\_{n-1} {\boldsymbol{Φ}}\_{n-1}\\] +\\[-\left(\frac{{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}}{{\boldsymbol{Ψ}}\_{n-1}}\right)={\boldsymbol{G}}\_{n-1}\\,{\boldsymbol{Φ}}\_{n-1}\\] Substituting Equation (8) into Equation (4) yields Equation (10): -\\[{\boldsymbol{G}}\_{2} {\boldsymbol{Φ}}\_{2}={\boldsymbol{Ψ}}\_{1} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\\] +\\[{\boldsymbol{G}}\_{2}\\,{\boldsymbol{Φ}}\_{2}={\boldsymbol{Ψ}}\_{1}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\\] which can be substituted into Equation (5) to get Equation (11): -\\[{\boldsymbol{G}}\_{3} {\boldsymbol{Φ}}\_{3}={\boldsymbol{Ψ}}\_{2} \left({\boldsymbol{Ψ}}\_{1} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{3}-{\boldsymbol{R}}\_{3}\\] +\\[{\boldsymbol{G}}\_{3}\\,{\boldsymbol{Φ}}\_{3}={\boldsymbol{Ψ}}\_{2}\\,\left({\boldsymbol{Ψ}}\_{1}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{3}-{\boldsymbol{R}}\_{3}\\] and so on until Equation (12) is obtained from Equation (7): -\\[{\boldsymbol{G}}\_{n-1} {\boldsymbol{Φ}}\_{n-1}={\boldsymbol{Ψ}}\_{n-2} \left({\boldsymbol{Ψ}}\_{n-3} \left({\boldsymbol{Ψ}}\_{1} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{n-2}-{\boldsymbol{R}}\_{n-2}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\\] +\\[{\boldsymbol{G}}\_{n-1}\\,{\boldsymbol{Φ}}\_{n-1}={\boldsymbol{Ψ}}\_{n-2}\\,\left({\boldsymbol{Ψ}}\_{n-3}\\,\left({\boldsymbol{Ψ}}\_{1}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-2}-{\boldsymbol{R}}\_{n-2}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\\] Equation (9) can then be substituted into the left-hand side of Equation (12), resulting in: -\\[-\left(\frac{{F\_{\text{S}}} {\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}}{{\boldsymbol{Ψ}}\_{n-1}}\right)={\boldsymbol{Ψ}}\_{n-2} \left({\boldsymbol{Ψ}}\_{n-3} \left({\boldsymbol{Ψ}}\_{1} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{n-2}-{\boldsymbol{R}}\_{n-2}\right)+{F\_{\text{S}}} {\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\\] +\\[-\left(\frac{{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}}{{\boldsymbol{Ψ}}\_{n-1}}\right)={\boldsymbol{Ψ}}\_{n-2}\\,\left({\boldsymbol{Ψ}}\_{n-3}\\,\left({\boldsymbol{Ψ}}\_{1}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-2}-{\boldsymbol{R}}\_{n-2}\right)+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\\] This can be rearranged by multiplying both sides by \\({\boldsymbol{Ψ}}\_{n-1}\\) and then distributing the multiplication of each \\(\boldsymbol{Ψ}\\) over addition to obtain: -\\[-\left({F\_{\text{S}}} {\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}\right)={\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{Ψ}}\_{1} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{Ψ}}\_{2} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{\boldsymbol{Ψ}}\_{n-1} \left({F\_{\text{S}}} {\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\right)\\] +\\[-\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n}-{\boldsymbol{R}}\_{n}\right)={\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{Ψ}}\_{1}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}\right)+{\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{Ψ}}\_{2}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{2}-{\boldsymbol{R}}\_{2}\right)+{\boldsymbol{Ψ}}\_{n-1}\\,\left({F\_{\text{S}}}\\,{\boldsymbol{T}}\_{n-1}-{\boldsymbol{R}}\_{n-1}\right)\\] The multiplication of the \\(\boldsymbol{Ψ}\\) terms can be further distributed over the subtractions, resulting in the equation having terms that each either contain an \\(\boldsymbol{R}\\) or a \\(\boldsymbol{T}\\). The equation can then be rearranged so terms containing an \\(\boldsymbol{R}\\) are on one side of the equality, and terms containing a \\(\boldsymbol{T}\\) are on the other. The multiplication by the factor of safety is common to all of the \\(\boldsymbol{T}\\) terms, and thus can be factored out, resulting in: -\\[{F\_{\text{S}}} \left({\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{Ψ}}\_{1} {\boldsymbol{T}}\_{1}+{\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{Ψ}}\_{2} {\boldsymbol{T}}\_{2}+{\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{T}}\_{n-1}+{\boldsymbol{T}}\_{n}\right)={\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{Ψ}}\_{1} {\boldsymbol{R}}\_{1}+{\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{Ψ}}\_{n-2} {\boldsymbol{Ψ}}\_{2} {\boldsymbol{R}}\_{2}+{\boldsymbol{Ψ}}\_{n-1} {\boldsymbol{R}}\_{n-1}+{\boldsymbol{R}}\_{n}\\] +\\[{F\_{\text{S}}}\\,\left({\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{Ψ}}\_{1}\\,{\boldsymbol{T}}\_{1}+{\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{Ψ}}\_{2}\\,{\boldsymbol{T}}\_{2}+{\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{T}}\_{n-1}+{\boldsymbol{T}}\_{n}\right)={\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{Ψ}}\_{1}\\,{\boldsymbol{R}}\_{1}+{\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{Ψ}}\_{n-2}\\,{\boldsymbol{Ψ}}\_{2}\\,{\boldsymbol{R}}\_{2}+{\boldsymbol{Ψ}}\_{n-1}\\,{\boldsymbol{R}}\_{n-1}+{\boldsymbol{R}}\_{n}\\] Isolating the factor of safety on the left-hand side and using compact notation for the products and sums yields Equation (13), which can also be seen in [IM:fctSfty](./SecIMs.md#IM:fctSfty): -\\[{F\_{\text{S}}}=\frac{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{R}}\_{i} \displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{R}}\_{n}}{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{T}}\_{i} \displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{T}}\_{n}}\\] +\\[{F\_{\text{S}}}=\frac{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{R}}\_{i}\\,\displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{R}}\_{n}}{\displaystyle\sum\_{i=1}^{n-1}{{\boldsymbol{T}}\_{i}\\,\displaystyle\prod\_{v=i}^{n-1}{{\boldsymbol{Ψ}}\_{v}}}+{\boldsymbol{T}}\_{n}}\\] \\({F\_{\text{S}}}\\) depends on the unknowns \\(λ\\) ([IM:nrmShrFor](./SecIMs.md#IM:nrmShrFor)) and \\(\boldsymbol{G}\\) ([IM:intsliceFs](./SecIMs.md#IM:intsliceFs)). @@ -140,19 +140,19 @@ Isolating the factor of safety on the left-hand side and using compact notation From the moment equilibrium of [GD:momentEql](./SecGDs.md#GD:momentEql) with the primary assumption for the Morgenstern-Price method of [A:Interslice-Norm-Shear-Forces-Linear](./SecAssumps.md#assumpINSFL) and associated definition [GD:normShrR](./SecGDs.md#GD:normShrR), Equation (14) can be derived: -\\[0=-{\boldsymbol{G}}\_{i} \left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1} \left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+λ \frac{{\boldsymbol{b}}\_{i}}{2} \left({\boldsymbol{G}}\_{i} {\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1} {\boldsymbol{f}}\_{i-1}\right)+\frac{-{K\_{\text{c}}} {\boldsymbol{W}}\_{i} {\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right) {\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right) {\boldsymbol{h}}\_{i}\\] +\\[0=-{\boldsymbol{G}}\_{i}\\,\left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1}\\,\left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+λ\\,\frac{{\boldsymbol{b}}\_{i}}{2}\\,\left({\boldsymbol{G}}\_{i}\\,{\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1}\\,{\boldsymbol{f}}\_{i-1}\right)+\frac{-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}\\,{\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}\\] Rearranging the equation in terms of \\(λ\\) leads to Equation (15): -\\[λ=\frac{-{\boldsymbol{G}}\_{i} \left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1} \left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+\frac{-{K\_{\text{c}}} {\boldsymbol{W}}\_{i} {\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right) {\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i} \sin\left({\boldsymbol{ω}}\_{i}\right) {\boldsymbol{h}}\_{i}}{-\frac{{\boldsymbol{b}}\_{i}}{2} \left({\boldsymbol{G}}\_{i} {\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1} {\boldsymbol{f}}\_{i-1}\right)}\\] +\\[λ=\frac{-{\boldsymbol{G}}\_{i}\\,\left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1}\\,\left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+\frac{-{K\_{\text{c}}}\\,{\boldsymbol{W}}\_{i}\\,{\boldsymbol{h}}\_{i}}{2}+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}+{\boldsymbol{Q}}\_{i}\\,\sin\left({\boldsymbol{ω}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}}{-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\left({\boldsymbol{G}}\_{i}\\,{\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1}\\,{\boldsymbol{f}}\_{i-1}\right)}\\] This equation can be simplified by applying assumptions [A:Seismic-Force](./SecAssumps.md#assumpSF) and [A:Surface-Load](./SecAssumps.md#assumpSL), which state that the seismic and external forces, respectively, are zero: -\\[λ=\frac{-{\boldsymbol{G}}\_{i} \left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1} \left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1} \left(\frac{1}{3} {\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2} \tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right) {\boldsymbol{h}}\_{i}}{-\frac{{\boldsymbol{b}}\_{i}}{2} \left({\boldsymbol{G}}\_{i} {\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1} {\boldsymbol{f}}\_{i-1}\right)}\\] +\\[λ=\frac{-{\boldsymbol{G}}\_{i}\\,\left({\boldsymbol{h}\_{\text{z},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{G}}\_{i-1}\\,\left({\boldsymbol{h}\_{\text{z},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)-{\boldsymbol{H}}\_{i}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i}}+\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{H}}\_{i-1}\\,\left(\frac{1}{3}\\,{\boldsymbol{h}\_{\text{z,w},i-1}}-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\tan\left({\boldsymbol{α}}\_{i}\right)\right)+{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)\\,{\boldsymbol{h}}\_{i}}{-\frac{{\boldsymbol{b}}\_{i}}{2}\\,\left({\boldsymbol{G}}\_{i}\\,{\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1}\\,{\boldsymbol{f}}\_{i-1}\right)}\\] Taking the summation of all slices, and applying [A:Edge-Slices](./SecAssumps.md#assumpES) to set \\({\boldsymbol{G}}\_{0}\\), \\({\boldsymbol{G}}\_{n}\\), \\({\boldsymbol{H}}\_{0}\\), and \\({\boldsymbol{H}}\_{n}\\) equal to zero, a general equation for the proportionality constant \\(λ\\) is developed in Equation (16), which combines [IM:nrmShrFor](./SecIMs.md#IM:nrmShrFor), [IM:nrmShrForNum](./SecIMs.md#IM:nrmShrForNum), and [IM:nrmShrForDen](./SecIMs.md#IM:nrmShrForDen): -\\[λ=\frac{\displaystyle\sum\_{i=1}^{n}{{\boldsymbol{b}}\_{i} \left({{\boldsymbol{F}\_{\text{x}}}^{\text{G}}}+{{\boldsymbol{F}\_{\text{x}}}^{\text{H}}}\right) \tan\left({\boldsymbol{α}}\_{i}\right)+{\boldsymbol{h}}\_{i} -2 {\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right)}}{\displaystyle\sum\_{i=1}^{n}{{\boldsymbol{b}}\_{i} \left({\boldsymbol{G}}\_{i} {\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1} {\boldsymbol{f}}\_{i-1}\right)}}\\] +\\[λ=\frac{\displaystyle\sum\_{i=1}^{n}{{\boldsymbol{b}}\_{i}\\,\left({{\boldsymbol{F}\_{\text{x}}}^{\text{G}}}+{{\boldsymbol{F}\_{\text{x}}}^{\text{H}}}\right)\\,\tan\left({\boldsymbol{α}}\_{i}\right)+{\boldsymbol{h}}\_{i}\\,-2\\,{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right)}}{\displaystyle\sum\_{i=1}^{n}{{\boldsymbol{b}}\_{i}\\,\left({\boldsymbol{G}}\_{i}\\,{\boldsymbol{f}}\_{i}+{\boldsymbol{G}}\_{i-1}\\,{\boldsymbol{f}}\_{i-1}\right)}}\\] Equation (16) for \\(λ\\) is a function of the unknown interslice normal forces \\(\boldsymbol{G}\\) ([IM:intsliceFs](./SecIMs.md#IM:intsliceFs)) which itself depends on the unknown factor of safety \\({F\_{\text{S}}}\\) ([IM:fctSfty](./SecIMs.md#IM:fctSfty)). @@ -169,7 +169,7 @@ Equation (16) for \\(λ\\) is a function of the unknown interslice normal forces |Output |\\({\boldsymbol{C}\_{\text{num}}}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[{\boldsymbol{C}\_{\text{num},i}}=\begin{cases}{\boldsymbol{b}}\_{1} \left({\boldsymbol{G}}\_{1}+{\boldsymbol{H}}\_{1}\right) \tan\left({\boldsymbol{α}}\_{1}\right), & i=1\\\\{\boldsymbol{b}}\_{i} \left({{\boldsymbol{F}\_{\text{x}}}^{\text{G}}}+{{\boldsymbol{F}\_{\text{x}}}^{\text{H}}}\right) \tan\left({\boldsymbol{α}}\_{i}\right)+\boldsymbol{h} -2 {\boldsymbol{U}\_{\text{g},i}} \sin\left({\boldsymbol{β}}\_{i}\right), & 2\leq{}i\leq{}n-1\\\\{\boldsymbol{b}}\_{n} \left({\boldsymbol{G}}\_{n-1}+{\boldsymbol{H}}\_{n-1}\right) \tan\left({\boldsymbol{α}}\_{n-1}\right), & i=n\end{cases}\\] | +|Equation |\\[{\boldsymbol{C}\_{\text{num},i}}=\begin{cases}{\boldsymbol{b}}\_{1}\\,\left({\boldsymbol{G}}\_{1}+{\boldsymbol{H}}\_{1}\right)\\,\tan\left({\boldsymbol{α}}\_{1}\right), & i=1\\\\{\boldsymbol{b}}\_{i}\\,\left({{\boldsymbol{F}\_{\text{x}}}^{\text{G}}}+{{\boldsymbol{F}\_{\text{x}}}^{\text{H}}}\right)\\,\tan\left({\boldsymbol{α}}\_{i}\right)+\boldsymbol{h}\\,-2\\,{\boldsymbol{U}\_{\text{g},i}}\\,\sin\left({\boldsymbol{β}}\_{i}\right), & 2\leq{}i\leq{}n-1\\\\{\boldsymbol{b}}\_{n}\\,\left({\boldsymbol{G}}\_{n-1}+{\boldsymbol{H}}\_{n-1}\right)\\,\tan\left({\boldsymbol{α}}\_{n-1}\right), & i=n\end{cases}\\] | |Description |
  • \\({\boldsymbol{C}\_{\text{num}}}\\) is the proportionality constant numerator (\\({\text{N}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{H}\\) is the interslice normal water forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{α}\\) is the base angles (\\({{}^{\circ}}\\))
  • \\({{\boldsymbol{F}\_{\text{x}}}^{\text{G}}}\\) is the sums of the interslice normal forces (\\({\text{N}}\\))
  • \\({{\boldsymbol{F}\_{\text{x}}}^{\text{H}}}\\) is the sums of the interslice normal water forces (\\({\text{N}}\\))
  • \\(\boldsymbol{h}\\) is the \\(y\\)-direction heights of slices (\\({\text{m}}\\))
  • \\({\boldsymbol{U}\_{\text{g}}}\\) is the surface hydrostatic forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{β}\\) is the surface angles (\\({{}^{\circ}}\\))
  • \\(n\\) is the number of slices (Unitless)
| |Notes |
  • \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB), \\(\boldsymbol{H}\\) is defined in [DD:intersliceWtrF](./SecDDs.md#DD:intersliceWtrF), \\(\boldsymbol{α}\\) is defined in [DD:angleA](./SecDDs.md#DD:angleA), \\(\boldsymbol{h}\\) is defined in [DD:slcHeight](./SecDDs.md#DD:slcHeight), \\({\boldsymbol{U}\_{\text{g}}}\\) is defined in [GD:srfWtrF](./SecGDs.md#GD:srfWtrF), and \\(\boldsymbol{β}\\) is defined in [DD:angleB](./SecDDs.md#DD:angleB).
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | @@ -192,7 +192,7 @@ See [IM:nrmShrFor](./SecIMs.md#IM:nrmShrFor) for the derivation of \\({\boldsymb |Output |\\({\boldsymbol{C}\_{\text{den}}}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[{\boldsymbol{C}\_{\text{den},i}}=\begin{cases}{\boldsymbol{b}}\_{1} {\boldsymbol{f}}\_{1} {\boldsymbol{G}}\_{1}, & i=1\\\\{\boldsymbol{b}}\_{i} \left({\boldsymbol{f}}\_{i} {\boldsymbol{G}}\_{i}+{\boldsymbol{f}}\_{i-1} {\boldsymbol{G}}\_{i-1}\right), & 2\leq{}i\leq{}n-1\\\\{\boldsymbol{b}}\_{n} {\boldsymbol{G}}\_{n-1} {\boldsymbol{f}}\_{n-1}, & i=n\end{cases}\\] | +|Equation |\\[{\boldsymbol{C}\_{\text{den},i}}=\begin{cases}{\boldsymbol{b}}\_{1}\\,{\boldsymbol{f}}\_{1}\\,{\boldsymbol{G}}\_{1}, & i=1\\\\{\boldsymbol{b}}\_{i}\\,\left({\boldsymbol{f}}\_{i}\\,{\boldsymbol{G}}\_{i}+{\boldsymbol{f}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}\right), & 2\leq{}i\leq{}n-1\\\\{\boldsymbol{b}}\_{n}\\,{\boldsymbol{G}}\_{n-1}\\,{\boldsymbol{f}}\_{n-1}, & i=n\end{cases}\\] | |Description |
  • \\({\boldsymbol{C}\_{\text{den}}}\\) is the proportionality constant denominator (\\({\text{N}}\\))
  • \\(i\\) is the index (Unitless)
  • \\(\boldsymbol{b}\\) is the base width of slices (\\({\text{m}}\\))
  • \\(\boldsymbol{f}\\) is the interslice normal to shear force ratio variation function (Unitless)
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(n\\) is the number of slices (Unitless)
| |Notes |
  • \\(\boldsymbol{b}\\) is defined in [DD:lengthB](./SecDDs.md#DD:lengthB) and \\(\boldsymbol{f}\\) is defined in [DD:ratioVariation](./SecDDs.md#DD:ratioVariation).
| |Source |[chen2005](./SecReferences.md#chen2005) and [karchewski2012](./SecReferences.md#karchewski2012) | @@ -215,7 +215,7 @@ See [IM:nrmShrFor](./SecIMs.md#IM:nrmShrFor) for the derivation of \\({\boldsymb |Output |\\(\boldsymbol{G}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[{\boldsymbol{G}}\_{i}=\begin{cases}\frac{{F\_{\text{S}}} {\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}}{{\boldsymbol{Φ}}\_{1}}, & i=1\\\\\frac{{\boldsymbol{Ψ}}\_{i-1} {\boldsymbol{G}}\_{i-1}+{F\_{\text{S}}} {\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}}{{\boldsymbol{Φ}}\_{i}}, & 2\leq{}i\leq{}n-1\\\\0, & i=0\lor{}i=n\end{cases}\\] | +|Equation |\\[{\boldsymbol{G}}\_{i}=\begin{cases}\frac{{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{1}-{\boldsymbol{R}}\_{1}}{{\boldsymbol{Φ}}\_{1}}, & i=1\\\\\frac{{\boldsymbol{Ψ}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}}{{\boldsymbol{Φ}}\_{i}}, & 2\leq{}i\leq{}n-1\\\\0, & i=0\lor{}i=n\end{cases}\\] | |Description |
  • \\(\boldsymbol{G}\\) is the interslice normal forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(i\\) is the index (Unitless)
  • \\({F\_{\text{S}}}\\) is the factor of safety (Unitless)
  • \\(\boldsymbol{T}\\) is the mobilized shear forces without the influence of interslice forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{R}\\) is the resistive shear forces without the influence of interslice forces (\\(\frac{\text{N}}{\text{m}}\\))
  • \\(\boldsymbol{Φ}\\) is the first function for incorporating interslice forces into shear force (Unitless)
  • \\(\boldsymbol{Ψ}\\) is the second function for incorporating interslice forces into shear force (Unitless)
  • \\(n\\) is the number of slices (Unitless)
| |Notes |
  • \\(\boldsymbol{T}\\) is defined in [GD:mobShearWO](./SecGDs.md#GD:mobShearWO), \\(\boldsymbol{R}\\) is defined in [GD:resShearWO](./SecGDs.md#GD:resShearWO), \\(\boldsymbol{Φ}\\) is defined in [DD:convertFunc1](./SecDDs.md#DD:convertFunc1), and \\(\boldsymbol{Ψ}\\) is defined in [DD:convertFunc2](./SecDDs.md#DD:convertFunc2).
| |Source |[chen2005](./SecReferences.md#chen2005) | @@ -225,11 +225,11 @@ See [IM:nrmShrFor](./SecIMs.md#IM:nrmShrFor) for the derivation of \\({\boldsymb This derivation is identical to the derivation for [IM:fctSfty](./SecIMs.md#IM:fctSfty) up until Equation (3) shown again below: -\\[{\boldsymbol{G}}\_{i} {\boldsymbol{Φ}}\_{i}={\boldsymbol{Ψ}}\_{i-1} {\boldsymbol{G}}\_{i-1} {\boldsymbol{Φ}}\_{i-1}+{F\_{\text{S}}} {\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}\\] +\\[{\boldsymbol{G}}\_{i}\\,{\boldsymbol{Φ}}\_{i}={\boldsymbol{Ψ}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}\\,{\boldsymbol{Φ}}\_{i-1}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}\\] A simple rearrangement of Equation (3) leads to Equation (17), also seen in [IM:intsliceFs](./SecIMs.md#IM:intsliceFs): -\\[{\boldsymbol{G}}\_{i}=\frac{{\boldsymbol{Ψ}}\_{i-1} {\boldsymbol{G}}\_{i-1}+{F\_{\text{S}}} {\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}}{{\boldsymbol{Φ}}\_{i}}\\] +\\[{\boldsymbol{G}}\_{i}=\frac{{\boldsymbol{Ψ}}\_{i-1}\\,{\boldsymbol{G}}\_{i-1}+{F\_{\text{S}}}\\,{\boldsymbol{T}}\_{i}-{\boldsymbol{R}}\_{i}}{{\boldsymbol{Φ}}\_{i}}\\] The cases shown in [IM:intsliceFs](./SecIMs.md#IM:intsliceFs) for when \\(i=0\\), \\(i=1\\), or \\(i=n\\) are derived by applying [A:Edge-Slices](./SecAssumps.md#assumpES), which says that \\({\boldsymbol{G}}\_{0}\\) and \\({\boldsymbol{G}}\_{n}\\) are zero, to Equation (17). \\(\boldsymbol{G}\\) depends on the unknowns \\({F\_{\text{S}}}\\) ([IM:fctSfty](./SecIMs.md#IM:fctSfty)) and \\(λ\\) ([IM:nrmShrFor](./SecIMs.md#IM:nrmShrFor)). diff --git a/code/stable/ssp/SRS/mdBook/src/SecTMs.md b/code/stable/ssp/SRS/mdBook/src/SecTMs.md index 7f2243036d..eeed927666 100644 --- a/code/stable/ssp/SRS/mdBook/src/SecTMs.md +++ b/code/stable/ssp/SRS/mdBook/src/SecTMs.md @@ -40,7 +40,7 @@ This section focuses on the general equations and laws that SSP is based on. |Refname |TM:mcShrStrgth | |:----------|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Mohr-Coulumb shear strength | -|Equation |\\[{τ^{\text{f}}}={σ\_{N}}' \tan\left(φ'\right)+c'\\] | +|Equation |\\[{τ^{\text{f}}}={σ\_{N}}'\\,\tan\left(φ'\right)+c'\\] | |Description|
  • \\({τ^{\text{f}}}\\) is the shear strength (\\({\text{Pa}}\\))
  • \\({σ\_{N}}'\\) is the effective normal stress (\\({\text{Pa}}\\))
  • \\(φ'\\) is the effective angle of friction (\\({{}^{\circ}}\\))
  • \\(c'\\) is the effective cohesion (\\({\text{Pa}}\\))
| |Notes |
  • In this model the shear strength \\({τ^{\text{f}}}\\) is proportional to the product of the effective normal stress \\({σ\_{N}}'\\) on the plane with its static friction in the angular form \\(\tan\left(φ'\right)\\). The \\({τ^{\text{f}}}\\) versus \\({σ\_{N}}'\\) relationship is not truly linear, but assuming the effective normal forces is strong enough, it can be approximated with a linear fit ([A:Surface-Base-Slice-between-Interslice-Straight-Lines](./SecAssumps.md#assumpSBSBISL)) where the effective cohesion \\(c'\\) represents the \\({τ^{\text{f}}}\\) intercept of the fitted line.
| |Source |[fredlund1977](./SecReferences.md#fredlund1977) | @@ -70,7 +70,7 @@ This section focuses on the general equations and laws that SSP is based on. |Refname |TM:NewtonSecLawMot | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's second law of motion | -|Equation |\\[\boldsymbol{F}=m \boldsymbol{a}\text{(}t\text{)}\\] | +|Equation |\\[\boldsymbol{F}=m\\,\boldsymbol{a}\text{(}t\text{)}\\] | |Description|
  • \\(\boldsymbol{F}\\) is the force (\\({\text{N}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(\boldsymbol{a}\text{(}t\text{)}\\) is the acceleration (\\(\frac{\text{m}}{\text{s}^{2}}\\))
| |Notes |
  • The net force \\(\boldsymbol{F}\\) on a body is proportional to the acceleration \\(\boldsymbol{a}\text{(}t\text{)}\\) of the body, where \\(m\\) denotes the mass of the body as the constant of proportionality.
| |Source |-- | diff --git a/code/stable/swhs/SRS/HTML/SWHS_SRS.html b/code/stable/swhs/SRS/HTML/SWHS_SRS.html index 4fd665a015..f57a170c58 100644 --- a/code/stable/swhs/SRS/HTML/SWHS_SRS.html +++ b/code/stable/swhs/SRS/HTML/SWHS_SRS.html @@ -1096,7 +1096,7 @@

Theoretical Models

@@ -1162,9 +1162,9 @@

Theoretical Models

@@ -1298,7 +1298,7 @@

Theoretical Models

- + @@ -1370,7 +1370,7 @@

General Definitions

@@ -1424,23 +1424,23 @@

Integrating TM:consThermE over a volume (V), we have:

- \[-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\] + \[-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

Applying Gauss's Divergence Theorem to the first term over the surface S of the volume, with q as the thermal flux vector for the surface and n̂ as a unit outward normal vector for a surface:

- \[-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\] + \[-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as:

- \[{q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\] + \[{q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

Where qin, qout, Ain, and Aout are explained in GD:rocTempSimp. The integral over the surface could be simplified because the thermal flux is assumed constant over Ain and Aout and 0 on all other surfaces. Outward flux is considered positive. Assuming ρ, C, and T are constant over the volume, which is true in our case by A:Constant-Water-Temp-Across-Tank, A:Temp-PCM-Constant-Across-Volume, A:Density-Water-PCM-Constant-over-Volume, and A:Specific-Heat-Energy-Constant-over-Volume, we have:

- \[ρ C V \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] + \[ρ\,C\,V\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]

Using the fact that ρ=m/V, Equation (2) can be written as:

- \[m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] + \[m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]
@@ -1464,7 +1464,7 @@

@@ -1533,7 +1533,7 @@

@@ -1608,7 +1608,7 @@

Data Definitions

- + @@ -1714,7 +1714,7 @@

Data Definitions

- + @@ -1770,7 +1770,7 @@

Data Definitions

@@ -1832,7 +1832,7 @@

Data Definitions

@@ -1896,7 +1896,7 @@

Data Definitions

@@ -1960,7 +1960,7 @@

Data Definitions

@@ -2083,7 +2083,9 @@

Data Definitions

- + @@ -2217,7 +2219,7 @@

Instance Models

@@ -2288,29 +2290,29 @@

Detailed derivation of the energy balance on water:

To find the rate of change of TW, we look at the energy balance on water. The volume being considered is the volume of water in the tank VW, which has mass mW and specific heat capacity, CW. Heat transfer occurs in the water from the heating coil as qC (GD:htFluxWaterFromCoil) and from the water into the PCM as qP (GD:htFluxPCMFromWater), over areas AC and AP, respectively. The thermal flux is constant over AC, since the temperature of the heating coil is assumed to not vary along its length (A:Temp-Heating-Coil-Constant-over-Length), and the thermal flux is constant over AP, since the temperature of the PCM is the same throughout its volume (A:Temp-PCM-Constant-Across-Volume) and the water is fully mixed (A:Constant-Water-Temp-Across-Tank). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (A:Perfect-Insulation-Tank). Since the assumption is made that no internal heat is generated (A:No-Internal-Heat-Generation-By-Water-PCM), g = 0. Therefore, the equation for GD:rocTempSimp can be written as:

- \[{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}} {A_{\text{C}}}-{q_{\text{P}}} {A_{\text{P}}}\] + \[{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}}\,{A_{\text{C}}}-{q_{\text{P}}}\,{A_{\text{P}}}\]

Using GD:htFluxWaterFromCoil for qC and GD:htFluxPCMFromWater for qP, this can be written as:

- \[{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)-{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\] + \[{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)-{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Dividing Equation (2) by mW CW, we obtain:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)-\frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)-\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Factoring the negative sign out of the second term of the right-hand side (RHS) of Equation (3) and multiplying it by hC AC / hC AC yields:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{C}}} {A_{\text{C}}}}{{h_{\text{C}}} {A_{\text{C}}}} \frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\,\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\]

Rearranging this equation gives us:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{P}}} {A_{\text{P}}}}{{h_{\text{C}}} {A_{\text{C}}}} \frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\,\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\]

By substituting τW (from DD:balanceDecayRate) and η (from DD:balanceDecayTime), this can be written as:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{η}{{τ_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{η}{{τ_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\]

Finally, factoring out \(\frac{1}{{τ_{\text{W}}}}\), we are left with the governing ODE for IM:eBalanceOnWtr:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}+η \left({T_{\text{P}}}-{T_{\text{W}}}\right)\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}+η\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\right)\]
@@ -2351,8 +2353,8 @@

Detailed derivation of the energy balance on water:

@@ -2428,19 +2430,19 @@

To find the rate of change of TP, we look at the energy balance on the PCM. The volume being considered is the volume of PCM (VP). The derivation that follows is initially for the solid PCM. The mass of phase change material is mP and the specific heat capacity of PCM as a solid is CPS. The heat flux into the PCM from water is qP (GD:htFluxPCMFromWater) over phase change material surface area AP. The thermal flux is constant over AP, since the temperature of the PCM is the same throughout its volume (A:Temp-PCM-Constant-Across-Volume) and the water is fully mixed (A:Constant-Water-Temp-Across-Tank). There is no heat flux output from the PCM. Assuming no volumetric heat generation per unit volume (A:No-Internal-Heat-Generation-By-Water-PCM), g = 0, the equation for GD:rocTempSimp can be written as:

- \[{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}} \frac{\,d{T_{\text{P}}}}{\,dt}={q_{\text{P}}} {A_{\text{P}}}\] + \[{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}\,\frac{\,d{T_{\text{P}}}}{\,dt}={q_{\text{P}}}\,{A_{\text{P}}}\]

Using GD:htFluxPCMFromWater for qP, this equation can be written as:

- \[{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}} \frac{\,d{T_{\text{P}}}}{\,dt}={h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\] + \[{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}\,\frac{\,d{T_{\text{P}}}}{\,dt}={h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Dividing by mP CPS we obtain:

- \[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\] + \[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

By substituting τPS (from DD:balanceSolidPCM), this can be written as:

- \[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{1}{{{τ_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right)\] + \[\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{1}{{{τ_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right)\]

Equation (4) applies for the solid PCM. In the case where all of the PCM is melted, the same derivation applies, except that CPS is replaced by CPL, and thus τPS is replaced by τPL. Although a small change in surface area would be expected with melting, this is not included, since the volume change of the PCM with melting is assumed to be negligible (A:Volume-Change-Melting-PCM-Negligible).

@@ -2485,7 +2487,7 @@

@@ -2573,8 +2575,8 @@

@@ -2894,11 +2896,11 @@

Properties of a Correct Solution

A correct solution must exhibit the law of conservation of energy. This means that the change in heat energy in the water should equal the difference between the total energy input from the heating coil and the energy output to the PCM. This can be shown as an equation by taking GD:htFluxWaterFromCoil and GD:htFluxPCMFromWater, multiplying each by their respective surface area of heat transfer, and integrating each over the simulation time, as follows:

- \[{E_{\text{W}}}=\int_{0}^{t}{{h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)}\,dt-\int_{0}^{t}{{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt\] + \[{E_{\text{W}}}=\int_{0}^{t}{{h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)}\,dt-\int_{0}^{t}{{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt\]

In addition, the change in heat energy in the PCM should equal the energy input to the PCM from the water. This can be expressed as

- \[{E_{\text{P}}}=\int_{0}^{t}{{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt\] + \[{E_{\text{P}}}=\int_{0}^{t}{{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt\]

Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as "sanity" checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than Ctol FR:Verify-Energy-Output-Follow-Conservation-of-Energy.

diff --git a/code/stable/swhs/SRS/Jupyter/SWHS_SRS.ipynb b/code/stable/swhs/SRS/Jupyter/SWHS_SRS.ipynb index 55123095ce..b8944dbc55 100644 --- a/code/stable/swhs/SRS/Jupyter/SWHS_SRS.ipynb +++ b/code/stable/swhs/SRS/Jupyter/SWHS_SRS.ipynb @@ -446,7 +446,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -505,7 +505,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -625,7 +625,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -688,7 +688,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -727,19 +727,19 @@ "\n", "\n", "Integrating [TM:consThermE](#TM:consThermE) over a volume ($V$), we have:\n", - "$$-\\int_{V}{∇\\cdot{}\\symbf{q}}\\,dV+\\int_{V}{g}\\,dV=\\int_{V}{ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", + "$$-\\int_{V}{∇\\cdot{}\\symbf{q}}\\,dV+\\int_{V}{g}\\,dV=\\int_{V}{ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", "\n", "Applying Gauss's Divergence Theorem to the first term over the surface $S$ of the volume, with $q$ as the thermal flux vector for the surface and $n̂$ as a unit outward normal vector for a surface:\n", - "$$-\\int_{S}{\\symbf{q}\\cdot{}\\symbf{\\hat{n}}}\\,dS+\\int_{V}{g}\\,dV=\\int_{V}{ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", + "$$-\\int_{S}{\\symbf{q}\\cdot{}\\symbf{\\hat{n}}}\\,dS+\\int_{V}{g}\\,dV=\\int_{V}{ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", "\n", "We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as:\n", - "$${q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V=\\int_{V}{ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", + "$${q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V=\\int_{V}{ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", "\n", "Where $q_in$, $q_out$, $A_in$, and $A_out$ are explained in [GD:rocTempSimp](#GD:rocTempSimp). The integral over the surface could be simplified because the thermal flux is assumed constant over $A_in$ and $A_out$ and $0$ on all other surfaces. Outward flux is considered positive. Assuming $ρ$, $C$, and $T$ are constant over the volume, which is true in our case by [A:Constant-Water-Temp-Across-Tank](#assumpCWTAT), [A:Temp-PCM-Constant-Across-Volume](#assumpTPCAV), [A:Density-Water-PCM-Constant-over-Volume](#assumpDWPCoV), and [A:Specific-Heat-Energy-Constant-over-Volume](#assumpSHECov), we have:\n", - "$$ρ C V \\frac{\\,dT}{\\,dt}={q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V$$\n", + "$$ρ\\,C\\,V\\,\\frac{\\,dT}{\\,dt}={q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V$$\n", "\n", "Using the fact that $ρ$=$m$/$V$, Equation (2) can be written as:\n", - "$$m C \\frac{\\,dT}{\\,dt}={q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V$$\n", + "$$m\\,C\\,\\frac{\\,dT}{\\,dt}={q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V$$\n", "
\n", "\n", "
Equation\n", - "$$p=γ h$$\n", + "$$p=γ\\,h$$\n", "
Equation\n", - "$${\\symbf{U}_{\\text{b},i}}={\\symbf{L}_{b,i}} {γ_{w}} \\frac{1}{2} \\begin{cases} {\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}, & {\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slip},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\gt{}{\\symbf{y}_{\\text{slip},i-1}}\\\\ 0, & {\\symbf{y}_{\\text{wt},i}}\\leq{}{\\symbf{y}_{\\text{slip},i}}\\land{}{\\symbf{y}_{\\text{wt},i-1}}\\leq{}{\\symbf{y}_{\\text{slip},i-1}} \\end{cases}$$\n", + "$${\\symbf{U}_{\\text{b},i}}={\\symbf{L}_{b,i}}\\,{γ_{w}}\\,\\frac{1}{2}\\,\\begin{cases} {\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slip},i-1}}, & {\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slip},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\gt{}{\\symbf{y}_{\\text{slip},i-1}}\\\\ 0, & {\\symbf{y}_{\\text{wt},i}}\\leq{}{\\symbf{y}_{\\text{slip},i}}\\land{}{\\symbf{y}_{\\text{wt},i-1}}\\leq{}{\\symbf{y}_{\\text{slip},i-1}} \\end{cases}$$\n", "
Equation\n", - "$${\\symbf{U}_{\\text{g},i}}={\\symbf{L}_{s,i}} {γ_{w}} \\frac{1}{2} \\begin{cases} {\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slope},i-1}}, & {\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slope},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\gt{}{\\symbf{y}_{\\text{slope},i-1}}\\\\ 0, & {\\symbf{y}_{\\text{wt},i}}\\leq{}{\\symbf{y}_{\\text{slope},i}}\\land{}{\\symbf{y}_{\\text{wt},i-1}}\\leq{}{\\symbf{y}_{\\text{slope},i-1}} \\end{cases}$$\n", + "$${\\symbf{U}_{\\text{g},i}}={\\symbf{L}_{s,i}}\\,{γ_{w}}\\,\\frac{1}{2}\\,\\begin{cases} {\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}+{\\symbf{y}_{\\text{wt},i-1}}-{\\symbf{y}_{\\text{slope},i-1}}, & {\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slope},i}}\\lor{}{\\symbf{y}_{\\text{wt},i-1}}\\gt{}{\\symbf{y}_{\\text{slope},i-1}}\\\\ 0, & {\\symbf{y}_{\\text{wt},i}}\\leq{}{\\symbf{y}_{\\text{slope},i}}\\land{}{\\symbf{y}_{\\text{wt},i-1}}\\leq{}{\\symbf{y}_{\\text{slope},i-1}} \\end{cases}$$\n", "
Equation\n", - "$$\\symbf{H}=\\begin{cases} \\frac{\\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}\\right)^{2}}{2} {γ_{w}}+\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}\\right)^{2} {γ_{w}}, & {\\symbf{y}_{\\text{wt},i}}\\geq{}{\\symbf{y}_{\\text{slope},i}}\\\\ \\frac{\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}\\right)^{2}}{2} {γ_{w}}, & {\\symbf{y}_{\\text{slope},i}}\\gt{}{\\symbf{y}_{\\text{wt},i}}\\land{}{\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slip},i}}\\\\ 0, & {\\symbf{y}_{\\text{wt},i}}\\leq{}{\\symbf{y}_{\\text{slip},i}} \\end{cases}$$\n", + "$$\\symbf{H}=\\begin{cases} \\frac{\\left({\\symbf{y}_{\\text{slope},i}}-{\\symbf{y}_{\\text{slip},i}}\\right)^{2}}{2}\\,{γ_{w}}+\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slope},i}}\\right)^{2}\\,{γ_{w}}, & {\\symbf{y}_{\\text{wt},i}}\\geq{}{\\symbf{y}_{\\text{slope},i}}\\\\ \\frac{\\left({\\symbf{y}_{\\text{wt},i}}-{\\symbf{y}_{\\text{slip},i}}\\right)^{2}}{2}\\,{γ_{w}}, & {\\symbf{y}_{\\text{slope},i}}\\gt{}{\\symbf{y}_{\\text{wt},i}}\\land{}{\\symbf{y}_{\\text{wt},i}}\\gt{}{\\symbf{y}_{\\text{slip},i}}\\\\ 0, & {\\symbf{y}_{\\text{wt},i}}\\leq{}{\\symbf{y}_{\\text{slip},i}} \\end{cases}$$\n", "
Equation\n", - "$${\\symbf{L}_{b}}={\\symbf{b}}_{i} \\sec\\left({\\symbf{α}}_{i}\\right)$$\n", + "$${\\symbf{L}_{b}}={\\symbf{b}}_{i}\\,\\sec\\left({\\symbf{α}}_{i}\\right)$$\n", "
Equation\n", - "$${\\symbf{L}_{s}}={\\symbf{b}}_{i} \\sec\\left({\\symbf{β}}_{i}\\right)$$\n", + "$${\\symbf{L}_{s}}={\\symbf{b}}_{i}\\,\\sec\\left({\\symbf{β}}_{i}\\right)$$\n", "
Equation\n", - "$$\\symbf{h}=\\frac{1}{2} \\left({{\\symbf{h}^{\\text{R}}}}_{i}+{{\\symbf{h}^{\\text{L}}}}_{i}\\right)$$\n", + "$$\\symbf{h}=\\frac{1}{2}\\,\\left({{\\symbf{h}^{\\text{R}}}}_{i}+{{\\symbf{h}^{\\text{L}}}}_{i}\\right)$$\n", "
Equation\n", - "$$\\symbf{f}=\\begin{cases} 1, & \\mathit{const\\_f}\\\\ \\sin\\left(π \\frac{{\\symbf{x}_{\\text{slip},i}}-{\\symbf{x}_{\\text{slip},0}}}{{\\symbf{x}_{\\text{slip},n}}-{\\symbf{x}_{\\text{slip},0}}}\\right), & \\neg{}\\mathit{const\\_f} \\end{cases}$$\n", + "$$\\symbf{f}=\\begin{cases} 1, & \\mathit{const\\_f}\\\\ \\sin\\left(π\\,\\frac{{\\symbf{x}_{\\text{slip},i}}-{\\symbf{x}_{\\text{slip},0}}}{{\\symbf{x}_{\\text{slip},n}}-{\\symbf{x}_{\\text{slip},0}}}\\right), & \\neg{}\\mathit{const\\_f} \\end{cases}$$\n", "
Equation\n", - "$$\\symbf{Φ}=\\left(λ {\\symbf{f}}_{i} \\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right) \\tan\\left(φ'\\right)-\\left(λ {\\symbf{f}}_{i} \\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right) {F_{\\text{S}}}$$\n", + "$$\\symbf{Φ}=\\left(λ\\,{\\symbf{f}}_{i}\\,\\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right)\\,\\tan\\left(φ'\\right)-\\left(λ\\,{\\symbf{f}}_{i}\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right)\\,{F_{\\text{S}}}$$\n", "
Equation\n", - "$$\\symbf{Ψ}=\\frac{\\left(λ {\\symbf{f}}_{i} \\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right) \\tan\\left(φ'\\right)-\\left(λ {\\symbf{f}}_{i} \\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right) {F_{\\text{S}}}}{{\\symbf{Φ}}_{i-1}}$$\n", + "$$\\symbf{Ψ}=\\frac{\\left(λ\\,{\\symbf{f}}_{i}\\,\\cos\\left({\\symbf{α}}_{i}\\right)-\\sin\\left({\\symbf{α}}_{i}\\right)\\right)\\,\\tan\\left(φ'\\right)-\\left(λ\\,{\\symbf{f}}_{i}\\,\\sin\\left({\\symbf{α}}_{i}\\right)+\\cos\\left({\\symbf{α}}_{i}\\right)\\right)\\,{F_{\\text{S}}}}{{\\symbf{Φ}}_{i-1}}$$\n", "
Equation\n", - "$${F_{\\text{S}}}=\\frac{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{R}}_{i} \\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{R}}_{n}}{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{T}}_{i} \\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{T}}_{n}}$$\n", + "$${F_{\\text{S}}}=\\frac{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{R}}_{i}\\,\\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{R}}_{n}}{\\displaystyle\\sum_{i=1}^{n-1}{{\\symbf{T}}_{i}\\,\\displaystyle\\prod_{v=i}^{n-1}{{\\symbf{Ψ}}_{v}}}+{\\symbf{T}}_{n}}$$\n", "
Equation\n", - "$${\\symbf{C}_{\\text{num},i}}=\\begin{cases} {\\symbf{b}}_{1} \\left({\\symbf{G}}_{1}+{\\symbf{H}}_{1}\\right) \\tan\\left({\\symbf{α}}_{1}\\right), & i=1\\\\ {\\symbf{b}}_{i} \\left({{\\symbf{F}_{\\text{x}}}^{\\text{G}}}+{{\\symbf{F}_{\\text{x}}}^{\\text{H}}}\\right) \\tan\\left({\\symbf{α}}_{i}\\right)+\\symbf{h} -2 {\\symbf{U}_{\\text{g},i}} \\sin\\left({\\symbf{β}}_{i}\\right), & 2\\leq{}i\\leq{}n-1\\\\ {\\symbf{b}}_{n} \\left({\\symbf{G}}_{n-1}+{\\symbf{H}}_{n-1}\\right) \\tan\\left({\\symbf{α}}_{n-1}\\right), & i=n \\end{cases}$$\n", + "$${\\symbf{C}_{\\text{num},i}}=\\begin{cases} {\\symbf{b}}_{1}\\,\\left({\\symbf{G}}_{1}+{\\symbf{H}}_{1}\\right)\\,\\tan\\left({\\symbf{α}}_{1}\\right), & i=1\\\\ {\\symbf{b}}_{i}\\,\\left({{\\symbf{F}_{\\text{x}}}^{\\text{G}}}+{{\\symbf{F}_{\\text{x}}}^{\\text{H}}}\\right)\\,\\tan\\left({\\symbf{α}}_{i}\\right)+\\symbf{h}\\,-2\\,{\\symbf{U}_{\\text{g},i}}\\,\\sin\\left({\\symbf{β}}_{i}\\right), & 2\\leq{}i\\leq{}n-1\\\\ {\\symbf{b}}_{n}\\,\\left({\\symbf{G}}_{n-1}+{\\symbf{H}}_{n-1}\\right)\\,\\tan\\left({\\symbf{α}}_{n-1}\\right), & i=n \\end{cases}$$\n", "
Equation\n", - "$${\\symbf{C}_{\\text{den},i}}=\\begin{cases} {\\symbf{b}}_{1} {\\symbf{f}}_{1} {\\symbf{G}}_{1}, & i=1\\\\ {\\symbf{b}}_{i} \\left({\\symbf{f}}_{i} {\\symbf{G}}_{i}+{\\symbf{f}}_{i-1} {\\symbf{G}}_{i-1}\\right), & 2\\leq{}i\\leq{}n-1\\\\ {\\symbf{b}}_{n} {\\symbf{G}}_{n-1} {\\symbf{f}}_{n-1}, & i=n \\end{cases}$$\n", + "$${\\symbf{C}_{\\text{den},i}}=\\begin{cases} {\\symbf{b}}_{1}\\,{\\symbf{f}}_{1}\\,{\\symbf{G}}_{1}, & i=1\\\\ {\\symbf{b}}_{i}\\,\\left({\\symbf{f}}_{i}\\,{\\symbf{G}}_{i}+{\\symbf{f}}_{i-1}\\,{\\symbf{G}}_{i-1}\\right), & 2\\leq{}i\\leq{}n-1\\\\ {\\symbf{b}}_{n}\\,{\\symbf{G}}_{n-1}\\,{\\symbf{f}}_{n-1}, & i=n \\end{cases}$$\n", "
Equation\n", - "$${\\symbf{G}}_{i}=\\begin{cases} \\frac{{F_{\\text{S}}} {\\symbf{T}}_{1}-{\\symbf{R}}_{1}}{{\\symbf{Φ}}_{1}}, & i=1\\\\ \\frac{{\\symbf{Ψ}}_{i-1} {\\symbf{G}}_{i-1}+{F_{\\text{S}}} {\\symbf{T}}_{i}-{\\symbf{R}}_{i}}{{\\symbf{Φ}}_{i}}, & 2\\leq{}i\\leq{}n-1\\\\ 0, & i=0\\lor{}i=n \\end{cases}$$\n", + "$${\\symbf{G}}_{i}=\\begin{cases} \\frac{{F_{\\text{S}}}\\,{\\symbf{T}}_{1}-{\\symbf{R}}_{1}}{{\\symbf{Φ}}_{1}}, & i=1\\\\ \\frac{{\\symbf{Ψ}}_{i-1}\\,{\\symbf{G}}_{i-1}+{F_{\\text{S}}}\\,{\\symbf{T}}_{i}-{\\symbf{R}}_{i}}{{\\symbf{Φ}}_{i}}, & 2\\leq{}i\\leq{}n-1\\\\ 0, & i=0\\lor{}i=n \\end{cases}$$\n", "
Equation - \[-∇\cdot{}\symbf{q}+g=ρ C \frac{\,\partial{}T}{\,\partial{}t}\] + \[-∇\cdot{}\symbf{q}+g=ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}\]
Equation \[E=\begin{cases} - {C^{\text{S}}} m ΔT, & T\lt{}{T_{\text{melt}}}\\ - {C^{\text{L}}} m ΔT, & {T_{\text{melt}}}\lt{}T\lt{}{T_{\text{boil}}}\\ - {C^{\text{V}}} m ΔT, & {T_{\text{boil}}}\lt{}T + {C^{\text{S}}}\,m\,ΔT, & T\lt{}{T_{\text{melt}}}\\ + {C^{\text{L}}}\,m\,ΔT, & {T_{\text{melt}}}\lt{}T\lt{}{T_{\text{boil}}}\\ + {C^{\text{V}}}\,m\,ΔT, & {T_{\text{boil}}}\lt{}T \end{cases}\]
Equation\[q\left(t\right)=h ΔT\left(t\right)\]\[q\left(t\right)=h\,ΔT\left(t\right)\]
Description
Equation - \[m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] + \[m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]
Equation - \[{q_{\text{C}}}={h_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)\] + \[{q_{\text{C}}}={h_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)\]
Equation - \[{q_{\text{P}}}={h_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)\] + \[{q_{\text{P}}}={h_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)\]
Equation\[{m_{\text{W}}}={V_{\text{W}}} {ρ_{\text{W}}}\]\[{m_{\text{W}}}={V_{\text{W}}}\,{ρ_{\text{W}}}\]
Description
Equation\[{V_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L\]\[{V_{\text{tank}}}=π\,\left(\frac{D}{2}\right)^{2}\,L\]
Description
Equation - \[{τ_{\text{W}}}=\frac{{m_{\text{W}}} {C_{\text{W}}}}{{h_{\text{C}}} {A_{\text{C}}}}\] + \[{τ_{\text{W}}}=\frac{{m_{\text{W}}}\,{C_{\text{W}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\]
Equation - \[η=\frac{{h_{\text{P}}} {A_{\text{P}}}}{{h_{\text{C}}} {A_{\text{C}}}}\] + \[η=\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\]
Equation - \[{{τ_{\text{P}}}^{\text{S}}}=\frac{{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}}}{{h_{\text{P}}} {A_{\text{P}}}}\] + \[{{τ_{\text{P}}}^{\text{S}}}=\frac{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}}{{h_{\text{P}}}\,{A_{\text{P}}}}\]
Equation - \[{{τ_{\text{P}}}^{\text{L}}}=\frac{{m_{\text{P}}} {{C_{\text{P}}}^{\text{L}}}}{{h_{\text{P}}} {A_{\text{P}}}}\] + \[{{τ_{\text{P}}}^{\text{L}}}=\frac{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{L}}}}{{h_{\text{P}}}\,{A_{\text{P}}}}\]
Equation\[ϕ=\frac{{Q_{\text{P}}}}{{H_{\text{f}}} {m_{\text{P}}}}\] + \[ϕ=\frac{{Q_{\text{P}}}}{{H_{\text{f}}}\,{m_{\text{P}}}}\] +
Description
Equation - \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)+η \left({T_{\text{P}}}\left(t\right)-{T_{\text{W}}}\left(t\right)\right)\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)+η\,\left({T_{\text{P}}}\left(t\right)-{T_{\text{W}}}\left(t\right)\right)\right)\]
Equation \[\frac{\,d{T_{\text{P}}}}{\,dt}=\begin{cases} - \frac{1}{{{τ_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ - \frac{1}{{{τ_{\text{P}}}^{\text{L}}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ + \frac{1}{{{τ_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ + \frac{1}{{{τ_{\text{P}}}^{\text{L}}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ 0, & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1 \end{cases}\]
Equation - \[{E_{\text{W}}}\left(t\right)={C_{\text{W}}} {m_{\text{W}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)\] + \[{E_{\text{W}}}\left(t\right)={C_{\text{W}}}\,{m_{\text{W}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)\]
Equation \[{E_{\text{P}}}=\begin{cases} - {{C_{\text{P}}}^{\text{S}}} {m_{\text{P}}} \left({T_{\text{P}}}\left(t\right)-{T_{\text{init}}}\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ - {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{\text{f}}} {m_{\text{P}}}+{{C_{\text{P}}}^{\text{L}}} {m_{\text{P}}} \left({T_{\text{P}}}\left(t\right)-{{T_{\text{melt}}}^{\text{P}}}\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ + {{C_{\text{P}}}^{\text{S}}}\,{m_{\text{P}}}\,\left({T_{\text{P}}}\left(t\right)-{T_{\text{init}}}\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ + {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{\text{f}}}\,{m_{\text{P}}}+{{C_{\text{P}}}^{\text{L}}}\,{m_{\text{P}}}\,\left({T_{\text{P}}}\left(t\right)-{{T_{\text{melt}}}^{\text{P}}}\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{Q_{\text{P}}}\left(t\right), & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1 \end{cases}\]
Equation\n", - "$$-∇\\cdot{}\\symbf{q}+g=ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}$$\n", + "$$-∇\\cdot{}\\symbf{q}+g=ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}$$\n", "
Equation\n", - "$$E=\\begin{cases} {C^{\\text{S}}} m ΔT, & T\\lt{}{T_{\\text{melt}}}\\\\ {C^{\\text{L}}} m ΔT, & {T_{\\text{melt}}}\\lt{}T\\lt{}{T_{\\text{boil}}}\\\\ {C^{\\text{V}}} m ΔT, & {T_{\\text{boil}}}\\lt{}T \\end{cases}$$\n", + "$$E=\\begin{cases} {C^{\\text{S}}}\\,m\\,ΔT, & T\\lt{}{T_{\\text{melt}}}\\\\ {C^{\\text{L}}}\\,m\\,ΔT, & {T_{\\text{melt}}}\\lt{}T\\lt{}{T_{\\text{boil}}}\\\\ {C^{\\text{V}}}\\,m\\,ΔT, & {T_{\\text{boil}}}\\lt{}T \\end{cases}$$\n", "
Equation\n", - "$$q\\left(t\\right)=h ΔT\\left(t\\right)$$\n", + "$$q\\left(t\\right)=h\\,ΔT\\left(t\\right)$$\n", "
Equation\n", - "$$m C \\frac{\\,dT}{\\,dt}={q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V$$\n", + "$$m\\,C\\,\\frac{\\,dT}{\\,dt}={q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V$$\n", "
\n", @@ -766,7 +766,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -830,7 +830,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -904,7 +904,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1031,7 +1031,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1092,7 +1092,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1154,7 +1154,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1216,7 +1216,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1278,7 +1278,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1407,7 +1407,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1555,7 +1555,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1607,25 +1607,25 @@ "\n", "\n", "To find the rate of change of $T_W$, we look at the energy balance on water. The volume being considered is the volume of water in the tank $V_W$, which has mass $m_W$ and specific heat capacity, $C_W$. Heat transfer occurs in the water from the heating coil as $q_C$ ([GD:htFluxWaterFromCoil](#GD:htFluxWaterFromCoil)) and from the water into the PCM as $q_P$ ([GD:htFluxPCMFromWater](#GD:htFluxPCMFromWater)), over areas $A_C$ and $A_P$, respectively. The thermal flux is constant over $A_C$, since the temperature of the heating coil is assumed to not vary along its length ([A:Temp-Heating-Coil-Constant-over-Length](#assumpTHCCoL)), and the thermal flux is constant over $A_P$, since the temperature of the PCM is the same throughout its volume ([A:Temp-PCM-Constant-Across-Volume](#assumpTPCAV)) and the water is fully mixed ([A:Constant-Water-Temp-Across-Tank](#assumpCWTAT)). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated ([A:Perfect-Insulation-Tank](#assumpPIT)). Since the assumption is made that no internal heat is generated ([A:No-Internal-Heat-Generation-By-Water-PCM](#assumpNIHGBWP)), $g = 0$. Therefore, the equation for [GD:rocTempSimp](#GD:rocTempSimp) can be written as:\n", - "$${m_{\\text{W}}} {C_{\\text{W}}} \\frac{\\,d{T_{\\text{W}}}}{\\,dt}={q_{\\text{C}}} {A_{\\text{C}}}-{q_{\\text{P}}} {A_{\\text{P}}}$$\n", + "$${m_{\\text{W}}}\\,{C_{\\text{W}}}\\,\\frac{\\,d{T_{\\text{W}}}}{\\,dt}={q_{\\text{C}}}\\,{A_{\\text{C}}}-{q_{\\text{P}}}\\,{A_{\\text{P}}}$$\n", "\n", "Using [GD:htFluxWaterFromCoil](#GD:htFluxWaterFromCoil) for $q_C$ and [GD:htFluxPCMFromWater](#GD:htFluxPCMFromWater) for $q_P$, this can be written as:\n", - "$${m_{\\text{W}}} {C_{\\text{W}}} \\frac{\\,d{T_{\\text{W}}}}{\\,dt}={h_{\\text{C}}} {A_{\\text{C}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)-{h_{\\text{P}}} {A_{\\text{P}}} \\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", + "$${m_{\\text{W}}}\\,{C_{\\text{W}}}\\,\\frac{\\,d{T_{\\text{W}}}}{\\,dt}={h_{\\text{C}}}\\,{A_{\\text{C}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)-{h_{\\text{P}}}\\,{A_{\\text{P}}}\\,\\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", "\n", "Dividing Equation (2) by $m_WC_W$, we obtain:\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}} {A_{\\text{C}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)-\\frac{{h_{\\text{P}}} {A_{\\text{P}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}}\\,{A_{\\text{C}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)-\\frac{{h_{\\text{P}}}\\,{A_{\\text{P}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", "\n", "Factoring the negative sign out of the second term of the right-hand side (RHS) of Equation (3) and multiplying it by $h_C$ $A_C$ / $h_C$ $A_C$ yields:\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}} {A_{\\text{C}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)+\\frac{{h_{\\text{C}}} {A_{\\text{C}}}}{{h_{\\text{C}}} {A_{\\text{C}}}} \\frac{{h_{\\text{P}}} {A_{\\text{P}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}}\\,{A_{\\text{C}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)+\\frac{{h_{\\text{C}}}\\,{A_{\\text{C}}}}{{h_{\\text{C}}}\\,{A_{\\text{C}}}}\\,\\frac{{h_{\\text{P}}}\\,{A_{\\text{P}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)$$\n", "\n", "Rearranging this equation gives us:\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}} {A_{\\text{C}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)+\\frac{{h_{\\text{P}}} {A_{\\text{P}}}}{{h_{\\text{C}}} {A_{\\text{C}}}} \\frac{{h_{\\text{C}}} {A_{\\text{C}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}}\\,{A_{\\text{C}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)+\\frac{{h_{\\text{P}}}\\,{A_{\\text{P}}}}{{h_{\\text{C}}}\\,{A_{\\text{C}}}}\\,\\frac{{h_{\\text{C}}}\\,{A_{\\text{C}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)$$\n", "\n", "By substituting $τ_W$ (from [DD:balanceDecayRate](#DD:balanceDecayRate)) and $η$ (from [DD:balanceDecayTime](#DD:balanceDecayTime)), this can be written as:\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)+\\frac{η}{{τ_{\\text{W}}}} \\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)+\\frac{η}{{τ_{\\text{W}}}}\\,\\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)$$\n", "\n", "Finally, factoring out $\\frac{1}{τ_W}$, we are left with the governing ODE for [IM:eBalanceOnWtr](#IM:eBalanceOnWtr):\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}+η \\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}+η\\,\\left({T_{\\text{P}}}-{T_{\\text{W}}}\\right)\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${q_{\\text{C}}}={h_{\\text{C}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)\\right)$$\n", + "$${q_{\\text{C}}}={h_{\\text{C}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)\\right)$$\n", "
Equation\n", - "$${q_{\\text{P}}}={h_{\\text{P}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right)$$\n", + "$${q_{\\text{P}}}={h_{\\text{P}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right)$$\n", "
Equation\n", - "$${m_{\\text{W}}}={V_{\\text{W}}} {ρ_{\\text{W}}}$$\n", + "$${m_{\\text{W}}}={V_{\\text{W}}}\\,{ρ_{\\text{W}}}$$\n", "
Equation\n", - "$${V_{\\text{tank}}}=π \\left(\\frac{D}{2}\\right)^{2} L$$\n", + "$${V_{\\text{tank}}}=π\\,\\left(\\frac{D}{2}\\right)^{2}\\,L$$\n", "
Equation\n", - "$${τ_{\\text{W}}}=\\frac{{m_{\\text{W}}} {C_{\\text{W}}}}{{h_{\\text{C}}} {A_{\\text{C}}}}$$\n", + "$${τ_{\\text{W}}}=\\frac{{m_{\\text{W}}}\\,{C_{\\text{W}}}}{{h_{\\text{C}}}\\,{A_{\\text{C}}}}$$\n", "
Equation\n", - "$$η=\\frac{{h_{\\text{P}}} {A_{\\text{P}}}}{{h_{\\text{C}}} {A_{\\text{C}}}}$$\n", + "$$η=\\frac{{h_{\\text{P}}}\\,{A_{\\text{P}}}}{{h_{\\text{C}}}\\,{A_{\\text{C}}}}$$\n", "
Equation\n", - "$${{τ_{\\text{P}}}^{\\text{S}}}=\\frac{{m_{\\text{P}}} {{C_{\\text{P}}}^{\\text{S}}}}{{h_{\\text{P}}} {A_{\\text{P}}}}$$\n", + "$${{τ_{\\text{P}}}^{\\text{S}}}=\\frac{{m_{\\text{P}}}\\,{{C_{\\text{P}}}^{\\text{S}}}}{{h_{\\text{P}}}\\,{A_{\\text{P}}}}$$\n", "
Equation\n", - "$${{τ_{\\text{P}}}^{\\text{L}}}=\\frac{{m_{\\text{P}}} {{C_{\\text{P}}}^{\\text{L}}}}{{h_{\\text{P}}} {A_{\\text{P}}}}$$\n", + "$${{τ_{\\text{P}}}^{\\text{L}}}=\\frac{{m_{\\text{P}}}\\,{{C_{\\text{P}}}^{\\text{L}}}}{{h_{\\text{P}}}\\,{A_{\\text{P}}}}$$\n", "
Equation\n", - "$$ϕ=\\frac{{Q_{\\text{P}}}}{{H_{\\text{f}}} {m_{\\text{P}}}}$$\n", + "$$ϕ=\\frac{{Q_{\\text{P}}}}{{H_{\\text{f}}}\\,{m_{\\text{P}}}}$$\n", "
Equation\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)+η \\left({T_{\\text{P}}}\\left(t\\right)-{T_{\\text{W}}}\\left(t\\right)\\right)\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)+η\\,\\left({T_{\\text{P}}}\\left(t\\right)-{T_{\\text{W}}}\\left(t\\right)\\right)\\right)$$\n", "
\n", @@ -1670,7 +1670,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1723,16 +1723,16 @@ "\n", "\n", "To find the rate of change of $T_P$, we look at the energy balance on the PCM. The volume being considered is the volume of PCM ($V_P$). The derivation that follows is initially for the solid PCM. The mass of phase change material is $m_P$ and the specific heat capacity of PCM as a solid is $C_P^S$. The heat flux into the PCM from water is $q_P$ ([GD:htFluxPCMFromWater](#GD:htFluxPCMFromWater)) over phase change material surface area $A_P$. The thermal flux is constant over $A_P$, since the temperature of the PCM is the same throughout its volume ([A:Temp-PCM-Constant-Across-Volume](#assumpTPCAV)) and the water is fully mixed ([A:Constant-Water-Temp-Across-Tank](#assumpCWTAT)). There is no heat flux output from the PCM. Assuming no volumetric heat generation per unit volume ([A:No-Internal-Heat-Generation-By-Water-PCM](#assumpNIHGBWP)), $g = 0$, the equation for [GD:rocTempSimp](#GD:rocTempSimp) can be written as:\n", - "$${m_{\\text{P}}} {{C_{\\text{P}}}^{\\text{S}}} \\frac{\\,d{T_{\\text{P}}}}{\\,dt}={q_{\\text{P}}} {A_{\\text{P}}}$$\n", + "$${m_{\\text{P}}}\\,{{C_{\\text{P}}}^{\\text{S}}}\\,\\frac{\\,d{T_{\\text{P}}}}{\\,dt}={q_{\\text{P}}}\\,{A_{\\text{P}}}$$\n", "\n", "Using [GD:htFluxPCMFromWater](#GD:htFluxPCMFromWater) for $q_P$, this equation can be written as:\n", - "$${m_{\\text{P}}} {{C_{\\text{P}}}^{\\text{S}}} \\frac{\\,d{T_{\\text{P}}}}{\\,dt}={h_{\\text{P}}} {A_{\\text{P}}} \\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", + "$${m_{\\text{P}}}\\,{{C_{\\text{P}}}^{\\text{S}}}\\,\\frac{\\,d{T_{\\text{P}}}}{\\,dt}={h_{\\text{P}}}\\,{A_{\\text{P}}}\\,\\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", "\n", "Dividing by $m_P$ $C_P^S$ we obtain:\n", - "$$\\frac{\\,d{T_{\\text{P}}}}{\\,dt}=\\frac{{h_{\\text{P}}} {A_{\\text{P}}}}{{m_{\\text{P}}} {{C_{\\text{P}}}^{\\text{S}}}} \\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{P}}}}{\\,dt}=\\frac{{h_{\\text{P}}}\\,{A_{\\text{P}}}}{{m_{\\text{P}}}\\,{{C_{\\text{P}}}^{\\text{S}}}}\\,\\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", "\n", "By substituting $τ_P^S$ (from [DD:balanceSolidPCM](#DD:balanceSolidPCM)), this can be written as:\n", - "$$\\frac{\\,d{T_{\\text{P}}}}{\\,dt}=\\frac{1}{{{τ_{\\text{P}}}^{\\text{S}}}} \\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{P}}}}{\\,dt}=\\frac{1}{{{τ_{\\text{P}}}^{\\text{S}}}}\\,\\left({T_{\\text{W}}}-{T_{\\text{P}}}\\right)$$\n", "\n", "Equation (4) applies for the solid PCM. In the case where all of the PCM is melted, the same derivation applies, except that $C_P^S$ is replaced by $C_P^L$, and thus $τ_P^S$ is replaced by $τ_P^L$. Although a small change in surface area would be expected with melting, this is not included, since the volume change of the PCM with melting is assumed to be negligible ([A:Volume-Change-Melting-PCM-Negligible](#assumpVCMPN)).\n", "\n", @@ -1782,7 +1782,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1867,7 +1867,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1968,10 +1968,10 @@ "\n", "\n", "A correct solution must exhibit the law of conservation of energy. This means that the change in heat energy in the water should equal the difference between the total energy input from the heating coil and the energy output to the PCM. This can be shown as an equation by taking [GD:htFluxWaterFromCoil](#GD:htFluxWaterFromCoil) and [GD:htFluxPCMFromWater](#GD:htFluxPCMFromWater), multiplying each by their respective surface area of heat transfer, and integrating each over the simulation time, as follows:\n", - "$${E_{\\text{W}}}=\\int_{0}^{t}{{h_{\\text{C}}} {A_{\\text{C}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)\\right)}\\,dt-\\int_{0}^{t}{{h_{\\text{P}}} {A_{\\text{P}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right)}\\,dt$$\n", + "$${E_{\\text{W}}}=\\int_{0}^{t}{{h_{\\text{C}}}\\,{A_{\\text{C}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)\\right)}\\,dt-\\int_{0}^{t}{{h_{\\text{P}}}\\,{A_{\\text{P}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right)}\\,dt$$\n", "\n", "In addition, the change in heat energy in the PCM should equal the energy input to the PCM from the water. This can be expressed as\n", - "$${E_{\\text{P}}}=\\int_{0}^{t}{{h_{\\text{P}}} {A_{\\text{P}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right)}\\,dt$$\n", + "$${E_{\\text{P}}}=\\int_{0}^{t}{{h_{\\text{P}}}\\,{A_{\\text{P}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right)}\\,dt$$\n", "\n", "Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as \"sanity\" checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than $C_tol$ [FR:Verify-Energy-Output-Follow-Conservation-of-Energy](#verifyEnergyOutput).\n", "\n", diff --git a/code/stable/swhs/SRS/PDF/SWHS_SRS.tex b/code/stable/swhs/SRS/PDF/SWHS_SRS.tex index 25e67d77ce..4dd1b82e59 100644 --- a/code/stable/swhs/SRS/PDF/SWHS_SRS.tex +++ b/code/stable/swhs/SRS/PDF/SWHS_SRS.tex @@ -446,7 +446,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - -∇\cdot{}\symbf{q}+g=ρ C \frac{\,\partial{}T}{\,\partial{}t} + -∇\cdot{}\symbf{q}+g=ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -486,9 +486,9 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} E=\begin{cases} - {C^{\text{S}}} m ΔT, & T\lt{}{T_{\text{melt}}}\\ - {C^{\text{L}}} m ΔT, & {T_{\text{melt}}}\lt{}T\lt{}{T_{\text{boil}}}\\ - {C^{\text{V}}} m ΔT, & {T_{\text{boil}}}\lt{}T + {C^{\text{S}}}\,m\,ΔT, & T\lt{}{T_{\text{melt}}}\\ + {C^{\text{L}}}\,m\,ΔT, & {T_{\text{melt}}}\lt{}T\lt{}{T_{\text{boil}}}\\ + {C^{\text{V}}}\,m\,ΔT, & {T_{\text{boil}}}\lt{}T \end{cases} \end{displaymath} \\ \midrule @@ -569,7 +569,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - q\left(t\right)=h ΔT\left(t\right) + q\left(t\right)=h\,ΔT\left(t\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -611,7 +611,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V + m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -641,27 +641,27 @@ \subsubsection{General Definitions} Integrating \hyperref[TM:consThermE]{TM:consThermE} over a volume ($V$), we have: \begin{displaymath} --\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} Applying Gauss's Divergence Theorem to the first term over the surface $S$ of the volume, with $\symbf{q}$ as the thermal flux vector for the surface and $\symbf{\hat{n}}$ as a unit outward normal vector for a surface: \begin{displaymath} --\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as: \begin{displaymath} -{q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +{q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} Where ${q_{\text{in}}}$, ${q_{\text{out}}}$, ${A_{\text{in}}}$, and ${A_{\text{out}}}$ are explained in \hyperref[GD:rocTempSimp]{GD:rocTempSimp}. The integral over the surface could be simplified because the thermal flux is assumed constant over ${A_{\text{in}}}$ and ${A_{\text{out}}}$ and $0$ on all other surfaces. Outward flux is considered positive. Assuming $ρ$, $C$, and $T$ are constant over the volume, which is true in our case by \hyperref[assumpCWTAT]{A:Constant-Water-Temp-Across-Tank}, \hyperref[assumpTPCAV]{A:Temp-PCM-Constant-Across-Volume}, \hyperref[assumpDWPCoV]{A:Density-Water-PCM-Constant-over-Volume}, and \hyperref[assumpSHECov]{A:Specific-Heat-Energy-Constant-over-Volume}, we have: \begin{displaymath} -ρ C V \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V +ρ\,C\,V\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V \end{displaymath} Using the fact that $ρ$=$m$/$V$, Equation (2) can be written as: \begin{displaymath} -m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V +m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V \end{displaymath} \medskip \noindent @@ -678,7 +678,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {q_{\text{C}}}={h_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right) + {q_{\text{C}}}={h_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -718,7 +718,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {q_{\text{P}}}={h_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right) + {q_{\text{P}}}={h_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -763,7 +763,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {m_{\text{W}}}={V_{\text{W}}} {ρ_{\text{W}}} + {m_{\text{W}}}={V_{\text{W}}}\,{ρ_{\text{W}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -838,7 +838,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {V_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L + {V_{\text{tank}}}=π\,\left(\frac{D}{2}\right)^{2}\,L \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -875,7 +875,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {τ_{\text{W}}}=\frac{{m_{\text{W}}} {C_{\text{W}}}}{{h_{\text{C}}} {A_{\text{C}}}} + {τ_{\text{W}}}=\frac{{m_{\text{W}}}\,{C_{\text{W}}}}{{h_{\text{C}}}\,{A_{\text{C}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -913,7 +913,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - η=\frac{{h_{\text{P}}} {A_{\text{P}}}}{{h_{\text{C}}} {A_{\text{C}}}} + η=\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{h_{\text{C}}}\,{A_{\text{C}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -951,7 +951,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {{τ_{\text{P}}}^{\text{S}}}=\frac{{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}}}{{h_{\text{P}}} {A_{\text{P}}}} + {{τ_{\text{P}}}^{\text{S}}}=\frac{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}}{{h_{\text{P}}}\,{A_{\text{P}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -989,7 +989,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {{τ_{\text{P}}}^{\text{L}}}=\frac{{m_{\text{P}}} {{C_{\text{P}}}^{\text{L}}}}{{h_{\text{P}}} {A_{\text{P}}}} + {{τ_{\text{P}}}^{\text{L}}}=\frac{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{L}}}}{{h_{\text{P}}}\,{A_{\text{P}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1066,7 +1066,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - ϕ=\frac{{Q_{\text{P}}}}{{H_{\text{f}}} {m_{\text{P}}}} + ϕ=\frac{{Q_{\text{P}}}}{{H_{\text{f}}}\,{m_{\text{P}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1155,7 +1155,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - \frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)+η \left({T_{\text{P}}}\left(t\right)-{T_{\text{W}}}\left(t\right)\right)\right) + \frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)+η\,\left({T_{\text{P}}}\left(t\right)-{T_{\text{W}}}\left(t\right)\right)\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1194,37 +1194,37 @@ \subsubsection{Instance Models} To find the rate of change of ${T_{\text{W}}}$, we look at the energy balance on water. The volume being considered is the volume of water in the tank ${V_{\text{W}}}$, which has mass ${m_{\text{W}}}$ and specific heat capacity, ${C_{\text{W}}}$. Heat transfer occurs in the water from the heating coil as ${q_{\text{C}}}$ (\hyperref[GD:htFluxWaterFromCoil]{GD:htFluxWaterFromCoil}) and from the water into the PCM as ${q_{\text{P}}}$ (\hyperref[GD:htFluxPCMFromWater]{GD:htFluxPCMFromWater}), over areas ${A_{\text{C}}}$ and ${A_{\text{P}}}$, respectively. The thermal flux is constant over ${A_{\text{C}}}$, since the temperature of the heating coil is assumed to not vary along its length (\hyperref[assumpTHCCoL]{A:Temp-Heating-Coil-Constant-over-Length}), and the thermal flux is constant over ${A_{\text{P}}}$, since the temperature of the PCM is the same throughout its volume (\hyperref[assumpTPCAV]{A:Temp-PCM-Constant-Across-Volume}) and the water is fully mixed (\hyperref[assumpCWTAT]{A:Constant-Water-Temp-Across-Tank}). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (\hyperref[assumpPIT]{A:Perfect-Insulation-Tank}). Since the assumption is made that no internal heat is generated (\hyperref[assumpNIHGBWP]{A:No-Internal-Heat-Generation-By-Water-PCM}), $g=0$. Therefore, the equation for \hyperref[GD:rocTempSimp]{GD:rocTempSimp} can be written as: \begin{displaymath} -{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}} {A_{\text{C}}}-{q_{\text{P}}} {A_{\text{P}}} +{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}}\,{A_{\text{C}}}-{q_{\text{P}}}\,{A_{\text{P}}} \end{displaymath} Using \hyperref[GD:htFluxWaterFromCoil]{GD:htFluxWaterFromCoil} for ${q_{\text{C}}}$ and \hyperref[GD:htFluxPCMFromWater]{GD:htFluxPCMFromWater} for ${q_{\text{P}}}$, this can be written as: \begin{displaymath} -{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)-{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right) +{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)-{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right) \end{displaymath} -Dividing Equation (2) by ${m_{\text{W}}} {C_{\text{W}}}$, we obtain: +Dividing Equation (2) by ${m_{\text{W}}}\,{C_{\text{W}}}$, we obtain: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)-\frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)-\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right) \end{displaymath} Factoring the negative sign out of the second term of the right-hand side (RHS) of Equation (3) and multiplying it by ${h_{\text{C}}}$ ${A_{\text{C}}}$ / ${h_{\text{C}}}$ ${A_{\text{C}}}$ yields: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{C}}} {A_{\text{C}}}}{{h_{\text{C}}} {A_{\text{C}}}} \frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\,\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right) \end{displaymath} Rearranging this equation gives us: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{P}}} {A_{\text{P}}}}{{h_{\text{C}}} {A_{\text{C}}}} \frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\,\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right) \end{displaymath} By substituting ${τ_{\text{W}}}$ (from \hyperref[DD:balanceDecayRate]{DD:balanceDecayRate}) and $η$ (from \hyperref[DD:balanceDecayTime]{DD:balanceDecayTime}), this can be written as: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{η}{{τ_{\text{W}}}} \left({T_{\text{P}}}-{T_{\text{W}}}\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)+\frac{η}{{τ_{\text{W}}}}\,\left({T_{\text{P}}}-{T_{\text{W}}}\right) \end{displaymath} Finally, factoring out $\frac{1}{{τ_{\text{W}}}}$, we are left with the governing ODE for \hyperref[IM:eBalanceOnWtr]{IM:eBalanceOnWtr}: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}+η \left({T_{\text{P}}}-{T_{\text{W}}}\right)\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}+η\,\left({T_{\text{P}}}-{T_{\text{W}}}\right)\right) \end{displaymath} \medskip \noindent @@ -1251,8 +1251,8 @@ \subsubsection{Instance Models} \\ \midrule Equation & \begin{displaymath} \frac{\,d{T_{\text{P}}}}{\,dt}=\begin{cases} - \frac{1}{{{τ_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ - \frac{1}{{{τ_{\text{P}}}^{\text{L}}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ + \frac{1}{{{τ_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ + \frac{1}{{{τ_{\text{P}}}^{\text{L}}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ 0, & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1 \end{cases} \end{displaymath} @@ -1294,22 +1294,22 @@ \subsubsection{Instance Models} To find the rate of change of ${T_{\text{P}}}$, we look at the energy balance on the PCM. The volume being considered is the volume of PCM (${V_{\text{P}}}$). The derivation that follows is initially for the solid PCM. The mass of phase change material is ${m_{\text{P}}}$ and the specific heat capacity of PCM as a solid is ${{C_{\text{P}}}^{\text{S}}}$. The heat flux into the PCM from water is ${q_{\text{P}}}$ (\hyperref[GD:htFluxPCMFromWater]{GD:htFluxPCMFromWater}) over phase change material surface area ${A_{\text{P}}}$. The thermal flux is constant over ${A_{\text{P}}}$, since the temperature of the PCM is the same throughout its volume (\hyperref[assumpTPCAV]{A:Temp-PCM-Constant-Across-Volume}) and the water is fully mixed (\hyperref[assumpCWTAT]{A:Constant-Water-Temp-Across-Tank}). There is no heat flux output from the PCM. Assuming no volumetric heat generation per unit volume (\hyperref[assumpNIHGBWP]{A:No-Internal-Heat-Generation-By-Water-PCM}), $g=0$, the equation for \hyperref[GD:rocTempSimp]{GD:rocTempSimp} can be written as: \begin{displaymath} -{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}} \frac{\,d{T_{\text{P}}}}{\,dt}={q_{\text{P}}} {A_{\text{P}}} +{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}\,\frac{\,d{T_{\text{P}}}}{\,dt}={q_{\text{P}}}\,{A_{\text{P}}} \end{displaymath} Using \hyperref[GD:htFluxPCMFromWater]{GD:htFluxPCMFromWater} for ${q_{\text{P}}}$, this equation can be written as: \begin{displaymath} -{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}} \frac{\,d{T_{\text{P}}}}{\,dt}={h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right) +{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}\,\frac{\,d{T_{\text{P}}}}{\,dt}={h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right) \end{displaymath} Dividing by ${m_{\text{P}}}$ ${{C_{\text{P}}}^{\text{S}}}$ we obtain: \begin{displaymath} -\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{{h_{\text{P}}} {A_{\text{P}}}}{{m_{\text{P}}} {{C_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right) +\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{{h_{\text{P}}}\,{A_{\text{P}}}}{{m_{\text{P}}}\,{{C_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right) \end{displaymath} By substituting ${{τ_{\text{P}}}^{\text{S}}}$ (from \hyperref[DD:balanceSolidPCM]{DD:balanceSolidPCM}), this can be written as: \begin{displaymath} -\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{1}{{{τ_{\text{P}}}^{\text{S}}}} \left({T_{\text{W}}}-{T_{\text{P}}}\right) +\frac{\,d{T_{\text{P}}}}{\,dt}=\frac{1}{{{τ_{\text{P}}}^{\text{S}}}}\,\left({T_{\text{W}}}-{T_{\text{P}}}\right) \end{displaymath} Equation (4) applies for the solid PCM. In the case where all of the PCM is melted, the same derivation applies, except that ${{C_{\text{P}}}^{\text{S}}}$ is replaced by ${{C_{\text{P}}}^{\text{L}}}$, and thus ${{τ_{\text{P}}}^{\text{S}}}$ is replaced by ${{τ_{\text{P}}}^{\text{L}}}$. Although a small change in surface area would be expected with melting, this is not included, since the volume change of the PCM with melting is assumed to be negligible (\hyperref[assumpVCMPN]{A:Volume-Change-Melting-PCM-Negligible}). @@ -1339,7 +1339,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - {E_{\text{W}}}\left(t\right)={C_{\text{W}}} {m_{\text{W}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right) + {E_{\text{W}}}\left(t\right)={C_{\text{W}}}\,{m_{\text{W}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -1392,8 +1392,8 @@ \subsubsection{Instance Models} \\ \midrule Equation & \begin{displaymath} {E_{\text{P}}}=\begin{cases} - {{C_{\text{P}}}^{\text{S}}} {m_{\text{P}}} \left({T_{\text{P}}}\left(t\right)-{T_{\text{init}}}\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ - {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{\text{f}}} {m_{\text{P}}}+{{C_{\text{P}}}^{\text{L}}} {m_{\text{P}}} \left({T_{\text{P}}}\left(t\right)-{{T_{\text{melt}}}^{\text{P}}}\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ + {{C_{\text{P}}}^{\text{S}}}\,{m_{\text{P}}}\,\left({T_{\text{P}}}\left(t\right)-{T_{\text{init}}}\right), & {T_{\text{P}}}\lt{}{{T_{\text{melt}}}^{\text{P}}}\\ + {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{H_{\text{f}}}\,{m_{\text{P}}}+{{C_{\text{P}}}^{\text{L}}}\,{m_{\text{P}}}\,\left({T_{\text{P}}}\left(t\right)-{{T_{\text{melt}}}^{\text{P}}}\right), & {T_{\text{P}}}\gt{}{{T_{\text{melt}}}^{\text{P}}}\\ {{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}+{Q_{\text{P}}}\left(t\right), & {T_{\text{P}}}={{T_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1 \end{cases} \end{displaymath} @@ -1419,7 +1419,7 @@ \subsubsection{Instance Models} ${E_{\text{P}}}$ for the melted PCM (${T_{\text{P}}}\gt{}{{{E_{\text{P}}}_{\text{melt}}}^{\text{init}}}$) is found using \hyperref[TM:sensHtE]{TM:sensHtE} for sensible heat of the liquid PCM plus the energy when melting starts, plus the energy required to melt all of the PCM. - The energy required to melt all of the PCM is ${H_{\text{f}}} {m_{\text{P}}}$ (${\text{J}}$) (from \hyperref[DD:htFusion]{DD:htFusion}). + The energy required to melt all of the PCM is ${H_{\text{f}}}\,{m_{\text{P}}}$ (${\text{J}}$) (from \hyperref[DD:htFusion]{DD:htFusion}). The change in temperature is ${T_{\text{P}}}-{{T_{\text{melt}}}^{\text{P}}}$ (${{}^{\circ}\text{C}}$). @@ -1448,7 +1448,7 @@ \subsubsection{Data Constraints} \\ ${A_{\text{C}}}$ & ${A_{\text{C}}}\gt{}0$ & ${A_{\text{C}}}\leq{}{{A_{\text{C}}}^{\text{max}}}$ & $0.12$ ${\text{m}^{2}}$ & 10$\%$ \\ -${A_{\text{P}}}$ & ${A_{\text{P}}}\gt{}0$ & ${V_{\text{P}}}\leq{}{A_{\text{P}}}\leq{}\frac{2}{{h_{\text{min}}}} {V_{\text{tank}}}$ & $1.2$ ${\text{m}^{2}}$ & 10$\%$ +${A_{\text{P}}}$ & ${A_{\text{P}}}\gt{}0$ & ${V_{\text{P}}}\leq{}{A_{\text{P}}}\leq{}\frac{2}{{h_{\text{min}}}}\,{V_{\text{tank}}}$ & $1.2$ ${\text{m}^{2}}$ & 10$\%$ \\ ${{C_{\text{P}}}^{\text{L}}}$ & ${{C_{\text{P}}}^{\text{L}}}\gt{}0$ & ${{{C_{\text{P}}}^{\text{L}}}_{\text{min}}}\lt{}{{C_{\text{P}}}^{\text{L}}}\lt{}{{{C_{\text{P}}}^{\text{L}}}_{\text{max}}}$ & $2270$ $\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}$ & 10$\%$ \\ @@ -1476,7 +1476,7 @@ \subsubsection{Data Constraints} \\ ${t_{\text{step}}}$ & $0\lt{}{t_{\text{step}}}\lt{}{t_{\text{final}}}$ & -- & $0.01$ ${\text{s}}$ & 10$\%$ \\ -${V_{\text{P}}}$ & $0\lt{}{V_{\text{P}}}\lt{}{V_{\text{tank}}}$ & ${V_{\text{P}}}\geq{}\mathit{MINFRACT} {V_{\text{tank}}}$ & $0.05$ ${\text{m}^{3}}$ & 10$\%$ +${V_{\text{P}}}$ & $0\lt{}{V_{\text{P}}}\lt{}{V_{\text{tank}}}$ & ${V_{\text{P}}}\geq{}\mathit{MINFRACT}\,{V_{\text{tank}}}$ & $0.05$ ${\text{m}^{3}}$ & 10$\%$ \\ ${ρ_{\text{P}}}$ & ${ρ_{\text{P}}}\gt{}0$ & ${{ρ_{\text{P}}}^{\text{min}}}\lt{}{ρ_{\text{P}}}\lt{}{{ρ_{\text{P}}}^{\text{max}}}$ & $1007$ $\frac{\text{kg}}{\text{m}^{3}}$ & 10$\%$ \\ @@ -1504,12 +1504,12 @@ \subsubsection{Properties of a Correct Solution} A correct solution must exhibit the law of conservation of energy. This means that the change in heat energy in the water should equal the difference between the total energy input from the heating coil and the energy output to the PCM. This can be shown as an equation by taking \hyperref[GD:htFluxWaterFromCoil]{GD:htFluxWaterFromCoil} and \hyperref[GD:htFluxPCMFromWater]{GD:htFluxPCMFromWater}, multiplying each by their respective surface area of heat transfer, and integrating each over the simulation time, as follows: \begin{displaymath} -{E_{\text{W}}}=\int_{0}^{t}{{h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)}\,dt-\int_{0}^{t}{{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt +{E_{\text{W}}}=\int_{0}^{t}{{h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)}\,dt-\int_{0}^{t}{{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt \end{displaymath} In addition, the change in heat energy in the PCM should equal the energy input to the PCM from the water. This can be expressed as \begin{displaymath} -{E_{\text{P}}}=\int_{0}^{t}{{h_{\text{P}}} {A_{\text{P}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt +{E_{\text{P}}}=\int_{0}^{t}{{h_{\text{P}}}\,{A_{\text{P}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{P}}}\left(t\right)\right)}\,dt \end{displaymath} Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as ``sanity'' checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than ${C_{\text{tol}}}$ \hyperref[verifyEnergyOutput]{FR:Verify-Energy-Output-Follow-Conservation-of-Energy}. diff --git a/code/stable/swhs/SRS/mdBook/src/SecCorSolProps.md b/code/stable/swhs/SRS/mdBook/src/SecCorSolProps.md index 812de2c188..cb53f39a86 100644 --- a/code/stable/swhs/SRS/mdBook/src/SecCorSolProps.md +++ b/code/stable/swhs/SRS/mdBook/src/SecCorSolProps.md @@ -15,10 +15,10 @@ The [Data Constraints Table](./SecCorSolProps.md#Table:OutDataConstraints) shows A correct solution must exhibit the law of conservation of energy. This means that the change in heat energy in the water should equal the difference between the total energy input from the heating coil and the energy output to the PCM. This can be shown as an equation by taking [GD:htFluxWaterFromCoil](./SecGDs.md#GD:htFluxWaterFromCoil) and [GD:htFluxPCMFromWater](./SecGDs.md#GD:htFluxPCMFromWater), multiplying each by their respective surface area of heat transfer, and integrating each over the simulation time, as follows: -\\[{E\_{\text{W}}}=\int\_{0}^{t}{{h\_{\text{C}}} {A\_{\text{C}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)\right)}\\,dt-\int\_{0}^{t}{{h\_{\text{P}}} {A\_{\text{P}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right)}\\,dt\\] +\\[{E\_{\text{W}}}=\int\_{0}^{t}{{h\_{\text{C}}}\\,{A\_{\text{C}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)\right)}\\,dt-\int\_{0}^{t}{{h\_{\text{P}}}\\,{A\_{\text{P}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right)}\\,dt\\] In addition, the change in heat energy in the PCM should equal the energy input to the PCM from the water. This can be expressed as -\\[{E\_{\text{P}}}=\int\_{0}^{t}{{h\_{\text{P}}} {A\_{\text{P}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right)}\\,dt\\] +\\[{E\_{\text{P}}}=\int\_{0}^{t}{{h\_{\text{P}}}\\,{A\_{\text{P}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right)}\\,dt\\] Equations (FIXME: Equation 7) and (FIXME: Equation 8) can be used as "sanity" checks to gain confidence in any solution computed by SWHS. The relative error between the results computed by SWHS and the results calculated from the RHS of these equations should be less than \\({C\_{\text{tol}}}\\) [FR:Verify-Energy-Output-Follow-Conservation-of-Energy](./SecFRs.md#verifyEnergyOutput). diff --git a/code/stable/swhs/SRS/mdBook/src/SecDDs.md b/code/stable/swhs/SRS/mdBook/src/SecDDs.md index 131c5c38c3..e9882cf99f 100644 --- a/code/stable/swhs/SRS/mdBook/src/SecDDs.md +++ b/code/stable/swhs/SRS/mdBook/src/SecDDs.md @@ -13,7 +13,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Mass of water | |Symbol |\\({m\_{\text{W}}}\\) | |Units |\\({\text{kg}}\\) | -|Equation |\\[{m\_{\text{W}}}={V\_{\text{W}}} {ρ\_{\text{W}}}\\] | +|Equation |\\[{m\_{\text{W}}}={V\_{\text{W}}}\\,{ρ\_{\text{W}}}\\] | |Description|
  • \\({m\_{\text{W}}}\\) is the mass of water (\\({\text{kg}}\\))
  • \\({V\_{\text{W}}}\\) is the volume of water (\\({\text{m}^{3}}\\))
  • \\({ρ\_{\text{W}}}\\) is the density of water (\\(\frac{\text{kg}}{\text{m}^{3}}\\))
| |Source |-- | |RefBy |[FR:Find-Mass](./SecFRs.md#findMass) | @@ -46,7 +46,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Volume of the cylindrical tank | |Symbol |\\({V\_{\text{tank}}}\\) | |Units |\\({\text{m}^{3}}\\) | -|Equation |\\[{V\_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L\\] | +|Equation |\\[{V\_{\text{tank}}}=π\\,\left(\frac{D}{2}\right)^{2}\\,L\\] | |Description|
  • \\({V\_{\text{tank}}}\\) is the volume of the cylindrical tank (\\({\text{m}^{3}}\\))
  • \\(π\\) is the ratio of circumference to diameter for any circle (Unitless)
  • \\(D\\) is the diameter of tank (\\({\text{m}}\\))
  • \\(L\\) is the length of tank (\\({\text{m}}\\))
| |Source |-- | |RefBy |[DD:waterVolume_pcm](./SecDDs.md#DD:waterVolume.pcm) and [FR:Find-Mass](./SecFRs.md#findMass) | @@ -62,7 +62,7 @@ This section collects and defines all the data needed to build the instance mode |Label |ODE parameter for water related to decay time | |Symbol |\\({τ\_{\text{W}}}\\) | |Units |\\({\text{s}}\\) | -|Equation |\\[{τ\_{\text{W}}}=\frac{{m\_{\text{W}}} {C\_{\text{W}}}}{{h\_{\text{C}}} {A\_{\text{C}}}}\\] | +|Equation |\\[{τ\_{\text{W}}}=\frac{{m\_{\text{W}}}\\,{C\_{\text{W}}}}{{h\_{\text{C}}}\\,{A\_{\text{C}}}}\\] | |Description|
  • \\({τ\_{\text{W}}}\\) is the ODE parameter for water related to decay time (\\({\text{s}}\\))
  • \\({m\_{\text{W}}}\\) is the mass of water (\\({\text{kg}}\\))
  • \\({C\_{\text{W}}}\\) is the specific heat capacity of water (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({h\_{\text{C}}}\\) is the convective heat transfer coefficient between coil and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({A\_{\text{C}}}\\) is the heating coil surface area (\\({\text{m}^{2}}\\))
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | |RefBy |[IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr) and [FR:Output-Input-Derived-Values](./SecFRs.md#outputInputDerivVals) | @@ -78,7 +78,7 @@ This section collects and defines all the data needed to build the instance mode |Label |ODE parameter related to decay rate | |Symbol |\\(η\\) | |Units |Unitless | -|Equation |\\[η=\frac{{h\_{\text{P}}} {A\_{\text{P}}}}{{h\_{\text{C}}} {A\_{\text{C}}}}\\] | +|Equation |\\[η=\frac{{h\_{\text{P}}}\\,{A\_{\text{P}}}}{{h\_{\text{C}}}\\,{A\_{\text{C}}}}\\] | |Description|
  • \\(η\\) is the ODE parameter related to decay rate (Unitless)
  • \\({h\_{\text{P}}}\\) is the convective heat transfer coefficient between PCM and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({A\_{\text{P}}}\\) is the phase change material surface area (\\({\text{m}^{2}}\\))
  • \\({h\_{\text{C}}}\\) is the convective heat transfer coefficient between coil and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({A\_{\text{C}}}\\) is the heating coil surface area (\\({\text{m}^{2}}\\))
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | |RefBy |[IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr) and [FR:Output-Input-Derived-Values](./SecFRs.md#outputInputDerivVals) | @@ -94,7 +94,7 @@ This section collects and defines all the data needed to build the instance mode |Label |ODE parameter for solid PCM | |Symbol |\\({{τ\_{\text{P}}}^{\text{S}}}\\) | |Units |\\({\text{s}}\\) | -|Equation |\\[{{τ\_{\text{P}}}^{\text{S}}}=\frac{{m\_{\text{P}}} {{C\_{\text{P}}}^{\text{S}}}}{{h\_{\text{P}}} {A\_{\text{P}}}}\\] | +|Equation |\\[{{τ\_{\text{P}}}^{\text{S}}}=\frac{{m\_{\text{P}}}\\,{{C\_{\text{P}}}^{\text{S}}}}{{h\_{\text{P}}}\\,{A\_{\text{P}}}}\\] | |Description|
  • \\({{τ\_{\text{P}}}^{\text{S}}}\\) is the ODE parameter for solid PCM (\\({\text{s}}\\))
  • \\({m\_{\text{P}}}\\) is the mass of phase change material (\\({\text{kg}}\\))
  • \\({{C\_{\text{P}}}^{\text{S}}}\\) is the specific heat capacity of PCM as a solid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({h\_{\text{P}}}\\) is the convective heat transfer coefficient between PCM and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({A\_{\text{P}}}\\) is the phase change material surface area (\\({\text{m}^{2}}\\))
| |Source |[lightstone2012](./SecReferences.md#lightstone2012) | |RefBy |[IM:eBalanceOnPCM](./SecIMs.md#IM:eBalanceOnPCM) and [FR:Output-Input-Derived-Values](./SecFRs.md#outputInputDerivVals) | @@ -110,7 +110,7 @@ This section collects and defines all the data needed to build the instance mode |Label |ODE parameter for liquid PCM | |Symbol |\\({{τ\_{\text{P}}}^{\text{L}}}\\) | |Units |\\({\text{s}}\\) | -|Equation |\\[{{τ\_{\text{P}}}^{\text{L}}}=\frac{{m\_{\text{P}}} {{C\_{\text{P}}}^{\text{L}}}}{{h\_{\text{P}}} {A\_{\text{P}}}}\\] | +|Equation |\\[{{τ\_{\text{P}}}^{\text{L}}}=\frac{{m\_{\text{P}}}\\,{{C\_{\text{P}}}^{\text{L}}}}{{h\_{\text{P}}}\\,{A\_{\text{P}}}}\\] | |Description|
  • \\({{τ\_{\text{P}}}^{\text{L}}}\\) is the ODE parameter for liquid PCM (\\({\text{s}}\\))
  • \\({m\_{\text{P}}}\\) is the mass of phase change material (\\({\text{kg}}\\))
  • \\({{C\_{\text{P}}}^{\text{L}}}\\) is the specific heat capacity of PCM as a liquid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({h\_{\text{P}}}\\) is the convective heat transfer coefficient between PCM and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({A\_{\text{P}}}\\) is the phase change material surface area (\\({\text{m}^{2}}\\))
| |Source |[lightstone2012](./SecReferences.md#lightstone2012) | |RefBy |[IM:eBalanceOnPCM](./SecIMs.md#IM:eBalanceOnPCM) and [FR:Output-Input-Derived-Values](./SecFRs.md#outputInputDerivVals) | @@ -143,7 +143,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Melt fraction | |Symbol |\\(ϕ\\) | |Units |Unitless | -|Equation |\\[ϕ=\frac{{Q\_{\text{P}}}}{{H\_{\text{f}}} {m\_{\text{P}}}}\\] | +|Equation |\\[ϕ=\frac{{Q\_{\text{P}}}}{{H\_{\text{f}}}\\,{m\_{\text{P}}}}\\] | |Description|
  • \\(ϕ\\) is the melt fraction (Unitless)
  • \\({Q\_{\text{P}}}\\) is the latent heat energy added to PCM (\\({\text{J}}\\))
  • \\({H\_{\text{f}}}\\) is the specific latent heat of fusion (\\(\frac{\text{J}}{\text{kg}}\\))
  • \\({m\_{\text{P}}}\\) is the mass of phase change material (\\({\text{kg}}\\))
| |Notes |
  • The value of \\(ϕ\\) is constrained to \\(0\leq{}ϕ\leq{}1\\).
  • [DD:htFusion](./SecDDs.md#DD:htFusion)
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | diff --git a/code/stable/swhs/SRS/mdBook/src/SecDataConstraints.md b/code/stable/swhs/SRS/mdBook/src/SecDataConstraints.md index 23bfabf064..eeb53907a7 100644 --- a/code/stable/swhs/SRS/mdBook/src/SecDataConstraints.md +++ b/code/stable/swhs/SRS/mdBook/src/SecDataConstraints.md @@ -7,7 +7,7 @@ The [Data Constraints Table](./SecDataConstraints.md#Table:InDataConstraints) sh |Var |Physical Constraints |Software Constraints |Typical Value |Uncert. | |:------------------------------------|:--------------------------------------------------------------|:-----------------------------------------------------------------------------------------------------------------------------------|:---------------------------------------------------------------|:----------| |\\({A\_{\text{C}}}\\) |\\({A\_{\text{C}}}\gt{}0\\) |\\({A\_{\text{C}}}\leq{}{{A\_{\text{C}}}^{\text{max}}}\\) |\\(0.12\\) \\({\text{m}^{2}}\\) |10\\(\\%\\)| -|\\({A\_{\text{P}}}\\) |\\({A\_{\text{P}}}\gt{}0\\) |\\({V\_{\text{P}}}\leq{}{A\_{\text{P}}}\leq{}\frac{2}{{h\_{\text{min}}}} {V\_{\text{tank}}}\\) |\\(1.2\\) \\({\text{m}^{2}}\\) |10\\(\\%\\)| +|\\({A\_{\text{P}}}\\) |\\({A\_{\text{P}}}\gt{}0\\) |\\({V\_{\text{P}}}\leq{}{A\_{\text{P}}}\leq{}\frac{2}{{h\_{\text{min}}}}\\,{V\_{\text{tank}}}\\) |\\(1.2\\) \\({\text{m}^{2}}\\) |10\\(\\%\\)| |\\({{C\_{\text{P}}}^{\text{L}}}\\) |\\({{C\_{\text{P}}}^{\text{L}}}\gt{}0\\) |\\({{{C\_{\text{P}}}^{\text{L}}}\_{\text{min}}}\lt{}{{C\_{\text{P}}}^{\text{L}}}\lt{}{{{C\_{\text{P}}}^{\text{L}}}\_{\text{max}}}\\)|\\(2270\\) \\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\) |10\\(\\%\\)| |\\({{C\_{\text{P}}}^{\text{S}}}\\) |\\({{C\_{\text{P}}}^{\text{S}}}\gt{}0\\) |\\({{{C\_{\text{P}}}^{\text{S}}}\_{\text{min}}}\lt{}{{C\_{\text{P}}}^{\text{S}}}\lt{}{{{C\_{\text{P}}}^{\text{S}}}\_{\text{max}}}\\)|\\(1760\\) \\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\) |10\\(\\%\\)| |\\({C\_{\text{W}}}\\) |\\({C\_{\text{W}}}\gt{}0\\) |\\({{C\_{\text{W}}}^{\text{min}}}\lt{}{C\_{\text{W}}}\lt{}{{C\_{\text{W}}}^{\text{max}}}\\) |\\(4186\\) \\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\) |10\\(\\%\\)| @@ -21,7 +21,7 @@ The [Data Constraints Table](./SecDataConstraints.md#Table:InDataConstraints) sh |\\({{T\_{\text{melt}}}^{\text{P}}}\\)|\\(0\lt{}{{T\_{\text{melt}}}^{\text{P}}}\lt{}{T\_{\text{C}}}\\)|-- |\\(44.2\\) \\({{}^{\circ}\text{C}}\\) |10\\(\\%\\)| |\\({t\_{\text{final}}}\\) |\\({t\_{\text{final}}}\gt{}0\\) |\\({t\_{\text{final}}}\lt{}{{t\_{\text{final}}}^{\text{max}}}\\) |\\(50000\\) \\({\text{s}}\\) |10\\(\\%\\)| |\\({t\_{\text{step}}}\\) |\\(0\lt{}{t\_{\text{step}}}\lt{}{t\_{\text{final}}}\\) |-- |\\(0.01\\) \\({\text{s}}\\) |10\\(\\%\\)| -|\\({V\_{\text{P}}}\\) |\\(0\lt{}{V\_{\text{P}}}\lt{}{V\_{\text{tank}}}\\) |\\({V\_{\text{P}}}\geq{}\mathit{MINFRACT} {V\_{\text{tank}}}\\) |\\(0.05\\) \\({\text{m}^{3}}\\) |10\\(\\%\\)| +|\\({V\_{\text{P}}}\\) |\\(0\lt{}{V\_{\text{P}}}\lt{}{V\_{\text{tank}}}\\) |\\({V\_{\text{P}}}\geq{}\mathit{MINFRACT}\\,{V\_{\text{tank}}}\\) |\\(0.05\\) \\({\text{m}^{3}}\\) |10\\(\\%\\)| |\\({ρ\_{\text{P}}}\\) |\\({ρ\_{\text{P}}}\gt{}0\\) |\\({{ρ\_{\text{P}}}^{\text{min}}}\lt{}{ρ\_{\text{P}}}\lt{}{{ρ\_{\text{P}}}^{\text{max}}}\\) |\\(1007\\) \\(\frac{\text{kg}}{\text{m}^{3}}\\) |10\\(\\%\\)| |\\({ρ\_{\text{W}}}\\) |\\({ρ\_{\text{W}}}\gt{}0\\) |\\({{ρ\_{\text{W}}}^{\text{min}}}\lt{}{ρ\_{\text{W}}}\leq{}{{ρ\_{\text{W}}}^{\text{max}}}\\) |\\(1000\\) \\(\frac{\text{kg}}{\text{m}^{3}}\\) |10\\(\\%\\)| diff --git a/code/stable/swhs/SRS/mdBook/src/SecGDs.md b/code/stable/swhs/SRS/mdBook/src/SecGDs.md index 0da70b0dd8..7f96493257 100644 --- a/code/stable/swhs/SRS/mdBook/src/SecGDs.md +++ b/code/stable/swhs/SRS/mdBook/src/SecGDs.md @@ -11,7 +11,7 @@ This section collects the laws and equations that will be used to build the inst |Refname |GD:rocTempSimp | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Simplified rate of change of temperature | -|Equation |\\[m C \frac{\\,dT}{\\,dt}={q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V\\] | +|Equation |\\[m\\,C\\,\frac{\\,dT}{\\,dt}={q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V\\] | |Description|
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(C\\) is the specific heat capacity (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(T\\) is the temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({q\_{\text{in}}}\\) is the heat flux input (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({A\_{\text{in}}}\\) is the surface area over which heat is transferred in (\\({\text{m}^{2}}\\))
  • \\({q\_{\text{out}}}\\) is the heat flux output (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({A\_{\text{out}}}\\) is the surface area over which heat is transferred out (\\({\text{m}^{2}}\\))
  • \\(g\\) is the volumetric heat generation per unit volume (\\(\frac{\text{W}}{\text{m}^{3}}\\))
  • \\(V\\) is the volume (\\({\text{m}^{3}}\\))
| |Source |-- | |RefBy |[GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp), [IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr), and [IM:eBalanceOnPCM](./SecIMs.md#IM:eBalanceOnPCM) | @@ -20,23 +20,23 @@ This section collects the laws and equations that will be used to build the inst Integrating [TM:consThermE](./SecTMs.md#TM:consThermE) over a volume (\\(V\\)), we have: -\\[-\int\_{V}{∇\cdot{}\boldsymbol{q}}\\,dV+\int\_{V}{g}\\,dV=\int\_{V}{ρ C \frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] +\\[-\int\_{V}{∇\cdot{}\boldsymbol{q}}\\,dV+\int\_{V}{g}\\,dV=\int\_{V}{ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] Applying Gauss's Divergence Theorem to the first term over the surface \\(S\\) of the volume, with \\(\boldsymbol{q}\\) as the thermal flux vector for the surface and \\(\boldsymbol{\hat{n}}\\) as a unit outward normal vector for a surface: -\\[-\int\_{S}{\boldsymbol{q}\cdot{}\boldsymbol{\hat{n}}}\\,dS+\int\_{V}{g}\\,dV=\int\_{V}{ρ C \frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] +\\[-\int\_{S}{\boldsymbol{q}\cdot{}\boldsymbol{\hat{n}}}\\,dS+\int\_{V}{g}\\,dV=\int\_{V}{ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as: -\\[{q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V=\int\_{V}{ρ C \frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] +\\[{q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V=\int\_{V}{ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] Where \\({q\_{\text{in}}}\\), \\({q\_{\text{out}}}\\), \\({A\_{\text{in}}}\\), and \\({A\_{\text{out}}}\\) are explained in [GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp). The integral over the surface could be simplified because the thermal flux is assumed constant over \\({A\_{\text{in}}}\\) and \\({A\_{\text{out}}}\\) and \\(0\\) on all other surfaces. Outward flux is considered positive. Assuming \\(ρ\\), \\(C\\), and \\(T\\) are constant over the volume, which is true in our case by [A:Constant-Water-Temp-Across-Tank](./SecAssumps.md#assumpCWTAT), [A:Temp-PCM-Constant-Across-Volume](./SecAssumps.md#assumpTPCAV), [A:Density-Water-PCM-Constant-over-Volume](./SecAssumps.md#assumpDWPCoV), and [A:Specific-Heat-Energy-Constant-over-Volume](./SecAssumps.md#assumpSHECov), we have: -\\[ρ C V \frac{\\,dT}{\\,dt}={q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V\\] +\\[ρ\\,C\\,V\\,\frac{\\,dT}{\\,dt}={q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V\\] Using the fact that \\(ρ\\)=\\(m\\)/\\(V\\), Equation (2) can be written as: -\\[m C \frac{\\,dT}{\\,dt}={q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V\\] +\\[m\\,C\\,\frac{\\,dT}{\\,dt}={q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V\\]
@@ -48,7 +48,7 @@ Using the fact that \\(ρ\\)=\\(m\\)/\\(V\\), Equation (2) can be written as: |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Heat flux into the water from the coil | |Units |\\(\frac{\text{W}}{\text{m}^{2}}\\) | -|Equation |\\[{q\_{\text{C}}}={h\_{\text{C}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)\right)\\] | +|Equation |\\[{q\_{\text{C}}}={h\_{\text{C}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)\right)\\] | |Description|
  • \\({q\_{\text{C}}}\\) is the heat flux into the water from the coil (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({h\_{\text{C}}}\\) is the convective heat transfer coefficient between coil and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{C}}}\\) is the temperature of the heating coil (\\({{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
| |Notes |
  • \\({q\_{\text{C}}}\\) is found by assuming that Newton's law of cooling applies ([A:Newton-Law-Convective-Cooling-Coil-Water](./SecAssumps.md#assumpLCCCW)). This law (defined in [TM:nwtnCooling](./SecTMs.md#TM:nwtnCooling)) is used on the surface of the heating coil.
  • [A:Temp-Heating-Coil-Constant-over-Time](./SecAssumps.md#assumpTHCCoT)
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | @@ -64,7 +64,7 @@ Using the fact that \\(ρ\\)=\\(m\\)/\\(V\\), Equation (2) can be written as: |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Heat flux into the PCM from water | |Units |\\(\frac{\text{W}}{\text{m}^{2}}\\) | -|Equation |\\[{q\_{\text{P}}}={h\_{\text{P}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right)\\] | +|Equation |\\[{q\_{\text{P}}}={h\_{\text{P}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right)\\] | |Description|
  • \\({q\_{\text{P}}}\\) is the heat flux into the PCM from water (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({h\_{\text{P}}}\\) is the convective heat transfer coefficient between PCM and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({T\_{\text{P}}}\\) is the temperature of the phase change material (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • \\({q\_{\text{P}}}\\) is found by assuming that Newton's law of cooling applies ([A:Law-Convective-Cooling-Water-PCM](./SecAssumps.md#assumpLCCWP)). This law (defined in [TM:nwtnCooling](./SecTMs.md#TM:nwtnCooling)) is used on the surface of the phase change material.
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | diff --git a/code/stable/swhs/SRS/mdBook/src/SecIMs.md b/code/stable/swhs/SRS/mdBook/src/SecIMs.md index cf2f6f9ad6..89e956023b 100644 --- a/code/stable/swhs/SRS/mdBook/src/SecIMs.md +++ b/code/stable/swhs/SRS/mdBook/src/SecIMs.md @@ -17,7 +17,7 @@ The goals [GS:Predict-Water-Temperature](./SecGoalStmt.md#waterTempGS), [GS:Pred |Output |\\({T\_{\text{W}}}\\) | |Input Constraints |\\[{T\_{\text{C}}}\gt{}{T\_{\text{init}}}\\] | |Output Constraints| | -|Equation |\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)+η \left({T\_{\text{P}}}\left(t\right)-{T\_{\text{W}}}\left(t\right)\right)\right)\\] | +|Equation |\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)+η\\,\left({T\_{\text{P}}}\left(t\right)-{T\_{\text{W}}}\left(t\right)\right)\right)\\] | |Description |
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\({τ\_{\text{W}}}\\) is the ODE parameter for water related to decay time (\\({\text{s}}\\))
  • \\({T\_{\text{C}}}\\) is the temperature of the heating coil (\\({{}^{\circ}\text{C}}\\))
  • \\(η\\) is the ODE parameter related to decay rate (Unitless)
  • \\({T\_{\text{P}}}\\) is the temperature of the phase change material (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • \\({T\_{\text{P}}}\\) is defined by [IM:eBalanceOnPCM](./SecIMs.md#IM:eBalanceOnPCM).
  • The input constraint \\({T\_{\text{init}}}\leq{}{T\_{\text{C}}}\\) comes from [A:Charging-Tank-No-Temp-Discharge](./SecAssumps.md#assumpCTNOD).
  • \\({τ\_{\text{W}}}\\) is calculated from [DD:balanceDecayRate](./SecDDs.md#DD:balanceDecayRate).
  • \\(η\\) is calculated from [DD:balanceDecayTime](./SecDDs.md#DD:balanceDecayTime).
  • The initial conditions for the ODE are \\({T\_{\text{W}}}\left(0\right)={T\_{\text{P}}}\left(0\right)={T\_{\text{init}}}\\) following [A:Same-Initial-Temp-Water-PCM](./SecAssumps.md#assumpSITWP).
  • The ODE applies as long as the water is in liquid form, \\(0\lt{}{T\_{\text{W}}}\lt{}100\\) (\\({{}^{\circ}\text{C}}\\)) where \\(0\\) (\\({{}^{\circ}\text{C}}\\)) and \\(100\\) (\\({{}^{\circ}\text{C}}\\)) are the melting and boiling point temperatures of water, respectively (from [A:Water-Always-Liquid](./SecAssumps.md#assumpWAL) and [A:Atmospheric-Pressure-Tank](./SecAssumps.md#assumpAPT)).
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | @@ -27,31 +27,31 @@ The goals [GS:Predict-Water-Temperature](./SecGoalStmt.md#waterTempGS), [GS:Pred To find the rate of change of \\({T\_{\text{W}}}\\), we look at the energy balance on water. The volume being considered is the volume of water in the tank \\({V\_{\text{W}}}\\), which has mass \\({m\_{\text{W}}}\\) and specific heat capacity, \\({C\_{\text{W}}}\\). Heat transfer occurs in the water from the heating coil as \\({q\_{\text{C}}}\\) ([GD:htFluxWaterFromCoil](./SecGDs.md#GD:htFluxWaterFromCoil)) and from the water into the PCM as \\({q\_{\text{P}}}\\) ([GD:htFluxPCMFromWater](./SecGDs.md#GD:htFluxPCMFromWater)), over areas \\({A\_{\text{C}}}\\) and \\({A\_{\text{P}}}\\), respectively. The thermal flux is constant over \\({A\_{\text{C}}}\\), since the temperature of the heating coil is assumed to not vary along its length ([A:Temp-Heating-Coil-Constant-over-Length](./SecAssumps.md#assumpTHCCoL)), and the thermal flux is constant over \\({A\_{\text{P}}}\\), since the temperature of the PCM is the same throughout its volume ([A:Temp-PCM-Constant-Across-Volume](./SecAssumps.md#assumpTPCAV)) and the water is fully mixed ([A:Constant-Water-Temp-Across-Tank](./SecAssumps.md#assumpCWTAT)). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated ([A:Perfect-Insulation-Tank](./SecAssumps.md#assumpPIT)). Since the assumption is made that no internal heat is generated ([A:No-Internal-Heat-Generation-By-Water-PCM](./SecAssumps.md#assumpNIHGBWP)), \\(g=0\\). Therefore, the equation for [GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp) can be written as: -\\[{m\_{\text{W}}} {C\_{\text{W}}} \frac{\\,d{T\_{\text{W}}}}{\\,dt}={q\_{\text{C}}} {A\_{\text{C}}}-{q\_{\text{P}}} {A\_{\text{P}}}\\] +\\[{m\_{\text{W}}}\\,{C\_{\text{W}}}\\,\frac{\\,d{T\_{\text{W}}}}{\\,dt}={q\_{\text{C}}}\\,{A\_{\text{C}}}-{q\_{\text{P}}}\\,{A\_{\text{P}}}\\] Using [GD:htFluxWaterFromCoil](./SecGDs.md#GD:htFluxWaterFromCoil) for \\({q\_{\text{C}}}\\) and [GD:htFluxPCMFromWater](./SecGDs.md#GD:htFluxPCMFromWater) for \\({q\_{\text{P}}}\\), this can be written as: -\\[{m\_{\text{W}}} {C\_{\text{W}}} \frac{\\,d{T\_{\text{W}}}}{\\,dt}={h\_{\text{C}}} {A\_{\text{C}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)-{h\_{\text{P}}} {A\_{\text{P}}} \left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] +\\[{m\_{\text{W}}}\\,{C\_{\text{W}}}\\,\frac{\\,d{T\_{\text{W}}}}{\\,dt}={h\_{\text{C}}}\\,{A\_{\text{C}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)-{h\_{\text{P}}}\\,{A\_{\text{P}}}\\,\left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] -Dividing Equation (2) by \\({m\_{\text{W}}} {C\_{\text{W}}}\\), we obtain: +Dividing Equation (2) by \\({m\_{\text{W}}}\\,{C\_{\text{W}}}\\), we obtain: -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}} {A\_{\text{C}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)-\frac{{h\_{\text{P}}} {A\_{\text{P}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}}\\,{A\_{\text{C}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)-\frac{{h\_{\text{P}}}\\,{A\_{\text{P}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] Factoring the negative sign out of the second term of the right-hand side (RHS) of Equation (3) and multiplying it by \\({h\_{\text{C}}}\\) \\({A\_{\text{C}}}\\) / \\({h\_{\text{C}}}\\) \\({A\_{\text{C}}}\\) yields: -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}} {A\_{\text{C}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)+\frac{{h\_{\text{C}}} {A\_{\text{C}}}}{{h\_{\text{C}}} {A\_{\text{C}}}} \frac{{h\_{\text{P}}} {A\_{\text{P}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}}\\,{A\_{\text{C}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)+\frac{{h\_{\text{C}}}\\,{A\_{\text{C}}}}{{h\_{\text{C}}}\\,{A\_{\text{C}}}}\\,\frac{{h\_{\text{P}}}\\,{A\_{\text{P}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\\] Rearranging this equation gives us: -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}} {A\_{\text{C}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)+\frac{{h\_{\text{P}}} {A\_{\text{P}}}}{{h\_{\text{C}}} {A\_{\text{C}}}} \frac{{h\_{\text{C}}} {A\_{\text{C}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}}\\,{A\_{\text{C}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)+\frac{{h\_{\text{P}}}\\,{A\_{\text{P}}}}{{h\_{\text{C}}}\\,{A\_{\text{C}}}}\\,\frac{{h\_{\text{C}}}\\,{A\_{\text{C}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\\] By substituting \\({τ\_{\text{W}}}\\) (from [DD:balanceDecayRate](./SecDDs.md#DD:balanceDecayRate)) and \\(η\\) (from [DD:balanceDecayTime](./SecDDs.md#DD:balanceDecayTime)), this can be written as: -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)+\frac{η}{{τ\_{\text{W}}}} \left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)+\frac{η}{{τ\_{\text{W}}}}\\,\left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\\] Finally, factoring out \\(\frac{1}{{τ\_{\text{W}}}}\\), we are left with the governing ODE for [IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr): -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}+η \left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}+η\\,\left({T\_{\text{P}}}-{T\_{\text{W}}}\right)\right)\\]
@@ -66,7 +66,7 @@ Finally, factoring out \\(\frac{1}{{τ\_{\text{W}}}}\\), we are left with the go |Output |\\({T\_{\text{P}}}\\) | |Input Constraints |\\[{{T\_{\text{melt}}}^{\text{P}}}\gt{}{T\_{\text{init}}}\\] | |Output Constraints| | -|Equation |\\[\frac{\\,d{T\_{\text{P}}}}{\\,dt}=\begin{cases}\frac{1}{{{τ\_{\text{P}}}^{\text{S}}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right), & {T\_{\text{P}}}\lt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\\frac{1}{{{τ\_{\text{P}}}^{\text{L}}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right), & {T\_{\text{P}}}\gt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\0, & {T\_{\text{P}}}={{T\_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1\end{cases}\\] | +|Equation |\\[\frac{\\,d{T\_{\text{P}}}}{\\,dt}=\begin{cases}\frac{1}{{{τ\_{\text{P}}}^{\text{S}}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right), & {T\_{\text{P}}}\lt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\\frac{1}{{{τ\_{\text{P}}}^{\text{L}}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{P}}}\left(t\right)\right), & {T\_{\text{P}}}\gt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\0, & {T\_{\text{P}}}={{T\_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1\end{cases}\\] | |Description |
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({T\_{\text{P}}}\\) is the temperature of the phase change material (\\({{}^{\circ}\text{C}}\\))
  • \\({{τ\_{\text{P}}}^{\text{S}}}\\) is the ODE parameter for solid PCM (\\({\text{s}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\({{τ\_{\text{P}}}^{\text{L}}}\\) is the ODE parameter for liquid PCM (\\({\text{s}}\\))
  • \\({{T\_{\text{melt}}}^{\text{P}}}\\) is the melting point temperature for PCM (\\({{}^{\circ}\text{C}}\\))
  • \\(ϕ\\) is the melt fraction (Unitless)
| |Notes |
  • \\({T\_{\text{W}}}\\) is defined by [IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr).
  • The input constraint \\({T\_{\text{init}}}\leq{}{{T\_{\text{melt}}}^{\text{P}}}\\) comes from [A:PCM-Initially-Solid](./SecAssumps.md#assumpPIS).
  • The temperature remains constant at \\({{T\_{\text{melt}}}^{\text{P}}}\\), even with the heating (or cooling), until the phase change has occurred for all of the material; that is as long as \\(0\lt{}ϕ\lt{}1\\). \\(ϕ\\) (from [DD:meltFrac](./SecDDs.md#DD:meltFrac)) is determined as part of the heat energy in the PCM, as given in ([IM:heatEInPCM](./SecIMs.md#IM:heatEInPCM)).
  • \\({{τ\_{\text{P}}}^{\text{S}}}\\) is calculated in [DD:balanceSolidPCM](./SecDDs.md#DD:balanceSolidPCM).
  • \\({{τ\_{\text{P}}}^{\text{L}}}\\) is calculated in [DD:balanceLiquidPCM](./SecDDs.md#DD:balanceLiquidPCM).
  • The initial conditions for the ODE are \\({T\_{\text{W}}}\left(0\right)={T\_{\text{P}}}\left(0\right)={T\_{\text{init}}}\\) following [A:Same-Initial-Temp-Water-PCM](./SecAssumps.md#assumpSITWP).
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | @@ -76,19 +76,19 @@ Finally, factoring out \\(\frac{1}{{τ\_{\text{W}}}}\\), we are left with the go To find the rate of change of \\({T\_{\text{P}}}\\), we look at the energy balance on the PCM. The volume being considered is the volume of PCM (\\({V\_{\text{P}}}\\)). The derivation that follows is initially for the solid PCM. The mass of phase change material is \\({m\_{\text{P}}}\\) and the specific heat capacity of PCM as a solid is \\({{C\_{\text{P}}}^{\text{S}}}\\). The heat flux into the PCM from water is \\({q\_{\text{P}}}\\) ([GD:htFluxPCMFromWater](./SecGDs.md#GD:htFluxPCMFromWater)) over phase change material surface area \\({A\_{\text{P}}}\\). The thermal flux is constant over \\({A\_{\text{P}}}\\), since the temperature of the PCM is the same throughout its volume ([A:Temp-PCM-Constant-Across-Volume](./SecAssumps.md#assumpTPCAV)) and the water is fully mixed ([A:Constant-Water-Temp-Across-Tank](./SecAssumps.md#assumpCWTAT)). There is no heat flux output from the PCM. Assuming no volumetric heat generation per unit volume ([A:No-Internal-Heat-Generation-By-Water-PCM](./SecAssumps.md#assumpNIHGBWP)), \\(g=0\\), the equation for [GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp) can be written as: -\\[{m\_{\text{P}}} {{C\_{\text{P}}}^{\text{S}}} \frac{\\,d{T\_{\text{P}}}}{\\,dt}={q\_{\text{P}}} {A\_{\text{P}}}\\] +\\[{m\_{\text{P}}}\\,{{C\_{\text{P}}}^{\text{S}}}\\,\frac{\\,d{T\_{\text{P}}}}{\\,dt}={q\_{\text{P}}}\\,{A\_{\text{P}}}\\] Using [GD:htFluxPCMFromWater](./SecGDs.md#GD:htFluxPCMFromWater) for \\({q\_{\text{P}}}\\), this equation can be written as: -\\[{m\_{\text{P}}} {{C\_{\text{P}}}^{\text{S}}} \frac{\\,d{T\_{\text{P}}}}{\\,dt}={h\_{\text{P}}} {A\_{\text{P}}} \left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] +\\[{m\_{\text{P}}}\\,{{C\_{\text{P}}}^{\text{S}}}\\,\frac{\\,d{T\_{\text{P}}}}{\\,dt}={h\_{\text{P}}}\\,{A\_{\text{P}}}\\,\left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] Dividing by \\({m\_{\text{P}}}\\) \\({{C\_{\text{P}}}^{\text{S}}}\\) we obtain: -\\[\frac{\\,d{T\_{\text{P}}}}{\\,dt}=\frac{{h\_{\text{P}}} {A\_{\text{P}}}}{{m\_{\text{P}}} {{C\_{\text{P}}}^{\text{S}}}} \left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] +\\[\frac{\\,d{T\_{\text{P}}}}{\\,dt}=\frac{{h\_{\text{P}}}\\,{A\_{\text{P}}}}{{m\_{\text{P}}}\\,{{C\_{\text{P}}}^{\text{S}}}}\\,\left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] By substituting \\({{τ\_{\text{P}}}^{\text{S}}}\\) (from [DD:balanceSolidPCM](./SecDDs.md#DD:balanceSolidPCM)), this can be written as: -\\[\frac{\\,d{T\_{\text{P}}}}{\\,dt}=\frac{1}{{{τ\_{\text{P}}}^{\text{S}}}} \left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] +\\[\frac{\\,d{T\_{\text{P}}}}{\\,dt}=\frac{1}{{{τ\_{\text{P}}}^{\text{S}}}}\\,\left({T\_{\text{W}}}-{T\_{\text{P}}}\right)\\] Equation (4) applies for the solid PCM. In the case where all of the PCM is melted, the same derivation applies, except that \\({{C\_{\text{P}}}^{\text{S}}}\\) is replaced by \\({{C\_{\text{P}}}^{\text{L}}}\\), and thus \\({{τ\_{\text{P}}}^{\text{S}}}\\) is replaced by \\({{τ\_{\text{P}}}^{\text{L}}}\\). Although a small change in surface area would be expected with melting, this is not included, since the volume change of the PCM with melting is assumed to be negligible ([A:Volume-Change-Melting-PCM-Negligible](./SecAssumps.md#assumpVCMPN)). @@ -109,7 +109,7 @@ This derivation does not consider the boiling of the PCM, as the PCM is assumed |Output |\\({E\_{\text{W}}}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[{E\_{\text{W}}}\left(t\right)={C\_{\text{W}}} {m\_{\text{W}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{init}}}\right)\\] | +|Equation |\\[{E\_{\text{W}}}\left(t\right)={C\_{\text{W}}}\\,{m\_{\text{W}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{init}}}\right)\\] | |Description |
  • \\({E\_{\text{W}}}\\) is the change in heat energy in the water (\\({\text{J}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({C\_{\text{W}}}\\) is the specific heat capacity of water (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({m\_{\text{W}}}\\) is the mass of water (\\({\text{kg}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{init}}}\\) is the initial temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • The above equation is derived using [TM:sensHtE](./SecTMs.md#TM:sensHtE).
  • The change in temperature is the difference between the temperature at time \\(t\\) (\\({\text{s}}\\)), \\({T\_{\text{W}}}\\) and the initial temperature, \\({T\_{\text{init}}}\\) (\\({{}^{\circ}\text{C}}\\)).
  • This equation applies as long as \\(0\lt{}{T\_{\text{W}}}\lt{}100\\)\\({{}^{\circ}\text{C}}\\) ([A:Water-Always-Liquid](./SecAssumps.md#assumpWAL), [A:Atmospheric-Pressure-Tank](./SecAssumps.md#assumpAPT)).
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | @@ -121,15 +121,15 @@ This derivation does not consider the boiling of the PCM, as the PCM is assumed
-|Refname |IM:heatEInPCM | -|:-----------------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| -|Label |Heat energy in the PCM | -|Input |\\({{T\_{\text{melt}}}^{\text{P}}}\\), \\({t\_{\text{final}}}\\), \\({T\_{\text{init}}}\\), \\({A\_{\text{P}}}\\), \\({h\_{\text{P}}}\\), \\({m\_{\text{P}}}\\), \\({{C\_{\text{P}}}^{\text{S}}}\\), \\({{C\_{\text{P}}}^{\text{L}}}\\), \\({T\_{\text{P}}}\\), \\({H\_{\text{f}}}\\), \\({{t\_{\text{melt}}}^{\text{init}}}\\) | -|Output |\\({E\_{\text{P}}}\\) | -|Input Constraints |\\[{{T\_{\text{melt}}}^{\text{P}}}\gt{}{T\_{\text{init}}}\\] | -|Output Constraints| | -|Equation |\\[{E\_{\text{P}}}=\begin{cases}{{C\_{\text{P}}}^{\text{S}}} {m\_{\text{P}}} \left({T\_{\text{P}}}\left(t\right)-{T\_{\text{init}}}\right), & {T\_{\text{P}}}\lt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\{{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}+{H\_{\text{f}}} {m\_{\text{P}}}+{{C\_{\text{P}}}^{\text{L}}} {m\_{\text{P}}} \left({T\_{\text{P}}}\left(t\right)-{{T\_{\text{melt}}}^{\text{P}}}\right), & {T\_{\text{P}}}\gt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\{{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}+{Q\_{\text{P}}}\left(t\right), & {T\_{\text{P}}}={{T\_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1\end{cases}\\] | -|Description |
  • \\({E\_{\text{P}}}\\) is the change in heat energy in the PCM (\\({\text{J}}\\))
  • \\({{C\_{\text{P}}}^{\text{S}}}\\) is the specific heat capacity of PCM as a solid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({m\_{\text{P}}}\\) is the mass of phase change material (\\({\text{kg}}\\))
  • \\({T\_{\text{P}}}\\) is the temperature of the phase change material (\\({{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({T\_{\text{init}}}\\) is the initial temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}\\) is the change in heat energy in the PCM at the instant when melting begins (\\({\text{J}}\\))
  • \\({H\_{\text{f}}}\\) is the specific latent heat of fusion (\\(\frac{\text{J}}{\text{kg}}\\))
  • \\({{C\_{\text{P}}}^{\text{L}}}\\) is the specific heat capacity of PCM as a liquid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({{T\_{\text{melt}}}^{\text{P}}}\\) is the melting point temperature for PCM (\\({{}^{\circ}\text{C}}\\))
  • \\({Q\_{\text{P}}}\\) is the latent heat energy added to PCM (\\({\text{J}}\\))
  • \\(ϕ\\) is the melt fraction (Unitless)
| -|Notes |
  • The above equation is derived using [TM:sensHtE](./SecTMs.md#TM:sensHtE) and [TM:latentHtE](./SecTMs.md#TM:latentHtE).
  • \\({E\_{\text{P}}}\\) for the solid PCM is found using [TM:sensHtE](./SecTMs.md#TM:sensHtE) for sensible heating, with the specific heat capacity of the solid PCM, \\({{C\_{\text{P}}}^{\text{S}}}\\) (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\)) and the change in the PCM temperature from the initial temperature (\\({{}^{\circ}\text{C}}\\)).
  • \\({E\_{\text{P}}}\\) for the melted PCM (\\({T\_{\text{P}}}\gt{}{{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}\\)) is found using [TM:sensHtE](./SecTMs.md#TM:sensHtE) for sensible heat of the liquid PCM plus the energy when melting starts, plus the energy required to melt all of the PCM.
  • The energy required to melt all of the PCM is \\({H\_{\text{f}}} {m\_{\text{P}}}\\) (\\({\text{J}}\\)) (from [DD:htFusion](./SecDDs.md#DD:htFusion)).
  • The change in temperature is \\({T\_{\text{P}}}-{{T\_{\text{melt}}}^{\text{P}}}\\) (\\({{}^{\circ}\text{C}}\\)).
  • \\({E\_{\text{P}}}\\) during melting of the PCM is found using the energy required at the instant melting of the PCM begins, \\({{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}\\) plus the latent heat energy added to the PCM, \\({Q\_{\text{P}}}\\) (\\({\text{J}}\\)) since the time when melting began \\({{t\_{\text{melt}}}^{\text{init}}}\\) (\\({\text{s}}\\)).
  • The heat energy for boiling of the PCM is not detailed, since the PCM is assumed to either be in a solid or liquid state ([A:No-Gaseous-State-PCM](./SecAssumps.md#assumpNGSP)) ([A:PCM-Initially-Solid](./SecAssumps.md#assumpPIS)).
| -|Source |[koothoor2013](./SecReferences.md#koothoor2013) | -|RefBy |[IM:eBalanceOnPCM](./SecIMs.md#IM:eBalanceOnPCM), [UC:No-Gaseous-State](./SecUCs.md#unlikeChgNGS), [FR:Output-Values](./SecFRs.md#outputValues), [FR:Find-Mass](./SecFRs.md#findMass), and [FR:Calculate-Values](./SecFRs.md#calcValues) | +|Refname |IM:heatEInPCM | +|:-----------------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| +|Label |Heat energy in the PCM | +|Input |\\({{T\_{\text{melt}}}^{\text{P}}}\\), \\({t\_{\text{final}}}\\), \\({T\_{\text{init}}}\\), \\({A\_{\text{P}}}\\), \\({h\_{\text{P}}}\\), \\({m\_{\text{P}}}\\), \\({{C\_{\text{P}}}^{\text{S}}}\\), \\({{C\_{\text{P}}}^{\text{L}}}\\), \\({T\_{\text{P}}}\\), \\({H\_{\text{f}}}\\), \\({{t\_{\text{melt}}}^{\text{init}}}\\) | +|Output |\\({E\_{\text{P}}}\\) | +|Input Constraints |\\[{{T\_{\text{melt}}}^{\text{P}}}\gt{}{T\_{\text{init}}}\\] | +|Output Constraints| | +|Equation |\\[{E\_{\text{P}}}=\begin{cases}{{C\_{\text{P}}}^{\text{S}}}\\,{m\_{\text{P}}}\\,\left({T\_{\text{P}}}\left(t\right)-{T\_{\text{init}}}\right), & {T\_{\text{P}}}\lt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\{{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}+{H\_{\text{f}}}\\,{m\_{\text{P}}}+{{C\_{\text{P}}}^{\text{L}}}\\,{m\_{\text{P}}}\\,\left({T\_{\text{P}}}\left(t\right)-{{T\_{\text{melt}}}^{\text{P}}}\right), & {T\_{\text{P}}}\gt{}{{T\_{\text{melt}}}^{\text{P}}}\\\\{{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}+{Q\_{\text{P}}}\left(t\right), & {T\_{\text{P}}}={{T\_{\text{melt}}}^{\text{P}}}\land{}0\lt{}ϕ\lt{}1\end{cases}\\] | +|Description |
  • \\({E\_{\text{P}}}\\) is the change in heat energy in the PCM (\\({\text{J}}\\))
  • \\({{C\_{\text{P}}}^{\text{S}}}\\) is the specific heat capacity of PCM as a solid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({m\_{\text{P}}}\\) is the mass of phase change material (\\({\text{kg}}\\))
  • \\({T\_{\text{P}}}\\) is the temperature of the phase change material (\\({{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({T\_{\text{init}}}\\) is the initial temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}\\) is the change in heat energy in the PCM at the instant when melting begins (\\({\text{J}}\\))
  • \\({H\_{\text{f}}}\\) is the specific latent heat of fusion (\\(\frac{\text{J}}{\text{kg}}\\))
  • \\({{C\_{\text{P}}}^{\text{L}}}\\) is the specific heat capacity of PCM as a liquid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({{T\_{\text{melt}}}^{\text{P}}}\\) is the melting point temperature for PCM (\\({{}^{\circ}\text{C}}\\))
  • \\({Q\_{\text{P}}}\\) is the latent heat energy added to PCM (\\({\text{J}}\\))
  • \\(ϕ\\) is the melt fraction (Unitless)
| +|Notes |
  • The above equation is derived using [TM:sensHtE](./SecTMs.md#TM:sensHtE) and [TM:latentHtE](./SecTMs.md#TM:latentHtE).
  • \\({E\_{\text{P}}}\\) for the solid PCM is found using [TM:sensHtE](./SecTMs.md#TM:sensHtE) for sensible heating, with the specific heat capacity of the solid PCM, \\({{C\_{\text{P}}}^{\text{S}}}\\) (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\)) and the change in the PCM temperature from the initial temperature (\\({{}^{\circ}\text{C}}\\)).
  • \\({E\_{\text{P}}}\\) for the melted PCM (\\({T\_{\text{P}}}\gt{}{{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}\\)) is found using [TM:sensHtE](./SecTMs.md#TM:sensHtE) for sensible heat of the liquid PCM plus the energy when melting starts, plus the energy required to melt all of the PCM.
  • The energy required to melt all of the PCM is \\({H\_{\text{f}}}\\,{m\_{\text{P}}}\\) (\\({\text{J}}\\)) (from [DD:htFusion](./SecDDs.md#DD:htFusion)).
  • The change in temperature is \\({T\_{\text{P}}}-{{T\_{\text{melt}}}^{\text{P}}}\\) (\\({{}^{\circ}\text{C}}\\)).
  • \\({E\_{\text{P}}}\\) during melting of the PCM is found using the energy required at the instant melting of the PCM begins, \\({{{E\_{\text{P}}}\_{\text{melt}}}^{\text{init}}}\\) plus the latent heat energy added to the PCM, \\({Q\_{\text{P}}}\\) (\\({\text{J}}\\)) since the time when melting began \\({{t\_{\text{melt}}}^{\text{init}}}\\) (\\({\text{s}}\\)).
  • The heat energy for boiling of the PCM is not detailed, since the PCM is assumed to either be in a solid or liquid state ([A:No-Gaseous-State-PCM](./SecAssumps.md#assumpNGSP)) ([A:PCM-Initially-Solid](./SecAssumps.md#assumpPIS)).
| +|Source |[koothoor2013](./SecReferences.md#koothoor2013) | +|RefBy |[IM:eBalanceOnPCM](./SecIMs.md#IM:eBalanceOnPCM), [UC:No-Gaseous-State](./SecUCs.md#unlikeChgNGS), [FR:Output-Values](./SecFRs.md#outputValues), [FR:Find-Mass](./SecFRs.md#findMass), and [FR:Calculate-Values](./SecFRs.md#calcValues) | diff --git a/code/stable/swhs/SRS/mdBook/src/SecTMs.md b/code/stable/swhs/SRS/mdBook/src/SecTMs.md index e18b915527..a2e1a1c672 100644 --- a/code/stable/swhs/SRS/mdBook/src/SecTMs.md +++ b/code/stable/swhs/SRS/mdBook/src/SecTMs.md @@ -11,7 +11,7 @@ This section focuses on the general equations and laws that SWHS is based on. |Refname |TM:consThermE | |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Conservation of thermal energy | -|Equation |\\[-∇\cdot{}\boldsymbol{q}+g=ρ C \frac{\\,\partial{}T}{\\,\partial{}t}\\] | +|Equation |\\[-∇\cdot{}\boldsymbol{q}+g=ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}\\] | |Description|
  • \\(∇\\) is the gradient (Unitless)
  • \\(\boldsymbol{q}\\) is the thermal flux vector (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\(g\\) is the volumetric heat generation per unit volume (\\(\frac{\text{W}}{\text{m}^{3}}\\))
  • \\(ρ\\) is the density (\\(\frac{\text{kg}}{\text{m}^{3}}\\))
  • \\(C\\) is the specific heat capacity (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(T\\) is the temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • The above equation gives the law of conservation of energy for transient heat transfer in a given material.
  • For this equation to apply, other forms of energy, such as mechanical energy, are assumed to be negligible in the system ([A:Thermal-Energy-Only](./SecAssumps.md#assumpTEO)).
| |Source |[Fourier Law of Heat Conduction and Heat Equation](http://www.efunda.com/formulae/heat_transfer/conduction/overview_cond.cfm) | @@ -26,7 +26,7 @@ This section focuses on the general equations and laws that SWHS is based on. |Refname |TM:sensHtE | |:----------|:--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Sensible heat energy | -|Equation |\\[E=\begin{cases}{C^{\text{S}}} m ΔT, & T\lt{}{T\_{\text{melt}}}\\\\{C^{\text{L}}} m ΔT, & {T\_{\text{melt}}}\lt{}T\lt{}{T\_{\text{boil}}}\\\\{C^{\text{V}}} m ΔT, & {T\_{\text{boil}}}\lt{}T\end{cases}\\] | +|Equation |\\[E=\begin{cases}{C^{\text{S}}}\\,m\\,ΔT, & T\lt{}{T\_{\text{melt}}}\\\\{C^{\text{L}}}\\,m\\,ΔT, & {T\_{\text{melt}}}\lt{}T\lt{}{T\_{\text{boil}}}\\\\{C^{\text{V}}}\\,m\\,ΔT, & {T\_{\text{boil}}}\lt{}T\end{cases}\\] | |Description|
  • \\(E\\) is the sensible heat (\\({\text{J}}\\))
  • \\({C^{\text{S}}}\\) is the specific heat capacity of a solid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(ΔT\\) is the change in temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({C^{\text{L}}}\\) is the specific heat capacity of a liquid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({C^{\text{V}}}\\) is the specific heat capacity of a vapour (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(T\\) is the temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{melt}}}\\) is the melting point temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{boil}}}\\) is the boiling point temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • Sensible heating occurs as long as the material does not reach a temperature where a phase change occurs. A phase change occurs if \\(T={T\_{\text{boil}}}\\) or \\(T={T\_{\text{melt}}}\\). If this is the case, refer to [TM:latentHtE](./SecTMs.md#TM:latentHtE).
| |Source |[Definition of Sensible Heat](http://en.wikipedia.org/wiki/Sensible_heat) | @@ -56,7 +56,7 @@ This section focuses on the general equations and laws that SWHS is based on. |Refname |TM:nwtnCooling | |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's law of cooling | -|Equation |\\[q\left(t\right)=h ΔT\left(t\right)\\] | +|Equation |\\[q\left(t\right)=h\\,ΔT\left(t\right)\\] | |Description|
  • \\(q\\) is the heat flux (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(h\\) is the convective heat transfer coefficient (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\(ΔT\\) is the change in temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • Newton's law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings.
  • \\(h\\) is assumed to be independent of \\(T\\) (from [A:Heat-Transfer-Coeffs-Constant](./SecAssumps.md#assumpHTCC)).
  • \\(ΔT\left(t\right)=T\left(t\right)-{T\_{\text{env}}}\left(t\right)\\) is the time-dependant thermal gradient between the environment and the object.
| |Source |[incroperaEtAl2007](./SecReferences.md#incroperaEtAl2007) (pg. 8) | diff --git a/code/stable/swhsnopcm/SRS/HTML/SWHSNoPCM_SRS.html b/code/stable/swhsnopcm/SRS/HTML/SWHSNoPCM_SRS.html index c2937901bb..3087c3de60 100644 --- a/code/stable/swhsnopcm/SRS/HTML/SWHSNoPCM_SRS.html +++ b/code/stable/swhsnopcm/SRS/HTML/SWHSNoPCM_SRS.html @@ -836,7 +836,7 @@

Theoretical Models

@@ -902,7 +902,7 @@

Theoretical Models

- + @@ -953,7 +953,7 @@

Theoretical Models

- + @@ -1025,7 +1025,7 @@

General Definitions

@@ -1079,23 +1079,23 @@

Integrating TM:consThermE over a volume (V), we have:

- \[-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\] + \[-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

Applying Gauss's Divergence Theorem to the first term over the surface S of the volume, with q as the thermal flux vector for the surface and n̂ as a unit outward normal vector for a surface:

- \[-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\] + \[-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as:

- \[{q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV\] + \[{q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV\]

Where qin, qout, Ain, and Aout are explained in GD:rocTempSimp. Assuming ρ, C, and T are constant over the volume, which is true in our case by A:Constant-Water-Temp-Across-Tank, A:Density-Water-Constant-over-Volume, and A:Specific-Heat-Energy-Constant-over-Volume, we have:

- \[ρ C V \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] + \[ρ\,C\,V\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]

Using the fact that ρ=m/V, Equation (2) can be written as:

- \[m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] + \[m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]
@@ -1119,7 +1119,7 @@

@@ -1197,7 +1197,7 @@

Data Definitions

- + @@ -1300,7 +1300,7 @@

Data Definitions

- + @@ -1356,7 +1356,7 @@

Data Definitions

@@ -1443,7 +1443,7 @@

Instance Models

@@ -1498,19 +1498,19 @@

Detailed derivation of the energy balance on water:

To find the rate of change of TW, we look at the energy balance on water. The volume being considered is the volume of water in the tank VW, which has mass mW and specific heat capacity, CW. Heat transfer occurs in the water from the heating coil as qC (GD:htFluxWaterFromCoil), over area AC. No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (A:Perfect-Insulation-Tank). Since the assumption is made that no internal heat is generated (A:No-Internal-Heat-Generation-By-Water), g = 0. Therefore, the equation for GD:rocTempSimp can be written as:

- \[{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}} {A_{\text{C}}}\] + \[{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}}\,{A_{\text{C}}}\]

Using GD:htFluxWaterFromCoil for qC, this can be written as:

- \[{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)\] + \[{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)\]

Dividing Equation (2) by mW CW, we obtain:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)\]

By substituting τW (from DD:balanceDecayRate), this can be written as:

- \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right)\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right)\]
@@ -1546,7 +1546,7 @@

Detailed derivation of the energy balance on water:

diff --git a/code/stable/swhsnopcm/SRS/Jupyter/SWHSNoPCM_SRS.ipynb b/code/stable/swhsnopcm/SRS/Jupyter/SWHSNoPCM_SRS.ipynb index 79cf553118..87178ba400 100644 --- a/code/stable/swhsnopcm/SRS/Jupyter/SWHSNoPCM_SRS.ipynb +++ b/code/stable/swhsnopcm/SRS/Jupyter/SWHSNoPCM_SRS.ipynb @@ -371,7 +371,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -430,7 +430,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -484,7 +484,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -547,7 +547,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -586,19 +586,19 @@ "\n", "\n", "Integrating [TM:consThermE](#TM:consThermE) over a volume ($V$), we have:\n", - "$$-\\int_{V}{∇\\cdot{}\\symbf{q}}\\,dV+\\int_{V}{g}\\,dV=\\int_{V}{ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", + "$$-\\int_{V}{∇\\cdot{}\\symbf{q}}\\,dV+\\int_{V}{g}\\,dV=\\int_{V}{ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", "\n", "Applying Gauss's Divergence Theorem to the first term over the surface $S$ of the volume, with $q$ as the thermal flux vector for the surface and $n̂$ as a unit outward normal vector for a surface:\n", - "$$-\\int_{S}{\\symbf{q}\\cdot{}\\symbf{\\hat{n}}}\\,dS+\\int_{V}{g}\\,dV=\\int_{V}{ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", + "$$-\\int_{S}{\\symbf{q}\\cdot{}\\symbf{\\hat{n}}}\\,dS+\\int_{V}{g}\\,dV=\\int_{V}{ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", "\n", "We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as:\n", - "$${q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V=\\int_{V}{ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", + "$${q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V=\\int_{V}{ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}}\\,dV$$\n", "\n", "Where $q_in$, $q_out$, $A_in$, and $A_out$ are explained in [GD:rocTempSimp](#GD:rocTempSimp). Assuming $ρ$, $C$, and $T$ are constant over the volume, which is true in our case by [A:Constant-Water-Temp-Across-Tank](#assumpCWTAT), [A:Density-Water-Constant-over-Volume](#assumpDWCoW), and [A:Specific-Heat-Energy-Constant-over-Volume](#assumpSHECoW), we have:\n", - "$$ρ C V \\frac{\\,dT}{\\,dt}={q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V$$\n", + "$$ρ\\,C\\,V\\,\\frac{\\,dT}{\\,dt}={q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V$$\n", "\n", "Using the fact that $ρ$=$m$/$V$, Equation (2) can be written as:\n", - "$$m C \\frac{\\,dT}{\\,dt}={q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V$$\n", + "$$m\\,C\\,\\frac{\\,dT}{\\,dt}={q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V$$\n", "
\n", "\n", "
Equation\n", - "$$\\frac{\\,d{T_{\\text{P}}}}{\\,dt}=\\begin{cases} \\frac{1}{{{τ_{\\text{P}}}^{\\text{S}}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right), & {T_{\\text{P}}}\\lt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ \\frac{1}{{{τ_{\\text{P}}}^{\\text{L}}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right), & {T_{\\text{P}}}\\gt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ 0, & {T_{\\text{P}}}={{T_{\\text{melt}}}^{\\text{P}}}\\land{}0\\lt{}ϕ\\lt{}1 \\end{cases}$$\n", + "$$\\frac{\\,d{T_{\\text{P}}}}{\\,dt}=\\begin{cases} \\frac{1}{{{τ_{\\text{P}}}^{\\text{S}}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right), & {T_{\\text{P}}}\\lt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ \\frac{1}{{{τ_{\\text{P}}}^{\\text{L}}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{P}}}\\left(t\\right)\\right), & {T_{\\text{P}}}\\gt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ 0, & {T_{\\text{P}}}={{T_{\\text{melt}}}^{\\text{P}}}\\land{}0\\lt{}ϕ\\lt{}1 \\end{cases}$$\n", "
Equation\n", - "$${E_{\\text{W}}}\\left(t\\right)={C_{\\text{W}}} {m_{\\text{W}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{init}}}\\right)$$\n", + "$${E_{\\text{W}}}\\left(t\\right)={C_{\\text{W}}}\\,{m_{\\text{W}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{init}}}\\right)$$\n", "
Equation\n", - "$${E_{\\text{P}}}=\\begin{cases} {{C_{\\text{P}}}^{\\text{S}}} {m_{\\text{P}}} \\left({T_{\\text{P}}}\\left(t\\right)-{T_{\\text{init}}}\\right), & {T_{\\text{P}}}\\lt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ {{{E_{\\text{P}}}_{\\text{melt}}}^{\\text{init}}}+{H_{\\text{f}}} {m_{\\text{P}}}+{{C_{\\text{P}}}^{\\text{L}}} {m_{\\text{P}}} \\left({T_{\\text{P}}}\\left(t\\right)-{{T_{\\text{melt}}}^{\\text{P}}}\\right), & {T_{\\text{P}}}\\gt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ {{{E_{\\text{P}}}_{\\text{melt}}}^{\\text{init}}}+{Q_{\\text{P}}}\\left(t\\right), & {T_{\\text{P}}}={{T_{\\text{melt}}}^{\\text{P}}}\\land{}0\\lt{}ϕ\\lt{}1 \\end{cases}$$\n", + "$${E_{\\text{P}}}=\\begin{cases} {{C_{\\text{P}}}^{\\text{S}}}\\,{m_{\\text{P}}}\\,\\left({T_{\\text{P}}}\\left(t\\right)-{T_{\\text{init}}}\\right), & {T_{\\text{P}}}\\lt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ {{{E_{\\text{P}}}_{\\text{melt}}}^{\\text{init}}}+{H_{\\text{f}}}\\,{m_{\\text{P}}}+{{C_{\\text{P}}}^{\\text{L}}}\\,{m_{\\text{P}}}\\,\\left({T_{\\text{P}}}\\left(t\\right)-{{T_{\\text{melt}}}^{\\text{P}}}\\right), & {T_{\\text{P}}}\\gt{}{{T_{\\text{melt}}}^{\\text{P}}}\\\\ {{{E_{\\text{P}}}_{\\text{melt}}}^{\\text{init}}}+{Q_{\\text{P}}}\\left(t\\right), & {T_{\\text{P}}}={{T_{\\text{melt}}}^{\\text{P}}}\\land{}0\\lt{}ϕ\\lt{}1 \\end{cases}$$\n", "
Equation - \[-∇\cdot{}\symbf{q}+g=ρ C \frac{\,\partial{}T}{\,\partial{}t}\] + \[-∇\cdot{}\symbf{q}+g=ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}\]
Equation\[E={C^{\text{L}}} m ΔT\]\[E={C^{\text{L}}}\,m\,ΔT\]
Description
Equation\[q\left(t\right)=h ΔT\left(t\right)\]\[q\left(t\right)=h\,ΔT\left(t\right)\]
Description
Equation - \[m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V\] + \[m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V\]
Equation - \[{q_{\text{C}}}={h_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)\] + \[{q_{\text{C}}}={h_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right)\]
Equation\[{m_{\text{W}}}={V_{\text{W}}} {ρ_{\text{W}}}\]\[{m_{\text{W}}}={V_{\text{W}}}\,{ρ_{\text{W}}}\]
Description
Equation\[{V_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L\]\[{V_{\text{tank}}}=π\,\left(\frac{D}{2}\right)^{2}\,L\]
Description
Equation - \[{τ_{\text{W}}}=\frac{{m_{\text{W}}} {C_{\text{W}}}}{{h_{\text{C}}} {A_{\text{C}}}}\] + \[{τ_{\text{W}}}=\frac{{m_{\text{W}}}\,{C_{\text{W}}}}{{h_{\text{C}}}\,{A_{\text{C}}}}\]
Equation - \[\frac{\,d{T_{\text{W}}}}{\,dt}+\frac{1}{{τ_{\text{W}}}} {{T_{\text{W}}}}=\frac{1}{{τ_{\text{W}}}} {T_{\text{C}}}\] + \[\frac{\,d{T_{\text{W}}}}{\,dt}+\frac{1}{{τ_{\text{W}}}}\,{{T_{\text{W}}}}=\frac{1}{{τ_{\text{W}}}}\,{T_{\text{C}}}\]
Equation - \[{E_{\text{W}}}\left(t\right)={C_{\text{W}}} {m_{\text{W}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)\] + \[{E_{\text{W}}}\left(t\right)={C_{\text{W}}}\,{m_{\text{W}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right)\]
Equation\n", - "$$-∇\\cdot{}\\symbf{q}+g=ρ C \\frac{\\,\\partial{}T}{\\,\\partial{}t}$$\n", + "$$-∇\\cdot{}\\symbf{q}+g=ρ\\,C\\,\\frac{\\,\\partial{}T}{\\,\\partial{}t}$$\n", "
Equation\n", - "$$E={C^{\\text{L}}} m ΔT$$\n", + "$$E={C^{\\text{L}}}\\,m\\,ΔT$$\n", "
Equation\n", - "$$q\\left(t\\right)=h ΔT\\left(t\\right)$$\n", + "$$q\\left(t\\right)=h\\,ΔT\\left(t\\right)$$\n", "
Equation\n", - "$$m C \\frac{\\,dT}{\\,dt}={q_{\\text{in}}} {A_{\\text{in}}}-{q_{\\text{out}}} {A_{\\text{out}}}+g V$$\n", + "$$m\\,C\\,\\frac{\\,dT}{\\,dt}={q_{\\text{in}}}\\,{A_{\\text{in}}}-{q_{\\text{out}}}\\,{A_{\\text{out}}}+g\\,V$$\n", "
\n", @@ -625,7 +625,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -701,7 +701,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -827,7 +827,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -888,7 +888,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -968,7 +968,7 @@ "\n", "\n", "\n", "\n", "\n", @@ -1010,16 +1010,16 @@ "\n", "\n", "To find the rate of change of $T_W$, we look at the energy balance on water. The volume being considered is the volume of water in the tank $V_W$, which has mass $m_W$ and specific heat capacity, $C_W$. Heat transfer occurs in the water from the heating coil as $q_C$ ([GD:htFluxWaterFromCoil](#GD:htFluxWaterFromCoil)), over area $A_C$. No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated ([A:Perfect-Insulation-Tank](#assumpPIT)). Since the assumption is made that no internal heat is generated ([A:No-Internal-Heat-Generation-By-Water](#assumpNIHGBW)), $g = 0$. Therefore, the equation for [GD:rocTempSimp](#GD:rocTempSimp) can be written as:\n", - "$${m_{\\text{W}}} {C_{\\text{W}}} \\frac{\\,d{T_{\\text{W}}}}{\\,dt}={q_{\\text{C}}} {A_{\\text{C}}}$$\n", + "$${m_{\\text{W}}}\\,{C_{\\text{W}}}\\,\\frac{\\,d{T_{\\text{W}}}}{\\,dt}={q_{\\text{C}}}\\,{A_{\\text{C}}}$$\n", "\n", "Using [GD:htFluxWaterFromCoil](#GD:htFluxWaterFromCoil) for $q_C$, this can be written as:\n", - "$${m_{\\text{W}}} {C_{\\text{W}}} \\frac{\\,d{T_{\\text{W}}}}{\\,dt}={h_{\\text{C}}} {A_{\\text{C}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)$$\n", + "$${m_{\\text{W}}}\\,{C_{\\text{W}}}\\,\\frac{\\,d{T_{\\text{W}}}}{\\,dt}={h_{\\text{C}}}\\,{A_{\\text{C}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)$$\n", "\n", "Dividing Equation (2) by $m_WC_W$, we obtain:\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}} {A_{\\text{C}}}}{{m_{\\text{W}}} {C_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{{h_{\\text{C}}}\\,{A_{\\text{C}}}}{{m_{\\text{W}}}\\,{C_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)$$\n", "\n", "By substituting $τ_W$ (from [DD:balanceDecayRate](#DD:balanceDecayRate)), this can be written as:\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}=\\frac{1}{{τ_{\\text{W}}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\right)$$\n", "
\n", "\n", "
Equation\n", - "$${q_{\\text{C}}}={h_{\\text{C}}} \\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)\\right)$$\n", + "$${q_{\\text{C}}}={h_{\\text{C}}}\\,\\left({T_{\\text{C}}}-{T_{\\text{W}}}\\left(t\\right)\\right)$$\n", "
Equation\n", - "$${m_{\\text{W}}}={V_{\\text{W}}} {ρ_{\\text{W}}}$$\n", + "$${m_{\\text{W}}}={V_{\\text{W}}}\\,{ρ_{\\text{W}}}$$\n", "
Equation\n", - "$${V_{\\text{tank}}}=π \\left(\\frac{D}{2}\\right)^{2} L$$\n", + "$${V_{\\text{tank}}}=π\\,\\left(\\frac{D}{2}\\right)^{2}\\,L$$\n", "
Equation\n", - "$${τ_{\\text{W}}}=\\frac{{m_{\\text{W}}} {C_{\\text{W}}}}{{h_{\\text{C}}} {A_{\\text{C}}}}$$\n", + "$${τ_{\\text{W}}}=\\frac{{m_{\\text{W}}}\\,{C_{\\text{W}}}}{{h_{\\text{C}}}\\,{A_{\\text{C}}}}$$\n", "
Equation\n", - "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}+\\frac{1}{{τ_{\\text{W}}}} {{T_{\\text{W}}}}=\\frac{1}{{τ_{\\text{W}}}} {T_{\\text{C}}}$$\n", + "$$\\frac{\\,d{T_{\\text{W}}}}{\\,dt}+\\frac{1}{{τ_{\\text{W}}}}\\,{{T_{\\text{W}}}}=\\frac{1}{{τ_{\\text{W}}}}\\,{T_{\\text{C}}}$$\n", "
\n", @@ -1063,7 +1063,7 @@ "\n", "\n", "\n", "\n", "\n", diff --git a/code/stable/swhsnopcm/SRS/PDF/SWHSNoPCM_SRS.tex b/code/stable/swhsnopcm/SRS/PDF/SWHSNoPCM_SRS.tex index 1870189ef9..ba644dadd0 100644 --- a/code/stable/swhsnopcm/SRS/PDF/SWHSNoPCM_SRS.tex +++ b/code/stable/swhsnopcm/SRS/PDF/SWHSNoPCM_SRS.tex @@ -356,7 +356,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - -∇\cdot{}\symbf{q}+g=ρ C \frac{\,\partial{}T}{\,\partial{}t} + -∇\cdot{}\symbf{q}+g=ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -395,7 +395,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - E={C^{\text{L}}} m ΔT + E={C^{\text{L}}}\,m\,ΔT \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -429,7 +429,7 @@ \subsubsection{Theoretical Models} \\ \midrule Equation & \begin{displaymath} - q\left(t\right)=h ΔT\left(t\right) + q\left(t\right)=h\,ΔT\left(t\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -471,7 +471,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V + m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -501,27 +501,27 @@ \subsubsection{General Definitions} Integrating \hyperref[TM:consThermE]{TM:consThermE} over a volume ($V$), we have: \begin{displaymath} --\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +-\int_{V}{∇\cdot{}\symbf{q}}\,dV+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} Applying Gauss's Divergence Theorem to the first term over the surface $S$ of the volume, with $\symbf{q}$ as the thermal flux vector for the surface and $\symbf{\hat{n}}$ as a unit outward normal vector for a surface: \begin{displaymath} --\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +-\int_{S}{\symbf{q}\cdot{}\symbf{\hat{n}}}\,dS+\int_{V}{g}\,dV=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as: \begin{displaymath} -{q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V=\int_{V}{ρ C \frac{\,\partial{}T}{\,\partial{}t}}\,dV +{q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V=\int_{V}{ρ\,C\,\frac{\,\partial{}T}{\,\partial{}t}}\,dV \end{displaymath} Where ${q_{\text{in}}}$, ${q_{\text{out}}}$, ${A_{\text{in}}}$, and ${A_{\text{out}}}$ are explained in \hyperref[GD:rocTempSimp]{GD:rocTempSimp}. Assuming $ρ$, $C$, and $T$ are constant over the volume, which is true in our case by \hyperref[assumpCWTAT]{A:Constant-Water-Temp-Across-Tank}, \hyperref[assumpDWCoW]{A:Density-Water-Constant-over-Volume}, and \hyperref[assumpSHECoW]{A:Specific-Heat-Energy-Constant-over-Volume}, we have: \begin{displaymath} -ρ C V \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V +ρ\,C\,V\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V \end{displaymath} Using the fact that $ρ$=$m$/$V$, Equation (2) can be written as: \begin{displaymath} -m C \frac{\,dT}{\,dt}={q_{\text{in}}} {A_{\text{in}}}-{q_{\text{out}}} {A_{\text{out}}}+g V +m\,C\,\frac{\,dT}{\,dt}={q_{\text{in}}}\,{A_{\text{in}}}-{q_{\text{out}}}\,{A_{\text{out}}}+g\,V \end{displaymath} \medskip \noindent @@ -538,7 +538,7 @@ \subsubsection{General Definitions} \\ \midrule Equation & \begin{displaymath} - {q_{\text{C}}}={h_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right) + {q_{\text{C}}}={h_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\left(t\right)\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -585,7 +585,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {m_{\text{W}}}={V_{\text{W}}} {ρ_{\text{W}}} + {m_{\text{W}}}={V_{\text{W}}}\,{ρ_{\text{W}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -659,7 +659,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {V_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L + {V_{\text{tank}}}=π\,\left(\frac{D}{2}\right)^{2}\,L \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -696,7 +696,7 @@ \subsubsection{Data Definitions} \\ \midrule Equation & \begin{displaymath} - {τ_{\text{W}}}=\frac{{m_{\text{W}}} {C_{\text{W}}}}{{h_{\text{C}}} {A_{\text{C}}}} + {τ_{\text{W}}}=\frac{{m_{\text{W}}}\,{C_{\text{W}}}}{{h_{\text{C}}}\,{A_{\text{C}}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -746,7 +746,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - \frac{\,d{T_{\text{W}}}}{\,dt}+\frac{1}{{τ_{\text{W}}}} {{T_{\text{W}}}}=\frac{1}{{τ_{\text{W}}}} {T_{\text{C}}} + \frac{\,d{T_{\text{W}}}}{\,dt}+\frac{1}{{τ_{\text{W}}}}\,{{T_{\text{W}}}}=\frac{1}{{τ_{\text{W}}}}\,{T_{\text{C}}} \end{displaymath} \\ \midrule Description & \begin{symbDescription} @@ -775,22 +775,22 @@ \subsubsection{Instance Models} To find the rate of change of ${T_{\text{W}}}$, we look at the energy balance on water. The volume being considered is the volume of water in the tank ${V_{\text{W}}}$, which has mass ${m_{\text{W}}}$ and specific heat capacity, ${C_{\text{W}}}$. Heat transfer occurs in the water from the heating coil as ${q_{\text{C}}}$ (\hyperref[GD:htFluxWaterFromCoil]{GD:htFluxWaterFromCoil}), over area ${A_{\text{C}}}$. No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated (\hyperref[assumpPIT]{A:Perfect-Insulation-Tank}). Since the assumption is made that no internal heat is generated (\hyperref[assumpNIHGBW]{A:No-Internal-Heat-Generation-By-Water}), $g=0$. Therefore, the equation for \hyperref[GD:rocTempSimp]{GD:rocTempSimp} can be written as: \begin{displaymath} -{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}} {A_{\text{C}}} +{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={q_{\text{C}}}\,{A_{\text{C}}} \end{displaymath} Using \hyperref[GD:htFluxWaterFromCoil]{GD:htFluxWaterFromCoil} for ${q_{\text{C}}}$, this can be written as: \begin{displaymath} -{m_{\text{W}}} {C_{\text{W}}} \frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}} {A_{\text{C}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right) +{m_{\text{W}}}\,{C_{\text{W}}}\,\frac{\,d{T_{\text{W}}}}{\,dt}={h_{\text{C}}}\,{A_{\text{C}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right) \end{displaymath} -Dividing Equation (2) by ${m_{\text{W}}} {C_{\text{W}}}$, we obtain: +Dividing Equation (2) by ${m_{\text{W}}}\,{C_{\text{W}}}$, we obtain: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}} {A_{\text{C}}}}{{m_{\text{W}}} {C_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{{h_{\text{C}}}\,{A_{\text{C}}}}{{m_{\text{W}}}\,{C_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right) \end{displaymath} By substituting ${τ_{\text{W}}}$ (from \hyperref[DD:balanceDecayRate]{DD:balanceDecayRate}), this can be written as: \begin{displaymath} -\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}} \left({T_{\text{C}}}-{T_{\text{W}}}\right) +\frac{\,d{T_{\text{W}}}}{\,dt}=\frac{1}{{τ_{\text{W}}}}\,\left({T_{\text{C}}}-{T_{\text{W}}}\right) \end{displaymath} \medskip \noindent @@ -814,7 +814,7 @@ \subsubsection{Instance Models} Output Constraints & \\ \midrule Equation & \begin{displaymath} - {E_{\text{W}}}\left(t\right)={C_{\text{W}}} {m_{\text{W}}} \left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right) + {E_{\text{W}}}\left(t\right)={C_{\text{W}}}\,{m_{\text{W}}}\,\left({T_{\text{W}}}\left(t\right)-{T_{\text{init}}}\right) \end{displaymath} \\ \midrule Description & \begin{symbDescription} diff --git a/code/stable/swhsnopcm/SRS/mdBook/src/SecDDs.md b/code/stable/swhsnopcm/SRS/mdBook/src/SecDDs.md index fb61da7c86..731e3e81c7 100644 --- a/code/stable/swhsnopcm/SRS/mdBook/src/SecDDs.md +++ b/code/stable/swhsnopcm/SRS/mdBook/src/SecDDs.md @@ -13,7 +13,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Mass of water | |Symbol |\\({m\_{\text{W}}}\\) | |Units |\\({\text{kg}}\\) | -|Equation |\\[{m\_{\text{W}}}={V\_{\text{W}}} {ρ\_{\text{W}}}\\] | +|Equation |\\[{m\_{\text{W}}}={V\_{\text{W}}}\\,{ρ\_{\text{W}}}\\] | |Description|
  • \\({m\_{\text{W}}}\\) is the mass of water (\\({\text{kg}}\\))
  • \\({V\_{\text{W}}}\\) is the volume of water (\\({\text{m}^{3}}\\))
  • \\({ρ\_{\text{W}}}\\) is the density of water (\\(\frac{\text{kg}}{\text{m}^{3}}\\))
| |Source |-- | |RefBy |[FR:Find-Mass](./SecFRs.md#findMass) | @@ -46,7 +46,7 @@ This section collects and defines all the data needed to build the instance mode |Label |Volume of the cylindrical tank | |Symbol |\\({V\_{\text{tank}}}\\) | |Units |\\({\text{m}^{3}}\\) | -|Equation |\\[{V\_{\text{tank}}}=π \left(\frac{D}{2}\right)^{2} L\\] | +|Equation |\\[{V\_{\text{tank}}}=π\\,\left(\frac{D}{2}\right)^{2}\\,L\\] | |Description|
  • \\({V\_{\text{tank}}}\\) is the volume of the cylindrical tank (\\({\text{m}^{3}}\\))
  • \\(π\\) is the ratio of circumference to diameter for any circle (Unitless)
  • \\(D\\) is the diameter of tank (\\({\text{m}}\\))
  • \\(L\\) is the length of tank (\\({\text{m}}\\))
| |Source |-- | |RefBy |[DD:waterVolume_nopcm](./SecDDs.md#DD:waterVolume.nopcm) and [FR:Find-Mass](./SecFRs.md#findMass) | @@ -62,7 +62,7 @@ This section collects and defines all the data needed to build the instance mode |Label |ODE parameter for water related to decay time | |Symbol |\\({τ\_{\text{W}}}\\) | |Units |\\({\text{s}}\\) | -|Equation |\\[{τ\_{\text{W}}}=\frac{{m\_{\text{W}}} {C\_{\text{W}}}}{{h\_{\text{C}}} {A\_{\text{C}}}}\\] | +|Equation |\\[{τ\_{\text{W}}}=\frac{{m\_{\text{W}}}\\,{C\_{\text{W}}}}{{h\_{\text{C}}}\\,{A\_{\text{C}}}}\\] | |Description|
  • \\({τ\_{\text{W}}}\\) is the ODE parameter for water related to decay time (\\({\text{s}}\\))
  • \\({m\_{\text{W}}}\\) is the mass of water (\\({\text{kg}}\\))
  • \\({C\_{\text{W}}}\\) is the specific heat capacity of water (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({h\_{\text{C}}}\\) is the convective heat transfer coefficient between coil and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({A\_{\text{C}}}\\) is the heating coil surface area (\\({\text{m}^{2}}\\))
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | |RefBy |[IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr) and [FR:Output-Input-Derived-Values](./SecFRs.md#outputInputDerivVals) | diff --git a/code/stable/swhsnopcm/SRS/mdBook/src/SecGDs.md b/code/stable/swhsnopcm/SRS/mdBook/src/SecGDs.md index ace8c85445..c0be2f3e28 100644 --- a/code/stable/swhsnopcm/SRS/mdBook/src/SecGDs.md +++ b/code/stable/swhsnopcm/SRS/mdBook/src/SecGDs.md @@ -11,7 +11,7 @@ This section collects the laws and equations that will be used to build the inst |Refname |GD:rocTempSimp | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Simplified rate of change of temperature | -|Equation |\\[m C \frac{\\,dT}{\\,dt}={q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V\\] | +|Equation |\\[m\\,C\\,\frac{\\,dT}{\\,dt}={q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V\\] | |Description|
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(C\\) is the specific heat capacity (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(T\\) is the temperature (\\({{}^{\circ}\text{C}}\\))
  • \\({q\_{\text{in}}}\\) is the heat flux input (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({A\_{\text{in}}}\\) is the surface area over which heat is transferred in (\\({\text{m}^{2}}\\))
  • \\({q\_{\text{out}}}\\) is the heat flux output (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({A\_{\text{out}}}\\) is the surface area over which heat is transferred out (\\({\text{m}^{2}}\\))
  • \\(g\\) is the volumetric heat generation per unit volume (\\(\frac{\text{W}}{\text{m}^{3}}\\))
  • \\(V\\) is the volume (\\({\text{m}^{3}}\\))
| |Source |-- | |RefBy |[GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp) and [IM:eBalanceOnWtr](./SecIMs.md#IM:eBalanceOnWtr) | @@ -20,23 +20,23 @@ This section collects the laws and equations that will be used to build the inst Integrating [TM:consThermE](./SecTMs.md#TM:consThermE) over a volume (\\(V\\)), we have: -\\[-\int\_{V}{∇\cdot{}\boldsymbol{q}}\\,dV+\int\_{V}{g}\\,dV=\int\_{V}{ρ C \frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] +\\[-\int\_{V}{∇\cdot{}\boldsymbol{q}}\\,dV+\int\_{V}{g}\\,dV=\int\_{V}{ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] Applying Gauss's Divergence Theorem to the first term over the surface \\(S\\) of the volume, with \\(\boldsymbol{q}\\) as the thermal flux vector for the surface and \\(\boldsymbol{\hat{n}}\\) as a unit outward normal vector for a surface: -\\[-\int\_{S}{\boldsymbol{q}\cdot{}\boldsymbol{\hat{n}}}\\,dS+\int\_{V}{g}\\,dV=\int\_{V}{ρ C \frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] +\\[-\int\_{S}{\boldsymbol{q}\cdot{}\boldsymbol{\hat{n}}}\\,dS+\int\_{V}{g}\\,dV=\int\_{V}{ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] We consider an arbitrary volume. The volumetric heat generation per unit volume is assumed constant. Then Equation (1) can be written as: -\\[{q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V=\int\_{V}{ρ C \frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] +\\[{q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V=\int\_{V}{ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}}\\,dV\\] Where \\({q\_{\text{in}}}\\), \\({q\_{\text{out}}}\\), \\({A\_{\text{in}}}\\), and \\({A\_{\text{out}}}\\) are explained in [GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp). Assuming \\(ρ\\), \\(C\\), and \\(T\\) are constant over the volume, which is true in our case by [A:Constant-Water-Temp-Across-Tank](./SecAssumps.md#assumpCWTAT), [A:Density-Water-Constant-over-Volume](./SecAssumps.md#assumpDWCoW), and [A:Specific-Heat-Energy-Constant-over-Volume](./SecAssumps.md#assumpSHECoW), we have: -\\[ρ C V \frac{\\,dT}{\\,dt}={q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V\\] +\\[ρ\\,C\\,V\\,\frac{\\,dT}{\\,dt}={q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V\\] Using the fact that \\(ρ\\)=\\(m\\)/\\(V\\), Equation (2) can be written as: -\\[m C \frac{\\,dT}{\\,dt}={q\_{\text{in}}} {A\_{\text{in}}}-{q\_{\text{out}}} {A\_{\text{out}}}+g V\\] +\\[m\\,C\\,\frac{\\,dT}{\\,dt}={q\_{\text{in}}}\\,{A\_{\text{in}}}-{q\_{\text{out}}}\\,{A\_{\text{out}}}+g\\,V\\]
@@ -48,7 +48,7 @@ Using the fact that \\(ρ\\)=\\(m\\)/\\(V\\), Equation (2) can be written as: |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Heat flux into the water from the coil | |Units |\\(\frac{\text{W}}{\text{m}^{2}}\\) | -|Equation |\\[{q\_{\text{C}}}={h\_{\text{C}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)\right)\\] | +|Equation |\\[{q\_{\text{C}}}={h\_{\text{C}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\left(t\right)\right)\\] | |Description|
  • \\({q\_{\text{C}}}\\) is the heat flux into the water from the coil (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\({h\_{\text{C}}}\\) is the convective heat transfer coefficient between coil and water (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{C}}}\\) is the temperature of the heating coil (\\({{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
| |Notes |
  • \\({q\_{\text{C}}}\\) is found by assuming that Newton's law of cooling applies ([A:Newton-Law-Convective-Cooling-Coil-Water](./SecAssumps.md#assumpLCCCW)). This law (defined in [TM:nwtnCooling](./SecTMs.md#TM:nwtnCooling)) is used on the surface of the heating coil.
  • [A:Temp-Heating-Coil-Constant-over-Time](./SecAssumps.md#assumpTHCCoT)
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | diff --git a/code/stable/swhsnopcm/SRS/mdBook/src/SecIMs.md b/code/stable/swhsnopcm/SRS/mdBook/src/SecIMs.md index 0772a04f6a..a42c5b6355 100644 --- a/code/stable/swhsnopcm/SRS/mdBook/src/SecIMs.md +++ b/code/stable/swhsnopcm/SRS/mdBook/src/SecIMs.md @@ -17,7 +17,7 @@ The goal [GS:Predict-Water-Temperature](./SecGoalStmt.md#waterTempGS) is met by |Output |\\({T\_{\text{W}}}\\) | |Input Constraints |\\[{T\_{\text{C}}}\geq{}{T\_{\text{init}}}\\] | |Output Constraints| | -|Equation |\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}+\frac{1}{{τ\_{\text{W}}}} {{T\_{\text{W}}}}=\frac{1}{{τ\_{\text{W}}}} {T\_{\text{C}}}\\] | +|Equation |\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}+\frac{1}{{τ\_{\text{W}}}}\\,{{T\_{\text{W}}}}=\frac{1}{{τ\_{\text{W}}}}\\,{T\_{\text{C}}}\\] | |Description |
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\({τ\_{\text{W}}}\\) is the ODE parameter for water related to decay time (\\({\text{s}}\\))
  • \\({T\_{\text{C}}}\\) is the temperature of the heating coil (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • \\({τ\_{\text{W}}}\\) is calculated from [DD:balanceDecayRate](./SecDDs.md#DD:balanceDecayRate).
  • The above equation applies as long as the water is in liquid form, \\(0\lt{}{T\_{\text{W}}}\lt{}100\\) (\\({{}^{\circ}\text{C}}\\)) where \\(0\\) (\\({{}^{\circ}\text{C}}\\)) and \\(100\\) (\\({{}^{\circ}\text{C}}\\)) are the melting and boiling point temperatures of water, respectively ([A:Water-Always-Liquid](./SecAssumps.md#assumpWAL)).
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) (with PCM removed) | @@ -27,19 +27,19 @@ The goal [GS:Predict-Water-Temperature](./SecGoalStmt.md#waterTempGS) is met by To find the rate of change of \\({T\_{\text{W}}}\\), we look at the energy balance on water. The volume being considered is the volume of water in the tank \\({V\_{\text{W}}}\\), which has mass \\({m\_{\text{W}}}\\) and specific heat capacity, \\({C\_{\text{W}}}\\). Heat transfer occurs in the water from the heating coil as \\({q\_{\text{C}}}\\) ([GD:htFluxWaterFromCoil](./SecGDs.md#GD:htFluxWaterFromCoil)), over area \\({A\_{\text{C}}}\\). No heat transfer occurs to the outside of the tank, since it has been assumed to be perfectly insulated ([A:Perfect-Insulation-Tank](./SecAssumps.md#assumpPIT)). Since the assumption is made that no internal heat is generated ([A:No-Internal-Heat-Generation-By-Water](./SecAssumps.md#assumpNIHGBW)), \\(g=0\\). Therefore, the equation for [GD:rocTempSimp](./SecGDs.md#GD:rocTempSimp) can be written as: -\\[{m\_{\text{W}}} {C\_{\text{W}}} \frac{\\,d{T\_{\text{W}}}}{\\,dt}={q\_{\text{C}}} {A\_{\text{C}}}\\] +\\[{m\_{\text{W}}}\\,{C\_{\text{W}}}\\,\frac{\\,d{T\_{\text{W}}}}{\\,dt}={q\_{\text{C}}}\\,{A\_{\text{C}}}\\] Using [GD:htFluxWaterFromCoil](./SecGDs.md#GD:htFluxWaterFromCoil) for \\({q\_{\text{C}}}\\), this can be written as: -\\[{m\_{\text{W}}} {C\_{\text{W}}} \frac{\\,d{T\_{\text{W}}}}{\\,dt}={h\_{\text{C}}} {A\_{\text{C}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)\\] +\\[{m\_{\text{W}}}\\,{C\_{\text{W}}}\\,\frac{\\,d{T\_{\text{W}}}}{\\,dt}={h\_{\text{C}}}\\,{A\_{\text{C}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)\\] -Dividing Equation (2) by \\({m\_{\text{W}}} {C\_{\text{W}}}\\), we obtain: +Dividing Equation (2) by \\({m\_{\text{W}}}\\,{C\_{\text{W}}}\\), we obtain: -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}} {A\_{\text{C}}}}{{m\_{\text{W}}} {C\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{{h\_{\text{C}}}\\,{A\_{\text{C}}}}{{m\_{\text{W}}}\\,{C\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)\\] By substituting \\({τ\_{\text{W}}}\\) (from [DD:balanceDecayRate](./SecDDs.md#DD:balanceDecayRate)), this can be written as: -\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}} \left({T\_{\text{C}}}-{T\_{\text{W}}}\right)\\] +\\[\frac{\\,d{T\_{\text{W}}}}{\\,dt}=\frac{1}{{τ\_{\text{W}}}}\\,\left({T\_{\text{C}}}-{T\_{\text{W}}}\right)\\]
@@ -54,7 +54,7 @@ By substituting \\({τ\_{\text{W}}}\\) (from [DD:balanceDecayRate](./SecDDs.md#D |Output |\\({E\_{\text{W}}}\\) | |Input Constraints | | |Output Constraints| | -|Equation |\\[{E\_{\text{W}}}\left(t\right)={C\_{\text{W}}} {m\_{\text{W}}} \left({T\_{\text{W}}}\left(t\right)-{T\_{\text{init}}}\right)\\] | +|Equation |\\[{E\_{\text{W}}}\left(t\right)={C\_{\text{W}}}\\,{m\_{\text{W}}}\\,\left({T\_{\text{W}}}\left(t\right)-{T\_{\text{init}}}\right)\\] | |Description |
  • \\({E\_{\text{W}}}\\) is the change in heat energy in the water (\\({\text{J}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\({C\_{\text{W}}}\\) is the specific heat capacity of water (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\({m\_{\text{W}}}\\) is the mass of water (\\({\text{kg}}\\))
  • \\({T\_{\text{W}}}\\) is the temperature of the water (\\({{}^{\circ}\text{C}}\\))
  • \\({T\_{\text{init}}}\\) is the initial temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • The above equation is derived using [TM:sensHtE](./SecTMs.md#TM:sensHtE).
  • The change in temperature is the difference between the temperature at time \\(t\\) (\\({\text{s}}\\)), \\({T\_{\text{W}}}\\) and the initial temperature, \\({T\_{\text{init}}}\\) (\\({{}^{\circ}\text{C}}\\)).
  • This equation applies as long as \\(0\lt{}{T\_{\text{W}}}\lt{}100\\)\\({{}^{\circ}\text{C}}\\) ([A:Water-Always-Liquid](./SecAssumps.md#assumpWAL), [A:Atmospheric-Pressure-Tank](./SecAssumps.md#assumpAPT)).
| |Source |[koothoor2013](./SecReferences.md#koothoor2013) | diff --git a/code/stable/swhsnopcm/SRS/mdBook/src/SecTMs.md b/code/stable/swhsnopcm/SRS/mdBook/src/SecTMs.md index d82b56fe60..d393ec1f2c 100644 --- a/code/stable/swhsnopcm/SRS/mdBook/src/SecTMs.md +++ b/code/stable/swhsnopcm/SRS/mdBook/src/SecTMs.md @@ -11,7 +11,7 @@ This section focuses on the general equations and laws that SWHSNoPCM is based o |Refname |TM:consThermE | |:----------|:-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Conservation of thermal energy | -|Equation |\\[-∇\cdot{}\boldsymbol{q}+g=ρ C \frac{\\,\partial{}T}{\\,\partial{}t}\\] | +|Equation |\\[-∇\cdot{}\boldsymbol{q}+g=ρ\\,C\\,\frac{\\,\partial{}T}{\\,\partial{}t}\\] | |Description|
  • \\(∇\\) is the gradient (Unitless)
  • \\(\boldsymbol{q}\\) is the thermal flux vector (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\(g\\) is the volumetric heat generation per unit volume (\\(\frac{\text{W}}{\text{m}^{3}}\\))
  • \\(ρ\\) is the density (\\(\frac{\text{kg}}{\text{m}^{3}}\\))
  • \\(C\\) is the specific heat capacity (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(T\\) is the temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • The above equation gives the law of conservation of energy for transient heat transfer in a given material.
  • For this equation to apply, other forms of energy, such as mechanical energy, are assumed to be negligible in the system ([A:Thermal-Energy-Only](./SecAssumps.md#assumpTEO)).
| |Source |[Fourier Law of Heat Conduction and Heat Equation](http://www.efunda.com/formulae/heat_transfer/conduction/overview_cond.cfm) | @@ -26,7 +26,7 @@ This section focuses on the general equations and laws that SWHSNoPCM is based o |Refname |TM:sensHtE | |:----------|:-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Sensible heat energy (no state change) | -|Equation |\\[E={C^{\text{L}}} m ΔT\\] | +|Equation |\\[E={C^{\text{L}}}\\,m\\,ΔT\\] | |Description|
  • \\(E\\) is the sensible heat (\\({\text{J}}\\))
  • \\({C^{\text{L}}}\\) is the specific heat capacity of a liquid (\\(\frac{\text{J}}{\text{kg}{}^{\circ}\text{C}}\\))
  • \\(m\\) is the mass (\\({\text{kg}}\\))
  • \\(ΔT\\) is the change in temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • \\(E\\) occurs as long as the material does not reach a temperature where a phase change occurs, as assumed in [A:Water-Always-Liquid](./SecAssumps.md#assumpWAL).
| |Source |[Definition of Sensible Heat](http://en.wikipedia.org/wiki/Sensible_heat) | @@ -41,7 +41,7 @@ This section focuses on the general equations and laws that SWHSNoPCM is based o |Refname |TM:nwtnCooling | |:----------|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------| |Label |Newton's law of cooling | -|Equation |\\[q\left(t\right)=h ΔT\left(t\right)\\] | +|Equation |\\[q\left(t\right)=h\\,ΔT\left(t\right)\\] | |Description|
  • \\(q\\) is the heat flux (\\(\frac{\text{W}}{\text{m}^{2}}\\))
  • \\(t\\) is the time (\\({\text{s}}\\))
  • \\(h\\) is the convective heat transfer coefficient (\\(\frac{\text{W}}{\text{m}^{2}{}^{\circ}\text{C}}\\))
  • \\(ΔT\\) is the change in temperature (\\({{}^{\circ}\text{C}}\\))
| |Notes |
  • Newton's law of cooling describes convective cooling from a surface. The law is stated as: the rate of heat loss from a body is proportional to the difference in temperatures between the body and its surroundings.
  • \\(h\\) is assumed to be independent of \\(T\\) (from [A:Heat-Transfer-Coeffs-Constant](./SecAssumps.md#assumpHTCC)).
  • \\(ΔT\left(t\right)=T\left(t\right)-{T\_{\text{env}}}\left(t\right)\\) is the time-dependant thermal gradient between the environment and the object.
| |Source |[incroperaEtAl2007](./SecReferences.md#incroperaEtAl2007) (pg. 8) |
Equation\n", - "$${E_{\\text{W}}}\\left(t\\right)={C_{\\text{W}}} {m_{\\text{W}}} \\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{init}}}\\right)$$\n", + "$${E_{\\text{W}}}\\left(t\\right)={C_{\\text{W}}}\\,{m_{\\text{W}}}\\,\\left({T_{\\text{W}}}\\left(t\\right)-{T_{\\text{init}}}\\right)$$\n", "