path: root/Master/texmf-dist/tex/latex/mecaso/mecaso.sty

%Mecaso, un package LaTeX pour les formules en mécanique du solide
%Conçu par : Youssef DERRAZI, version 1.0
%Distribué sous la license GNU Public License V3.0

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{mecaso}[2021/08/27 v1.0]

\RequirePackage{amsmath}
\RequirePackage{amssymb}
\RequirePackage{mathrsfs}

%Commands :
\newcommand{\R}{\mathbb{R}}

%Derivative on line \Dex[R]{u(t)} :
\newcommand{\Dex}[2][R]{\frac{d}{dt}{#2}\bigg|_{#1}}

%Derivative on fraction \Din[R]{u(t)} :
\newcommand{\Din}[2][R]{\frac{d{#2}}{dt}\bigg|_{#1}}

%Vector representation \Vct{AB} :
\newcommand{\Vct}[1]{\overrightarrow{#1}}

%################################### A-Kinematics :

%Rotation vector definition \Rotation[R]{S} :
\newcommand{\Rotation}[2][R]{\Vct{\omega}_{#2/#1}}

%Velocity vector definition \Vel[R]{A}{S} :
\newcommand{\Vel}[3][R]{\Vct{v}_{#2\in #3/#1}}

%Transport formula for velocities \VARK[R]{A}{B}{S} :
\newcommand{\VARK}[4][R]{\Vel[#1]{#3}{#4}=\Vel[#1]{#2}{#4}+\Rotation[#1]{#4}\times\Vct{#2 #3}}

%################################### B-Statics :

%Force vector definition \Force[S]{Ext} :
\newcommand{\Force}[2][S]{\Vct{F}_{#2\rightarrow #1}}

%Moment vector definition with force precision \Moment[S]{Ext}{A} :
\newcommand{\Moment}[3][S]{\Vct{M}_{#3}(\Vct{F}_{#2\rightarrow #1})}

%Moment vector definition without force precision \Gmoment{Force}{A} :
\newcommand{\Gmoment}[2]{\Vct{M}_{#2}(#1)}

%Transport formula for moments with force precision \VARS[S]{A}{B}{Ext} :
\newcommand{\VARS}[4][S]{\Moment[#1]{#4}{#3}=\Moment[#1]{#4}{#2}+\Force[#1]{#4}\times\Vct{#2 #3}}

%Transport formula for moments without force precision \GVARS{Force}{A}{B} :
\newcommand{\GVARS}[3]{\Gmoment{#1}{#3}=\Gmoment{#1}{#2}+{#1}\times\Vct{#2 #3}}

%Volumetric force resultant \Vnet[S]{f}{Ext} :
\newcommand{\Vnet}[3][S]{\iiint_{M \in #1}\Vct{#2}_{#3\rightarrow #1}(M)dm}

%Volumetric force resultant with explicit volume \VVnet[S]{f}{Ext} :
\newcommand{\VVnet}[3][S]{\iiint_{M \in #1}\rho(M)\Vct{#2}_{#3\rightarrow #1}(M)dV}

%Surface force resultant \Snet[S]{T}{Ext} :
\newcommand{\Snet}[3][S]{\iint_{M \in \partial #1}\Vct{#2}_{#3\rightarrow #1}(M)dS}

%Surface force resultant with explicit normal \SSnet[S]{T}{Ext}{n} :
\newcommand{\SSnet}[4][S]{\iint_{M \in \partial #1}{#2}_{#3\rightarrow #1}(M)\Vct{#4}dS}

%################################### C-Kinetics :

%Momentum vector definition \Psr[R]{S} :
\newcommand{\Psr}[2][R]{\Vct{p}_{#2/#1}}

%Momentum vector 2nd definition \Mv[R]{S} :
\newcommand{\Mv}[2][R]{m_{#2}\Vel[#1]{G}{#2}}

%Angular momentum integral def \Lmi[R]{A}{S}:
\newcommand{\Lmi}[3][R]{\int_{M \in {#3}}\Vct{#2 M}\times \Vel[#1]{M}{#3} dm}

%Angular momentum definition \Lm[R]{A}{S} :
\newcommand{\Lm}[3][R]{\Vct{L}_{#2, #3/#1}}

%Transport formula for angular momenta \VARC[R]{A}{B}{S} :
\newcommand{\VARC}[4][R]{\Lm[#1]{#3}{#4}=\Lm[#1]{#2}{#4}+\Psr[#1]{#4}\times\Vct{#2 #3}}

%Alternate transport formula for angular momenta \AVARC[R]{A}{B}{S} :
\newcommand{\AVARC}[4][R]{\Lm[#1]{#3}{#4}=\Lm[#1]{#2}{#4}+\Mv[#1]{#4}\times\Vct{#2 #3}}

%Inertia tensor definition \Inertia[R]{A}{S} :
\newcommand{\Inertia}[3][R]{\overline{\overline{I}}_{#2}(#3/#1)}

%Angular momentum general definition on arbitrary point \LA[R]{A}{S} :
\newcommand{\LA}[3][R]{m_{#3}\Vct{#2 G}\times \Vel[#1]{G}{#3}+\Inertia[#1]{#2}{#3} \cdot \Rotation[#1]{#3}}

%Angular momentum general definition on fixed point and/or center of gravity \LF[R]{A}{S} :
\newcommand{\LF}[3][R]{\Inertia[#1]{#2}{#3} \cdot \Rotation[#1]{#3}}

%Transport inertia tensor from G to A \Ip[S]{G}{A} :
\newcommand{\Ip}[3][S]{\overline{\overline{I}}_{\Vct{#3 #2}}(#1)}

%Huygens formula \Huygens[R]{G}{A}{S} :
\newcommand{\Huygens}[4][R]{\Inertia[#1]{#3}{#4}=\Inertia[#1]{#2}{#4}+ \Ip[#4]{#2}{#3}}

%Standard inertia tensor explicit expression \InertiaE[R]{A}{S} :
\newcommand{\InertiaE}[3][R]{\begin{Bmatrix}
A_{#3} & -D_{#3} & -E_{#3} \\ 
-D_{#3} & B_{#3} & -F_{#3} \\ 
-E_{#3} & -F_{#3} & C_{#3}
\end{Bmatrix}_{ (#3 / #1) }}

%################################### C-Dynamics :

%Acceleration vector definition \Acc[R]{A}{S} :
\newcommand{\Acc}[3][R]{\Vct{\Gamma}_{#2\in #3/#1}}

%Acceleration vector differential definition \Accd[R]{A}{S} :
\newcommand{\Accd}[3][R]{\Dex[#1]{\Vel[#1]{#2}{#3}}}

%Coriolis acceleration vector \Coriolis{R_0}{R_1}{A}{S} :
\newcommand{\Coriolis}[4]{2\Rotation[#1]{#2}\times\Vel[#2]{#3}{#4}}

%Transport acceleration vector \Entr{R_0}{R_1}{O}{A} :
\newcommand{\Entr}[4]{\Acc[#1]{#3}{#2}+\Rotation[#1]{#2}\times(\Rotation[#1]{#2}\times \Vct{#3 #4})+\Dex[#1]{\Rotation[#1]{#2}}\times\Vct{#3 #4}}

%Acceleration momentum vector definition \Am[R]{S} :
\newcommand{\Am}[2][R]{\Vct{d}_{#2/#1}}

%Acceleration momentum vector 2nd definition \Amg[R]{S} :
\newcommand{\Amg}[2][R]{m_{#2}\Acc[#1]{G}{#2}}

%Dynamic momentum integral def \IDelta[R]{A}{S}:
\newcommand{\IDelta}[3][R]{\int_{M \in {#3}}\Vct{#2 M}\times \Acc[#1]{M}{#3} dm}

%Dynamic momentum definition \ADelta[R]{A}{S} :
\newcommand{\ADelta}[3][R]{\Vct{\delta}_{#2, #3/#1}}

%Transport formula for dynamic momenta \VARD[R]{A}{B}{S} :
\newcommand{\VARD}[4][R]{\ADelta[#1]{#3}{#4}=\ADelta[#1]{#2}{#4}+\Am[#1]{#4}\times\Vct{#2 #3}}

%Alternate transport formula for dynamic momenta \AARD[R]{A}{B}{S} :
\newcommand{\AVARD}[4][R]{\ADelta[#1]{#3}{#4}=\ADelta[#1]{#2}{#4}+\Amg[#1]{#4}\times\Vct{#2 #3}}

%Dynamic momentum general expression \EDelta[R]{A}{S} :
\newcommand{\EDelta}[3][R]{\Dex[#1]{\Lm[#1]{#2}{#3}}+m_{#3}\Vct{v}_{#2/#1}\times\Vct{v}_{G/#1}}

%Dynamic momentum general expression for fixed point or center of gravity \FDelta[R]{G}{S} :
\newcommand{\FDelta}[3][R]{\Dex[#1]{\Lm[#1]{#2}{#3}}}

%################################### D-Screw field representation :

%Kinematic screw field on a point \Ktor[R]{A}{S}:
\newcommand{\Ktor}[3][R]{\begin{Bmatrix}
\mathscr{V}_{({#3}/{#1})}
\end{Bmatrix}=
\begin{Bmatrix}
\Rotation[#3]{#1} \\ 
\Vel[#1]{#2}{#3}
\end{Bmatrix}_{#2}}

%Explicit kinematic screw field for a support \EKtor[R]{A}{S} :
\newcommand{\EKtor}[3][R]{\begin{Bmatrix}
\mathscr{V}_{({#3}/{#1})}
\end{Bmatrix}=
\begin{Bmatrix}
\omega_{x,({#3}/{#1})} & v_{x,(#2\in #3/#1)} \\ 
\omega_{y,({#3}/{#1})} & v_{y,(#2\in #3/#1)} \\ 
\omega_{z,({#3}/{#1})} & v_{z,(#2\in #3/#1)}
\end{Bmatrix}_{#2}}

%Static screw field on a point \Stor[S]{Ext}{A}:
\newcommand{\Stor}[3][S]{\begin{Bmatrix}
\mathscr{T}_{({#2}\rightarrow{#1})}
\end{Bmatrix}=
\begin{Bmatrix}
\Force[#1]{#2} \\ 
\Moment[#1]{#2}{#3}
\end{Bmatrix}_{#3}}

%Kinetic screw field on a point \Ctor[R]{A}{S}:
\newcommand{\Ctor}[3][R]{\begin{Bmatrix}
\mathscr{C}_{({#3}/{#1})}
\end{Bmatrix}=
\begin{Bmatrix}
\Psr[#3]{#1} \\ 
\Lm[#1]{#2}{#3}
\end{Bmatrix}_{#2}}

%Dynamic screw field on a point \Dtor[R]{A}{S}:
\newcommand{\Dtor}[3][R]{\begin{Bmatrix}
\mathscr{D}_{({#3}/{#1})}
\end{Bmatrix}=
\begin{Bmatrix}
\Am[#3]{#1} \\ 
\ADelta[#1]{#2}{#3}
\end{Bmatrix}_{#2}}

%Explicit static screw field for a support \Ltor{S_1}{S_2}{A} :
\newcommand{\Ltor}[3]{\begin{Bmatrix}
\mathscr{L}_{({#1}\rightarrow{#2})}
\end{Bmatrix}=
\begin{Bmatrix}
X_{({#1}\rightarrow{#2})} & L_{({#1}\rightarrow{#2})} \\ 
Y_{({#1}\rightarrow{#2})} & M_{({#1}\rightarrow{#2})} \\ 
Z_{({#1}\rightarrow{#2})} & N_{({#1}\rightarrow{#2})}
\end{Bmatrix}_{#3}}

%Explicit static screw field for a planar support \Lptor{S_1}{S_2}{A} :
\newcommand{\Lptor}[3]{\begin{Bmatrix}
\mathscr{L}_{({#1}\rightarrow{#2})}
\end{Bmatrix}=
\begin{Bmatrix}
X_{({#1}\rightarrow{#2})} & 0 \\ 
Y_{({#1}\rightarrow{#2})} & 0 \\ 
0 & M_{({#1}\rightarrow{#2})}
\end{Bmatrix}_{#3}}

%Empty screw field template on a point \Ftor[S/R]{W}{A}{B}{O}:
\newcommand{\Ftor}[5][Ref]{\begin{Bmatrix}
\mathscr{#2}_{#1}
\end{Bmatrix}=
\begin{Bmatrix}
{#3} \\ 
{#4}
\end{Bmatrix}_{#5}}

%Empty explicit screw field template on a point \LFtor[S/R]{W}{X}{Y}{Z}{X}{Y}{Z}{0}:
\newcommand{\LFtor}[9][Ref]{\begin{Bmatrix}
\mathscr{#2}_{#1}
\end{Bmatrix}=
\begin{Bmatrix}
{#3} & {#6} \\ 
{#4} & {#7} \\ 
{#5} & {#8}
\end{Bmatrix}_{#9}}

%################################### E-Energy and power :

%Power delivered from ext to S \Pow[R]{S}{Ext} :
\newcommand{\Pow}[3][R]{P_{(#3 \rightarrow #2/#1)}}

%Power delivered from ext to S, screw expression \Ptor[R]{S}{Ext} :
\newcommand{\Ptor}[3][R]{\begin{Bmatrix}
\mathscr{T}_{({#3}\rightarrow{#2})}
\end{Bmatrix} \otimes \begin{Bmatrix}
\mathscr{V}_{({#2}/{#1})}
\end{Bmatrix}}

%Power delivered from ext to S, explicit expression \Power[R]{A}{S}{Ext} :
\newcommand{\Power}[4][R]{\Force[#3]{#4}\cdot\Vel[#1]{#2}{#3} + \Gmoment{\Force[#3]{#4}}{#2}\cdot \Rotation[#1]{#3}}

%Power delivered from ext to S, integral expression \IPow[R]{S}{f} :
\newcommand{\IPow}[3][R]{\int_{M \in #2} \Vct{#3}(M)\cdot \Vel[#1]{M}{#2} dm}

%Internal support power \LPow{S_1}{S_2} :
\newcommand{\LPow}[2]{P_{(#1 \leftrightarrow  #2)}}

%Internal support power, screw definition \PLtor{S_1}{S_2} :
\newcommand{\PLtor}[2]{\begin{Bmatrix}
\mathscr{T}_{({#1}\rightarrow{#2})}
\end{Bmatrix} \otimes \begin{Bmatrix}
\mathscr{V}_{({#2}/{#1})}
\end{Bmatrix}}

%Kinetic energy, integral expression \KE[R]{S}:
\newcommand{\KE}[2][R]{ \frac{1}{2}\int_{M \in #2} \Vel[#1]{M}{#2}^{2}dm }

%Kinetic energy, screw expression \KEtor[R]{S}:
\newcommand{\KEtor}[2][R]{\frac{1}{2} \begin{Bmatrix}
\mathscr{C}_{(#2 / #1)}
\end{Bmatrix} \otimes \begin{Bmatrix}
\mathscr{V}_{(#2 / #1)}
\end{Bmatrix}}

%Kinetic energy, translation only \KineticT[R]{A}{S}:
\newcommand{\KineticT}[3][R]{\frac{1}{2}m_{#3}\Vel[#1]{#2}{#3}^{2}}

%Kinetic energy, rotation about center only \KineticR[R]{C}{S}:
\newcommand{\KineticR}[3][R]{\frac{1}{2}\Rotation[#1]{#3} \cdot (\Inertia[#1]{#2}{#3} \cdot \Rotation[#1]{#3})}

%Kinetic energy, general expression \Kinetic[R]{A}{S}:
\newcommand{\Kinetic}[3][R]{\KineticT[#1]{#2}{#3} + \KineticR[#1]{#2}{#3} + m_{#3} \Rotation[#1]{#3} \cdot( \Vct{#2 G} \times \Vel[#1]{#2}{#3}) }

%Kinetic energy written in G \KineticG[R]{S}:
\newcommand{\KineticG}[2][R]{\KineticT[#1]{G}{#2} + \KineticR[#1]{G}{#2}}

\endinput