From 4741a8ef84b9fe7edd116988e2482b96e6df1063 Mon Sep 17 00:00:00 2001 From: taban Date: Thu, 7 May 2026 18:22:28 +0200 Subject: [PATCH] ajout des contenus dans les annexes --- latex/contents/annexes.tex | 124 ++++++++++++++++++++++++-- latex/main.pdf | Bin 572878 -> 573334 bytes latex/template/table_des_matieres.tex | 2 +- 3 files changed, 119 insertions(+), 7 deletions(-) diff --git a/latex/contents/annexes.tex b/latex/contents/annexes.tex index 0fb729c..df08436 100644 --- a/latex/contents/annexes.tex +++ b/latex/contents/annexes.tex @@ -14,12 +14,124 @@ \pagenumbering{Alph} \section{Annexe} -\subsection{Question 1} - - - - - +\subsection{Question 2} +\vspace*{-2em} +\begin{align*} + \left\{\begin{aligned} + x_1 &\approx \phi^Tx_{1d}(t) \\ + x_2 &\approx \phi^Tx_{2d}(t) + \end{aligned}\right. & & + \left\{\begin{aligned} + e_1 &\approx \phi^Te_{1d}(t) \\ + e_2 &\approx \phi^Te_{2d}(t) + \end{aligned}\right. +\end{align*} + +En utilisant la première ligne de l'équation 1, on trouve : + +\begin{equation*} + \int\phi(\zeta)d\zeta\times\phi^T\dot{x_{1d}} = \int\phi(\zeta)d\zeta\times\frac{\partial^2}{\partial\zeta^2} = \ddot{\phi}(\zeta)^Te_{2d} +\end{equation*} + +\begin{equation} + \underbrace{\int_{0}^{L}\phi(\zeta)\phi(\zeta)^Td\zeta}_\text{E}\times\dot{x_{1d}} = \left(\int_{0}^{L}\phi(\zeta)\ddot{\phi}(\zeta)d\zeta\right)e_{2d} +\end{equation} + +On applique plusieurs fois de l’intégration par partie (IPP) : + +\begin{equation*} + \begin{aligned} + \int_{0}^{L}\phi(\zeta)\ddot{\phi}(\zeta)^T d\zeta &= [\phi(\zeta)\dot{\phi}(\zeta)^T]_{0}^{L} - \int_{0}^{L}\dot{\phi}(\zeta)\dot{\phi}(\zeta)^T d\zeta \\ + &= \phi(L)\dot{\phi}(L)^T - \underbrace{\phi(0)\dot{\phi}(0)^T }_\text{=0} - \dot{\phi}(L)\phi(L)^T + \underbrace{\dot{\phi(0)}\phi(0)^T }_\text{=0} + \int_{0}^{L} \ddot{\phi}(\zeta)\phi(\zeta)^T d\zeta \\ + &= \phi(L)\dot{\phi}(L)^T - \dot{\phi}(L)\phi(L)^T + \int_{0}^{L} \ddot{\phi}(\zeta)\phi(\zeta)^T d\zeta = D + \end{aligned} +\end{equation*} + +On continue avec ça en remplaçant dans l'équation 2 : $\Rightarrow \boxed{E\dot{x}_{1d} = De_{2d}}$ \\ + + +Puis on a dans la deuxième ligne de l'équation 1 : + +\begin{equation*} + \begin{aligned} + \phi^T(\zeta)\dot{x}_{2d} &= - \frac{\partial^2}{\partial\zeta^2} (\phi(\zeta)^T e_{1d}) - q(\zeta, t) \\ + \int_{0}^{L} \phi^T(\zeta)d\zeta\times\dot{x}_{2d} &= -e_{1d}\int_{0}^{L}\phi(\zeta)\ddot{\phi}(\zeta)^T d\zeta - \underbrace{\int_{0}^{L}\phi(\zeta)q(\zeta,t)d\zeta}_{=F_{ext}} + \end{aligned} +\end{equation*} + +Puis on trouve D : + +\begin{equation*} + \begin{aligned} + & D = \phi(L)\dot{\phi}(L)^T - \phi(L)\dot{\phi}(L)^T + \int_{0}^{L}\ddot{\phi}(\zeta)\phi(\zeta)^Td\zeta \\ + \Rightarrow & D^T = \dot{\phi}(L)\phi(L)^T - \dot{\phi}(L)\phi(L)^T + \int_{0}^{L}\phi(\zeta)\ddot{\phi}(\zeta)d\zeta \\ + \Rightarrow & \boxed{\int_{0}^{L}\underbrace{\phi(\zeta)}\ddot{\phi}(\zeta)^Td\zeta} = D^T - \dot{\phi}(L)\phi(L)^T + \phi(L)\dot{\phi}(L)^T \\ + \Rightarrow & E\dot{x}_{2d} = -e_{1d}D^T + \overbrace{e_{1d}\dot{\phi}(L)\phi(L)^T}^{=0} - e_{1d}\phi(L)\dot{\phi}(L)^T - F_{ext} + \end{aligned} +\end{equation*} + +Or on sait que : + +\begin{align*} + \Rightarrow\left\{\begin{aligned} + \frac{de_1}{d\zeta} = u(t) &\Rightarrow \dot{\phi}(L)^T e_{1d} = u(t) \\ + e_1 (L,t) = 0 &\Rightarrow \phi(L)^T e_{1d} = 0 + \end{aligned}\right. & & + \Rightarrow & & + \boxed{E\dot{x}_{2d} = - e_{1d}D^T - \phi(L)u(t) - F_{ext}} +\end{align*} + +Puis on a ceci : + +\begin{align*} + \left\{\begin{aligned} + y(t) &= -e_2 (L,t) \\ + e_2 (L,t) &\approx \phi(L)^T e_{2d} + \end{aligned}\right. & & + \Rightarrow & & + \boxed{y(t) = -\phi(L)^T e_{2d}} +\end{align*} + + +\subsection{Question 3} +On a : + +\begin{align*} + \left\{\begin{aligned} + e_1 = EIx_1 = x_1 &\Rightarrow e_{1d} = x_{1d} \\ + e_2 = \frac{1}{\rho(\zeta)}x_2 = x_2 &\Rightarrow e_{2d} = x_{2d} + \end{aligned}\right. & & + \Rightarrow & & + \left\{\begin{aligned} + E\dot{X}_{1d} &= Dx_{2d} \\ + E\dot{X}_{2d} &= -D^T x_{1d} - \phi(L)u(t) - 0 \\ + y &= -\phi(L)^T x_{2d} + \end{aligned}\right. +\end{align*} + +Ce qui nous donne : + +\begin{equation*} + A = \begin{pmatrix} + 0 & E^{-1}D \\ + -E^{-1}D^T & 0 + \end{pmatrix}, \ B = \begin{pmatrix} + 0 \\ + -E^{-1}\phi(L) + \end{pmatrix}, \ C = \begin{pmatrix} + 0 & -\phi(L)^T + \end{pmatrix},\ pour \ \left\{\begin{aligned} + \zeta = L = 1 \\ + \phi(1) = \begin{pmatrix} + 0 \\ + 1 \\ + 0 \\ + 0 + \end{pmatrix} + \end{aligned}\right. +\end{equation*} + +Puis application numérique. diff --git a/latex/main.pdf b/latex/main.pdf index fbd9405a8e5fc43592b354bd9fe59b921c72dbeb..3574091e2b86a6867181c0e76993494707588c7b 100644 GIT binary patch delta 25755 zcmV(zK<2;B`y!VAB7n33J5d8UH#L)?+9`iqS(Dqw5q{UNK#~WMC2>6WA(yIRNoz+g z#g668OC(o{;!+ALyQIWJl5_m`d@}=p7;pfRGIzapT&mInV56sJy8G*62CSg?#|lBH z5V)FL-FbJ15n-4z@h8HSDB2o$v6E7H{zItDV2qE2G9JkIsMZ zeDRME+eUOO;UvcO);_rj|1VyNF`Zeh%+=0X~Kmn zMvzG8Qn(ziQANjyW8xTgJU0tT43>Y43u26_C!r!GerQlyVk{=XFk8zb>BH>0Frt_$ 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b/latex/template/table_des_matieres.tex @@ -4,4 +4,4 @@ \newpage \pagenumbering{arabic} % Numérotation en chiffres romains (i, ii, iii, ...) -\setcounter{page}{1}1 \ No newline at end of file +\setcounter{page}{1} \ No newline at end of file