137 lines
4.3 KiB
TeX
137 lines
4.3 KiB
TeX
%\newpage
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%\addtocontents{toc}{\protect\setcounter{tocdepth}{0}} % Désactivation de la table des matières
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% Personnalisation de la table des annexes
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%\renewcommand{\stctitle}{} % Titre (issue with previous subsection showing up)
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%\renewcommand\thesubsection{A\Alph{section}} % Numérotation
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%\renewcommand{\stcSSfont}{} % Police normale, pas en gras
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%\mtcsetrules{secttoc}{off} % Désactivation des lignes en haut et en bas de la table
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% Affichage de la table des annexes
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%\secttoc
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\newpage
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\pagenumbering{Alph}
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\section{Annexe}
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\subsection{Question 2}
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\vspace*{-2em}
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\begin{align*}
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\left\{\begin{aligned}
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x_1 &\approx \phi^Tx_{1d}(t) \\
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x_2 &\approx \phi^Tx_{2d}(t)
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\end{aligned}\right. & &
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\left\{\begin{aligned}
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e_1 &\approx \phi^Te_{1d}(t) \\
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e_2 &\approx \phi^Te_{2d}(t)
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\end{aligned}\right.
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\end{align*}
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En utilisant la première ligne de l'équation 1, on trouve :
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\begin{equation*}
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\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}
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\end{equation*}
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\begin{equation}
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\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}
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\end{equation}
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On applique plusieurs fois de l’intégration par partie (IPP) :
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\begin{equation*}
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\begin{aligned}
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\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 \\
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&= \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 \\
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&= \phi(L)\dot{\phi}(L)^T - \dot{\phi}(L)\phi(L)^T + \int_{0}^{L} \ddot{\phi}(\zeta)\phi(\zeta)^T d\zeta = D
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\end{aligned}
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\end{equation*}
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On continue avec ça en remplaçant dans l'équation 2 : $\Rightarrow \boxed{E\dot{x}_{1d} = De_{2d}}$ \\
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Puis on a dans la deuxième ligne de l'équation 1 :
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\begin{equation*}
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\begin{aligned}
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\phi^T(\zeta)\dot{x}_{2d} &= - \frac{\partial^2}{\partial\zeta^2} (\phi(\zeta)^T e_{1d}) - q(\zeta, t) \\
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\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}}
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\end{aligned}
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\end{equation*}
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Puis on trouve D :
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\begin{equation*}
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\begin{aligned}
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& D = \phi(L)\dot{\phi}(L)^T - \phi(L)\dot{\phi}(L)^T + \int_{0}^{L}\ddot{\phi}(\zeta)\phi(\zeta)^Td\zeta \\
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\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 \\
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\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 \\
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\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}
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\end{aligned}
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\end{equation*}
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Or on sait que :
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\begin{align*}
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\Rightarrow\left\{\begin{aligned}
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\frac{de_1}{d\zeta} = u(t) &\Rightarrow \dot{\phi}(L)^T e_{1d} = u(t) \\
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e_1 (L,t) = 0 &\Rightarrow \phi(L)^T e_{1d} = 0
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\end{aligned}\right. & &
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\Rightarrow & &
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\boxed{E\dot{x}_{2d} = - e_{1d}D^T - \phi(L)u(t) - F_{ext}}
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\end{align*}
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Puis on a ceci :
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\begin{align*}
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\left\{\begin{aligned}
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y(t) &= -e_2 (L,t) \\
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e_2 (L,t) &\approx \phi(L)^T e_{2d}
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\end{aligned}\right. & &
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\Rightarrow & &
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\boxed{y(t) = -\phi(L)^T e_{2d}}
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\end{align*}
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\subsection{Question 3}
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On a :
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\begin{align*}
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\left\{\begin{aligned}
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e_1 = EIx_1 = x_1 &\Rightarrow e_{1d} = x_{1d} \\
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e_2 = \frac{1}{\rho(\zeta)}x_2 = x_2 &\Rightarrow e_{2d} = x_{2d}
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\end{aligned}\right. & &
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\Rightarrow & &
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\left\{\begin{aligned}
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E\dot{X}_{1d} &= Dx_{2d} \\
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E\dot{X}_{2d} &= -D^T x_{1d} - \phi(L)u(t) - 0 \\
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y &= -\phi(L)^T x_{2d}
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\end{aligned}\right.
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\end{align*}
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Ce qui nous donne :
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\begin{equation*}
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A = \begin{pmatrix}
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0 & E^{-1}D \\
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-E^{-1}D^T & 0
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\end{pmatrix}, \ B = \begin{pmatrix}
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0 \\
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-E^{-1}\phi(L)
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\end{pmatrix}, \ C = \begin{pmatrix}
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0 & -\phi(L)^T
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\end{pmatrix},\ pour \ \left\{\begin{aligned}
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\zeta = L = 1 \\
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\phi(1) = \begin{pmatrix}
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0 \\
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1 \\
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0 \\
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0
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\end{pmatrix}
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\end{aligned}\right.
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\end{equation*}
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Puis application numérique.
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