From 17314eb9df935f6b07768090c21014d9b33b6305 Mon Sep 17 00:00:00 2001 From: taban Date: Sun, 19 Apr 2026 02:17:30 +0200 Subject: [PATCH] =?UTF-8?q?Premiere=20partie=20finie,=20bisous=20=C3=A0=20?= =?UTF-8?q?tous?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- latex/contents.tex | 153 +++++++++++++++++++++++++++++++++++++++++++++ latex/main.pdf | Bin 205411 -> 227548 bytes 2 files changed, 153 insertions(+) diff --git a/latex/contents.tex b/latex/contents.tex index 1a37de7..bf52f8f 100644 --- a/latex/contents.tex +++ b/latex/contents.tex @@ -110,19 +110,172 @@ Puis on a dans la deuxième ligne de l'équation 1 : \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 + \end{aligned} +\end{equation*} + +Donc : + +\begin{equation*} + 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{equation*} + +Or on sait que : + +\begin{equation} + \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. +\end{equation} + +Donc : + +\begin{equation*} + \boxed{E\dot{x}_{2d} = - e_{1d}D^T - \phi(L)u(t) - F_{ext}} +\end{equation*} + +Puis on a : + +\begin{equation*} + \left\{\begin{aligned} + y(t) &= -e_2 (L,t) \\ + e_2 (L,t) &\approx \phi(L)^T e_{2d} + \end{aligned}\right. +\end{equation*} + +Ce qui nous donne l'approximation : + +\begin{equation*} + \boxed{y(t) = -\phi(L)^T e_{2d}} +\end{equation*} \subsection{Exercice 3} +On a : + +\begin{equation*} + \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. +\end{equation*} + +Donc : + +\begin{equation*} + \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{equation*} + +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} +\end{equation*} + +Pour : + +\begin{equation*} + \left\{\begin{aligned} + \zeta = L = 1 \\ + \phi(1) = \begin{pmatrix} + 0 \\ + 1 \\ + 0 \\ + 0 + \end{pmatrix} + \end{aligned}\right. \Rightarrow d\acute{e}cla \ A, \ B, \ C \ en \ MATLAB +\end{equation*} \subsection{Exercice 4} +\begin{equation*} + \omega(p, t) = C_\omega(p)x_d(t) +\end{equation*} + +\begin{equation*} + C_\omega(p) = \begin{bmatrix} + \frac{p^2(2p^3-5p^2+10)}{20} & -\frac{p^4(2p-5)}{20} & \frac{p^3(3p^2-10p+10)}{60} & \frac{p^4(3p-5)}{60} & 0 & 0 & 0 & 0 + \end{bmatrix} +\end{equation*} + +\begin{equation*} + \phi(L)^T = \begin{bmatrix} + 2p^3 - 3p^2 + 1 & 3p^2 - 2p^3 & p^3 - 2p^2 + p & p^3 - p^2 + \end{bmatrix} +\end{equation*} \subsection{Exercice 5} +\begin{align*} + y = -\phi(L)^Te_{2d} && y = -e_2(L, t) & & e_2(\zeta, t) \approx \phi(\zeta)^Te_{2d}(t) +\end{align*} + +\begin{align*} + x_1 = \frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t)\approx\phi(\zeta)^2x_{1d} & & x_2(\zeta,t) = \rho(\zeta)\frac{\partial\omega}{\partial t}(\zeta,t) \approx\phi(\zeta)^Tx_{2d} +\end{align*} + +\begin{align*} + \begin{bmatrix} + \dot{x}_{1d} \\ \dot{x}_{1d} + \end{bmatrix} = A \begin{bmatrix} + x_{1d} \\ x_{2d} + \end{bmatrix} + Bu & & y = C \begin{bmatrix} + x_{1d} \\ x_{2d} + \end{bmatrix} +\end{align*} \subsection{Exercice 6} +\begin{equation*} + \begin{bmatrix} + x_1 \\ x_2 + \end{bmatrix} = \begin{bmatrix} + \frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t) \\ + \rho(\zeta)\frac{\partial\omega}{\partial t}(\zeta,t) + \end{bmatrix} = \begin{bmatrix} + \phi(\zeta)^T & 0000 \\ + 0000 & \phi(\zeta)^T + \end{bmatrix} \begin{bmatrix} + x_{1d} \\ x_{2d} + \end{bmatrix} +\end{equation*} +\begin{equation*} + \frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t) = \underbrace{\begin{bmatrix} + \phi(\zeta)^T & 0 & 0 & 0 & 0 + \end{bmatrix}}_{C.I.=0+0}x_{d}(t) +\end{equation*} +\begin{equation*} + \begin{aligned} + \frac{\partial\omega}{\partial\zeta}(\zeta,t) &= \int\frac{\partial^2\omega}{\partial\zeta^2}(\zeta,t)d\zeta = x_d(t)\int\begin{bmatrix} + \phi(\zeta)^T & 0 & 0 & 0 & 0 + \end{bmatrix}d\zeta + C_1(t) \\ + \omega(\zeta,t) &= \int\int\frac{\partial^2\omega}{\partial^2\zeta^2}(\zeta,t)d\zeta^2 = x_d(t)\int\int\begin{bmatrix} + \phi(\zeta)^T & 0 & 0 & 0 & 0 + \end{bmatrix}d\zeta^2 + \underbrace{\int C_1(t)d\zeta + C_2(t)}_{\zeta C_1(t) + C_2(t)} + \end{aligned} +\end{equation*} + +$C_1$ et $C_2$ ????????????????? \section{Retour de sortie} diff --git a/latex/main.pdf b/latex/main.pdf index 8498ee635324a14c6b63c9b032710f8e1262a143..d01af9c38ed096cd21b8c47d1f343bffb47432bf 100644 GIT binary patch delta 101910 zcmV)pK%2kg#0=c=4X}Eum%-=)E0d+F8Gq7g(z*jBbY&u;OQ2f6{cLQqCdCo0)5PMS2 zvm#$7lXt5ge-AORvsId>+0a3rFRI5Lu-&0U!K)=C2wwRK|1^OZ5D8Z_rT|G>j!4H5 z8ICb*|{HOOXNWD(4v(c#Lu2Jq-fV8wAfZe_D5=v99PxdWYZ}kk10CbqIk0 z`7DrHhX4V;KR^cI4#Hkaj60P0!xg5mN-8lyL%!EvfAO9P8Z?d6C(f~6c_A_g`yte7 zF2n|5KZJUqUGUR|pdGQ==~{RNop5nyWCtxxEaJp#-O z!hQ(#e*)|og!>TEmY`mLeM7Kc5NcP#U|Rvz2(zL zut?0a<@!d74u5NIiFbnQVO=%Rr3ikMNtEOTuht9RJRLRs ze~*$Pt&8nC>5fC_(f*3|@BRoYCf&obx1h_~PClTxj`Rn;! z^}?>6>ScyAoS{tUpkepxSJiLdszt%^PZWxArSR+Y>fqynV9E*O%3DktEp7Z=-mIRe zF1+~Ll`M{L4t~ZzUi^ZKe{KG+L1!)O-<=%1cx``tAvg;>h~nfuehLkoY=hvT(X43h?>zljV^YZ8Eyt?-7(YurHfyX3xlw&6NVkQ;yMwEEoa>FZN)du#= zd4L9n%Moxs!_-R8FG4F<15*mW{ z3Zdz+L^EMRnFMmq;{JcE 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