23 lines
727 B
TeX
23 lines
727 B
TeX
\textbf{Question 15}
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\begin{center}
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Soit $q(\zeta, t) = q_0(t)$ et $F_{ext}=-\int_0^L\Phi(\zeta)q(\zeta,t)d\zeta$
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$\Leftrightarrow F_{ext}=-\int_0^L\Phi(t)q_0(t)$\\
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Or $E\dot{x}_{2d}=-D^Tx_{1d}(t)-\Phi(L)u(t)-F_{ext}$
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$\Leftrightarrow$
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$\dot{x}_{2d}=-E^{-1}D^Tx_{1d}(t)-E^{-1}\Phi(L)u(t)+ E^{-1}\int_0^L\Phi(\zeta)d\zeta q_0(t)$\\
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D'où $\dot{x}_d(t)=Ax_d(t)+Bu(t)+ \begin{bmatrix}
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0_{\rm I\!R_{_{4\times1}}} \\ E^{-1}\int_0^L\Phi(\zeta)d\zeta
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\end{bmatrix}q_0(t)$
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\end{center}
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Après avoir calculé, nous retrouvons la matrice $B_p$ sous la forme:
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\begin{equation}
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B_p =
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\begin{bmatrix}
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0 \\ -E^{-1}\int_0^L \Phi(\zeta)d\zeta
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\end{bmatrix}
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=
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\begin{bmatrix}
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0 & 0 & 0 & 0 & 1 & 1 & 0 & 0
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\end{bmatrix}^{T}
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\end{equation}
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