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Aufgabenstellung

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Lösung mit Maxima

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$\underset{\_}{\tilde{Q}}=\left[\begin{array}{c}{W}_0\\ {\mathrm{\Phi }}_0\\ W_1\\ {\mathrm{\Phi }}_1\\ W_2\\ {\mathrm{\Phi }}_2,\\ {\mathrm{\Psi }}_1,\\ {\mathrm{\Psi }}_2\end{array}\right]$

$\underset{\_}{\delta \tilde{Q}}=\left[\begin{array}{c}\delta W_0\\ \delta {\mathrm{\Phi }}_0\\ \delta W_1\\ \delta {\mathrm{\Phi }}_1\\ \delta W_2\\ \delta {\mathrm{\Phi }}_2,\\ \delta {\mathrm{\Psi }}_1,\\ \delta {\mathrm{\Psi }}_2\end{array}\right]$

$\underset{\_}{Y}=\left[\begin{array}{c}\underset{\_}{Q}\\ \underset{\_}{\dot{Q}}\end{array}\right]$

$\begin{array}{ll}\underset{\_}{\dot{Y}}& =\underset{\_}{f}\left(\underset{\_}{Y}\right)\\ & =\left[\begin{array}{l}\underset{\_}{\dot{Q}}\\ -{\underset{\_}{\underset{\_}{M}}}^{-1}\cdot \underset{\_}{\underset{\_}{K}}\cdot \underset{\_}{Q}+{\underset{\_}{\underset{\_}{M}}}^{-1}\cdot \underset{\_}{P}\end{array}\right]\end{array}$

$\underset{\_}{\underset{\_}{M}}\cdot \underset{\_}{\ddot{Q}}+\underset{\_}{\underset{\_}{K}}\cdot \underset{\_}{Q}=\underset{\_}{P}(\underset{\_}{Q},\underset{\_}{\dot{Q}})$

$\begin{array}{lcc}\delta W& =& \delta W^a-\delta \mathrm{\Pi }\\ & \stackrel{!}{=}& 0\end{array}$

$\delta \mathrm{\Pi }=\delta {\mathrm{\Pi }}_G+\delta {\mathrm{\Pi }}_S$

$\delta {\mathrm{\Pi }}_S=K_S\cdot ({\mathrm{\Psi }}_2-{\mathrm{\Psi }}_1)\cdot (\delta {\mathrm{\Psi }}_2-\delta {\mathrm{\Psi }}_1)$

$\delta {\mathrm{\Pi }}_G=\delta {\mathrm{\Pi }}_{G1}+\delta {\mathrm{\Pi }}_{G2}$

$\delta {\mathrm{\Pi }}_{Gi}=\underset{0}{\overset{{\ell }_i}{\int }}{M}_i(x)\cdot \delta {w_i}^{″}(x)dx$

$M_i(x)=E\cdot I(x)\cdot {w_i}^{″}(x)$

$w_i(x)={\underset{\_}{Q}}_i^T\cdot \underset{\_}{\phi }$

$\underset{\_}{\phi }=\left[\begin{array}{c}(\xi -1{)}^2\cdot (2\cdot \xi +1)\\ {\ell }_i\cdot \xi \cdot (\xi -1{)}^2\\ -{\xi }^2\cdot (2\xi -3)\\ {\ell }_i\cdot {\xi }^2\cdot (\xi -1)\end{array}\right]$

$\delta {\mathrm{\Pi }}_{Gi}=\underset{0}{\overset{{\ell }_i}{\int }}E\cdot I_i(x)\cdot (W_{i-1}{\phi }_1\cdot \delta W_{i-1}{\phi }_1+{\mathrm{\Phi }}_{i-1}{\phi }_2\cdot \delta W_{i-1}{\phi }_1+W_i{\phi }_3\cdot \delta W_{i-1}{\phi }_1+\dots +{\mathrm{\Phi }}_i{\phi }_4\cdot \delta {\mathrm{\Phi }}_i{\phi }_4)$

$\delta {\mathrm{\Pi }}_{Gi}=\delta Q_i^T\cdot {\underset{\_}{\underset{\_}{K}}}_i\cdot Q_i$

$k_{j,k}^i=\underset{0}{\overset{{\ell }_i}{\int }}E\cdot I_i(x){\phi }_j\cdot {\phi }_{k}dx$

$I_i(x)=\frac{\pi }{64}(D_i(x{)}^4-d_i(x{)}^4)$

$\begin{array}{ccccc}{D}_i(x)& =& D_{i-1}\cdot ({\xi }_1-1)& +& D_i\cdot {\xi }_1\\ d_i(x)& =& d_{i-1}\cdot ({\xi }_1-1)& +& d_i\cdot {\xi }_1\end{array}$

${\xi }_1=\frac{x}{{\ell }_1}$

${\xi }_2=\frac{(x-H)}{{\ell }_2}$

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