水中線状構造物：状態方程式

● $\displaystyle{\bf Derivation \#3a}$
$\displaystyle{\eta(\xi,\tau) =\displaystyle\sum_{i=1}^\infty q_i(\tau)\phi_i(\xi)\simeq q^T(\tau)\phi(\xi)}$
$\displaystyle{ q(\tau)= \left[\begin{array}{c} q_1(\tau) \\ \vdots\\ q_N(\tau) \\ \end{array}\right],\ \phi(\xi)= \left[\begin{array}{c} \phi_1(\xi) \\ \vdots\\ \phi_N(\xi) \\ \end{array}\right] }$

for $i=1,\cdots,N$
$\displaystyle{\sum_{j=1}^N {m^y_{ij}} \times\ddot{q}_j +\sum_{j=1}^N {c^y_{ij}} \times\dot{q}_j +\sum_{j=1}^N {k^y_{ij}} \times q_j }$
$\displaystyle{+\sum_{j,k,\ell=1}^N{B^y_{ijk\ell}} \times q_j q_k q_\ell+\sum_{j,k,\ell=1}^N{D^y_{ijk\ell}} \times q_j q_k \dot{q}_\ell {+\sum_{j,k,\ell=1}^N{E^y_{ijk\ell}} \times q_j \dot{q}_k \dot{q}_\ell+\sum_{j,k,\ell=1}^N F^y_{ijk\ell} \times q_j q_k \ddot{q}_\ell}$
$\displaystyle{+\sum_{j,k,\ell=1}^N{H^y_{ijk\ell}} \times q_j p_k p_\ell+\sum_{j,k,\ell=1}^N{L^y_{ijk\ell}} \times q_j p_k \dot{p}_\ell +\sum_{j,k,\ell=1}^N{M^y_{ijk\ell}} \times q_j \dot{p}_k \dot{p}_\ell+\sum_{j,k,\ell=1}^N N^y_{ijk\ell} \times q_j p_k \ddot{p}_\ell}$
$\displaystyle{=\int_0^1\phi(\xi)\frac{L^3}{EI} f_yd\xi}$

● $\displaystyle{\bf Derivation \#3b}$
$\displaystyle{\zeta(\xi,\tau)=\displaystyle\sum_{i=1}^\infty p_i(\tau)\psi_i(\xi)\simeq p^T(\tau)\psi(\xi)}$
$\displaystyle{ p(\tau)= \left[\begin{array}{c} p_1(\tau) \\ \vdots\\ p_N(\tau) \\ \end{array}\right],\ \psi(\xi)= \left[\begin{array}{c} \psi_1(\xi) \\ \vdots\\ \psi_N(\xi) \\ \end{array}\right] }$

for $i=1,\cdots,N$
$\displaystyle{\sum_{j=1}^N {m^z_{ij}} \times\ddot{p}_j +\sum_{j=1}^N {c^z_{ij}} \times\dot{p}_j +\sum_{j=1}^N {k^z_{ij}} \times p_j }$
$\displaystyle{+\sum_{j,k,\ell=1}^N{B^z_{ijk\ell}} \times p_j p_k p_\ell +\sum_{j,k,\ell=1}^N{D^z_{ijk\ell}} \times p_j p_k \dot{p}_\ell +\sum_{j,k,\ell=1}^N{E^z_{ijk\ell}} \times p_j \dot{p}_k \dot{p}_\ell +\sum_{j,k,\ell=1}^N F^z_{ijk\ell} \times p_j p_k \ddot{p}_\ell}$
$\displaystyle{+\sum_{j,k,\ell=1}^N{H^z_{ijk\ell}} \times p_j q_k q_\ell +\sum_{j,k,\ell=1}^N{L^z_{ijk\ell}} \times p_j q_k \dot{q}_\ell +\sum_{j,k,\ell=1}^N{M^z_{ijk\ell}} \times p_j \dot{q}_k \dot{q}_\ell +\sum_{j,k,\ell=1}^N N^z_{ijk\ell} \times p_j q_k \ddot{q}_\ell}$
$\displaystyle{=\int_0^1\psi(\xi)\frac{L^3}{EI} f_zd\xi}$

● $\displaystyle{\bf Derivation \#4}$

$\displaystyle{\left[\begin{array}{cc} M_{qq} & M_{qp} \\ M_{pq} & M_{pp} \\ \end{array}\right] \left[\begin{array}{c} \ddot{q} \\ \ddot{p} \end{array}\right] +\left[\begin{array}{cc} D_{qq} & D_{qp} \\ D_{pq} & D_{pp} \\ \end{array}\right] \left[\begin{array}{c} \dot{q} \\ \dot{p} \end{array}\right]}$
$\displaystyle{ +\left[\begin{array}{cc} K_{qq} & K_{qp} \\ K_{pq} & K_{pp} \\ \end{array}\right] \left[ \begin{array}{c} q \\ p \end{array} \right] = \left[\begin{array}{c} \int_0^1\phi(\xi)\frac{L^3}{EI} f_yd\xi \\ \int_0^1\psi(\xi)\frac{L^3}{EI} f_zd\xi \end{array}\right]}$

$\displaystyle{M_{qq}=[m^y_{ij} + \sum_{l,k=1}^NF^y_{ilkj}q_lq_k]_{i,j=1,\cdots,N}}$
$\displaystyle{M_{qp}=[\sum_{l,k=1}^N N^y_{ilkj} q_l p_k]_{i,j=1,\cdots,N}}$
$\displaystyle{M_{pq}=[\sum_{l,k=1}^N N^z_{ilkj} p_l q_k]_{i,j=1,\cdots,N}}$
$\displaystyle{M_{pp}=[m^z_{ij} + \sum_{l,k=1}^N F^z_{ilkj}p_l p_k]_{i,j=1,\cdots,N}}$
$\displaystyle{D_{qq}=[c^y_{ij} + \sum_{l,k=1}^ND^y_{ilkj}q_lq_k+ \sum_{l,k=1}^NE^y_{ilkj}q_l\dot{q}_k]_{i,j=1,\cdots,N}}$
$\displaystyle{D_{qp}=[\sum_{l,k=1}^NL^y_{ilkj}q_lp_k+\sum_{l,k=1}^NM^y_{ilkj}q_l\dot{p}_k]_{i,j=1,\cdots,N}}$
$\displaystyle{D_{pq}=[\displaystyle\sum_{l,k=1}^NL^z_{ilkj}p_lq_k+\displaystyle\sum_{l,k=1}^NM^z_{ilkj}p_l\dot{q}_k]_{i,j=1,\cdots,N} }$
$\displaystyle{D_{pp}=[c^z_{ij} + \displaystyle\sum_{l,k=1}^ND^z_{ilkj}p_lp_k+ \displaystyle\sum_{l,k=1}^NE^z_{ilkj}p_l\dot{p}_k]_{i,j=1,\cdots,N} }$
$\displaystyle{K_{qq}=[k^y_{ij} + \displaystyle\sum_{l,k=1}^NB^y_{ilkj}q_kq_l+ \displaystyle\sum_{l,k=1}^NH^y_{ilkj}p_kp_l]_{i,j=1,\cdots,N}}$
$\displaystyle{K_{pp}=[k^z_{ij} + \displaystyle\sum_{l,k=1}^NB^z_{ilkj}p_kp_l+ \displaystyle\sum_{l,k=1}^NH^z_{ilkj}q_kq_l]_{i,j=1,\cdots,N}}$

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水中線状構造物：状態方程式