水中線状構造物：振動子

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

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$\displaystyle{ V_R(s,t)= \left[\begin{array}{c} V(s,t)-\dot{y}(s,t)\\ -\dot{z}(s,t) \end{array}\right] }$
$\displaystyle{||V_R(s,t)||^2=(V(s,t)-\dot{y}(s,t))^2+\dot{z}^2(s,t)}$

$\displaystyle{ \left[\begin{array}{c} f_y(s,t) \\ f_z(s,t) \end{array}\right] =-C_a\rho_eA_e \left[\begin{array}{c} \ddot{y}(s,t) \\ \ddot{z}(s,t) \end{array}\right] }$
$\displaystyle{ +C_{Ds}\cdot\frac{1}{2}\rho_eD_e||V_R(s,t)|| \left[\begin{array}{cc} V(s,t)-\dot{y}(s,t) \\ -\dot{z}(s,t) \end{array}\right] }$
$\displaystyle{+ \frac{1}{||V_R(s,t)||} \left[\begin{array}{cc} V(s,t)-\dot{y}(s,t) & \dot{z}(s,t)\\ -\dot{z}(s,t)& V(s,t)-\dot{y}(s,t) \end{array}\right] \left[\begin{array}{c} f_{Dv}(s,t) \\ f_{L}(s,t) \end{array}\right] }$
$\displaystyle{ \left[\begin{array}{c} f_{Dv}(s,t) \\ f_{L}(s,t) \end{array}\right] = \frac{1}{2}\rho_eD_e||V_R(s,t)||^2 \left[\begin{array}{c} \frac{1}{2}C_{D0}a(s,t) \\ \frac{1}{2}C_{L0}b(s,t) \end{array}\right] }$

$V(s,t)>>\sqrt{\dot{y}^2(s,t)+\dot{z}^2(s,t)}$
$\displaystyle{||V_R(s,t)||^2=(V(s,t)-\dot{y}(s,t))^2+\dot{z}^2(s,t)\simeq V^2(s,t)}$
$\displaystyle{ \frac{1}{||V_R(s,t)||} \left[\begin{array}{cc} V(s,t)-\dot{y}(s,t) & \dot{z}(s,t)\\ -\dot{z}(s,t)& V(s,t)-\dot{y}(s,t) \end{array}\right] \simeq \left[\begin{array}{cc} 1 & \frac{\dot{z}(s,t)}{V(s,t)}\\ -\frac{\dot{z}(s,t)}{V(s,t)} & 1 \end{array}\right] }$

$\displaystyle{ \left[\begin{array}{c} f_y(s,t) \\ f_z(s,t) \end{array}\right] =-C_a\rho_eA_e \left[\begin{array}{c} \ddot{y}(s,t) \\ \ddot{z}(s,t) \end{array}\right] }$
$\displaystyle{+C_{Ds}\cdot\frac{1}{2}\rho_eD_e \left[\begin{array}{c} |V(s,t)-\dot{y}(s,t)|(V(s,t)-\dot{y}(s,t)) \\ -|\dot{z}(s,t)|\dot{z}(s,t) \end{array}\right] }$
$\displaystyle{+ \left[\begin{array}{cc} 1 & \frac{\dot{z}(s,t)}{V(s,t)}\\ -\frac{\dot{z}(s,t)}{V(s,t)} & 1 \end{array}\right] \left[\begin{array}{c} f_{Dv}(s,t) \\ f_{L}(s,t) \end{array}\right] }$
$\displaystyle{ \left[\begin{array}{c} f_{Dv}(s,t) \\ f_{L}(s,t) \end{array}\right] = \frac{1}{2}\rho_eD_eV^2(s,t) \left[\begin{array}{c} \frac{1}{2}C_{D0}a(s,t) \\ \frac{1}{2}C_{L0}b(s,t) \end{array}\right] }$

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$\displaystyle{\ddot{a}(s,t)+2\epsilon_y\Omega_f(a(s,t)^2-1)\dot{a}(s,t)+4\Omega_f^2a(s,t)=\frac{A_y}{D}\ddot{y}(s,t)}$
$\displaystyle{\ddot{b}(s,t)+ \epsilon_z\Omega_f(b(s,t)^2-1)\dot{b}(s,t)+ \Omega_f^2b(s,t)=\frac{A_z}{D}\ddot{z}(s,t)}$

$\displaystyle{\alpha(\xi,\tau) =\displaystyle\sum_{i=1}^\infty q_{ai}(\tau)\phi_i(\xi)\simeq q_a^T(\tau)\phi(\xi)}$
$\displaystyle{ q_a(\tau)= \left[\begin{array}{c} q_{a1}(\tau) \\ \vdots\\ q_{aN}(\tau) \\ \end{array}\right],\ \phi(\xi)= \left[\begin{array}{c} \phi_1(\xi) \\ \vdots\\ \phi_N(\xi) \\ \end{array}\right] }$

$\displaystyle{\beta(\xi,\tau)=\displaystyle\sum_{i=1}^\infty p_{bi}(\tau)\psi_i(\xi)\simeq p_b^T(\tau)\psi(\xi)}$
$\displaystyle{ p_b(\tau)= \left[\begin{array}{c} p_{b1}(\tau) \\ \vdots\\ p_{bN}(\tau) \\ \end{array}\right],\ \psi(\xi)= \left[\begin{array}{c} \psi_1(\xi) \\ \vdots\\ \psi_N(\xi) \\ \end{array}\right] }$

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$\displaystyle{{(9)_y:f_y(s,t)=-C_a\rho_eA_e(s,t)\ddot{y}(s,t) }}$
$\displaystyle{{+\frac{1}{2}C_{Ds}\rho_eD_e|V(s,t)-\dot{y}(s,t)|(V(s,t)-\dot{y}(s,t)) }}$
$\displaystyle{{+\frac{1}{4}C_{D0}\rho_eD_eV^2(s,t)a(s,t)+\frac{1}{4}C_{L0}\rho_eD_eV(s,t)\dot{z}(s,t)b(s,t) }}$

$\displaystyle{{(9)_\eta:\frac{L^3}{EI}f_y(s,t) =-\underbrace{C_a\frac{\rho_eA_e}{m+m_i}}_{\chi_2}\underbrace{\frac{d^2\frac{y}{L}}{d\frac{t}{L^4{\frac{m+m_i}{EI}}}}}_{\ddot{\eta}} }}$
$\displaystyle{{+\underbrace{\frac{1}{2}C_{Ds}\frac{\rho_eD_e}{m+m_i}}_{\chi_3} |\underbrace{\frac{V}{\frac{1}{L}\sqrt{\frac{EI}{m+m_i}}}}_{v}- \underbrace{\frac{d\frac{y}{L}}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\eta}}| (\underbrace{\frac{V}{\frac{1}{L}\sqrt{\frac{EI}{m+m_i}}}}_{v}- \underbrace{\frac{d\frac{y}{L}}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\eta}}) }}$
$\displaystyle{{+\underbrace{\frac{1}{4}C_{D0}\frac{\rho_eD_e}{m+m_i}}_{\chi_4}\underbrace{\frac{V^2}{\frac{1}{L^2}\frac{EI}{m+m_i}}}_{v^2}{a} +\underbrace{\frac{1}{4}C_{L0}\frac{\rho_eD_e}{m+m_i}}_{\chi_5}\underbrace{\frac{V}{\frac{1}{L}\sqrt{\frac{EI}{m+m_i}}}}_{v} \underbrace{\frac{d\frac{z}{L}}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\zeta}}{b} }}$
$\displaystyle{{=-\chi_2\ddot{\eta}+\chi_3|v-\dot{\eta}|(v-\dot{\eta})+\chi_4v^2\alpha+\chi_5v\dot{\eta}\beta }}$
$\displaystyle{{=-\chi_2\ddot{\eta}-\chi_3|v-\dot{\eta}|\dot{\eta}+\chi_3|v-\dot{\eta}|v+\chi_4v^2\alpha+\chi_5v\dot{\eta}\beta }}$

$\displaystyle{{\int_0^1\phi(\xi)\times(9)_\eta d\xi=\int_0^1\phi(-\chi_2\ddot{\eta}-\chi_3|v-\dot{\eta}|\dot{\eta}+\chi_3|v-\dot{\eta}|v+\chi_4v^2\alpha+\chi_5v\dot{\eta}\beta)d\xi }}$
$\displaystyle{{=\int_0^1\phi(-\chi_2\underbrace{\phi^T\ddot{q}}_{\ddot{\eta}}-\chi_3|v-\dot{\eta}|\underbrace{\phi^T\dot{q}}_{\dot{\eta}} +\chi_3|v-\dot{\eta}|v+\chi_4\underbrace{(\phi^Tq_v)^2}_{v^2}\underbrace{\phi^T{q}_a}_{\alpha} +\chi_5\underbrace{\phi^Tq_v}_{v}\underbrace{\phi^T\dot{q}}_{\dot{\eta}}\underbrace{\psi^T{p}_b}_{\beta})d\xi }}$
$\displaystyle{{=-\chi_2\int_0^1\phi\phi^T\ddot{q}d\xi-\chi_3\int_0^1|v-\dot{\eta}|\phi\phi^T\dot{q}d\xi+\chi_3\int_0^1\phi|v-\dot{\eta}|vd\xi }}$
$\displaystyle{{+\chi_4\int_0^1 q_v^T\phi\phi^Tq_v\phi\phi^T{q}_ad\xi+\chi_5\int_0^1 q_v^T\phi\phi^T\dot{q}\phi\psi^T{p}_bd\xi }}$

for $i=1,\cdots,N$
$\displaystyle{{\int_0^1\phi_i(\xi)\times(9)_\eta d\xi }}$
$\displaystyle{{= -\chi_2\underbrace{\sum_{j=1}^N \int_0^1 \phi_i \phi_j d\xi\times \ddot{q}_j}_{\ddot{q}_i} -\chi_3\sum_{j=1}^N\int_0^1|v-\dot{\eta}|\phi_i\phi_jd\xi\times \dot{q}_j+\chi_3\int_0^1\phi_i|v-\dot{\eta}|vd\xi }}$
$\displaystyle{{+\chi_4\sum_{j,k,l=1}^N \underbrace{\int_0^1 \phi_k\phi_l\phi_i\phi_j d\xi}_{S^y_{ijkl}}\times {q}_{aj}q_{vk}q_{vl} +\chi_5\sum_{j,k,l=1}^N \underbrace{\int_0^1 \phi_k\phi_l\phi_i \psi_j d\xi}_{S^y_{ijkl}}\times {p}_{bj}q_{vk}\dot{q}_{l} }}$
$\displaystyle{{=-\chi_2\ddot{q}_j-\chi_3\sum_{j=1}^N\int_0^1|v-\dot{\eta}|\phi_i\phi_jd\xi\times \dot{q}_j +\chi_3\int_0^1\phi_i|v-\dot{\eta}|vd\xi }}$
$\displaystyle{{+\chi_4\sum_{j,k,l=1}^N {S^y_{ijkl}}\times {q}_{aj}q_{vk}q_{vl}+\chi_5\sum_{j,k,l=1}^N {S^y_{ijkl}}\times {p}_{bj}q_{vk}\dot{q}_{l} }}$

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$\displaystyle{{(9)_z:f_z(s,t)=-C_a\rho_eA_e(s,t)\ddot{z}(s,t) }}$
$\displaystyle{{-\frac{1}{2}C_{Ds}\rho_eD_e|\dot{z}(s,t)|\dot{z}(s,t) }}$
$\displaystyle{{-\frac{1}{4}C_{D0}\rho_eD_eV(s,t)\dot{z}(s,t)a(s,t)+\frac{1}{4}C_{L0}\rho_eD_eV^2(s,t)b(s,t) }}$

$\displaystyle{{(9)_\zeta:\frac{L^3}{EI}f_y(s,t) =-\underbrace{C_a\frac{\rho_eA_e}{m+m_i}}_{\chi_2}\underbrace{\frac{d^2\frac{z}{L}}{d\frac{t}{L^4{\frac{m+m_i}{EI}}}}}_{\ddot{\zeta}} }}$
$\displaystyle{{-\underbrace{\frac{1}{2}C_{Ds}\frac{\rho_eD_e}{m+m_i}}_{\chi_3}|\underbrace{\frac{d\frac{z}{L}}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\zeta}}|\underbrace{\frac{d\frac{z}{L}}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\zeta}} }}$
$\displaystyle{{-\underbrace{\frac{1}{4}C_{D0}\frac{\rho_eD_e}{m+m_i}}_{\chi_4}\underbrace{\frac{V}{\frac{1}{L}\sqrt{\frac{EI}{m+m_i}}}}_{v} \underbrace{\frac{d\frac{z}{L}}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\zeta}}{a}+\underbrace{\frac{1}{4}C_{L0}\frac{\rho_eD_e}{m+m_i}}_{\chi_5}\underbrace{\frac{V^2}{\frac{1}{L^2}\frac{EI}{m+m_i}}}_{v^2}{b} }}$
$\displaystyle{{=-\chi_2\ddot{\zeta}-\chi_3|\dot{\zeta}|\dot{\zeta}-\chi_4v\dot{\zeta}\alpha+\chi_5v^2\beta }}$

$\displaystyle{{\int_0^1\psi(\xi)\times(9)_\zeta d\xi=\int_0^1\psi(-\chi_2\ddot{\zeta}-\chi_3|\dot{\zeta}|\dot{\zeta}-\chi_4v\dot{\zeta}\alpha+\chi_5v^2\beta)d\xi }}$
$\displaystyle{{=\int_0^1\psi(-\chi_2\underbrace{\psi^T\ddot{p}}_{\ddot{\zeta}}-\chi_3|\dot{\zeta}|\underbrace{\psi^T\dot{p}}_{\dot{\zeta}} -\chi_4\underbrace{\psi^Tp_v}_{v}\underbrace{\psi^T\dot{p}}_{\dot{\zeta}}\underbrace{\phi^T{q}_a}_{\alpha}+\chi_5\underbrace{(\psi^Tp_v)^2}_{v^2}\underbrace{\psi^T{p}_b}_{\beta})d\xi }}$
$\displaystyle{{=\int_0^1(-\chi_2\psi\psi^T\ddot{p}-\chi_3|\dot{\zeta}|\psi^T\dot{p} -\chi_4p_v^T\psi\psi^T\dot{p}\psi\phi^T{q}_a+\chi_5p_v^T\psi\psi^Tp_v\psi\psi^T{p}_b)d\xi }}$
$\displaystyle{{=-\chi_2\int_0^1\psi\psi^T\ddot{p}d\xi-\chi_3\int_0^1|\dot{\zeta}|\psi^T\dot{p}d\xi}}$
$\displaystyle{{-\chi_4\int_0^1p_v^T\psi\psi^T\dot{p}\psi\phi^T{q}_ad\xi+\chi_5\int_0^1p_v^T\psi\psi^Tp_v\psi\psi^T{p}_bd\xi }}$

for $i=1,\cdots,N$
$\displaystyle{{\int_0^1\psi_i(\xi)\times(9)_\zeta d\xi }}$
$\displaystyle{{= -\chi_2 \sum_{j=1}^N\int_0^1 \psi_i\psi_jd\xi\times \ddot{p}_j -\chi_3\sum_{j=1}^N\int_0^1|\dot{\zeta}|\psi_i\psi_jd\xi\times \dot{p}_j }}$
$\displaystyle{{-\chi_4\sum_{j,k,l=1}^N \int_0^1 \psi_k\psi_l\psi_i\phi_j d\xi\times {q}_{aj}p_{vk}\dot{p}_{l} +\chi_5\sum_{j,k,l=1}^N \int_0^1 \psi_k\psi_l\psi_i \psi_j d\xi\times {p}_{bj}p_{vk}p_{vl} }}$
$\displaystyle{{= -\chi_2 \underbrace{\sum_{j=1}^N\int_0^1 \psi_i\psi_jd\xi\times \ddot{p}_j}_{\ddot{p}_i} -\chi_3\sum_{j=1}^N\int_0^1|\dot{\zeta}|\psi_i\psi_jd\xi\times \dot{p}_j }}$
$\displaystyle{{-\chi_4\sum_{j,k,l=1}^N \underbrace{\int_0^1 \psi_k\psi_l\psi_i\phi_j d\xi}_{S^z_{ijkl}}\times {q}_{aj}p_{vk}\dot{p}_{l} +\chi_5\sum_{j,k,l=1}^N \underbrace{\int_0^1 \psi_k\psi_l\psi_i \psi_j d\xi}_{S^z_{ijkl}}\times {p}_{bj}p_{vk}p_{vl} }}$
$\displaystyle{{= -\chi_2 {\ddot{p}_i}-\chi_3\sum_{j=1}^N\int_0^1|\dot{\zeta}|\psi_i\psi_jd\xi\times \dot{p}_j }}$
$\displaystyle{{-\chi_4\sum_{j,k,l=1}^N {S^z_{ijkl}}\times {q}_{aj}p_{vk}\dot{p}_{l}+\chi_5\sum_{j,k,l=1}^N {S^z_{ijkl}}\times {p}_{bj}p_{vk}p_{vl} }}$

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$\displaystyle{(10)_a:\ddot{a}(s,t)+2\epsilon_y\Omega_f(a(s,t)^2-1)\dot{a}(s,t)+4\Omega_f^2a(s,t)=\frac{A_y}{D}\ddot{y}(s,t)}$
$\displaystyle{(10)_\alpha:\underbrace{\frac{d^2a}{d\frac{t}{L^4{\frac{m+m_i}{EI}}}}}_{\ddot{\alpha}} +2\epsilon_ym^*\underbrace{\frac{\Omega_f}{\frac{m^*}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\omega_f} (\underbrace{a^2}_{\alpha^2}-1)\underbrace{\frac{da}{d\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\alpha}}}$
$\displaystyle{+4m^{*2}\underbrace{\frac{\Omega_f^2}{\frac{m^{*2}}{L^4{\frac{m+m_i}{EI}}} }}_{\omega_f^2} \underbrace{a}_\alpha={A_y}\frac{L}{D}\underbrace{\frac{d^2\frac{y}{L}}{d\frac{t}{L^4{\frac{m+m_i}{EI}}}}}_{\ddot{\eta}}}$
$\displaystyle{\ddot{\alpha}+\underbrace{2\epsilon_y m^*\omega_f}_{c_1}(\alpha^2-1)\dot{\alpha}+\underbrace{4m^{*2}\omega_f^2}_{c_2}\alpha=\underbrace{A_y\frac{L}{D}}_{c_0}\ddot{\eta}}$

$\displaystyle{\int_0^1\phi(\xi)\times(10)_\alpha d\xi:\int_0^1\phi(\ddot{\alpha}+c_1(\alpha^2-1)\dot{\alpha}+c_2\alpha- c_0\ddot{\eta}) d\xi=0}$
$\displaystyle{\int_0^1\phi(\underbrace{\phi^T\ddot{q}_a}_{\ddot{\alpha}}+c_1(\underbrace{(\phi^Tq_a)^2}_{\alpha^2}-1)\underbrace{\phi^T\dot{q}_a}_{\dot{\alpha}}+c_2\underbrace{\phi^Tq_a}_{\alpha}- c_0\underbrace{\phi^T\ddot{q}}_{\ddot{\eta}}) d\xi=0}$
$\displaystyle{\int_0^1(\phi\phi^T\ddot{q}_a+c_1(q_a^T\phi\phi^Tq_a-1)\phi\phi^T\dot{q}_a+c_2\phi\phi^Tq_a- c_0\phi\phi^T\ddot{q}) d\xi=0}$

for $i=1,\cdots,N$
$\displaystyle{\underbrace{\sum_{j=1}^N\int_0^1 \phi_i\phi_jd\xi\times \ddot{q}_{aj}}_{\ddot{q}_{ai}} +c_1\sum_{j,k,l=1}^N\underbrace{\int_0^1 \phi_i\phi_j\phi_k\phi_l d\xi}_{S^y_{ijkl}}\times \dot{q}_{aj}q_{ak}q_{al} -c_1\underbrace{\sum_{j=1}^N\int_0^1 \phi_i\phi_j d\xi\times \dot{q}_{aj}}_{\dot{q}_{ai}} }$
$\displaystyle{+c_2\underbrace{\sum_{j=1}^N\int_0^1 \phi_i\phi_jd\xi\times {q}_{aj}}_{{q}_{ai}} -c_0\underbrace{\sum_{j=1}^N\int_0^1 \phi_i\phi_jd\xi\times \ddot{q}_{j}}_{\ddot{q}_{i}}=0 }$
$\displaystyle{{\ddot{q}_{ai}} +c_1\sum_{j,k,l=1}^N S^y_{ijkl}\times \dot{q}_{aj}q_{ak}q_{al}-c_1{\dot{q}_{ai}} }$ $\displaystyle{+c_2{{q}_{ai}} -c_0{\ddot{q}_{i}}=0 }$

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$\displaystyle{(10)_b:\ddot{b}(s,t)+ \epsilon_z\Omega_f(b(s,t)^2-1)\dot{b}(s,t)+ \Omega_f^2b(s,t)=\frac{A_z}{D}\ddot{z}(s,t)}$
$\displaystyle{(10)_\beta:\underbrace{\frac{d^2{b}}{d\frac{t}{L^4{\frac{m+m_i}{EI}}}}}_{\ddot{\beta}} +\epsilon_z m^*\underbrace{\frac{\Omega_f}{\frac{m^*}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\omega_f} (\underbrace{{b^2}}_{\beta^2}-{1})\underbrace{\frac{d{b}}{d\frac{m^*}{L^2\sqrt{\frac{m+m_i}{EI}}}}}_{\dot{\beta}}}$
$\displaystyle{+m^{*2}\underbrace{\frac{\Omega_f^2}{\frac{m^{*2}}{L^4{\frac{m+m_i}{EI}}}}}_{\omega_f^2} \underbrace{{b}}_\beta ={A_z}\frac{L}{D}\underbrace{\frac{d^2\frac{z}{L}}{d\frac{t}{L^4{\frac{m+m_i}{EI}}}}}_{\ddot{\zeta}}}$
$\displaystyle{\ddot{\beta}+\underbrace{\epsilon_z m^*\omega_f}_{d_1}(\beta^2-1)\dot{\beta}+\underbrace{m^{*2}\omega_f^2}_{d_2}\beta=\underbrace{A_z\frac{L}{D}}_{d_0}\ddot{\zeta}}$

$\displaystyle{\int_0^1\psi(\xi)\times(10)_\beta d\xi:\int_0^1\psi(\ddot{\beta}+d_1(\beta^2-1)\dot{\beta}+d_2\beta- d_0\ddot{\zeta}) d\xi=0}$
$\displaystyle{\int_0^1\psi(\underbrace{\psi^T\ddot{p}_b}_{\ddot{\beta}}+d_1(\underbrace{(\psi^Tp_b)^2}_{\beta^2}-1)\underbrace{\psi^T\dot{p}_b}_{\dot{\beta}} +d_2\underbrace{\psi^Tq_a}_{\beta}- d_0\underbrace{\psi^T\ddot{q}}_{\ddot{\zeta}}) d\xi=0}$
$\displaystyle{\int_0^1(\psi\psi^T\ddot{p}_b+d_1(p_b^T\psi\psi^Tp_b-1)\psi\psi^T\dot{p}_b+d_2\psi\psi^Tp_b- d_0\psi\psi^T\ddot{p}) d\xi=0}$

for $i=1,\cdots,N$
$\displaystyle{\underbrace{\sum_{j=1}^N\int_0^1 \psi_i\psi_jd\xi\times \ddot{p}_{bj}}_{\ddot{p}_{bi}} +d_1\sum_{j,k,l=1}^N\underbrace{\int_0^1 \psi_i\psi_j\psi_k\psi_l d\xi}_{S^z_{ijkl}}\times \dot{p}_{bj}p_{bk}p_{bl} -d_1\underbrace{\sum_{j=1}^N\int_0^1 \psi_i\psi_j d\xi\times \dot{p}_{bj}}_{\dot{p}_{bi}} }$
$\displaystyle{+d_2\underbrace{\sum_{j=1}^N\int_0^1 \psi_i\psi_jd\xi\times {p}_{bj}}_{{p}_{bi}} -d_0\underbrace{\sum_{j=1}^N\int_0^1 \psi_i\psi_jd\xi\times \ddot{p}_{j}}_{\ddot{p}_{i}}=0 }$
$\displaystyle{{\ddot{p}_{bi}} +d_1\sum_{j,k,l=1}^N S^z_{ijkl}\times \dot{p}_{bj}p_{bk}p_{bl} -d_1{\dot{p}_{bi}} }$ $\displaystyle{+d_2{{p}_{bi}} -d_0{\ddot{p}_{i}}=0 }$

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$\displaystyle{ \left[\begin{array}{cc} M_{qq}+M_{qq}^a & M_{qp} \\ M_{pq} & M_{pp}+M_{pp}^b \\ \end{array}\right] \left[\begin{array}{c} \ddot{q} \\ \ddot{p} \end{array}\right] +\left[\begin{array}{cc} D_{qq}+D_{qq}^a & D_{qp} \\ D_{pq} & D_{pp}+D_{qq}^b \\ \end{array}\right] \left[\begin{array}{c} \dot{q} \\ \dot{p} \end{array}\right]}$
$\displaystyle{ +\left[\begin{array}{cc} K_{qq} & 0 \\ 0 & K_{pp} \\ \end{array}\right] \left[ \begin{array}{c} q \\ p \end{array} \right] = \left[\begin{array}{c} f_1 \\ 0 \end{array}\right] +\left[\begin{array}{cc} K_{q_aq_a} & 0 \\ 0 & K_{p_bp_b} \\ \end{array}\right] \left[ \begin{array}{c} q_a \\ p_b \end{array} \right]}$
$\displaystyle{ \left[\begin{array}{c} \ddot{q}_a \\ \ddot{p}_b \end{array}\right] +\left[\begin{array}{cc} {\cal D}_{qq}^a & 0 \\ 0 & {\cal D}_{pp}^b \\ \end{array}\right] \left[\begin{array}{c} \dot{q}_a \\ \dot{p}_b \end{array}\right] + \left[\begin{array}{cc} c_2I_N & 0 \\ 0 & d_2I_N \\ \end{array}\right] \left[ \begin{array}{c} q_a \\ p_b \end{array} \right]}$
$\displaystyle{= \left[\begin{array}{cc} c_oI_N & 0 \\ 0 & d_0I_N \\ \end{array}\right] \left[\begin{array}{c} \ddot{q} \\ \ddot{p} \end{array}\right] }$

$\displaystyle{M_{qq}^a=\chi_2I_N}$
$\displaystyle{M_{qq}^b=\chi_2I_N}$
$\displaystyle{D_{qq}^a=[\chi_3\sum_{j=1}^N\int_0^1|v-\dot{\eta}|\phi_i\phi_jd\xi]_{i,j=1,\cdots,N}}$
$\displaystyle{D_{qq}^b=[\chi_3\sum_{j=1}^N\int_0^1|\dot{\eta}|\psi_i\psi_jd\xi]_{i,j=1,\cdots,N} }$
$\displaystyle{f_1=[\chi_3\int_0^1\phi_i|v-\dot{\eta}|vd\xi]_{i=1,\cdots,N}}$
$\displaystyle{K_{q_aq_a}=[\chi_4 \displaystyle\sum_{l,k=1}^NS^y_{ijkl}q_{vk}q_{vl}]_{i,j=1,\cdots,N}}$
$\displaystyle{K_{q_ap_b}=[-\chi_5 \displaystyle\sum_{l,k=1}^NS^y_{ijkl}q_{vk}\dot{q}_{l}]_{i,j=1,\cdots,N}}$
$\displaystyle{K_{p_bq_a}=[\chi_4 \displaystyle\sum_{l,k=1}^NS^z_{ijkl}p_{vk}p_{vl}]_{i,j=1,\cdots,N}}$
$\displaystyle{K_{p_bp_b}=[\chi_5 \displaystyle\sum_{l,k=1}^NS^z_{ijkl}p_{vk}\dot{p}_{l}]_{i,j=1,\cdots,N}}$
$\displaystyle{{\cal D}_{qq}^a=[c_1\displaystyle\sum_{l,k=1}^NS^y_{ijkl}q_{ak}q_{al}]_{i,j=1,\cdots,N}-c_1I_N}$
$\displaystyle{{\cal D}_{pp}^b=[d_1\displaystyle\sum_{l,k=1}^NS^z_{ijkl}p_{bk}p_{bl}]_{i,j=1,\cdots,N}-d_1I_N}$

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水中線状構造物：振動子