水中線状構造物：運動方程式

● $\displaystyle{\bf Derivation \#1}$
$\displaystyle{\int_{t_1}^{t_2}\int_0^L(\{\cdot\}_y\delta y+\{\cdot\}_z\delta z)dsdt=0}$

●パイプ
$\displaystyle{\int_{t_1}^{t_2}\int_0^L \{ }$
$\displaystyle{\{ }$
$\displaystyle{(1)_y:+(m+m_i+m_b\delta_D(s-L)) (\ddot y+ y'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds) }$
$\displaystyle{(2)_y:-y''\int_s^L (m+m_i+m_b\delta_D(s-L))\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds }$
$\displaystyle{(3)_y:+((m+m_i-m_e)g+(m_bg+\Delta P_L x_L')\delta_D(s-L))(y'+{1\over 2}y'^3+{1\over 2}y'z'^2) }$
$\displaystyle{(4)_y:-(((m+m_i-m_e)g}$ $\displaystyle{{-m_i\dot U} }$ $\displaystyle{)(L-s)+m_bg+\Delta P_L x_L)}$ $\displaystyle{\times(y''+{3\over 2}y'^2y''+{1\over 2}y''z'^2+y'z'z'') }$
$\displaystyle{{(5)_y:-{1\over 2}m_i\dot Uy''\int_s^L (y'^2+z'^2)ds }}$
$\displaystyle{{(6)_y:+m_iU^2 (y'^2y''+ y'z'z'' +y'' - y''\int_s^L (y'y''+z'z'')ds )　}}$
$\displaystyle{{(7)_y:+2 m_iU(\dot{y}'+y'^2\dot{y}'+y'z'\dot{z}'- y''\int_s^L(y'\dot{y}'+z'\dot{z}')ds) }}$
$\displaystyle{{(8)_y:+EI(y''''+y''^3+4y'y''y'''+y'^2y''''+y''z''^2+y''z'z'''+3y'z''z'''+y'z'z'''') }}$
$\displaystyle{(9)_y:-f_y }$
$\displaystyle{\}_y\delta y }$
$\displaystyle{+\{ }$
$\displaystyle{(1)_z:+((m+m_i+m_b\delta_D(s-L))(\ddot z+ z'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds) }$
$\displaystyle{(2)_z:-z''\int_s^L (m+m_i+m_b\delta_D(s-L))\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds }$
$\displaystyle{(3)_z:+((m+m_i-m_e)g+(m_bg+\Delta P_L x_L')\delta_D(s-L))(z'+{1\over 2}z'^3+{1\over 2}z'y'^2) }$
$\displaystyle{(4)_z:-(((m+m_i-m_e)g}$ $\displaystyle{{-m_i\dot U} }$ $\displaystyle{)(L-s)+m_bg+\Delta P_L x_L')}$ $\displaystyle{\times(z''+{3\over 2}z'^2z''+{1\over 2}z''y'^2+z'y'y'') }$
$\displaystyle{{(5)_z:-{1\over 2}m_i\dot Uz''\int_s^L (y'^2+z'^2)ds }}$
$\displaystyle{{(6)_z:+m_iU^2 (z'^2z''+ y'y''z' +z'' - z''\int_s^L (y'y''+z'z'')ds ) }}$
$\displaystyle{{(7)_z:+2 m_iU(\dot{z}'+z'^2\dot{z}'+z'y'\dot{y}'- z''\int_s^L(y'\dot{y}'+z'\dot{z}')ds) }}$
$\displaystyle{{(8)_z:+EI(z''''+z''^3+4z'z''z'''+z'^2z''''+z''y''^2+z''y'y'''+3z'y''y'''+z'y'y'''') }}$
$\displaystyle{(9)_z:-f_z }$
$\displaystyle{\}_z \delta z }$
$\displaystyle{\} dsdt=0}$

●ワイヤ
$\displaystyle{\int_{t_1}^{t_2}\int_0^L \{ }$
$\displaystyle{\{ }$
$\displaystyle{(1)_y:+m+m_b\delta_D(s-L)) (\ddot y+ y'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds }$
$\displaystyle{(2)_y:-y''\int_s^L (m+m_b\delta_D(s-L))\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds }$
$\displaystyle{(3)_y:+(mg+m_bg\delta_D(s-L))(y'+{1\over 2}y'^3+{1\over 2}y'z'^2) }$
$\displaystyle{(4)_y:-(mg(L-s)+m_bg)(y''+{3\over 2}y'^2y''+{1\over 2}y''z'^2+y'z'z'') }$
$\displaystyle{(9)_y:-f_y }$
$\displaystyle{\}_y\delta y }$
$\displaystyle{+\{ }$
$\displaystyle{(1)_z:+(m+m_b\delta_D(s-L))(\ddot z+ z'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds }$
$\displaystyle{(2)_z:-z''\int_s^L (m+m_b\delta_D(s-L))\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds }$
$\displaystyle{(3)_z:+(mg+m_bg\delta_D(s-L))(z'+{1\over 2}z'^3+{1\over 2}z'y'^2) }$
$\displaystyle{(4)_z:-(mg(L-s)+m_bg)(z''+{3\over 2}z'^2z''+{1\over 2}z''y'^2+z'y'y'') }$
$\displaystyle{(9)_z:-f_z }$
$\displaystyle{\}_z \delta z }$
$\displaystyle{\} dsdt=0}$

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水中線状構造物：運動方程式