水中線状構造物：無次元化

投稿日時: 2018年8月15日投稿者: cacsd

● $\displaystyle{\bf Derivation \#2}$
$\displaystyle{\int_{\tau_1}^{\tau_2}\int_0^1(\{\cdot\}_\eta\delta \eta(\xi,\tau)+\{\cdot\}_\zeta\delta \zeta(\xi,\tau))d\xi d\tau=0}$

●パイプ
$\displaystyle{\xi=\frac{s}{L} }$ , $\displaystyle{\eta=\frac{y}{L} }$ , $\displaystyle{\zeta=\frac{z}{L} }$
$\displaystyle{\tau=\frac{t}{L^2\sqrt{\frac{m+m_i}{EI}}} }$ , $\displaystyle{u=\frac{U}{\frac{1}{L}\sqrt{\frac{EI}{m_i}}} }$ , $\displaystyle{v=\frac{V}{\frac{1}{L}\sqrt{\frac{EI}{m+m_i}}} }$
$\displaystyle{\Gamma_m=\frac{m_b}{(m+m_i-m_e)L} }$ , $\displaystyle{\Gamma_p=\frac{m_bg+\Delta P_Lx_L'}{(m+m_i-m_e)gL} }$
$\displaystyle{\beta=\frac{m_i}{m+m_i} }$ , $\displaystyle{\gamma=(m+m_i-m_e)g\frac{L^3}{EI} }$

$\displaystyle{\{\cdot\}_\eta=\frac{L^3}{EI}\times\{\cdot\}_y=(1)_\eta+(2)_\eta+(3)_\eta+(4)_\eta+(5)_\eta+(6)_\eta+(7)_\eta+(8)_\eta+(9)_\eta }$
$\displaystyle{(1)_\eta: (1+\Gamma_m\delta_D(\xi-1))(\ddot{\eta}+ \eta' \int_0^\xi{(\dot{\eta'}^2 + \dot{\zeta}'^2 + \eta'\ddot{\eta}' + \zeta'\ddot{\zeta}')}d\xi ) }$
$\displaystyle{(2)_\eta: -\eta'' \int_\xi^1(1+\Gamma_m\delta_D(\xi-1))\int_0^\xi{(\dot{\eta'}^2 + \dot{\zeta}'^2 + \eta'\ddot{\eta}' + \zeta'\ddot{\zeta}')}d\xi d\xi }$
$\displaystyle{(3)_\eta: +\gamma(1+\Gamma_p\delta_D(\xi-1))(\eta'+{1\over 2}\eta'^3+{1\over 2}\eta'\zeta'^2) }$
$\displaystyle{(4)_\eta: -(\gamma(1-\xi+\Gamma_p)}$ $\displaystyle{{-\sqrt{\beta}\dot{u}(1-\xi)} }$ $\displaystyle{)(\eta''+{3\over 2}\eta'^2\eta''+{1\over 2}\eta''\zeta'^2+\eta'\zeta'\zeta'') }$
$\displaystyle{{(5)_\eta: -\sqrt{\beta}\dot{u}\eta''\int_\xi^1 (\eta'^2+\zeta'^2)d\xi }}$
$\displaystyle{{(6)_\eta: +u^2 (\eta'^2\eta''+ \eta'\zeta'\zeta'' +\eta'' - \eta''\int_\xi^1 (\eta'\eta''+\zeta'\zeta'')d\xi ) }}$
$\displaystyle{{(7)_\eta: +2 \sqrt{\beta}u(\dot{\eta}'+\eta'^2\dot{\eta}'+\eta'\zeta'\dot{\zeta}'- \eta''\int_\xi^1(\eta'\dot{\eta}'+\zeta'\dot{\zeta}')d\xi) }}$
$\displaystyle{{(8)_\eta: +\eta''''+\eta''^3+4\eta'\eta''\eta'''+\eta'^2\eta''''+\eta''\zeta''^2+\eta''\zeta'\zeta'''+3\eta'\zeta''\zeta'''+\eta'\zeta'\zeta'''' }}$
$\displaystyle{(9)_\eta: -\frac{L^3}{EI} f_y }$

$\displaystyle{\{\cdot\}_\zeta=\frac{L^3}{EI}\times\{\cdot\}_z=(1)_\zeta+(2)_\zeta+(3)_\zeta+(4)_\zeta+(5)_\zeta+(6)_\zeta+(7)_\zeta+(8)_\zeta+(9)_\zeta }$
$\displaystyle{(1)_\zeta:(1+\Gamma_m\delta_D(\xi-1))(\ddot{\zeta}+ \zeta' \int_0^\xi{(\dot{\zeta'}^2 + \dot{\eta}'^2 + \zeta'\ddot{\zeta}' + \eta'\ddot{\eta}')}d\xi ) }$
$\displaystyle{(2)_\zeta:-\zeta'' \int_\xi^1(1+\Gamma_m\delta_D(\xi-1))\int_0^\xi{(\dot{\zeta'}^2 + \dot{\eta}'^2 + \zeta'\ddot{\zeta}' + \eta'\ddot{\eta}')}d\xi d\xi }$
$\displaystyle{(3)_\zeta:+\gamma(1+\Gamma_p\delta_D(\xi-1))(\zeta'+{1\over 2}\zeta'^3+{1\over 2}\zeta'\eta'^2) }$
$\displaystyle{(4)_\zeta:-(\gamma(1-\xi+\Gamma_p)}$ $\displaystyle{{-\sqrt{\beta}\dot{u}(1-\xi)} }$ $\displaystyle{)(\zeta''+{3\over 2}\zeta'^2\zeta''+{1\over 2}\zeta''\eta'^2+\zeta'\eta'\eta'') }$
$\displaystyle{{(5)_\zeta:-\sqrt{\beta}\dot{u}\zeta''\int_\xi^1 (\zeta'^2+\eta'^2)d\xi }}$
$\displaystyle{{(6)_\zeta:+u^2 (\zeta'^2\zeta''+ \zeta'\eta'\eta'' +\zeta'' - \zeta''\int_\xi^1 (\zeta'\zeta''+\eta'\eta'')d\xi )　}}$
$\displaystyle{{(7)_\zeta:+2 \sqrt{\beta}u(\dot{\zeta}'+\zeta'^2\dot{\zeta}'+\zeta'\eta'\dot{\eta}'- \zeta''\int_\xi^1(\zeta'\dot{\zeta}'+\eta'\dot{\eta}')d\xi) }}$
$\displaystyle{{(8)_\zeta:+\zeta''''+\zeta''^3+4\zeta'\zeta''\zeta'''+\zeta'^2\zeta''''+\zeta''\eta''^2+\zeta''\eta'\eta'''+3\zeta'\eta''\eta'''+\zeta'\eta'\eta'''' }}$
$\displaystyle{(9)_\eta: -\frac{L^3}{EI}\times f_z }$

●ワイヤ
$\displaystyle{\xi=\frac{s}{L}}$ , $\displaystyle{\eta=\frac{y}{L}}$ , $\displaystyle{\zeta=\frac{z}{L}}$
$\displaystyle{\tau=\frac{t}{\sqrt{\frac{L}{g}}}$ , $\displaystyle{v=\frac{V}{\sqrt{Lg}}}$
$\displaystyle{\Gamma_m=\Gamma_p=\frac{m_b}{mL}}$ , $\displaystyle{\gamma=1}$

$\displaystyle{\{\cdot\}_\eta=\frac{1}{mg}\times\{\cdot\}_y=(1)_\eta+(2)_\eta+(3)_\eta+(4)_\eta+(9)_\eta}$
$\displaystyle{(1)_\eta: (1+\Gamma_m\delta_D(\xi-1))(\ddot{\eta}+ \eta' \int_0^\xi{(\dot{\eta'}^2 + \dot{\zeta}'^2 + \eta'\ddot{\eta}' + \zeta'\ddot{\zeta}')}d\xi )}$
$\displaystyle{(2)_\eta: -\eta'' \int_\xi^1(1+\Gamma_m\delta_D(\xi-1))\int_0^\xi{(\dot{\eta'}^2 + \dot{\zeta}'^2 + \eta'\ddot{\eta}' + \zeta'\ddot{\zeta}')}d\xi d\xi }$
$\displaystyle{(3)_\eta: +\gamma(1+\Gamma_p\delta_D(\xi-1))(\eta'+{1\over 2}\eta'^3+{1\over 2}\eta'\zeta'^2)}$
$\displaystyle{(4)_\eta: -(\gamma(1-\xi+\Gamma_p) )(\eta''+{3\over 2}\eta'^2\eta''+{1\over 2}\eta''\zeta'^2+\eta'\zeta'\zeta'')}$
$\displaystyle{(9)_\eta: -\frac{1}{mg} f_y }$

$\displaystyle{\{\cdot\}_\zeta=\frac{1}{mg}\times\{\cdot\}_z=(1)_\zeta+(2)_\zeta+(3)_\zeta+(4)_\zeta+(9)_\zeta}$
$\displaystyle{(1)_\zeta:(1+\Gamma_m\delta_D(\xi-1))(\ddot{\zeta}+ \zeta' \int_0^\xi{(\dot{\zeta'}^2 + \dot{\eta}'^2 + \zeta'\ddot{\zeta}' + \eta'\ddot{\eta}')}d\xi )}$
$\displaystyle{(2)_\zeta:-\zeta'' \int_\xi^1(1+\Gamma_m\delta_D(\xi-1))\int_0^\xi{(\dot{\zeta'}^2 + \dot{\eta}'^2 + \zeta'\ddot{\zeta}' + \eta'\ddot{\eta}')}d\xi d\xi }$
$\displaystyle{(3)_\zeta:+\gamma(1+\Gamma_p\delta_D(\xi-1))(\zeta'+{1\over 2}\zeta'^3+{1\over 2}\zeta'\eta'^2)}$
$\displaystyle{(4)_\zeta:-(\gamma(1-\xi+\Gamma_p) )(\zeta''+{3\over 2}\zeta'^2\zeta''+{1\over 2}\zeta''\eta'^2+\zeta'\eta'\eta'')}$
$\displaystyle{(9)_\eta: -\frac{1}{mg} f_z }$

水中線状構造物：運動方程式

投稿日時: 2018年8月15日投稿者: cacsd

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

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

水中線状構造物：ハミルトンの原理

投稿日時: 2018年8月15日投稿者: cacsd

● $\displaystyle{\bf Derivation \#0}$
$\displaystyle{\underline{\int_{t_1}^{t_2}(\delta T-\delta V+\delta W)dt=} }$ $\displaystyle{{\underline{\int_{t_1}^{t_2}m_iU(\dot{r}_L+U\tau_L)\delta r_Ldt}} }$
$\displaystyle{\delta T-\delta V+\delta W=(\delta T_p+\delta T_f+\delta T_b)-(\delta V_p+\delta V_b+\delta V_s)+(\delta W_{bx}+\delta W_f)}$

$\displaystyle{T_p={1\over 2}m\int_0^L v_p^Tv_p ds={1\over 2}m\int_0^L (\dot{x}^2+\dot{y}^2+\dot{z}^2) ds}$
$\displaystyle{{T_f={1\over 2}m_i\int_0^L v_f^2ds={1\over 2}m_i\int_0^L ((\dot{x}+Ux')^2+(\dot{y}+Uy')^2+(\dot{z}+Uz')^2) ds} }$
$\displaystyle{T_b={1\over 2}m_b v_b^Tv_b ds={1\over 2}m_b (\dot{x}_L^2+\dot{y}_L^2+\dot{z}_L^2) ds={1\over 2}m_b\int_0^L \delta_D(s-L)(\dot{x}^2+\dot{y}^2+\dot{z}^2) ds}$
$\displaystyle{V_p=(m+m_i-m_e)g\int_0^Lxds}$
$\displaystyle{V_b=m_bgx_L=m_bg\int_0^L \delta_D(s-L)x ds}$
$\displaystyle{{V_s=\frac{1}{2}EI\int_0^L\kappa^2ds\approx\frac{1}{2}EI\int_0^L(y'^2y''^2+z'^2z''^2+2y'y''z'z''+y''^2+z''^2)ds }}$
$\displaystyle{W_{bx}=P_L x_L'\delta x_L=P_L x_L'\int_0^L \delta_D(s-L)\delta x ds}$
$\displaystyle{W_f=f_y y+f_z z}$

$\displaystyle{\int_{t_1}^{t_2}(\delta T_p+\delta T_f)dt }$
$\displaystyle{=- (m+m_i)\int_{t_1}^{t_2}\int_0^L(\ddot y+y'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds)\delta y dsdt }$
$\displaystyle{- (m+m_i)\int_{t_1}^{t_2}\int_0^L(\ddot z+z'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds)\delta z dsdt }$
$\displaystyle{+ (m+m_i)\int_{t_1}^{t_2}\int_0^Ly''\int_s^L\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds\delta y dsdt }$
$\displaystyle{+ (m+m_i)\int_{t_1}^{t_2}\int_0^Lz''\int_s^L\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds\delta z dsdt }$
$\displaystyle{{-\int_{t_1}^{t_2}\int_0^Lm_i\dot U(L-s)(y''+{3\over 2}y'^2y''+{1\over 2}y''z'^2+y'z'z'')\delta ydsdt }}$
$\displaystyle{{-\int_{t_1}^{t_2}\int_0^Lm_i\dot U(L-s)(z''+{3\over 2}z'^2z''+{1\over 2}z''y'^2+z'y'y'')\delta zdsdt }}$
$\displaystyle{{+{1\over 2}\int_{t_1}^{t_2}\int_0^Lm_i\dot Uy''\int_s^L (y'^2+z'^2)ds\delta ydsdt }}$
$\displaystyle{{+{1\over 2}\int_{t_1}^{t_2}\int_0^Lm_i\dot Uz''\int_s^L (y'^2+z'^2)ds\delta zdsdt }}$
$\displaystyle{{-2\int_{t_1}^{t_2}\int_0^L m_iU(\dot{y}'+y'^2\dot{y}'+y'z'\dot{z}'- y''\int_s^L(y'\dot{y}'+z'\dot{z}')ds)\delta ydsdt }}$
$\displaystyle{{-2\int_{t_1}^{t_2}\int_0^L m_iU(\dot{z}'+z'^2\dot{z}'+z'y'\dot{y}'- z''\int_s^L(y'\dot{y}'+z'\dot{z}')ds)\delta zdsdt }}$
$\displaystyle{{+\int_{t_1}^{t_2}m_iU(\dot x_L\delta x_L+\dot y_L\delta y_L+\dot z_L\delta z_L)dt }}$

$\displaystyle{ \int_{t_1}^{t_2}\delta T_bdt }$
$\displaystyle{ =-m_b\int_{t_1}^{t_2}\int_0^L\delta_D(s-L)(\ddot y+y'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds)\delta y dsdt }$
$\displaystyle{ -m_b\int_{t_1}^{t_2}\int_0^L\delta_D(s-L)(\ddot z+z'\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})ds)\delta z dsdt }$
$\displaystyle{ +m_b\int_{t_1}^{t_2}\int_0^Ly''\int_s^L\delta_D(s-L)\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds\delta y dsdt }$
$\displaystyle{ +m_b\int_{t_1}^{t_2}\int_0^Lz''\int_s^L\delta_D(s-L)\int_0^s({\dot{y'}}^2+{\dot{z'}}^2+y'\ddot{y'}+z'\ddot{z'})dsds\delta z dsdt }$

$\displaystyle{ \int_{t_1}^{t_2}\delta V_pdt }$
$\displaystyle{ =-(m+m_i-m_e)g\int_{t_1}^{t_2}\int_0^L(y'+{1\over 2}y'^3+{1\over 2}y'z'^2)\delta ydsdt }$
$\displaystyle{ -(m+m_i-m_e)g\int_{t_1}^{t_2}\int_0^L(z'+{1\over 2}z'^3+{1\over 2}z'y'^2)\delta zdsdt }$
$\displaystyle{ +(m+m_i-m_e)g\int_{t_1}^{t_2}\int_0^L(L-s)(y''+{3\over 2}y'^2y''+{1\over 2}y''z'^2+y'z'z'')\delta ydsdt }$
$\displaystyle{ +(m+m_i-m_e)g\int_{t_1}^{t_2}\int_0^L(L-s)(z''+{3\over 2}z'^2z''+{1\over 2}z''y'^2+z'y'y'')\delta zdsdt }$

$\displaystyle{ \int_{t_1}^{t_2}\delta V_bdt }$
$\displaystyle{ =-m_bg\int_{t_1}^{t_2}\int_0^L\delta_D(s-L)(y'+{1\over 2}y'^3+{1\over 2}y'z'^2)\delta y dsdt }$
$\displaystyle{ -m_bg\int_{t_1}^{t_2}\int_0^L\delta_D(s-L)(z'+{1\over 2}z'^3+{1\over 2}z'y'^2)\delta z dsdt }$
$\displaystyle{ +m_bg\int_{t_1}^{t_2}\int_0^L(y''+{3\over 2}y'^2y''+{1\over 2}y''z'^2+y'z'z'')\delta yds dt }$
$\displaystyle{ +m_bg\int_{t_1}^{t_2}\int_0^L(z''+{3\over 2}z'^2z''+{1\over 2}z''y'^2+z'y'y'')\delta zds dt }$

$\displaystyle{{\int_{t_1}^{t_2}\delta V_sdt }}$
$\displaystyle{{= EI\int_{t_1}^{t_2}\int_0^L(y''''+y''^3+4y'y''y'''+y'^2y'''')\delta ydsdt }}$
$\displaystyle{{+EI\int_{t_1}^{t_2}\int_0^L(y''z''^2+y''z'z'''+3y'z''z'''+y'z'z'''')\delta ydsdt }}$
$\displaystyle{{+EI\int_{t_1}^{t_2}\int_0^L(z''''+z''^3+4z'z''z'''+z'^2z'''')\delta zdsdt }}$
$\displaystyle{{+EI\int_{t_1}^{t_2}\int_0^L(z''y''^2+z''y'y'''+3z'y''y'''+z'y'y'''')\delta zdsdt }}$

$\displaystyle{\int_{t_1}^{t_2}\delta W_{bx}dt }$
$\displaystyle{=-\int_{t_1}^{t_2}\int_0^L\Delta P_L x_L'\delta_D(s-L)(y'+{1\over 2}y'^3+{1\over 2}y'z'^2)\delta y dsdt }$
$\displaystyle{-\int_{t_1}^{t_2}\int_0^L\Delta P_L x_L'\delta_D(s-L)(z'+{1\over 2}z'^3+{1\over 2}z'y'^2)\delta z dsdt }$
$\displaystyle{+\int_{t_1}^{t_2}\int_0^L\Delta P_L x_L'(y''+{3\over 2}y'^2y''+{1\over 2}y''z'^2+y'z'z'')\delta yds dt }$
$\displaystyle{+\int_{t_1}^{t_2}\int_0^L\Delta P_L x_L'(z''+{3\over 2}z'^2z''+{1\over 2}z''y'^2+z'y'y'')\delta zds dt }$

$\displaystyle{ \int_{t_1}^{t_2}\delta W_fdt=\int_{t_1}^{t_2}\int_{0}^{L}(f_y\delta y+f_z\delta z)dsdt }$

$\displaystyle{{\int_{t_1}^{t_2}m_iU(\dot{r}_L+U\tau_L)\delta r_Ldt }}$
$\displaystyle{{=\int_{t_1}^{t_2}m_iU(\dot x_L\delta x_L+\dot y_L\delta y_L+\dot z_L\delta z_L)dt }}$
$\displaystyle{{+\int_{t_1}^{t_2}m_iU^2\int_0^L (y'^2y''+ y'z'z'' +y'' - y''\int_s^L (y'y''+z'z'')ds )\delta ydsdt }}$
$\displaystyle{{+\int_{t_1}^{t_2}m_iU^2\int_0^L (z'^2z''+ y'y''z' +z'' - z''\int_s^L (y'y''+z'z'')ds )\delta zdsdt }}$

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水中線状構造物：無次元化

水中線状構造物：運動方程式

水中線状構造物：ハミルトンの原理