水中線状構造物：ハミルトンの原理

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

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

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

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

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

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

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

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

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

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

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水中線状構造物：ハミルトンの原理