水中線状構造物：無次元化

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

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

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

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

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水中線状構造物：無次元化