水中線状構造物：モード関数

投稿日時: 投稿者:

ビームのモード関数

●パイプ（固定・質点付自由）
\displaystyle{\ddot{\eta}(\xi,\tau)+\eta''''(\xi,\tau)=0}
\displaystyle{\cdot\Downarrow \eta(\xi,\tau)=\phi(\xi)r(\tau) }
\displaystyle{\phi(\xi)\ddot{r}(\tau)+\phi''''(\xi)r(\tau)=0 }
\displaystyle{\cdot\Downarrow \ddot{r}(\tau)=-\Omega^2r(\tau) }
\displaystyle{-\phi(\xi)\Omega^2r(\tau)+\phi''''(\xi)r(\tau)=0 }
\displaystyle{\cdot\Downarrow}
\displaystyle{\phi''''(\xi)-\Omega^2\phi(\xi)=0 }
\displaystyle{\cdot\Downarrow \omega^4=\Omega^2}
\displaystyle{\phi''''(\xi)-\omega^4\phi(\xi)=0 }
\displaystyle{\cdot\Downarrow}
\displaystyle{\phi(\xi)=c_1C_{\omega\xi}+c_2S_{\omega\xi}+c_3C^h_{\omega\xi}+c_4S^h_{\omega\xi}}
\displaystyle{\phi'(\xi)=\omega(-c_1S_{\omega\xi}+c_2C_{\omega\xi}+c_3S^h_{\omega\xi}+c_4C^h_{\omega\xi})}
\displaystyle{\phi''(\xi)=\omega^2(-c_1C_{\omega\xi}-c_2S_{\omega\xi}+c_3C^h_{\omega\xi}+c_4S^h_{\omega\xi})}
\displaystyle{\phi'''(\xi)=\omega^3(c_1S_{\omega\xi}-c_2C_{\omega\xi}+c_3S^h_{\omega\xi}+c_4C^h_{\omega\xi})}

\displaystyle{\phi(0)=\phi'(0)=0,\ \phi''(1) =0,\ \phi'''(1) =-\beta\omega^4\phi(1)}
\displaystyle{\cdot\Downarrow}
\displaystyle{c_1+c_3=0, c_2+c_4=0 \Rightarrow c_3=-c_1, c_4=-c_2}
\displaystyle{\cdot\Downarrow}
\displaystyle{\phi''(\xi)=\omega^2(-c_1C_{\omega\xi}-c_2S_{\omega\xi}-c_1C^h_{\omega\xi}-c_2S^h_{\omega\xi})}
\displaystyle{\phi'''(\xi)=\omega^3(c_1S_{\omega\xi}-c_2C_{\omega\xi}-c_1S^h_{\omega\xi}-c_2C^h_{\omega\xi})}
\displaystyle{\underline{\phi(\xi)=c_1(C_{\omega\xi}-C^h_{\omega\xi})+c_2(S_{\omega\xi}-S^h_{\omega\xi})} }
\displaystyle{\cdot\Downarrow}
\displaystyle{c_1(C_{\omega}+C^h_{\omega})+c_2(S_{\omega}+S^h_{\omega})=0}
\displaystyle{c_1((S_{\omega}-S^h_{\omega})+\beta\omega(C_{\omega}-C^h_{\omega}))+c_2(-(C_{\omega}+C^h_{\omega})+\beta\omega(S_{\omega}-S^h_{\omega}))=0}
\displaystyle{\cdot\Downarrow}
\displaystyle{(C_{\omega}+C^h_{\omega})^2+(S^2_{\omega}-S^{h2}_{\omega})-\beta\omega(C_{\omega}+C^h_{\omega})(S_{\omega}-S^h_{\omega})+\beta\omega(C_{\omega}-C^h_{\omega})(S_{\omega}+S^h_{\omega})=0}
\displaystyle{\cdot\Downarrow}
\displaystyle{\underline{(1+C^h_{\omega}C_{\omega})+\beta\omega(C_{\omega}S^h_{\omega}-S_{\omega}C^h_{\omega})=0} }

●ワイヤ（ピン・質点付自由）
\displaystyle{(1+\Gamma_m\delta_D(\xi-1))\ddot{\eta}+\gamma(\xi-1-\Gamma_m)\eta''+\gamma(1+\Gamma_m\delta_D(\xi-1))\eta' =0}
\displaystyle{\cdot\Downarrow }
\displaystyle{\ddot{\eta}(\xi,\tau)+\gamma(\xi-1-\Gamma_m)\eta''(\xi,\tau)+\gamma\eta'(\xi,\tau)=0}
\displaystyle{\ddot{\eta}(1,\tau)+\gamma\eta'(1,\tau) =0 }
\displaystyle{\cdot\Downarrow \eta(\xi,\tau)=\phi(\xi)q(\tau)}
\displaystyle{(\xi-1-\Gamma_m)\phi''(\xi)q(\tau) + \phi'(\xi)q(\tau)+\gamma^{-1}\phi(\xi)\ddot{q}(\tau)=0}
\displaystyle{\phi(1)\ddot{q}(\tau)+\gamma\phi'(1)q(\tau) =0 }
\displaystyle{\cdot\Downarrow \ddot{q}(\tau)=-\Omega^2q(\tau)}
\displaystyle{(1+\Gamma_m-\xi )\phi''(\xi) - \phi'(\xi) + \underbrace{\frac{\Omega^2}{\gamma}}_{\lambda^2}\phi(\xi)=0,\ \phi'(1)=\underbrace{\frac{\Omega^2}{\gamma}}_{\lambda^2}\phi(1) }
\displaystyle{\cdot\Downarrow \sigma=2\lambda\sqrt{1+\Gamma_m-\xi}}
\displaystyle{\frac{\sigma^2}{(2\lambda)^2}(\frac{d^2\phi}{d\sigma^2}(\frac{d\sigma}{d\xi})^2+\frac{d\phi}{d\sigma} \frac{d^2\sigma}{d\xi^2})-\frac{d\phi}{d\sigma}\frac{d\sigma}{d\xi} +\lambda^2\phi=0,\ \phi'(1)=\lambda^2\phi(1)}
\displaystyle{\cdot\Downarrow \frac{d\sigma}{d\xi}=-\frac{2\lambda^2}{\sigma}, \frac{d^2\sigma}{d\xi^2}=-\frac{(2\lambda^2)^2}{\sigma^3}}
\displaystyle{\frac{\sigma^2}{(2\lambda)^2}(\frac{d^2\phi}{d\sigma^2}(-\frac{2\lambda^2}{\sigma})^2+\frac{d\phi}{d\sigma} (-\frac{(2\lambda^2)^2}{\sigma^3}))-\frac{d\phi}{d\sigma}(-\frac{2\lambda^2}{\sigma}) +\lambda^2\phi=0}
\displaystyle{\phi'(1)=\lambda^2\phi(1)}
\displaystyle{\cdot\Downarrow }
\displaystyle{\underline{\frac{d^2\phi}{d\sigma^2}+\frac{1}{\sigma}\frac{d\phi}{d\sigma} +\phi=0,\ \phi'(1)=\lambda^2\phi(1)} }
\displaystyle{\cdot\Downarrow }
\displaystyle{\underline{\phi(\xi)=c_1J_0(\sigma)+c_2Y_0(\sigma),\ \sigma(\xi)=2\lambda\sqrt{1+\Gamma_m-\xi}} }
\displaystyle{\cdot\Downarrow }
\displaystyle{\phi'(\xi)=-(c_1J_1(\sigma)+c_2Y_1(\sigma))\sigma' =\frac{2\lambda^2}{\sigma}(c_1J_1(\sigma)+c_2Y_1(\sigma))}

\displaystyle{\phi(0)=0,\quad \phi'(1)=\lambda^2\phi(1)}
\displaystyle{\cdot\Downarrow \phi(0)=c_1J_0(\sigma_0)+c_2Y_0(\sigma_0),\ \sigma_0=2\lambda\sqrt{1+\Gamma_m}}
\displaystyle{\cdot\Downarrow \phi(1)=c_1J_0(\sigma_1)+c_2Y_0(\sigma_1),\ \sigma_1=2\lambda\sqrt{\Gamma_m}}
\displaystyle{\cdot\Downarrow \phi'(1)=\frac{2\lambda^2}{\sigma_1}(c_1J_1(\sigma_1)+c_2Y_1(\sigma_1))}
\displaystyle{c_1J_0(\sigma_0)+c_2Y_0(\sigma_0)=0}
\displaystyle{\frac{2\lambda^2}{\sigma_1}(c_1J_1(\sigma_1)+c_2Y_1(\sigma_1))= \lambda^2(c_1J_0(\sigma_1)+c_2Y_0(\sigma_1))}
\displaystyle{\cdot\Downarrow }
\displaystyle{c_1J_0(\sigma_0)+c_2Y_0(\sigma_0)=0}
\displaystyle{c_1(2J_1(\sigma_1)-\sigma_1J_0(\sigma_1))+c_2(2Y_1(\sigma_1)-\sigma_1Y_0(\sigma_1))=0}
\displaystyle{\cdot\Downarrow }
\displaystyle{\underline{J_0(\sigma_0)(2Y_1(\sigma_1)-\sigma_1Y_0(\sigma_1))-Y_0(\sigma_0)(2J_1(\sigma_1)-\sigma_1J_0(\sigma_1))=0} }

function y=freqeq(x)
s0=2*x*sqrt(1+gamma);
s1=2*x*sqrt(gamma);
y=besselj(0,s0).*(2*bessely(1,s1)-s1.*bessely(0,s1))…
-bessely(0,s0).*(2*besselj(1,s1)-s1.*besselj(0,s1));
endfunction
x=linspace(0.01,10,1000)’;
clf(0);scf(0);plot(x,freqeq(x)),mtlb_grid,mtlb_axis([0 10 -0.1 0.1])
w=locate(3);
lambda=[];
for i=1:3
x=fsolve(w(1,i),freqeq); lambda=[lambda x];
end
freqeq(lambda)
//—–
for i=1:3
x=lambda(i);
s0=2*x*sqrt(1+gamma);
s1=2*x*sqrt(gamma);
M=[besselj(0,s0) bessely(0,s0);
2*besselj(1,s1)-s1*besselj(0,s1) 2*bessely(1,s1)-s1*bessely(0,s1)];
[U,S,V]=svd(M);
c1(i)=V(1,2);
c2(i)=V(2,2);
end
function y=phi(i,x)
s=2*lambda(i).*sqrt(1+gamma-x);
y=c1(i)*besselj(0,s)+c2(i)*bessely(0,s);
endfunction
xi=(0.01:0.01:1)’;
y1=phi(1,xi);
y2=phi(2,xi);
y3=phi(3,xi);
clf(1);scf(1);plot(xi,y1,xi,y2,xi,y3),mtlb_grid

mode0