バネマスダンパー系，γ,ω0による比較

バネマスダンパー系

\(\gamma, \omega_0\)による比較

$$\begin{eqnarray} \ddot{x}(t) &+&\frac{c}{m}\dot{x}(t) &+&\frac{k}{m}x(t) &=&0 \\ \frac{\mathrm{d^2}}{\mathrm{d^2}t}x(t) &+&\frac{c}{m}\frac{\mathrm{d}}{\mathrm{d}t}x(t) &+&\frac{k}{m}x(t) &=&0 \end{eqnarray}$$ $$\begin{eqnarray} \gamma&=&\frac{c}{2m} \\\omega_0^2&=&\frac{k}{m} \\\omega^2&=&\left|\gamma^2-\omega_0^2\right| \\&=&\left|\frac{\omega_0^2}{\omega_0^2}\left(\gamma^2-\omega_0^2\right)\right| \\&=&\omega_0^2\left|\left(\frac{\gamma}{\omega_0}\right)^2-1\right| \\&=&\omega_0^2\left|\zeta^2-1\right| \;\ldots\;\zeta=\frac{\gamma}{\omega_0} \\\omega&=&\sqrt{\omega_0^2\left|\zeta^2-1\right|} \\&=&\omega_0\sqrt{\left|\zeta^2-1\right|} \end{eqnarray}$$ $$\begin{eqnarray} x(t)&=& &\left.x_0\cos{\left(\omega_0 t\right)}\right. &\left.+\frac{v_0 }{\omega_0} \sin{\left(\omega_0 t\right)}\right. &\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/0.html}{\gamma=0} &\href{https://shikitenkai.blogspot.com/2021/04/0.html}{単振動}&\zeta=0 \\&=& e^{-\gamma t} &\left[x_0 \cos{\left(\omega t\right)}\right. &\left.+\frac{v_0 +\gamma x_0 }{\omega}\sin{\left(\omega t\right)}\right] &\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/00.html}{0\lt\gamma\lt\omega_0} &\href{https://shikitenkai.blogspot.com/2021/04/00.html}{減衰振動}&0\lt\zeta\lt1 \\&=&e^{-\gamma t} &\left[x_0\right. &\left.+\left(v_0+\gamma x_0\right)t\right] &\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/0_56.html}{\gamma=\omega_0} &\href{https://shikitenkai.blogspot.com/2021/04/0_56.html}{臨界減衰}&\zeta=1 \\&=&e^{-\gamma t} &\left[x_0 \cosh\left(\omega t\right)\right. &\left.+\frac{v_0 +\gamma x_0 }{\omega}\sinh{\left(\omega t \right)}\right] &\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/0_17.html}{\omega_0\lt\gamma} &\href{https://shikitenkai.blogspot.com/2021/04/0_17.html}{過減衰}&1\lt\zeta \end{eqnarray}$$