バネマスダンパー系，過減衰(ω0<γ)の場合

バネマスダンパー系

\(\omega_0\lt\gamma\)

$$\begin{eqnarray} \lambda_{1,2}&=&-\gamma\pm\sqrt{\gamma^2-\omega_0^2} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/12.html}{\lambda_{1,2}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}} \\&=&-\gamma\pm\sqrt{\omega^2} \;\ldots\;\omega^2=\left|\gamma^2-\omega_0^2\right|,\;\gamma^2-\omega_0^2=\omega^2 \\&=&-\gamma\pm\omega \end{eqnarray}$$ $$\begin{eqnarray} C_{1,2} &=&\left.\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\gamma^2-\omega_0^2}}\right|_{\gamma\gt\omega_0} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/c1-c2.html}{ C_{1,2} =\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\gamma^2-\omega_0^2}} } \\&=&\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\omega^2}} \\&=&\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\omega}\;\ldots\;ただし\omega\ne0 \end{eqnarray}$$ $$\begin{eqnarray} x(t)&=&C_1 e^{\lambda_1 t}+C_2 e^{\lambda_2 t} \\&=&C_1 e^{\left(-\gamma+\omega\right)t}+C_2 e^{\left(-\gamma-\omega\right)t} \\&=&C_1 e^{-\gamma t}e^{\omega t}+C_2 e^{-\gamma t}e^{-\omega t} \\&=&e^{-\gamma t}\left\{C_1 e^{\omega t}+C_2 e^{-\omega t}\right\} \\&=&e^{-\gamma t}\left[ \left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}\right\}e^{\omega t} +\left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}\right\}e^{-\omega t} \right] \\&=&e^{-\gamma t}\left\{ \frac{x_0 }{2}e^{\omega t}+\frac{v_0 +\gamma x_0 }{2\omega}e^{\omega t} +\frac{x_0 }{2}e^{-\omega t}-\frac{v_0 +\gamma x_0 }{2\omega}e^{-\omega t} \right\} \\&=&e^{-\gamma t}\left\{ \frac{x_0 }{2}\left(e^{\omega t}+e^{-\omega t}\right)+\frac{v_0 +\gamma x_0 }{2\omega}\left(e^{\omega t}-e^{-\omega t}\right) \right\} \\&=&e^{-\gamma t}\left\{ x_0 \frac{e^{\omega t}+e^{-\omega t}}{2}+\frac{v_0 +\gamma x_0 }{\omega}\frac{e^{\omega t}-e^{-\omega t}}{2} \right\} \\&=&e^{-\gamma t}\left\{ x_0 \cosh\left(\omega t\right)+\frac{v_0 +\gamma x_0 }{\omega}\sinh{\left(\omega t \right)} \right\} \\&&\;\ldots\;\cosh{\left(\omega t\right)}=\frac{e^{\omega t}+e^{-\omega t}}{2}, \sinh{\left(\omega t \right)}=\frac{e^{\omega t}-e^{-\omega t}}{2} \\&=&\color{red}{x_0 e^{-\gamma t}\left\{ \cosh\left(\omega t\right)+\frac{\gamma }{\omega}\sinh{\left(\omega t \right)} \right\}} \color{blue}{+v_0 e^{-\gamma t}\left\{ \frac{1}{\omega}\sinh{\left(\omega t \right)} \right\}} \;\ldots\;\color{red}{初期位置x_0による過減衰}\color{black}{},\;\color{blue}{初期速度v_0による過減衰} \\&=& x_0 e^{-\gamma t}\left\{ \cosh\left(\omega t\right)+\frac{\zeta }{\sqrt{\zeta^2-1}}\sinh{\left(\omega t \right)} \right\} +v_0 e^{-\gamma t}\left\{ \frac{1}{\omega}\sinh{\left(\omega t \right)} \right\} \\&&\;\ldots\;\frac{\gamma}{\omega} =\frac{\gamma}{\sqrt{\left|\gamma^2-\omega_0^2\right|}} =\frac{\gamma}{\sqrt{\frac{\omega_0^2}{\omega_0^2}\left|\gamma^2-\omega_0^2\right|}} =\frac{\gamma}{\sqrt{\omega_0^2\left|\frac{\gamma^2}{\omega_0^2}-\frac{\omega_0^2}{\omega_0^2}\right|}} =\frac{\gamma}{\omega_0\sqrt{\left|\zeta^2-1\right|}} =\frac{\zeta}{\sqrt{\left|\zeta^2-1\right|}} \\&&\;\ldots\;\omega=\sqrt{\left|\gamma^2-\omega_0^2\right|},\;\zeta=\frac{\gamma}{\omega_0} \end{eqnarray}$$