バネマスダンパー系，減衰振動(0<γ<ω0)の場合

バネマスダンパー系

\(0\lt\gamma\lt\omega_0\)

$$\begin{eqnarray} \lambda_{1,2}&=&\left.-\gamma\pm\sqrt{\gamma^2-\omega_0^2}\right|_{0\lt\gamma\lt\omega_0} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/12.html}{\lambda_{1,2}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}} \\&=&-\gamma\pm\sqrt{\gamma^2-\omega_0^2} \\&=&-\gamma\pm\sqrt{-\omega^2} \;\ldots\;\omega^2=\left|\gamma^2-\omega_0^2\right|,\;\gamma^2-\omega_0^2=-\omega^2 \\&=&-\gamma\pm\omega\sqrt{-1} \\&=&-\gamma\pm\omega i \end{eqnarray}$$ $$\begin{eqnarray} C_{1,2} &=&\left.\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\gamma^2-\omega_0^2}}\right|_{0\lt\gamma\lt\omega_0} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/c1-c2.html}{ C_{1,2} =\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\gamma^2-\omega_0^2}} } \\&=&\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\omega\sqrt{-1}} \\&=&\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\omega i}\frac{i}{i} \\&=&\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\omega\cdot(-1)}i \\&=&\frac{x_0 }{2}\mp\frac{v_0 +\gamma x_0 }{2\omega}i\;\ldots\;ただし\omega\ne0 \end{eqnarray}$$ $$\begin{eqnarray} x(t)&=&C_1 e^{\lambda_1 t}+C_2 e^{\lambda_2 t} \\&=&C_1 e^{\left(-\gamma+\omega i\right) t}+C_2 e^{\left(-\gamma-\omega i\right) t} \\&=&C_1 e^{-\gamma t}e^{\omega i t}+C_2 e^{-\gamma t}e^{-\omega i t} \\&=&e^{-\gamma t}\left\{C_1 e^{\omega i t}+C_2 e^{-\omega i t}\right\} \\\left\{C_1 e^{\omega i t}+C_2 e^{-\omega i t}\right\}&=& \left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}i\right\}e^{\omega i t} +\left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}i\right\}e^{-\omega i t} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}i\right\} \left\{ \cos{\left(\omega t\right)}+i\sin{\left(\omega t\right)}\right\} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}i\right\} \left\{ \cos{\left(-\omega t\right)}+i\sin{\left(-\omega t\right)}\right\} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/eix.html}{e^{ix}= \cos{\left(x\right)}+i \sin{\left(x\right)}} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}i\right\} \left\{ \cos{\left(\omega t\right)}+i\sin{\left(\omega t\right)}\right\} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}i\right\} \left\{ \cos{\left(\omega t\right)}-i\sin{\left(\omega t\right)}\right\} \\&&\;\ldots\;\cos{\left(-\theta\right)}=\cos{\left(\theta\right)},\;\sin{\left(-\theta\right)}=-\sin{\left(\theta\right)} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}i\right\}\cos{\left(\omega t\right)} +i\left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}i\right\}\sin{\left(\omega t\right)} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}i\right\}\cos{\left(\omega t\right)} -i\left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}i\right\}\sin{\left(\omega t\right)} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 +\gamma x_0 }{2\omega}i\right\}\cos{\left(\omega t\right)} +\left\{\frac{x_0 }{2}i+\frac{v_0 +\gamma x_0 }{2\omega}\right\}\sin{\left(\omega t\right)} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 +\gamma x_0 }{2\omega}i\right\}\cos{\left(\omega t\right)} -\left\{\frac{x_0 }{2}i-\frac{v_0 +\gamma x_0 }{2\omega}\right\}\sin{\left(\omega t\right)} \\&&\;\ldots\;i\cdot i=-1 \\&=&\left\{\frac{x_0 }{2}\color{red}{-\frac{v_0 +\gamma x_0 }{2\omega}i}\color{black}{+\frac{x_0 }{2}}\color{red}{+\frac{v_0 +\gamma x_0 }{2\omega}i}\right\}\cos{\left(\omega t\right)} \\&&+\left\{\color{blue}{\frac{x_0 }{2}i}\color{black}{+\frac{v_0 +\gamma x_0 }{2\omega}}\color{blue}{-\frac{x_0 }{2}i}\color{black}{+\frac{v_0 +\gamma x_0 }{2\omega}}\right\}\sin{\left(\omega t\right)} \\&=&\left\{\frac{x_0 }{2}+\frac{x_0 }{2}\right\}\cos{\left(\omega t\right)}+\left\{\frac{v_0 +\gamma x_0 }{2\omega}+\frac{v_0 +\gamma x_0 }{2\omega}\right\}\sin{\left(\omega t\right)} \\&=&x_0 \cos{\left(\omega t\right)}+\frac{v_0 +\gamma x_0 }{\omega}\sin{\left(\omega t\right)} \end{eqnarray}$$ $$\begin{eqnarray} x(t)&=&e^{-\gamma t}\left\{C_1 e^{\omega i t}+C_2 e^{-\omega i t}\right\} \\&=&e^{-\gamma t}\left\{ x_0 \cos{\left(\omega t\right)}+\frac{v_0 +\gamma x_0 }{\omega}\sin{\left(\omega t\right)} \right\} \\&=& \color{red}{ x_0 e^{-\gamma t}\left\{ \cos{\left(\omega t\right)}+\frac{\gamma}{\omega}\sin{\left(\omega t\right)} \right\} } \color{blue}{ +v_0e^{-\gamma t}\left\{ \frac{1}{\omega}\sin{\left(\omega t\right)} \right\} }\color{black}{} \;\ldots\;\color{red}{初期位置による減衰振動}\color{black}{},\;\color{blue}{初期速度による減衰振動} \\&=& x_0 e^{-\gamma t}\left\{ \cos{\left(\omega t\right)}+\frac{\zeta}{\sqrt{\zeta^2-1}}\sin{\left(\omega t\right)} \right\} +v_0e^{-\gamma t}\left\{ \frac{1}{\omega}\sin{\left(\omega t\right)} \right\} \\&&\;\ldots\;\frac{\gamma}{\omega} =\frac{\gamma}{\sqrt{\left|\gamma^2-\omega_0^2\right|}} =\frac{\gamma}{\sqrt{\frac{\omega_0^2}{\omega_0^2}\left|\gamma^2-\omega_0^2\right|}} =\frac{\gamma}{\sqrt{\omega_0^2\left|\frac{\gamma^2}{\omega_0^2}-\frac{\omega_0^2}{\omega_0^2}\right|}} =\frac{\gamma}{\omega_0\sqrt{\left|\zeta^2-1\right|}} =\frac{\zeta}{\sqrt{\left|\zeta^2-1\right|}} \\&&\;\ldots\;\omega=\sqrt{\left|\gamma^2-\omega_0^2\right|},\;\zeta=\frac{\gamma}{\omega_0} \end{eqnarray}$$