バネマスダンパー系，単振動(γ=0 / バネマス系)の場合

バネマスダンパー系

\(\gamma=0\)

$$\begin{eqnarray} \lambda_{1,2}&=&\left.-\gamma\pm\sqrt{\gamma^2-\omega_0^2}\right|_{\gamma=0} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/12.html}{\lambda_{1,2}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}} \\&=&-0\pm\sqrt{0^2-\omega_0^2} \\&=&\pm\sqrt{-\omega_0^2} \\&=&\pm\omega_0\sqrt{-1} \\&=&\pm\omega_0i \end{eqnarray}$$ $$\begin{eqnarray} C_{1,2} &=&\left.\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\gamma^2-\omega_0^2}}\right|_{\gamma=0} \;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/c1-c2.html}{ C_{1,2} =\frac{x_0 }{2}\pm\frac{v_0 +\gamma x_0 }{2\sqrt{\gamma^2-\omega_0^2}} } \\&=&\frac{x_0 }{2}\pm\frac{v_0 +0 x_0 }{2\sqrt{0^2-\omega_0^2}} \\&=&\frac{x_0 }{2}\pm\frac{v_0 }{2\omega_0\sqrt{-1}} \\&=&\frac{x_0 }{2}\pm\frac{v_0 }{2\omega_0i}\frac{i}{i} \\&=&\frac{x_0 }{2}\pm\frac{v_0 }{2\omega_0\cdot(-1)}i \\&=&\frac{x_0 }{2}\mp\frac{v_0 }{2\omega_0}i\;\ldots\;ただし\omega_0\ne0 \end{eqnarray}$$ $$\begin{eqnarray} x(t)&=&C_1 e^{\lambda_1 t}+C_2 e^{\lambda_2 t} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 }{2\omega_0}i\right\}e^{\omega_0i t} +\left\{\frac{x_0 }{2}+\frac{v_0 }{2\omega_0}i\right\}e^{-\omega_0i t} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 }{2\omega_0}i\right\} \left\{ \cos{\left(\omega_0 t\right)}+i\sin{\left(\omega_0 t\right)}\right\} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 }{2\omega_0}i\right\} \left\{ \cos{\left(-\omega_0 t\right)}+i\sin{\left(-\omega_0 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 }{2\omega_0}i\right\} \left\{ \cos{\left(\omega_0 t\right)}+i\sin{\left(\omega_0 t\right)}\right\} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 }{2\omega_0}i\right\} \left\{ \cos{\left(\omega_0 t\right)}-i\sin{\left(\omega_0 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 }{2\omega_0}i\right\}\cos{\left(\omega_0 t\right)} +i \left\{\frac{x_0 }{2}-\frac{v_0 }{2\omega_0}i\right\} \sin{\left(\omega_0 t\right)} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 }{2\omega_0}i\right\} \cos{\left(\omega_0 t\right)} -i \left\{\frac{x_0 }{2}+\frac{v_0 }{2\omega_0}i\right\}\sin{\left(\omega_0 t\right)} \\&=&\left\{\frac{x_0 }{2}-\frac{v_0 }{2\omega_0}i\right\}\cos{\left(\omega_0 t\right)} +\left\{\frac{x_0 }{2}i+\frac{v_0 }{2\omega_0}\right\} \sin{\left(\omega_0 t\right)} \\&&+\left\{\frac{x_0 }{2}+\frac{v_0 }{2\omega_0}i\right\} \cos{\left(\omega_0 t\right)} -\left\{\frac{x_0 }{2}i-\frac{v_0 }{2\omega_0}\right\} \sin{\left(\omega_0 t\right)} \\&&\;\ldots\;i\cdot i=-1 \\&=&\left\{\frac{x_0 }{2}\color{red}{ -\frac{v_0 }{2\omega_0}i }\color{black}{ +\frac{x_0 }{2} }\color{red}{ +\frac{v_0 }{2\omega_0}i}\right\} \cos{\left(\omega_0 t\right)} \\&&+\left\{\color{blue}{\frac{x_0 }{2}i}\color{black}{+\frac{v_0 }{2\omega_0}}\color{blue}{-\frac{x_0 }{2}i}\color{black}{+\frac{v_0 }{2\omega_0}}\right\} \sin{\left(\omega_0 t\right)} \\&=&\left\{\frac{x_0 }{2}+\frac{x_0 }{2}\right\}\cos{\left(\omega t\right)}+\left\{\frac{v_0 }{2\omega_0}+\frac{v_0 }{2\omega_0}\right\}\sin{\left(\omega t\right)} \\&=&x_0 \cos{\left(\omega_0 t\right)}+\frac{v_0 }{\omega_0} \sin{\left(\omega_0 t\right)} \\&=&\color{red}{x_0 \cos{\left(\omega_0 t\right)}}\color{blue}{+v_0 \frac{1}{\omega_0} \sin{\left(\omega_0 t\right)}}\color{black}{} \;\ldots\;\color{red}{初期位置による単振動}\color{black}{},\;\color{blue}{初期速度による単振動} \end{eqnarray}$$