バネマスダンパー系
\(\gamma=\omega_0\)
$$\begin{eqnarray}
\lambda_{1,2}&=&\left.-\gamma\pm\sqrt{\gamma^2-\omega_0^2}\right|_{\gamma=\omega_0}
\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/12.html}{\lambda_{1,2}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}}
\\&=&-\omega_0\pm\sqrt{\omega_0^2-\omega_0^2}
\\&=&-\omega_0
\end{eqnarray}$$
$$\begin{eqnarray}
x(t)&=&C(t) e^{-\omega_0 t}
\;\ldots\;定数変化法
\\\dot{x}&=&C'(t)e^{-\omega_0 t}-\omega_0 C(t) e^{-\omega_0 t}&\;\ldots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post.html}{(fg)'=f'g+fg'}
\\\ddot{x}&=&\left\{
C''(t)e^{-\omega_0 t}- \omega_0 C'(t) e^{-\omega_0 t}
\right\}
+\left\{
-\omega_0 C(t)' e^{-\omega_0 t} + \omega_0^2 C(t) e^{-\omega_0 t}
\right\}&\;\ldots\;\href{https://shikitenkai.blogspot.com/2020/02/blog-post.html}{(fg)'=f'g+fg'}
\\&=&C''(t)e^{-\omega_0 t}-2\omega_0 C'(t) e^{-\omega_0 t}+\omega_0^2 C(t) e^{-\omega_0 t}
\end{eqnarray}$$
$$\begin{eqnarray}
\left.\ddot{x}+2\gamma\dot{x}+\omega_0^2 x\right|_{\gamma=\omega_0}&=&0
\\\ddot{x}+2\omega_0\dot{x}+\omega_0^2 x&=&0
\\\left[C''(t)e^{-\omega_0 t}-2\omega_0 C'(t) e^{-\omega_0 t}+\omega_0^2 C(t) e^{-\omega_0 t}\right]
+2\omega_0\left[C'(t)e^{-\omega_0 t}-\omega_0 C(t) e^{-\omega_0 t}\right]
+\omega_0^2 C(t) e^{-\omega_0 t} &=&0
\\C''(t)e^{-\omega_0 t}
\color{red} {-2\omega_0 C'(t)e^{-\omega_0 t}}
\color{blue}{+ \omega_0^2 C(t) e^{-\omega_0 t}}
\color{red} {+2\omega_0 C'(t)e^{-\omega_0 t}}
\color{blue}{-2\omega_0^2 C(t) e^{-\omega_0 t}}
\color{blue}{+ \omega_0^2 C(t) e^{-\omega_0 t}}
&=&0
\\C''(t)e^{-\omega_0 t} &=&0
\\C''(t)&=&0\;\ldots\;e^{-\omega_0 t}\gt 0
\end{eqnarray}$$
$$\begin{eqnarray}
C''(t)&=&0
\\C'(t)&=&C_1
\\C(t)&=&C_1 t + C_2
\\x(t)&=&\left(C_1 t + C_2\right)e^{-\omega_0 t}
\end{eqnarray}$$
$$\begin{eqnarray}
x(0)&=&\left.\left(C_1 t + C_2\right)e^{-\omega_0 t}\right|_{t=0}
\\&=&\left(C_1 0 + C_2\right)e^{-\omega_0 0}
\\&=&C_2\cdot 1=C_2
\\C_2&=&x(0)=x_0
\\x'(0)&=&\left.\left\{C_1e^{-\omega_0 t}-\omega_0\left(C_1 t + C_2\right)e^{-\omega_0 t}\right\}\right|_{t=0,C_2=x_0}
\\&=&C_1e^{-\omega_0 0}-\omega_0\left(C_1 0 + x_0\right)e^{-\omega_0 0}
\\&=&C_1\cdot1-\omega_0\left(x_0\right)\cdot1
\\&=&C_1-\omega_0x_0
\\C_1&=&x'(0)+\omega_0x_0=v_0+\omega_0x_0
\end{eqnarray}$$
$$\begin{eqnarray}
x(t)&=&C(t)e^{-\omega_0 t}
\\&=&\left(C_1 t+C_2\right)e^{-\omega_0 t}
\\&=&\left\{\left(v_0+\omega_0x_0\right)t+x_0\right\}e^{-\omega_0 t}
\\&=&\left\{\left(v_0+\gamma x_0\right)t+x_0\right\}e^{-\gamma t}\;\ldots\;\omega_0=\gamma
\\&=&e^{-\gamma t}\left\{x_0+\left(v_0+\gamma x_0\right)t\right\}
\\&=&\color{red}{x_0e^{-\gamma t} \left(1+\gamma t\right)}
\color{blue}{+v_0e^{-\gamma t}\left( t\right)}\color{black}{}
\;\ldots\;\color{red}{初期位置x_0による臨界減衰}\color{black}{},\;\color{blue}{初期速度v_0による臨界減衰}
\end{eqnarray}$$
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