バネマスダンパー系
運動方程式
$$\begin{eqnarray}
m\ddot{x}
&+&c\dot{x}
&+&kx
&=&Fu(t)
\\
\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&\frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\frac{k}{m}x
&=&\frac{F}{m}u(t)
\\
\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\omega_0^2x
&=&\frac{F}{m}u(t)
\;\cdots\;\gamma=\frac{c}{2m},\;\omega_0^2=\frac{k}{m}
\end{eqnarray}$$
ここで入力\(u\)を単位ステップ凾数とする．
$$\begin{eqnarray}
u(t) &=& \left\{ \begin{array}{l}1 \space(t \geq 0)\\ 0\space(t\lt0) \end{array} \right.
\end{eqnarray}$$
ラプラス変換
$$\begin{eqnarray}
\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t}
\right.&+&\left.2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
\right.&+&\left.\omega_0^2x \right]&=\mathfrak{L}\left[\frac{F}{m}u(t)\right]
\\\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t} \right]
&+&\mathfrak{L}\left[ 2\gamma\frac{\mathrm{d}x}{\mathrm{d}t} \right]
&+&\mathfrak{L}\left[ \omega_0^2 x\right]
&=\mathfrak{L}\left[\frac{F}{m}u(t)\right]
\\\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t} \right]
&+&2\gamma\mathfrak{L}\left[ \frac{\mathrm{d}x}{\mathrm{d}t} \right]
&+&\omega_0^2\mathfrak{L}\left[ x\right]
&=\frac{F}{m}\mathfrak{L}\left[u(t)\right]
\\
s^2X-sx_0 -v_0
&+&2\gamma\left(sX-x_0 \right)
&+&\omega_0^2X
&=\frac{F}{m}U
\\
\\&&&&&\;\ldots\;\mathfrak{L}\left[x\right]=X
\\&&&&&\;\ldots\;\mathfrak{L}\left[ \frac{\mathrm{d}x}{\mathrm{d}t}\right]
=s^2X-x_0,\;x_0=x(0)
\\&&&&&\;\ldots\;\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t}\right]
=s^2X-sx_0 -v_0,\;v_0=x'(0)
\\&&&&&\;\ldots\;\mathfrak{L}\left[u(t)\right]=U
\end{eqnarray}$$
\(X\)について解く
$$\begin{eqnarray}
s^2X+2\gamma Xs+\omega_0^2X
&=&
sx_0 +v_0
+2\gamma x_0 +\frac{F}{m}U
\\
\left(s^2+2\gamma s+\omega_0^2\right)X
&=&
sx_0 +v_0 +2\gamma x_0 +\frac{F}{m}U
\\
X&=&\frac{sx_0 +v_0 +2\gamma x_0 }{s^2+2\gamma s+\omega_0^2}+\frac{F}{m}\frac{1}{s^2+2\gamma s+\omega_0^2}U
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}+\frac{F}{m}\frac{1}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}U
\end{eqnarray}$$
部分分数分解 準備
$$\begin{eqnarray}
U&=&\href{https://shikitenkai.blogspot.com/2021/05/blog-post.html}{1/s}\;\ldots\;単位ステップ凾数のラプラス変換,
\\X&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}+\frac{F}{m}\frac{1}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}U
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}+\frac{F}{m}\frac{1}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}\frac{1}{s}
\\&=&\frac{C_1 }{s-\lambda_1}+\frac{C_2 }{s-\lambda_2}
+\frac{F}{m}\left\{
\frac{C_3}{s-\lambda_1}
+\frac{C_4}{s-\lambda_2}
+\frac{C_5}{s}
\right\}
\\&=&\frac{C_1 \left(s-\lambda_2\right)+C_2 \left(s-\lambda_1\right)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\left\{ \frac{
C_3\left(s-\lambda_2\right)s+C_4\left(s-\lambda_1\right)s+C_5\left(s-\lambda_1\right)\left(s-\lambda_2\right)
}{ s\left(s-\lambda_1\right)\left(s-\lambda_2\right)} \right\}
\\&=&\frac{C_1 \left(s-\lambda_2\right)+C_2 \left(s-\lambda_1\right)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\left\{ \frac{
C_3s^2-C_3\lambda_2s+C_4s^2-C_4\lambda_1s+C_5\left\{s^2-\left(\lambda_1+\lambda_2\right)s+\lambda_1\lambda_2\right\}
}{ s\left(s-\lambda_1\right)\left(s-\lambda_2\right)} \right\}
\\&=&\frac{C_1 \left(s-\lambda_2\right)+C_2 \left(s-\lambda_1\right)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\left\{ \frac{
C_3s^2-C_3\lambda_2s+C_4s^2-C_4\lambda_1s+C_5s^2-C_5\left(\lambda_1+\lambda_2\right)s+C_5\lambda_1\lambda_2
}{ s\left(s-\lambda_1\right)\left(s-\lambda_2\right)} \right\}
\\&=&\frac{(C_1 +C_2 )s-(C_1 \lambda_2+C_2 \lambda_1)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\left[ \frac{
\left(C_3+C_4+C_5\right)s^2
+\left\{-C_3\lambda_2-C_4\lambda_1-C_5(\lambda_1+\lambda_2)\right\}s
+C_5\lambda_1\lambda_2
}
{ s\left(s-\lambda_1\right)\left(s-\lambda_2\right)} \right]
\end{eqnarray}$$
部分分数分解 第1項分子の比較
$$\begin{eqnarray}
sx_0 +v_0 +2\gamma x_0&=&\left(C_1 +C_2 \right)s-\left(C_1 \lambda_2+C_2 \lambda_1\right)
\end{eqnarray}$$
$$\left\{\begin{eqnarray}
x_0&=&C_1 +C_2
\\v_0 +2\gamma x_0&=&-\left(C_1 \lambda_2+C_2 \lambda_1\right)
\end{eqnarray}\right.$$
部分分数分解 \(C_2\)
$$\begin{eqnarray}
x_0&=&C_1+C_2
\\C_1&=&x_0-C_2
\\v_0+2\gamma x_0&=&-\left\{\left(x_0-C_2\right)\lambda_2+C_2\lambda_1\right\}
\\&=&-\lambda_2x_0+C_2\lambda_2-C_2\lambda_1
\\v_0+2\gamma x_0+\lambda_2 x_0&=&C_2\left(\lambda_2-\lambda_1\right)
\\C_2 &=&\frac{v_0+2\gamma x_0+\lambda_2 x_0}{\lambda_2-\lambda_1}
\\&&\;\ldots\;\lambda_{1,2}=\frac{-2\gamma\pm\sqrt{\left(2\gamma\right)^2-4\cdot1\cdot\omega_0^2}}{2\cdot1}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}=-\gamma\pm\xi
\\&&\;\ldots\;\lambda_2-\lambda_1=-\gamma-\xi -\left(-\gamma+\xi \right)=-2\xi
\\&=&\frac{v_0+2\gamma x_0+\left(-\gamma-\xi \right) x_0}{-2\xi }
\\&=&\frac{v_0+\gamma x_0-\xi x_0}{-2\xi }
\\&=&\frac{v_0+\gamma x_0}{-2\xi }-\frac{\xi x_0}{-2\xi }
\\&=&\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }
\end{eqnarray}$$
部分分数分解 \(C_1\)
$$\begin{eqnarray}
C_1&=&x_0-C_2
\\&=&x_0-\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\right)
\\&=&\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }
\end{eqnarray}$$
部分分数分解 第2項分子の比較
$$\begin{eqnarray}
1&=&\left(C_3+C_4+C_5\right)s^2
+\left\{-C_3\lambda_2-C_4\lambda_1-C_5(\lambda_1+\lambda_2)\right\}s
+C_5\lambda_1\lambda_2
\end{eqnarray}$$
$$\left\{\begin{eqnarray}
0&=&C_3+C_4+C_5
\\0&=&-C_3\lambda_2-C_4\lambda_1-C_5(\lambda_1+\lambda_2)
\\1&=&C_5\lambda_1\lambda_2
\end{eqnarray}\right.$$
部分分数分解 \(C_5\)
$$\begin{eqnarray}
1&=&C_5\lambda_1\lambda_2\;\ldots\;第3式
\\C_5&=&\frac{1}{\lambda_1\lambda_2}
\\&=&\frac{1}{\gamma^2-\xi^2}
\\&&\;\ldots\;\lambda_{1,2}=-\gamma\pm\xi
\\&&\;\ldots\;\lambda_1\lambda_2=\left(-\gamma+\xi\right)\left(-\gamma-\xi\right)=\left(-\gamma\right)^2-\xi^2=\gamma^2-\xi^2
\\&=&\frac{1}{\omega_0^2}
\\&&\;\ldots\;\gamma^2-\xi^2=\gamma^2-\left(\sqrt{\gamma^2-\omega_0^2}\right)^2=\gamma^2-\gamma^2+\omega_0^2=\omega_0^2
\end{eqnarray}$$
部分分数分解 \(C_3,\;C_4\)
$$\begin{eqnarray}
0-0&=&(C_3+C_4+C_5)\lambda_2-C_3\lambda_2-C_4\lambda_1-C_5(\lambda_1+\lambda_2)
\;\ldots\;第1式\cdot\lambda_2+第2式
\\0&=&\color{red}{C_3\lambda_2}\color{black}{+C_4\lambda_2}\color{blue}{+C_5\lambda_2}\color{red}{-C_3\lambda_2}\color{black}{-C_4\lambda_1}-C_5\lambda_1\color{blue}{-C_5\lambda_2}
\\0&=&C_4(\lambda_2-\lambda_1)-C_5\lambda_1
\\C_4(\lambda_2-\lambda_1)&=&C_5\lambda_1
\\C_4&=&C_5\frac{\lambda_1}{\lambda_2-\lambda_1}
\\&=&\frac{1}{\lambda_1\lambda_2}\frac{\lambda_1}{\lambda_2-\lambda_1}
\\&=&\frac{1}{\lambda_2}\frac{1}{\lambda_2-\lambda_1}
\\&=&\frac{1}{\lambda_2^2-\lambda_1\lambda_2}
\\&=&\frac{1}{\left(-\gamma-\xi\right)^2-\left(-\gamma+\xi\right)\left(-\gamma-\xi\right)}
\\&=&\frac{1}{\gamma^2+2\gamma\xi+\xi^2-\left(\gamma^2-\xi^2\right)}
\\&=&\frac{1}{\gamma^2+2\gamma\xi+\xi^2-\gamma^2+\xi^2}
\\&=&\frac{1}{2\gamma\xi+2\xi^2}
\\&=&\frac{1}{2\xi\left(\gamma+\xi\right)}
\\&=&\frac{1}{2\xi\left(\gamma+\xi\right)}\frac{\gamma-\xi}{\gamma-\xi}
\\&=&\frac{\gamma-\xi}{2\xi\left(\gamma^2-\xi^2\right)}
\\&=&\frac{\gamma-\xi}{2\xi\omega_0^2}
\\&=&\frac{1}{\omega_0^2}\frac{\gamma-\xi}{2\xi}
\\0&=&C_3+C_4+C_5\;\ldots\;第1式
\\C_3&=&-C_4-C_5
\\&=&-\frac{1}{\omega_0^2}\frac{\gamma-\xi}{2\xi}-\frac{1}{\omega_0^2}
\\&=&\frac{-1}{\omega_0^2}\left(\frac{\gamma-\xi}{2\xi}+1\right)
\\&=&\frac{-1}{\omega_0^2}\left(\frac{\gamma-\xi}{2\xi}+\frac{2\xi}{2\xi}\right)
\\&=&\frac{-1}{\omega_0^2}\left(\frac{\gamma+\xi}{2\xi}\right)
\end{eqnarray}$$
部分分数分解 まとめる
$$\begin{eqnarray}
X&=&\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_1}
+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_2}
+\frac{F}{m}\left\{
\frac{\frac{-1}{\omega_0^2}\left(\frac{\gamma+\xi}{2\xi}\right)}{s-\lambda_1}
+\frac{\frac{1}{\omega_0^2}\frac{\gamma-\xi}{2\xi}}{s-\lambda_2}
+ \frac{\frac{1}{\omega_0^2}}{s}
\right\}
\end{eqnarray}$$
逆ラプラス変換
$$\begin{eqnarray}
\mathfrak{L}^{-1}\left[X\right]&=&\mathfrak{L}^{-1}\left[
C_1 \frac{1}{s-\lambda_1}+C_2 \frac{1}{s-\lambda_2}
+\frac{F}{m}\left\{
C_3 \frac{1}{s-\lambda_1}+C_4 \frac{1}{s-\lambda_2}+C_5\frac{1}{s}
\right\}
\right]
\\&=&C_1 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_2 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
+\frac{F}{m}C_3\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+\frac{F}{m}C_4\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
+\frac{F}{m}C_5\mathfrak{L}^{-1}\left[\frac{1}{s}\right]
\\&=&\left\{
C_1 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_2 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\right\}
+\frac{F}{m}\left\{
C_3\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_4\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\right\}
+\left\{
\frac{F}{m}C_5\mathfrak{L}^{-1}\left[\frac{1}{s}\right]
\right\}
\end{eqnarray}$$
逆ラプラス変換 第1項
\(\gamma \lt \omega_0(\xiが虚数の場合)\)
$$\begin{eqnarray}
\\&&C_1 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_2 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\\&=&C_1 e^{\lambda_1 t}+C_2 e^{\lambda_2 t}
\;\ldots\;\mathfrak{L}^{-1}\left[ \frac{1}{s+a} \right]=e^{-at}
\\&=&\left(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\right) e^{\lambda_1 t}
+\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\right) e^{\lambda_2 t}
\\&=&\left(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\right) e^{\left(-\gamma+\xi\right) t}
+\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\right) e^{\left(-\gamma-\xi\right) t}
\\&&\;\ldots\;\lambda_{1,2}
=-\frac{c}{2m}\pm\sqrt{\left(\frac{c}{2m}\right)^2-\left(\sqrt{\frac{k}{m}}\right)^2}
=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}
=-\gamma\pm\xi
\\&=&\left(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\right) e^{-\gamma t}e^{\xi t}
+\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\right) e^{-\gamma t}e^{-\xi t}
\;\ldots\;a^{A+B}=a^Aa^B
\\&=& e^{-\gamma t}\left\{
\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}
\right\}
\\&&\;\ldots\;\gamma \lt \omega_0(\xiが虚数の場合),\;\xi=\sqrt{\gamma^2-\omega_0^2}=\sqrt{\left|\gamma^2-\omega_0^2\right|}\;i=\omega i
\\&=& e^{-\gamma t}\left[
\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\}
\right]
\\&=& e^{-\gamma t}\left[
\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\}
\right]
\\&&\;\ldots\;\cos{\left(-\omega t\right)}=\cos{\left(\omega t\right)},\;\sin{\left(-\omega t\right)}=-\sin{\left(\omega t\right)}
\\&=& e^{-\gamma t}\left[
\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}+\frac{v_0+\gamma x_0}{2\omega i }\right)i\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}-\frac{v_0+\gamma x_0}{2\omega i }\right)i\sin{\left(\omega t\right)}
\right]
\\&=& e^{-\gamma t}\left[
\left\{
\left(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i }\right)
+\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i }\right)
\right\}\cos{\left(\omega t\right)}
+\left\{
\left(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i }\right)
-\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i }\right)
\right\}i\sin{\left(\omega t\right)}
\right]
\\&=& e^{-\gamma t}\left\{
x_0\cos{\left(\omega t\right)}
+\frac{v_0+\gamma x_0}{\omega i }i\sin{\left(\omega t\right)}
\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\}
\;\ldots\;\frac{i}{i}=1
\end{eqnarray}$$
逆ラプラス変換 第2項
\(\gamma \lt \omega_0(\xiが虚数の場合)\)
$$\begin{eqnarray}
\\&&\frac{F}{m}\left\{
C_3\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_4\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\right\}
\\&=&\frac{F}{m}\left(C_3 e^{\lambda_1 t}+C_4 e^{\lambda_2 t}\right)
\;\ldots\;\mathfrak{L}^{-1}\left[ \frac{1}{s+a} \right]=e^{-at}
\\&=&\frac{F}{m}
\left\{
-\frac{1}{\omega_0^2}\left(\frac{\gamma+\xi}{2\xi}\right)e^{\lambda_1 t}
+\frac{1}{\omega_0^2}\left(\frac{\gamma-\xi}{2\xi}\right) e^{\lambda_2 t}
\right\}
\\&=&\frac{F}{m}\frac{1}{\omega_0^2}
\left\{
-\left(\frac{\gamma}{2\xi}+\frac{\xi}{2\xi}\right)e^{\lambda_1 t}
+\left(\frac{\gamma}{2\xi}-\frac{\xi}{2\xi}\right)e^{\lambda_2 t}
\right\}
\\&=&\frac{F}{k}
\left\{
-\left(\frac{\gamma}{2\xi}+\frac{1}{2}\right)e^{\lambda_1 t}
+\left(\frac{\gamma}{2\xi}-\frac{1}{2}\right)e^{\lambda_2 t}
\right\}
\\&&\;\ldots\;\omega_0=\sqrt{\frac{k}{m}},\;\omega_0^2=\frac{k}{m},\;m\omega_0^2=k
\\&=&\frac{F}{k}
\left\{
-\left(\frac{\gamma}{2\xi}+\frac{1}{2}\right)e^{\left(-\gamma+\xi\right) t}
+\left(\frac{\gamma}{2\xi}-\frac{1}{2}\right)e^{\left(-\gamma-\xi\right) t}
\right\}
\\&&\;\ldots\;\lambda_{1,2}
=-\frac{c}{2m}\pm\sqrt{\left(\frac{c}{2m}\right)^2-\left(\sqrt{\frac{k}{m}}\right)^2}
=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}
=-\gamma\pm\xi
\\&=&\frac{F}{k}
\left\{
-\left(\frac{\gamma}{2\xi}+\frac{1}{2}\right)e^{-\gamma t}e^{\xi t}
+\left(\frac{\gamma}{2\xi}-\frac{1}{2}\right)e^{-\gamma t}e^{-\xi t}
\right\}
\\&=&\frac{F}{k}e^{-\gamma t}
\left\{
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)e^{\omega i t}
+\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)e^{-\omega i t}
\right\}
\\&&\;\ldots\;\gamma \lt \omega_0(\xiが虚数の場合),\;\xi=\sqrt{\gamma^2-\omega_0^2}=\sqrt{\left|\gamma^2-\omega_0^2\right|}\;i=\omega i
\\&=&\frac{F}{k}e^{-\gamma t}
\left[
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)\left\{\cos{\left(\omega t\right)}+i\sin{\left(\omega t\right)}\right\}
+\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)\left\{\cos{\left(-\omega t\right)}+i\sin{\left(-\omega t\right)}\right\}
\right]
\\&=&\frac{F}{k}e^{-\gamma t}
\left[
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)\left\{\cos{\left(\omega t\right)}+i\sin{\left(\omega t\right)}\right\}
+\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)\left\{\cos{\left(\omega t\right)}-i\sin{\left(\omega t\right)}\right\}
\right]
\\&&\;\ldots\;\cos{\left(-\omega t\right)}=\cos{\left(\omega t\right)},\;\sin{\left(-\omega t\right)}=-\sin{\left(\omega t\right)}
\\&=&\frac{F}{k}e^{-\gamma t}
\left[
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)\cos{\left(\omega t\right)}
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)i\sin{\left(\omega t\right)}
+\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)\cos{\left(\omega t\right)}
-\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)i\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{k}e^{-\gamma t}
\left[
\left\{
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)
+\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)
\right\}\cos{\left(\omega t\right)}
+\left\{
-\left(\frac{\gamma}{2\omega i}+\frac{1}{2}\right)
-\left(\frac{\gamma}{2\omega i}-\frac{1}{2}\right)
\right\}i\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{k}e^{-\gamma t}
\left\{
-\cos{\left(\omega t\right)}
-\frac{\gamma}{\omega i}i\sin{\left(\omega t\right)}
\right\}
\\&=&-\frac{F}{k}e^{-\gamma t}
\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}\sin{\left(\omega t\right)}
\right\}
\;\ldots\;\frac{i}{i}=1
\end{eqnarray}$$
$$\begin{eqnarray}
\frac{\gamma}{\omega}
&=&\frac{\gamma}{\sqrt{\left|\gamma^2-\omega_0^2\right|}}
\;\ldots\;\omega=\sqrt{\left|\gamma^2-\omega_0^2\right|}
\\&=&\frac{\gamma}{\sqrt{
\frac{\omega_0^2}{\omega_0^2}
\left|
\gamma^2-\omega_0^2
\right|
}}
\\&=&\frac{\gamma}{\sqrt{
\omega_0^2
\left|
\frac{\gamma^2}{\omega_0^2}-\frac{\omega_0^2}{\omega_0^2}
\right|
}}
\\&=&\frac{\gamma}{\omega_0\sqrt{\left|\zeta^2-1\right|}}
\;\ldots\;\zeta=\frac{\gamma}{\omega_0}
\\&=&\frac{\zeta}{\sqrt{\left|\zeta^2-1\right|}}
\;\ldots\;\zeta=\frac{\gamma}{\omega_0}
\end{eqnarray}$$
逆ラプラス変換 第3項
$$\begin{eqnarray}
\\&&\frac{F}{m}C_5\mathfrak{L}^{-1}\left[\frac{1}{s}\right]
\\&=&\frac{F}{m}\frac{1}{\omega_0^2}\mathfrak{L}^{-1}\left[\frac{1}{s}\right]
\\&=&\frac{F}{m}\frac{1}{\omega_0^2}\cdot1
\;\ldots\;\mathfrak{L}^{-1}\left[ \frac{1}{s} \right]=1
\\&=&\frac{F}{k}
\\&&\;\ldots\;\omega_0=\sqrt{\frac{k}{m}},\;\omega_0^2=\frac{k}{m},\;m\omega_0^2=k
\end{eqnarray}$$
逆ラプラス変換 第1,2,3項
$$\begin{eqnarray}
x(t)&=&\left\{
C_1 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_2 \mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\right\}
+\frac{F}{m}\left\{
C_3\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+C_4\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\right\}
+\left\{
\frac{F}{m}C_5\mathfrak{L}^{-1}\left[\frac{1}{s}\right]
\right\}
\\&=& e^{-\gamma t}\left\{
x_0\cos{\left(\omega t\right)}
+\frac{v_0+\gamma x_0}{\omega}\sin{\left(\omega t\right)}
\right\}
-\frac{F}{k}e^{-\gamma t}
\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}\sin{\left(\omega t\right)}
\right\}
+\frac{F}{k}
\\&=& e^{-\gamma t}\left\{
x_0\cos{\left(\omega t\right)}
+\frac{v_0+\gamma x_0}{\omega}\sin{\left(\omega t\right)}
\right\}
+\frac{F}{k}\left[1-e^{-\gamma t}
\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}\sin{\left(\omega t\right)}
\right\}
\right]
\\&=&
\color{red}{
x_0e^{-\gamma t}\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}\sin{\left(\omega t\right)}
\right\}}
\color{blue}{
+ v_0e^{-\gamma t}\left\{
\frac{1}{\omega}\sin{\left(\omega t\right)}
\right\}
}
\color{green}{
+\frac{F}{k}\left[1-e^{-\gamma t}
\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}\sin{\left(\omega t\right)}
\right\}
\right]
}
\\&&\color{black}{\;\ldots\;}
\color{red}{第1項:初期位置による振動}
,\;\color{blue}{第2項:初期速度による振動}
,\;\color{green}{第3項:ステップ入力による振動}
\\&=&
\color{green}{
\frac{F}{k}
}
\color{red}{
+\left(x_0
-\frac{F}{k}\right)e^{-\gamma t}
\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}\sin{\left(\omega t\right)}
\right\}
}
\color{blue}{
+ v_0e^{-\gamma t}\left\{
\frac{1}{\omega}\sin{\left(\omega t\right)}
\right\}
}
\\&&\color{black}{\;\ldots\;}
\color{green}{第1項:ステップ入力により釣り合い位置が変化}
,\;\color{red}{第2項:初期位置と釣り合い位置の差を改めて初期位置として減衰振動}
,\;\color{blue}{第3項:初期速度による振動}
\end{eqnarray}$$
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