バネマスダンパー系
運動方程式
$$\begin{eqnarray}
m\ddot{x}
&+&c\dot{x}
&+&kx
&=&F\sin{\left(\omega_f t\right)}
\\
\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&\frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\frac{k}{m}x
&=&\frac{F}{m}\sin{\left(\omega_f t\right)}
\\
\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\omega_0^2x
&=&\frac{F}{m}\sin{\left(\omega_f t\right)}
\;\cdots\;\gamma=\frac{c}{2m},\;\omega_0^2=\frac{k}{m}
\end{eqnarray}$$
ラプラス変換
$$\begin{eqnarray}
\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t}
\right.&+&\left.2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
\right.&+&\left.\omega_0^2x \right]&=\mathfrak{L}\left[\frac{F}{m}\sin{\left(\omega_f t\right)}\right]
\\\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t} \right]
&+&\mathfrak{L}\left[ 2\gamma\frac{\mathrm{d}x}{\mathrm{d}t} \right]
&+&\mathfrak{L}\left[ \omega_0^2 x\right]
&=\mathfrak{L}\left[\frac{F}{m}\sin{\left(\omega_f t\right)}\right]
\\\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t} \right]
&+&2\gamma\mathfrak{L}\left[ \frac{\mathrm{d}x}{\mathrm{d}t} \right]
&+&\omega_0^2\mathfrak{L}\left[ x\right]
&=\mathfrak{L}\left[\frac{F}{m}\sin{\left(\omega_f t\right)}\right]
\\
s^2X-sx_0 -v_0
&+&2\gamma\left(sX-x_0 \right)
&+&\omega_0^2X
&=\frac{F}{m}\frac{\omega}{s^2+\omega_f^2}
\\
\\&&&&&\;\ldots\;\mathfrak{L}\left[x\right]=X
\\&&&&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/fracmathrmdfmathrmdt.html}{\mathfrak{L}\left[ \frac{\mathrm{d}x}{\mathrm{d}t}\right]
=s^2X-x_0,\;x_0=x(0)}
\\&&&&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/blog-post_62.html}{\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t}\right]
=s^2X-sx_0 -v_0,\;v_0=x'(0)}
\\&&&&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/sin.html}{\mathfrak{L}\left[\sin{\left(\omega_f t\right)}\right]=\frac{\omega_f}{s^2+\omega_f^2}}
\end{eqnarray}$$
Xについて解く
$$\begin{eqnarray}
s^2X+2\gamma sX+\omega_0^2X
&=&
sx_0 +v_0
+2\gamma x_0 +\frac{F}{m}\frac{\omega_f}{s^2+\omega_f^2}
\\
\left(s^2+2\gamma s+\omega_0^2\right)X
&=&
sx_0 +v_0 +2\gamma x_0 +\frac{F}{m}\frac{\omega_f}{s^2+\omega_f^2}
\\
X&=&\frac{sx_0 +v_0 +2\gamma x_0 }{s^2+2\gamma s+\omega_0^2}+\frac{F}{m}\frac{1}{s^2+2\gamma s+\omega_0^2}\frac{\omega_f}{s^2+\omega_f^2}
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}+\frac{F}{m}\frac{1}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}\frac{\omega_f}{s^2+\omega_f^2}
\end{eqnarray}$$
部分分数分解 準備
$$\begin{eqnarray}
\\X&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}+\frac{F}{m}\frac{1}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}\frac{\omega_f}{s^2+\omega_f^2}
\\&=&\frac{C_1 }{s-\lambda_1}+\frac{C_2 }{s-\lambda_2}
+\frac{F}{m}\left\{
\frac{C_3}{s-\lambda_1}
+\frac{C_4}{s-\lambda_2}
+\frac{C_5s+C_6}{s^2+\omega_f^2}
\right\}
\\&=&\frac{C_1 \left(s-\lambda_2\right)+C_2 \left(s-\lambda_1\right)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
\\&&+\scriptsize{\frac{F}{m}\left\{ \frac{
C_3\left(s-\lambda_2\right)\left(s^2+\omega_f^2\right)+C_4\left(s-\lambda_1\right)\left(s^2+\omega_f^2\right)+\left(C_5s+C_6\right)\left(s-\lambda_1\right)\left(s-\lambda_2\right)
}{ s\left(s-\lambda_1\right)\left(s-\lambda_2\right)\left(s^2+\omega_f^2\right)} \right\}
}
\\&=&\frac{C_1 \left(s-\lambda_2\right)+C_2 \left(s-\lambda_1\right)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
\\&&+\scriptsize{
\frac{F}{m}\left\{
\frac{
C_3\left(s^3+\omega_f^2s-\lambda_2 s^2 -\lambda_2\omega_f^2\right)
+C_4\left(s^3+\omega_f^2s-\lambda_1 s^2 -\lambda_1\omega_f^2\right)
+C_5\left\{s^3-\left(\lambda_1+\lambda_2\right)s^2+\lambda_1\lambda_2 s\right\}
+C_6\left\{s^2-\left(\lambda_1+\lambda_2\right)s+\lambda_1\lambda_2\right\}
}{ \left(s-\lambda_1\right)\left(s-\lambda_2\right)\left(s^2+\omega_f^2\right)}
\right\}
}
\\&=&\frac{(C_1 +C_2 )s-(C_1 \lambda_2+C_2 \lambda_1)}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
\\&&+\small{\frac{F}{m}\left[ \frac{
\left(C_3+C_4+C_5\right)s^3
+\left\{-C_3\lambda_2-C_4\lambda_1-C_5(\lambda_1+\lambda_2)+C_6\right\}s^2
+\left\{C_3\omega_f^2+C_4\omega_f^2+C_5\lambda_1\lambda_2-C_6(\lambda_1+\lambda_2)\right\}s
+\left(-C_3\lambda_2\omega_f^2 -C_4\lambda_1\omega_f^2 +C_6\lambda_1\lambda_2\right)
}
{ \left(s-\lambda_1\right)\left(s-\lambda_2\right)\left(s^2+\omega_f^2\right)} \right]
}
\end{eqnarray}$$
部分分数分解 第1項分子の係数比較
$$\begin{eqnarray}
sx_0 +v_0 +2\gamma x_0&=&\left(C_1 +C_2 \right)s-\left(C_1 \lambda_2+C_2 \lambda_1\right)
\end{eqnarray}$$
$$\left\{\begin{eqnarray}
x_0&=&C_1 +C_2
\\v_0 +2\gamma x_0&=&-\left(C_1 \lambda_2+C_2 \lambda_1\right)
\end{eqnarray}\right.$$
部分分数分解 \(C_2\)
$$\begin{eqnarray}
x_0&=&C_1+C_2
\\C_1&=&x_0-C_2
\\v_0+2\gamma x_0&=&-\left\{\left(x_0-C_2\right)\lambda_2+C_2\lambda_1\right\}
\\&=&-\lambda_2x_0+C_2\lambda_2-C_2\lambda_1
\\v_0+2\gamma x_0+\lambda_2 x_0&=&C_2\left(\lambda_2-\lambda_1\right)
\\C_2 &=&\frac{v_0+2\gamma x_0+\lambda_2 x_0}{\lambda_2-\lambda_1}
\\&&\;\ldots\;\lambda_{1,2}=\frac{-2\gamma\pm\sqrt{\left(2\gamma\right)^2-4\cdot1\cdot\omega_0^2}}{2\cdot1}=-\gamma\pm\sqrt{\gamma^2-\omega_0}=-\gamma\pm\xi
\\&&\;\ldots\;\lambda_2-\lambda_1=-\gamma-\xi -\left(-\gamma+\xi \right)=-2\xi
\\&=&\frac{v_0+2\gamma x_0+\left(-\gamma-\xi \right) x_0}{-2\xi }
\\&=&\frac{v_0+\gamma x_0-\xi x_0}{-2\xi }
\\&=&\frac{v_0+\gamma x_0}{-2\xi }-\frac{\xi x_0}{-2\xi }
\\&=&\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }
\end{eqnarray}$$
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項分子の係数比較
$$\begin{eqnarray}
\omega_f&=&\left(C_3+C_4+C_5\right)s^3
+\left\{-C_3\lambda_2-C_4\lambda_1-C_5(\lambda_1+\lambda_2)+C_6\right\}s^2
+\left\{C_3\omega_f^2+C_4\omega_f^2+C_5\lambda_1\lambda_2-C_6(\lambda_1+\lambda_2)\right\}s
+\left(-C_3\lambda_2\omega_f^2 -C_4\lambda_1\omega_f^2 +C_6\lambda_1\lambda_2\right)
\end{eqnarray}$$
$$\left\{\begin{eqnarray}
0&=&&C_3&+&C_4&+&C_5&
\\0&=&-\lambda_2&C_3&-\lambda_1&C_4&-(\lambda_1+\lambda_2)&C_5&+&C_6
\\0&=&\omega_f^2&C_3&+\omega_f^2&C_4&+\lambda_1\lambda_2&C_5&-(\lambda_1+\lambda_2)&C_6
\\\omega_f&=&-\lambda_2\omega_f^2&C_3&-\lambda_1\omega_f^2&C_4&&&+\lambda_1\lambda_2&C_6
\end{eqnarray}\right.$$
行列とベクトルで表現すると以下のようになる．
$$\begin{eqnarray}
\begin{bmatrix}
0\\0\\0\\\omega_f
\end{bmatrix}
&=&
\begin{bmatrix}
1&1&1&0
\\-\lambda_2&-\lambda_1&-(\lambda_1+\lambda_2)&1
\\\omega_f^2&\omega_f^2&\lambda_1\lambda_2&-(\lambda_1+\lambda_2)
\\-\lambda_2\omega_f^2&-\lambda_1\omega_f^2&0&\lambda_1\lambda_2
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\end{eqnarray}$$
行列とベクトルを以下の文字で表すとする．
$$\begin{eqnarray}
\boldsymbol{y}=\begin{bmatrix}
0\\0\\0\\\omega_f
\end{bmatrix}
,\;\boldsymbol{A}=\begin{bmatrix}
1&1&1&0
\\-\lambda_2&-\lambda_1&-(\lambda_1+\lambda_2)&1
\\\omega_f^2&\omega_f^2&\lambda_1\lambda_2&-(\lambda_1+\lambda_2)
\\-\lambda_2\omega_f^2&-\lambda_1\omega_f^2&0&\lambda_1\lambda_2
\end{bmatrix}
,\;\boldsymbol{x}=\begin{bmatrix}C_3\\C_4\\C_5\\C_6\end{bmatrix}
\end{eqnarray}$$
これを\(\boldsymbol{x}\)について解く．
$$\begin{eqnarray}
\\\boldsymbol{y}&=&\boldsymbol{A}\boldsymbol{x}
\\\boldsymbol{A}^{-1}\boldsymbol{y}&=&\boldsymbol{A}^{-1}\boldsymbol{A}\boldsymbol{x}
\\\boldsymbol{x}&=&\boldsymbol{A}^{-1}\boldsymbol{y}
\\\boldsymbol{x}&=&\frac{\tilde{\boldsymbol{A}}}{\left|\boldsymbol{A}\right|}\boldsymbol{y}
\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/05/blog-post_96.html}{\boldsymbol{A}^{-1}=\frac{\tilde{\boldsymbol{A}}}{\left|\boldsymbol{A}\right|}},\;\href{https://shikitenkai.blogspot.com/2021/05/blog-post_3.html}{\tilde{\boldsymbol{A}}は余因子行列}
\\\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}&=&
\frac{1}{\left|\boldsymbol{A}\right|}
\begin{bmatrix}
(-1)^{1+1}\left|\boldsymbol{M}_{11}\right|
& (-1)^{1+2}\left|\boldsymbol{M}_{21}\right|
& (-1)^{1+3}\left|\boldsymbol{M}_{31}\right|
& (-1)^{1+4}\left|\boldsymbol{M}_{41}\right|
\\(-1)^{2+1}\left|\boldsymbol{M}_{12}\right|
& (-1)^{2+2}\left|\boldsymbol{M}_{22}\right|
& (-1)^{2+3}\left|\boldsymbol{M}_{32}\right|
& (-1)^{2+4}\left|\boldsymbol{M}_{42}\right|
\\(-1)^{3+1}\left|\boldsymbol{M}_{13}\right|
& (-1)^{3+2}\left|\boldsymbol{M}_{23}\right|
& (-1)^{3+3}\left|\boldsymbol{M}_{33}\right|
& (-1)^{3+4}\left|\boldsymbol{M}_{43}\right|
\\(-1)^{4+1}\left|\boldsymbol{M}_{14}\right|
& (-1)^{4+2}\left|\boldsymbol{M}_{24}\right|
& (-1)^{4+3}\left|\boldsymbol{M}_{34}\right|
& (-1)^{4+4}\left|\boldsymbol{M}_{44}\right|
\end{bmatrix}
\begin{bmatrix}
0\\0\\0\\\omega_f
\end{bmatrix}
=
\frac{\omega_f}{\left|\boldsymbol{A}\right|}
\begin{bmatrix}
-\left|\boldsymbol{M}_{41}\right|
\\\;\;\;\left|\boldsymbol{M}_{42}\right|
\\-\left|\boldsymbol{M}_{43}\right|
\\\;\;\;\left|\boldsymbol{M}_{44}\right|
\end{bmatrix}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/05/blog-post_3.html}{\boldsymbol{M}_{ij}:元の行列\boldsymbol{A}からi行とj列を除いた行列(添え字の順序に注意)}
\end{eqnarray}$$
\(\left|\boldsymbol{M}_{3*}\right|\)及び\(\left|\boldsymbol{A}\right|\)の計算
$$\begin{eqnarray}
\left|\boldsymbol{M}_{41}\right|
&=&\begin{vmatrix}
1&1&0
\\-\lambda_1&-\left(\lambda_1+\lambda_2\right)&1
\\\omega_f^2&\lambda_1\lambda_2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
=\begin{vmatrix}
-\left(\lambda_1+\lambda_2\right)&1
\\\lambda_1\lambda_2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
-\begin{vmatrix}
-\lambda_1&1
\\\omega_f^2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
\\&=&\left\{-\left(\lambda_1+\lambda_2\right)\right\}\cdot\left\{-\left(\lambda_1+\lambda_2\right)\right\}-1\cdot\lambda_1\lambda_2
-\left[
-\lambda_1\cdot\left\{-\left(\lambda_1+\lambda_2\right)\right\}-1\cdot\omega_f^2
\right]
\\&=&\cancel{\lambda_1^2}\cancel{+2\lambda_1\lambda_2}+\lambda_2^2\cancel{-\lambda_1\lambda_2}\cancel{-\lambda_1^2}\cancel{-\lambda_1\lambda_2}+\omega_f^2
\\&=&\lambda_2^2+\omega_f^2
\\
\left|\boldsymbol{M}_{42}\right|
&=&\begin{vmatrix}
1&1&0
\\-\lambda_2&-\left(\lambda_1+\lambda_2\right)&1
\\\omega_f^2&\lambda_1\lambda_2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
=\begin{vmatrix}
-\left(\lambda_1+\lambda_2\right)&1
\\\lambda_1\lambda_2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
-\begin{vmatrix}
-\lambda_2&1
\\\omega_f^2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
\\&=&\left\{-\left(\lambda_1+\lambda_2\right)\right\}\cdot\left\{-\left(\lambda_1+\lambda_2\right)\right\}-1\cdot\lambda_1\lambda_2
-\left[
-\lambda_2\cdot\left\{-\left(\lambda_1+\lambda_2\right)\right\}-1\cdot\omega_f^2
\right]
\\&=&\lambda_1^2\cancel{+2\lambda_1\lambda_2}\cancel{+\lambda_2^2}\cancel{-\lambda_1\lambda_2}\cancel{-\lambda_1\lambda_2}\cancel{-\lambda_2^2}+\omega_f^2
\\&=&\lambda_1^2+\omega_f^2
\\
\left|\boldsymbol{M}_{43}\right|
&=&\begin{vmatrix}
1&1&0
\\-\lambda_2&-\lambda_1&1
\\\omega_f^2&\omega_f^2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
=\begin{vmatrix}
-\lambda_1&1
\\\omega_f^2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
-\begin{vmatrix}
-\lambda_2&1
\\\omega_f^2&-\left(\lambda_1+\lambda_2\right)
\end{vmatrix}
\\&=&-\lambda_1\cdot\left\{-\left(\lambda_1+\lambda_2\right)\right\}-1\cdot\omega_f^2
-\left[
-\lambda_2\cdot\left\{-\left(\lambda_1+\lambda_2\right)\right\}-1\cdot\omega_f^2
\right]
\\&=&\lambda_1^2\cancel{+\lambda_1\lambda_2}\cancel{-\omega_f^2}\cancel{-\lambda_1\lambda_2}-\lambda_2^2\cancel{+\omega_f}
\\&=&\lambda_1^2-\lambda_2^2
\\&=&\left(\lambda_1+\lambda_2\right)\left(\lambda_1-\lambda_2\right)
\\
\left|\boldsymbol{M}_{44}\right|
&=&\begin{vmatrix}
1&1&1
\\-\lambda_2&-\lambda_1&-(\lambda_1+\lambda_2)
\\\omega_f^2&\omega_f^2&\lambda_1\lambda_2
\end{vmatrix}
=\begin{vmatrix}
-\lambda_1&-(\lambda_1+\lambda_2)
\\\omega_f^2&\lambda_1\lambda_2
\end{vmatrix}
-\begin{vmatrix}
-\lambda_2&-(\lambda_1+\lambda_2)
\\\omega_f^2&\lambda_1\lambda_2
\end{vmatrix}
+\begin{vmatrix}
-\lambda_2&-\lambda_1
\\\omega_f^2&\omega_f^2
\end{vmatrix}
\\&=&
-\lambda_1\cdot\lambda_1\lambda_2-\left\{-\left(\lambda_1+\lambda_2\right)\right\}\cdot\omega_f^2
-\left[-\lambda_2\cdot\lambda_1\lambda_2-\left\{-\left(\lambda_1+\lambda_2\right)\right\}\cdot\omega_f^2\right]
+\left(-\lambda_2\right)\omega_f^2-\left(-\lambda_1\right)\omega_f^2
\\&=&
-\lambda_1^2\lambda_2\cancel{+\lambda_1\omega_f^2}\cancel{+\lambda_2\omega_f^2}
+\lambda_1\lambda_2^2\cancel{-\lambda_1\omega_f^2}\cancel{-\lambda_2\omega_f^2}
-\lambda_2\omega_f^2+\lambda_1\omega_f^2
\\&=&-\lambda_1^2\lambda_2+\lambda_1\lambda_2^2-\lambda_2\omega_f^2+\lambda_1\omega_f^2
\\&=&-\lambda_1\lambda_2\left(\lambda_1-\lambda_2\right)+\left(\lambda_1-\lambda_2\right)\omega_f^2
\\&=&-\left(\lambda_1\lambda_2-\omega_f^2\right)\left(\lambda_1-\lambda_2\right)
\\
\\\left|\boldsymbol{A}\right|
&=&
\color{red}{a_{41}(-1)^{4+1}\left|\boldsymbol{M}_{41}\right|}
\color{blue}{+a_{42}(-1)^{4+2}\left|\boldsymbol{M}_{42}\right|}
\color{green}{+a_{43}(-1)^{4+3}\left|\boldsymbol{M}_{43}\right|}
\color{magenta}{+a_{44}(-1)^{4+4}\left|\boldsymbol{M}_{44}\right|}
\\&=&
\color{red}{-\lambda_2\omega_f^2\cdot-1\cdot\left\{\left(\lambda_2^2+\omega_f^2\right)\right\}}
\color{blue}{-\lambda_1\omega_f^2\cdot1\cdot\left\{\left(\lambda_1^2+\omega_f^2\right)\right\}}
\color{green}{+0\cdot-1\cdot\left\{\left(\lambda_1^2-\lambda_2^2\right)\right\}}
\color{magenta}{+\lambda_1\lambda_2\cdot1\cdot\left\{\left(\omega_f^2-\lambda_1\lambda_2\right)\left(\lambda_1-\lambda_2\right)\right\}}
\\&=&
\color{red}{\lambda_2\omega_f^2\left(\lambda_2^2+\omega_f^2\right)}
\color{blue}{-\lambda_1\omega_f^2\left(\lambda_1^2+\omega_f^2\right)}
\color{magenta}{+\lambda_1\lambda_2\left\{
\left(\omega_f^2-\lambda_1\lambda_2\right)\left(\lambda_1-\lambda_2\right)
\right\}
}
\\&=&
\color{red}{\lambda_2\omega_f^2\left(\lambda_2^2+\omega_f^2\right)}
\color{blue}{-\lambda_1\omega_f^2\left(\lambda_1^2+\omega_f^2\right)}
\color{magenta}{
+\lambda_1\lambda_2\left(\omega_f^2-\lambda_1\lambda_2\right)\left(\lambda_1-\lambda_2\right)
}
\\&=&
\color{red}{\lambda_2\omega_f^2\left(\lambda_2^2+\omega_f^2\right)}
\color{blue}{-\lambda_1\omega_f^2\left(\lambda_1^2+\omega_f^2\right)}
\color{magenta}{
-\lambda_1^2\lambda_2^2\left(\lambda_1-\lambda_2\right)
+\lambda_1\lambda_2\omega_f^2\left(\lambda_1-\lambda_2\right)
}
\\&=&
\lambda_2\omega_f^2\left(\lambda_2^2+\omega_f^2+\lambda_1^2\right)
-\lambda_1\omega_f^2\left(\lambda_1^2+\omega_f^2+\lambda_2^2\right)
-\lambda_1^2\lambda_2^2\left(\lambda_1-\lambda_2\right)
\\&=&
\left(\lambda_2\omega_f^2-\lambda_1\omega_f^2\right)\left(\lambda_1^2+\lambda_2^2+\omega_f^2\right)
-\lambda_1^2\lambda_2^2\left(\lambda_1-\lambda_2\right)
\\&=&
-\omega_f^2\left(\lambda_1-\lambda_2\right)\left(\lambda_1^2+\lambda_2^2+\omega_f^2\right)
-\lambda_1^2\lambda_2^2\left(\lambda_1-\lambda_2\right)
\\&=&
-\left(\lambda_1-\lambda_2\right)\left(\omega_f^2\left(\lambda_1^2+\lambda_2^2+\omega_f^2\right)+\lambda_1^2\lambda_2^2\right)
\\&=&
-\left(\lambda_1-\lambda_2\right)\left(\lambda_1^2\omega_f^2+\lambda_2^2\omega_f^2+\omega_f^4+\lambda_1^2\lambda_2^2\right)
\\&=&
-\left(\lambda_1-\lambda_2\right)\left(\lambda_1^2+\omega_f^2\right)\left(\lambda_2^2+\omega_f^2\right)
\end{eqnarray}$$
部分分数分解 \(C_3,\;C_4,\;C_5,\;C_6\)
$$\begin{eqnarray}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}&=&
\frac{\omega_f}{\left|\boldsymbol{A}\right|}
\begin{bmatrix}
-\left|\boldsymbol{M}_{41}\right|
\\\;\;\;\left|\boldsymbol{M}_{42}\right|
\\-\left|\boldsymbol{M}_{43}\right|
\\\;\;\;\left|\boldsymbol{M}_{44}\right|
\end{bmatrix}
=
\omega_f
\begin{bmatrix}
\cancel{-}\frac{\cancel{\left(\lambda_2^2+\omega_f^2\right)}}
{\cancel{-}\left(\lambda_1-\lambda_2\right)\left(\lambda_1^2+\omega_f^2\right)\cancel{\left(\lambda_2^2+\omega_f^2\right)}}
\\\frac{\cancel{\left(\lambda_1^2+\omega_f^2\right)}}
{-\left(\lambda_1-\lambda_2\right)\cancel{\left(\lambda_1^2+\omega_f^2\right)}\left(\lambda_2^2+\omega_f^2\right)}
\\\cancel{-}\frac{\left(\lambda_1+\lambda_2\right)\cancel{\left(\lambda_1-\lambda_2\right)}}
{\cancel{-}\cancel{\left(\lambda_1-\lambda_2\right)}\left(\lambda_1^2+\omega_f^2\right)\left(\lambda_2^2+\omega_f^2\right)}
\\\frac{\cancel{-}\left(\lambda_1\lambda_2-\omega_f^2\right)\cancel{\left(\lambda_1-\lambda_2\right)}}
{\cancel{-}\cancel{\left(\lambda_1-\lambda_2\right)}\left(\lambda_1^2+\omega_f^2\right)\left(\lambda_2^2+\omega_f^2\right)}
\end{bmatrix}
=
\begin{bmatrix}
\frac{\omega_f}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)}
\\\frac{-\omega_f}{(\lambda_1 - \lambda_2)(\lambda_2^2 + \omega_f^2)}
\\\frac{\omega_f(\lambda_1 + \lambda_2)}{(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}
\\\frac{\omega_f(\lambda_1\lambda_2 - \omega_f^2)}{(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}
\end{bmatrix}
\end{eqnarray}$$
部分分数分解 まとめる
$$\begin{eqnarray}
X&=&
\frac{C_1}{s-\lambda_1}
+\frac{C_2}{s-\lambda_2}
+\frac{F}{m}\left(
\frac{C_3}{s-\lambda_1}
+\frac{C_4}{s-\lambda_2}
+\frac{C_5s+C_6}{s^2+\omega_f^2}
\right)
\\&=&
\left(
\frac{C_1}{s-\lambda_1}
+\frac{C_2}{s-\lambda_2}
\right)
+\frac{F}{m}\left(
\frac{C_3}{s-\lambda_1}
+\frac{C_4}{s-\lambda_2}
\right)
+\frac{F}{m}\left(
\frac{C_5s+C_6}{s^2+\omega_f^2}
\right)
\\&=&
\left(
\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_1}
+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_2}
\right)
+\frac{F}{m}\left(
\frac{\frac{\omega_f}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)}}{s-\lambda_1}
+\frac{\frac{-\omega_f}{(\lambda_1 - \lambda_2)(\lambda_2^2 + \omega_f^2)}}{s-\lambda_2}
\right)
+\frac{F}{m}\left(
\frac{\frac{\omega_f(\lambda_1 + \lambda_2)}{(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\;s
+\frac{\omega_f(\lambda_1\lambda_2 - \omega_f^2)}{(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}}{s^2+\omega_f^2}
\right)
\\&=&
\left\{
\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi}}{s-\lambda_1}
+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi}}{s-\lambda_2}
\right\}
\\&&+\frac{F}{m}\frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\frac{\omega_f(\lambda_2^2 + \omega_f^2)}{s-\lambda_1}
+\frac{-\omega_f(\lambda_1^2 + \omega_f^2)}{s-\lambda_2}
\right\}
\\&&+\frac{F}{m}\frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}
\frac{\omega_f\left(\lambda_1 + \lambda_2\right)\left(\lambda_1 - \lambda_2\right)s
+\omega_f\left(\lambda_1\lambda_2 - \omega_f^2\right)\left(\lambda_1 - \lambda_2\right)}
{s^2+\omega_f^2}
\end{eqnarray}$$
\(\lambda\)に関する幾つかの式を先に計算しておく
$$\begin{eqnarray}
\lambda_{1,2}&=&-\gamma\pm\sqrt{\gamma^2-\omega_0^2}
\\&=&-\gamma\pm\xi
\\\omega&=&\sqrt{\left|\gamma^2-\omega_0^2\right|}
\\\xi&=&\omega i\;\ldots\;\left(\gamma\lt\omega_0 の時\right)
\\\omega^2&=&-\left(\gamma^2-\omega_0^2\right)
\\&=&\omega_0^2-\gamma^2
\\\lambda_1 + \lambda_2&=&\left(-\gamma+\xi\right)+\left(-\gamma-\xi\right)
\\&=&-2\gamma
\\\lambda_1 - \lambda_2&=&\left(-\gamma+\xi\right)-\left(-\gamma-\xi\right)
\\&=&2\xi
\\
\\\lambda_1 \lambda_2&=&\left(-\gamma+\xi\right)\left(-\gamma-\xi\right)
\\&=&(-\gamma)^2-\xi^2
\\&=&\gamma^2-\xi^2
\\
\\\lambda_1^2&=&\left(-\gamma+\xi\right)^2
\\&=&(-\gamma)^2+2(-\gamma)\xi+\xi^2
\\&=&\gamma^2-2\gamma\xi+\xi^2
\\&=&\left(\gamma-\xi\right)^2
\\
\\\lambda_2^2&=&\left(-\gamma-\xi\right)^2
\\&=&(-\gamma)^2-2(-\gamma)\xi+\xi^2
\\&=&\gamma^2+2\gamma\xi+\xi^2
\\&=&\left(\gamma+\xi\right)^2
\\
\\\lambda_1^2+\lambda_2^2&=&(\gamma^2-2\gamma\xi+\xi^2)+(\gamma^2+2\gamma\xi+\xi^2)
\\&=&2(\gamma^2+\xi^2)
\\
\\\lambda_2^2-\lambda_1^2&=&(\gamma^2+2\gamma\xi+\xi^2)-(\gamma^2-2\gamma\xi+\xi^2)
\\&=&4\gamma\xi
\\
\\
\\
\\\left(\lambda_1^2 + \omega_f^2\right)\left(\lambda_2^2 + \omega_f^2\right)
&=&\lambda_1^2\lambda_2^2+\left(\lambda_1^2+\lambda_2^2\right)\omega_f^2+\omega_f^4
\\&=&\left(\gamma-\xi\right)^2\left(\gamma+\xi\right)^2+2(\gamma^2+\xi^2)\omega_f^2+\omega_f^4
\\&=&\left\{\left(\gamma-\xi\right)\left(\gamma+\xi\right)\right\}^2+2\gamma^2\omega_f^2+2\xi^2\omega_f^2+\omega_f^4
\\&=&\left(\gamma^2-\xi^2\right)^2+2\gamma^2\omega_f^2+2(\gamma^2-\omega_0^2)\omega_f^2+\omega_f^4
\\&=&\left(\omega_0^2\right)^2+2\gamma^2\omega_f^2+2\gamma^2\omega_f^2-2\omega_0^2\omega_f^2+\omega_f^4
\\&=&\omega_0^4+4\gamma^2\omega_f^2-2\omega_0^2\omega_f^2+\omega_f^4
\\&=&\left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2
\end{eqnarray}$$
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
\\&=& e^{-\gamma t}\left\{
x_0\cos{\left(\omega t\right)}
+\frac{v_0}{\omega }\sin{\left(\omega t\right)}
+\frac{\gamma x_0}{\omega }\sin{\left(\omega t\right)}
\right\}
\\&=& x_0e^{-\gamma t}\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma }{\omega }\sin{\left(\omega t\right)}
\right\}
+v_0e^{-\gamma t}\left\{
\frac{1}{\omega }\sin{\left(\omega t\right)}
\right\}
\;\ldots\;初期位置x_0による項と初期速度v_0による項
\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}
&&\mathfrak{L}^{-1}\left[ \frac{F}{m}\frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\frac{\omega_f(\lambda_2^2 + \omega_f^2)}{s-\lambda_1}
+\frac{-\omega_f(\lambda_1^2 + \omega_f^2)}{s-\lambda_2}
\right\}
\right]
\\&=&\frac{F}{m}\frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\omega_f(\lambda_2^2 + \omega_f^2)\mathfrak{L}^{-1}\left[ \frac{1}{s-\lambda_1} \right]
-\omega_f(\lambda_1^2 + \omega_f^2)\mathfrak{L}^{-1}\left[ \frac{1}{s-\lambda_2} \right]
\right\}
\\&=&\frac{F}{m}\frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)e^{\lambda_1 t}
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)e^{\lambda_2 t}
\right\}
\\&=&\frac{F}{m}\frac{1}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)e^{(-\gamma+\omega i) t}
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)e^{(-\gamma-\omega i) t}
\right\}
\\&=&\frac{F}{m}\frac{1}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)e^{-\gamma t}e^{\omega i t}
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)e^{-\gamma t}e^{-\omega i t}
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)e^{\omega i t}
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)e^{-\omega i t}
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)\left(\cos{\left(\omega t\right)}+i\sin{\left(\omega t\right)}\right)
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)\left(\cos{\left(-\omega t\right)}+i\sin{\left(-\omega t\right)}\right)
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)\left(\cos{\left(\omega t\right)}+i\sin{\left(\omega t\right)}\right)
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)\left(\cos{\left(\omega t\right)}-i\sin{\left(\omega t\right)}\right)
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\left(\omega_f\lambda_2^2 + \omega_f^3\right)\cos{\left(\omega t\right)}
+\left(\omega_f\lambda_2^2 + \omega_f^3\right)i\sin{\left(\omega t\right)}
-\left(\omega_f\lambda_1^2 + \omega_f^3\right)\cos{\left(\omega t\right)}
+\left(\omega_f\lambda_1^2 + \omega_f^3\right)i\sin{\left(\omega t\right)}
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
\left\{\left(\omega_f\lambda_2^2 + \omega_f^3\right)-\left(\omega_f\lambda_1^2 + \omega_f^3\right)\right\}\cos{\left(\omega t\right)}
+\left\{\left(\omega_f\lambda_2^2 + \omega_f^3\right)+\left(\omega_f\lambda_1^2 + \omega_f^3\right)\right\}i\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
\omega_f\left\{\left(\lambda_2^2 + \omega_f^2\right)-\left(\lambda_1^2 + \omega_f^2\right)\right\}\cos{\left(\omega t\right)}
+\omega_f\left\{\left(\lambda_2^2 + \omega_f^2\right)+\left(\lambda_1^2 + \omega_f^2\right)\right\}i\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\omega_f\left(\lambda_2^2 - \lambda_1^2\right)\cos{\left(\omega t\right)}
+\omega_f\left(\lambda_1^2 + \lambda_2^2 + 2\omega_f^2\right)i\sin{\left(\omega t\right)}
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
4\gamma\omega_f\cos{\left(\omega t\right)}
+\omega_f\left\{
2\left(\gamma^2+\xi^2\right)
+ 2\omega_f^2
\right\}i\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{2 \xi \left(2\gamma\omega_f\right)+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
\omega_f\left(4\gamma\xi\right)\cos{\left(\omega t\right)}
+2\omega_f i\left\{
\left(\gamma^2+\xi^2\right)
+ \omega_f^2
\right\}\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
\frac{4\gamma\omega_f\xi}{2\xi}\cos{\left(\omega t\right)}
+\frac{2\omega_f i}{2\xi}
\left\{
\gamma^2
\left(\gamma^2-\omega_0^2\right)
+ \omega_f^2
\right\}\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
2\gamma\omega_f\cos{\left(\omega t\right)}
+\frac{\omega_f i}{\xi}\left(2\gamma^2-\omega_0^2 + \omega_f^2\right)\sin{\left(\omega t\right)}
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
2\gamma\omega_f\cos{\left(\omega t\right)}
+\frac{\omega_f i}{\omega i}\left\{2\gamma^2-\left(\omega_0 ^2 - \omega_f^2\right)\right\}\sin{\left(\omega t\right)}
\right]
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}
\left[
2\gamma\omega_f\cos{\left(\omega t\right)}
+\frac{\omega_f}{\omega}\left\{
2\gamma^2-\left(\omega_0 ^2 - \omega_f^2\right)
\right\}\sin{\left(\omega t\right)}
\right]
\end{eqnarray}$$
逆ラプラス変換 第3項
$$\begin{eqnarray}
&&\mathfrak{L}^{-1}\left[\frac{F}{m} \frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\frac{
\omega_f\left(\lambda_1 + \lambda_2\right)\left(\lambda_1 - \lambda_2\right)s
+\omega_f\left(\lambda_1\lambda_2 - \omega_f^2\right)\left(\lambda_1 - \lambda_2\right)
}
{s^2+\omega_f^2}
\right\}
\right]
\\&=&\frac{F}{m} \frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}
\mathfrak{L}^{-1}\left[
\frac{\omega_f\left(\lambda_1 + \lambda_2\right)\left(\lambda_1 - \lambda_2\right)s}{s^2+\omega_f^2}
+\frac{\omega_f\left(\lambda_1\lambda_2 - \omega_f^2\right)\left(\lambda_1 - \lambda_2\right)}{s^2+\omega_f^2}
\right]
\\&=&\frac{F}{m}\frac{1}{(\lambda_1 - \lambda_2)(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\omega_f\left(\lambda_1 + \lambda_2\right)\left(\lambda_1 - \lambda_2\right)\mathfrak{L}^{-1}\left[ \frac{s}{s^2+\omega_f^2} \right]
+\left(\lambda_1\lambda_2 - \omega_f^2\right)\left(\lambda_1 - \lambda_2\right)\mathfrak{L}^{-1}\left[ \frac{\omega_f}{s^2+\omega_f^2} \right]
\right\}
\\&=&\frac{F}{m}\frac{1}{(\lambda_1^2 + \omega_f^2)(\lambda_2^2 + \omega_f^2)}\left\{
\omega_f\left(\lambda_1 + \lambda_2\right)\mathfrak{L}^{-1}\left[ \frac{s}{s^2+\omega_f^2} \right]
+\left(\lambda_1\lambda_2 - \omega_f^2\right)\mathfrak{L}^{-1}\left[ \frac{\omega_f}{s^2+\omega_f^2} \right]
\right\}
\\&=&\frac{F}{m}\frac{e^{-\gamma t}}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
\omega_f(–2\gamma)\cos{\left(\omega_f t\right)}
+\left\{\left(\omega_0^2\right)-\omega_f^2\right\}\sin{\left(\omega_f t\right)}
\right\}
\\&&\;\ldots\;\lambda_1 + \lambda_2
=-2\gamma
\\&&\;\ldots\;\lambda_1 - \lambda_2
=2\xi
=2\omega i
\;\left(\omega=\sqrt{\left|\gamma^2-\omega_0^2\right|},\;\gamma\lt\omega_0\right)
\\&&\;\ldots\;\lambda_1\lambda_2
=\gamma^2-\xi^2
=\gamma^2-(\omega i)^2
=\gamma^2+\omega^2
=\omega_0^2
\\&=&\frac{F}{m}\frac{1}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}\left\{
-2\gamma\omega_f\cos{\left(\omega_f t\right)}
+\left(\omega_0^2-\omega_f^2\right)\sin{\left(\omega_f t\right)}
\right\}
\end{eqnarray}$$
逆ラプラス変換 第1,2,3項
$$\begin{eqnarray}
x(t)&=&
\color{red}{
x_0e^{-\gamma t}\left\{
\cos{\left(\omega t\right)}
+\frac{\gamma }{\omega }\sin{\left(\omega t\right)}
\right\}
}&\ldots初期位置による振動\;振幅にe^{-\gamma t}があるのでt\rightarrow\inftyで消える
\\&+&
\color{blue}{
v_0e^{-\gamma t}\left\{
\frac{1}{\omega }\sin{\left(\omega t\right)}
\right\}
}&\ldots初期速度による振動\;振幅にe^{-\gamma t}があるのでt\rightarrow\inftyで消える
\\&+&
\color{green}{
\frac{F}{m}\frac{e^{-\gamma t}}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}
\left\{
2\gamma\omega_f\cos{\left(\omega t\right)}
+\frac{\omega_f}{\omega}\left(2\gamma^2-(\omega_0 ^2 - \omega_f^2)\right)\sin{\left(\omega t\right)}
\right\}
}&\ldots過渡応答による振動\;振幅にe^{-\gamma t}があるのでt\rightarrow\inftyで消える
\\&+&
\color{magenta}{
\frac{F}{m}\frac{1}{\left(2\gamma\omega_f\right)^2+\left(\omega_0^2-\omega_f^2\right)^2}\left\{
-2\gamma\omega_f\cos{\left(\omega_f t\right)}
+\left(\omega_0^2-\omega_f^2\right)\sin{\left(\omega_f t\right)}
\right\}
}&\ldots強制振動による振動\;振幅にe^{-\gamma t}がないのでt\rightarrow\inftyでも残る
\end{eqnarray}$$
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