バネマスダンパー系
運動方程式
$$\begin{eqnarray}
m\ddot{x}
&+&c\dot{x}
&+&kx
&=&F\cos{\left(\omega_f t\right)}
\\
\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&\frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\frac{k}{m}x
&=&\frac{F}{m}\cos{\left(\omega_f t\right)}
\\
\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\omega_0^2x
&=&\frac{F}{m}\cos{\left(\omega_f t\right)}
\;\cdots\;\gamma=\frac{c}{2m},\;\omega_0^2=\frac{k}{m}
\end{eqnarray}$$
ラプラス変換
$$\begin{eqnarray}
\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t}
\right.&+&\left.2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
\right.&+&\left.\omega_0^2x \right]&=\mathfrak{L}\left[\frac{F}{m}\cos{\left(\omega_f t\right)}\right]
\\\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t} \right]
&+&\mathfrak{L}\left[ 2\gamma\frac{\mathrm{d}x}{\mathrm{d}t} \right]
&+&\mathfrak{L}\left[ \omega_0^2 x\right]
&=\mathfrak{L}\left[\frac{F}{m}\cos{\left(\omega_f t\right)}\right]
\\\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t} \right]
&+&2\gamma\mathfrak{L}\left[ \frac{\mathrm{d}x}{\mathrm{d}t} \right]
&+&\omega_0^2\mathfrak{L}\left[ x\right]
&=\mathfrak{L}\left[\frac{F}{m}\cos{\left(\omega_f t\right)}\right]
\\
s^2X-sx_0 -v_0
&+&2\gamma\left(sX-x_0 \right)
&+&\omega_0^2X
&=\frac{F}{m}\frac{s}{s^2+\omega_f^2}
\\
\\&&&&&\;\ldots\;\mathfrak{L}\left[x\right]=X
\\&&&&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/fracmathrmdfmathrmdt.html}{\mathfrak{L}\left[ \frac{\mathrm{d}x}{\mathrm{d}t}\right]
=s^2X-x_0,\;x_0=x(0)}
\\&&&&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/blog-post_62.html}{\mathfrak{L}\left[ \frac{\mathrm{d^2}x}{\mathrm{d^2}t}\right]
=s^2X-sx_0 -v_0,\;v_0=x'(0)}
\\&&&&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/05/cos.html}{\mathfrak{L}\left[\cos{\left(\omega_f t\right)}\right]=\frac{s}{s^2+\omega_f^2}}
\end{eqnarray}$$
Xについて解く
$$\begin{eqnarray}
s^2X+2\gamma Xs+\omega_0^2X
&=&
sx_0 +v_0
+2\gamma x_0 +\frac{F}{m}\frac{s}{s^2+\omega_f^2}
\\
\left(s^2+2\gamma s+\omega_0^2\right)X
&=&
sx_0 +v_0 +2\gamma x_0 +\frac{F}{m}\frac{s}{s^2+\omega_f^2}
\\
X&=&\frac{sx_0 +v_0 +2\gamma x_0 }{s^2+2\gamma s+\omega_0^2}
+\frac{F}{m}\frac{1}{s^2+2\gamma s+\omega_0^2}\frac{s}{s^2+\omega_f^2}
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\frac{1}{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}\frac{s}{s^2+\omega_f^2}
\\&&\;\ldots\;\left(s-\lambda_1\right)\left(s-\lambda_2\right)=s^2+2\gamma s+\omega_0^2
\\&&\;\ldots\;\lambda_{1,2}=\href{https://shikitenkai.blogspot.com/2020/11/blog-post.html}{\frac{-2\gamma\pm\sqrt{\left(2\gamma\right)^2-4\cdot1\cdot\omega_0^2}}{2\cdot1}}=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}=-\gamma\pm\xi
\end{eqnarray}$$
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{s}{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}{s^2-\left(\lambda_1+\lambda_2\right)s+\lambda_1\lambda_2}\frac{s}{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{s}{s^4-\left(\lambda_1+\lambda_2\right)s^3+\lambda_1\lambda_2s^2+\omega_f ^2s^2-\omega_f ^2\left(\lambda_1+\lambda_2\right)s+\omega_f ^2\lambda_1\lambda_2}
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\frac{s}{s^4-\left(\lambda_1+\lambda_2\right)s^3+\left(\lambda_1\lambda_2+\omega_f ^2\right)s^2-\omega_f ^2\left(\lambda_1+\lambda_2\right)s+\omega_f ^2\lambda_1\lambda_2}
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\frac{s}{\left\{\left(s-a\right)^2+b^2\right\}\left\{\left(s-c\right)^2+d^2\right\}}
\\&=&\frac{sx_0 +v_0 +2\gamma x_0 }{\left(s-\lambda_1\right)\left(s-\lambda_2\right)}
+\frac{F}{m}\left\{
C_A\frac{s-a+b}{\left(s-a\right)^2+b^2} + C_B \frac{s-c+d}{\left(s-c\right)^2+d^2}
\right\}
\\&=&\frac{C_1}{s-\lambda_1}+\frac{C_2}{s-\lambda_2}
+\frac{F}{m}\left\{
C_3 \frac{s-a}{\left(s-a\right)^2+b^2}
+ C_4 \frac{b} {\left(s-a\right)^2+b^2}
+ C_5 \frac{s-c}{\left(s-c\right)^2+d^2}
+ C_6 \frac{d} {\left(s-c\right)^2+d^2}
\right\}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/e-at.html}{\mathfrak{L}\left[ e^{a t} \right]=\frac{1}{s-a}}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/05/eatcos.html}{\mathfrak{L}\left[ e^{a t}\cos{\left( b t \right) } \right]=\frac{s-a}{(s-a)^2+b^2}}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/05/eatsin.html}{\mathfrak{L}\left[ e^{a t}\sin{\left( b t \right) } \right]=\frac{b}{(s-a)^2+b^2}}
\\&=&\frac{C_1\left(s-\lambda_2\right)+C_2\left(s-\lambda_1\right)}{\left(s-\lambda_2\right)\left(s-\lambda_1\right)}
+\frac{F}{m}\left\{
\frac{C_3\left(s-a\right)+C_4b}{\left(s-a\right)^2+b^2} + \frac{C_5\left(s-c\right)+C_6d}{\left(s-c\right)^2+d^2}
\right\}
\\&=&\frac{\left(C_1+C_2\right)s-\left(C_1\lambda_2+C_2\lambda_1\right)}{\left(s-\lambda_2\right)\left(s-\lambda_1\right)}
+\frac{F}{m}\frac{
\left\{C_3\left(s-a\right)+C_4b\right\}\left\{\left(s-c\right)^2+d^2\right\}
+\left\{C_5\left(s-c\right)+C_6d\right\}\left\{\left(s-a\right)^2+b^2\right\}
}{\left\{\left(s-a\right)^2+b^2\right\}\left\{\left(s-c\right)^2+d^2\right\}}
\end{eqnarray}$$
部分分数分解 第1項分子の係数比較
$$\begin{eqnarray}
sx_0 +v_0 +2\gamma x_0&=&\left(C_1 +C_2 \right)s-\left(C_1 \lambda_2+C_2 \lambda_1\right)
\end{eqnarray}$$
$$\left\{\begin{eqnarray}
x_0&=&C_1 +C_2
\\v_0 +2\gamma x_0&=&-\left(C_1 \lambda_2+C_2 \lambda_1\right)
\end{eqnarray}\right.$$
解く
$$\begin{eqnarray}
\begin{bmatrix}
x_0\\v_0 +2\gamma x_0
\end{bmatrix}
&=&
\begin{bmatrix}
1&1
\\-\lambda_2&-\lambda_1
\end{bmatrix}
\begin{bmatrix}
C_1\\C_2
\end{bmatrix}
\\&=&\boldsymbol{A}
\begin{bmatrix}
C_1\\C_2
\end{bmatrix}
\\\boldsymbol{A}^{-1}
&=&\href{https://shikitenkai.blogspot.com/2021/05/blog-post_96.html}{\frac{1}{\begin{vmatrix}\boldsymbol{A}\end{vmatrix}}\tilde{\boldsymbol{A}}}
\\&=&\frac{1}{\begin{vmatrix}\boldsymbol{A}\end{vmatrix}}
\href{https://shikitenkai.blogspot.com/2021/05/blog-post_3.html}{
\begin{bmatrix}
(-1)^{1+1}\left|\boldsymbol{M}_{11}\right|
&(-1)^{1+2}\left|\boldsymbol{M}_{21}\right|
\\
(-1)^{2+1}\left|\boldsymbol{M}_{12}\right|
&(-1)^{2+2}\left|\boldsymbol{M}_{22}\right|
\end{bmatrix}}
\\&=&
\frac{1}{\left(1\cdot-\lambda_1\right)-\left(1\cdot-\lambda_2\right)}
\begin{bmatrix}
(-1)^2\cdot-\lambda_1&(-1)^3\cdot1
\\(-1)^3\cdot-\lambda_2&(-1)^2\cdot1
\end{bmatrix}
\\&=&
\frac{1}{\lambda_2-\lambda_1}
\begin{bmatrix}
-\lambda_1&-1
\\\lambda_2&1
\end{bmatrix}
\\&=&
\frac{1}{(-\gamma-\xi)-(-\gamma+\xi)}
\begin{bmatrix}
-(-\gamma+\xi)&-1
\\(-\gamma-\xi)&1
\end{bmatrix}
\;\ldots\;\lambda_{1,2}=-\gamma\pm\xi
\\&=&
\frac{1}{-2\xi}
\begin{bmatrix}
\gamma-\xi&-1
\\-\gamma-\xi&1
\end{bmatrix}
\\\begin{bmatrix}
C_1\\C_2
\end{bmatrix}
&=&\boldsymbol{A}^{-1}
\begin{bmatrix}
x_0\\v_0 +2\gamma x_0
\end{bmatrix}
\\&=&\frac{1}{-2\xi}
\begin{bmatrix}
\gamma-\xi&-1
\\-\gamma-\xi&1
\end{bmatrix}
\begin{bmatrix}
x_0\\v_0 +2\gamma x_0
\end{bmatrix}
\\&=&\frac{1}{-2\xi}
\begin{bmatrix}
(\gamma-\xi) \cdot x_0 + -1 \cdot (v_0 +2\gamma x_0)
\\(-\gamma-\xi) \cdot x_0 + 1 \cdot (v_0 +2\gamma x_0)
\end{bmatrix}
\\&=&\frac{1}{-2\xi}
\begin{bmatrix}
\gamma x_0-\xi x_0 - v_0 - 2\gamma x_0
\\-\gamma x_0-\xi x_0 + v_0 +2\gamma x_0
\end{bmatrix}
\\&=&\frac{1}{-2\xi}
\begin{bmatrix}
-\xi x_0 - v_0 - \gamma x_0
\\-\xi x_0 + v_0 + \gamma x_0
\end{bmatrix}
\\&=&
\begin{bmatrix}
\frac{x_0}{2} + \frac{v_0 + \gamma x_0}{2\xi}
\\\frac{x_0}{2} - \frac{v_0 + \gamma x_0}{2\xi}
\end{bmatrix}
\end{eqnarray}$$
\(C_1, C_2\)
$$\begin{eqnarray}
\begin{bmatrix}
C_1\\C_2
\end{bmatrix}&=&
\begin{bmatrix}
\frac{x_0}{2} + \frac{v_0 + \gamma x_0}{2\xi}
\\\frac{x_0}{2} - \frac{v_0 + \gamma x_0}{2\xi}
\end{bmatrix}
\end{eqnarray}$$
部分分数分解 第2項分母の係数比較
$$\begin{eqnarray}
\\&&s^4-\left(\lambda_1+\lambda_2\right)s^3
+\left(\lambda_1\lambda_2+\omega_f ^2\right)s^2
-\omega_f ^2\left(\lambda_1+\lambda_2\right)s
+\omega_f ^2\lambda_1\lambda_2
\\&=&\left\{\left(s-a\right)^2+b^2\right\}\left\{\left(s-c\right)^2+d^2\right\}
\\&=&\left(s^2-2as+a^2+b^2\right)\left(s^2-2cs+c^2+d^2\right)
\\&=&s^4-2cs^3+(c^2+d^2)s^2-2as^3+4ac s^2-2as(c^2+d^2)+(a^2+b^2)s^2-2(a^2+b^2)cs+(a^2+b^2)(c^2+d^2)
\\&=&s^4-2(a+c)s^3+(a^2+b^2+c^2+d^2+4ac)s^2-2(ac^2+ad^2+a^2c+b^2c)s+a^2c^2+a^2d^2+b^2c^2+b^2d^2
\end{eqnarray}$$
$$
\left\{\begin{eqnarray}
-2(a+c)&=&-\left(\lambda_1+\lambda_2\right)
\\a^2+b^2+c^2+d^2+4ac&=&\lambda_1\lambda_2+\omega_f ^2
\\-2(ac^2+ad^2+a^2c+b^2c)&=&-\omega_f ^2\left(\lambda_1+\lambda_2\right)
\\a^2c^2+a^2d^2+b^2c^2+b^2d^2&=&\omega_f ^2\lambda_1\lambda_2
\end{eqnarray}
\right.$$
解く
$$\begin{eqnarray}
a&=&0を仮定
\\c&=&\frac{\lambda_1+\lambda_2}{2}-a&\;\ldots\;一つ目の式より
\\&=&\frac{\lambda_1+\lambda_2}{2}-0&\;\ldots\;a=0を代入
\\&=&\frac{\lambda_1+\lambda_2}{2}
\\\omega_f^2\frac{\lambda_1+\lambda_2}{2}&=&ac^2+ad^2+a^2c+b^2c&\;\ldots\;三つ目の式より
\\&=&0\cdot c^2+0\cdot d^2+0^2\cdot c+b^2c&\;\ldots\;a=0を代入
\\&=&b^2c
\\&=&b^2\frac{\lambda_1+\lambda_2}{2}&\;\ldots\;c=\frac{\lambda_1+\lambda_2}{2}を代入
\\\omega_f^2&=&b^2&\;\ldots\;両辺\frac{\lambda_1+\lambda_2}{2}で割る
\\b&=&\omega_f&\;\ldots\;角周波数なので正を利用
\\a^2c^2+a^2d^2+b^2c^2+b^2d^2&=&\omega_f^2\lambda_1\lambda_2&\;\ldots\;四つ目の式
\\0^2c^2+0^2d^2+b^2c^2+b^2d^2&=&&\;\ldots\;a=0を代入
\\b^2\left(c^2+d^2\right)&=&&
\\\omega_f^2\left(c^2+d^2\right)&=&&\;\ldots\;b=\omega_fを代入
\\c^2+d^2&=&\lambda_1\lambda_2\;\ldots\;両辺\omega_f^2で割る
\\d^2&=&\lambda_1\lambda_2-c^2
\\&=&\lambda_1\lambda_2-\left(\frac{\lambda_1+\lambda_2}{2}\right)^2&\;\ldots\;c=\frac{\lambda_1+\lambda_2}{2}を代入
\\&=&\frac{1}{4}\left\{4\lambda_1\lambda_2-\left(\lambda_1+\lambda_2\right)^2\right\}
\\&=&\frac{1}{4}\left(4\lambda_1\lambda_2-\lambda_1^2-2\lambda_1\lambda_2-\lambda_2^2\right)
\\&=&-\frac{1}{4}\left(\lambda_1^2-2\lambda_1\lambda_2+\lambda_2^2\right)
\\&=&-\frac{\left(\lambda_1-\lambda_2\right)^2}{4}
\\&=&-\left(\frac{\left(\lambda_1-\lambda_2\right)}{2}\right)^2
\\d&=&\sqrt{-\left(\frac{\left(\lambda_1-\lambda_2\right)}{2}\right)^2}&\;\ldots\;角周波数なので正を利用
\\&=& \sqrt{-1}\frac{\lambda_1-\lambda_2}{2}
\\&=& i\frac{\lambda_1-\lambda_2}{2}
\end{eqnarray}$$
$$ \left\{
\begin{eqnarray}
a&=&0
\\b&=&\omega_f
\\c&=&\frac{\lambda_1+\lambda_2}{2}=\frac{(-\gamma+\xi)+(-\gamma-\xi)}{2}=\frac{-2\gamma}{2}=-\gamma
\\d&=&i\frac{\lambda_1-\lambda_2}{2}=i\frac{(-\gamma+\xi)-(-\gamma-\xi)}{2}=i\frac{2\xi}{2}=i\xi
\end{eqnarray}
\right.$$
部分分数分解 第2項分子の係数比較
$$\begin{eqnarray}
s &=&\left\{C_3\left(s-a\right)+C_4b\right\}\left\{\left(s-c\right)^2+d^2\right\}
+\left\{C_5\left(s-c\right)+C_6d\right\}\left\{\left(s-a\right)^2+b^2\right\}
\\&=&\left\{C_3s-C_3a+C_4b\right\}\left\{s^2-2cs+c^2+d^2\right\}
+\left\{C_5s-C_5c+C_6d\right\}\left\{s^2-2as+a^2+b^2\right\}
\\&=& C_3s^3 -2C_3cs^2 +C_3(c^2+d^2)s
-C_3as^2 +2C_3acs -C_3a(c^2+d^2)
+C_4bs^2 -2C_4bcs +C_4b(c^2+d^2)
\\&& +C_5s^3 -2C_5as^2 +C_5(a^2+b^2)s
-C_5cs^2 +2C_5acs -C_5c(a^2+b^2)
+C_6ds^2 -2C_6ads +C_6d(a^2+b^2)
\\&=&C_3s^3 +C_5s^3
\\&&-2C_3cs^2-C_3as^2+C_4bs^2 -2C_5as^2-C_5cs^2+C_6ds^2
\\&&+C_3(c^2+d^2)s+2C_3acs-2C_4bcs +C_5(a^2+b^2)s+2C_5acs-2C_6ads
\\&&-C_3a(c^2+d^2)+C_4b(c^2+d^2) -C_5c(a^2+b^2)+C_6d(a^2+b^2)
\\&=&(C_3+C_5)s^3
\\&&+(-2C_3c-C_3a+C_4b-2C_5a-C_5c+C_6d)s^2
\\&&+(C_3(c^2+d^2)+2C_3ac-2C_4bc+C_5(a^2+b^2)+2C_5ac-2C_6ad)s
\\&&+\left\{-C_3a(c^2+d^2)+C_4b(c^2+d^2)-C_5c(a^2+b^2)+C_6d(a^2+b^2)\right\}
\\&=&(C_3+C_5)s^3
\\&&+(C_3(-2c-a)+C_4b+C_5(-2a-c)+C_6d)s^2
\\&&+(C_3((c^2+d^2)+2ac)+C_4(-2bc)+C_5((a^2+b^2)+2ac)+C_6(-2ad))s
\\&&+\left\{C_3(-a(c^2+d^2))+C_4b(c^2+d^2)+C_5(-c(a^2+b^2))+C_6d(a^2+b^2)\right\}
\end{eqnarray}$$
$$\left\{
\begin{eqnarray}
0&=&C_3+C_5
\\0&=&C_3(-2c-a)+C_4b+C_5(-2a-c)+C_6d
\\1&=&C_3(c^2+d^2+2ac)+C_4(-2bc)+C_5((a^2+b^2)+2ac)+C_6(-2ad)
\\0 &=&C_3(-a(c^2+d^2))+C_4b(c^2+d^2)+C_5(-c(a^2+b^2))+C_6d(a^2+b^2)
\end{eqnarray}
\right.$$
解く
$$\begin{eqnarray}
\begin{bmatrix}
0\\0\\1\\0
\end{bmatrix}
&=&
\begin{bmatrix}
1&0&1&0
\\-2c-a&b&-2a-c&d
\\c^2+d^2+2ac&-2bc&(a^2+b^2)+2ac&-2ad
\\-a(c^2+d^2)&b(c^2+d^2)&-c(a^2+b^2)&d(a^2+b^2)
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\\&=&
\begin{bmatrix}
1&0&1&0
\\-2c-0&b&-2\cdot0-c&d
\\c^2+d^2+2\cdot 0\cdot c&-2bc&(0^2+b^2)+2\cdot 0\cdot c&-2\cdot 0\cdot d
\\-0(c^2+d^2)&b(c^2+d^2)&-c(0^2+b^2)&d(0^2+b^2)
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\\&=&
\begin{bmatrix}
1&0&1&0
\\-2c&b &-c&d
\\c^2+d^2&-2bc&b^2&0
\\0&b(c^2+d^2)&-b^2c&b^2d
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\\&=&
\begin{bmatrix}
1&0&1&0
\\-2(-\gamma)&(\omega_f) &-(-\gamma)&(i\xi)
\\(-\gamma)^2+(i\xi)^2&-2(\omega_f)(-\gamma)&(\omega_f)^2&0
\\0&(\omega_f)((-\gamma)^2+(i\xi)^2)&-(\omega_f)^2(-\gamma)&(\omega_f)^2(i\xi)
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\\&=&
\begin{bmatrix}
1&0&1&0
\\2\gamma&\omega_f&\gamma&i\xi
\\\gamma^2-\xi^2&2\gamma\omega_f&\omega_f^2&0
\\0&\omega_f(\gamma^2-\xi^2)&\gamma\omega_f^2&i\xi\omega_f^2
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\\&=&
\begin{bmatrix}
1&0&1&0
\\2\gamma&\omega_f&\gamma&i\xi
\\\omega_0^2&2\gamma\omega_f&\omega_f^2&0
\\0&\omega_0^2\omega_f&\gamma\omega_f^2&i\xi\omega_f^2
\end{bmatrix}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}\;\ldots\;\omega_0^2=\gamma^2-\xi^2\;\;(\xi^2=\gamma^2-\omega_0^2より)
\\&=&\boldsymbol{A}\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
\end{eqnarray}$$
$$\begin{eqnarray}
\left|\boldsymbol{M}_{31}\right|
&=&\begin{vmatrix}
0&1&0
\\\omega_f&\gamma&i\xi
\\\omega_0^2\omega_f&\gamma\omega_f^2&i\xi\omega_f^2
\end{vmatrix}
\\&=&-\begin{vmatrix}
\omega_f&i\xi
\\\omega_0^2\omega_f&i\xi\omega_f^2
\end{vmatrix}
\\&=&-\left\{
\omega_f \cdot i\xi\omega_f^2
-i\xi \cdot \omega_0^2\omega_f
\right\}
\\&=&-\left\{
i\xi\omega_f^3 - i\xi\omega_0^2\omega_f
\right\}
\\&=&-i\xi\omega_f\left(\omega_f^2 - \omega_0^2\right)
\\&=&i\xi\omega_f\left(\omega_0^2 - \omega_f^2\right)
\\
\\\left|\boldsymbol{M}_{32}\right|
&=&\begin{vmatrix}
1&1&0
\\2\gamma&\gamma&i\xi
\\0&\gamma\omega_f^2&i\xi\omega_f^2
\end{vmatrix}
\\&=&\begin{vmatrix}
\gamma&i\xi
\\\gamma\omega_f^2&i\xi\omega_f^2
\end{vmatrix}
-\begin{vmatrix}
2\gamma&i\xi
\\0&i\xi\omega_f^2
\end{vmatrix}
\\&=&
(
\gamma \cdot i\xi\omega_f^2
- i\xi \cdot \gamma\omega_f^2
)
-(
2\gamma \cdot i\xi\omega_f^2
-i\xi\cdot 0
)
\\&=&i\gamma\xi\omega_f^2 - i\gamma\xi\omega_f^2 - i2\gamma\xi\omega_f^2
\\&=&-i2\gamma\xi\omega_f^2
\\
\\\left|\boldsymbol{M}_{33}\right|
&=&\begin{vmatrix}
1&0&0
\\2\gamma&\omega_f&i\xi
\\0&\omega_0^2\omega_f&i\xi\omega_f^2
\end{vmatrix}
\\&=&\begin{vmatrix}
\omega_f&i\xi
\\\omega_0^2\omega_f&i\xi\omega_f^2
\end{vmatrix}
\\&=&\omega_f \cdot i\xi\omega_f^2 - i\xi \cdot \omega_0^2\omega_f
\\&=& i\xi\omega_f^3- i\xi\omega_0^2\omega_f
\\&=& i\xi\omega_f(\omega_f^2-\omega_0^2)
\\&=& -i\xi\omega_f(\omega_0^2-\omega_f^2)
\\
\\\left|\boldsymbol{M}_{34}\right|
&=&\begin{vmatrix}
1&0&1
\\2\gamma&\omega_f&\gamma
\\0&\omega_0^2\omega_f&\gamma\omega_f^2
\end{vmatrix}
\\&=&\begin{vmatrix}
\omega_f&\gamma
\\\omega_0^2\omega_f&\gamma\omega_f^2
\end{vmatrix}
+\begin{vmatrix}
2\gamma&\omega_f
\\0&\omega_0^2\omega_f
\end{vmatrix}
\\&=&
\omega_f \cdot \gamma\omega_f^2
- \gamma \cdot \omega_0^2\omega_f
+2\gamma \cdot \omega_0^2\omega_f
-\omega_f \cdot 0
\\&=&\gamma\omega_f^3-\gamma\omega_0^2\omega_f+2\gamma\omega_0^2\omega_f
\\&=&\gamma\omega_f(\omega_0^2 + \omega_f^2)
\\
\\
\\
\\&&(-1)^{3+1}a_{31}\left|\boldsymbol{M}_{31}\right|+(-1)^{3+2}a_{32}\left|\boldsymbol{M}_{32}\right|+(-1)^{3+3}a_{33}\left|\boldsymbol{M}_{33}\right|+(-1)^{3+4}a_{34}\left|\boldsymbol{M}_{34}\right|
\\&=& \omega_0^2 \cdot ( i\xi\omega_f\left(\omega_0^2 - \omega_f^2\right) )
\\&& -2\gamma\omega_f \cdot ( -i2\gamma\xi\omega_f^2 )
\\&& +\omega_f^2 \cdot ( -i\xi\omega_f\left(\omega_0^2 - \omega_f^2\right) )
\\&& -0 \cdot ( \gamma\omega_f(\omega_0^2 + \omega_f^2) )
\\&=&i\xi\omega_0^2\omega_f\left(\omega_0^2 - \omega_f^2\right)
+i4\gamma^2\xi\omega_f^3
-i\xi\omega_f^3\left(\omega_0^2 - \omega_f^2\right)
-0
\\&=&i\xi\omega_f\left(\omega_0^2 - \omega_f^2\right)\left(\omega_0^2-\omega_f^2\right)
+i4\gamma^2\xi\omega_f^3
\\&=&i\xi\omega_f\left\{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2\right\}
\end{eqnarray}$$
$$\begin{eqnarray}
(-1)^{1+3}\frac{\left|\boldsymbol{M}_{31}\right|}{\left|A\right|}
&=&\frac{i\xi\omega_f\left(\omega_0^2 - \omega_f^2\right) }
{i\xi\omega_f\left\{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2\right\}}
\\&=&\frac{\omega_0^2 - \omega_f^2}
{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2}
\\(-1)^{2+3}\frac{\left|\boldsymbol{M}_{32}\right|}{\left|A\right|}
&=&-1\cdot\frac{-i2\gamma\xi\omega_f^2}
{i\xi\omega_f\left\{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2\right\}}
\\&=&\frac{2\gamma\omega_f }
{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2}
\\(-1)^{3+3}\frac{\left|\boldsymbol{M}_{33}\right|}{\left|A\right|}
&=&\frac{-i\xi\omega_f\left(\omega_0^2 - \omega_f^2\right)}
{i\xi\omega_f\left\{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2\right\}}
\\&=&\frac{-\left(\omega_0^2 - \omega_f^2\right)}
{\left(\omega_0^2+\omega_f^2\right)^2-\left(2\gamma\omega_f\right)^2}
\\(-1)^{4+3}\frac{\left|\boldsymbol{M}_{34}\right|}{\left|A\right|}
&=&-1\cdot\frac{\gamma\omega_f(\omega_0^2 + \omega_f^2) }
{i\xi\omega_f\left\{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2\right\}}
\\&=&\frac{-\gamma(\omega_0^2 + \omega_f^2)}
{i\xi\left\{\left(\omega_0^2 - \omega_f^2\right)^2-(2\gamma\omega_f)^2\right\}}
\end{eqnarray}$$
$$\begin{eqnarray}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
&=&\boldsymbol{A}^{-1} \begin{bmatrix}
0\\0\\1\\0
\end{bmatrix}
\\&=&\begin{bmatrix}
(-1)^{1+1}\frac{\left|\boldsymbol{M}_{11}\right|}{\left|A\right|}
&(-1)^{1+2}\frac{\left|\boldsymbol{M}_{21}\right|}{\left|A\right|}
&(-1)^{1+3}\frac{\left|\boldsymbol{M}_{31}\right|}{\left|A\right|}
&(-1)^{1+4}\frac{\left|\boldsymbol{M}_{41}\right|}{\left|A\right|}
\\(-1)^{2+1}\frac{\left|\boldsymbol{M}_{12}\right|}{\left|A\right|}
&(-1)^{2+2}\frac{\left|\boldsymbol{M}_{22}\right|}{\left|A\right|}
&(-1)^{2+3}\frac{\left|\boldsymbol{M}_{32}\right|}{\left|A\right|}
&(-1)^{2+4}\frac{\left|\boldsymbol{M}_{42}\right|}{\left|A\right|}
\\(-1)^{3+1}\frac{\left|\boldsymbol{M}_{13}\right|}{\left|A\right|}
&(-1)^{3+2}\frac{\left|\boldsymbol{M}_{23}\right|}{\left|A\right|}
&(-1)^{3+3}\frac{\left|\boldsymbol{M}_{33}\right|}{\left|A\right|}
&(-1)^{3+4}\frac{\left|\boldsymbol{M}_{43}\right|}{\left|A\right|}
\\(-1)^{4+1}\frac{\left|\boldsymbol{M}_{14}\right|}{\left|A\right|}
&(-1)^{4+2}\frac{\left|\boldsymbol{M}_{24}\right|}{\left|A\right|}
&(-1)^{4+3}\frac{\left|\boldsymbol{M}_{34}\right|}{\left|A\right|}
&(-1)^{4+4}\frac{\left|\boldsymbol{M}_{44}\right|}{\left|A\right|}
\end{bmatrix}
\begin{bmatrix}
0\\0\\1\\0
\end{bmatrix}
\\&=&
\begin{bmatrix}
(-1)^{1+3}\frac{\left|\boldsymbol{M}_{31}\right|}{\left|A\right|}\cdot 1
\\(-1)^{2+3}\frac{\left|\boldsymbol{M}_{32}\right|}{\left|A\right|}\cdot 1
\\(-1)^{3+3}\frac{\left|\boldsymbol{M}_{33}\right|}{\left|A\right|}\cdot 1
\\(-1)^{4+3}\frac{\left|\boldsymbol{M}_{34}\right|}{\left|A\right|}\cdot 1
\end{bmatrix}
=
\begin{bmatrix}
\frac{\omega_0^2 - \omega_f^2}{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2}
\\\frac{2\gamma\omega_f }{\left(\omega_0^2 - \omega_f^2\right)^2+(2\gamma\omega_f)^2}
\\\frac{-\left(\omega_0^2 - \omega_f^2\right)}{\left(\omega_0^2 - \omega_f^2\right)^2-\left(2\gamma\omega_f\right)^2}
\\\frac{-\gamma(\omega_f^2 + \omega_0^2)}{i\xi\left\{\left(\omega_0^2 - \omega_f^2\right)^2-(2\gamma\omega_f)^2\right\}}
\end{bmatrix}
\\&=&
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2-\left(2\gamma\omega_f\right)^2}
\begin{bmatrix}
\omega_0^2 - \omega_f^2\\2\gamma\omega_f\\-\left(\omega_0^2 - \omega_f^2\right)\\-\frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2)
\end{bmatrix}
\end{eqnarray}$$
\(C_3,C_4,C_5,C_6\)
$$\begin{eqnarray}
\begin{bmatrix}
C_3\\C_4\\C_5\\C_6
\end{bmatrix}
&=&
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2-\left(2\gamma\omega_f\right)^2}
\begin{bmatrix}
\omega_0^2 - \omega_f^2\\2\gamma\omega_f\\-\left(\omega_0^2 - \omega_f^2\right)\\-\frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2)
\end{bmatrix}
\end{eqnarray}$$
求まった係数を用いて表す
$$\begin{eqnarray}
X&=&\frac{C_1}{s-\lambda_1}+\frac{C_2}{s-\lambda_2}
+\frac{F}{m}\left\{
C_3 \frac{s-a}{\left(s-a\right)^2+b^2}
+ C_4 \frac{b} {\left(s-a\right)^2+b^2}
+ C_5 \frac{s-c}{\left(s-c\right)^2+d^2}
+ C_6 \frac{d} {\left(s-c\right)^2+d^2}
\right\}
\\&=&
\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_2}
\\&&+\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \frac{s-0} {\left(s-0\right)^2+\omega_f^2}
+ 2\gamma\omega_f \frac{\omega_f} {\left(s-0\right)^2+\omega_f^2}
- \left(\omega_0^2 - \omega_f^2\right) \frac{s-(-\gamma)} {\left(s-(-\gamma)\right)^2+(i\xi)^2}
- \frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2) \frac{(i\xi)} {\left(s-(-\gamma)\right)^2+(i\xi)^2}
\right\}
\\&=&
\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-\lambda_2}
\\&&+\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \frac{s-0} {\left(s-0\right)^2+\omega_f^2}
+ 2\gamma\omega_f \frac{\omega_f} {\left(s-0\right)^2+\omega_f^2}
\right\}
\\&&+\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
- \left(\omega_0^2 - \omega_f^2\right) \frac{s-(-\gamma)} {\left(s-(-\gamma)\right)^2+(i\xi)^2}
- \frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2) \frac{(i\xi)} {\left(s-(-\gamma)\right)^2+(i\xi)^2}
\right\}
\end{eqnarray}$$
第1,2項の逆ラプラス変換
$$\begin{eqnarray}
&&\mathfrak{L}^{-1}\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]
\\&=&
\left(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\right)\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_1}\right]
+\left(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\right)\mathfrak{L}^{-1}\left[\frac{1}{s-\lambda_2}\right]
\\&=&
\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}=-\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\;e^{A+B}=e^Ae^B
\\&=&e^{-\gamma t}\left\{
\frac{x_0}{2} e^{\xi t} +\frac{v_0+\gamma x_0}{2\xi } e^{\xi t}
+\frac{x_0}{2} e^{-\xi t} -\frac{v_0+\gamma x_0}{2\xi } e^{-\xi t}
\right\}
\\&=&e^{-\gamma t}\left\{
\frac{x_0}{2} \left(e^{\xi t}+e^{-\xi t}\right)
+\frac{v_0+\gamma x_0}{2\xi } \left(e^{\xi t}-e^{-\xi t}\right)
\right\}
\\&=&e^{-\gamma t}\left\{
\frac{x_0}{2} \left(e^{i\omega t}+e^{-i\omega t}\right)
+\frac{v_0+\gamma x_0}{2i\omega } \left(e^{i\omega t}-e^{-i\omega t}\right)
\right\}
\;\ldots\;\xi=\sqrt{\gamma^2-\omega_0^2}=\sqrt{-1}\sqrt{\left|\gamma^2-\omega^2\right|}=i\omega\;(\gamma \lt \omega_0の場合)
\\&=&e^{-\gamma t}\left\{
\frac{x_0}{2} 2\cos{\left(\omega t\right)}
+\frac{v_0+\gamma x_0}{2i\omega } 2i\sin{\left(\omega t\right)}
\right\}
\;\ldots\;\cos{\left(\theta\right)}=\frac{e^{i\theta}+e^{-i\theta}}{2},\;\sin{\left(\theta\right)}=\frac{e^{i\theta}+e^{-i\theta}}{2i}
\\&=&e^{-\gamma t}\left\{
x_0 \cos{\left(\omega t\right)}
+\frac{v_0+\gamma x_0}{\omega } \sin{\left(\omega t\right)}
\right\}
\\&=&
x_0 e^{-\gamma t} \left\{
\cos{\left(\omega t\right)} + \frac{\gamma}{\omega } \sin{\left(\omega t\right)}
\right\}
+v_0 e^{-\gamma t} \frac{1}{\omega } \sin{\left(\omega t\right)}
\end{eqnarray}$$
第3項の逆ラプラス変換
$$\begin{eqnarray}
&&\mathfrak{L}^{-1}\left[
\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \frac{s-0} {\left(s-0\right)^2+\omega_f^2}
+ 2\gamma\omega_f \frac{\omega_f} {\left(s-0\right)^2+\omega_f^2}
\right\}
\right]
\\&=&
\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) e^{0 t}\cos{\left(\omega_f t\right)}
+ 2\gamma\omega_f e^{0 t}\sin{\left(\omega_f t\right)}
\right\}
\\&=&
\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(\omega_f t\right)}
+ 2\gamma\omega_f \sin{\left(\omega_f t\right)}
\right\}
\end{eqnarray}$$
第4項の逆ラプラス変換
$$\begin{eqnarray}
&&\mathfrak{L}^{-1}\left[
\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
- \left(\omega_0^2 - \omega_f^2\right) \frac{s-(-\gamma)} {\left(s-(-\gamma)\right)^2+(i\xi)^2}
- \frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2) \frac{(i\xi)} {\left(s-(-\gamma)\right)^2+(i\xi)^2}
\right\}
\right]
\\&=&
\frac{F}{m}\frac{- \left(\omega_0^2 - \omega_f^2\right)} {\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\mathfrak{L}^{-1}\left[\frac{s-(-\gamma)} {\left(s-(-\gamma)\right)^2+(i\xi)^2} \right]
+\frac{F}{m}\frac{- \frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2)} {\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\mathfrak{L}^{-1}\left[\frac{(i\xi)} {\left(s-(-\gamma)\right)^2+(i\xi)^2} \right]
\\&=&
\frac{F}{m}\frac{- \left(\omega_0^2 - \omega_f^2\right)} {\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
e^{-\gamma t}\cos{\left(i\xi t\right)}
+\frac{F}{m}\frac{- \frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2)} {\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
e^{-\gamma t}\sin{\left(i\xi t\right)}
\\&=&
\frac{F}{m}
\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
- \left(\omega_0^2 - \omega_f^2\right) e^{-\gamma t}\cos{\left(i\xi t\right)}
- \frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2) e^{-\gamma t}\sin{\left(i\xi t\right)}
\right\}
\\&=&
\frac{F}{m}
\frac{-e^{-\gamma t}}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(i\xi t\right)}
+\frac{\gamma}{i\xi}(\omega_0^2 + \omega_f^2) \sin{\left(i\xi t\right)}
\right\}
\\&=&
\frac{F}{m}
\frac{-e^{-\gamma t}}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(i\cdot i\omega t\right)}
+\frac{\gamma}{i\cdot i\omega}(\omega_0^2 + \omega_f^2) \sin{\left(i\cdot i\omega t\right)}
\right\}\;\ldots\;\xi=i\omega
\\&=&
\frac{F}{m}
\frac{-e^{-\gamma t}}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(-\omega t\right)}
+(-1)\frac{\gamma}{\omega}(\omega_0^2 + \omega_f^2) \sin{\left(-\omega t\right)}
\right\}\;\ldots\;i\cdot i=-1
\\&=&
\frac{F}{m}
\frac{-e^{-\gamma t}}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(\omega t\right)}
+(-1)\frac{\gamma}{\omega}(\omega_0^2 + \omega_f^2) (-1) \sin{\left(\omega t\right)}
\right\}\;\ldots\;\cos{\left(-\theta\right)}=\cos{\left(\theta\right)},\;\sin{\left(-\theta\right)}=-\sin{\left(\theta\right)}
\\&=&
\frac{F}{m}
\frac{-e^{-\gamma t}}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}(\omega_0^2 + \omega_f^2) \sin{\left(\omega t\right)}
\right\}\;\ldots\;-1\cdot -1=1
\end{eqnarray}$$
全体の逆ラプラス変換(1,2,4,3項の順に入れ替え)
$$\begin{eqnarray}
\mathfrak{L}^{-1}\left[X\right]=x\left(t\right)
&=& x_0 e^{-\gamma t} \left\{
\cos{\left(\omega t\right)} + \frac{\gamma}{\omega } \sin{\left(\omega t\right)}
\right\}
\\&&+v_0 e^{-\gamma t} \frac{1}{\omega } \sin{\left(\omega t\right)}
\\&&+\frac{F}{m}\frac{-e^{-\gamma t}}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(\omega t\right)}
+\frac{\gamma}{\omega}(\omega_0^2 + \omega_f^2) \sin{\left(\omega t\right)}
\right\}
\\&&+\frac{F}{m}\frac{1}{\left(\omega_0^2-\omega_f^2\right)^2+\left(2\gamma\omega_f\right)^2}
\left\{
\left(\omega_0^2 - \omega_f^2\right) \cos{\left(\omega_f t\right)}
+ 2\gamma\omega_f \sin{\left(\omega_f t\right)}
\right\}
\end{eqnarray}$$
0 件のコメント:
コメントを投稿