バネマスダンパー系，運動方程式，ラプラス変換，逆ラプラス変換，cos凾数

バネマスダンパー系

運動方程式

$$\begin{array}{rclrclr}m\ddot{x}& +& c\dot{x}& +& kx& =& F\cos \left({\omega }_{f}t\right)\\ \frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& \frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}& +& \frac{k}{m}x& =& \frac{F}{m}\cos \left({\omega }_{f}t\right)\\ \frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& 2\gamma \frac{\mathrm{d}x}{\mathrm{d}t}& +& {\omega }_0^{2}x& =& \frac{F}{m}\cos \left({\omega }_{f}t\right)\cdots \gamma =\frac{c}{2m},{\omega }_0^2=\frac{k}{m}\end{array}$$

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ラプラス変換

$$\begin{array}{rclrcl}\mathfrak{L}[\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& 2\gamma \frac{\mathrm{d}x}{\mathrm{d}t}& +& {\omega }_0^{2}x]& =\mathfrak{L}[\frac{F}{m}\cos \left({\omega }_{f}t\right)]\\ \mathfrak{L}\left[\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}\right]& +& \mathfrak{L}\left[2\gamma \frac{\mathrm{d}x}{\mathrm{d}t}\right]& +& \mathfrak{L}\left[{\omega }_0^{2}x\right]& =\mathfrak{L}[\frac{F}{m}\cos \left({\omega }_{f}t\right)]\\ \mathfrak{L}\left[\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}\right]& +& 2\gamma \mathfrak{L}\left[\frac{\mathrm{d}x}{\mathrm{d}t}\right]& +& {\omega }_0^2\mathfrak{L}\left[x\right]& =\mathfrak{L}[\frac{F}{m}\cos \left({\omega }_{f}t\right)]\\ s^{2}X-s{x}_0-v_0& +& 2\gamma (sX-x_0)& +& {\omega }_0^{2}X& =\frac{F}{m}\frac{s}{s^2+{\omega }_f^2}\\ \\ & & & & & \dots \mathfrak{L}\left[x\right]=X\\ & & & & & \dots \mathfrak{L}\left[\frac{\mathrm{d}x}{\mathrm{d}t}\right]=s^{2}X-x_0,x_0=x(0)\\ & & & & & \dots \mathfrak{L}\left[\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}\right]=s^{2}X-s{x}_0-v_0,v_0=x^{\prime }(0)\\ & & & & & \dots \mathfrak{L}[\cos \left(\omega t\right)]=\frac{s}{s^2+{\omega }_f^2}\end{array}$$

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Xについて解く

$$\begin{array}{rcl}{s}^{2}X+2\gamma Xs+{\omega }_0^{2}X& =& s{x}_0+v_0+2\gamma x_0+\frac{F}{m}\frac{s}{s^2+{\omega }_f^2}\\ (s^2+2\gamma s+{\omega }_0^2)X& =& s{x}_0+v_0+2\gamma x_0+\frac{F}{m}\frac{s}{s^2+{\omega }_f^2}\\ X& =& \frac{s{x}_0+v_0+2\gamma x_0}{s^2+2\gamma s+{\omega }_0^2}+\frac{F}{m}\frac{1}{s^2+2\gamma s+{\omega }_0^2}\frac{s}{s^2+{\omega }_f^2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{(s-{\lambda }_1)(s-{\lambda }_2)}\frac{s}{s^2+{\omega }_f^2}\end{array}$$

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部分分数分解 準備

$$\begin{array}{r}\\ X& =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{(s-{\lambda }_1)(s-{\lambda }_2)}\frac{s}{s^2+{\omega }_f^2}\\ & =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}\{\frac{C_3}{s-{\lambda }_1}+\frac{C_4}{s-{\lambda }_2}+\frac{C_{5}s+C_6}{s^2+{\omega }_f^2}\}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}\\ & & +\frac{F}{m}\left\{\frac{C_3(s-{\lambda }_2)(s^2+{\omega }_f^2)+C_4(s-{\lambda }_1)(s^2+{\omega }_f^2)+(C_{5}s+C_6)(s-{\lambda }_1)(s-{\lambda }_2)}{s(s-{\lambda }_1)(s-{\lambda }_2)(s^2+{\omega }_f^2)}\right\}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}\\ & & +\frac{F}{m}\left\{\frac{C_3(s^3+{\omega }_f^{2}s-{\lambda }_2{s}^2-{\lambda }_2{\omega }_f^2)+C_4(s^3+{\omega }_f^{2}s-{\lambda }_1{s}^2-{\lambda }_1{\omega }_f^2)+C_5\{s^3-({\lambda }_1+{\lambda }_2)s^2+{\lambda }_1{\lambda }_{2}s\}+C_6\{s^2-({\lambda }_1+{\lambda }_2)s+{\lambda }_1{\lambda }_2\}}{(s-{\lambda }_1)(s-{\lambda }_2)(s^2+{\omega }_f^2)}\right\}\\ & =& \frac{(C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}\\ & & +\frac{F}{m}\left[\frac{(C_3+C_4+C_5)s^3+\{-C_3{\lambda }_2-C_4{\lambda }_1-C_5({\lambda }_1+{\lambda }_2)+C_6\}{s}^2+\{C_3{\omega }_f^2+C_4{\omega }_f^2+C_5{\lambda }_1{\lambda }_2-C_6({\lambda }_1+{\lambda }_2)\}s+(-C_3{\lambda }_2{\omega }_f^2-C_4{\lambda }_1{\omega }_f^2+C_6{\lambda }_1{\lambda }_2)}{(s-{\lambda }_1)(s-{\lambda }_2)(s^2+{\omega }_f^2)}\right]\end{array}$$

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部分分数分解 第1項分子の係数比較

$$\begin{array}{rcl}s{x}_0+v_0+2\gamma x_0& =& (C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)\end{array}$$ $$\{\begin{array}{rcl}{x}_0& =& C_1+C_2\\ v_0+2\gamma x_0& =& -(C_1{\lambda }_2+C_2{\lambda }_1)\end{array}$$

部分分数分解 $C_2$

$$\begin{array}{rcl}{x}_0& =& C_1+C_2\\ C_1& =& x_0-C_2\\ v_0+2\gamma x_0& =& -\{(x_0-C_2){\lambda }_2+C_2{\lambda }_1\}\\ & =& -{\lambda }_2{x}_0+C_2{\lambda }_2-C_2{\lambda }_1\\ v_0+2\gamma x_0+{\lambda }_2{x}_0& =& C_2({\lambda }_2-{\lambda }_1)\\ C_2& =& \frac{v_0+2\gamma x_0+{\lambda }_2{x}_0}{{\lambda }_2-{\lambda }_1}\\ & & \dots {\lambda }_{1,2}=\frac{-2\gamma \pm \sqrt{{\left(2\gamma \right)}^2-4\cdot 1\cdot {\omega }_0^2}}{2\cdot 1}=-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0}=-\gamma \pm \xi \\ & & \dots {\lambda }_2-{\lambda }_1=-\gamma -\xi -(-\gamma +\xi )=-2\xi \\ & =& \frac{v_0+2\gamma x_0+(-\gamma -\xi )x_0}{-2\xi }\\ & =& \frac{v_0+\gamma x_0-\xi x_0}{-2\xi }\\ & =& \frac{v_0+\gamma x_0}{-2\xi }-\frac{\xi x_0}{-2\xi }\\ & =& \frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\end{array}$$

部分分数分解 $C_1$

$$\begin{array}{rcl}{C}_1& =& x_0-C_2\\ & =& x_0-(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })\\ & =& \frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\end{array}$$

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部分分数分解 第2項分子の係数比較

$$\begin{array}{rcl}s& =& (C_3+C_4+C_5)s^3+\{-C_3{\lambda }_2-C_4{\lambda }_1-C_5({\lambda }_1+{\lambda }_2)+C_6\}{s}^2+\{C_3{\omega }_f^2+C_4{\omega }_f^2+C_5{\lambda }_1{\lambda }_2-C_6({\lambda }_1+{\lambda }_2)\}s+(-C_3{\lambda }_2{\omega }_f^2-C_4{\lambda }_1{\omega }_f^2+C_6{\lambda }_1{\lambda }_2)\end{array}$$ $$\{\begin{array}{rclrclrcl}0& =& & C_3& +& C_4& +& C_5& \\ 0& =& -{\lambda }_2& C_3& -{\lambda }_1& C_4& -({\lambda }_1+{\lambda }_2)& C_5& +& C_6\\ 1& =& {\omega }_f^2& C_3& +{\omega }_f^2& C_4& +{\lambda }_1{\lambda }_2& C_5& -({\lambda }_1+{\lambda }_2)& C_6\\ 0& =& -{\lambda }_2{\omega }_f^2& C_3& -{\lambda }_1{\omega }_f^2& C_4& & & +{\lambda }_1{\lambda }_2& C_6\end{array}$$ 行列とベクトルで表現すると以下のようになる． $$\begin{array}{rcl}\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right]& =& \left[\begin{array}{cccc}1& 1& 1& 0\\ -{\lambda }_2& -{\lambda }_1& -({\lambda }_1+{\lambda }_2)& 1\\ {\omega }_f^2& {\omega }_f^2& {\lambda }_1{\lambda }_2& -({\lambda }_1+{\lambda }_2)\\ -{\lambda }_2{\omega }_f^2& -{\lambda }_1{\omega }_f^2& 0& {\lambda }_1{\lambda }_2\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\end{array}$$ 行列とベクトルを以下の文字で表すとする． $$\begin{array}{r}\mathit{y}=\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right],\mathit{A}=\left[\begin{array}{cccc}1& 1& 1& 0\\ -{\lambda }_2& -{\lambda }_1& -({\lambda }_1+{\lambda }_2)& 1\\ {\omega }_f^2& {\omega }_f^2& {\lambda }_1{\lambda }_2& -({\lambda }_1+{\lambda }_2)\\ -{\lambda }_2{\omega }_f^2& -{\lambda }_1{\omega }_f^2& 0& {\lambda }_1{\lambda }_2\end{array}\right],\mathit{x}=\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\end{array}$$ これを$\mathit{x}$について解く． $$\begin{array}{r}\\ \mathit{y}& =& \mathit{A}\mathit{x}\\ {\mathit{A}}^{-1}\mathit{y}& =& {\mathit{A}}^{-1}\mathit{A}\mathit{x}\\ \mathit{x}& =& {\mathit{A}}^{-1}\mathit{y}\\ \mathit{x}& =& \frac{\tilde{\mathit{A}}}{\left|\mathit{A}\right|}\mathit{y}\\ & & \dots {\mathit{A}}^{-1}=\frac{\tilde{\mathit{A}}}{\left|\mathit{A}\right|},\tilde{\mathit{A}}\mathrm{は}\mathrm{余}\mathrm{因}\mathrm{子}\mathrm{行}\mathrm{列}\\ \left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& \frac{1}{\left|\mathit{A}\right|}\left[\begin{array}{cccc}(-1{)}^{1+1}\left|{\mathit{M}}_{11}\right|& (-1{)}^{1+2}\left|{\mathit{M}}_{21}\right|& (-1{)}^{1+3}\left|{\mathit{M}}_{31}\right|& (-1{)}^{1+4}\left|{\mathit{M}}_{41}\right|\\ (-1{)}^{2+1}\left|{\mathit{M}}_{12}\right|& (-1{)}^{2+2}\left|{\mathit{M}}_{22}\right|& (-1{)}^{2+3}\left|{\mathit{M}}_{32}\right|& (-1{)}^{2+4}\left|{\mathit{M}}_{42}\right|\\ (-1{)}^{3+1}\left|{\mathit{M}}_{13}\right|& (-1{)}^{3+2}\left|{\mathit{M}}_{23}\right|& (-1{)}^{3+3}\left|{\mathit{M}}_{33}\right|& (-1{)}^{3+4}\left|{\mathit{M}}_{43}\right|\\ (-1{)}^{4+1}\left|{\mathit{M}}_{14}\right|& (-1{)}^{4+2}\left|{\mathit{M}}_{24}\right|& (-1{)}^{4+3}\left|{\mathit{M}}_{34}\right|& (-1{)}^{4+4}\left|{\mathit{M}}_{44}\right|\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right]=\frac{1}{\left|\mathit{A}\right|}\left[\begin{array}{c}\left|{\mathit{M}}_{31}\right|\\ -\left|{\mathit{M}}_{32}\right|\\ \left|{\mathit{M}}_{33}\right|\\ -\left|{\mathit{M}}_{34}\right|\end{array}\right]\\ & & \dots {\mathit{M}}_{ij}:\mathrm{元}\mathrm{の}\mathrm{行}\mathrm{列}\mathit{A}\mathrm{か}\mathrm{ら}i\mathrm{行}\mathrm{と}j\mathrm{列}\mathrm{を}\mathrm{除}\mathrm{い}\mathrm{た}\mathrm{行}\mathrm{列}(\mathrm{添}\mathrm{え}\mathrm{字}\mathrm{の}\mathrm{順}\mathrm{序}\mathrm{に}\mathrm{注}\mathrm{意})\end{array}$$

$\left|{\mathit{M}}_{3\ast }\right|$及び$\left|\mathit{A}\right|$の計算

$$\begin{array}{rcl}\left|{\mathit{M}}_{31}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ -{\lambda }_1& -({\lambda }_1+{\lambda }_2)& 1\\ -{\lambda }_1{\omega }_f^2& 0& {\lambda }_1{\lambda }_2\end{array}\right|=\left|\begin{array}{cc}-({\lambda }_1+{\lambda }_2)& 1\\ 0& {\lambda }_1{\lambda }_2\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_1& 1\\ -{\lambda }_1{\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|\\ & =& -({\lambda }_1+{\lambda }_2)\cdot {\lambda }_1{\lambda }_2-1\cdot 0-\{(-{\lambda }_1)\cdot {\lambda }_1{\lambda }_2-1\cdot (-{\lambda }_1{\omega }_f^2)\}\\ & =& -{\lambda }_1^2{\lambda }_2-{\lambda }_1{\lambda }_2^2+{\lambda }_1^2{\lambda }_2-{\lambda }_1{\omega }_f^2\\ & =& -{\lambda }_1{\lambda }_2^2-{\lambda }_1{\omega }_f^2\\ & =& -{\lambda }_1({\lambda }_2^2+{\omega }_f^2)\\ \left|{\mathit{M}}_{32}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ -{\lambda }_2& -({\lambda }_1+{\lambda }_2)& 1\\ -{\lambda }_2{\omega }_f^2& 0& {\lambda }_1{\lambda }_2\end{array}\right|=\left|\begin{array}{cc}-({\lambda }_1+{\lambda }_2)& 1\\ 0& {\lambda }_1{\lambda }_2\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_2& 1\\ -{\lambda }_2{\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|\\ & =& -({\lambda }_1+{\lambda }_2)\cdot {\lambda }_1{\lambda }_2-1\cdot 0-\{(-{\lambda }_2)\cdot {\lambda }_1{\lambda }_2-1\cdot (-{\lambda }_2{\omega }_f^2)\}\\ & =& -{\lambda }_1^2{\lambda }_2-{\lambda }_1{\lambda }_1^2+{\lambda }_1{\lambda }_2^2-{\lambda }_2{\omega }_f^2\\ & =& -{\lambda }_1^2{\lambda }_2-{\lambda }_2{\omega }_f^2\\ & =& -{\lambda }_2({\lambda }_1^2+{\omega }_f^2)\\ \left|{\mathit{M}}_{33}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ -{\lambda }_2& -{\lambda }_1& 1\\ -{\lambda }_2{\omega }_f^2& -{\lambda }_1{\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|=\left|\begin{array}{cc}-{\lambda }_1& 1\\ -{\lambda }_1{\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_2& 1\\ -{\lambda }_2{\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|\\ & =& -{\lambda }_1\cdot {\lambda }_1{\lambda }_2-1\cdot (-{\lambda }_1{\omega }_f^2)-\{-{\lambda }_2\cdot {\lambda }_1{\lambda }_2-1\cdot (-{\lambda }_2{\omega }_f^2)\}\\ & =& -{\lambda }_1^2{\lambda }_2+{\lambda }_1{\omega }_f^2+{\lambda }_1{\lambda }_2^2-{\lambda }_2{\omega }_f^2\\ & =& -{\lambda }_1{\lambda }_2({\lambda }_1-{\lambda }_2)+{\omega }_f^2({\lambda }_1-{\lambda }_2)\\ & =& (-{\lambda }_1{\lambda }_2+{\omega }_f^2)({\lambda }_1-{\lambda }_2)\\ & =& -({\lambda }_1-{\lambda }_2)({\lambda }_1{\lambda }_2-{\omega }_f^2)\\ \left|{\mathit{M}}_{34}\right|& =& \left|\begin{array}{ccc}1& 1& 1\\ -{\lambda }_2& -{\lambda }_1& -({\lambda }_1+{\lambda }_2)\\ -{\lambda }_2{\omega }_f^2& -{\lambda }_1{\omega }_f^2& 0\end{array}\right|=\left|\begin{array}{cc}-{\lambda }_1& -({\lambda }_1+{\lambda }_2)\\ -{\lambda }_1{\omega }_f^2& 0\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_2& -({\lambda }_1+{\lambda }_2)\\ -{\lambda }_2{\omega }_f^2& 0\end{array}\right|+\left|\begin{array}{cc}-{\lambda }_2& -{\lambda }_1\\ -{\lambda }_2{\omega }_f^2& -{\lambda }_1{\omega }_f^2\end{array}\right|\\ & =& -{\lambda }_1\cdot 0-\{-({\lambda }_1+{\lambda }_2)\}\cdot (-{\lambda }_1{\omega }_f^2)-[-{\lambda }_2\cdot 0-\{-({\lambda }_1+{\lambda }_2)\}\cdot (-{\lambda }_2{\omega }_f^2)]+(-{\lambda }_2)(-{\lambda }_1{\omega }_f^2)-(-{\lambda }_1)(-{\lambda }_2{\omega }_f^2)\\ & =& -{\lambda }_1^2{\omega }_f^2-{\lambda }_1{\lambda }_2{\omega }_f^2+{\lambda }_1{\lambda }_2{\omega }_f^2+{\lambda }_2^2{\omega }_f^2+{\lambda }_1{\lambda }_2{\omega }_f^2-{\lambda }_1{\lambda }_2{\omega }_f^2\\ & =& (-{\lambda }_1^2+{\lambda }_2^2){\omega }_f^2\\ & =& -{\omega }_f^2({\lambda }_1^2-{\lambda }_2^2)\\ & =& -{\omega }_f^2({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)\\ \\ \left|\mathit{A}\right|& =& a_{31}(-1{)}^{3+1}\left|{\mathit{M}}_{31}\right|+a_{32}(-1{)}^{3+2}\left|{\mathit{M}}_{32}\right|+a_{33}(-1{)}^{3+3}\left|{\mathit{M}}_{33}\right|+a_{34}(-1{)}^{3+4}\left|{\mathit{M}}_{34}\right|\\ & =& {\omega }_f^2\cdot 1\cdot \{-{\lambda }_1({\lambda }_2^2+{\omega }_f^2)\}+{\omega }_f^2\cdot -1\cdot \{-{\lambda }_2({\lambda }_1^2+{\omega }_f^2)\}+{\lambda }_1{\lambda }_2\cdot 1\cdot \{-({\lambda }_1-{\lambda }_2)({\lambda }_1{\lambda }_2-{\omega }_f^2)\}-({\lambda }_1+{\lambda }_2)\cdot -1\cdot \{-{\omega }_f^2({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)\}\\ & =& -{\omega }_f^2{\lambda }_1({\lambda }_2^2+{\omega }_f^2)+{\omega }_f^2{\lambda }_2({\lambda }_1^2+{\omega }_f^2)-{\lambda }_1{\lambda }_2({\lambda }_1-{\lambda }_2)({\lambda }_1{\lambda }_2-{\omega }_f^2)-{\omega }_f^2{({\lambda }_1+{\lambda }_2)}^2({\lambda }_1-{\lambda }_2)\\ & =& -{\omega }_f^2{\lambda }_1({\lambda }_2^2+{\omega }_f^2)+{\omega }_f^2{\lambda }_2({\lambda }_1^2+{\omega }_f^2)-{\lambda }_1^3{\lambda }_2^2\cancel{+{\omega }_f^2{\lambda }_1^2{\lambda }_2}+{\lambda }_1^2{\lambda }_2^3\cancel{-{\omega }_f^2{\lambda }_1{\lambda }_2^2}-{\omega }_f^2{\lambda }_1^3\cancel{-{\omega }_f^2{\lambda }_1^2{\lambda }_2}\cancel{+{\omega }_f^2{\lambda }_1{\lambda }_2^2}+{\omega }_f^2{\lambda }_2^3\\ & & \dots {(A+B)}^2(A-B)=A^3+A^{2}B-A{B}^2-B^3\\ & =& -{\omega }_f^2{\lambda }_1({\lambda }_2^2+{\omega }_f^2)+{\omega }_f^2{\lambda }_2({\lambda }_1^2+{\omega }_f^2)-{\lambda }_1^3({\lambda }_2^2+{\omega }_f^2)+{\lambda }_2^3({\lambda }_1^2+{\omega }_f^2)\\ & =& (-{\omega }_f^2{\lambda }_1-{\lambda }_1^3)({\lambda }_2^2+{\omega }_f^2)+({\omega }_f^2{\lambda }_2+{\lambda }_2^3)({\lambda }_1^2+{\omega }_f^2)\\ & =& -{\lambda }_1({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)+{\lambda }_2({\lambda }_2^2+{\omega }_f^2)({\lambda }_1^2+{\omega }_f^2)\\ & =& -({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)\end{array}$$

部分分数分解 $C_3,C_4,C_5,C_6$

$$\begin{array}{rcl}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& \frac{1}{\left|\mathit{A}\right|}\left[\begin{array}{c}\left|{\mathit{M}}_{31}\right|\\ -\left|{\mathit{M}}_{32}\right|\\ \left|{\mathit{M}}_{33}\right|\\ -\left|{\mathit{M}}_{34}\right|\end{array}\right]=\left[\begin{array}{c}\frac{-{\lambda }_1\cancel{({\lambda }_2^2+{\omega }_f^2)}}{-({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)\cancel{({\lambda }_2^2+{\omega }_f^2)}}\\ \frac{-1\cdot -{\lambda }_2\cancel{({\lambda }_1^2+{\omega }_f^2)}}{-({\lambda }_1-{\lambda }_2)\cancel{({\lambda }_1^2+{\omega }_f^2)}({\lambda }_2^2+{\omega }_f^2)}\\ \frac{-\cancel{({\lambda }_1-{\lambda }_2)}({\lambda }_1{\lambda }_2-{\omega }_f^2)}{-\cancel{({\lambda }_1-{\lambda }_2)}({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\\ \frac{-1\cdot -{\omega }_f^2({\lambda }_1+{\lambda }_2)\cancel{({\lambda }_1-{\lambda }_2)}}{-\cancel{({\lambda }_1-{\lambda }_2)}({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\end{array}\right]=\left[\begin{array}{c}\frac{{\lambda }_1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)}\\ \frac{-{\lambda }_2}{({\lambda }_1-{\lambda }_2)({\lambda }_2^2+{\omega }_f^2)}\\ \frac{({\lambda }_1{\lambda }_2-{\omega }_f^2)}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\\ \frac{-{\omega }_f^2({\lambda }_1+{\lambda }_2)}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\end{array}\right]\end{array}$$

部分分数分解 まとめる

$$\begin{array}{rcl}X& =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}(\frac{C_3}{s-{\lambda }_1}+\frac{C_4}{s-{\lambda }_2}+\frac{C_{5}s+C_6}{s^2+{\omega }_f^2})\\ & =& (\frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2})+\frac{F}{m}(\frac{C_3}{s-{\lambda }_1}+\frac{C_4}{s-{\lambda }_2})+\frac{F}{m}\left(\frac{C_{5}s+C_6}{s^2+{\omega }_f^2}\right)\\ & =& (\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2})\\ & & +\frac{F}{m}\{\frac{\frac{{\lambda }_1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)}}{s-{\lambda }_1}+\frac{\frac{-{\lambda }_2}{({\lambda }_1-{\lambda }_2)({\lambda }_2^2+{\omega }_f^2)}}{s-{\lambda }_2}\}\\ & & +\frac{F}{m}\frac{\frac{({\lambda }_1{\lambda }_2-{\omega }_f^2)}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}s+\frac{-{\omega }_f^2({\lambda }_1+{\lambda }_2)}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}}{s^2+{\omega }_f^2}\\ & =& (\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2})\\ & & +\frac{F}{m}\frac{1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\{\frac{{\lambda }_1({\lambda }_2^2+{\omega }_f^2)}{s-{\lambda }_1}+\frac{-{\lambda }_2({\lambda }_1^2+{\omega }_f^2)}{s-{\lambda }_2}\}\\ & & +\frac{F}{m}\frac{1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\frac{({\lambda }_1{\lambda }_2-{\omega }_f^2)({\lambda }_1-{\lambda }_2)s-{\omega }_f^2({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)}{s^2+{\omega }_f^2}\end{array}$$

逆ラプラス変換 第1項

$\gamma <{\omega }_0(\xi \mathrm{が}\mathrm{虚}\mathrm{数}\mathrm{の}\mathrm{場}\mathrm{合})$ $$\begin{array}{r}\\ & & 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}\dots {\mathfrak{L}}^{-1}\left[\frac{1}{s+a}\right]=e^{-at}\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{{\lambda }_{1}t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{{\lambda }_{2}t}\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{(-\gamma +\xi )t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{(-\gamma -\xi )t}\\ & & \dots {\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{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{-\gamma t}{e}^{\xi t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{-\gamma t}{e}^{-\xi t}\dots a^{A+B}=a^A{a}^B\\ & =& e^{-\gamma t}\{(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})e^{\omega it}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})e^{-\omega it}\}\\ & & \dots \gamma <{\omega }_0(\xi \mathrm{が}\mathrm{虚}\mathrm{数}\mathrm{の}\mathrm{場}\mathrm{合}),\xi =\sqrt{{\gamma }^2-{\omega }_0^2}=\sqrt{|{\gamma }^2-{\omega }_0^2|}i=\omega i\\ & =& e^{-\gamma t}[(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})\{\cos \left(\omega t\right)+i\sin \left(\omega t\right)\}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})\{\cos (-\omega t)+i\sin (-\omega t)\}]\\ & =& e^{-\gamma t}[(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})\{\cos \left(\omega t\right)+i\sin \left(\omega t\right)\}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})\{\cos \left(\omega t\right)-i\sin \left(\omega t\right)\}]\\ & & \dots \cos (-\omega t)=\cos \left(\omega t\right),\sin (-\omega t)=-\sin \left(\omega t\right)\\ & =& e^{-\gamma t}[(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})\cos \left(\omega t\right)+(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})i\sin \left(\omega t\right)+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})\cos \left(\omega t\right)-(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})i\sin \left(\omega t\right)]\\ & =& e^{-\gamma t}[\{(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})\}\cos \left(\omega t\right)+\{(\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega i})-(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega i})\}i\sin \left(\omega t\right)]\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega i}i\sin \left(\omega t\right)\}\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\}\dots \frac{i}{i}=1\\ & =& e^{-\gamma t}\{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)\}\\ & =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}+v_0{e}^{-\gamma t}\{\frac{1}{\omega }\sin \left(\omega t\right)\}\dots \mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}{x}_0\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{項}\mathrm{と}\mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}{v}_0\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{項}\end{array}$$

逆ラプラス変換 第2項

$\gamma <{\omega }_0(\xi \mathrm{が}\mathrm{虚}\mathrm{数}\mathrm{の}\mathrm{場}\mathrm{合})$ $$\begin{array}{rcl}& & {\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)}\{\frac{{\lambda }_1({\lambda }_2^2+{\omega }_f^2)}{s-{\lambda }_1}+\frac{-{\lambda }_2({\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)}\{{\lambda }_1({\lambda }_2^2+{\omega }_f^2){\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]-{\lambda }_2({\lambda }_1^2+{\omega }_f^2){\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\}\\ & =& \frac{F}{m}\frac{1}{2\xi \{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{({\lambda }_1{\lambda }_2^2+{\lambda }_1{\omega }_f^2)e^{{\lambda }_{1}t}-({\lambda }_1^2{\lambda }_2+{\lambda }_2{\omega }_f^2)e^{{\lambda }_{2}t}\}\\ & & \dots {\lambda }_1+{\lambda }_2=-2\gamma ,{\lambda }_1-{\lambda }_2=2\xi ,{\lambda }_1{\lambda }_2=(-\gamma +\xi )(-\gamma -\xi )=(-\gamma {)}^2-{\xi }^2={\omega }_0^2\\ & =& \frac{F}{m}\frac{1}{2\xi \{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{({\lambda }_1{\lambda }_2^2+{\lambda }_1{\omega }_f^2)e^{(-\gamma +\xi )t}-({\lambda }_1^2{\lambda }_2+{\lambda }_2{\omega }_f^2)e^{(-\gamma -\xi )t}\}\\ & =& \frac{F}{m}\frac{1}{2\xi \{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{({\lambda }_1{\lambda }_2^2+{\lambda }_1{\omega }_f^2)e^{-\gamma t}{e}^{\xi t}-({\lambda }_1^2{\lambda }_2+{\lambda }_2{\omega }_f^2)e^{-\gamma t}{e}^{-\xi t}\}\\ & & \dots e^{A+B}=e^A{e}^B\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{({\lambda }_1{\lambda }_2^2+{\lambda }_1{\omega }_f^2)e^{\omega it}-({\lambda }_1^2{\lambda }_2+{\lambda }_2{\omega }_f^2)e^{-\omega it}\}\\ & & \dots \gamma <{\omega }_0(\xi \mathrm{が}\mathrm{虚}\mathrm{数}\mathrm{の}\mathrm{場}\mathrm{合}),\xi =\sqrt{{\gamma }^2-{\omega }_0^2}=\sqrt{|{\gamma }^2-{\omega }_0^2|}i=\omega i\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}[({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)\{\cos \left(\omega t\right)+i\sin \left(\omega t\right)\}-({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)\{\cos (-\omega t)+i\sin (-\omega t)\}]\\ & & \dots e^{ix}=\cos \left(\omega t\right)+i\sin \left(\omega t\right)\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)\cos \left(\omega t\right)+({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)i\sin \left({\omega }_{0}t\right)-({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)\cos (-\omega t)-({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)i\sin (-\omega t)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)\cos \left(\omega t\right)+({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)i\sin \left(\omega t\right)-({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)\cos \left(\omega t\right)+({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)i\sin \left(\omega t\right)\}\\ & & \dots \cos (-\omega t)=\cos \left(\omega t\right),\sin (-\omega t)=-\sin \left(\omega t\right)\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}[\{({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)-({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)\}\cos \left(\omega t\right)+\{({\lambda }_2{\omega }_0^2+{\lambda }_1{\omega }_f^2)+({\lambda }_1{\omega }_0^2+{\lambda }_2{\omega }_f^2)\}i\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{-({\lambda }_1-{\lambda }_2)({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+({\lambda }_1+{\lambda }_2)({\omega }_0^2+{\omega }_f^2)i\sin \left(\omega t\right)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\omega i\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{-\left(2\xi \right)({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+(-2\gamma )({\omega }_0^2+{\omega }_f^2)i\sin \left(\omega t\right)\}\\ & & \dots {\lambda }_1+{\lambda }_2=-2\gamma ,{\lambda }_1-{\lambda }_2=2\xi \\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-\frac{2\omega i}{2\omega i}({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+\frac{-2\gamma }{2\omega i}({\omega }_0^2+{\omega }_f^2)i\sin \left(\omega t\right)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)-\frac{\gamma }{\omega }({\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}\dots \frac{i}{i}=1\\ & =& \frac{F}{m}\frac{-e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+\frac{\gamma }{\omega }({\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}\\ & =& \frac{F}{m}\frac{-e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+\frac{\gamma }{{\omega }_0\sqrt{{\zeta }^2-1}}({\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}\\ & & \dots \omega =\sqrt{{\gamma }^2-{\omega }_0^2}=\sqrt{{\omega }_0^2(\frac{{\gamma }^2}{{\omega }_0^2}-\frac{{\omega }_0^2}{{\omega }_0^2})}={\omega }_0\sqrt{{\zeta }^2-1},\zeta =\frac{\gamma }{{\omega }_0}\\ & =& \frac{F}{m}\frac{-e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+\frac{\zeta }{\sqrt{{\zeta }^2-1}}({\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}\end{array}$$

逆ラプラス変換 第3項

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}\left[\frac{F}{m}\frac{1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\left\{\frac{({\lambda }_1{\lambda }_2-{\omega }_f^2)({\lambda }_1-{\lambda }_2)s-{\omega }_f^2({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)}{s^2+{\omega }_f^2}\right\}\right]\\ & =& \frac{F}{m}\frac{1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}{\mathfrak{L}}^{-1}[\frac{({\lambda }_1{\lambda }_2-{\omega }_f^2)({\lambda }_1-{\lambda }_2)s}{s^2+{\omega }_f^2}-\frac{{\omega }_f^2({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)}{s^2+{\omega }_f^2}]\\ & =& \frac{F}{m}\frac{1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\{({\lambda }_1{\lambda }_2-{\omega }_f^2)({\lambda }_1-{\lambda }_2){\mathfrak{L}}^{-1}\left[\frac{s}{s^2+{\omega }_f^2}\right]-{\omega }_f({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2){\mathfrak{L}}^{-1}\left[\frac{{\omega }_f}{s^2+{\omega }_f^2}\right]\}\\ & =& \frac{F}{m}\frac{1}{2\xi \{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\{(({\omega }_0^2)-{\omega }_f^2)\left(2\xi \right)\cos \left({\omega }_{f}t\right)-{\omega }_f(–2\gamma )(2\xi )\sin \left({\omega }_{f}t\right)\}\\ & & \dots {\lambda }_1+{\lambda }_2=-2\gamma ,{\lambda }_1-{\lambda }_2=2\xi ,{\lambda }_1{\lambda }_2=(-\gamma +\xi )(-\gamma -\xi )=(-\gamma {)}^2-{\xi }^2={\omega }_0^2\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\frac{2\xi }{2\xi }\cos \left({\omega }_{f}t\right)+2\gamma {\omega }_f\frac{2\xi }{2\xi }\sin \left({\omega }_{f}t\right)\}\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left({\omega }_{f}t\right)+2\gamma {\omega }_f\sin \left({\omega }_{f}t\right)\}\end{array}$$

逆ラプラス変換 第1,2,3項

$$\begin{array}{rclr}x(t)& =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}& \dots \mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動}\mathrm{振}\mathrm{幅}\mathrm{に}{e}^{-\gamma t}\mathrm{が}\mathrm{あ}\mathrm{る}\mathrm{の}\mathrm{で}t\to \mathrm{\infty }\mathrm{で}\mathrm{消}\mathrm{え}\mathrm{る}\\ & +& v_0{e}^{-\gamma t}\{\frac{1}{\omega }\sin \left(\omega t\right)\}& \dots \mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動}\mathrm{振}\mathrm{幅}\mathrm{に}{e}^{-\gamma t}\mathrm{が}\mathrm{あ}\mathrm{る}\mathrm{の}\mathrm{で}t\to \mathrm{\infty }\mathrm{で}\mathrm{消}\mathrm{え}\mathrm{る}\\ & +& \frac{F}{m}\frac{-e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+\frac{\gamma }{\omega }({\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}& \dots \mathrm{過}\mathrm{渡}\mathrm{応}\mathrm{答}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動}\mathrm{振}\mathrm{幅}\mathrm{に}{e}^{-\gamma t}\mathrm{が}\mathrm{あ}\mathrm{る}\mathrm{の}\mathrm{で}t\to \mathrm{\infty }\mathrm{で}\mathrm{消}\mathrm{え}\mathrm{る}\\ & +& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left({\omega }_{f}t\right)+2\gamma {\omega }_f\sin \left({\omega }_{f}t\right)\}& \dots \mathrm{強}\mathrm{制}\mathrm{振}\mathrm{動}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動}\mathrm{振}\mathrm{幅}\mathrm{に}{e}^{-\gamma t}\mathrm{が}\mathrm{な}\mathrm{い}\mathrm{の}\mathrm{で}t\to \mathrm{\infty }\mathrm{で}\mathrm{も}\mathrm{残}\mathrm{る}\end{array}$$