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，sin凾数

バネマスダンパー系

運動方程式

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

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ラプラス変換

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

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Xについて解く

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

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部分分数分解 準備

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

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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

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

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

$$\begin{array}{rcl}\left|{\mathit{M}}_{41}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ -{\lambda }_1& -({\lambda }_1+{\lambda }_2)& 1\\ {\omega }_f^2& {\lambda }_1{\lambda }_2& -({\lambda }_1+{\lambda }_2)\end{array}\right|=\left|\begin{array}{cc}-({\lambda }_1+{\lambda }_2)& 1\\ {\lambda }_1{\lambda }_2& -({\lambda }_1+{\lambda }_2)\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_1& 1\\ {\omega }_f^2& -({\lambda }_1+{\lambda }_2)\end{array}\right|\\ & =& \{-({\lambda }_1+{\lambda }_2)\}\cdot \{-({\lambda }_1+{\lambda }_2)\}-1\cdot {\lambda }_1{\lambda }_2-[-{\lambda }_1\cdot \{-({\lambda }_1+{\lambda }_2)\}-1\cdot {\omega }_f^2]\\ & =& \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|{\mathit{M}}_{42}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ -{\lambda }_2& -({\lambda }_1+{\lambda }_2)& 1\\ {\omega }_f^2& {\lambda }_1{\lambda }_2& -({\lambda }_1+{\lambda }_2)\end{array}\right|=\left|\begin{array}{cc}-({\lambda }_1+{\lambda }_2)& 1\\ {\lambda }_1{\lambda }_2& -({\lambda }_1+{\lambda }_2)\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_2& 1\\ {\omega }_f^2& -({\lambda }_1+{\lambda }_2)\end{array}\right|\\ & =& \{-({\lambda }_1+{\lambda }_2)\}\cdot \{-({\lambda }_1+{\lambda }_2)\}-1\cdot {\lambda }_1{\lambda }_2-[-{\lambda }_2\cdot \{-({\lambda }_1+{\lambda }_2)\}-1\cdot {\omega }_f^2]\\ & =& {\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|{\mathit{M}}_{43}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ -{\lambda }_2& -{\lambda }_1& 1\\ {\omega }_f^2& {\omega }_f^2& -({\lambda }_1+{\lambda }_2)\end{array}\right|=\left|\begin{array}{cc}-{\lambda }_1& 1\\ {\omega }_f^2& -({\lambda }_1+{\lambda }_2)\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_2& 1\\ {\omega }_f^2& -({\lambda }_1+{\lambda }_2)\end{array}\right|\\ & =& -{\lambda }_1\cdot \{-({\lambda }_1+{\lambda }_2)\}-1\cdot {\omega }_f^2-[-{\lambda }_2\cdot \{-({\lambda }_1+{\lambda }_2)\}-1\cdot {\omega }_f^2]\\ & =& {\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\\ & =& ({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)\\ \left|{\mathit{M}}_{44}\right|& =& \left|\begin{array}{ccc}1& 1& 1\\ -{\lambda }_2& -{\lambda }_1& -({\lambda }_1+{\lambda }_2)\\ {\omega }_f^2& {\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|=\left|\begin{array}{cc}-{\lambda }_1& -({\lambda }_1+{\lambda }_2)\\ {\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|-\left|\begin{array}{cc}-{\lambda }_2& -({\lambda }_1+{\lambda }_2)\\ {\omega }_f^2& {\lambda }_1{\lambda }_2\end{array}\right|+\left|\begin{array}{cc}-{\lambda }_2& -{\lambda }_1\\ {\omega }_f^2& {\omega }_f^2\end{array}\right|\\ & =& -{\lambda }_1\cdot {\lambda }_1{\lambda }_2-\{-({\lambda }_1+{\lambda }_2)\}\cdot {\omega }_f^2-[-{\lambda }_2\cdot {\lambda }_1{\lambda }_2-\{-({\lambda }_1+{\lambda }_2)\}\cdot {\omega }_f^2]+(-{\lambda }_2){\omega }_f^2-(-{\lambda }_1){\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({\lambda }_1-{\lambda }_2)+({\lambda }_1-{\lambda }_2){\omega }_f^2\\ & =& -({\lambda }_1{\lambda }_2-{\omega }_f^2)({\lambda }_1-{\lambda }_2)\\ \\ \left|\mathit{A}\right|& =& a_{41}(-1{)}^{4+1}\left|{\mathit{M}}_{41}\right|+a_{42}(-1{)}^{4+2}\left|{\mathit{M}}_{42}\right|+a_{43}(-1{)}^{4+3}\left|{\mathit{M}}_{43}\right|+a_{44}(-1{)}^{4+4}\left|{\mathit{M}}_{44}\right|\\ & =& -{\lambda }_2{\omega }_f^2\cdot -1\cdot \left\{({\lambda }_2^2+{\omega }_f^2)\right\}-{\lambda }_1{\omega }_f^2\cdot 1\cdot \left\{({\lambda }_1^2+{\omega }_f^2)\right\}+0\cdot -1\cdot \left\{({\lambda }_1^2-{\lambda }_2^2)\right\}+{\lambda }_1{\lambda }_2\cdot 1\cdot \left\{({\omega }_f^2-{\lambda }_1{\lambda }_2)({\lambda }_1-{\lambda }_2)\right\}\\ & =& {\lambda }_2{\omega }_f^2({\lambda }_2^2+{\omega }_f^2)-{\lambda }_1{\omega }_f^2({\lambda }_1^2+{\omega }_f^2)+{\lambda }_1{\lambda }_2\left\{({\omega }_f^2-{\lambda }_1{\lambda }_2)({\lambda }_1-{\lambda }_2)\right\}\\ & =& {\lambda }_2{\omega }_f^2({\lambda }_2^2+{\omega }_f^2)-{\lambda }_1{\omega }_f^2({\lambda }_1^2+{\omega }_f^2)+{\lambda }_1{\lambda }_2({\omega }_f^2-{\lambda }_1{\lambda }_2)({\lambda }_1-{\lambda }_2)\\ & =& {\lambda }_2{\omega }_f^2({\lambda }_2^2+{\omega }_f^2)-{\lambda }_1{\omega }_f^2({\lambda }_1^2+{\omega }_f^2)-{\lambda }_1^2{\lambda }_2^2({\lambda }_1-{\lambda }_2)+{\lambda }_1{\lambda }_2{\omega }_f^2({\lambda }_1-{\lambda }_2)\\ & =& {\lambda }_2{\omega }_f^2({\lambda }_2^2+{\omega }_f^2+{\lambda }_1^2)-{\lambda }_1{\omega }_f^2({\lambda }_1^2+{\omega }_f^2+{\lambda }_2^2)-{\lambda }_1^2{\lambda }_2^2({\lambda }_1-{\lambda }_2)\\ & =& ({\lambda }_2{\omega }_f^2-{\lambda }_1{\omega }_f^2)({\lambda }_1^2+{\lambda }_2^2+{\omega }_f^2)-{\lambda }_1^2{\lambda }_2^2({\lambda }_1-{\lambda }_2)\\ & =& -{\omega }_f^2({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\lambda }_2^2+{\omega }_f^2)-{\lambda }_1^2{\lambda }_2^2({\lambda }_1-{\lambda }_2)\\ & =& -({\lambda }_1-{\lambda }_2)({\omega }_f^2({\lambda }_1^2+{\lambda }_2^2+{\omega }_f^2)+{\lambda }_1^2{\lambda }_2^2)\\ & =& -({\lambda }_1-{\lambda }_2)({\lambda }_1^2{\omega }_f^2+{\lambda }_2^2{\omega }_f^2+{\omega }_f^4+{\lambda }_1^2{\lambda }_2^2)\\ & =& -({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)\end{array}$$

部分分数分解 $C_3,C_4,C_5,C_6$

$$\begin{array}{rcl}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& \frac{{\omega }_f}{\left|\mathit{A}\right|}\left[\begin{array}{c}-\left|{\mathit{M}}_{41}\right|\\ \left|{\mathit{M}}_{42}\right|\\ -\left|{\mathit{M}}_{43}\right|\\ \left|{\mathit{M}}_{44}\right|\end{array}\right]={\omega }_f\left[\begin{array}{c}\cancel{-}\frac{\cancel{({\lambda }_2^2+{\omega }_f^2)}}{\cancel{-}({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)\cancel{({\lambda }_2^2+{\omega }_f^2)}}\\ \frac{\cancel{({\lambda }_1^2+{\omega }_f^2)}}{-({\lambda }_1-{\lambda }_2)\cancel{({\lambda }_1^2+{\omega }_f^2)}({\lambda }_2^2+{\omega }_f^2)}\\ \cancel{-}\frac{({\lambda }_1+{\lambda }_2)\cancel{({\lambda }_1-{\lambda }_2)}}{\cancel{-}\cancel{({\lambda }_1-{\lambda }_2)}({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\\ \frac{\cancel{-}({\lambda }_1{\lambda }_2-{\omega }_f^2)\cancel{({\lambda }_1-{\lambda }_2)}}{\cancel{-}\cancel{({\lambda }_1-{\lambda }_2)}({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\end{array}\right]=\left[\begin{array}{c}\frac{{\omega }_f}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)}\\ \frac{-{\omega }_f}{({\lambda }_1-{\lambda }_2)({\lambda }_2^2+{\omega }_f^2)}\\ \frac{{\omega }_f({\lambda }_1+{\lambda }_2)}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\\ \frac{{\omega }_f({\lambda }_1{\lambda }_2-{\omega }_f^2)}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\end{array}\right]\end{array}$$

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部分分数分解 まとめる

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

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$\lambda$に関する幾つかの式を先に計算しておく

$$\begin{array}{rcl}{\lambda }_{1,2}& =& -\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\\ & =& -\gamma \pm \xi \\ \omega & =& \sqrt{|{\gamma }^2-{\omega }_0^2|}\\ \xi & =& \omega i\dots (\gamma <{\omega }_0\mathrm{の}\mathrm{時})\\ {\omega }^2& =& -({\gamma }^2-{\omega }_0^2)\\ & =& {\omega }_0^2-{\gamma }^2\\ {\lambda }_1+{\lambda }_2& =& (-\gamma +\xi )+(-\gamma -\xi )\\ & =& -2\gamma \\ {\lambda }_1-{\lambda }_2& =& (-\gamma +\xi )-(-\gamma -\xi )\\ & =& 2\xi \\ \\ {\lambda }_1{\lambda }_2& =& (-\gamma +\xi )(-\gamma -\xi )\\ & =& (-\gamma {)}^2-{\xi }^2\\ & =& {\gamma }^2-{\xi }^2\\ \\ {\lambda }_1^2& =& {(-\gamma +\xi )}^2\\ & =& (-\gamma {)}^2+2(-\gamma )\xi +{\xi }^2\\ & =& {\gamma }^2-2\gamma \xi +{\xi }^2\\ & =& {(\gamma -\xi )}^2\\ \\ {\lambda }_2^2& =& {(-\gamma -\xi )}^2\\ & =& (-\gamma {)}^2-2(-\gamma )\xi +{\xi }^2\\ & =& {\gamma }^2+2\gamma \xi +{\xi }^2\\ & =& {(\gamma +\xi )}^2\\ \\ {\lambda }_1^2+{\lambda }_2^2& =& ({\gamma }^2-2\gamma \xi +{\xi }^2)+({\gamma }^2+2\gamma \xi +{\xi }^2)\\ & =& 2({\gamma }^2+{\xi }^2)\\ \\ {\lambda }_2^2-{\lambda }_1^2& =& ({\gamma }^2+2\gamma \xi +{\xi }^2)-({\gamma }^2-2\gamma \xi +{\xi }^2)\\ & =& 4\gamma \xi \\ \\ \\ \\ ({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)& =& {\lambda }_1^2{\lambda }_2^2+({\lambda }_1^2+{\lambda }_2^2){\omega }_f^2+{\omega }_f^4\\ & =& {(\gamma -\xi )}^2{(\gamma +\xi )}^2+2({\gamma }^2+{\xi }^2){\omega }_f^2+{\omega }_f^4\\ & =& {\left\{(\gamma -\xi )(\gamma +\xi )\right\}}^2+2{\gamma }^2{\omega }_f^2+2{\xi }^2{\omega }_f^2+{\omega }_f^4\\ & =& {({\gamma }^2-{\xi }^2)}^2+2{\gamma }^2{\omega }_f^2+2({\gamma }^2-{\omega }_0^2){\omega }_f^2+{\omega }_f^4\\ & =& {\left({\omega }_0^2\right)}^2+2{\gamma }^2{\omega }_f^2+2{\gamma }^2{\omega }_f^2-2{\omega }_0^2{\omega }_f^2+{\omega }_f^4\\ & =& {\omega }_0^4+4{\gamma }^2{\omega }_f^2-2{\omega }_0^2{\omega }_f^2+{\omega }_f^4\\ & =& \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2\end{array}$$

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{{\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)}\{{\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]\}\\ & =& \frac{F}{m}\frac{1}{({\lambda }_1-{\lambda }_2)({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)e^{{\lambda }_{1}t}-({\omega }_f{\lambda }_1^2+{\omega }_f^3)e^{{\lambda }_{2}t}\}\\ & =& \frac{F}{m}\frac{1}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)e^{(-\gamma +\omega i)t}-({\omega }_f{\lambda }_1^2+{\omega }_f^3)e^{(-\gamma -\omega i)t}\}\\ & =& \frac{F}{m}\frac{1}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)e^{-\gamma t}{e}^{\omega it}-({\omega }_f{\lambda }_1^2+{\omega }_f^3)e^{-\gamma t}{e}^{-\omega it}\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)e^{\omega it}-({\omega }_f{\lambda }_1^2+{\omega }_f^3)e^{-\omega it}\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)(\cos \left(\omega t\right)+i\sin \left(\omega t\right))-({\omega }_f{\lambda }_1^2+{\omega }_f^3)(\cos (-\omega t)+i\sin (-\omega t))\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)(\cos \left(\omega t\right)+i\sin \left(\omega t\right))-({\omega }_f{\lambda }_1^2+{\omega }_f^3)(\cos \left(\omega t\right)-i\sin \left(\omega t\right))\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)\cos \left(\omega t\right)+({\omega }_f{\lambda }_2^2+{\omega }_f^3)i\sin \left(\omega t\right)-({\omega }_f{\lambda }_1^2+{\omega }_f^3)\cos \left(\omega t\right)+({\omega }_f{\lambda }_1^2+{\omega }_f^3)i\sin \left(\omega t\right)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}[\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)-({\omega }_f{\lambda }_1^2+{\omega }_f^3)\}\cos \left(\omega t\right)+\{({\omega }_f{\lambda }_2^2+{\omega }_f^3)+({\omega }_f{\lambda }_1^2+{\omega }_f^3)\}i\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}[{\omega }_f\{({\lambda }_2^2+{\omega }_f^2)-({\lambda }_1^2+{\omega }_f^2)\}\cos \left(\omega t\right)+{\omega }_f\{({\lambda }_2^2+{\omega }_f^2)+({\lambda }_1^2+{\omega }_f^2)\}i\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}\{{\omega }_f({\lambda }_2^2-{\lambda }_1^2)\cos \left(\omega t\right)+{\omega }_f({\lambda }_1^2+{\lambda }_2^2+2{\omega }_f^2)i\sin \left(\omega t\right)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}[4\gamma {\omega }_f\cos \left(\omega t\right)+{\omega }_f\{2({\gamma }^2+{\xi }^2)+2{\omega }_f^2\}i\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{2\xi \left(2\gamma {\omega }_f\right)+{({\omega }_0^2-{\omega }_f^2)}^2}[{\omega }_f\left(4\gamma \xi \right)\cos \left(\omega t\right)+2{\omega }_{f}i\{({\gamma }^2+{\xi }^2)+{\omega }_f^2\}\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{\left(2\gamma {\omega }_f\right)}^2+{({\omega }_0^2-{\omega }_f^2)}^2}[\frac{4\gamma {\omega }_f\xi }{2\xi }\cos \left(\omega t\right)+\frac{2{\omega }_{f}i}{2\xi }\{{\gamma }^2({\gamma }^2-{\omega }_0^2)+{\omega }_f^2\}\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{\left(2\gamma {\omega }_f\right)}^2+{({\omega }_0^2-{\omega }_f^2)}^2}\{2\gamma {\omega }_f\cos \left(\omega t\right)+\frac{{\omega }_{f}i}{\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{\left(2\gamma {\omega }_f\right)}^2+{({\omega }_0^2-{\omega }_f^2)}^2}[2\gamma {\omega }_f\cos \left(\omega t\right)+\frac{{\omega }_{f}i}{\omega i}\{2{\gamma }^2-({\omega }_0^2-{\omega }_f^2)\}\sin \left(\omega t\right)]\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{\left(2\gamma {\omega }_f\right)}^2+{({\omega }_0^2-{\omega }_f^2)}^2}[2\gamma {\omega }_f\cos \left(\omega t\right)+\frac{{\omega }_f}{\omega }\{2{\gamma }^2-({\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{{\omega }_f({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)s+{\omega }_f({\lambda }_1{\lambda }_2-{\omega }_f^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{{\omega }_f({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2)s}{s^2+{\omega }_f^2}+\frac{{\omega }_f({\lambda }_1{\lambda }_2-{\omega }_f^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)}\{{\omega }_f({\lambda }_1+{\lambda }_2)({\lambda }_1-{\lambda }_2){\mathfrak{L}}^{-1}\left[\frac{s}{s^2+{\omega }_f^2}\right]+({\lambda }_1{\lambda }_2-{\omega }_f^2)({\lambda }_1-{\lambda }_2){\mathfrak{L}}^{-1}\left[\frac{{\omega }_f}{s^2+{\omega }_f^2}\right]\}\\ & =& \frac{F}{m}\frac{1}{({\lambda }_1^2+{\omega }_f^2)({\lambda }_2^2+{\omega }_f^2)}\{{\omega }_f({\lambda }_1+{\lambda }_2){\mathfrak{L}}^{-1}\left[\frac{s}{s^2+{\omega }_f^2}\right]+({\lambda }_1{\lambda }_2-{\omega }_f^2){\mathfrak{L}}^{-1}\left[\frac{{\omega }_f}{s^2+{\omega }_f^2}\right]\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{\left(2\gamma {\omega }_f\right)}^2+{({\omega }_0^2-{\omega }_f^2)}^2}\{{\omega }_f(–2\gamma )\cos \left({\omega }_{f}t\right)+\{\left({\omega }_0^2\right)-{\omega }_f^2\}\sin \left({\omega }_{f}t\right)\}\\ & & \dots {\lambda }_1+{\lambda }_2=-2\gamma \\ & & \dots {\lambda }_1-{\lambda }_2=2\xi =2\omega i(\omega =\sqrt{|{\gamma }^2-{\omega }_0^2|},\gamma <{\omega }_0)\\ & & \dots {\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+{({\omega }_0^2-{\omega }_f^2)}^2}\{-2\gamma {\omega }_f\cos \left({\omega }_{f}t\right)+({\omega }_0^2-{\omega }_f^2)\sin \left({\omega }_{f}t\right)\}\end{array}$$

逆ラプラス変換 第1,2,3項

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