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，単位ステップ凾数

バネマスダンパー系

運動方程式

$$\begin{array}{rclrclr}m\ddot{x}& +& c\dot{x}& +& kx& =& Fu(t)\\ \frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& \frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}& +& \frac{k}{m}x& =& \frac{F}{m}u(t)\\ \frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& 2\gamma \frac{\mathrm{d}x}{\mathrm{d}t}& +& {\omega }_0^{2}x& =& \frac{F}{m}u(t)\cdots \gamma =\frac{c}{2m},{\omega }_0^2=\frac{k}{m}\end{array}$$ ここで入力$u$を単位ステップ凾数とする． $$\begin{array}{rcl}u(t)& =& \{\begin{array}{c}1(t\ge 0)\\ 0(t<0)\end{array}\end{array}$$

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ラプラス変換

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

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$X$について解く

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

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部分分数分解 準備

$$\begin{array}{rcl}U& =& 1/s\dots \mathrm{単}\mathrm{位}\mathrm{ス}\mathrm{テ}\mathrm{ッ}\mathrm{プ}\mathrm{凾}\mathrm{数}\mathrm{の}\mathrm{ラ}\mathrm{プ}\mathrm{ラ}\mathrm{ス}\mathrm{変}\mathrm{換},\\ X& =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{(s-{\lambda }_1)(s-{\lambda }_2)}U\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{(s-{\lambda }_1)(s-{\lambda }_2)}\frac{1}{s}\\ & =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}\{\frac{C_3}{s-{\lambda }_1}+\frac{C_4}{s-{\lambda }_2}+\frac{C_5}{s}\}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\left\{\frac{C_3(s-{\lambda }_2)s+C_4(s-{\lambda }_1)s+C_5(s-{\lambda }_1)(s-{\lambda }_2)}{s(s-{\lambda }_1)(s-{\lambda }_2)}\right\}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\left\{\frac{C_3{s}^2-C_3{\lambda }_{2}s+C_4{s}^2-C_4{\lambda }_{1}s+C_5\{s^2-({\lambda }_1+{\lambda }_2)s+{\lambda }_1{\lambda }_2\}}{s(s-{\lambda }_1)(s-{\lambda }_2)}\right\}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\left\{\frac{C_3{s}^2-C_3{\lambda }_{2}s+C_4{s}^2-C_4{\lambda }_{1}s+C_5{s}^2-C_5({\lambda }_1+{\lambda }_2)s+C_5{\lambda }_1{\lambda }_2}{s(s-{\lambda }_1)(s-{\lambda }_2)}\right\}\\ & =& \frac{(C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\left[\frac{(C_3+C_4+C_5)s^2+\{-C_3{\lambda }_2-C_4{\lambda }_1-C_5({\lambda }_1+{\lambda }_2)\}s+C_5{\lambda }_1{\lambda }_2}{s(s-{\lambda }_1)(s-{\lambda }_2)}\right]\end{array}$$

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部分分数分解 第1項分子の比較

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

部分分数分解  $C_2$

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

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項分子の比較

$$\begin{array}{rcl}1& =& (C_3+C_4+C_5)s^2+\{-C_3{\lambda }_2-C_4{\lambda }_1-C_5({\lambda }_1+{\lambda }_2)\}s+C_5{\lambda }_1{\lambda }_2\end{array}$$ $$\{\begin{array}{rcl}0& =& C_3+C_4+C_5\\ 0& =& -C_3{\lambda }_2-C_4{\lambda }_1-C_5({\lambda }_1+{\lambda }_2)\\ 1& =& C_5{\lambda }_1{\lambda }_2\end{array}$$

部分分数分解 $C_5$

$$\begin{array}{rcl}1& =& C_5{\lambda }_1{\lambda }_2\dots \mathrm{第}3\mathrm{式}\\ C_5& =& \frac{1}{{\lambda }_1{\lambda }_2}\\ & =& \frac{1}{{\gamma }^2-{\xi }^2}\\ & & \dots {\lambda }_{1,2}=-\gamma \pm \xi \\ & & \dots {\lambda }_1{\lambda }_2=(-\gamma +\xi )(-\gamma -\xi )={(-\gamma )}^2-{\xi }^2={\gamma }^2-{\xi }^2\\ & =& \frac{1}{{\omega }_0^2}\\ & & \dots {\gamma }^2-{\xi }^2={\gamma }^2-{\left(\sqrt{{\gamma }^2-{\omega }_0^2}\right)}^2={\gamma }^2-{\gamma }^2+{\omega }_0^2={\omega }_0^2\end{array}$$

部分分数分解 $C_3,C_4$

$$\begin{array}{rcl}0-0& =& (C_3+C_4+C_5){\lambda }_2-C_3{\lambda }_2-C_4{\lambda }_1-C_5({\lambda }_1+{\lambda }_2)\dots \mathrm{第}1\mathrm{式}\cdot {\lambda }_2+\mathrm{第}2\mathrm{式}\\ 0& =& C_3{\lambda }_2+C_4{\lambda }_2+C_5{\lambda }_2-C_3{\lambda }_2-C_4{\lambda }_1-C_5{\lambda }_1-C_5{\lambda }_2\\ 0& =& C_4({\lambda }_2-{\lambda }_1)-C_5{\lambda }_1\\ C_4({\lambda }_2-{\lambda }_1)& =& C_5{\lambda }_1\\ C_4& =& C_5\frac{{\lambda }_1}{{\lambda }_2-{\lambda }_1}\\ & =& \frac{1}{{\lambda }_1{\lambda }_2}\frac{{\lambda }_1}{{\lambda }_2-{\lambda }_1}\\ & =& \frac{1}{{\lambda }_2}\frac{1}{{\lambda }_2-{\lambda }_1}\\ & =& \frac{1}{{\lambda }_2^2-{\lambda }_1{\lambda }_2}\\ & =& \frac{1}{{(-\gamma -\xi )}^2-(-\gamma +\xi )(-\gamma -\xi )}\\ & =& \frac{1}{{\gamma }^2+2\gamma \xi +{\xi }^2-({\gamma }^2-{\xi }^2)}\\ & =& \frac{1}{{\gamma }^2+2\gamma \xi +{\xi }^2-{\gamma }^2+{\xi }^2}\\ & =& \frac{1}{2\gamma \xi +2{\xi }^2}\\ & =& \frac{1}{2\xi (\gamma +\xi )}\\ & =& \frac{1}{2\xi (\gamma +\xi )}\frac{\gamma -\xi }{\gamma -\xi }\\ & =& \frac{\gamma -\xi }{2\xi ({\gamma }^2-{\xi }^2)}\\ & =& \frac{\gamma -\xi }{2\xi {\omega }_0^2}\\ & =& \frac{1}{{\omega }_0^2}\frac{\gamma -\xi }{2\xi }\\ 0& =& C_3+C_4+C_5\dots \mathrm{第}1\mathrm{式}\\ C_3& =& -C_4-C_5\\ & =& -\frac{1}{{\omega }_0^2}\frac{\gamma -\xi }{2\xi }-\frac{1}{{\omega }_0^2}\\ & =& \frac{-1}{{\omega }_0^2}(\frac{\gamma -\xi }{2\xi }+1)\\ & =& \frac{-1}{{\omega }_0^2}(\frac{\gamma -\xi }{2\xi }+\frac{2\xi }{2\xi })\\ & =& \frac{-1}{{\omega }_0^2}\left(\frac{\gamma +\xi }{2\xi }\right)\end{array}$$

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部分分数分解 まとめる

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

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逆ラプラス変換

$$\begin{array}{rcl}{\mathfrak{L}}^{-1}\left[X\right]& =& {\mathfrak{L}}^{-1}[C_1\frac{1}{s-{\lambda }_1}+C_2\frac{1}{s-{\lambda }_2}+\frac{F}{m}\{C_3\frac{1}{s-{\lambda }_1}+C_4\frac{1}{s-{\lambda }_2}+C_5\frac{1}{s}\}]\\ & =& C_1{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+C_2{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]+\frac{F}{m}{C}_3{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+\frac{F}{m}{C}_4{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]+\frac{F}{m}{C}_5{\mathfrak{L}}^{-1}\left[\frac{1}{s}\right]\\ & =& \{C_1{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+C_2{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\}+\frac{F}{m}\{C_3{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+C_4{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\}+\left\{\frac{F}{m}{C}_5{\mathfrak{L}}^{-1}\left[\frac{1}{s}\right]\right\}\end{array}$$

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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\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}{r}\\ & & \frac{F}{m}\{C_3{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+C_4{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\}\\ & =& \frac{F}{m}(C_3{e}^{{\lambda }_{1}t}+C_4{e}^{{\lambda }_{2}t})\dots {\mathfrak{L}}^{-1}\left[\frac{1}{s+a}\right]=e^{-at}\\ & =& \frac{F}{m}\{-\frac{1}{{\omega }_0^2}\left(\frac{\gamma +\xi }{2\xi }\right)e^{{\lambda }_{1}t}+\frac{1}{{\omega }_0^2}\left(\frac{\gamma -\xi }{2\xi }\right)e^{{\lambda }_{2}t}\}\\ & =& \frac{F}{m}\frac{1}{{\omega }_0^2}\{-(\frac{\gamma }{2\xi }+\frac{\xi }{2\xi })e^{{\lambda }_{1}t}+(\frac{\gamma }{2\xi }-\frac{\xi }{2\xi })e^{{\lambda }_{2}t}\}\\ & =& \frac{F}{k}\{-(\frac{\gamma }{2\xi }+\frac{1}{2})e^{{\lambda }_{1}t}+(\frac{\gamma }{2\xi }-\frac{1}{2})e^{{\lambda }_{2}t}\}\\ & & \dots {\omega }_0=\sqrt{\frac{k}{m}},{\omega }_0^2=\frac{k}{m},m{\omega }_0^2=k\\ & =& \frac{F}{k}\{-(\frac{\gamma }{2\xi }+\frac{1}{2})e^{(-\gamma +\xi )t}+(\frac{\gamma }{2\xi }-\frac{1}{2})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{F}{k}\{-(\frac{\gamma }{2\xi }+\frac{1}{2})e^{-\gamma t}{e}^{\xi t}+(\frac{\gamma }{2\xi }-\frac{1}{2})e^{-\gamma t}{e}^{-\xi t}\}\\ & =& \frac{F}{k}{e}^{-\gamma t}\{-(\frac{\gamma }{2\omega i}+\frac{1}{2})e^{\omega it}+(\frac{\gamma }{2\omega i}-\frac{1}{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}{k}{e}^{-\gamma t}[-(\frac{\gamma }{2\omega i}+\frac{1}{2})\{\cos \left(\omega t\right)+i\sin \left(\omega t\right)\}+(\frac{\gamma }{2\omega i}-\frac{1}{2})\{\cos (-\omega t)+i\sin (-\omega t)\}]\\ & =& \frac{F}{k}{e}^{-\gamma t}[-(\frac{\gamma }{2\omega i}+\frac{1}{2})\{\cos \left(\omega t\right)+i\sin \left(\omega t\right)\}+(\frac{\gamma }{2\omega i}-\frac{1}{2})\{\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)\\ & =& \frac{F}{k}{e}^{-\gamma t}[-(\frac{\gamma }{2\omega i}+\frac{1}{2})\cos \left(\omega t\right)-(\frac{\gamma }{2\omega i}+\frac{1}{2})i\sin \left(\omega t\right)+(\frac{\gamma }{2\omega i}-\frac{1}{2})\cos \left(\omega t\right)-(\frac{\gamma }{2\omega i}-\frac{1}{2})i\sin \left(\omega t\right)]\\ & =& \frac{F}{k}{e}^{-\gamma t}[\{-(\frac{\gamma }{2\omega i}+\frac{1}{2})+(\frac{\gamma }{2\omega i}-\frac{1}{2})\}\cos \left(\omega t\right)+\{-(\frac{\gamma }{2\omega i}+\frac{1}{2})-(\frac{\gamma }{2\omega i}-\frac{1}{2})\}i\sin \left(\omega t\right)]\\ & =& \frac{F}{k}{e}^{-\gamma t}\{-\cos \left(\omega t\right)-\frac{\gamma }{\omega i}i\sin \left(\omega t\right)\}\\ & =& -\frac{F}{k}{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}\dots \frac{i}{i}=1\end{array}$$ $$\begin{array}{rcl}\frac{\gamma }{\omega }& =& \frac{\gamma }{\sqrt{|{\gamma }^2-{\omega }_0^2|}}\dots \omega =\sqrt{|{\gamma }^2-{\omega }_0^2|}\\ & =& \frac{\gamma }{\sqrt{\frac{{\omega }_0^2}{{\omega }_0^2}|{\gamma }^2-{\omega }_0^2|}}\\ & =& \frac{\gamma }{\sqrt{{\omega }_0^2|\frac{{\gamma }^2}{{\omega }_0^2}-\frac{{\omega }_0^2}{{\omega }_0^2}|}}\\ & =& \frac{\gamma }{{\omega }_0\sqrt{|{\zeta }^2-1|}}\dots \zeta =\frac{\gamma }{{\omega }_0}\\ & =& \frac{\zeta }{\sqrt{|{\zeta }^2-1|}}\dots \zeta =\frac{\gamma }{{\omega }_0}\end{array}$$

逆ラプラス変換  第3項

$$\begin{array}{r}\\ & & \frac{F}{m}{C}_5{\mathfrak{L}}^{-1}\left[\frac{1}{s}\right]\\ & =& \frac{F}{m}\frac{1}{{\omega }_0^2}{\mathfrak{L}}^{-1}\left[\frac{1}{s}\right]\\ & =& \frac{F}{m}\frac{1}{{\omega }_0^2}\cdot 1\dots {\mathfrak{L}}^{-1}\left[\frac{1}{s}\right]=1\\ & =& \frac{F}{k}\\ & & \dots {\omega }_0=\sqrt{\frac{k}{m}},{\omega }_0^2=\frac{k}{m},m{\omega }_0^2=k\end{array}$$

逆ラプラス変換  第1,2,3項

$$\begin{array}{rcl}x(t)& =& \{C_1{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+C_2{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\}+\frac{F}{m}\{C_3{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+C_4{\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\}+\left\{\frac{F}{m}{C}_5{\mathfrak{L}}^{-1}\left[\frac{1}{s}\right]\right\}\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\}-\frac{F}{k}{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}+\frac{F}{k}\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\}+\frac{F}{k}[1-e^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\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)\}+\frac{F}{k}[1-e^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}]\\ & & \dots \mathrm{第}1\mathrm{項}:\mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動},\mathrm{第}2\mathrm{項}:\mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動},\mathrm{第}3\mathrm{項}:\mathrm{ス}\mathrm{テ}\mathrm{ッ}\mathrm{プ}\mathrm{入}\mathrm{力}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動}\\ & =& \frac{F}{k}+(x_0-\frac{F}{k})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{第}1\mathrm{項}:\mathrm{ス}\mathrm{テ}\mathrm{ッ}\mathrm{プ}\mathrm{入}\mathrm{力}\mathrm{に}\mathrm{よ}\mathrm{り}\mathrm{釣}\mathrm{り}\mathrm{合}\mathrm{い}\mathrm{位}\mathrm{置}\mathrm{が}\mathrm{変}\mathrm{化},\mathrm{第}2\mathrm{項}:\mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}\mathrm{と}\mathrm{釣}\mathrm{り}\mathrm{合}\mathrm{い}\mathrm{位}\mathrm{置}\mathrm{の}\mathrm{差}\mathrm{を}\mathrm{改}\mathrm{め}\mathrm{て}\mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}\mathrm{と}\mathrm{し}\mathrm{て}\mathrm{減}\mathrm{衰}\mathrm{振}\mathrm{動},\mathrm{第}3\mathrm{項}:\mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{振}\mathrm{動}\end{array}$$