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，その2

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

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

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Xについて解く

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

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部分分数分解 準備

$$\begin{array}{rcl}X& =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{(s-{\lambda }_1)(s-{\lambda }_2)}\frac{{\omega }_f}{s^2+{\omega }_f^2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{s^2-({\lambda }_1+{\lambda }_2)s+{\lambda }_1{\lambda }_2}\frac{{\omega }_f}{s^2+{\omega }_f^2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{{\omega }_f}{s^4-({\lambda }_1+{\lambda }_2)s^3+{\lambda }_1{\lambda }_2{s}^2+{\omega }_f^2{s}^2-{\omega }_f^2({\lambda }_1+{\lambda }_2)s+{\omega }_f^2{\lambda }_1{\lambda }_2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{{\omega }_f}{s^4-({\lambda }_1+{\lambda }_2)s^3+({\lambda }_1{\lambda }_2+{\omega }_f^2)s^2-{\omega }_f^2({\lambda }_1+{\lambda }_2)s+{\omega }_f^2{\lambda }_1{\lambda }_2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{{\omega }_f}{\{{(s-a)}^2+b^2\}\{{(s-c)}^2+d^2\}}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\{C_A\frac{s-a+b}{{(s-a)}^2+b^2}+C_B\frac{s-c+d}{{(s-c)}^2+d^2}\}\\ & =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}\{C_3\frac{s-a}{{(s-a)}^2+b^2}+C_4\frac{b}{{(s-a)}^2+b^2}+C_5\frac{s-c}{{(s-c)}^2+d^2}+C_6\frac{d}{{(s-c)}^2+d^2}\}\\ & & \dots \mathfrak{L}\left[e^{at}\right]=\frac{1}{s-a}\\ & & \dots \mathfrak{L}[e^{at}\cos \left(bt\right)]=\frac{s-a}{(s-a{)}^2+b^2}\\ & & \dots \mathfrak{L}[e^{at}\sin \left(bt\right)]=\frac{b}{(s-a{)}^2+b^2}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_2)(s-{\lambda }_1)}+\frac{F}{m}\{\frac{C_3(s-a)+C_{4}b}{{(s-a)}^2+b^2}+\frac{C_5(s-c)+C_{6}d}{{(s-c)}^2+d^2}\}\\ & =& \frac{(C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)}{(s-{\lambda }_2)(s-{\lambda }_1)}+\frac{F}{m}\frac{\{C_3(s-a)+C_{4}b\}\{{(s-c)}^2+d^2\}+\{C_5(s-c)+C_{6}d\}\{{(s-a)}^2+b^2\}}{\{{(s-a)}^2+b^2\}\{{(s-c)}^2+d^2\}}\end{array}$$

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部分分数分解 第1項分子の係数比較

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

解く

$$\begin{array}{rcl}\left[\begin{array}{c}{x}_0\\ v_0+2\gamma x_0\end{array}\right]& =& \left[\begin{array}{cc}1& 1\\ -{\lambda }_2& -{\lambda }_1\end{array}\right]\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]\\ & =& \mathit{A}\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]\\ {\mathit{A}}^{-1}& =& \frac{1}{\left|\begin{array}{c}\mathit{A}\end{array}\right|}\tilde{\mathit{A}}\\ & =& \frac{1}{\left|\begin{array}{c}\mathit{A}\end{array}\right|}\left[\begin{array}{cc}(-1{)}^{1+1}\left|{\mathit{M}}_{11}\right|& (-1{)}^{1+2}\left|{\mathit{M}}_{21}\right|\\ (-1{)}^{2+1}\left|{\mathit{M}}_{12}\right|& (-1{)}^{2+2}\left|{\mathit{M}}_{22}\right|\end{array}\right]\\ & =& \frac{1}{(1\cdot -{\lambda }_1)-(1\cdot -{\lambda }_2)}\left[\begin{array}{cc}(-1{)}^2\cdot -{\lambda }_1& (-1{)}^3\cdot 1\\ (-1{)}^3\cdot -{\lambda }_2& (-1{)}^2\cdot 1\end{array}\right]\\ & =& \frac{1}{{\lambda }_2-{\lambda }_1}\left[\begin{array}{cc}-{\lambda }_1& -1\\ {\lambda }_2& 1\end{array}\right]\\ & =& \frac{1}{(-\gamma -\xi )-(-\gamma +\xi )}\left[\begin{array}{cc}-(-\gamma +\xi )& -1\\ (-\gamma -\xi )& 1\end{array}\right]\dots {\lambda }_{1,2}=-\gamma \pm \xi \\ & =& \frac{1}{-2\xi }\left[\begin{array}{cc}\gamma -\xi & -1\\ -\gamma -\xi & 1\end{array}\right]\\ \left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]& =& {\mathit{A}}^{-1}\left[\begin{array}{c}{x}_0\\ v_0+2\gamma x_0\end{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{cc}\gamma -\xi & -1\\ -\gamma -\xi & 1\end{array}\right]\left[\begin{array}{c}{x}_0\\ v_0+2\gamma x_0\end{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{c}(\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{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{c}\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{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{c}-\xi x_0-v_0-\gamma x_0\\ -\xi x_0+v_0+\gamma x_0\end{array}\right]\\ & =& \left[\begin{array}{c}\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}\right]\end{array}$$

$C_1,C_2$

$$\begin{array}{rcl}\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]& =& \left[\begin{array}{c}\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\\ \frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\end{array}\right]\end{array}$$

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部分分数分解 第2項分母の係数比較

$$\begin{array}{r}\\ & & s^4-({\lambda }_1+{\lambda }_2)s^3+({\lambda }_1{\lambda }_2+{\omega }_f^2)s^2-{\omega }_f^2({\lambda }_1+{\lambda }_2)s+{\omega }_f^2{\lambda }_1{\lambda }_2\\ & =& \{{(s-a)}^2+b^2\}\{{(s-c)}^2+d^2\}\\ & =& (s^2-2as+a^2+b^2)(s^2-2cs+c^2+d^2)\\ & =& s^4-2c{s}^3+(c^2+d^2)s^2-2a{s}^3+4ac{s}^2-2as(c^2+d^2)+(a^2+b^2)s^2-2(a^2+b^2)cs+(a^2+b^2)(c^2+d^2)\\ & =& s^4-2(a+c)s^3+(a^2+b^2+c^2+d^2+4ac)s^2-2(a{c}^2+a{d}^2+a^{2}c+b^{2}c)s+a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2\end{array}$$ $$\{\begin{array}{rcl}-2(a+c)& =& -({\lambda }_1+{\lambda }_2)\\ a^2+b^2+c^2+d^2+4ac& =& {\lambda }_1{\lambda }_2+{\omega }_f^2\\ -2(a{c}^2+a{d}^2+a^{2}c+b^{2}c)& =& -{\omega }_f^2({\lambda }_1+{\lambda }_2)\\ a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2& =& {\omega }_f^2{\lambda }_1{\lambda }_2\end{array}$$

解く

$$\begin{array}{rcl}a& =& 0\mathrm{を}\mathrm{仮}\mathrm{定}\\ c& =& \frac{{\lambda }_1+{\lambda }_2}{2}-a& \dots \mathrm{一}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{よ}\mathrm{り}\\ & =& \frac{{\lambda }_1+{\lambda }_2}{2}-0& \dots a=0\mathrm{を}\mathrm{代}\mathrm{入}\\ & =& \frac{{\lambda }_1+{\lambda }_2}{2}\\ {\omega }_f^2\frac{{\lambda }_1+{\lambda }_2}{2}& =& a{c}^2+a{d}^2+a^{2}c+b^{2}c& \dots \mathrm{三}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{よ}\mathrm{り}\\ & =& 0\cdot c^2+0\cdot d^2+0^2\cdot c+b^{2}c& \dots a=0\mathrm{を}\mathrm{代}\mathrm{入}\\ & =& b^{2}c\\ & =& b^2\frac{{\lambda }_1+{\lambda }_2}{2}& \dots c=\frac{{\lambda }_1+{\lambda }_2}{2}\mathrm{を}\mathrm{代}\mathrm{入}\\ {\omega }_f^2& =& b^2& \dots \mathrm{両}\mathrm{辺}\frac{{\lambda }_1+{\lambda }_2}{2}\mathrm{で}\mathrm{割}\mathrm{る}\\ b& =& {\omega }_f& \dots \mathrm{角}\mathrm{周}\mathrm{波}\mathrm{数}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{正}\mathrm{を}\mathrm{利}\mathrm{用}(\mathrm{本}\mathrm{来}\mathrm{は}{A}^2=B^2\mathrm{な}\mathrm{ら}A=\pm B)\\ a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2& =& {\omega }_f^2{\lambda }_1{\lambda }_2& \dots \mathrm{四}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\\ 0^2{c}^2+0^2{d}^2+b^2{c}^2+b^2{d}^2& =& & \dots a=0\mathrm{を}\mathrm{代}\mathrm{入}\\ b^2(c^2+d^2)& =& & \\ {\omega }_f^2(c^2+d^2)& =& & \dots b={\omega }_f\mathrm{を}\mathrm{代}\mathrm{入}\\ c^2+d^2& =& {\lambda }_1{\lambda }_2\dots \mathrm{両}\mathrm{辺}{\omega }_f^2\mathrm{で}\mathrm{割}\mathrm{る}\\ d^2& =& {\lambda }_1{\lambda }_2-c^2\\ & =& {\lambda }_1{\lambda }_2-{\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)}^2& \dots c=\frac{{\lambda }_1+{\lambda }_2}{2}\mathrm{を}\mathrm{代}\mathrm{入}\\ & =& \frac{1}{4}\{4{\lambda }_1{\lambda }_2-{({\lambda }_1+{\lambda }_2)}^2\}\\ & =& \frac{1}{4}(4{\lambda }_1{\lambda }_2-{\lambda }_1^2-2{\lambda }_1{\lambda }_2-{\lambda }_2^2)\\ & =& -\frac{1}{4}({\lambda }_1^2-2{\lambda }_1{\lambda }_2+{\lambda }_2^2)\\ & =& -\frac{{({\lambda }_1-{\lambda }_2)}^2}{4}\\ & =& -{\left(\frac{({\lambda }_1-{\lambda }_2)}{2}\right)}^2\\ d& =& \sqrt{-{\left(\frac{({\lambda }_1-{\lambda }_2)}{2}\right)}^2}& \dots \mathrm{角}\mathrm{周}\mathrm{波}\mathrm{数}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{正}\mathrm{を}\mathrm{利}\mathrm{用}\\ & =& \sqrt{-1}\frac{{\lambda }_1-{\lambda }_2}{2}\\ & =& i\frac{{\lambda }_1-{\lambda }_2}{2}\end{array}$$

$$\{\begin{array}{rcl}a& =& 0(\mathrm{仮}\mathrm{定})\\ b& =& {\omega }_f\\ c& =& \frac{{\lambda }_1+{\lambda }_2}{2}=\frac{(-\gamma +\xi )+(-\gamma -\xi )}{2}=\frac{-2\gamma }{2}=-\gamma \\ d& =& i\frac{{\lambda }_1-{\lambda }_2}{2}=i\frac{(-\gamma +\xi )-(-\gamma -\xi )}{2}=i\frac{2\xi }{2}=i\xi \end{array}$$

部分分数分解 第2項分子の係数比較

$$\begin{array}{rcl}{\omega }_f& =& \{C_3(s-a)+C_{4}b\}\{{(s-c)}^2+d^2\}+\{C_5(s-c)+C_{6}d\}\{{(s-a)}^2+b^2\}\\ & =& \{C_{3}s-C_{3}a+C_{4}b\}\{s^2-2cs+c^2+d^2\}+\{C_{5}s-C_{5}c+C_{6}d\}\{s^2-2as+a^2+b^2\}\\ & =& C_3{s}^3-2C_{3}c{s}^2+C_3(c^2+d^2)s-C_{3}a{s}^2+2C_{3}acs-C_{3}a(c^2+d^2)+C_{4}b{s}^2-2C_{4}bcs+C_{4}b(c^2+d^2)\\ & & +C_5{s}^3-2C_{5}a{s}^2+C_5(a^2+b^2)s-C_{5}c{s}^2+2C_{5}acs-C_{5}c(a^2+b^2)+C_{6}d{s}^2-2C_{6}ads+C_{6}d(a^2+b^2)\\ & =& C_3{s}^3+C_5{s}^3\\ & & -2C_{3}c{s}^2-C_{3}a{s}^2+C_{4}b{s}^2-2C_{5}a{s}^2-C_{5}c{s}^2+C_{6}d{s}^2\\ & & +C_3(c^2+d^2)s+2C_{3}acs-2C_{4}bcs+C_5(a^2+b^2)s+2C_{5}acs-2C_{6}ads\\ & & -C_{3}a(c^2+d^2)+C_{4}b(c^2+d^2)-C_{5}c(a^2+b^2)+C_{6}d(a^2+b^2)\\ & =& (C_3+C_5)s^3\\ & & +(-2C_{3}c-C_{3}a+C_{4}b-2C_{5}a-C_{5}c+C_{6}d)s^2\\ & & +(C_3(c^2+d^2)+2C_{3}ac-2C_{4}bc+C_5(a^2+b^2)+2C_{5}ac-2C_{6}ad)s\\ & & +\{-C_{3}a(c^2+d^2)+C_{4}b(c^2+d^2)-C_{5}c(a^2+b^2)+C_{6}d(a^2+b^2)\}\\ & =& (C_3+C_5)s^3\\ & & +(C_3(-2c-a)+C_{4}b+C_5(-2a-c)+C_{6}d)s^2\\ & & +(C_3((c^2+d^2)+2ac)+C_4(-2bc)+C_5((a^2+b^2)+2ac)+C_6(-2ad))s\\ & & +\{C_3(-a(c^2+d^2))+C_{4}b(c^2+d^2)+C_5(-c(a^2+b^2))+C_{6}d(a^2+b^2)\}\end{array}$$ $$\{\begin{array}{rcl}0& =& C_3+C_5\\ 0& =& C_3(-2c-a)+C_{4}b+C_5(-2a-c)+C_{6}d\\ 0& =& C_3(c^2+d^2+2ac)+C_4(-2bc)+C_5((a^2+b^2)+2ac)+C_6(-2ad)\\ {\omega }_f& =& C_3(-a(c^2+d^2))+C_{4}b(c^2+d^2)+C_5(-c(a^2+b^2))+C_{6}d(a^2+b^2)\end{array}$$

解く

$$\begin{array}{rcl}\left[\begin{array}{c}0\\ 0\\ 0\\ {\omega }_f\end{array}\right]& =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2c-a& b& -2a-c& d\\ c^2+d^2+2ac& -2bc& (a^2+b^2)+2ac& -2ad\\ -a(c^2+d^2)& b(c^2+d^2)& -c(a^2+b^2)& d(a^2+b^2)\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2c-0& b& -2\cdot 0-c& d\\ c^2+d^2+2\cdot 0\cdot c& -2bc& (0^2+b^2)+2\cdot 0\cdot c& -2\cdot 0\cdot d\\ -0(c^2+d^2)& b(c^2+d^2)& -c(0^2+b^2)& d(0^2+b^2)\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2c& b& -c& d\\ c^2+d^2& -2bc& b^2& 0\\ 0& b(c^2+d^2)& -b^{2}c& b^{2}d\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2(-\gamma )& ({\omega }_f)& -(-\gamma )& (i\xi )\\ (-\gamma {)}^2+(i\xi {)}^2& -2({\omega }_f)(-\gamma )& ({\omega }_f{)}^2& 0\\ 0& ({\omega }_f)((-\gamma {)}^2+(i\xi {)}^2)& -({\omega }_f{)}^2(-\gamma )& ({\omega }_f{)}^2(i\xi )\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ 2\gamma & {\omega }_f& \gamma & i\xi \\ {\gamma }^2-{\xi }^2& 2\gamma {\omega }_f& {\omega }_f^2& 0\\ 0& {\omega }_f({\gamma }^2-{\xi }^2)& \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ 2\gamma & {\omega }_f& \gamma & i\xi \\ {\omega }_0^2& 2\gamma {\omega }_f& {\omega }_f^2& 0\\ 0& {\omega }_0^2{\omega }_f& \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\dots {\xi }^2={\gamma }^2-{\omega }_0^2\to {\omega }_0^2={\gamma }^2-{\xi }^2\\ & =& \mathit{A}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\end{array}$$ $$\begin{array}{rcl}\left|{\mathit{M}}_{41}\right|& =& \left|\begin{array}{ccc}0& 1& 0\\ {\omega }_f& \gamma & i\xi \\ 2\gamma {\omega }_f& {\omega }_f^2& 0\end{array}\right|\\ & =& -\left|\begin{array}{cc}{\omega }_f& i\xi \\ 2\gamma {\omega }_f& 0\end{array}\right|\\ & =& -\{{\omega }_f\cdot 0-i\xi \cdot 2\gamma {\omega }_f\}\\ & =& i2\gamma \xi {\omega }_f\\ \\ \left|{\mathit{M}}_{42}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ 2\gamma & \gamma & i\xi \\ {\omega }_0^2& {\omega }_f^2& 0\end{array}\right|\\ & =& \left|\begin{array}{cc}\gamma & i\xi \\ {\omega }_f^2& 0\end{array}\right|-\left|\begin{array}{cc}2\gamma & i\xi \\ {\omega }_0^2& 0\end{array}\right|\\ & =& (\gamma \cdot 0-i\xi \cdot {\omega }_f^2)-(2\gamma \cdot 0-i\xi \cdot {\omega }_0^2)\\ & =& -i\xi {\omega }_f^2+i\xi {\omega }_0^2\\ & =& i\xi ({\omega }_0^2-{\omega }_f^2)\\ \\ \left|{\mathit{M}}_{43}\right|& =& \left|\begin{array}{ccc}1& 0& 0\\ 2\gamma & {\omega }_f& i\xi \\ {\omega }_0^2& 2\gamma {\omega }_f& 0\end{array}\right|\\ & =& \left|\begin{array}{cc}{\omega }_f& i\xi \\ 2\gamma {\omega }_f& 0\end{array}\right|\\ & =& {\omega }_f\cdot 0-i\xi \cdot 2\gamma {\omega }_f\\ & =& -i2\gamma \xi {\omega }_f\\ \\ \left|{\mathit{M}}_{44}\right|& =& \left|\begin{array}{ccc}1& 0& 1\\ 2\gamma & {\omega }_f& \gamma \\ {\omega }_0^2& 2\gamma {\omega }_f& {\omega }_f^2\end{array}\right|\\ & =& \left|\begin{array}{cc}{\omega }_f& \gamma \\ 2\gamma {\omega }_f& {\omega }_f^2\end{array}\right|+\left|\begin{array}{cc}2\gamma & {\omega }_f\\ {\omega }_0^2& 2\gamma {\omega }_f\end{array}\right|\\ & =& {\omega }_f\cdot {\omega }_f^2-\gamma \cdot 2\gamma {\omega }_f+2\gamma \cdot 2\gamma {\omega }_f-{\omega }_f\cdot {\omega }_0^2\\ & =& {\omega }_f^3-2{\gamma }^2{\omega }_f+4{\gamma }^2{\omega }_f-{\omega }_0^2{\omega }_f\\ & =& {\omega }_f^3+2{\gamma }^2{\omega }_f-{\omega }_0^2{\omega }_f\\ & =& {\omega }_f(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\\ \\ & & (-1{)}^{4+1}{a}_{41}\left|{\mathit{M}}_{41}\right|+(-1{)}^{4+2}{a}_{42}\left|{\mathit{M}}_{42}\right|+(-1{)}^{4+3}{a}_{43}\left|{\mathit{M}}_{43}\right|+(-1{)}^{4+4}{a}_{44}\left|{\mathit{M}}_{44}\right|\\ & =& -1\cdot 0\cdot (i2\gamma \xi {\omega }_f)\\ & & +1\cdot ({\omega }_0^2{\omega }_f)\cdot (i\xi ({\omega }_0^2-{\omega }_f^2))\\ & & -1\cdot (\gamma {\omega }_f^2)\cdot (-i2\gamma \xi {\omega }_f)\\ & & +1\cdot (i\xi {\omega }_f^2)\cdot ({\omega }_f(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2))\\ & =& 0+i\xi {\omega }_0^2{\omega }_f({\omega }_0^2-{\omega }_f^2)+i2{\gamma }^2\xi {\omega }_f^3+i\xi {\omega }_f^3(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\\ & =& i\xi {\omega }_f\{{\omega }_0^2({\omega }_0^2-{\omega }_f^2)+2{\gamma }^2{\omega }_f^2+{\omega }_f^2(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\}\\ & =& i\xi {\omega }_f({\omega }_0^4-{\omega }_0^2{\omega }_f^2+2{\gamma }^2{\omega }_f^2+2{\gamma }^2{\omega }_f^2-{\omega }_0^2{\omega }_f^2+{\omega }_f^4)\\ & =& i\xi {\omega }_f({\omega }_0^4-2{\omega }_0^2{\omega }_f^2+{\omega }_f^4+4{\gamma }^2{\omega }_f^2)\\ & =& i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}\end{array}$$ $$\begin{array}{rcl}(-1{)}^{1+4}\frac{\left|{\mathit{M}}_{41}\right|}{\left|A\right|}& =& -1\cdot \frac{i2\gamma \xi {\omega }_f}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\\ & =& \frac{-2\gamma }{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\\ (-1{)}^{2+4}\frac{\left|{\mathit{M}}_{42}\right|}{\left|A\right|}& =& \frac{i\xi ({\omega }_0^2-{\omega }_f^2)}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\\ & =& \frac{{\omega }_0^2-{\omega }_f^2}{{\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\\ (-1{)}^{3+4}\frac{\left|{\mathit{M}}_{43}\right|}{\left|A\right|}& =& -1\cdot \frac{-i2\gamma \xi {\omega }_f}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\\ & =& \frac{2\gamma }{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\\ (-1{)}^{4+4}\frac{\left|{\mathit{M}}_{44}\right|}{\left|A\right|}& =& \frac{{\omega }_f(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\\ & =& \frac{2{\gamma }^2-{\omega }_0^2+{\omega }_f^2}{i\xi \{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\end{array}$$ $$\begin{array}{rcl}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& {\mathit{A}}^{-1}\left[\begin{array}{c}0\\ 0\\ 0\\ {\omega }_f\end{array}\right]\\ & =& \left[\begin{array}{cccc}(-1{)}^{1+1}\frac{\left|{\mathit{M}}_{11}\right|}{\left|A\right|}& (-1{)}^{1+2}\frac{\left|{\mathit{M}}_{21}\right|}{\left|A\right|}& (-1{)}^{1+3}\frac{\left|{\mathit{M}}_{31}\right|}{\left|A\right|}& (-1{)}^{1+4}\frac{\left|{\mathit{M}}_{41}\right|}{\left|A\right|}\\ (-1{)}^{2+1}\frac{\left|{\mathit{M}}_{12}\right|}{\left|A\right|}& (-1{)}^{2+2}\frac{\left|{\mathit{M}}_{22}\right|}{\left|A\right|}& (-1{)}^{2+3}\frac{\left|{\mathit{M}}_{32}\right|}{\left|A\right|}& (-1{)}^{2+4}\frac{\left|{\mathit{M}}_{42}\right|}{\left|A\right|}\\ (-1{)}^{3+1}\frac{\left|{\mathit{M}}_{13}\right|}{\left|A\right|}& (-1{)}^{3+2}\frac{\left|{\mathit{M}}_{23}\right|}{\left|A\right|}& (-1{)}^{3+3}\frac{\left|{\mathit{M}}_{33}\right|}{\left|A\right|}& (-1{)}^{3+4}\frac{\left|{\mathit{M}}_{43}\right|}{\left|A\right|}\\ (-1{)}^{4+1}\frac{\left|{\mathit{M}}_{14}\right|}{\left|A\right|}& (-1{)}^{4+2}\frac{\left|{\mathit{M}}_{24}\right|}{\left|A\right|}& (-1{)}^{4+3}\frac{\left|{\mathit{M}}_{34}\right|}{\left|A\right|}& (-1{)}^{4+4}\frac{\left|{\mathit{M}}_{44}\right|}{\left|A\right|}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 0\\ {\omega }_f\end{array}\right]\\ & =& \left[\begin{array}{c}(-1{)}^{1+4}\frac{\left|{\mathit{M}}_{41}\right|}{\left|A\right|}\cdot {\omega }_f\\ (-1{)}^{2+4}\frac{\left|{\mathit{M}}_{42}\right|}{\left|A\right|}\cdot {\omega }_f\\ (-1{)}^{3+4}\frac{\left|{\mathit{M}}_{43}\right|}{\left|A\right|}\cdot {\omega }_f\\ (-1{)}^{4+4}\frac{\left|{\mathit{M}}_{44}\right|}{\left|A\right|}\cdot {\omega }_f\end{array}\right]=\left[\begin{array}{c}\frac{-2\gamma {\omega }_f}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\\ \frac{{\omega }_f({\omega }_0^2-{\omega }_f^2)}{{\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\\ \frac{2\gamma {\omega }_f}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\\ \frac{{\omega }_f(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)}{i\xi \{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2\}}\end{array}\right]\\ & =& \frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\left[\begin{array}{c}-2\gamma {\omega }_f\\ {\omega }_0^2-{\omega }_f^2\\ 2\gamma {\omega }_f\\ \frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\end{array}\right]\end{array}$$

$C_3,C_4,C_5,C_6$

$$\begin{array}{rcl}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& \frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\left[\begin{array}{c}-2\gamma {\omega }_f\\ {\omega }_0^2-{\omega }_f^2\\ 2\gamma {\omega }_f\\ \frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\end{array}\right]\end{array}$$

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求まった係数を用いて表す

$$\begin{array}{rcl}X& =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}\{C_3\frac{s-a}{{(s-a)}^2+b^2}+C_4\frac{b}{{(s-a)}^2+b^2}+C_5\frac{s-c}{{(s-c)}^2+d^2}+C_6\frac{d}{{(s-c)}^2+d^2}\}\\ & =& \frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-2\gamma {\omega }_f\frac{s-0}{{(s-0)}^2+{\omega }_f^2}+({\omega }_0^2-{\omega }_f^2)\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}+2\gamma {\omega }_f\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}+\frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\}\\ & =& \frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-2\gamma {\omega }_f\frac{s-0}{{(s-0)}^2+{\omega }_f^2}+({\omega }_0^2-{\omega }_f^2)\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}\}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{+2\gamma {\omega }_f\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}+\frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\}\end{array}$$

第1,2項の逆ラプラス変換

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}[\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2}]\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }){\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }){\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{{\lambda }_{1}t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{{\lambda }_{2}t}\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{(-\gamma +\xi )t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{(-\gamma -\xi )t}\dots {\lambda }_{1,2}=-\gamma \pm \xi \\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{-\gamma t}{e}^{\xi t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{-\gamma t}{e}^{-\xi t}\dots e^{A+B}=e^A{e}^B\\ & =& e^{-\gamma t}\{\frac{x_0}{2}{e}^{\xi t}+\frac{v_0+\gamma x_0}{2\xi }{e}^{\xi t}+\frac{x_0}{2}{e}^{-\xi t}-\frac{v_0+\gamma x_0}{2\xi }{e}^{-\xi t}\}\\ & =& e^{-\gamma t}\{\frac{x_0}{2}(e^{\xi t}+e^{-\xi t})+\frac{v_0+\gamma x_0}{2\xi }(e^{\xi t}-e^{-\xi t})\}\\ & =& e^{-\gamma t}\{\frac{x_0}{2}(e^{i\omega t}+e^{-i\omega t})+\frac{v_0+\gamma x_0}{2i\omega }(e^{i\omega t}-e^{-i\omega t})\}\dots \xi =\sqrt{{\gamma }^2-{\omega }_0^2}=\sqrt{-1}\sqrt{|{\gamma }^2-{\omega }^2|}=i\omega (\gamma <{\omega }_0\mathrm{の}\mathrm{場}\mathrm{合})\\ & =& e^{-\gamma t}\{\frac{x_0}{2}2\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{2i\omega }2i\sin \left(\omega t\right)\}\dots \cos \left(\theta \right)=\frac{e^{i\theta }+e^{-i\theta }}{2},\sin \left(\theta \right)=\frac{e^{i\theta }+e^{-i\theta }}{2i}\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\}\\ & =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}+v_0{e}^{-\gamma t}\frac{1}{\omega }\sin \left(\omega t\right)\end{array}$$

第3項の逆ラプラス変換

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}\left[\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-2\gamma {\omega }_f\frac{s-0}{{(s-0)}^2+{\omega }_f^2}+({\omega }_0^2-{\omega }_f^2)\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}\}\right]\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}(-2\gamma {\omega }_f){\mathfrak{L}}^{-1}\left[\frac{s-0}{{(s-0)}^2+{\omega }_f^2}\right]+\frac{F}{m}\frac{{\omega }_0^2-{\omega }_f^2}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}({\omega }_0^2-{\omega }_f^2){\mathfrak{L}}^{-1}\left[\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}\right]\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}(-2\gamma {\omega }_f)e^{0t}\cos \left({\omega }_{f}t\right)+\frac{F}{m}\frac{{\omega }_0^2-{\omega }_f^2}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}({\omega }_0^2-{\omega }_f^2)e^{0t}\sin \left({\omega }_{f}t\right)\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-2\gamma {\omega }_f{e}^{0t}\cos \left({\omega }_{f}t\right)+({\omega }_0^2-{\omega }_f^2)e^{0t}\sin \left({\omega }_{f}t\right)\}\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-2\gamma {\omega }_f\cos \left({\omega }_{f}t\right)+({\omega }_0^2-{\omega }_f^2)\sin \left({\omega }_{f}t\right)\}\dots e^0=1\end{array}$$

第4項の逆ラプラス変換

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}\left[\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{2\gamma {\omega }_f\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}+\frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\}\right]\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\left(2\gamma {\omega }_f\right){\mathfrak{L}}^{-1}\left[\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\right]+\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{\frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\}{\mathfrak{L}}^{-1}\left[\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\right]\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\left(2\gamma {\omega }_f\right)e^{-\gamma t}\cos \left(i\xi t\right)+\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{\frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\}{e}^{-\gamma t}\sin \left(i\xi t\right)\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{2\gamma {\omega }_f\cos \left(i\xi t\right)+\frac{{\omega }_f}{i\xi }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\sin \left(i\xi t\right)\}\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{2\gamma {\omega }_f\cos (i\cdot i\omega t)+\frac{{\omega }_f}{i\cdot i\omega }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\sin (i\cdot i\omega t)\}\dots \xi =i\omega \\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{2\gamma {\omega }_f\cos (-\omega t)+(-1)\frac{{\omega }_f}{\omega }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)\sin (-\omega t)\}\dots i\cdot i=-1\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{2\gamma {\omega }_f\cos \left(\omega t\right)+(-1)\frac{{\omega }_f}{\omega }(2{\gamma }^2-{\omega }_0^2+{\omega }_f^2)(-1)\sin \left(\omega t\right)\}\dots \cos (-\theta )=\cos \left(\theta \right),\sin (-\theta )=-\sin \left(\theta \right)\\ & =& \frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^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 -1\cdot -1=1\end{array}$$

全体の逆ラプラス変換(1,2,4,3項の順に入れ替え)

$$\begin{array}{rcl}{\mathfrak{L}}^{-1}\left[X\right]=x\left(t\right)& =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}\\ & & +v_0{e}^{-\gamma t}\frac{1}{\omega }\sin \left(\omega t\right)\\ & & +\frac{F}{m}\frac{e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{2\gamma {\omega }_f\cos \left(\omega t\right)+\frac{{\omega }_f}{\omega }(2{\gamma }^2-({\omega }_0^2-{\omega }_f^2))\sin \left(\omega t\right)\}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-2\gamma {\omega }_f\cos \left({\omega }_{f}t\right)+({\omega }_0^2-{\omega }_f^2)\sin \left({\omega }_{f}t\right)\}\end{array}$$

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

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

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，その2

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

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

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Xについて解く

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

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部分分数分解 準備

$$\begin{array}{rcl}X& =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{(s-{\lambda }_1)(s-{\lambda }_2)}\frac{s}{s^2+{\omega }_f^2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{1}{s^2-({\lambda }_1+{\lambda }_2)s+{\lambda }_1{\lambda }_2}\frac{s}{s^2+{\omega }_f^2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{s}{s^4-({\lambda }_1+{\lambda }_2)s^3+{\lambda }_1{\lambda }_2{s}^2+{\omega }_f^2{s}^2-{\omega }_f^2({\lambda }_1+{\lambda }_2)s+{\omega }_f^2{\lambda }_1{\lambda }_2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{s}{s^4-({\lambda }_1+{\lambda }_2)s^3+({\lambda }_1{\lambda }_2+{\omega }_f^2)s^2-{\omega }_f^2({\lambda }_1+{\lambda }_2)s+{\omega }_f^2{\lambda }_1{\lambda }_2}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\frac{s}{\{{(s-a)}^2+b^2\}\{{(s-c)}^2+d^2\}}\\ & =& \frac{s{x}_0+v_0+2\gamma x_0}{(s-{\lambda }_1)(s-{\lambda }_2)}+\frac{F}{m}\{C_A\frac{s-a+b}{{(s-a)}^2+b^2}+C_B\frac{s-c+d}{{(s-c)}^2+d^2}\}\\ & =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}\{C_3\frac{s-a}{{(s-a)}^2+b^2}+C_4\frac{b}{{(s-a)}^2+b^2}+C_5\frac{s-c}{{(s-c)}^2+d^2}+C_6\frac{d}{{(s-c)}^2+d^2}\}\\ & & \dots \mathfrak{L}\left[e^{at}\right]=\frac{1}{s-a}\\ & & \dots \mathfrak{L}[e^{at}\cos \left(bt\right)]=\frac{s-a}{(s-a{)}^2+b^2}\\ & & \dots \mathfrak{L}[e^{at}\sin \left(bt\right)]=\frac{b}{(s-a{)}^2+b^2}\\ & =& \frac{C_1(s-{\lambda }_2)+C_2(s-{\lambda }_1)}{(s-{\lambda }_2)(s-{\lambda }_1)}+\frac{F}{m}\{\frac{C_3(s-a)+C_{4}b}{{(s-a)}^2+b^2}+\frac{C_5(s-c)+C_{6}d}{{(s-c)}^2+d^2}\}\\ & =& \frac{(C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)}{(s-{\lambda }_2)(s-{\lambda }_1)}+\frac{F}{m}\frac{\{C_3(s-a)+C_{4}b\}\{{(s-c)}^2+d^2\}+\{C_5(s-c)+C_{6}d\}\{{(s-a)}^2+b^2\}}{\{{(s-a)}^2+b^2\}\{{(s-c)}^2+d^2\}}\end{array}$$

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部分分数分解 第1項分子の係数比較

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

解く

$$\begin{array}{rcl}\left[\begin{array}{c}{x}_0\\ v_0+2\gamma x_0\end{array}\right]& =& \left[\begin{array}{cc}1& 1\\ -{\lambda }_2& -{\lambda }_1\end{array}\right]\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]\\ & =& \mathit{A}\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]\\ {\mathit{A}}^{-1}& =& \frac{1}{\left|\begin{array}{c}\mathit{A}\end{array}\right|}\tilde{\mathit{A}}\\ & =& \frac{1}{\left|\begin{array}{c}\mathit{A}\end{array}\right|}\left[\begin{array}{cc}(-1{)}^{1+1}\left|{\mathit{M}}_{11}\right|& (-1{)}^{1+2}\left|{\mathit{M}}_{21}\right|\\ (-1{)}^{2+1}\left|{\mathit{M}}_{12}\right|& (-1{)}^{2+2}\left|{\mathit{M}}_{22}\right|\end{array}\right]\\ & =& \frac{1}{(1\cdot -{\lambda }_1)-(1\cdot -{\lambda }_2)}\left[\begin{array}{cc}(-1{)}^2\cdot -{\lambda }_1& (-1{)}^3\cdot 1\\ (-1{)}^3\cdot -{\lambda }_2& (-1{)}^2\cdot 1\end{array}\right]\\ & =& \frac{1}{{\lambda }_2-{\lambda }_1}\left[\begin{array}{cc}-{\lambda }_1& -1\\ {\lambda }_2& 1\end{array}\right]\\ & =& \frac{1}{(-\gamma -\xi )-(-\gamma +\xi )}\left[\begin{array}{cc}-(-\gamma +\xi )& -1\\ (-\gamma -\xi )& 1\end{array}\right]\dots {\lambda }_{1,2}=-\gamma \pm \xi \\ & =& \frac{1}{-2\xi }\left[\begin{array}{cc}\gamma -\xi & -1\\ -\gamma -\xi & 1\end{array}\right]\\ \left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]& =& {\mathit{A}}^{-1}\left[\begin{array}{c}{x}_0\\ v_0+2\gamma x_0\end{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{cc}\gamma -\xi & -1\\ -\gamma -\xi & 1\end{array}\right]\left[\begin{array}{c}{x}_0\\ v_0+2\gamma x_0\end{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{c}(\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{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{c}\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{array}\right]\\ & =& \frac{1}{-2\xi }\left[\begin{array}{c}-\xi x_0-v_0-\gamma x_0\\ -\xi x_0+v_0+\gamma x_0\end{array}\right]\\ & =& \left[\begin{array}{c}\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}\right]\end{array}$$

$C_1,C_2$

$$\begin{array}{rcl}\left[\begin{array}{c}{C}_1\\ C_2\end{array}\right]& =& \left[\begin{array}{c}\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }\\ \frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }\end{array}\right]\end{array}$$

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部分分数分解 第2項分母の係数比較

$$\begin{array}{r}\\ & & s^4-({\lambda }_1+{\lambda }_2)s^3+({\lambda }_1{\lambda }_2+{\omega }_f^2)s^2-{\omega }_f^2({\lambda }_1+{\lambda }_2)s+{\omega }_f^2{\lambda }_1{\lambda }_2\\ & =& \{{(s-a)}^2+b^2\}\{{(s-c)}^2+d^2\}\\ & =& (s^2-2as+a^2+b^2)(s^2-2cs+c^2+d^2)\\ & =& s^4-2c{s}^3+(c^2+d^2)s^2-2a{s}^3+4ac{s}^2-2as(c^2+d^2)+(a^2+b^2)s^2-2(a^2+b^2)cs+(a^2+b^2)(c^2+d^2)\\ & =& s^4-2(a+c)s^3+(a^2+b^2+c^2+d^2+4ac)s^2-2(a{c}^2+a{d}^2+a^{2}c+b^{2}c)s+a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2\end{array}$$ $$\{\begin{array}{rcl}-2(a+c)& =& -({\lambda }_1+{\lambda }_2)\\ a^2+b^2+c^2+d^2+4ac& =& {\lambda }_1{\lambda }_2+{\omega }_f^2\\ -2(a{c}^2+a{d}^2+a^{2}c+b^{2}c)& =& -{\omega }_f^2({\lambda }_1+{\lambda }_2)\\ a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2& =& {\omega }_f^2{\lambda }_1{\lambda }_2\end{array}$$

解く

$$\begin{array}{rcl}a& =& 0\mathrm{を}\mathrm{仮}\mathrm{定}\\ c& =& \frac{{\lambda }_1+{\lambda }_2}{2}-a& \dots \mathrm{一}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{よ}\mathrm{り}\\ & =& \frac{{\lambda }_1+{\lambda }_2}{2}-0& \dots a=0\mathrm{を}\mathrm{代}\mathrm{入}\\ & =& \frac{{\lambda }_1+{\lambda }_2}{2}\\ {\omega }_f^2\frac{{\lambda }_1+{\lambda }_2}{2}& =& a{c}^2+a{d}^2+a^{2}c+b^{2}c& \dots \mathrm{三}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\mathrm{よ}\mathrm{り}\\ & =& 0\cdot c^2+0\cdot d^2+0^2\cdot c+b^{2}c& \dots a=0\mathrm{を}\mathrm{代}\mathrm{入}\\ & =& b^{2}c\\ & =& b^2\frac{{\lambda }_1+{\lambda }_2}{2}& \dots c=\frac{{\lambda }_1+{\lambda }_2}{2}\mathrm{を}\mathrm{代}\mathrm{入}\\ {\omega }_f^2& =& b^2& \dots \mathrm{両}\mathrm{辺}\frac{{\lambda }_1+{\lambda }_2}{2}\mathrm{で}\mathrm{割}\mathrm{る}\\ b& =& {\omega }_f& \dots \mathrm{角}\mathrm{周}\mathrm{波}\mathrm{数}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{正}\mathrm{を}\mathrm{利}\mathrm{用}\\ a^2{c}^2+a^2{d}^2+b^2{c}^2+b^2{d}^2& =& {\omega }_f^2{\lambda }_1{\lambda }_2& \dots \mathrm{四}\mathrm{つ}\mathrm{目}\mathrm{の}\mathrm{式}\\ 0^2{c}^2+0^2{d}^2+b^2{c}^2+b^2{d}^2& =& & \dots a=0\mathrm{を}\mathrm{代}\mathrm{入}\\ b^2(c^2+d^2)& =& & \\ {\omega }_f^2(c^2+d^2)& =& & \dots b={\omega }_f\mathrm{を}\mathrm{代}\mathrm{入}\\ c^2+d^2& =& {\lambda }_1{\lambda }_2\dots \mathrm{両}\mathrm{辺}{\omega }_f^2\mathrm{で}\mathrm{割}\mathrm{る}\\ d^2& =& {\lambda }_1{\lambda }_2-c^2\\ & =& {\lambda }_1{\lambda }_2-{\left(\frac{{\lambda }_1+{\lambda }_2}{2}\right)}^2& \dots c=\frac{{\lambda }_1+{\lambda }_2}{2}\mathrm{を}\mathrm{代}\mathrm{入}\\ & =& \frac{1}{4}\{4{\lambda }_1{\lambda }_2-{({\lambda }_1+{\lambda }_2)}^2\}\\ & =& \frac{1}{4}(4{\lambda }_1{\lambda }_2-{\lambda }_1^2-2{\lambda }_1{\lambda }_2-{\lambda }_2^2)\\ & =& -\frac{1}{4}({\lambda }_1^2-2{\lambda }_1{\lambda }_2+{\lambda }_2^2)\\ & =& -\frac{{({\lambda }_1-{\lambda }_2)}^2}{4}\\ & =& -{\left(\frac{({\lambda }_1-{\lambda }_2)}{2}\right)}^2\\ d& =& \sqrt{-{\left(\frac{({\lambda }_1-{\lambda }_2)}{2}\right)}^2}& \dots \mathrm{角}\mathrm{周}\mathrm{波}\mathrm{数}\mathrm{な}\mathrm{の}\mathrm{で}\mathrm{正}\mathrm{を}\mathrm{利}\mathrm{用}\\ & =& \sqrt{-1}\frac{{\lambda }_1-{\lambda }_2}{2}\\ & =& i\frac{{\lambda }_1-{\lambda }_2}{2}\end{array}$$

$$\{\begin{array}{rcl}a& =& 0\\ b& =& {\omega }_f\\ c& =& \frac{{\lambda }_1+{\lambda }_2}{2}=\frac{(-\gamma +\xi )+(-\gamma -\xi )}{2}=\frac{-2\gamma }{2}=-\gamma \\ d& =& i\frac{{\lambda }_1-{\lambda }_2}{2}=i\frac{(-\gamma +\xi )-(-\gamma -\xi )}{2}=i\frac{2\xi }{2}=i\xi \end{array}$$

部分分数分解 第2項分子の係数比較

$$\begin{array}{rcl}s& =& \{C_3(s-a)+C_{4}b\}\{{(s-c)}^2+d^2\}+\{C_5(s-c)+C_{6}d\}\{{(s-a)}^2+b^2\}\\ & =& \{C_{3}s-C_{3}a+C_{4}b\}\{s^2-2cs+c^2+d^2\}+\{C_{5}s-C_{5}c+C_{6}d\}\{s^2-2as+a^2+b^2\}\\ & =& C_3{s}^3-2C_{3}c{s}^2+C_3(c^2+d^2)s-C_{3}a{s}^2+2C_{3}acs-C_{3}a(c^2+d^2)+C_{4}b{s}^2-2C_{4}bcs+C_{4}b(c^2+d^2)\\ & & +C_5{s}^3-2C_{5}a{s}^2+C_5(a^2+b^2)s-C_{5}c{s}^2+2C_{5}acs-C_{5}c(a^2+b^2)+C_{6}d{s}^2-2C_{6}ads+C_{6}d(a^2+b^2)\\ & =& C_3{s}^3+C_5{s}^3\\ & & -2C_{3}c{s}^2-C_{3}a{s}^2+C_{4}b{s}^2-2C_{5}a{s}^2-C_{5}c{s}^2+C_{6}d{s}^2\\ & & +C_3(c^2+d^2)s+2C_{3}acs-2C_{4}bcs+C_5(a^2+b^2)s+2C_{5}acs-2C_{6}ads\\ & & -C_{3}a(c^2+d^2)+C_{4}b(c^2+d^2)-C_{5}c(a^2+b^2)+C_{6}d(a^2+b^2)\\ & =& (C_3+C_5)s^3\\ & & +(-2C_{3}c-C_{3}a+C_{4}b-2C_{5}a-C_{5}c+C_{6}d)s^2\\ & & +(C_3(c^2+d^2)+2C_{3}ac-2C_{4}bc+C_5(a^2+b^2)+2C_{5}ac-2C_{6}ad)s\\ & & +\{-C_{3}a(c^2+d^2)+C_{4}b(c^2+d^2)-C_{5}c(a^2+b^2)+C_{6}d(a^2+b^2)\}\\ & =& (C_3+C_5)s^3\\ & & +(C_3(-2c-a)+C_{4}b+C_5(-2a-c)+C_{6}d)s^2\\ & & +(C_3((c^2+d^2)+2ac)+C_4(-2bc)+C_5((a^2+b^2)+2ac)+C_6(-2ad))s\\ & & +\{C_3(-a(c^2+d^2))+C_{4}b(c^2+d^2)+C_5(-c(a^2+b^2))+C_{6}d(a^2+b^2)\}\end{array}$$ $$\{\begin{array}{rcl}0& =& C_3+C_5\\ 0& =& C_3(-2c-a)+C_{4}b+C_5(-2a-c)+C_{6}d\\ 1& =& C_3(c^2+d^2+2ac)+C_4(-2bc)+C_5((a^2+b^2)+2ac)+C_6(-2ad)\\ 0& =& C_3(-a(c^2+d^2))+C_{4}b(c^2+d^2)+C_5(-c(a^2+b^2))+C_{6}d(a^2+b^2)\end{array}$$

解く

$$\begin{array}{rcl}\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right]& =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2c-a& b& -2a-c& d\\ c^2+d^2+2ac& -2bc& (a^2+b^2)+2ac& -2ad\\ -a(c^2+d^2)& b(c^2+d^2)& -c(a^2+b^2)& d(a^2+b^2)\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2c-0& b& -2\cdot 0-c& d\\ c^2+d^2+2\cdot 0\cdot c& -2bc& (0^2+b^2)+2\cdot 0\cdot c& -2\cdot 0\cdot d\\ -0(c^2+d^2)& b(c^2+d^2)& -c(0^2+b^2)& d(0^2+b^2)\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2c& b& -c& d\\ c^2+d^2& -2bc& b^2& 0\\ 0& b(c^2+d^2)& -b^{2}c& b^{2}d\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ -2(-\gamma )& ({\omega }_f)& -(-\gamma )& (i\xi )\\ (-\gamma {)}^2+(i\xi {)}^2& -2({\omega }_f)(-\gamma )& ({\omega }_f{)}^2& 0\\ 0& ({\omega }_f)((-\gamma {)}^2+(i\xi {)}^2)& -({\omega }_f{)}^2(-\gamma )& ({\omega }_f{)}^2(i\xi )\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ 2\gamma & {\omega }_f& \gamma & i\xi \\ {\gamma }^2-{\xi }^2& 2\gamma {\omega }_f& {\omega }_f^2& 0\\ 0& {\omega }_f({\gamma }^2-{\xi }^2)& \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\\ & =& \left[\begin{array}{cccc}1& 0& 1& 0\\ 2\gamma & {\omega }_f& \gamma & i\xi \\ {\omega }_0^2& 2\gamma {\omega }_f& {\omega }_f^2& 0\\ 0& {\omega }_0^2{\omega }_f& \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right]\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\dots {\omega }_0^2={\gamma }^2-{\xi }^2({\xi }^2={\gamma }^2-{\omega }_0^2\mathrm{よ}\mathrm{り})\\ & =& \mathit{A}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]\end{array}$$ $$\begin{array}{rcl}\left|{\mathit{M}}_{31}\right|& =& \left|\begin{array}{ccc}0& 1& 0\\ {\omega }_f& \gamma & i\xi \\ {\omega }_0^2{\omega }_f& \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right|\\ & =& -\left|\begin{array}{cc}{\omega }_f& i\xi \\ {\omega }_0^2{\omega }_f& i\xi {\omega }_f^2\end{array}\right|\\ & =& -\{{\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|{\mathit{M}}_{32}\right|& =& \left|\begin{array}{ccc}1& 1& 0\\ 2\gamma & \gamma & i\xi \\ 0& \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right|\\ & =& \left|\begin{array}{cc}\gamma & i\xi \\ \gamma {\omega }_f^2& i\xi {\omega }_f^2\end{array}\right|-\left|\begin{array}{cc}2\gamma & i\xi \\ 0& i\xi {\omega }_f^2\end{array}\right|\\ & =& (\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|{\mathit{M}}_{33}\right|& =& \left|\begin{array}{ccc}1& 0& 0\\ 2\gamma & {\omega }_f& i\xi \\ 0& {\omega }_0^2{\omega }_f& i\xi {\omega }_f^2\end{array}\right|\\ & =& \left|\begin{array}{cc}{\omega }_f& i\xi \\ {\omega }_0^2{\omega }_f& i\xi {\omega }_f^2\end{array}\right|\\ & =& {\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|{\mathit{M}}_{34}\right|& =& \left|\begin{array}{ccc}1& 0& 1\\ 2\gamma & {\omega }_f& \gamma \\ 0& {\omega }_0^2{\omega }_f& \gamma {\omega }_f^2\end{array}\right|\\ & =& \left|\begin{array}{cc}{\omega }_f& \gamma \\ {\omega }_0^2{\omega }_f& \gamma {\omega }_f^2\end{array}\right|+\left|\begin{array}{cc}2\gamma & {\omega }_f\\ 0& {\omega }_0^2{\omega }_f\end{array}\right|\\ & =& {\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|{\mathit{M}}_{31}\right|+(-1{)}^{3+2}{a}_{32}\left|{\mathit{M}}_{32}\right|+(-1{)}^{3+3}{a}_{33}\left|{\mathit{M}}_{33}\right|+(-1{)}^{3+4}{a}_{34}\left|{\mathit{M}}_{34}\right|\\ & =& {\omega }_0^2\cdot (i\xi {\omega }_f({\omega }_0^2-{\omega }_f^2))\\ & & -2\gamma {\omega }_f\cdot (-i2\gamma \xi {\omega }_f^2)\\ & & +{\omega }_f^2\cdot (-i\xi {\omega }_f({\omega }_0^2-{\omega }_f^2))\\ & & -0\cdot (\gamma {\omega }_f({\omega }_0^2+{\omega }_f^2))\\ & =& i\xi {\omega }_0^2{\omega }_f({\omega }_0^2-{\omega }_f^2)+i4{\gamma }^2\xi {\omega }_f^3-i\xi {\omega }_f^3({\omega }_0^2-{\omega }_f^2)-0\\ & =& i\xi {\omega }_f({\omega }_0^2-{\omega }_f^2)({\omega }_0^2-{\omega }_f^2)+i4{\gamma }^2\xi {\omega }_f^3\\ & =& i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2\}\end{array}$$ $$\begin{array}{rcl}(-1{)}^{1+3}\frac{\left|{\mathit{M}}_{31}\right|}{\left|A\right|}& =& \frac{i\xi {\omega }_f({\omega }_0^2-{\omega }_f^2)}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2\}}\\ & =& \frac{{\omega }_0^2-{\omega }_f^2}{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2}\\ (-1{)}^{2+3}\frac{\left|{\mathit{M}}_{32}\right|}{\left|A\right|}& =& -1\cdot \frac{-i2\gamma \xi {\omega }_f^2}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2\}}\\ & =& \frac{2\gamma {\omega }_f}{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2}\\ (-1{)}^{3+3}\frac{\left|{\mathit{M}}_{33}\right|}{\left|A\right|}& =& \frac{-i\xi {\omega }_f({\omega }_0^2-{\omega }_f^2)}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2\}}\\ & =& \frac{-({\omega }_0^2-{\omega }_f^2)}{{({\omega }_0^2+{\omega }_f^2)}^2-{\left(2\gamma {\omega }_f\right)}^2}\\ (-1{)}^{4+3}\frac{\left|{\mathit{M}}_{34}\right|}{\left|A\right|}& =& -1\cdot \frac{\gamma {\omega }_f({\omega }_0^2+{\omega }_f^2)}{i\xi {\omega }_f\{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2\}}\\ & =& \frac{-\gamma ({\omega }_0^2+{\omega }_f^2)}{i\xi \{{({\omega }_0^2-{\omega }_f^2)}^2-(2\gamma {\omega }_f{)}^2\}}\end{array}$$ $$\begin{array}{rcl}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& {\mathit{A}}^{-1}\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right]\\ & =& \left[\begin{array}{cccc}(-1{)}^{1+1}\frac{\left|{\mathit{M}}_{11}\right|}{\left|A\right|}& (-1{)}^{1+2}\frac{\left|{\mathit{M}}_{21}\right|}{\left|A\right|}& (-1{)}^{1+3}\frac{\left|{\mathit{M}}_{31}\right|}{\left|A\right|}& (-1{)}^{1+4}\frac{\left|{\mathit{M}}_{41}\right|}{\left|A\right|}\\ (-1{)}^{2+1}\frac{\left|{\mathit{M}}_{12}\right|}{\left|A\right|}& (-1{)}^{2+2}\frac{\left|{\mathit{M}}_{22}\right|}{\left|A\right|}& (-1{)}^{2+3}\frac{\left|{\mathit{M}}_{32}\right|}{\left|A\right|}& (-1{)}^{2+4}\frac{\left|{\mathit{M}}_{42}\right|}{\left|A\right|}\\ (-1{)}^{3+1}\frac{\left|{\mathit{M}}_{13}\right|}{\left|A\right|}& (-1{)}^{3+2}\frac{\left|{\mathit{M}}_{23}\right|}{\left|A\right|}& (-1{)}^{3+3}\frac{\left|{\mathit{M}}_{33}\right|}{\left|A\right|}& (-1{)}^{3+4}\frac{\left|{\mathit{M}}_{43}\right|}{\left|A\right|}\\ (-1{)}^{4+1}\frac{\left|{\mathit{M}}_{14}\right|}{\left|A\right|}& (-1{)}^{4+2}\frac{\left|{\mathit{M}}_{24}\right|}{\left|A\right|}& (-1{)}^{4+3}\frac{\left|{\mathit{M}}_{34}\right|}{\left|A\right|}& (-1{)}^{4+4}\frac{\left|{\mathit{M}}_{44}\right|}{\left|A\right|}\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\\ 0\end{array}\right]\\ & =& \left[\begin{array}{c}(-1{)}^{1+3}\frac{\left|{\mathit{M}}_{31}\right|}{\left|A\right|}\cdot 1\\ (-1{)}^{2+3}\frac{\left|{\mathit{M}}_{32}\right|}{\left|A\right|}\cdot 1\\ (-1{)}^{3+3}\frac{\left|{\mathit{M}}_{33}\right|}{\left|A\right|}\cdot 1\\ (-1{)}^{4+3}\frac{\left|{\mathit{M}}_{34}\right|}{\left|A\right|}\cdot 1\end{array}\right]=\left[\begin{array}{c}\frac{{\omega }_0^2-{\omega }_f^2}{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2}\\ \frac{2\gamma {\omega }_f}{{({\omega }_0^2-{\omega }_f^2)}^2+(2\gamma {\omega }_f{)}^2}\\ \frac{-({\omega }_0^2-{\omega }_f^2)}{{({\omega }_0^2-{\omega }_f^2)}^2-{\left(2\gamma {\omega }_f\right)}^2}\\ \frac{-\gamma ({\omega }_f^2+{\omega }_0^2)}{i\xi \{{({\omega }_0^2-{\omega }_f^2)}^2-(2\gamma {\omega }_f{)}^2\}}\end{array}\right]\\ & =& \frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2-{\left(2\gamma {\omega }_f\right)}^2}\left[\begin{array}{c}{\omega }_0^2-{\omega }_f^2\\ 2\gamma {\omega }_f\\ -({\omega }_0^2-{\omega }_f^2)\\ -\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)\end{array}\right]\end{array}$$

$C_3,C_4,C_5,C_6$

$$\begin{array}{rcl}\left[\begin{array}{c}{C}_3\\ C_4\\ C_5\\ C_6\end{array}\right]& =& \frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2-{\left(2\gamma {\omega }_f\right)}^2}\left[\begin{array}{c}{\omega }_0^2-{\omega }_f^2\\ 2\gamma {\omega }_f\\ -({\omega }_0^2-{\omega }_f^2)\\ -\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)\end{array}\right]\end{array}$$

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求まった係数を用いて表す

$$\begin{array}{rcl}X& =& \frac{C_1}{s-{\lambda }_1}+\frac{C_2}{s-{\lambda }_2}+\frac{F}{m}\{C_3\frac{s-a}{{(s-a)}^2+b^2}+C_4\frac{b}{{(s-a)}^2+b^2}+C_5\frac{s-c}{{(s-c)}^2+d^2}+C_6\frac{d}{{(s-c)}^2+d^2}\}\\ & =& \frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\frac{s-0}{{(s-0)}^2+{\omega }_f^2}+2\gamma {\omega }_f\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}-({\omega }_0^2-{\omega }_f^2)\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}-\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\}\\ & =& \frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\frac{s-0}{{(s-0)}^2+{\omega }_f^2}+2\gamma {\omega }_f\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}\}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{-({\omega }_0^2-{\omega }_f^2)\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}-\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\}\end{array}$$

第1,2項の逆ラプラス変換

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}[\frac{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_1}+\frac{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }}{s-{\lambda }_2}]\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi }){\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_1}\right]+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi }){\mathfrak{L}}^{-1}\left[\frac{1}{s-{\lambda }_2}\right]\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{{\lambda }_{1}t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{{\lambda }_{2}t}\\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{(-\gamma +\xi )t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{(-\gamma -\xi )t}\dots {\lambda }_{1,2}=-\gamma \pm \xi \\ & =& (\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\xi })e^{-\gamma t}{e}^{\xi t}+(\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\xi })e^{-\gamma t}{e}^{-\xi t}\dots e^{A+B}=e^A{e}^B\\ & =& e^{-\gamma t}\{\frac{x_0}{2}{e}^{\xi t}+\frac{v_0+\gamma x_0}{2\xi }{e}^{\xi t}+\frac{x_0}{2}{e}^{-\xi t}-\frac{v_0+\gamma x_0}{2\xi }{e}^{-\xi t}\}\\ & =& e^{-\gamma t}\{\frac{x_0}{2}(e^{\xi t}+e^{-\xi t})+\frac{v_0+\gamma x_0}{2\xi }(e^{\xi t}-e^{-\xi t})\}\\ & =& e^{-\gamma t}\{\frac{x_0}{2}(e^{i\omega t}+e^{-i\omega t})+\frac{v_0+\gamma x_0}{2i\omega }(e^{i\omega t}-e^{-i\omega t})\}\dots \xi =\sqrt{{\gamma }^2-{\omega }_0^2}=\sqrt{-1}\sqrt{|{\gamma }^2-{\omega }^2|}=i\omega (\gamma <{\omega }_0\mathrm{の}\mathrm{場}\mathrm{合})\\ & =& e^{-\gamma t}\{\frac{x_0}{2}2\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{2i\omega }2i\sin \left(\omega t\right)\}\dots \cos \left(\theta \right)=\frac{e^{i\theta }+e^{-i\theta }}{2},\sin \left(\theta \right)=\frac{e^{i\theta }+e^{-i\theta }}{2i}\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\}\\ & =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}+v_0{e}^{-\gamma t}\frac{1}{\omega }\sin \left(\omega t\right)\end{array}$$

第3項の逆ラプラス変換

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}\left[\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\frac{s-0}{{(s-0)}^2+{\omega }_f^2}+2\gamma {\omega }_f\frac{{\omega }_f}{{(s-0)}^2+{\omega }_f^2}\}\right]\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)e^{0t}\cos \left({\omega }_{f}t\right)+2\gamma {\omega }_f{e}^{0t}\sin \left({\omega }_{f}t\right)\}\\ & =& \frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left({\omega }_{f}t\right)+2\gamma {\omega }_f\sin \left({\omega }_{f}t\right)\}\end{array}$$

第4項の逆ラプラス変換

$$\begin{array}{rcl}& & {\mathfrak{L}}^{-1}\left[\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{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}-\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\}\right]\\ & =& \frac{F}{m}\frac{-({\omega }_0^2-{\omega }_f^2)}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}{\mathfrak{L}}^{-1}\left[\frac{s-(-\gamma )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\right]+\frac{F}{m}\frac{-\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}{\mathfrak{L}}^{-1}\left[\frac{(i\xi )}{{(s-(-\gamma ))}^2+(i\xi {)}^2}\right]\\ & =& \frac{F}{m}\frac{-({\omega }_0^2-{\omega }_f^2)}{{({\omega }_0^2-{\omega }_f^2)}^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)}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}{e}^{-\gamma t}\sin \left(i\xi 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)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)\}\\ & =& \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(i\xi t\right)+\frac{\gamma }{i\xi }({\omega }_0^2+{\omega }_f^2)\sin \left(i\xi 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 (i\cdot i\omega t)+\frac{\gamma }{i\cdot i\omega }({\omega }_0^2+{\omega }_f^2)\sin (i\cdot i\omega t)\}\dots \xi =i\omega \\ & =& \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 (-\omega t)+(-1)\frac{\gamma }{\omega }({\omega }_0^2+{\omega }_f^2)\sin (-\omega t)\}\dots i\cdot 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)+(-1)\frac{\gamma }{\omega }({\omega }_0^2+{\omega }_f^2)(-1)\sin \left(\omega t\right)\}\dots \cos (-\theta )=\cos \left(\theta \right),\sin (-\theta )=-\sin \left(\theta \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 -1\cdot -1=1\end{array}$$

全体の逆ラプラス変換(1,2,4,3項の順に入れ替え)

$$\begin{array}{rcl}{\mathfrak{L}}^{-1}\left[X\right]=x\left(t\right)& =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}\\ & & +v_0{e}^{-\gamma t}\frac{1}{\omega }\sin \left(\omega t\right)\\ & & +\frac{F}{m}\frac{-e^{-\gamma t}}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left(\omega t\right)+\frac{\gamma }{\omega }({\omega }_0^2+{\omega }_f^2)\sin \left(\omega t\right)\}\\ & & +\frac{F}{m}\frac{1}{{({\omega }_0^2-{\omega }_f^2)}^2+{\left(2\gamma {\omega }_f\right)}^2}\{({\omega }_0^2-{\omega }_f^2)\cos \left({\omega }_{f}t\right)+2\gamma {\omega }_f\sin \left({\omega }_{f}t\right)\}\end{array}$$

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

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