バネマスダンパー系，C1, C2 を求める

バネマスダンパー系

$C_1,C_2\mathrm{を}\mathrm{求}\mathrm{め}\mathrm{る}$

$$\begin{array}{r}\frac{s{x}_0+v_0+\frac{c}{m}{x}_0}{s^2+s\frac{c}{m}+\frac{k}{m}}=\frac{(C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)}{(s-{\lambda }_1)(s-{\lambda }_2)}\end{array}$$ $$\begin{array}{r}\\ s{x}_0+v_0+\frac{c}{m}{x}_0& =& (C_1+C_2)s-(C_1{\lambda }_2+C_2{\lambda }_1)\cdots \mathrm{分}\mathrm{子}\mathrm{に}\mathrm{つ}\mathrm{い}\mathrm{て}\mathrm{考}\mathrm{え}\mathrm{る}\end{array}$$ $$\{\begin{array}{rcl}{C}_1+C_2& =& x_0\\ -(C_1{\lambda }_2+C_2{\lambda }_1)& =& v_0+\frac{c}{m}{x}_0\\ & =& v_0+2\gamma x_0\end{array}\dots s\mathrm{の}\mathrm{係}\mathrm{数}\mathrm{を}\mathrm{比}\mathrm{較}\mathrm{し}\mathrm{連}\mathrm{立}\mathrm{さ}\mathrm{せ}\mathrm{る}$$ $$\begin{array}{rcl}{C}_1& =& x_0-C_2\\ -(C_1{\lambda }_2+C_2{\lambda }_1)& =& v_0+2\gamma x_0\\ C_1{\lambda }_2+C_2{\lambda }_1& =& -(v_0+2\gamma x_0)\\ (x_0-C_2){\lambda }_2+C_2{\lambda }_1& =& -v_0-2\gamma x_0\\ x_0{\lambda }_2-C_2{\lambda }_2+C_2{\lambda }_1& =& -v_0-2\gamma x_0\\ C_2({\lambda }_1-{\lambda }_2)& =& -v_0-2\gamma x_0-{\lambda }_2{x}_0\\ C_2({\lambda }_1-{\lambda }_2)& =& -v_0-x_0(2\gamma +{\lambda }_2)\\ \\ C_2& =& \frac{-v_0-x_0(2\gamma +{\lambda }_2)}{{\lambda }_1-{\lambda }_2}\\ & =& -\frac{v_0+x_0\{2\gamma +(-\gamma -\sqrt{{\gamma }^2-{\omega }_0^2})\}}{(-\gamma +\sqrt{{\gamma }^2-{\omega }_0^2})-(-\gamma -\sqrt{{\gamma }^2-{\omega }_0^2})}\\ & =& -\frac{v_0+x_0(\gamma -\sqrt{{\gamma }^2-{\omega }_0^2})}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\\ & =& -\frac{v_0+\gamma x_0-x_0\sqrt{{\gamma }^2-{\omega }_0^2}}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\\ & =& -\frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}+\frac{x_0\sqrt{{\gamma }^2-{\omega }_0^2}}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\\ & =& -\frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}+\frac{x_0}{2}\\ & =& \frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\\ \\ C_1& =& x_0-C_2\\ & =& x_0-\{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\}\\ & =& \frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\end{array}$$ $$\begin{array}{rcl}{C}_{1,2}& =& \frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\dots \mathrm{た}\mathrm{だ}\mathrm{し}\gamma \ne {\omega }_0\end{array}$$