バネマスダンパー系，過減衰(ω0<γ)の場合

バネマスダンパー系

${\omega }_0<\gamma$

$$\begin{array}{rcl}{\lambda }_{1,2}& =& -\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\dots {\lambda }_{1,2}=-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\\ & =& -\gamma \pm \sqrt{{\omega }^2}\dots {\omega }^2=|{\gamma }^2-{\omega }_0^2|,{\gamma }^2-{\omega }_0^2={\omega }^2\\ & =& -\gamma \pm \omega \end{array}$$ $$\begin{array}{rcl}{C}_{1,2}& =& {\frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}|}_{\gamma >{\omega }_0}\dots C_{1,2}=\frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\\ & =& \frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\sqrt{{\omega }^2}}\\ & =& \frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\omega }\dots \mathrm{た}\mathrm{だ}\mathrm{し}\omega \ne 0\end{array}$$ $$\begin{array}{rcl}x(t)& =& C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}\\ & =& C_1{e}^{(-\gamma +\omega )t}+C_2{e}^{(-\gamma -\omega )t}\\ & =& C_1{e}^{-\gamma t}{e}^{\omega t}+C_2{e}^{-\gamma t}{e}^{-\omega t}\\ & =& e^{-\gamma t}\{C_1{e}^{\omega t}+C_2{e}^{-\omega t}\}\\ & =& e^{-\gamma t}[\{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega }\}{e}^{\omega t}+\{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega }\}{e}^{-\omega t}]\\ & =& e^{-\gamma t}\{\frac{x_0}{2}{e}^{\omega t}+\frac{v_0+\gamma x_0}{2\omega }{e}^{\omega t}+\frac{x_0}{2}{e}^{-\omega t}-\frac{v_0+\gamma x_0}{2\omega }{e}^{-\omega t}\}\\ & =& e^{-\gamma t}\{\frac{x_0}{2}(e^{\omega t}+e^{-\omega t})+\frac{v_0+\gamma x_0}{2\omega }(e^{\omega t}-e^{-\omega t})\}\\ & =& e^{-\gamma t}\{x_0\frac{e^{\omega t}+e^{-\omega t}}{2}+\frac{v_0+\gamma x_0}{\omega }\frac{e^{\omega t}-e^{-\omega t}}{2}\}\\ & =& e^{-\gamma t}\{x_0\cosh \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sinh \left(\omega t\right)\}\\ & & \dots \cosh \left(\omega t\right)=\frac{e^{\omega t}+e^{-\omega t}}{2},\sinh \left(\omega t\right)=\frac{e^{\omega t}-e^{-\omega t}}{2}\\ & =& x_0{e}^{-\gamma t}\{\cosh \left(\omega t\right)+\frac{\gamma }{\omega }\sinh \left(\omega t\right)\}+v_0{e}^{-\gamma t}\{\frac{1}{\omega }\sinh \left(\omega t\right)\}\dots \mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}{x}_0\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{過}\mathrm{減}\mathrm{衰},\mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}{v}_0\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{過}\mathrm{減}\mathrm{衰}\\ & =& x_0{e}^{-\gamma t}\{\cosh \left(\omega t\right)+\frac{\zeta }{\sqrt{{\zeta }^2-1}}\sinh \left(\omega t\right)\}+v_0{e}^{-\gamma t}\{\frac{1}{\omega }\sinh \left(\omega t\right)\}\\ & & \dots \frac{\gamma }{\omega }=\frac{\gamma }{\sqrt{|{\gamma }^2-{\omega }_0^2|}}=\frac{\gamma }{\sqrt{\frac{{\omega }_0^2}{{\omega }_0^2}|{\gamma }^2-{\omega }_0^2|}}=\frac{\gamma }{\sqrt{{\omega }_0^2|\frac{{\gamma }^2}{{\omega }_0^2}-\frac{{\omega }_0^2}{{\omega }_0^2}|}}=\frac{\gamma }{{\omega }_0\sqrt{|{\zeta }^2-1|}}=\frac{\zeta }{\sqrt{|{\zeta }^2-1|}}\\ & & \dots \omega =\sqrt{|{\gamma }^2-{\omega }_0^2|},\zeta =\frac{\gamma }{{\omega }_0}\end{array}$$