バネマスダンパー系，減衰振動(0<γ<ω0)の場合

バネマスダンパー系

$0<\gamma <{\omega }_0$

$$\begin{array}{rcl}{\lambda }_{1,2}& =& {-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}|}_{0<\gamma <{\omega }_0}\dots {\lambda }_{1,2}=-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\\ & =& -\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\\ & =& -\gamma \pm \sqrt{-{\omega }^2}\dots {\omega }^2=|{\gamma }^2-{\omega }_0^2|,{\gamma }^2-{\omega }_0^2=-{\omega }^2\\ & =& -\gamma \pm \omega \sqrt{-1}\\ & =& -\gamma \pm \omega i\end{array}$$ $$\begin{array}{rcl}{C}_{1,2}& =& {\frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}|}_{0<\gamma <{\omega }_0}\dots C_{1,2}=\frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\sqrt{{\gamma }^2-{\omega }_0^2}}\\ & =& \frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\omega \sqrt{-1}}\\ & =& \frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\omega i}\frac{i}{i}\\ & =& \frac{x_0}{2}\pm \frac{v_0+\gamma x_0}{2\omega \cdot (-1)}i\\ & =& \frac{x_0}{2}\mp \frac{v_0+\gamma x_0}{2\omega }i\dots \mathrm{た}\mathrm{だ}\mathrm{し}\omega \ne 0\end{array}$$ $$\begin{array}{rcl}x(t)& =& C_1{e}^{{\lambda }_{1}t}+C_2{e}^{{\lambda }_{2}t}\\ & =& C_1{e}^{(-\gamma +\omega i)t}+C_2{e}^{(-\gamma -\omega i)t}\\ & =& C_1{e}^{-\gamma t}{e}^{\omega it}+C_2{e}^{-\gamma t}{e}^{-\omega it}\\ & =& e^{-\gamma t}\{C_1{e}^{\omega it}+C_2{e}^{-\omega it}\}\\ \{C_1{e}^{\omega it}+C_2{e}^{-\omega it}\}& =& \{\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}\\ & =& \{\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)\}\dots e^{ix}=\cos \left(x\right)+i\sin \left(x\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)\}\\ & & +\{\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 (-\theta )=\cos \left(\theta \right),\sin (-\theta )=-\sin \left(\theta \right)\\ & =& \{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega }i\}\cos \left(\omega t\right)+i\{\frac{x_0}{2}-\frac{v_0+\gamma x_0}{2\omega }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\{\frac{x_0}{2}+\frac{v_0+\gamma x_0}{2\omega }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}i+\frac{v_0+\gamma x_0}{2\omega }\}\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}i-\frac{v_0+\gamma x_0}{2\omega }\}\sin \left(\omega t\right)\\ & & \dots i\cdot i=-1\\ & =& \left\{\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\right\}\cos \left(\omega t\right)\\ & & +\left\{\frac{x_0}{2}i+\frac{v_0+\gamma x_0}{2\omega }-\frac{x_0}{2}i+\frac{v_0+\gamma x_0}{2\omega }\right\}\sin \left(\omega t\right)\\ & =& \{\frac{x_0}{2}+\frac{x_0}{2}\}\cos \left(\omega t\right)+\{\frac{v_0+\gamma x_0}{2\omega }+\frac{v_0+\gamma x_0}{2\omega }\}\sin \left(\omega t\right)\\ & =& x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\end{array}$$ $$\begin{array}{rcl}x(t)& =& e^{-\gamma t}\{C_1{e}^{\omega it}+C_2{e}^{-\omega it}\}\\ & =& e^{-\gamma t}\{x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)\}\\ & =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\gamma }{\omega }\sin \left(\omega t\right)\}+v_0{e}^{-\gamma t}\{\frac{1}{\omega }\sin \left(\omega t\right)\}\dots \mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{減}\mathrm{衰}\mathrm{振}\mathrm{動},\mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{減}\mathrm{衰}\mathrm{振}\mathrm{動}\\ & =& x_0{e}^{-\gamma t}\{\cos \left(\omega t\right)+\frac{\zeta }{\sqrt{{\zeta }^2-1}}\sin \left(\omega t\right)\}+v_0{e}^{-\gamma t}\{\frac{1}{\omega }\sin \left(\omega t\right)\}\\ & & \dots \frac{\gamma }{\omega }=\frac{\gamma }{\sqrt{|{\gamma }^2-{\omega }_0^2|}}=\frac{\gamma }{\sqrt{\frac{{\omega }_0^2}{{\omega }_0^2}|{\gamma }^2-{\omega }_0^2|}}=\frac{\gamma }{\sqrt{{\omega }_0^2|\frac{{\gamma }^2}{{\omega }_0^2}-\frac{{\omega }_0^2}{{\omega }_0^2}|}}=\frac{\gamma }{{\omega }_0\sqrt{|{\zeta }^2-1|}}=\frac{\zeta }{\sqrt{|{\zeta }^2-1|}}\\ & & \dots \omega =\sqrt{|{\gamma }^2-{\omega }_0^2|},\zeta =\frac{\gamma }{{\omega }_0}\end{array}$$