バネマスダンパー系，単振動(γ=0 / バネマス系)の場合

バネマスダンパー系

$\gamma =0$

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