バネマスダンパー系，γ,ω0による比較

バネマスダンパー系

$\gamma ,{\omega }_0$による比較

$$\begin{array}{rclrclr}\ddot{x}(t)& +& \frac{c}{m}\dot{x}(t)& +& \frac{k}{m}x(t)& =& 0\\ \frac{{\mathrm{d}}^2}{{\mathrm{d}}^{2}t}x(t)& +& \frac{c}{m}\frac{\mathrm{d}}{\mathrm{d}t}x(t)& +& \frac{k}{m}x(t)& =& 0\end{array}$$ $$\begin{array}{rcl}\gamma & =& \frac{c}{2m}\\ {\omega }_0^2& =& \frac{k}{m}\\ {\omega }^2& =& |{\gamma }^2-{\omega }_0^2|\\ & =& \left|\frac{{\omega }_0^2}{{\omega }_0^2}({\gamma }^2-{\omega }_0^2)\right|\\ & =& {\omega }_0^2|{\left(\frac{\gamma }{{\omega }_0}\right)}^2-1|\\ & =& {\omega }_0^2|{\zeta }^2-1|\dots \zeta =\frac{\gamma }{{\omega }_0}\\ \omega & =& \sqrt{{\omega }_0^2|{\zeta }^2-1|}\\ & =& {\omega }_0\sqrt{|{\zeta }^2-1|}\end{array}$$ $$\begin{array}{rclrclrc}x(t)& =& & x_0\cos \left({\omega }_{0}t\right)& +\frac{v_0}{{\omega }_0}\sin \left({\omega }_{0}t\right)& \dots \gamma =0& \mathrm{単}\mathrm{振}\mathrm{動}& \zeta =0\\ & =& e^{-\gamma t}& [x_0\cos \left(\omega t\right)& +\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)]& \dots 0<\gamma <{\omega }_0& \mathrm{減}\mathrm{衰}\mathrm{振}\mathrm{動}& 0<\zeta <1\\ & =& e^{-\gamma t}& [x_0& +(v_0+\gamma x_0)t]& \dots \gamma ={\omega }_0& \mathrm{臨}\mathrm{界}\mathrm{減}\mathrm{衰}& \zeta =1\\ & =& e^{-\gamma t}& [x_0\cosh \left(\omega t\right)& +\frac{v_0+\gamma x_0}{\omega }\sinh \left(\omega t\right)]& \dots {\omega }_0<\gamma & \mathrm{過}\mathrm{減}\mathrm{衰}& 1<\zeta \end{array}$$