バネマスダンパー系，過減衰の式からの他の式の導出

バネマスダンパー系

運動方程式

$$\begin{array}{rclrclr}m\frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& c\frac{\mathrm{d}x}{\mathrm{d}t}& +& kx& =& 0\dots m:\mathrm{マ}\mathrm{ス},c:\mathrm{ダ}\mathrm{ン}\mathrm{パ}\mathrm{ー},\mathrm{バ}\mathrm{ネ}\\ \frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& \frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}& +& \frac{k}{m}x& =& 0\\ \frac{{\mathrm{d}}^{2}x}{{\mathrm{d}}^{2}t}& +& 2\gamma \frac{\mathrm{d}x}{\mathrm{d}t}& +& {\omega }_0^{2}x& =& 0\dots \gamma =\frac{c}{2m},{\omega }_0=\sqrt{\frac{k}{m}}\end{array}$$

単振動，振動減衰，臨界減衰，過減衰

$\gamma =\frac{c}{2m},{\omega }_0=\sqrt{\frac{k}{m}},x_0=x\left(0\right),v_0={\frac{\mathrm{d}x(t)}{\mathrm{d}t}|}_{t=0}$

	$\gamma$	$\zeta =\frac{\gamma }{{\omega }_0}$	$\xi ={\omega }_0\sqrt{{\zeta }^2-1}$	$e^{-\gamma t}[x_0\cosh \left(\xi t\right)+\frac{v_0+\gamma x_0}{\xi }\sinh \left(\xi t\right)]$	$x(t)$
単振動	$0$	$0$	${\omega }_{0}i$	$e^{-0t}[x_0\cosh \left({\omega }_{0}it\right)+\frac{v_0+0x_0}{{\omega }_{0}i}\sinh \left({\omega }_{0}it\right)]$	$x_0\cos \left({\omega }_{0}t\right)+\frac{v_0}{{\omega }_0}\sin \left({\omega }_{0}t\right)$
振動減衰	$0<\gamma <{\omega }_0$	$0<\zeta <1$	$\omega i$ $\mathrm{た}\mathrm{だ}\mathrm{し}0<\omega <1$	$e^{-\gamma t}[x_0\cosh \left(\omega it\right)+\frac{v_0+\gamma x_0}{\omega i}\sinh \left(\omega it\right)]$	$e^{-\gamma t}[x_0\cos \left(\omega it\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)]$
臨界減衰	${\omega }_0$	$1$	$0$	$e^{-\gamma t}\left[x_0(1+\frac{1}{2!}{\left(\xi t\right)}^2+\cdots )+\frac{v_0+\gamma x_0}{\xi }(\xi t+\frac{1}{3!}{\left(\xi t\right)}^3+\cdots )\right]$	$e^{-\gamma t}[x_0+(v_0+\gamma x_0)t]$
過減衰	${\omega }_0<\gamma$	$1<\zeta$	$\omega$ $\mathrm{た}\mathrm{だ}\mathrm{し}1<\omega$	$e^{-\gamma t}[x_0\cosh \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sinh \left(\omega t\right)]$	$e^{-\gamma t}[x_0\cosh \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sinh \left(\omega t\right)]$

過減衰の式

$$\begin{array}{rcl}x(t)& =& e^{-\gamma t}[x_0\cosh \left(\xi t\right)+\frac{v_0+\gamma x_0}{\xi }\sinh \left(\xi t\right)]\dots (\mathrm{導}\mathrm{出})\end{array}$$

${\omega }_0<\gamma$

$$\begin{array}{rcl}x(t)& =& {e^{-\gamma t}[x_0\cosh \left(\xi t\right)+\frac{v_0+\gamma x_0}{\xi }\sinh \left(\xi t\right)]|}_{1<\gamma ,\xi =\omega }\\ & & \dots {\omega }_0<\gamma ,1<\zeta ,\xi ={\omega }_0\sqrt{{\zeta }^2-1}=\omega \\ & & \dots \omega ={\omega }_0\sqrt{|{\zeta }^2-1|}\\ & =& e^{-\gamma t}[x_0\cosh \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sinh \left(\omega t\right)]\end{array}$$

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

$$\begin{array}{r}\\ x(t)& =& {e^{-\gamma t}[x_0\cosh \left(\xi t\right)+\frac{v_0+\gamma x_0}{\xi }\sinh \left(\xi t\right)]|}_{0<\gamma <1,\xi =\omega i}\\ & & \dots 0<\gamma <{\omega }_0,0<\zeta <1,\xi ={\omega }_0\sqrt{{\zeta }^2-1}=\omega i\\ & & \dots \omega ={\omega }_0\sqrt{|{\zeta }^2-1|}\\ & =& e^{-\gamma t}[x_0\cosh \left(\omega it\right)+\frac{v_0+\gamma x_0}{\omega i}\sinh \left(\omega it\right)]\\ & & \dots \cosh \left(ix\right)=\cos \left(x\right),\sinh \left(ix\right)=i\sin \left(x\right)\\ & =& e^{-\gamma t}[x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega i}i\sin \left(\omega t\right)]\\ & =& e^{-\gamma t}[x_0\cos \left(\omega t\right)+\frac{v_0+\gamma x_0}{\omega }\sin \left(\omega t\right)]\end{array}$$

$\gamma =0$

$$\begin{array}{r}\\ x(t)& =& {e^{-\gamma t}[x_0\cosh \left(\xi t\right)+\frac{v_0+\gamma x_0}{\xi }\sinh \left(\xi t\right)]|}_{\gamma =0,\xi ={\omega }_{0}i}\\ & & \dots \gamma =0,\zeta =0,\xi ={\omega }_0\sqrt{0-1}={\omega }_{0}i\\ & & \dots \omega ={\omega }_0\sqrt{|{\zeta }^2-1|}\\ & =& e^{-0t}[x_0\cosh \left({\omega }_{0}it\right)+\frac{v_0+0x_0}{{\omega }_{0}i}\sinh \left({\omega }_{0}it\right)]\\ & =& x_0\cos \left({\omega }_{0}t\right)+\frac{v_0}{{\omega }_{0}i}i\sin \left({\omega }_{0}t\right)\\ & & \dots \cosh \left(ix\right)=\cos \left(x\right),\sinh \left(ix\right)=i\sin \left(x\right)\\ & & \dots e^0=1\\ & =& x_0\cos \left({\omega }_{0}t\right)+\frac{v_0}{{\omega }_0}\sin \left({\omega }_{0}t\right)\end{array}$$

$\gamma \to {\omega }_0$

$$\begin{array}{r}\\ x(t)& =& \underset{\gamma \to {\omega }_0,\xi \to 0}{lim}{e}^{-\gamma t}[x_0\cosh \left(\xi t\right)+\frac{v_0+\gamma x_0}{\xi }\sinh \left(\xi t\right)]\\ & & \dots \gamma \to {\omega }_0,\zeta \to 1,\xi \to 0\\ & =& \underset{\gamma \to {\omega }_0,\xi \to 0}{lim}{e}^{-\gamma t}[x_0(1+\frac{1}{2!}{\left(\xi t\right)}^2+\cdots )+\frac{v_0+\gamma x_0}{\xi }(\xi t+\frac{1}{3!}{\left(\xi t\right)}^3+\cdots )]\\ & & \dots \cosh \left(\xi t\right)=\frac{1}{0!}{\left(\xi t\right)}^0+\frac{1}{2!}{\left(\xi t\right)}^2+\cdots ,\sinh \left(\xi t\right)=\frac{1}{1!}{\left(\xi t\right)}^1+\frac{1}{3!}{\left(\xi t\right)}^3+\cdots \\ & =& \underset{\gamma \to {\omega }_0,\xi \to 0}{lim}{e}^{-\gamma t}[x_0(1+\frac{1}{2!}{\left(\xi t\right)}^2+\cdots )+(v_0+\gamma x_0)(t+\frac{1}{3!}{\xi }^2{t}^3+\cdots )]\\ & =& e^{-{\omega }_{0}t}[x_0\cdot \left(1\right)+(v_0+{\omega }_0{x}_0)\cdot \left(t\right)]\\ & =& e^{-{\omega }_{0}t}[x_0+(v_0+{\omega }_0{x}_0)t]\\ & =& e^{-\gamma t}[x_0+(v_0+\gamma x_0)t]\end{array}$$