バネマスダンパー系
運動方程式
$$\begin{eqnarray}
m\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&c\frac{\mathrm{d}x}{\mathrm{d}t}
&+&kx
&=&0
\;\ldots\;m:マス,\;c:ダンパー,\;バネ
\\\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&\frac{c}{m}\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\frac{k}{m}x
&=&0
\\\frac{\mathrm{d^2}x}{\mathrm{d^2}t}
&+&2\gamma\frac{\mathrm{d}x}{\mathrm{d}t}
&+&\omega_0^2x
&=&0
\;\ldots\;\gamma=\frac{c}{2m},\;\omega_0=\sqrt{\frac{k}{m}}
\end{eqnarray}$$
単振動，振動減衰，臨界減衰，過減衰
\(\gamma=\frac{c}{2m},\;\omega_0=\sqrt{\frac{k}{m}},\;x_0=x\left(0\right),\;v_0=\left.\frac{\mathrm{d}x(t)}{\mathrm{d}t}\right|_{t=0}\)
|
\(\gamma\) |
\(\zeta=\frac{\gamma}{\omega_0}\) |
\(\xi=\omega_0\sqrt{\zeta^2-1}\) |
\(e^{-\gamma t}\left[x_0 \cosh\left(\xi t\right)+\frac{v_0 +\gamma x_0 }{\xi}\sinh{\left(\xi t \right)}\right]\) |
\(x(t)\) |
| 単振動 |
\(0\) |
\(0\) |
\(\omega_0i\) |
\(e^{-\color{red}{0}\color{black}{t}}\left[x_0 \cosh\left(\color{red}{\omega_0 i}\color{black}{t}\right)+\frac{v_0 +\color{red}{0}\color{black}{ x_0} }{\color{red}{\omega_0 i}} \sinh{\left(\color{red}{\omega_0 i}\color{black}{t}\right)}\right]\) |
\(x_0 \cos\left(\omega_0 t\right)+\frac{v_0}{\omega_0 } \sin{\left(\omega_0 t \right)}\) |
| 振動減衰 |
\(0\lt\gamma\lt\omega_0\) |
\(0\lt\zeta\lt1\) |
\(\omega i\)
\(ただし0\lt\omega\lt1\) |
\(e^{-\color{red}{\gamma}\color{black}{t}}\left[x_0 \cosh\left(\color{red}{\omega i}\color{black}{t}\right)+\frac{v_0 +\color{red}{\gamma}\color{black}{x_0} }{\color{red}{\omega i}}\sinh{\left(\color{red}{\omega i}\color{black}{t}\right)}\right]\) |
\(e^{-\gamma t}\left[x_0 \cos\left(\omega i t \right)+\frac{v_0 +\gamma x_0 }{\omega } \sin{\left(\omega t \right)}\right]\) |
| 臨界減衰 |
\(\omega_0\) |
\(1\) |
\(0\) |
\(e^{-\color{red}{\gamma}\color{black}{t}}\left[x_0 \color{red}{\left(1+\frac{1}{2!}\left(\xi t\right)^2+\cdots\right)}
\color{black}{+}\frac{v_0 +\color{red}{\gamma}\color{black}{x_0} }{\xi} \color{red}{\left( \xi t + \frac{1}{3!}\left(\xi t\right)^3 + \cdots\right)}
\right]\) |
\(e^{-\gamma t}\left[x_0 +\left(v_0 +\gamma x_0\right) t\right]\) |
| 過減衰 |
\(\omega_0\lt\gamma\) |
\(1\lt\zeta\) |
\(\omega\)
\(ただし1\lt\omega\) |
\(e^{-\color{red}{\gamma}\color{black}{t}}\left[x_0 \cosh\left(\color{red}{\omega}\color{black}{t}\right)+\frac{v_0 +\color{red}{\gamma}\color{black}{x_0} }{\color{red}{\omega}}\sinh{\left(\color{red}{\omega}\color{black}{t}\right)}\right]\) |
\(e^{-\gamma t}\left[x_0 \cosh\left(\omega t\right)+\frac{v_0 +\gamma x_0 }{\omega}\sinh{\left(\omega t \right)}\right]\) |
過減衰の式
$$\begin{eqnarray}
x(t)&=&\href{https://shikitenkai.blogspot.com/2021/04/0_17.html}{e^{-\gamma t}\left[x_0 \cosh\left(\xi t\right)+\frac{v_0 +\gamma x_0 }{\xi}\sinh{\left(\xi t \right)}\right]}\;\ldots\;(\href{https://shikitenkai.blogspot.com/2021/04/0_17.html}{導出})
\end{eqnarray}$$
\(\omega_0\lt\gamma\)
$$\begin{eqnarray}
x(t)&=&\left.e^{-\gamma t}\left[x_0 \cosh\left(\xi t\right)+\frac{v_0 +\gamma x_0 }{\xi}\sinh{\left(\xi t \right)}\right]\right|_{1\lt\gamma,\;\xi=\omega}
\\&&\;\ldots\;\omega_0\lt\gamma,\;1\lt\zeta,\;\xi=\omega_0\sqrt{\zeta^2-1}=\omega
\\&&\;\ldots\;\omega=\omega_0\sqrt{\left|\zeta^2-1\right|}
\\&=&e^{-\gamma t}\left[x_0 \cosh\left(\omega t\right)+\frac{v_0 +\gamma x_0 }{\omega}\sinh{\left(\omega t \right)}\right]
\end{eqnarray}$$
\(0\lt\gamma\lt\omega_0\)
$$\begin{eqnarray}
\\x(t)&=&\left.e^{-\gamma t}\left[x_0 \cosh\left(\xi t\right)+\frac{v_0 +\gamma x_0 }{\xi}\sinh{\left(\xi t \right)}\right]\right|_{0\lt\gamma\lt1,\;\xi=\omega i}
\\&&\;\ldots\;0\lt\gamma\lt\omega_0,\;0\lt\zeta\lt1,\;\xi=\omega_0\sqrt{\zeta^2-1}=\omega i
\\&&\;\ldots\;\omega=\omega_0\sqrt{\left|\zeta^2-1\right|}
\\&=&e^{-\gamma t}\left[x_0 \cosh\left(\omega i t\right)+\frac{v_0 +\gamma x_0 }{\omega i}\sinh{\left(\omega i t \right)}\right]
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/coshi-x-sinhi-x-cosh-sinh.html}{\cosh{\left(i x\right)}=\cos{\left(x\right)},\;\sinh{\left(i x\right)}=i\sin{\left(x\right)}}
\\&=&e^{-\gamma t}\left[x_0 \cos\left(\omega t\right)+\frac{v_0 +\gamma x_0 }{\omega i} i \sin{\left(\omega t \right)}\right]
\\&=&e^{-\gamma t}\left[x_0 \cos\left(\omega t\right)+\frac{v_0 +\gamma x_0 }{\omega } \sin{\left(\omega t \right)}\right]
\end{eqnarray}$$
\(\gamma=0\)
$$\begin{eqnarray} \\x(t)&=&\left.e^{-\gamma t}\left[x_0 \cosh\left(\xi t\right)+\frac{v_0 +\gamma x_0 }{\xi}\sinh{\left(\xi t \right)}\right]\right|_{\gamma=0,\;\xi=\omega_0 i}
\\&&\;\ldots\;\gamma=0,\;\zeta=0,\;\xi=\omega_0\sqrt{0-1}=\omega_0 i
\\&&\;\ldots\;\omega=\omega_0\sqrt{\left|\zeta^2-1\right|}
\\&=&e^{-0 t}\left[x_0 \cosh\left(\omega_0 i t\right)+\frac{v_0 +0 x_0 }{\omega_0 i} \sinh{\left(\omega_0 i t \right)}\right]
\\&=&x_0 \cos\left(\omega_0 t\right)+\frac{v_0 }{\omega_0 i} i \sin{\left(\omega_0 t \right)}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/coshi-x-sinhi-x-cosh-sinh.html}{\cosh{\left(i x\right)}=\cos{\left(x\right)},\;\sinh{\left(i x\right)}=i\sin{\left(x\right)}}
\\&&\;\ldots\;e^0=1
\\&=&x_0 \cos\left(\omega_0 t\right)+\frac{v_0}{\omega_0 } \sin{\left(\omega_0 t \right)}
\end{eqnarray}$$
\(\gamma\rightarrow\omega_0\)
$$\begin{eqnarray}
\\x(t)&=&\lim_{\gamma\rightarrow\omega_0,\;\xi \rightarrow 0}{e^{-\gamma t}\left[x_0 \cosh\left(\xi t\right)+\frac{v_0 +\gamma x_0 }{\xi}\sinh{\left(\xi t \right)}\right]}
\\&&\;\ldots\;\gamma\rightarrow\omega_0,\;\zeta\rightarrow1,\;\xi\rightarrow0
\\&=&\lim_{\gamma\rightarrow\omega_0,\;\xi \rightarrow 0}
{e^{-\gamma t}\left[
x_0 \left(1+\frac{1}{2!}\left(\xi t\right)^2+\cdots\right)
+\frac{v_0 +\gamma x_0 }{\xi} \left( \xi t + \frac{1}{3!}\left(\xi t\right)^3 + \cdots\right)
\right]}
\\&&\;\ldots\;\href{https://shikitenkai.blogspot.com/2021/04/coshx.html}{\cosh\left(\xi t\right)=\frac{1}{0!}\left(\xi t\right)^0+\frac{1}{2!}\left(\xi t\right)^2+\cdots}
,\;\href{https://shikitenkai.blogspot.com/2021/04/sinhx.html}{\sinh{\left(\xi t \right)}=\frac{1}{1!}\left(\xi t\right)^1 + \frac{1}{3!}\left(\xi t\right)^3 + \cdots}
\\&=&\lim_{\gamma\rightarrow\omega_0,\;\xi \rightarrow 0}
{e^{-\gamma t}\left[
x_0 \left(1+\frac{1}{2!}\left(\xi t\right)^2+\cdots\right)
+\left(v_0 +\gamma x_0\right) \left( t + \frac{1}{3!}\xi^2 t^3 + \cdots\right)
\right]}
\\&=&e^{-\omega_0 t}\left[
x_0 \cdot \left(1\right)
+\left(v_0 +\omega_0 x_0\right) \cdot \left( t \right)
\right]
\\&=&e^{-\omega_0 t}\left[x_0 +\left(v_0 +\omega_0 x_0\right) t\right]
\\&=&e^{-\gamma t}\left[x_0 +\left(v_0 +\gamma x_0\right) t\right]
\end{eqnarray}$$
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