状態空間表現，バネマスダンパー系，状態行列の固有値

状態空間表現

　 $$\{\begin{array}{rclrc}\frac{\mathrm{d}}{\mathrm{d}t}\mathit{x}& =& \mathit{A}\mathit{x}& +& \mathit{B}\mathit{u}\dots \mathit{A}:\mathrm{状}\mathrm{態}\mathrm{行}\mathrm{列},\mathit{B}:\mathrm{入}\mathrm{力}\mathrm{行}\mathrm{列},\mathit{x}:\mathrm{状}\mathrm{態}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル},\mathit{u}:\mathrm{入}\mathrm{力}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\\ \mathit{y}& =& \mathit{C}\mathit{x}& +& \mathit{D}\mathit{u}\dots \mathit{C}:\mathrm{出}\mathrm{力}\mathrm{行}\mathrm{列},\mathit{D}:\mathrm{直}\mathrm{達}\mathrm{行}\mathrm{列},\mathit{y}:\mathrm{出}\mathrm{力}\mathrm{ベ}\mathrm{ク}\mathrm{ト}\mathrm{ル}\end{array}$$

バネマスダンパー系(入力なし)

$$\{\begin{array}{rcl}\frac{\mathrm{d}}{\mathrm{d}t}\left[\begin{array}{c}x\\ v\end{array}\right]& =& \left[\begin{array}{cc}0& 1\\ -\frac{k}{m}& -\frac{c}{m}\end{array}\right]\left[\begin{array}{c}x\\ v\end{array}\right]\dots k:\mathrm{バ}\mathrm{ネ},m:\mathrm{マ}\mathrm{ス},c:\mathrm{ダ}\mathrm{ン}\mathrm{パ}\mathrm{ー},x:\mathrm{位}\mathrm{置},v=\frac{\mathrm{d}x}{\mathrm{d}t}:\mathrm{速}\mathrm{度}\\ \left[\begin{array}{c}y\end{array}\right]& =& \left[\begin{array}{cc}1& 0\end{array}\right]\left[\begin{array}{c}x\\ v\end{array}\right]\end{array}$$

状態行列の固有値

$$\begin{array}{rcl}|\lambda \mathit{I}-\mathit{A}|& =& 0\\ \left|\begin{array}{cc}\lambda -0& 1\\ -\frac{k}{m}& \lambda -(-\frac{c}{m})\end{array}\right|& =& \lambda \{\lambda -(-\frac{c}{m})\}-(-\frac{k}{m})\\ & =& \lambda (\lambda +\frac{c}{m})+\frac{k}{m}\\ & =& {\lambda }^2+\frac{c}{m}\lambda +\frac{k}{m}\\ {\lambda }^2+\frac{c}{m}\lambda +\frac{k}{m}& =& 0\\ {\lambda }_{1,2}& =& \frac{-\frac{c}{m}\pm \sqrt{{\left(\frac{c}{m}\right)}^2-4\frac{k}{m}}}{2}\\ & =& -\frac{c}{2m}\pm \sqrt{\frac{c^2}{4m^2}-\frac{k}{m}}\\ & =& -\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\dots \gamma =\frac{c}{2m},{\omega }_0=\sqrt{\frac{k}{m}}\\ & =& -\gamma \pm \frac{{\omega }_0}{{\omega }_0}\sqrt{{\gamma }^2-{\omega }_0^2}\\ & =& -\gamma \pm {\omega }_0\sqrt{\frac{1}{{\omega }_0^2}({\gamma }^2-{\omega }_0^2)}\\ & =& -\gamma \pm {\omega }_0\sqrt{\frac{{\gamma }^2}{{\omega }_0^2}-\frac{{\omega }_0^2}{{\omega }_0^2}}\\ & =& -\gamma \pm {\omega }_0\sqrt{\frac{{\gamma }^2}{{\omega }_0^2}-1}\\ & =& -\gamma \pm {\omega }_0\sqrt{{\left(\frac{\gamma }{{\omega }_0}\right)}^2-1}\\ & =& -\gamma \pm {\omega }_0\sqrt{{\zeta }^2-1}\dots \zeta =\frac{\gamma }{{\omega }_0}\\ & =& -\gamma \pm \xi \dots \xi ={\omega }_0\sqrt{{\zeta }^2-1}\end{array}$$