バネマスダンパー系，臨界減衰(γ=ω0)の場合

バネマスダンパー系

$\gamma ={\omega }_0$

$$\begin{array}{rcl}{\lambda }_{1,2}& =& {-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}|}_{\gamma ={\omega }_0}\dots {\lambda }_{1,2}=-\gamma \pm \sqrt{{\gamma }^2-{\omega }_0^2}\\ & =& -{\omega }_0\pm \sqrt{{\omega }_0^2-{\omega }_0^2}\\ & =& -{\omega }_0\end{array}$$ $$\begin{array}{rcl}x(t)& =& C(t)e^{-{\omega }_{0}t}\dots \mathrm{定}\mathrm{数}\mathrm{変}\mathrm{化}\mathrm{法}\\ \dot{x}& =& C^{\prime }(t)e^{-{\omega }_{0}t}-{\omega }_{0}C(t)e^{-{\omega }_{0}t}& \dots (fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }\\ \ddot{x}& =& \{C^{″}(t)e^{-{\omega }_{0}t}-{\omega }_0{C}^{\prime }(t)e^{-{\omega }_{0}t}\}+\{-{\omega }_{0}C(t{)}^{\prime }{e}^{-{\omega }_{0}t}+{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}\}& \dots (fg{)}^{\prime }=f^{\prime }g+f{g}^{\prime }\\ & =& C^{″}(t)e^{-{\omega }_{0}t}-2{\omega }_0{C}^{\prime }(t)e^{-{\omega }_{0}t}+{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}\end{array}$$ $$\begin{array}{rcl}{\ddot{x}+2\gamma \dot{x}+{\omega }_0^{2}x|}_{\gamma ={\omega }_0}& =& 0\\ \ddot{x}+2{\omega }_0\dot{x}+{\omega }_0^{2}x& =& 0\\ [C^{″}(t)e^{-{\omega }_{0}t}-2{\omega }_0{C}^{\prime }(t)e^{-{\omega }_{0}t}+{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}]+2{\omega }_0[C^{\prime }(t)e^{-{\omega }_{0}t}-{\omega }_{0}C(t)e^{-{\omega }_{0}t}]+{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}& =& 0\\ C^{″}(t)e^{-{\omega }_{0}t}-2{\omega }_0{C}^{\prime }(t)e^{-{\omega }_{0}t}+{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}+2{\omega }_0{C}^{\prime }(t)e^{-{\omega }_{0}t}-2{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}+{\omega }_0^{2}C(t)e^{-{\omega }_{0}t}& =& 0\\ C^{″}(t)e^{-{\omega }_{0}t}& =& 0\\ C^{″}(t)& =& 0\dots e^{-{\omega }_{0}t}>0\end{array}$$ $$\begin{array}{rcl}{C}^{″}(t)& =& 0\\ C^{\prime }(t)& =& C_1\\ C(t)& =& C_{1}t+C_2\\ x(t)& =& (C_{1}t+C_2)e^{-{\omega }_{0}t}\end{array}$$ $$\begin{array}{rcl}x(0)& =& {(C_{1}t+C_2)e^{-{\omega }_{0}t}|}_{t=0}\\ & =& (C_{1}0+C_2)e^{-{\omega }_{0}0}\\ & =& C_2\cdot 1=C_2\\ C_2& =& x(0)=x_0\\ x^{\prime }(0)& =& {\{C_1{e}^{-{\omega }_{0}t}-{\omega }_0(C_{1}t+C_2)e^{-{\omega }_{0}t}\}|}_{t=0,C_2=x_0}\\ & =& C_1{e}^{-{\omega }_{0}0}-{\omega }_0(C_{1}0+x_0)e^{-{\omega }_{0}0}\\ & =& C_1\cdot 1-{\omega }_0\left(x_0\right)\cdot 1\\ & =& C_1-{\omega }_0{x}_0\\ C_1& =& x^{\prime }(0)+{\omega }_0{x}_0=v_0+{\omega }_0{x}_0\end{array}$$ $$\begin{array}{rcl}x(t)& =& C(t)e^{-{\omega }_{0}t}\\ & =& (C_{1}t+C_2)e^{-{\omega }_{0}t}\\ & =& \{(v_0+{\omega }_0{x}_0)t+x_0\}{e}^{-{\omega }_{0}t}\\ & =& \{(v_0+\gamma x_0)t+x_0\}{e}^{-\gamma t}\dots {\omega }_0=\gamma \\ & =& e^{-\gamma t}\{x_0+(v_0+\gamma x_0)t\}\\ & =& x_0{e}^{-\gamma t}(1+\gamma t)+v_0{e}^{-\gamma t}\left(t\right)\dots \mathrm{初}\mathrm{期}\mathrm{位}\mathrm{置}{x}_0\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{臨}\mathrm{界}\mathrm{減}\mathrm{衰},\mathrm{初}\mathrm{期}\mathrm{速}\mathrm{度}{v}_0\mathrm{に}\mathrm{よ}\mathrm{る}\mathrm{臨}\mathrm{界}\mathrm{減}\mathrm{衰}\end{array}$$