Second-Order Linear ODE (3) - Free Oscillaion

6. Free Oscillation of a Mass Spring System - Undamped

 

Small Damping & Short Time → Disregard Damping

$ my''+ky=0 $ (k: spring constant)

$ y''+\frac{k}{m}y=0 $

$ y''+\omega _{0}^{2}y=0 $ ($ \omega _{0}=\sqrt{\frac{k}{m}} $)

$ y=e^{\lambda t} \rightarrow \lambda ^{2}+\omega _{0}^{2}=0 \rightarrow \lambda =\pm \omega _{0}i $

$ \therefore y(t)=A\cos \omega _{0}t+B\sin \omega _{0}t $

   $ =\sqrt{A^{2}+B^{2}}(\cos \omega _{0}t\frac{A}{\sqrt{A^{2}+B^{2}}}+\sin \omega _{0}t\frac{B}{\sqrt{A^{2}+B^{2}}})={\sqrt{A^{2}+B^{2}}}\cos (\omega _{0}t-\delta ) $

 

7. Free Oscillation of a Mass Spring System - Damped

 

$ my''+cy'+ky=0 (c,k,m>0) $

$ y''+\frac{c}{m}y'+\frac{k}{m}y=0 $

$ y=e^{\lambda t} \rightarrow \lambda^{2}+\frac{c}{m}\lambda+\frac{k}{m}=0 $

$ \lambda =-\frac{c}{2m}\pm \frac{\sqrt{c^{2}-4mk}}{2m}=-\alpha \pm \beta (\alpha >0, \beta >0) $

 

1) Two Distinct Real Root(Overdamping)

$ c^{2}>4mk $

$ \frac{c}{2m}>\frac{\sqrt{c^{2}-4mk}}{2m} \rightarrow \alpha > \beta $

$ \lambda _{1}=-\alpha +\beta, \lambda _{2}=-\alpha -\beta $

$ \therefore y(t)=c_{1}e^{(-\alpha +\beta)t}+c_{2}e^{(-\alpha -\beta)t} $

$ \lim_{t\to \infty}y=0 $

 

2) Real Double Root(Critical Damping)

$ c^{2}=4mk $

$ \lambda _{1}=\lambda _{2}=-\alpha $

$ \therefore y(t)=(c_{1}+c_{2}t)e^{-\alpha t} $

$ \lim_{t\to \infty}y=0 $

 

3) Complex Root(Underdamping)

$ c^{2}<4mk $

$ \lambda =-\frac{c}{2m}\pm \frac{\sqrt{c^{2}-4mk}}{2m}=-\alpha \pm i\omega ^{*} $ ($ \omega ^{*}=\frac{\sqrt{c^{2}-4mk}}{2m} $)

$ \therefore y(t)=e^{-\alpha t}(\cos \omega ^{*}t+\sin \omega ^{*}t) $

$ \lim_{t\to \infty}y=0 $