Time Response of Second-order Systems

$\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)}$

$G(s)=\frac{\omega ^2_n}{s(s+2\zeta \omega_n)}$

$\frac{C(s)}{R(s)}=\frac{\left (\frac{\omega ^2_n}{s(s+2\zeta \omega_n)} \right )}{1+ \left ( \frac{\omega ^2_n}{s(s+2\zeta \omega_n)} \right )}=\frac{\omega _n^2}{s^2+2\zeta \omega _ns+\omega _n^2}$

$s^2+2\zeta \omega _ns+\omega _n^2=0$

$s=\frac{-2\omega \zeta _n\pm \sqrt{(2\zeta\omega _n)^2-4\omega _n^2}}{2}=\frac{-2(\zeta\omega _n\pm \omega _n\sqrt{\zeta ^2-1})}{2}$

$\Rightarrow s=-\zeta \omega_n \pm \omega _n\sqrt{\zeta ^2-1}$

$C(s)=\left ( \frac{\omega _n^2}{s^2+2\zeta\omega_ns+\omega_n^2} \right )R(s)$

$R(s)=\frac{1}{s}$

$\frac{C(s)}{R(s)}=\frac{\omega _n^2}{s^2+2\zeta\omega_ns+\omega_n^2}$

$\frac{C(s)}{R(s)}=\frac{\omega_n^2}{s^2+\omega_n^2}$

$\Rightarrow C(s)=\left( \frac{\omega_n^2}{s^2+\omega_n^2} \right )R(s)$

$R(s) = \frac{1}{s}$

$C(s)=\left( \frac{\omega_n^2}{s^2+\omega_n^2} \right )\left( \frac{1}{s} \right )=\frac{\omega_n^2}{s(s^2+\omega_n^2)}$

$c(t)=\left ( 1-\cos(\omega_n t) \right )u(t)$

$/delta = 1$

$\frac{C(s)}{R(s)}=\frac{\omega_n^2}{s^2+2\omega_ns+\omega_n^2}$

$\Rightarrow C(s)=\left( \frac{\omega_n^2}{(s+\omega_n)^2} \right)R(s)$

$C(s)=\left( \frac{\omega_n^2}{(s+\omega_n)^2} \right)\left ( \frac{1}{s} \right)=\frac{\omega_n^2}{s(s+\omega_n)^2}$

$C(s)=\frac{\omega_n^2}{s(s+\omega_n)^2}=\frac{A}{s}+\frac{B}{s+\omega_n}+\frac{C}{(s+\omega_n)^2}$

$C(s)=\frac{1}{s}-\frac{1}{s+\omega_n}-\frac{\omega_n}{(s+\omega_n)^2}$

$c(t)=(1-e^{-\omega_nt}-\omega _nte^{-\omega_nt})u(t)$

$s^2+2\zeta\omega_ns+\omega_n^2=\left \{ s^2+2(s)(\zeta \omega_n)+(\zeta \omega_n)^2 \right \}+\omega_n^2-(\zeta\omega_n)^2$

$=(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)$

$\frac{C(s)}{R(s)}=\frac{\omega_n^2}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}$

$\Rightarrow C(s)=\left( \frac{\omega_n^2}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)} \right )R(s)$

$C(s)=\left( \frac{\omega_n^2}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)} \right )\left( \frac{1}{s} \right )=\frac{\omega_n^2}{s\left ((s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2) \right)}$

$C(s)=\frac{\omega_n^2}{s\left ((s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2) \right)}=\frac{A}{s}+\frac{Bs+C}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}$

$C(s)=\frac{1}{s}-\frac{s+2\zeta\omega_n}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}$

$C(s)=\frac{1}{s}-\frac{s+\zeta\omega_n}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}-\frac{\zeta\omega_n}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}$

$C(s)=\frac{1}{s}-\frac{(s+\zeta\omega_n)}{(s+\zeta\omega_n)^2+(\omega_n\sqrt{1-\zeta^2})^2}-\frac{\zeta}{\sqrt{1-\zeta^2}}\left ( \frac{\omega_n\sqrt{1-\zeta^2}}{(s+\zeta\omega_n)^2+(\omega_n\sqrt{1-\zeta^2})^2} \right )$

$C(s)=\frac{1}{s}-\frac{(s+\zeta\omega_n)}{(s+\zeta\omega_n)^2+\omega_d^2}-\frac{\zeta}{\sqrt{1-\zeta^2}}\left ( \frac{\omega_d}{(s+\zeta\omega_n)^2+\omega_d^2} \right )$

$c(t)=\left ( 1-e^{-\zeta \omega_nt}\cos(\omega_dt)-\frac{\zeta}{\sqrt{1-\zeta^2}}e^{-\zeta\omega_nt}\sin(\omega_dt) \right )u(t)$

$c(t)=\left ( 1-\frac{e^{-\zeta\omega_nt}}{\sqrt{1-\zeta^2}}\left ( (\sqrt{1-\zeta^2})\cos(\omega_dt)+\zeta \sin(\omega_dt) \right ) \right )u(t)$

$\sqrt{1-\zeta^2}=\sin(\theta)$

$c(t)=\left ( 1-\frac{e^{-\zeta\omega_nt}}{\sqrt{1-\zeta^2}}(\sin(\theta)\cos(\omega_dt)+\cos(\theta)\sin(\omega_dt)) \right )u(t)$

$\Rightarrow c(t)=\left ( 1-\left ( \frac{e^{-\zeta\omega_nt}}{\sqrt{1-\zeta^2}} \right )\sin(\omega_dt+\theta) \right )u(t)$

$s^2+2\zeta\omega_ns+\omega_n^2=\left \{ s^2+2(s)(\zeta\omega_n)+(\zeta\omega_n)^2 \right \}+\omega_n^2-(\zeta\omega_n)^2$

$=\left ( s+\zeta\omega_n \right )^2-\omega_n^2\left ( \zeta^2-1 \right )$

$\frac{C(s)}{R(s)}=\frac{\omega_n^2}{(s+\zeta\omega_n)^2-\omega_n^2(\zeta^2-1)}$

$\Rightarrow C(s)=\left ( \frac{\omega_n^2}{(s+\zeta\omega_n)^2-\omega_n^2(\zeta^2-1)} \right )R(s)$

$C(s)=\left ( \frac{\omega_n^2}{(s+\zeta\omega_n)^2-(\omega_n\sqrt{\zeta^2-1})^2} \right )\left ( \frac{1}{s} \right )=\frac{\omega_n^2}{s(s+\zeta\omega_n+\omega_n\sqrt{\zeta^2-1})(s+\zeta\omega_n-\omega_n\sqrt{\zeta^2-1})}$

$C(s)=\frac{\omega_n^2}{s(s+\zeta\omega_n+\omega_n\sqrt{\zeta^2-1})(s+\zeta\omega_n-\omega_n\sqrt{\zeta^2-1})}$

$=\frac{A}{s}+\frac{B}{s+\zeta\omega_n+\omega_n\sqrt{\zeta^2-1}}+\frac{C}{s+\zeta\omega_n-\omega_n\sqrt{\zeta^2-1}}$

$\frac{1}{2(\zeta+\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})}$ and $\frac{-1}{2(\zeta-\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})}$ $C(s)$.

$C(s)=\frac{1}{s}+\frac{1}{2(\zeta+\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})}\left ( \frac{1}{s+\zeta\omega_n+\omega_n\sqrt{\zeta^2-1}} \right )-\left ( \frac{1}{2(\zeta-\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})} \right )\left ( \frac{1}{s+\zeta\omega_n-\omega_n\sqrt{\zeta^2-1}} \right )$

$c(t)=\left ( 1+\left ( \frac{1}{2(\zeta+\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})} \right )e^{-(\zeta\omega_n+\omega_n\sqrt{\zeta^2-1})t}-\left ( \frac{1}{2(\zeta-\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})} \right )e^{-(\zeta\omega_n-\omega_n\sqrt{\zeta^2-1})t} \right )u(t)$