[tex]\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)}[/tex]
[tex]G(s)=\frac{\omega ^2_n}{s(s+2\zeta \omega_n)}[/tex]
[tex]\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}[/tex]
[tex]s^2+2\zeta \omega _ns+\omega _n^2=0[/tex]
[tex]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}[/tex]
[tex]\Rightarrow s=-\zeta \omega_n \pm \omega _n\sqrt{\zeta ^2-1}[/tex]
[tex]C(s)=\left ( \frac{\omega _n^2}{s^2+2\zeta\omega_ns+\omega_n^2} \right )R(s)[/tex]
[tex]R(s)=\frac{1}{s}[/tex]
[tex]\frac{C(s)}{R(s)}=\frac{\omega _n^2}{s^2+2\zeta\omega_ns+\omega_n^2}[/tex]
[tex]\frac{C(s)}{R(s)}=\frac{\omega_n^2}{s^2+\omega_n^2}[/tex]
[tex]\Rightarrow C(s)=\left( \frac{\omega_n^2}{s^2+\omega_n^2} \right )R(s)[/tex]
[tex]R(s) = \frac{1}{s}[/tex]
[tex]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)}[/tex]
[tex]c(t)=\left ( 1-\cos(\omega_n t) \right )u(t)[/tex]
[tex]/delta = 1[/tex]
[tex]\frac{C(s)}{R(s)}=\frac{\omega_n^2}{s^2+2\omega_ns+\omega_n^2}[/tex]
[tex]\Rightarrow C(s)=\left( \frac{\omega_n^2}{(s+\omega_n)^2} \right)R(s)[/tex]
[tex]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}[/tex]
[tex]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}[/tex]
[tex]C(s)=\frac{1}{s}-\frac{1}{s+\omega_n}-\frac{\omega_n}{(s+\omega_n)^2}[/tex]
[tex]c(t)=(1-e^{-\omega_nt}-\omega _nte^{-\omega_nt})u(t)[/tex]
[tex]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[/tex]
[tex]=(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)[/tex]
[tex]\frac{C(s)}{R(s)}=\frac{\omega_n^2}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}[/tex]
[tex]\Rightarrow C(s)=\left( \frac{\omega_n^2}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)} \right )R(s)[/tex]
[tex]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)}[/tex]
[tex]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)}[/tex]
[tex]C(s)=\frac{1}{s}-\frac{s+2\zeta\omega_n}{(s+\zeta\omega_n)^2+\omega_n^2(1-\zeta^2)}[/tex]
[tex]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)}[/tex]
[tex]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 )[/tex]
[tex]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 )[/tex]
[tex]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)[/tex]
[tex]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)[/tex]
[tex]\sqrt{1-\zeta^2}=\sin(\theta)[/tex]
[tex]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)[/tex]
[tex]\Rightarrow c(t)=\left ( 1-\left ( \frac{e^{-\zeta\omega_nt}}{\sqrt{1-\zeta^2}} \right )\sin(\omega_dt+\theta) \right )u(t)[/tex]
[tex]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[/tex]
[tex]=\left ( s+\zeta\omega_n \right )^2-\omega_n^2\left ( \zeta^2-1 \right )[/tex]
[tex]\frac{C(s)}{R(s)}=\frac{\omega_n^2}{(s+\zeta\omega_n)^2-\omega_n^2(\zeta^2-1)}[/tex]
[tex]\Rightarrow C(s)=\left ( \frac{\omega_n^2}{(s+\zeta\omega_n)^2-\omega_n^2(\zeta^2-1)} \right )R(s)[/tex]
[tex]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})}[/tex]
[tex]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})}[/tex]
[tex]=\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}}[/tex]
[tex]\frac{1}{2(\zeta+\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})}[/tex] and [tex]\frac{-1}{2(\zeta-\sqrt{\zeta^2-1})(\sqrt{\zeta^2-1})}[/tex] [tex]C(s)[/tex].
[tex]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 )[/tex]
[tex]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)[/tex]
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