The generalized standard form of transfer function of the Second-order System is
[tex]\frac{{C(s)}}{{R(s)}} = \frac{{\omega _n^2}}{{{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}[/tex]
where [tex]\zeta[/tex]=damping factor (or damping ratio)
and [tex]\omega_n[/tex]=undamped natural frequency.
The characteristic equation of this system is
[tex]{s^2} + 2\zeta {\omega _n}s + \omega _n^2 = 0[/tex]
The roots of this characteristic equation are
[tex]{s^2} + 2\zeta {\omega _n}s + \omega _n^2 = (s - {s_1})(s - {s_2})[/tex]
For [tex]\zeta \prec 1[/tex],
[tex]\begin{array}{l}{s_1},{s_2} = - \zeta {\omega _n} \pm j{\omega _n}\sqrt {(1 - {\zeta ^2})} \\ = - \zeta {\omega _n} \pm j{\omega _d}\end{array}[/tex]
where [tex]{\omega _d} = {\omega _n}\sqrt {(1 - {\zeta ^2}}[/tex] is called the damped natural frequency.
For the unit step input [tex]R(s)=\frac{1}{s}[/tex], the output response is given by
[tex]C(s) = \frac{{\omega _n^2}}{{s\left[ {s + \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]\left[ {s + \zeta {\omega _n} + j{\omega _n}\sqrt {(1 -{\zeta ^2})} } \right]}}[/tex]
The Laplace Inverse of above equation is obatained by the method of residues as
[tex]c(t) = {\left. {{{\omega _n^2} \over {{s^2} + 2\zeta {\omega _n}s + \omega _n^2}}} \right|_{s = 0}} + 2{\rm{Re}}\left[ {{{\left. {{{\omega _n^2} \over {s\left[ {s + \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]}}} \right|}_{s = - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} }}{e^{\left[ { - \zeta {\omega _n} - j{\omega _n}\sqrt {(1 - {\zeta ^2})} } \right]t}}} \right][/tex]
[tex]= 1 - \frac{{{e^{ - \zeta {\omega _n}{t_r}}}}}{{\sqrt {(1 - {\zeta ^2}} }}\sin \left[ {{\omega _n}\sqrt {(1 - {\zeta ^2})} {t_r} + {{\tan }^{ - 1}}\frac{{\sqrt{(1 - {\zeta ^2})} }}{\zeta }} \right][/tex]