A unity feedback second order control system is characterized by the open loop transfer function \(\left( s \right) = \frac{K}{{s\left( {Js + B} \right)}}\)

J = moment of inertia, B = damping constant and K = system gain

The transient response specification which is not affected by system gain variation is

The open-loop transfer function of a unity feedback 2^nd order system is,

\(G\left( s \right) = \frac{k}{{s\left( {Js + B} \right)}}\;\) 

Closed-loop transfer function:

\(H\left( s \right) = \frac{k}{{s\left( {Js + B} \right) + k}}\) 

\(H\left( s \right) = \frac{{k/J}}{{{s^2} + \frac{B}{J}s + \frac{k}{J}}}\) 

Now, compare it with the standard equation:

\(H\left( s \right) = \frac{{{\omega _n}}}{{{s^2} + 2\xi {\omega _n}s + \omega _n^2}}\) 

Now, settling time

\( = \frac{1}{{\xi {\omega _n}}} = \frac{1}{{\frac{B}{{2J}}}} = \frac{{2J}}{B}\)