Question

The unit step response of a second order system is

$c\left(t\right)=1+0.3e^{-40t}-1.2e^{-30t}$

The natural frequency of the system is

• A. 40 rad/sec
• B. $\sqrt{1200}$ rad/sec
• C. 1200 rad/sec
• D. 70 rad/sec

Answer 1

The correct option is B

$\sqrt{1200}$ rad/sec

(b)

Given unit step response is

$c\left(t\right)=1+0.3e^{-40t}-1.2e^{-30t}$

Impulse response $h\left(t\right)$

$h\left(t\right)=\frac{dc\left(t\right)}{dt}=-40\times 0.3e^{-40t}+1.2\times 30e^{-30t}$

$h\left(t\right)=-12e^{-40t}+36e^{-30t}$

$H\left(s\right)=\frac{-12}{s+40}+\frac{36}{s+30}$

$H\left(s\right)=\frac{-12s-360+36s+1440}{s^2+70s+1200}=\frac{24s+1080}{s^2+70s+1200}$

${\omega }_n^2=1200\Rightarrow {\omega }_n=\sqrt{1200}$ rad/sec