Question

The magnitude of frequency response of an underdamped second order system is $5$ at $0rad/sec$ and peaks to $\frac{10}{\sqrt{3}}$ at $5\sqrt{2}rad/sec.$ The transfer function of the system is

• A. $\frac{500}{s^2+10s+100}$
• B. $\frac{375}{s^2+5s+75}$
• C. $\frac{720}{s^2+12s+144}$
• D. $\frac{1125}{s^2+25s+225}$

Answer 1

The correct option is A $\frac{500}{s^2+10s+100}$

$M_r=\frac{5}{2\xi \sqrt{1-{\xi }^2}}=\frac{10}{\sqrt{3}}$
$or\xi =\frac{1}{2}$
Also,
${\omega }_r={\omega }_n\sqrt{1-2Z^2}$
$5\sqrt{2}={\omega }_n\sqrt{1-2\times \left(\frac{1}{4}\right)}$
$or{\omega }_n=10rad/sec$
Only option (a) satisfies the conditions