Question

The damped natural frequency of an underdamped system is given by

• A. \(\omega_n\)
• B. \(\omega_n\zeta^2\)
• C. \(\omega_n\sqrt{1+\zeta^2}\)
• D. \(\omega_n\sqrt{1-\zeta^2}\)

Hint:

Important second-order system relations: \[ \omega_d=\omega_n\sqrt{1-\zeta^2} \] \[ T_p=\frac{\pi}{\omega_d} \] \[ M_p=e^{-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}} \]

Answer 1

The Correct Option is D

Concept: For a standard second-order system, \[ G(s)=\frac{\omega_n^2}{s^2+2\zeta\omega_n s+\omega_n^2} \] the oscillation frequency decreases because of damping.

Step 1: Write the damped natural frequency formula.
For an underdamped system, \[ \omega_d = \omega_n\sqrt{1-\zeta^2} \] where \[ \omega_n = \text{natural frequency} \] and \[ \zeta = \text{damping ratio}. \]

Step 2: Interpret the expression.
Since \[ 0<\zeta<1, \] the damped frequency is always smaller than the natural frequency.

Step 3: Final result.
\[ \boxed{\omega_d=\omega_n\sqrt{1-\zeta^2}} \]