Question

A second order control system has a damping ratio \(\zeta = 0.8\) and the natural frequency of oscillation \(\omega_n = 12\) rad/sec. Determine the damped frequency of oscillation.

• A. 12 rad/sec.
• B. 15 rad/sec.
• C. 9.6 rad/sec.
• D. 7.2 rad/sec.

Hint:

Remember the relationship between the three key frequencies in a second-order system: the natural frequency (\(\omega_n\)), the damped frequency (\(\omega_d\)), and the resonant frequency (\(\omega_r\)). The damped frequency is always less than the natural frequency for an underdamped system (\(0<\zeta<1\)).

Answer 1

The Correct Option is D

Step 1: Recall the formula for the damped frequency of oscillation (\(\omega_d\)). For a second-order system, the damped frequency is related to the natural frequency (\(\omega_n\)) and the damping ratio (\(\zeta\)) by the following equation: \[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \]
Step 2: Substitute the given values into the formula. Given \(\zeta = 0.8\) and \(\omega_n = 12\) rad/sec. \[ \omega_d = 12 \sqrt{1 - (0.8)^2} \] \[ \omega_d = 12 \sqrt{1 - 0.64} \] \[ \omega_d = 12 \sqrt{0.36} \]
Step 3: Calculate the final value. \[ \omega_d = 12 \times 0.6 = 7.2 \text{ rad/sec} \]