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 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.
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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 (\(0<\zeta<1\)).
Updated On: Feb 18, 2026
- 12 rad/sec.
- 15 rad/sec.
- 9.6 rad/sec.
- 7.2 rad/sec.
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The Correct Option is D
Solution and Explanation
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. \[ \omega_d = 12 \times 0.6 = 7.2 \text{ rad/sec} \]
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. \[ \omega_d = 12 \times 0.6 = 7.2 \text{ rad/sec} \]
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