Question

Find the frequency of sustained oscillations for the following system when \( K = 48 \): \[ G(s)H(s) = \frac{K}{s(s^2 + 7s + 4)} \]

• A. \( \sqrt{6} \ \text{rad/sec} \)
• B. \( 2 \ \text{rad/sec} \)
• C. \( \sqrt{2} \ \text{rad/sec} \)
• D. \( \sqrt{8} \ \text{rad/sec} \)

Hint:

Sustained oscillations correspond to purely imaginary roots → use Routh array and check when a row becomes zero.

Answer 1

The Correct Option is D

To find the frequency of sustained oscillations, we apply the Routh-Hurwitz criterion. Form the characteristic equation: \[ 1 + G(s)H(s) = 0 \Rightarrow s(s^2 + 7s + 4) + 48 = 0 \Rightarrow s^3 + 7s^2 + 4s + 48 = 0 \] Construct the Routh array and find the value of \( \omega \) where a sign change leads to imaginary roots. This results in: \[ \omega = \sqrt{8} \ \text{rad/sec} \]