For the closed-loop system with \(G_p(s) = \frac{14.4}{s(1 + 0.1s)}\) and \(G_c(s) = 1\), the unit-step response shows damped oscillations. The damped natural frequency is \(\underline{\hspace{2cm}}\) rad/s. (Round off to 2 decimal places.)
For the closed-loop system with \(G_p(s) = \frac{14.4}{s(1 + 0.1s)}\) and \(G_c(s) = 1\), the unit-step response shows damped oscillations. The damped natural frequency is \(\underline{\hspace{2cm}}\) rad/s. (Round off to 2 decimal places.)
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Correct Answer: 10.8
Solution and Explanation
Open-loop transfer function:
\[ G(s) = \frac{14.4}{s(1 + 0.1s)} \]
Closed-loop characteristic equation:
\[ 1 + G(s) = 0 \]
\[ 1 + \frac{14.4}{s(1 + 0.1s)} = 0 \]
\[ s(1 + 0.1s) + 14.4 = 0 \]
\[ 0.1s^2 + s + 14.4 = 0 \]
Divide throughout by 0.1:
\[ s^2 + 10s + 144 = 0 \]
Compare with standard second-order system:
\[ s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 \]
Matching coefficients:
\[ \omega_n^2 = 144 \;\Rightarrow\; \omega_n = 12 \]
\[ 2\zeta\omega_n = 10 \;\Rightarrow\; \zeta = \frac{10}{24} = 0.4167 \]
Damped natural frequency:
\[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \]
\[ \omega_d = 12 \sqrt{1 - (0.4167)^2} \]
\[ \omega_d = 12 \sqrt{0.8264} = 12 \times 0.908 = 10.90\ \text{rad/s} \]
The value lies within the expected range of 10.80 to 11.00 rad/s.
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