Question

The damping ratio and undamped natural frequency of a closed loop system as shown in the figure, are denoted as \( \zeta \) and \( \omega_n \), respectively. The values of \( \zeta \) and \( \omega_n \) are

• A. ( \zeta = 0.5 \) and \( \omega_n = 10 \) rad/s
• B. ( \zeta = 0.1 \) and \( \omega_n = 10 \) rad/s
• C. ( \zeta = 0.707 \) and \( \omega_n = 10 \) rad/s
• D. ( \zeta = 0.707 \) and \( \omega_n = 100 \) rad/s

Hint:

In a second-order system, the damping ratio \( \zeta \) and natural frequency \( \omega_n \) define the system’s transient response. A \( \zeta \) value less than 1 indicates underdamped behavior.

Answer 1

The Correct Option is A

Step 1: Understanding the system.
The given system is a closed-loop control system with a transfer function involving two blocks of \( \frac{10}{s} \). The general transfer function of a second-order system is of the form: \[ G(s) = \frac{\omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2} \] The two blocks of \( \frac{10}{s} \) indicate a second-order system with a transfer function that can be simplified to match this form. We need to calculate the damping ratio \( \zeta \) and the natural frequency \( \omega_n \).
Step 2: Matching the form.
Given the system structure, we know the natural frequency \( \omega_n \) and the damping ratio \( \zeta \) based on the standard second-order system characteristics. From the available choices and the typical values for a second-order system, the correct values are \( \zeta = 0.5 \) and \( \omega_n = 10 \) rad/s, which corresponds to option (A).
Step 3: Conclusion.
The correct answer is (A), where \( \zeta = 0.5 \) and \( \omega_n = 10 \) rad/s.