Question

For critically damped system, the value of the damping ratio $\zeta$ is:

• A. 0
• B. $>1$
• C. $<1$
• D. 1

Hint:

A critically damped system returns to its steady-state position in the shortest possible time without oscillating about it.
This is highly desired in many physical systems, such as analog meters.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the damping ratio ($\zeta$) that corresponds to a critically damped second-order system.

Step 2: Key Formula or Approach:
The roots of the characteristic equation for a second-order system are given by:
\[ s_{1,2} = -\zeta\omega_n \pm \omega_n \sqrt{\zeta^2 - 1} \]

Step 3: Detailed Explanation:

• The value of the damping ratio $\zeta$ determines the nature of the transient response of the system:

• Undamped: $\zeta = 0$.

• Underdamped: $0 < \zeta < 1$.

• Critically Damped: $\zeta = 1$. The roots are real and equal ($s_1 = s_2 = -\omega_n$). This represents the fastest response without any overshoot.

• Overdamped: $\zeta > 1$. The roots are real and distinct.

Step 4: Final Answer:
For a critically damped system, the damping ratio is exactly $1$, which corresponds to Option (D).