Question

For an underdamped system: A. Damping ratio is unity
B. Motion is periodic
C. System stop within least possible time
D. System vibrate at damped natural frequency

• A. C and D only
• B. A and C only
• C. B and D only
• D. A, B and D only

Hint:

Underdamped system always oscillates with decaying amplitude at damped natural frequency.

Answer 1

The Correct Option is C

Concept: A second-order system is classified based on damping ratio \(\zeta\):
• \(\zeta = 0\): Undamped
• \(0 < \zeta < 1\): Underdamped
• \(\zeta = 1\): Critically damped
• \(\zeta > 1\): Overdamped Underdamped systems exhibit oscillatory motion with gradually decreasing amplitude due to energy dissipation.

Step 1: Analyze statement A.
Damping ratio unity corresponds to critically damped system, not underdamped.
Hence, A is incorrect.

Step 2: Analyze statement B.
Underdamped system exhibits oscillations:
• Motion is sinusoidal in nature
• Amplitude decays exponentially with time Thus motion is periodic in nature (though decaying).
Hence, B is correct.

Step 3: Analyze statement C.
System stopping in least time corresponds to critically damped system:
• Underdamped system takes longer due to oscillations Hence, C is incorrect.

Step 4: Analyze statement D.
Underdamped system oscillates with damped natural frequency: \[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \] Hence, D is correct.

Step 5: Final conclusion.
Only B and D are correct. \[ \boxed{\text{Answer: B and D only}} \]