If a system is underdamped, its time-domain response exhibits which behavior?

• A. No oscillation
• B. Oscillations with increasing amplitude
• C. Oscillations with decreasing amplitude
• D. Immediate steady state

Answer: (C)

In an underdamped system, the damping is light enough that the system overshoots and oscillates, but the damping slowly removes energy so each swing is smaller than the last. That means the time-domain response shows oscillations whose amplitude decays over time, eventually settling at the steady value. The decaying envelope is tied to the damping factor: the oscillations occur at a damped frequency and shrink as e^{-ζω_n t}.

If there were no oscillations, the system would be overdamped or critically damped. If the amplitude grew, the system would be unstable. If there were immediate steady state, there would be no transient, which isn’t the case here.