Which damping case has oscillatory decay with decreasing amplitude?

• A. Underdamped
• B. Overdamped
• C. Critically Damped
• D. Undamped

Answer: (A)

Oscillations with shrinking amplitude occur when the system is underdamped, meaning damping is light enough for the motion to continue past the equilibrium while the peaks decay over time. In a typical second-order damped oscillator, the solution has an exponentially decaying envelope with a sinusoidal term: x(t) ≈ e^{-ζ ω_n t} cos(ω_d t) (and similar for other initial phases), where ω_n is the natural frequency, ω_d is the damped frequency, and ζ is the damping ratio. The exponential factor e^{-ζ ω_n t} causes the amplitude to decrease with each cycle, producing oscillatory decay.

If damping is stronger (ζ ≥ 1), the motion does not cross the equilibrium and simply returns to rest without oscillating (overdamped and critically damped). If there is no damping (ζ = 0), you get a pure undamped oscillation with constant amplitude.