Underdamped vibrations — in a single-degree-of-freedom viscously damped system, how does the amplitude of free vibration vary with time when the system is underdamped?

Introduction / Context: Real mechanical systems usually possess some damping. When damping is light (underdamped), oscillations persist but their envelope decays. Recognizing the correct decay law is essential for estimating settling time and vibration severity.

Given Data / Assumptions:

• Single mass–spring–damper with damping ratio 0 < ζ < 1.
• No external forcing (free vibration).
• Undamped natural frequency ω_n=√(k/m).

Concept / Approach: The displacement solution is x(t)=X₀e^−ζω_ntcos(ω_dt+φ), where ω_d=ω_n√(1−ζ²). The amplitude envelope X(t)=X₀e^−ζω_nt decays exponentially with time.

Step-by-Step Solution:

Verification / Alternative check: Log-decrement δ=ln(x_i/x_i+1)=2πζ/√(1−ζ²) confirms an exponential decay trend across cycles.

Why Other Options Are Wrong:
Linear increase/decrease — decay is not linear in time.
Increases exponentially — contradicts dissipative energy loss.

Common Pitfalls: Confusing forced resonance amplitude growth with free-decay behavior; mixing underdamped decay with critically damped non-oscillatory return.

Final Answer: decreases exponentially with time.