Questions: Resonance and Damping in Forced Vibrations

Question 1

An undamped spring-mass system is driven at exactly its natural frequency ω₀. What happens to the amplitude of oscillation over time?

• A. It reaches a finite steady-state value determined by the magnitude of the forcing
• B. It oscillates at ω₀ but with the same amplitude as the unforced system
• C. It grows without bound, increasing linearly with time
• D. It immediately becomes infinite at the first instant of forcing

Answer: (C)

At pure resonance with no damping, the standard particular solution guess (A·cos(ω₀t)) fails because cos(ω₀t) already appears in the homogeneous solution. The correct particular solution is y_p = (F₀/2mω₀)t·sin(ω₀t) — note the factor of t. This means amplitude grows linearly without bound, never reaching a steady state. Option A describes what happens away from resonance; option D is wrong because the growth is gradual, not instantaneous.

Question 2

A damped oscillator is driven near its natural frequency. Compared to an identical undamped oscillator at the same driving frequency, the steady-state amplitude is:

• A. Larger — the damping adds energy to the system near resonance
• B. Always finite and smaller — damping limits the peak and prevents blow-up
• C. Identical far from resonance but still infinite exactly at ω₀
• D. Zero — the damping force exactly cancels the driving force

Answer: (B)

With damping, the steady-state amplitude is A = F₀/√((k − mω²)² + (cω)²). The damping term (cω)² in the denominator is always positive, so the denominator is never zero — the amplitude is always finite, even at ω = ω₀. Damping reduces amplitude everywhere (not just at resonance) and prevents the unbounded growth seen in the undamped case. The peak shifts slightly below ω₀ and flattens as damping increases.

Question 3

Without damping, the steady-state solution formula for a forced oscillator breaks down when ω = ω₀ because the particular solution must include a factor of t to capture the growing amplitude.

• T. True
• F. False

Answer: True

Correct. When ω = ω₀, the standard undetermined-coefficients guess A·cos(ω₀t) + B·sin(ω₀t) fails because these are already solutions to the homogeneous equation. The correct particular solution multiplies by t, giving y_p = (F₀/2mω₀)t·sin(ω₀t). This t factor is what produces the linearly growing amplitude characteristic of pure resonance — the system absorbs energy every cycle with no mechanism to shed it.

Question 4

Adding damping to a forced oscillator shifts the resonance peak to a frequency higher than the natural frequency ω₀.

• T. True
• F. False

Answer: False

The resonance peak shifts slightly BELOW ω₀ for underdamped systems, not above. The peak occurs at ω_peak = √(ω₀² − c²/2m²), which is less than ω₀. As damping increases, the peak shifts further below ω₀ and becomes flatter, until at critical damping the response has no distinct resonance peak at all.

Question 5

Physically, why does damping prevent unbounded amplitude growth at resonance?

Model answer: At resonance, the driving force is perfectly in phase with the velocity, so it does maximum positive work on the system every cycle — continuously adding energy. Without damping, this energy accumulates indefinitely and amplitude grows without bound. With damping, energy is dissipated every cycle (proportional to velocity and the damping coefficient). The steady state is reached when energy input from the driving force exactly balances energy dissipation by the damper — a finite amplitude where these two rates are equal.

The key insight is that resonance is an energy balance problem. Damping doesn't just reduce the amplitude of oscillation — it provides the only mechanism by which the system can reach equilibrium under sustained forcing. Larger damping dissipates more energy per cycle, which limits the amplitude more strongly. Without any dissipation pathway, there is no equilibrium and the amplitude must grow forever.