A damped mechanical system has damping ratio ζ = 0.1 and is being driven at its natural frequency. An engineer reduces the damping ratio to ζ = 0.05. The steady-state resonant amplitude will:
ADouble, because the dynamic magnification factor M = 1/(2ζ) is inversely proportional to ζ
BStay the same, because the natural frequency did not change
CDecrease, because less damping means the system returns to rest faster
DBecome infinite, because any reduction in damping leads to unbounded amplitude
At resonance, M = 1/(2ζ). When ζ halves from 0.1 to 0.05, M doubles from 5 to 10. This inverse relationship is the key formula for resonant amplitude. The misconception in option C reverses the logic — less damping means energy dissipates more slowly, so steady-state amplitude builds higher, not lower. Option D is only true for the theoretical ζ = 0 case; any finite damping produces a finite resonant amplitude.
Question 2 Multiple Choice
For a damped forced oscillator, where does the maximum steady-state amplitude actually occur?
AExactly at the undamped natural frequency ω_n, regardless of damping ratio
BAt a driving frequency slightly below ω_n, at ω = ω_n√(1 − 2ζ²)
CAt the damped natural frequency ω_d = ω_n√(1 − ζ²)
DAt a driving frequency slightly above ω_n, because damping shifts the peak upward
The resonant peak in the frequency response of a damped forced system occurs at ω = ω_n√(1 − 2ζ²), which is slightly below both ω_n and ω_d. This is a common source of confusion: for lightly damped systems the shift is negligible, but for ζ > 1/√2 ≈ 0.707, the peak disappears entirely and there is no resonance. The undamped natural frequency ω_n is where phase lag equals 90°, not where amplitude peaks.
Question 3 True / False
For an underdamped system, the damped natural frequency ω_d equals the undamped natural frequency ω_n.
TTrue
FFalse
Answer: False
The damped natural frequency is ω_d = ω_n√(1 − ζ²), which is always less than ω_n for any positive damping ratio. The difference is small for lightly damped systems (e.g., ζ = 0.1 gives ω_d ≈ 0.995 ω_n), but becomes significant as ζ approaches 1. At critical damping (ζ = 1), ω_d = 0 — the system no longer oscillates at all. This distinction matters when designing systems to avoid specific resonant frequencies.
Question 4 True / False
At resonance, the steady-state response of a forced damped oscillator lags the driving force by exactly 90°, regardless of the value of the damping ratio ζ.
TTrue
FFalse
Answer: True
The 90° phase lag at ω = ω_n is a universal property of the damped forced oscillator, independent of damping level. This is because the phase angle φ = arctan(2ζr / (1 − r²)) evaluated at r = ω/ω_n = 1 gives arctan(2ζ·0) = arctan(∞) ... wait, actually at r=1: φ = arctan(2ζ·1 / (1−1²)) = arctan(2ζ/0) = 90° for all ζ > 0. The phase is always 90° at the undamped natural frequency — it is the amplitude peak that shifts, not the 90° phase crossing.
Question 5 Short Answer
Explain why a mechanical system can experience harmful resonance even when damping is present, and what determines the severity of the resonant response.
Think about your answer, then reveal below.
Model answer: Damping reduces resonant amplitude but does not eliminate resonance for ζ < 1/√2. The resonant amplitude is M = 1/(2ζ), so lightly damped systems (small ζ) still experience very large amplification near ω_n. For example, ζ = 0.05 gives M = 10 — ten times the static deflection. Whether this is harmful depends on the design requirements: a bridge or aircraft wing with insufficient damping can fail structurally because the resonant amplitude exceeds material limits, even though the vibration is not growing without bound.
The Tacoma Narrows Bridge is the canonical example: it had some damping, but not enough to reduce the resonant amplitude below the threshold for structural failure. The key insight is that 'damped' does not mean 'safe at resonance' — it only means 'finite amplitude at resonance.' Engineers use frequency response plots to evaluate whether the resonant peak, even with damping included, stays within acceptable displacement or stress limits. Adding more damping (via tuned mass dampers, viscoelastic materials, or active control) flattens the peak.