Questions: Mechanical Resonance and Frequency Response

Question 1

An engineer gradually increases the frequency of a periodic force applied to a lightly damped structure. As driving frequency approaches the structure's natural frequency from below, what happens to response amplitude?

• A. It decreases, because the system has less time to respond to each cycle
• B. It stays roughly constant until the frequencies match, then drops sharply
• C. It increases dramatically, potentially reaching destructive levels even for small force magnitudes
• D. It increases linearly with driving frequency, reaching a maximum well above the natural frequency

Answer: (C)

The magnification factor MF = 1/|1−(Ω/ωₙ)²| grows as Ω approaches ωₙ. As the denominator approaches zero, the magnification grows without bound in an undamped system. Even with light damping (ζ=0.01), the peak amplitude at resonance is 1/(2ζ) = 50 times the static deflection. The key insight is that force magnitude and response amplitude decouple near resonance — a tiny periodic force can produce catastrophic oscillation if its frequency matches the natural frequency.

Question 2

A lightly damped metal structure has a damping ratio ζ = 0.02. When driven at its natural frequency, its steady-state amplitude is approximately how many times its static deflection?

• A. 2 times — the damping ratio directly limits amplification
• B. 25 times — from the resonance formula 1/(2ζ) = 1/(0.04)
• C. 50 times — from the resonance formula 1/(2ζ) = 1/(0.02)
• D. 100 times — because the damping ratio squares at resonance

Answer: (B)

At resonance (Ω = ωₙ) in a damped system, the magnification factor peaks at 1/(2ζ). With ζ = 0.02: MF = 1/(2×0.02) = 1/0.04 = 25. Option C would apply if ζ = 0.01. The formula 1/(2ζ) means lighter damping gives larger resonant amplification — a ζ = 0.01 structure would reach 50× at resonance.

Question 3

Adding mass to a mechanical structure generally increases its risk of resonance with a fixed driving frequency.

• T. True
• F. False

Answer: False

Adding mass lowers the natural frequency (ωₙ = √(k/m) — increasing m decreases ωₙ). Whether this increases or decreases resonance risk depends entirely on where ωₙ was relative to the excitation frequency before the change. If the excitation is above ωₙ, adding mass moves ωₙ further from the excitation — reducing risk. If the excitation is below ωₙ, adding mass moves ωₙ toward the excitation — increasing risk. The relationship is not monotonic; what matters is the ratio Ω/ωₙ.

Question 4

At resonance, the steady-state displacement response of a damped system lags exactly 90° behind the driving force.

• T. True
• F. False

Answer: True

The phase lag between response and driving force depends on Ω/ωₙ: near zero frequency the phase lag ≈ 0° (in phase); at resonance (Ω = ωₙ) the phase lag is exactly 90° regardless of damping level; above resonance the phase lag → 180° (out of phase). The 90° phase at resonance is observable experimentally and is sometimes used as the operational definition of resonance — it occurs at the exact natural frequency, while the amplitude peak shifts slightly with damping.

Question 5

Explain why the Tacoma Narrows Bridge collapsed under relatively light wind forces, connecting your explanation to the concept of resonance.

Model answer: The bridge had a torsional natural frequency. Wind flowing past the bridge created vortices that shed alternately from each side at a frequency that matched (or approached) this natural frequency. Even though the periodic aerodynamic forces from vortex shedding were small, the magnification factor near resonance amplified them dramatically. Each cycle added energy that was not dissipated by the bridge's low damping, so oscillation amplitude grew progressively over minutes until the structure failed. Small force × large magnification factor = catastrophic response.

The key lesson: resonance decouples force magnitude from response magnitude. Engineers now calculate natural frequencies and ensure they differ from expected excitation frequencies, or add damping to limit the magnification factor.