A lightly damped oscillator is driven with constant force amplitude. Its steady-state (driving frequency in Hz, displacement amplitude in mm) readings are (2.8, 2), (3.2, 5), (3.6, 20), (4, 4), (4.4, 2). (a) Estimate the resonance frequency and state the resolution limitation. (b) Describe the response to greater damping. (c) Explain how to improve the frequency scan and why steady-state readings are required.

Question

Accepted Answer

Measured maximum near 3.6 Hz, with true peak unresolved between nearby readings; greater damping lowers and broadens the resonance peak. The largest sampled amplitude is 20 mm at 3.6 Hz, so this is the best sampled estimate of resonance. The neighbouring frequencies are 0.4 Hz away; the exact peak cannot be located from these coarse readings alone. Near resonance the periodic driving force transfers energy efficiently to the oscillator. For light damping the resonant frequency is close to its natural frequency, and a comparatively large steady-state displacement develops. Increasing damping removes more mechanical energy during each cycle. At the same driving-force amplitude the resonance peak becomes lower and broader, and its frequency may shift slightly; a narrower, taller peak is not the expected effect. Take more closely spaced frequencies around the observed maximum while keeping driving amplitude fixed. Wait at each frequency for transients to decay, because a reading taken during build-up depends on the initial state rather than solely on that frequency's steady response. Repeat to estimate measurement spread.

A driven oscillator frequency scan 9

What a complete response should do

Frequency-response readings

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Frequency-response readings — supplied values
Driving frequency / Hz	Amplitude / mm
2.8	2
3.2	5
3.6	20
4	4
4.4	2