Past exam of the physics A course of the University of Cambridge / 2017 / Paper 2 / Section B / B6 / Resonance and high-frequency asymptote

To find the displacement peak, minimize the squared denominator:

\frac{d}{d ω} [(k - m ω^{2})^{2} + b^{2} ω^{2}] = 2 ω [- 2 m (k - m ω^{2}) + b^{2}] .

The nonzero minimum gives

ω_{a} = \frac{k}{m} - \frac{b ^{2}}{2 m ^{2}}, b^{2} < 2 mk .

For stronger damping there is no positive-frequency displacement maximum. When

ω ≫ ω_{0}

and

b ≪ mω

,

∣ R (ω) ∣ ≃ \frac{1}{m ω ^{2}}, R ≃ - \frac{1}{m ω ^{2}} .

The inertia then dominates the restoring and damping forces.

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