Damping and Resonance

A resistive force (friction, air or fluid resistance) removes energy from an oscillator, so its amplitude decays: damping. Light damping gives slowly decaying oscillations; critical damping returns the system to equilibrium in the shortest time without oscillating; heavy damping returns it slowly without oscillating. When a periodic driving force matches the natural frequency, the amplitude becomes a maximum: resonance. More damping lowers and broadens the resonance peak.

Cambridge 9702 Paper 4Edexcel YPH11 WPH15AQA 7408 Paper 1≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Explain damping as energy loss to a resistive force.
Describe and sketch light, critical and heavy damping.
Define natural frequency, forced oscillation and resonance.
Describe how damping changes a resonance curve; give useful and harmful examples.

Pause and think

Soldiers are traditionally ordered to break step when crossing a bridge. Why?

Section 2 of 14

Core idea

Start hereA bridge that danced

In 2000 London's Millennium Bridge began swaying as crowds walked in step with its natural frequency. Pushing anything at its natural frequency builds up huge oscillations: resonance. Engineers added dampers to fix it.

The big idea

Damping removes energy; amplitude falls. Forced oscillation at the natural frequency gives resonance; damping lowers and broadens the peak.

1. Types of damping

Light, critical and heavy damping

Car suspensions are designed to be near critical damping: they settle quickly after a bump without bouncing.

Light: oscillates, amplitude decays slowly.
Critical: returns fastest without oscillating.
Heavy: returns slowly without oscillating.

Worked exampleLosing 10% amplitude per cycle

A lightly damped oscillator.

Goal: Energy lost per cycle; amplitude after 5 cycles.
Strategy: Energy ∝ amplitude².

$0.9^2=0.81$
Energy fraction kept.
$19\%\text{ lost per cycle}$
$1-0.81$ .
$0.9^5=0.59$
Amplitude after 5 cycles.

Answer19% per cycle; 59% of the original amplitude.

✓ CheckExponential decay: same fraction each cycle ✓.

2. Resonance

Resonance curves: amplitude against driving frequency

Lighter damping gives a taller, sharper peak at the natural frequency. More damping lowers the peak and shifts it slightly lower.

Why resonance builds up

Drawing diagram…

More damping: balance reached sooner, so a lower, broader peak.

Useful: microwave heating, radio tuning, MRI.
Harmful: bridges, machinery; prevented by damping or shifting the natural frequency.

Common misconceptionDamping and frequency Heavier damping also shifts the peak slightly to lower frequency.

Pause and think

Why are car suspensions critically damped?

Summary

Light, critical, heavy.
Resonance at the natural frequency.

Resonance: Large amplitude when driving frequency equals natural frequency.

HistoryThe wobbly bridge. London's Millennium Bridge opened in June

2000

and closed two days later. As people walked, the bridge swayed sideways slightly; pedestrians instinctively fell into step with the sway, driving it at its natural frequency. It reopened in

2002

with

91

dampers fitted.

Fun factBreaking a glass with your voice. A wine glass rings at its natural frequency when tapped. A singer holding that exact note loudly enough can drive it at resonance until the amplitude of vibration cracks the glass. It needs very light damping and a sustained, precise pitch.

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