Resonance occurs when driving frequency matches natural frequency.fd 1.00 Hz · f0 1.00 Hz · amplitude 0.00 m

3D Resonance

Natural frequency, forced vibration, damping, phase, and energy

Workspace

Driven-oscillator laboratory

Experiment controls

Oscillating mass1.0 kg

Spring stiffness40 N/m

Damping ratio0.08

Driving frequency1.00 Hz

Driving force2.0 N

Simulation speed1×

Camera zoom1×

Oscillator sound

Visual experiments

Live response

Natural frequency1.01 Hz

Amplitude0.00 m

Frequency ratio0.99

Phase lag0°

Equations

f₀ = (1 ÷ 2π) √(k ÷ m)

mx″ + cx′ + kx = F₀ sin(2πfdt)

A = (F₀ ÷ k) ÷ √((1 − r²)² + (2ζr)²)

Match the driving frequency to the natural frequency and observe the response.

Object focus

Tap the driver, spring, mass, damper, response curve, or example apparatus.

Teacher presets

Teacher narration

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Narrator

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Chapters

Learning objectives

Define resonance as frequency-matched energy transfer.
Calculate natural frequency for a spring–mass system.
Predict amplitude from frequency ratio and damping.
Connect phase, energy, and steady-state response.
Recognize resonance in sound, structures, and coupled oscillators.

Key relationships

Natural frequency: f₀ = √(k/m) ÷ 2π.

Frequency ratio: r = fd ÷ f₀.

Resonance: largest response occurs near r = 1.

Damping: increasing ζ lowers and broadens the peak.

Suggested experiments

Sweep the driver slowly through the natural frequency.
Increase damping while frequencies remain matched.
Double the mass and locate the new resonance.
Compare tuning forks, pendulums, and a bridge model.

Questions for exploration

Why does amplitude need time to build?
How does stiffness move the resonant frequency?
Why is damping important in buildings and machines?

Seven-question resonance check

Answer all seven questions.

Teacher whiteboard

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