Driven Harmonic Oscillator & Resonance

A mass-spring-damper system driven by a periodic force. Watch the amplitude explode as the driving frequency approaches the natural frequency.

ω₀ = 1.00 rad/s ω_drive = 1.00 rad/s Amplitude = — Phase = —
Natural freq ω₀: 1.00
Drive freq ω: 1.00
Damping γ: 0.10
Drive amp F₀: 1.00

Resonance Physics

Equation of motion: ẍ + 2γẋ + ω₀²x = F₀cos(ωt). The steady-state amplitude is A = F₀/√((ω₀²−ω²)²+(2γω)²). At resonance (ω=ω₀), amplitude peaks at F₀/(2γω₀). The lower canvas shows the amplitude curve across all driving frequencies — the sharp peak at ω₀ narrows as damping decreases. Phase lag φ = arctan(2γω/(ω₀²−ω²)) shifts from 0° to 180° through resonance.