SHM Resonance Lab

Damped harmonic oscillators are described by a characteristic equation...

This tool is really about timing. Every mass on a spring has one natural frequency ω₀ at which it 'wants' to wobble. Push it at that rhythm — like pumping a playground swing at just the right moment — and each small shove adds energy, so the amplitude X grows far beyond what a steady force would give. The magnification M measures that boost, and only damping keeps it from running away. At resonance the motion lags the force by exactly 90°.

Draw the resonance curve and you might expect its tallest point at ω = ω₀. It isn't: the displacement amplitude actually peaks slightly lower, at ω_r = ω₀√(1 − 2ζ²). Stranger still, once the damping ratio passes ζ = 1/√2 ≈ 0.707 — well before critical damping at ζ = 1 — that peak vanishes entirely and the curve just slides downhill. So a system can be lightly damped and yet have no resonant peak at all.