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🌊 Free Vibration Decay Explorer
Pull a structure aside and let it go — no ongoing force, just its own mass, stiffness, and damping deciding what happens next. Change the damping ratio and watch the decay curve shift from bouncy oscillation, to the fastest possible settle (critical damping), to a slow crawl back to zero (overdamped). No score, no target — just the physics every damped system in engineering follows.
System
Regime
Natural period Tn—
Damped period Td—
—
Live Decay Curve
This simulation uses the exact closed-form free-vibration
solution (
engine/dynamics.js → sdofFreeVibration, Chopra
§2.2) — not a numerical integration, an exact analytical formula. Independently
re-derived and cross-checked before publishing: at ζ=6.32%, u(0.05s)=−0.819391,
u(0.1s)=0.671263, u(0.2s)=0.450223 — matched the engine to 6 decimal places.Method
Chopra §2.2Underdamped: u(t)=e^(−ζωnt)·[u0·cos(ωdt) + ((v0+ζωnu0)/ωd)·sin(ωdt)], with ωd=ωn√(1−ζ²).
Regimesζ<1: underdamped (oscillates while decaying). ζ=1: critically damped (fastest non-oscillating return to zero). ζ>1: overdamped (slower, non-oscillating return).