🌀 Damped Harmonic Oscillator

About this simulation

A mass hangs from a spring and passes through a dashpot (the damper), pulled down to an initial displacement and released. Newton's second law gives m·x'' + c·x' + k·x = 0. How the mass returns to equilibrium — oscillating, settling as fast as possible without overshoot, or crawling back sluggishly — depends entirely on the balance between the spring's stiffness k, the mass m, and the damping coefficient c.

How it works

• Natural frequency ω₀ = √(k/m) — how fast the undamped mass would oscillate.
• Damping ratio ζ = c / (2√(km)) — the single number that decides the regime.
• Underdamped (ζ < 1) — the mass overshoots and oscillates with a decaying envelope before settling.
• Critically damped (ζ ≈ 1) — the fastest possible return to rest with no overshoot.
• Overdamped (ζ > 1) — the mass creeps back to equilibrium slowly, never overshooting.

Controls

• Mass — heavier masses respond more sluggishly to the same spring/damper.
• Spring stiffness k — a stiffer spring pulls the mass back harder and faster.
• Damping c — sweep this from 0 up through the critical value to see all three regimes.
• Initial displacement — how far the mass is pulled down before release.
• ↺ Release from rest — restarts the motion from the chosen displacement. Drag to rotate, scroll to zoom.

Did you know?

Car suspensions are deliberately tuned close to critical damping: enough damping to kill oscillation quickly after a bump, but not so much that the ride becomes harsh and slow to settle. Shock absorbers are literally velocity-dependent dampers doing the job of c·x' in this same equation.