DAMPED
ma + cv + kx = 0 at every drawn frame — ring, settle, or crawl.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
The spring-mass system returns — with a leak. A dashpot now drains energy from every move the block makes, and the only question is the manner of the death: does the oscillation ring away, settle cleanly, or crawl home?
The drain
One new term in Newton’s law: a force proportional to velocity, always opposed to it, stealing energy at the rate cv². Two rates now race — the spring’s natural ω₀ against the drain’s γ — and their contest has exactly three outcomes.
Solve
Underdamped rings at a slightly lowered ω_d inside a shrinking envelope; critical is the razor edge — fastest return, zero overshoot — and overdamped is two dueling exponentials where the slow one always wins. Every regime has a full closed-form solution; nothing here will be integrated.
Release
Released from rest at A. The dashed envelope ±A′e^{−γt} bounds the ringing — the energy the spring can no longer refill. Slide c up to c_crit (shown in the outputs) and watch the ringing vanish.
Audit
The audit worth the whole module: ma + cv + kx evaluated at 61 samples in every regime comes back zero to machine precision — the drawn curve does not approximate the differential equation, it IS its solution. Doors, shock absorbers, and seismographs are all tuned somewhere on this one slider.