Illustrated Physics
physics, drawn
Concept

Limit cycle — where it appears

An isolated closed orbit that a self-sustained oscillation settles onto, with an amplitude fixed by the system itself rather than by how it started. Clocks, heartbeats, lasers and singing flames run on limit cycles, fed by a steady source and limited by a nonlinearity.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

The same steady pull, twice. Spring force divided by the normal load, against time, with the far end drawn away at 100 µm/s in both traces and every property of the contact the same. The soft holder at 0.4 k_c produces events every 3.75 s, each of which reaches 0.94 m/s — 9.4e+3 times the speed it is being pulled at. The stiff one at 1.6 k_c settles to a straight line and stays on it.

The chatter a stiffer holder removes

A brake squeals, a bow sounds a string, a fault slips in jerks. The usual explanation is that static friction exceeds kinetic friction — and that explanation, taken seriously, predicts the jerking would happen no matter how the thing were held. It does not, and what decides is a length nobody mentions.

mechanics · Friction A whisper that grows into a note. The amplitude of the tube's lowest mode against time, in units of the time sound takes to cross the tube (a period is two), started from a small disturbance, with the gauze at a quarter of the length, at 0.12 of it, and at three-quarters; heat-transfer strength K = 1. At L/4 the disturbance grows some fortyfold over about thirty-five periods and then stops growing, at an amplitude of 0.83 in these units, where the air's to-and-fro speed at the gauze is comparable to the steady upward draught and the heat transfer, which goes as the square root of the air's speed, can no longer follow it in proportion. At 0.12 the gauze is near the edge of the singing range and the sound grows only slowly. At 3L/4 the same disturbance dies away within a few periods.

The tube that sings when heated low down

Hold a wide metal tube upright, push a red-hot wire gauze a quarter of the way up inside it, and take the flame away. The tube begins to howl — a loud, steady note at its lowest resonance, lasting until the gauze cools. Put the gauze a quarter of the way down from the top instead and the tube is silent; lay the tube on its side and it stops. Heat is driving a sound, and whether it drives or damps depends on where the heat is released and when, in a rule Lord Rayleigh stated in 1878 that now governs rocket engines and power-station turbines.

waves · Resonance

Named alongside it

The objects these essays reach for when they reach for this one.

Coefficient of frictionCombustion instabilityContactDampingFrictionHeat transferHopf bifurcationInstabilityRayleigh's criterionRelaxationResonanceSelf excited oscillation

All concepts