ESCAPEMENTS

Controls

Rotate view	left-drag
Pan	right-drag
Zoom	scroll wheel
Play / pause	spacebar
Restart	R
Escapement	1 · 2 · 3
Step 20 ms	. (period)
Chapters	← → arrow keys

What is simulated

A real pendulum — I θ̈ = −m g L sin θ − c θ̇ + M, integrated at 2 kHz, with sin θ intact so circular error is present rather than assumed. For the anchor escapements the wheel angle is not integrated while a tooth is in contact: it is solved from the requirement that the tooth tip lie on the pallet face, and the torque delivered to the pendulum follows from virtual work, M = T · dφ/dθ. A locking arc concentric with the anchor arbor therefore gives dφ/dθ = 0 exactly — “dead-beat” falls out of the geometry instead of being asserted.

The charts are measurements, not formulas: each point is a full run of the same integrator, solving for the amplitude at which the escapement’s input balances the pendulum’s losses, then averaging the period over seventy seconds.

Sources

Mechanisms follow Henry T. Brown, 507 Mechanical Movements (19th ed., Brown & Seward, 1901; Dover reprint 2005): figure 288 recoil, 289 repose or dead-beat, 311 double three-legged gravity — the escapement Edmund Beckett Denison (Lord Grimthorpe) designed for the Great Clock at Westminster. Brown’s own text for 288/289 describes the distinction the simulation reproduces: the dead-beat faces are “cut to a curve concentric to the axis… consequently, during the time one of the teeth is against the pallet the wheel remains perfectly at rest.”

Pallet proportions are conventional rather than copied from any particular movement: a 30-tooth wheel, pallets embracing 7½ teeth, and a face inclination solved for a mechanical advantage of about 3. Drawn pendulum length is foreshortened with the usual break mark; the angle is exact.

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