Pendulum Clock: Gear Train and Escapement

Gear trains. Two meshing gears must have the same tooth size (module), so the teeth passing the contact point per second are equal: N_Aω_A = -N_Bω_B (the minus sign is the reversal of the direction). The speed ratio is therefore a ratio of tooth counts, and with a lossless mesh the torque ratio is the inverse: τ_B = (N_B/N_A) τ_A η, so power τω is conserved apart from the mesh loss (here η = 0.93 per mesh). This movement has barrel 90 / pinion 12, centre wheel 80 / pinion 10, third wheel 75 / pinion 10 and a 30-tooth escape wheel: 7.5 × 8 × 7.5 = 450. The escape wheel turns once a minute, the centre arbor once an hour (1/60), and the 12:1 motion work (10/40 × 10/30) turns the hour hand once in twelve hours.

Pendulum. A real pendulum is a rigid body: ω₀² = M g d / I with I the moment of inertia of rod and bob about the pivot. For large swings the period grows: T = T₀/AGM(1, cos(θ₀/2)) ≈ T₀(1 + θ₀²/16), the “circular error” that makes a clock with a bigger swing run slower. The rod length changes with temperature, L = L₂₀(1 + α(T - 20)), so a steel rod makes a clock lose about 0.5 α × 86400 ≈ 0.5 s per day for each degree of warming; Invar nearly removes the effect.

Escapement. The escape wheel is held back by the anchor and released once per beat (every zero crossing of the pendulum, 1 s for a 2 s pendulum), advancing half a tooth pitch (π/30). The work given to the pendulum per beat is W = η_imp τ_esc π/30, delivered here as a kick of kinetic energy at the bottom of the swing. Friction and air drag remove πE/Q per beat, so the swing settles where W = πE/Q, giving θ₀² = 2 W Q /(π I ω₀²). If the steady swing is below the lock-up angle (0.6 deg) the anchor cannot release the wheel and the clock stops.

Honest simplifications. The tooth profile is a simple trapezoid, not a true involute, and the gears are rigid and frictionless apart from a constant efficiency per mesh. The impulse efficiency (0.6), the Q factor, the 20 mm drum and the 0.6 deg lock-up are plausible round numbers, not measurements of a particular clock. The escapement is reduced to an energy kick at the bottom of the swing: recoil, drop and the dependence of the impulse on the swing angle are ignored, as are the pendulum’s suspension spring and air buoyancy. The visual swing can be exaggerated for readability (the physics always uses the true angle), and the animated tick of the escape wheel is smoothed. The rate figure in seconds per day assumes the train is designed for a 1 s beat.

• Press Hot summer day (steel rod): the steel rod grows by about 12 ppm per kelvin, the clock slows by several seconds per day. Then Hot day, Invar rod: the nickel alloy barely moves and the error nearly vanishes.
• Press Weight too light (stalls): the escapement can no longer keep the swing above the lock-up angle and the clock stops; the red banner appears. Push the pendulum restarts it only if the weight is heavy enough.
• Raise the weight and watch the steady swing grow in the lower-right graph while the rate drifts slow, because a wide swing lengthens the period (the circular error in the graph beside it).
• Use Advance 1 hour and read the drift on the clock-error graph; Regulate finds the length that makes the beat exactly one second.