Clockwork - 2D gear train and pendulum escapement

The pendulum sets the rate

For small swings a pendulum of length L in gravity g has period T = 2π√(L/g). A length of 0.994 m gives T = 2 s, one beat per second. For a finite amplitude θ0 the exact period is T = 4√(L/g) K(sin(θ0/2)) with the complete elliptic integral K, or T ≈ T0(1 + θ02/16 + 11θ04/3072). A larger swing means a longer period: the circular error.

Escapement and gear train

At each centre crossing the anchor lets the escape wheel advance half a tooth. With 30 teeth that is 60 beats per revolution, so a 2 s pendulum turns the escape wheel once per minute (the seconds hand). The train 8/64 and 8/60 gives (8/64)(8/60) = 1/60, a minute hand at one turn per hour; 10/30 and 12/48 add 1/12 for the hour hand. Meshing gears must share the same tooth pitch; two gears of pitch radius r = m z/2 have their centres m(z1 + z2)/2 apart and turn in opposite directions with speed ratio z1/z2.

Temperature

A rod lengthens by ΔL/L = α ΔT, so the period grows by αΔT/2 and the clock loses 86400 · αΔT/2 s per day. Steel (12e-6 /K) loses about 10 s per day per 20 K. Invar (1.2e-6 /K) is ten times better, and Harrison's gridiron of brass and steel rods cancels most of the expansion.

Honest simplifications: an ideal escapement restores the amplitude exactly at each centre crossing; there is no friction in the train, no recoil error, no swing-amplitude change from drive torque and no suspension spring. The pendulum is a point-mass bob on a massless rod. The gear train is the standard textbook 60:1 and 12:1 chain, not the 3D sim's decorative gear set.

• Select the Moon preset: g is a sixth, the period grows by 2.4 and the clock runs very slow.
• Raise the temperature with a steel rod, then switch to Invar.
• Increase the amplitude and watch the clock lose time.