Cuckoo Clock Lab (2D)

What this clock does

A cuckoo clock is two clocks in one case. The going train (left weight) drives the hands and, through an escape wheel and anchor, keeps a pendulum swinging. The striking train (right weight) is released every half hour; at the hour it calls once per hour on the dial, counted by a rack and snail, and its lever pulls the bird out and lifts two bellows that blow two stopped whistle pipes.

1. Pendulum and rate

The pendulum obeys θ'' = −(g/L) sinθ − (ω0/Q) θ' + kick with ω₀ = √(g/L). The exact period for amplitude A is T = 2π√(L/g) / AGM(1, cos(A/2)) (arithmetic-geometric mean), a little longer than the small-angle value: the "circular error". The clock counts one second per nominal period T_nom = 1 s, so the rate is 86400 (Tnom/T − 1) s/day (plus = gains). A rod of material α changes length by L(1 + αΔT): about −43200 α s/day per kelvin.

2. Escapement energy balance

Each beat the weight falls a small height h and the escapement hands the pendulum Ebeat = η M g h as an impulse while it passes through the centre arc. Viscous loss per beat is π m L² ω0² A² / (2Q), so the amplitude settles at A = √(2 Q Ebeat / (π m L² ω0²)). A heavier weight or a shorter run time gives a bigger swing. Mean power is M g h × 2 beats per second, here only a few hundred microwatts.

3. Rack and snail

The snail on the hour wheel has 12 steps; the rack tail falls onto the step of the current hour and the rack drops by (h − 1) tooth pitches. The gathering pallet then lifts the rack one tooth per cuckoo until it is back on its hook, so the number of calls is exactly h. Half hours call once. The day total is 2(1+2+...+12) + 24 = 180 calls.

4. Bellows and stopped pipes

A closed (stopped) pipe of length L and radius r resonates at f1 = c / (4 (L + 0.6133 r)) with odd harmonics 3f₁, 5f₁... and speed of sound c = 331.3 + 0.606 T m/s. The weighted bellows gives a steady pressure p = m g / A; air leaves through the flue slot at Q = Cd Aslot √(2p/ρ), so a note lasts t = V / Q. The short pipe (higher) calls "cuck", the long pipe "oo".

Model checks on this page

• Free-swing RK4 period against the exact AGM period, and the simulated steady amplitude against the energy-balance formula.
• A finite-difference solution of the 1D wave equation in each pipe against f₁ = c/(4(L+0.6133r)).
• Rack-and-snail counts for all 12 hours (sum 78) and a 24-hour run of the strike scheduler (180 calls), plus the bellows drain time against V/Q.

Honest simplifications

• The anchor and pendulum are one rigid body with the same angle; the real crutch and fork have clearance and a lock/drop that is not modelled.
• The escapement is a constant impulse over ±2.5° of the swing; frictional and recoil effects are folded into the efficiency η and the quality factor Q.
• Air damping is linear (a single Q); real pendulums add a quadratic term.
• Whistles are modelled as ideal stopped pipes with an end correction; the jet, the edge tone and the mouth geometry are not simulated, so the pitch ignores blowing pressure. The call sound is synthesised from the computed pitches.
• The 3D train wheels show the going-train gear ratios (48-8, 40-8, 32-8 and a 30-tooth escape wheel, 30 s per revolution); the chain wheel and the weights' travel are scaled for display.
• Not any specific maker's clock; numbers are typical orders of magnitude.

What to try

• Press Set to 11:59:40, switch to the Strike train camera and watch the rack fall onto the snail step for 12 and gather twelve cuckoos.
• Pick Cold cellar with the brass rod, then with the invar rod, and compare the rate.
• Shorten the short pipe by 10 mm and read the new pitch; the wave-equation check follows.
• Raise the time speed to 3600x to watch a whole day of 180 cuckoos.

• Press Set to 11:59:40 and watch the rack fall onto the snail step for 12 and gather twelve cuckoos, one tooth each.
• Open the Cold cellar presets: the brass rod shortens and the clock gains; the invar rod barely moves.
• Raise the going weight or shorten the run time: the swing grows and the rate shifts by the circular error.
• Shorten the short pipe by 10 mm and read the new pitch; the finite-difference check follows.
• Raise the time speed to 3600x to see a whole day: 180 cuckoos.