Clock motion work

Behind the hands of a clock, two pairs of gears turn the hour hand exactly 12 times slower than the minute hand, and bring it back onto the same axis. Speed it up and fade the dial to watch them.

How it works

1. The minute hand sits on the cannon pinion (18 teeth), which turns once an hour.
2. It drives the minute wheel (54 teeth): a third of the speed.
3. A minute pinion (18 teeth) on the same arbor drives the hour wheel (72 teeth): a quarter of that.
4. The hour wheel rides on the cannon pinion's tube, so the hour hand shares the minute hand's centre.

ω_hour = ω_minute × (18/54) × (18/72) = ω_minute / 12

For the hour wheel to come back to the centre, both pairs need the same centre distance. With module m, the first pair is (18 + 54)·m/2 = 36m apart. The second pair has 18 + 72 = 90 teeth, so its teeth are made smaller, with module 0.8m: 90 × 0.8m / 2 = 36m. This is called a reverted train.

Real clocks use their own tooth counts and often cycloidal teeth; we use involute gears with at least 17 teeth so the drawing is exact. The arithmetic is the same: the two ratios multiply to 1/12.