A directly meshing 24-tooth input and 40-tooth output first return together after five input turns and three opposite output turns. Mark a direction on each gear and compare it with that gear’s own start. The input comes home after every complete turn, but the output usually does not. A simultaneous return requires both accumulated turn counts to be whole numbers.
Build a small reunion experiment
The new Gear Reunion Bench isolates this question with two retained vertical shafts on a rigid frame. The red top bush identifies the 24-tooth input; the blue bush identifies the 40-tooth output. Both gears sit at the same height, with their centres four hole spacings, or thirty-two millimetres, apart. The estimated assembly time is twenty minutes. Follow the illustrated sequence rather than adding both gears before fitting the shaft retainers.
This is an original educational model inspired by the gearing section of Brick Technology’s credited multi-dial clock demonstration. That video shows several count scales and a much more involved machine, with an escapement and winding arrangements. Our bench does not recreate its clock, calendar or power system. It gives you a short, observable cycle before attempting a machine whose outputs might take much longer to return.
A separate public builder discussion asks whether a precise 365-to-one ratio can be constructed. That is a useful reminder to work with exact tooth counts instead of assuming that any desired decimal ratio follows from having many gears. This guide answers a narrower question about repeats in one pair. It does not claim to solve the calendar construction or certify any clock’s long-term accuracy.
Watch both markers, not only the input
Place a tiny removable dot on the top face of each gear, away from its teeth. Sketch each dot’s starting direction on separate paper beside the frame. Do not place paper, stickers or pointers between the moving teeth. Starting directions need not be parallel: you are looking for each marker to return to its own orientation.
Before turning the real model, predict what happens after one, two and five input turns. Try the lab’s turn counter and compare both arrows with their dashed starting directions. Negative turns let you inspect reversal. Virtual tooth choices are comparisons only; changing gear sizes on a physical model requires checking the centre distance, support and mesh again.
Interactive: When do both gear markers return?. Predict the first joint orientation return, scrub signed input turns, compare virtual integer tooth counts and reset. A 24T/40T pair returns after five input and three opposite output turns; ideal counts, not clock timing or measured wear.
The two drawing circles are orientation counters separated for readability. They are not a scale drawing of our mesh, and their arrows omit the initial tooth-seating offset. This distinction keeps the counting question clear: an initial phase belongs to setup, while accumulated rotation describes what happened after setup. The lab reports turns, degrees and tooth counts. It does not report measured speed, force, friction or wear.
Why the return takes five turns
Every tooth that passes through the mesh on the input corresponds to one tooth passing on the output. One input revolution sends twenty-four teeth past the contact. The output therefore makes twenty-four fortieths of a revolution, or three fifths, in the opposite direction. After two input turns it has made six fifths; after three, nine fifths. None of those is a whole output turn.
After five input revolutions, one hundred and twenty teeth have passed. Divide that count by twenty-four and you get five input turns. Divide it by forty and you get three output turns. One hundred and twenty is the smallest positive common multiple of the two tooth counts. This is why the first simultaneous return is a whole-cycle question as well as a gear-ratio question.
You can find the same answer by dividing both tooth counts by their greatest common divisor, eight. The reduced pair is three to five. Read those numbers carefully: the input makes five revolutions while the larger output makes three. Reversing the motion changes the signs, not the size of the joint cycle. Returning to the start by undoing your last turn is a different experiment from completing a positive cycle continuously.
Record a fair physical trial
Hold only the stationary base on a level table and turn gently. Keep the same input, viewing side and reference directions for three trials. Record at each complete input return; count output revolutions separately rather than guessing from a blurred marker. Do not brake an output while its neighbour is turning.
| Input turns | Ideal opposite output turns | Both home? | Your observation | Loose part or uncertainty |
|---|---|---|---|---|
| 1 | 0.6 | No | Record | Record |
| 2 | 1.2 | No | Record | Record |
| 3 | 1.8 | No | Record | Record |
| 4 | 2.4 | No | Record | Record |
| 5 | 3 | Yes | Record | Record |
The observed columns are blank because we have not physically tested this model. Hand-set landmarks, marker width, perspective and backlash affect how precisely you judge a return. Pause at a landmark, write the observation and record uncertainty. A repeated orientation does not establish even tooth loading or predict how long a gear will last. Those questions need contact and material evidence beyond this counting exercise.
Fix support problems before trusting a count
Each shaft turns in a round frame bearing. Seated half bushes immediately above and below the frame stop the whole shaft escaping vertically. A spacer carries the gear and a cap retains it from above. Four separately retained corner feet support the frame. Review the axle guide, bush guide and frame guide to identify which real part performs each job.
If a shaft rises, a cap comes loose or the gears rub, stop. Check retainer seating, equal gear height and the four-gap centre distance. Turn the entire keyed output assembly gently to seat teeth in gaps; do not force tooth against tooth. Our finite geometry and assembly checks examine the authored model, not manufacturing tolerances or a load rating. The animation cannot supply a missing bearing or retainer.
For a comparison, Gear Friends adds an idler, changing the last gear’s direction. The Compound Counting Bench uses shared-shaft stages and a much longer count. First learn to separate a single input return, an output return and a return of the whole configuration. That habit makes larger repeating mechanisms easier to explain without confusing their motion with elapsed time.
Interactive: Check yourself. A few quick questions on the ideas in this guide, each with an explanation.
- A 24T input turns once. Has its 40T output returned to its own starting orientation?
- What is the first positive joint return for this pair?
- Does the cycle count say how many seconds the experiment takes?
- What should you do if the moving shaft lifts?
