Which tooth meets which

Two gears in mesh, turning. Mark one tooth on the pinion with a dab of paint and watch which teeth of the wheel it touches.

The expectation — mine, before doing it — is that over enough revolutions it touches all of them. It does not, in general, and the number it touches is decided by a piece of arithmetic that appears nowhere in any drawing of the pair.

The count #

Number the wheel’s teeth $0$ to $z_2 - 1$ and start the marked pinion tooth against wheel tooth 0. One full turn of the pinion later, the wheel has advanced by $z_1$ teeth, so the marked tooth is now against wheel tooth $z_1 \bmod z_2$. Another turn and it is against $2z_1 \bmod z_2$, and so on.

The set of wheel teeth it ever touches is therefore the set of multiples of $z_1$ modulo $z_2$ — which is the subgroup generated by $z_1$ in the cyclic group of order $z_2$, and that subgroup has

$$\frac{z_2}{\gcd(z_1, z_2)}$$

elements. The pattern repeats after that many turns of the pinion.

For 20 in 40 the gcd is 20, so the pinion tooth meets $40/20 = 2$ wheel teeth and the whole mesh repeats every two pinion turns. For 21 in 40 the gcd is 1 and the tooth meets all forty, with the pattern repeating after forty turns.

The library computes it both ways. hunting walks the mesh — accumulates $(\text{partner} + z_1) \bmod z_2$ until a partner repeats, and counts the distinct partners collected — and then compares that count with $z_2/\gcd$. The walk knows no number theory; the formula does no walking. They agree on every pair tried, which is a check that can lose in either direction: a walk that terminated early would under-count, and a gcd computed wrongly would disagree.

One tooth, in every case, is the difference between a mesh that repeats in three turns and one that repeats in fifty-four.

Why anybody cares #

This is a kinematic statement — which surface touches which surface, and when — and its consequences are about wear and about error, both of which are matters of which pairs of surfaces spend their lives together.

A fault is inherited by its partners and by nobody else. A pinion tooth cut a little thick, or nicked in handling, does its damage to the wheel teeth it meets. With a gcd of 20 that is two teeth out of forty, so those two get all of it and the other thirty-eight get none; the wheel wears into a pattern with a step in it, and the pair becomes progressively worse at running with any other partner. With coprime counts the damage is spread over all forty, which is the same total wear distributed forty ways.

The error pattern repeats at the period. Any departure from the nominal — a tooth out of place, an eccentric bore — produces a positional error in the output that repeats every $z_2/\gcd$ pinion turns. A short period means a strongly repetitive error at a low harmonic; a long period means the same total error spread over a long cycle. The tolerance work in this site’s practice field is about how large the errors are; this is about how they are arranged in time, and the two are independent.

The design response is the hunting tooth: choose the counts so that the gcd is 1, usually by adding a single tooth to one wheel, which changes the ratio slightly and the pattern completely. It is one of the oldest rules in the trade and it is exactly the statement that $\gcd(z_1, z_2) = 1$.

The name is worth explaining because it is descriptive rather than decorative. The extra tooth is called a hunting tooth because it makes each pinion tooth hunt round the wheel rather than settling on a partner: after every revolution it has moved on by one, so it works its way through the whole wheel before returning. A pair without one is sometimes called a repeating pair, and the two names name the same fact from opposite ends.

What repeats, and when #

The period is worth a paragraph of its own, because it is the quantity a measurement actually sees.

The pattern of which tooth meets which repeats after $z_2/\gcd$ turns of the pinion, which is also $z_1/\gcd$ turns of the wheel. For 20 in 40 that is two pinion turns and one wheel turn; for 21 in 40 it is forty and twenty-one.

Now suppose one pinion tooth is a thousandth of a millimetre thick. The output’s position picks up an error every time that tooth is in mesh — once per pinion turn — and the pattern of which wheel teeth were involved repeats at the longer period. So a transmission-error trace of the pair shows a once-per-pinion-turn feature riding on a once-per-cycle envelope, and the envelope’s length is $z_2/\gcd$ pinion turns.

That is why the hunting rule is stated in terms of the pattern rather than in terms of a magnitude. The error’s size is what the tooth’s manufacture decides. Its arrangement in time is what the gcd decides, and the two are independent: a beautifully made pair with a gcd of 20 has a small error repeating twice per two turns, and a badly made coprime pair has a large error spread over forty.

The site’s own gearset fails it #

The default planetary in this library — the one drawn in a dozen figures across two fields — is a 24-tooth sun in a 72-tooth ring, which needs planets of

$$z_P = \frac{72 - 24}{2} = 24.$$

Twenty-four planets against a twenty-four-tooth sun. $\gcd(24, 24) = 24$, so

$$\frac{z_{\text{sun}}}{\gcd} = 1.$$

A given planet tooth meets exactly one sun tooth, and meets it every single revolution, for the life of the gearset. Not two, not a few: one. Three planets, three sun teeth doing all the work against them, and twenty-one sun teeth that never touch a planet at all except through a different planet’s own single partner.

The planet-to-ring mesh is barely better: $\gcd(24, 72) = 24$, so a planet tooth meets three of the ring’s seventy-two.

That gearset is not a mistake — it is the textbook example, 24 and 72 are chosen because they are round and divide by three, and every essay that uses it uses it to make a point about ratios, where none of this matters. But it is the kind of thing that is worth finding out about one’s own default, and the reason it went unnoticed for two fields is instructive: nothing about it is visible. The ratio is right, the mesh is conjugate, the assembly conditions pass, the figures draw. The defect is entirely in a pattern of incidence that no gate on this site looks at and no picture of a gear pair can show.

And the Fibonacci gearset passes it, for free #

The equal-step gearsets two essays ago came out of a quadratic whose root is the golden ratio, and their tooth counts are consecutive Fibonacci numbers: a sun of 55 in a ring of 89, with planets of 17.

Consecutive Fibonacci numbers are coprime — that is the standard consequence of the recurrence, since $\gcd(F_n, F_{n+1}) = \gcd(F_{n-1}, F_n) = \dots = \gcd(1,1) = 1$ — and 17 is not a Fibonacci number but is coprime to both 55 and 89 anyway.

So on that gearset:

Every tooth meets every tooth at both meshes. The gearset that was chosen to make its ratio steps equal turns out also to be fully hunting, and neither property was asked for by the other. That is a coincidence in the sense that no argument connects them, and it is not much of a coincidence in the sense that both are consequences of picking numbers with no common factors — which is what avoiding a rational relationship means in both cases.

How common is it #

Take the 506 planetary designs that survive the assembly conditions and ask how many are coprime at both meshes.

238 of 506 — 47.0%. Slightly fewer than half. So a planetary picked at random from a plausible catalogue is more likely than not to have at least one mesh in which the teeth do not all meet, and the ones that fail badly — like 24 in 24 — are exactly the round-numbered ones a designer reaches for first.

The two conditions pull against each other, which is the practical difficulty. The assembly condition wants $z_S + z_R$ divisible by the planet count, which is a reason to choose counts with small factors; the hunting condition wants the sun, planet and ring pairwise coprime, which is a reason not to. A gearset satisfying both is being asked for numbers that share a factor collectively and none pairwise, and that is a real constraint rather than a coincidence of tastes.

Inside a gearset #

A planetary has more meshes than a pair, and the condition applies to each of them separately.

For the Simpson gearset used in this field — sun 34, rings 74, planets 20 — the two meshes are 34 against 20 and 20 against 74, and both have a gcd of 2. So a planet tooth meets 17 of the sun’s 34 and 37 of the ring’s 74, in each case exactly half, and the pattern repeats after 37 turns of the planet. Not disastrous, and not what anybody chose: it is what happens when three even numbers are picked for other reasons.

For the Ravigneaux — suns 23 and 50, ring 74, planets 30 and 12 — the four meshes have gcds of 1, 6, 2 and 2. The one with a gcd of 6 is the short planet against the long one, where a tooth on the 30 meets two of the 12’s twelve. That mesh is an idler pair whose tooth counts do not affect any ratio, so nothing about the gearbox’s gears is at stake — and yet it is the mesh with the most concentrated wear pattern in the whole gearset, which is a nice illustration of a parameter being free in one respect and not in another.

The lesson for a designer is that a gearset has several of these conditions rather than one, and that the free parameters — the ones the ratios do not see — are the ones nobody thinks to choose well.

What the extra tooth costs #

The hunting tooth is described everywhere as free, and it is nearly free, but the price is exact and it decides which mechanisms may use it.

Adding one tooth to a wheel of $z_2$ changes the ratio by $1/z_2$ of itself. On the 24-in-72 pair that is 1.39%: a reduction of 3.0000 becomes 3.0417. In a gearbox, a hoist, a winch or a differential that is nothing whatever — the ratio was a round number by convenience rather than by requirement, and no downstream quantity notices four parts in three hundred.

In a timing mechanism it is fatal, and this is the case worth naming because it inverts the rule completely. A clock’s train has to divide one revolution of the escape wheel into an exact number of hours, so its ratios are not approximately anything: they are the factorisation of the numbers the dial demands, and every tooth count in the train is chosen to make integers come out. A clock is a factorisation, and a factorisation has no slack in it. Adding a tooth to make a gcd come out at one would move the clock, which is the one thing a clock may not do.

So a clock is a mechanism obliged to accept the worst case of this essay’s arithmetic. Its counts are chosen for divisibility, divisibility is exactly what makes a large gcd, and the same tooth therefore meets the same partner for a century of running. That is not a design oversight; it is a consequence of the requirement, and it explains a piece of horological practice that looks superstitious from outside — that a clock’s wheels are made better than a gearbox’s, in harder material, cut more accurately and polished, because they cannot be given the statistical protection that a hunting tooth provides. Where the ratio is a count, the wear has to be engineered out of the surfaces, because it cannot be spread over them.

The same reasoning sorts the rest of the field’s mechanisms. A positioning drive whose ratio has to be exact for the sake of a controller’s arithmetic is in the clock’s position. An indexing mechanism dividing a circle into a required number of parts is too. A traction drive, a transmission, a reduction for a motor — none of these has an exact ratio requirement, all of them may take the extra tooth, and all of them do.

That also gives the gcd condition its proper standing among the field’s arithmetic. It is not a rule every gearset should satisfy; it is a rule every gearset that can satisfy it should, and the ones that cannot are identifiable in advance by a question that has nothing to do with gears — does anything downstream require the ratio to be exact? A designer who can answer that has already decided whether the hunting tooth is available, before looking at a single tooth count.

Where it stops mattering #

Three cases where the gcd is irrelevant, since a rule with no exceptions is usually a rule that has not been examined.

A pair that is only ever run one way and briefly. A hoist, a jack, a hand-cranked mechanism: a few hundred cycles in a lifetime, and no wear pattern has time to develop. The gcd is a statement about the long run and there is no long run.

A pair with a large deliberate backlash and no positional duty. If nothing downstream cares where the output is to a fraction of a tooth, an error repeating twice per two turns is no worse than the same error spread over forty. The wear argument still applies; the accuracy argument does not.

An epicyclic’s planet-to-ring mesh where several planets share the ring. Here the accounting changes shape rather than going away: each planet has its own orbit of ring teeth, and whether the three orbits are the same set or three different sets is decided by yet another divisibility — this time involving the planet count as well. The site has not enumerated that case, and it is named here as owed rather than answered.

The general point is that the condition is about incidence over time, so it matters exactly when time matters, and the mechanisms in this fleet where it matters most are the ones that run continuously and are expected to stay accurate — which is to say, the ones this field is about.

The same shape as the assembly condition #

It is worth saying plainly what these two conditions have in common, because they are the two arithmetic facts in this whole subject and they behave identically.

Both are divisibility statements about tooth counts. Both are invisible in every drawing, every ratio calculation and every stress calculation. Both are unaffected by module, by size, by material and by tolerance — a perfectly made gearset has exactly the same gcd as a badly made one. And both are the sort of thing that produces a mechanism that works and then behaves oddly: a gearset that will not take its third planet, or one that wears into a pattern nobody can explain.

The site’s habit is that every claim gets a test it could fail, and the tests here are of the same kind: the walk against the arithmetic for this one, and the divisibility against a survey for the other. What neither has is a drawing that would reveal it, and that is the reason both essays exist.

One further consequence of the same trade, and it is the reason the rule is stated for the pinion rather than for the wheel. Adding a tooth to the pinion changes the ratio by $1/z_1$ of itself, which on a twenty-four-tooth pinion is 4.2% — three times the disturbance of adding one to a seventy-two-tooth wheel. So where the hunting tooth is available at all, it should be added to the larger member, and where the pair is close to one-to-one there is no larger member and the cheapest option costs the same either way. That is a small piece of practice with an exact justification, and it falls out of the same fraction that priced the clock’s refusal.