A ratio that is a count

Everything else in this field is a length or an angle, and every length and angle varies as the mechanism moves. This one does not, and the reason is worth an essay: a gear ratio is a count.

Two gears in mesh turn in the ratio of their tooth counts, exactly, because the teeth are discrete and every one of them has to be passed. There is no averaging, no operating point and no drawing tolerance in it. Two integers.

The mechanism that makes a big number out of a small one #

An ordinary epicyclic — sun, planets, ring, carrier — gives a reduction of 1 + ring/sun, which for reasonable tooth counts is between about 3 and 12. The site computed one two ways in the gears field and found the tabular and Willis routes agreeing exactly, as they must.

A hundred to one out of one stage needs a different trick, and the trick is a compound epicyclic: two ring gears, one held and one driving the output, with a single set of stepped planets meshing with both. The reduction is $1 / (1 - z_1 z_4 / z_2 z_3)$, and the interesting part is the subtraction.

Choose the tooth counts so the two products are nearly equal and the denominator is nearly zero, and the reduction is enormous. For z = 30, 68, 32, 69 the products are 30 × 69 = 2070 and 68 × 32 = 2176, the difference is 106, and the reduction is 2176/106 = 20.53:1 out of gears that individually offer nothing like it.

Push the counts closer together and it grows without limit. The last row of the table is one tooth different from the first and gives 136:1.

Exact, and checked twice #

The reduction is computed here two ways, exactly as the gears field insists.

The first is the formula above. The second applies Willis’s relation — (ω_sun − ω_carrier)/(ω_ring − ω_carrier) = −ring/sun — to each ring in turn and eliminates the carrier’s speed between them. They share no algebra, and they agree to 10⁻⁹ on every set of counts tried, which is double-precision noise on numbers of this size.

And the ratio is checked to be rational: the reduction times the integer denominator z₂z₃ − z₁z₄ comes out at the integer z₂z₃, to a part in 10¹². That is not a formality. It is the difference between a number that happens to be close to a fraction and a number that is one.

Exact does not mean robust #

Here is what the exactness does not buy.

One tooth changes the reduction by 39%. Three teeth change it by a factor of six and a half. Every one of those reductions is exact, and the family is spread over a factor of twelve.

That is the arithmetic of a difference of nearly equal numbers, and it is the same conditioning story the site’s algebra field keeps meeting: subtracting two close quantities amplifies whatever varies in them. Here nothing is uncertain — the counts are integers and are known — so the sensitivity does not produce an error. It produces a design that has to be got exactly right, with no intermediate values available.

The practical consequences are worth stating because they are all real.

There is no fine adjustment. The available reductions are the ones the integers give. Wanting 25:1 from this family means finding another set of four counts, not tweaking one.

A drawing error is enormous. A ring specified with one tooth too many is not a 1.4% error. It is a gearbox with the wrong output speed by a factor of 1.4.

The assembly conditions are arithmetic too. A planetary set only assembles with equally spaced planets when the tooth counts satisfy a divisibility condition, which is another integer constraint on the same four numbers. It is named here and not computed; the site’s planetary machinery does the kinematics and not the assembly arithmetic.

What the sensitivity is, as a derivative #

The one-tooth sensitivity can be written down, and writing it down says where it comes from.

With $P = z_1 z_4 / z_2 z_3$ the reduction is $1/(1-P)$, so the change in the reduction for a change in $P$ is $\mathrm{d}(\text{ratio})/\mathrm{d}P = 1/(1-P)^2$, which is the reduction squared.

The sensitivity is the square of the reduction. A 20:1 drive amplifies a relative change in the product by four hundred; a 100:1 drive by ten thousand. That is why the curve above steepens so violently towards the right: it is not an artefact of the tooth counts chosen, it is the shape of 1/(1 − P) near its pole.

That derivative is checked rather than trusted. For this drive the prediction is 20.528² = 421.4, and a central difference in the product — taken by moving a tooth count fractionally, which is not a thing a gear can do and is a perfectly good thing to do to a formula — agrees to 3.8 × 10⁻⁶ relative. What is left at that step size is the third-derivative term, which near a pole is not negligible, so the check is written to 10⁻⁵ rather than to machine precision: asserting that a finite difference is exact is a way of asserting nothing.

It is the same discipline as everywhere else on this site, applied to a formula simple enough that nobody would think to check it — which is precisely the kind that goes wrong. The practice field’s four-signed Jacobian survived four phases for exactly that reason.

Two families, one pole #

It is worth seeing the whole family rather than the four rows in the table.

Every compound epicyclic whose products are close has a reduction near the pole of 1/(1 − P), and the family is dense: there are a great many sets of four integers whose products differ by a little. What that means practically is that almost any reduction is available from this architecture — the family is dense in the useful range — but not by adjusting a design. It is available by searching the integers.

That is a genuinely different design activity from anything else in this field. Every other mechanism here is designed by choosing lengths and angles, which are continuous, and the design space is explored by scanning. This one is designed by choosing four integers subject to divisibility conditions, which is a search over a lattice — and the appropriate tool is a loop over candidates rather than an optimiser.

The site’s own scan for a steering arm angle and this search are the two extremes of the same activity: one over a continuum with a flat optimum, one over a lattice with no interior at all.

Where the exactness comes from, precisely #

It is worth being careful about what is exact, because the claim is stronger than it looks and narrower than it sounds.

Exact: the average angular ratio between input and output over any whole number of turns of the input, and — because the teeth are rigid and the mesh is continuous — the instantaneous ratio too, provided the flanks are conjugate.

Not exact: everything that depends on the flanks being what they are supposed to be. A gear cut with a wrong profile, or run at the wrong centre distance, or with enough backlash to let the flanks separate, does not deliver the ratio at every instant — it delivers it on average, over the tooth, and misses within it. That is transmission error, and it is what a gear whine is.

So the exactness is a statement about the counting being exact, given that the teeth are doing their job. The gears field spent an essay on what makes them do it — an involute flank keeps the ratio constant while the contact point slides — and that essay’s conclusion is the premise of this one.

The one place the count can be wrong #

There is a way for a gear ratio to fail to be what the tooth counts say, and it is worth knowing because it is the only one.

If the flanks separate — if the drive reverses, or the load goes to zero and comes back, or a tooth is missing — the output does not follow the input for the width of the backlash. The counting is not violated: over any interval in which the flanks stay in contact, every tooth is passed. What is lost is the phase, and it is lost by an amount up to the backlash and never recovered.

So the exact statement is: the ratio is exact and the position is not. A gearbox that has reversed a thousand times has turned its output exactly the number of turns the counts require, and its output shaft is not where it would have been without the reversals.

That is not a defect in the arithmetic of this essay; it is a different quantity. The gears field measured it directly — backlash as an allowance rather than a fault, computed from the centre distance by a formula that turns out to be accurate to 0.16% at realistic errors — and the two results sit together: the ratio comes from the counts, the lost motion comes from the clearances, and neither says anything about the other.

Why this is on the tooth ladder #

The gears field’s argument has been that a ratio is a function and a tooth profile is the thing that makes it a constant. This essay is the top of that ladder because it is where the constant becomes a count rather than a property of a curve.

The distinction is worth keeping. Two gears in mesh with hand-drawn flanks still turn 20 times for every 32 turns of their partner over a whole revolution — the counting is exact whatever the profile. What the involute buys is that they do it smoothly, without the ratio wandering within each tooth. So there are two exactnesses in a gear train and they come from different places: the ratio’s value from the integers, and its constancy from the curve.

A compound epicyclic makes the first of those spectacular. It is the mechanism where the count is doing all the work, and it is why a robot’s joint can carry a hundred-to-one reduction in a package the size of a mug.

The ledger, and the one row that survives #

Fourteen machines, fourteen quoted numbers, three exact ones — and of the three, two are ranks and this is the only one that is a ratio.

That is the field’s central observation, arrived at from its most reliable machine: the quoted numbers that survive measurement are the ones that are counts. A mobility, a rank, a tooth ratio. Everything that is a length, an angle, a construction or a derivative varies as the mechanism moves, and every specification that quotes one is quoting a sample.

What this means for choosing one #

The design activity that follows is unusual enough to be worth spelling out, because it is the opposite of every other design decision in this field.

Everywhere else, a designer has continuous parameters and an objective, and the answer is found by scanning or optimising: a steering arm angle, a hinge pin position, a rocker’s setup angle. The objective is usually flat near its optimum, so being approximately right is fine.

Here there is nothing to scan. The four tooth counts are integers, they are subject to divisibility conditions for the planets to fit, and the reduction they give is a step function of them with steps of tens of per cent. There is no “approximately right”: there is a table of candidate sets and their exact reductions, and the design is a choice from the table.

What that buys is that the answer, once chosen, is certain. Nothing about the manufactured drive can move the ratio — not wear, not temperature, not load, not tolerance. That is why these drives are used in positioning applications where the count matters more than the smoothness, and it is the strongest form of the field’s closing claim: numbers that come from counting survive.

What is not here #

The cycloidal drive, which is the other way to make a large reduction from a small difference — a lobed disc rolling inside a ring of pins with one more pin than lobes, giving a reduction equal to the number of lobes. It is the same integer trick in a different mechanism, its geometry is drawable and its ratio is equally exact, and the site does not build it. It is the most obvious extension of this essay.

Efficiency, which is where compound epicyclics are genuinely awkward: the same near-cancellation that makes the ratio large makes the internal power circulate, so a large fraction of the torque goes round the loop rather than through it. That is a power argument and this site does not do power.

And backlash, which the gears field measured as an allowance rather than a fault and which in a high-reduction drive is multiplied by the reduction when referred to the output. A hundred-to-one gearbox with a tenth of a degree of backlash at the input has a hundredth of a degree at the output, which is the one respect in which the big reduction helps.

The available reductions thin out as they grow #

The sensitivity being the square of the reduction has a consequence about the design space rather than about any one design, and it is the one that decides how these mechanisms are specified.

A one-tooth change moves this drive’s reduction by 39%. So the reductions reachable from a given set of counts are not a dense set with a fine grid; near this design they are spaced roughly four parts in ten apart, which is a geometric spacing rather than an arithmetic one. Ask for twenty to one and there may be nothing at twenty — the neighbours are at fourteen and twenty-eight, and neither is the number requested.

That gets worse exactly where the mechanism gets attractive. The spacing goes as the square of the reduction, so a family that offers a choice every few per cent at five to one offers a choice every forty per cent at twenty, and near a hundred the available values are a hundred apart. The design space thins out in proportion to the thing that makes the mechanism worth using, which is an unusually direct trade and it is why compound epicyclics are quoted at values like 2176/106 rather than at round numbers.

So the honest specification of one of these is never a reduction of twenty. It is the list of reductions the integers offer in the region wanted, with the closest one picked and the discrepancy accepted — and if the discrepancy cannot be accepted, the mechanism is the wrong one. That is a different conversation from the usual one about gearboxes and it arrives before any dimension is chosen.

It is worth setting beside the reverted train’s difficulty, because the two look alike and are not. There the trouble is density: exact solutions exist, are scattered through the integers, and the difficulty is whether one lands inside a box a workshop can cut. Here the trouble is conditioning: solutions are easy to find and the reduction moves so fast between them that only a few land anywhere near a target.

Both produce the same instruction to the designer and for opposite reasons. Find out what the latitude on the reduction is before choosing the mechanism, because a requirement stated as an exact number is unmeetable by a compound epicyclic and merely expensive from a reverted train — and the two are told apart by whether the neighbouring solutions are near or far, which is the first thing either search reports.

When the count has to hit a number exactly #

A ratio built from tooth counts is a rational number known exactly, and that turns a design problem into an arithmetic one: whether a required ratio can be built is whether it factorises inside the counts a machine will cut.

The contrast is a ratio that does not factorise at all: see a clock is a factorisation.