The module is a size, the ratio is a shape

A four-bar has four lengths and three recoverable combinations of them. A spur gear pair has one length, and the arithmetic that follows is unusually clean.

It is worth setting up carefully rather than assuming, because a gear pair has one length is not how anybody describes one. A drawing carries a pitch diameter, an outside diameter, a root diameter, a tooth thickness, a centre distance and a face width, which is six lengths on the page. Every one of them is the module multiplied by something that has no dimension: the pitch diameter is the module times the tooth count, the addendum is the module, the tooth thickness at the pitch circle is half the circular pitch. Six numbers, one length, five conventions.

One length, two integers and an angle #

A spur pair is specified by four numbers and only one of them has a dimension.

The module is a length: the pitch diameter divided by the tooth count, in millimetres per tooth. Everything with a size in it comes from here.

The two tooth counts are integers. The pressure angle is an angle. Neither carries a length, and there is nothing else.

So scaling a gear pair means one thing: multiplying the module. There is no other length to scale, and no possibility of scaling some lengths and not others.

That makes the sorting exceptionally sharp. Every quantity a gear pair produces is a function of the module, the two counts and the pressure angle, and its dependence on the module is either proportional or absent.

What scales #

module 2 → 4       from        to
centre distance     60        120
base pitch       5.904     11.809
contact length   9.655     19.309

Fitted exponents: 1.000000, 1.000000, 1.000000. Every one of them a length, doubling when the module doubles.

That is what a designer expects and it is worth having measured rather than assumed, because the fit is also a test of the code: a quantity computed with a stray absolute constant in it would come back at an exponent that is not an integer, and none did.

The measurement, and the control #

Both columns are produced by the same probe the scaling survey uses, and the procedure is worth stating once for a pair rather than for a linkage.

Take the pair at modules from 1.6 to 3.2, compute the quantity at each, and fit the log of the value against the log of the module. Then perturb one thing that is not the module — the pressure angle, or one tooth count — and check the quantity responds.

The second step is what stops the table being trivial. Without it, every quantity that came back with exponent zero would be indistinguishable from a constant, and two of them genuinely are constants in the module: the ratio does not move under the module perturbation at all, by exactly zero rather than by a small number.

That reading is the finding rather than an artefact. A quantity whose response to the only available length is exactly zero is not a shape, and the probe reports it as such rather than filing it with the transmission angle.

What does not #

module 2 → 4       from        to
ratio                2          2
contact ratio  1.63519    1.63519

The ratio is 40/20 and contains no length whatever. The contact ratio — how many tooth pairs are in mesh on average — is 1.63519 at both, to every digit either side can carry.

Their fitted exponents are zero and their response to the perturbation is also zero. That matters: the survey’s rule is that a quantity is a shape when scaling does not move it and something else does, and here the only thing to change is the module.

So the ratio and the contact ratio are not shapes. They are counts, in the third class: determined by the two integers and the pressure angle, and independent of the pair’s only length.

The pressure angle is the shape that is left #

Saying a gear pair has no shapes needs one qualification, and it is where the pair’s design freedom actually lives.

The pressure angle is a parameter of the pair and it is dimensionless, so it is unchanged by scaling. Every dimensionless quantity a pair produces is a function of it and of the two counts — the contact ratio, the tooth’s proportions, the fraction of the flank that is usable, how nearly the pinion undercuts.

So the pair does have a continuous dimensionless parameter and quantities that respond to it. Change the pressure angle from 20° to 25° and the contact ratio falls, the base circle grows, and the smallest tooth count that avoids undercutting drops.

What it does not have is a dimensionless parameter built out of lengths, which is what a shape means in the survey’s sense. A four-bar’s three invariants are ratios of lengths; a gear pair’s dimensionless parameter is an angle that was specified as an angle.

The distinction is between a ratio that emerges and one that was chosen, and it is the reason the sorting is two columns rather than three: everything dimensionless here was an input.

Why the sorting is cleaner than a linkage’s #

A four-bar has four lengths, so scaling is one direction in a four-dimensional space and there are three other directions to move in. A quantity can be invariant under the scaling and sensitive to the other three, which is what a shape is, and the four-bar has six of them.

A gear pair has one length, so scaling is the whole of the length space. A quantity is either proportional to the module or independent of it, and there is no third option and no room for a shape.

That is why the table has two columns and no middle. A one-parameter length space admits no shapes, and every dimensionless quantity a gear pair produces is a function of two integers and an angle.

The consequence for measurement is striking. Measure a gear pair’s motion — the ratio of the two shaft angles — and what comes back is the ratio, exactly, and nothing else at all. The motion contains no other information, because everything else that varies is proportional to a module the motion cannot see.

An involute flank and a centre distance are both properties a pair of gears has while it is working. The two cases that follow are where it stops working, and both of them turn on the size rather than on the ratio — which is the whole of this essay’s claim, arriving as a failure rather than as an argument.

Backlash is a size and a nuisance #

The most practically important length in a gear pair, and it behaves as the sorting predicts.

Run the twenty-tooth and forty-tooth pair at a centre distance of 61 rather than the nominal 60 and the backlash is 0.784 in the same units as the module. Double the module and run at 122 rather than 120 and it doubles.

So backlash is a length, exponent one, and a pair scaled up while its centre-distance error is held constant has proportionally less backlash — the same statement a tolerance band makes and for the same reason.

That is the practical warning of this essay. A gearbox designed at module 1 and rebuilt at module 3 with the same absolute machining tolerances is a proportionally better gearbox, and one rebuilt with the same proportional tolerances is exactly as good. Which of the two happens is decided by how the drawing was written and not by the design.

A profile shift is a fraction of the module #

The clearest case of the two columns interacting, and it is the one a gear designer uses constantly.

A profile shift moves the cutter out or in, and it is specified as a coefficient — a dimensionless number, typically between −0.5 and +0.5 — that is multiplied by the module to give the actual displacement.

That convention is not an accident. Specifying the shift as a coefficient makes the pair’s behaviour scale-free: a pair at module 2 with a shift of +0.3 and a pair at module 4 with the same +0.3 are similar figures, with everything doubled, and their contact ratios and undercut margins are identical.

Specifying it in millimetres would break that: two pairs with the same millimetre shift and different modules are not similar, and their dimensionless properties differ.

So gear practice has already made the choice this essay is about, and made it the right way round. Everything dimensionless is specified dimensionlessly and everything with a size carries the module explicitly. That is why gear standards transfer between sizes at all, and it is a convention worth noticing precisely because it is invisible when it works.

The contrast with linkage practice is sharp. A four-bar’s lengths are specified absolutely and its tolerances absolutely, so nothing about a four-bar’s specification transfers to a scaled copy — and the site’s own tolerance figures are, unstated, figures for a machine of one particular size.

What a measurement of a gear pair can find #

Applying the field’s own question to the mesh.

Measuring the two shaft angles recovers the ratio and nothing else. It is the purest possible instance of the angle-only limit: not merely no size, but no other information at all, because a gear pair’s whole behaviour in angle is one integer ratio.

Measuring the centre distance recovers a length, which pins the module given the counts — and the counts are obtained by looking at the gear.

And measuring the transmission error — the departure of the actual output angle from the exact ratio — recovers something neither of those does. It is an angle, so it carries no size, and it is sensitive to the flank geometry in a way the ratio is not. Measuring it is how a real gear is inspected, and it is the one measurement on a gear pair whose content is not exhausted by two integers.

A train has no lengths at all #

Push the argument up one level, to a gear train, and the sorting becomes even starker.

A train’s overall ratio is a product of tooth-count ratios. It is a rational number built from integers, it contains no module and no length, and it is unchanged not only by scaling but by changing the module of any stage — a train with a fine first stage and a coarse second one has the same ratio as one built entirely at either.

So a train’s whole functional specification is a set of integers, and everything with a size in it — centre distances, shaft spacings, the box the train fits in — is separate and decided afterwards.

That is why a clock’s train is a factorisation problem and not a geometry problem: the site’s own essay on it works entirely in integers, and it can, because nothing about the required ratio depends on how big the clock is.

A gear train is the one object on this site whose specification is purely combinatorial. Its lengths exist and none of them is in the answer, which puts it at the far end of a spectrum whose other end is the bodies field, where every quantity is a length and nothing is a count.

The involute’s own scaling #

One more quantity worth putting through the probe, because it is where the pair’s identity lives.

The involute flank of a base circle of radius rb is a curve whose every point is at a distance proportional to rb. Scale the module and rb scales, and the flank is the same curve at a different size — a similarity, exactly.

So the shape of an involute is decided by nothing at all: every involute is similar to every other involute. What distinguishes two flanks is the base radius, which is a length, and the pressure angle, which decides where the base circle sits relative to the pitch circle.

That is why conjugate action holds at any centre distance: the property is scale-free, and moving the centres changes the operating pressure angle without changing which curves are in contact.

The involute is the one curve in the subject whose whole family is one shape, and the module is the only thing that says which member of it.

Where the two columns meet #

There is one quantity that sits awkwardly and it is worth putting where it belongs, because it is the pair’s most consequential number.

The operating pressure angle — the angle of the line of action when the pair runs at a centre distance other than the nominal — is dimensionless, so it belongs in the right-hand column. But it is a function of the centre distance, which is a length, and of the base radii, which are lengths.

The resolution is that it is a function of their ratio. Run the pair at 61 rather than 60 and the operating pressure angle changes; run a pair of twice the module at 122 rather than 120 and it changes by exactly the same amount, because the ratio of the centre distance to the base radii is the same.

So the operating pressure angle is a genuine shape in the survey’s sense: built out of lengths, invariant under scaling them together, and responsive to changing their proportions. It is the one such quantity a gear pair has, and it exists only because the centre distance is a second length that need not be at its nominal value.

A pair running at its nominal centre has one length and no shapes; a pair running off-centre has two lengths and one shape between them. That is the sharpest form of the essay’s arithmetic and it says exactly where a gear pair’s design freedom comes from.

What a gear inspector actually measures #

The theory above says a gear pair’s motion carries one number. Real gear inspection measures a great deal more, and the difference is instructive rather than contradictory.

An inspector measures the departures: how far each flank is from the perfect involute, how far each tooth’s spacing is from uniform, how far the helix is from straight. Every one of those is a length, and none of them is a property of the nominal pair at all — they are properties of the manufactured object relative to a nominal that the two counts and the module already specify.

So inspection is not identification of the design. It is measurement of the error from a design that is known, and the quantities it produces are all in the size column.

That division is worth naming because it is the same one the whole metrology field draws. A dimension on a drawing is a demand and a dimension of a machine is a measurement; a gear’s nominal geometry is entirely a demand, expressed in two integers and a module, and everything measurable about a real gear is a departure from it.

A gear is the case where the demand is fully specified by integers, so the whole of the measurement is error. A four-bar is the opposite: the demand is four real numbers and the measurement recovers three of them.

What gear practice already settled #

The interesting thing about this field is not what the survey found in it. It is that the field had already arrived at the right answer, decades ago, and wrote it into its standards.

Every absolute quantity a gear pair has is proportional to the module: tooth thickness, addendum, dedendum, backlash, contact length, the lot. So a gearbox’s accuracy relative to its own size is decided entirely by whether its tolerances were written in millimetres or in modules — a drawing convention deciding a design property, which is the same finding the tolerance field produced from the other end of the site.

Linkage practice writes tolerances in millimetres. Gear practice does not. Profile shifts are quoted as coefficients, tooth proportions as fractions of the module, quality grades as a formula in the module and the pitch diameter. The standards are written so that a design transfers between sizes, which is exactly what expressing dimensionless quantities dimensionlessly buys, and it was not arrived at by anybody asking what scales.

It was arrived at because gears are made in families and interchangeability is the whole business. A gear cutter cuts one module; a designer picks a module and a tooth count and gets a diameter; two gears mesh if their modules match. The convention is a manufacturing fact that happens to be dimensionally correct, and the dimensional correctness is a consequence rather than a motive.

That is worth carrying to the rest of the site, because it is an existence proof. The recommendation the survey keeps producing — quote the dimensionless form and the absolute one, and say which is which — sounds like tidiness when it is made about coupler curves and tolerance bands. Here is a field where an entire industry adopted it, for its own reasons, and the result is that a gear result written down in 1950 still applies to a gear cut tomorrow at a different size.

The linkage fields have no equivalent and it shows. A coupler curve figure quotes coordinates, a tolerance study quotes ±0.01, a clearance quotes 0.432 — every one of them a statement about the one machine it was computed on, none of them wrong, and none of them transferable without work the reader has to do. Gear practice would have divided by the module before printing.

The one place the analogy breaks is worth naming: a gear ratio is two integers divided, and no linkage quantity is. The ratio is exact, brittle, and incapable of being slightly wrong; what can be slightly wrong is the transmission error, which is the ratio’s own margin and is where a real gear’s quality actually lives. So even here the rule holds — the count wants its margin printed beside it — and gear practice prints that too, as a quality grade.