The angle the standard left free

The involute is forced. Demand that two shafts hold an exactly constant speed ratio while their teeth slide over one another, and the curve is essentially determined — not chosen, not preferred, determined. That is the strongest statement the gears field makes, and it leaves something out.

An involute has a base circle, and nothing said so far fixes how large it is relative to the pitch circle. The ratio between them is a cosine, and the angle whose cosine it is has a name: the pressure angle. It is the angle between the line of action and the common tangent to the pitch circles, it is the one free parameter in the whole construction, and every gear standard ever written has had to put a number on it.

What the angle actually is #

Two circles carry a gear pair: the pitch circles, which roll on each other without slipping and are fixed by the tooth counts and the module, and the base circles, from which the involutes are unwound. The pressure angle α is what relates them, through $r_b = r\cos\alpha$.

That single relation propagates into everything. The base pitch — the spacing of contacts along the line of action, and the divisor in every contact ratio here — is $\pi m \cos\alpha$. The distance between the two base tangency points is $a\sin\alpha$. The roll angle at which a flank reaches any given radius depends on $r_b$ and so on α. Change the one number and every derived length in the field moves.

What does not move is the ratio. Two gears turn at the ratio of their tooth counts whatever the pressure angle is, and they hold it constant whatever the pressure angle is, because both of those are consequences of the involute rather than of which involute. So the angle is genuinely free in the sense that matters: no value of it breaks the law the field is built on.

The honest definition is the cutter’s flank angle #

There is a second way to say what α is, and it is the one a machine shop would give.

Every gear in a family is cut by one rack, and that rack’s teeth are straight-sided. The angle its flanks make with the perpendicular to its pitch line is the pressure angle. Nothing about involutes appears in that statement at all — a straight edge, rolled against a blank, generates an involute as its envelope, and the angle of the straight edge is what decides which involute.

That is the definition with the fewest moving parts, and it explains why the number is a standard rather than a preference. A workshop owns cutters, and a cutter is a piece of steel with an angle ground into it. Two gears mesh if the same cutter would produce both; a shop with 20° cutters can make any gear of any count that will run with any other gear it has ever made, and a 25° gear in the same box is not a slightly different gear but a member of a different family. The pressure angle is where the geometry touches the stock room, which is a harder constraint than any of the trades below and is why so few values of it exist.

It also explains the shape of the undercutting condition. A straight cutter tip sweeps a straight line as the rack rolls; whether that line reaches inside the base circle is a question about the angle of the flank and the size of the blank, and $2/\sin^2\alpha$ is its answer. The sine is the cutter’s, not the gear’s.

The first thing it buys: small pinions #

Undercutting is what happens when the generating rack’s tip reaches past the base circle and removes flank that the mesh needs. The threshold is $N \ge 2/\sin^2\alpha$, and the sine in the denominator makes it violent.

At 14.5° the threshold is 31.90 teeth. At 20° it is 17.10. At 25° it is 11.20. That is a factor of nearly three across a range of ten and a half degrees, and it is the entire reason anybody moved off the older standard: a machine designer who wants a fourteen-tooth pinion cannot have one at 14.5° without profile shift, and can have one at 20° with nothing.

Thirty-two teeth is a rounder number than it looks #

The old standard’s floor is worth a moment of arithmetic, because it is the sort of thing that looks like a coincidence and is not.

$\sin 14.5° = 0.250380$. That is a quarter to three figures, and $2/(1/4)^2$ is exactly 32. So the older standard’s undercutting floor is thirty-two teeth, exactly, under an approximation good to a part in six hundred — and the exact figure is 31.9029, which rounds to the same integer from the convenient side, as the seventeen-tooth rule does at the newer angle.

The angle whose sine is exactly a quarter is 14.4775°, which is not 14.5°. So the standard is a rounded version of a number chosen to make a sine come out round, which is what a constant looks like when the arithmetic behind it had to be done by hand. The arithmetic works out; what cannot be said is that is why the angle was picked, and the difference between those two statements is one this field keeps having to make.

The 14.5° standard is the older one, and its thirty-two-tooth floor is why gear catalogues of that era are full of pinions that look larger than the drive needs. It survives in some replacement parts and in a few imperial families, and a pair cut at one angle cannot mesh with a pair cut at the other — which is the whole content of interchangeability, since two gears mesh exactly when the same cutter would produce both.

The second thing it spends: contact ratio #

Contact ratio is how much of the line of action lies between the two tip circles, divided by the base pitch. Below one, the drive periodically stops being driven.

Raising α steepens the line of action. That shortens the part of it lying inside both tip circles — from 5.716 mm at 14.5° to 4.118 mm at 25° for a 20–32 pair at module 1 — while the base pitch shrinks more slowly, from 3.042 to 2.847. The quotient falls: 1.879, then 1.612, then 1.446.

The two curves fall together. That is the trade, and its shape is what this essay is about: both of the quantities a designer wants move the same way under the one knob that is free. A small undercutting floor and a large contact ratio cannot be had at once, because the first wants a large α and the second wants a small one.

There is no optimum, and the reason is that neither curve turns #

A quantity being traded against another does not by itself mean there is no best answer. A sum of two costs usually has a minimum somewhere in the middle. What rules that out here is that neither curve has a stationary point anywhere in the useful range: the floor falls monotonically from 46.27 teeth at 12° to 8.00 at 30°, the contact ratio falls monotonically from 2.092 to 1.344, and a monotone pair has no interior optimum to find.

Two worked rules make that concrete, and they disagree.

Take the largest angle that keeps the contact ratio above 1.4 on the thinnest pair the catalogue will sell. On a 20–40 pair the sweep crosses 1.4 at 27.35°, where the floor has fallen to 9.48 teeth. That rule licenses a ten-tooth pinion and a drive with barely more than one and a third tooth pairs carrying at any instant.

Take the smallest angle at which a twelve-tooth pinion is not undercut. That wants $2/\sin^2\alpha \le 12$, which is 24.095°, where the contact ratio is 1.487.

Both rules are reasonable, both are stated in the vocabulary of the field, and they land three and a quarter degrees apart — with the standard sitting four degrees below the lower of them. A third rule that asked for the largest contact ratio a seventeen-tooth pinion allows would land on 20° almost exactly, which is a good demonstration of how a requirement picks the answer: the seventeen was already the 20° floor, so the rule is the choice restated.

So the choice has to come from outside the geometry — from a requirement rather than from a stationary point. Setting a floor on contact ratio and taking the largest angle that clears it is one such rule; demanding that a twelve-tooth pinion be cuttable and taking the smallest angle that allows it is another; and the two rules give different answers, which is what it looks like when a standard is a decision rather than a discovery.

The tooth itself changes shape, in two directions at once #

The two quantities above are about the pair. A third is about one gear, and it splits.

The tooth’s thickness at the pitch circle is fixed by the standard at half the circular pitch — that is what makes any two gears of a family mesh — so the angle cannot touch it. Above and below the pitch line it can. At 14.5° a twenty-tooth tooth carries 1.628 modules of thickness at its base circle and 0.866 at its tip; at 25° it carries 1.967 and 0.510.

The tooth gets thicker where it is held and thinner where it reaches. In one direction that is a gain — a broader root is a stronger root, though how much stronger needs a modulus that is not carried here and every argument here survives with every force unknown. In the other direction it is a loss, and the loss ends at a wall.

A tooth’s tip thickness reaches zero. The flank is an involute unwinding from the base circle, its two sides converge as they climb, and at a large enough pressure angle they meet below the addendum circle. A twenty-tooth gear at module 1 comes to a point at 35.98°; a twelve-tooth gear does it at 34.61° and a forty-tooth gear at 37.04°. Beyond it the tooth as specified does not exist.

That wall is worth having because it is the first thing in this argument that is not a trade. The floor and the contact ratio are quantities a designer weighs; the pointed tooth is a count of zero. It also sits closer to the standard than the shape of the two curves suggests: the useful range is bounded above at about 30° by contact ratio and at about 36° by the tooth ceasing to be a tooth, which is not much of a gap.

Where twenty degrees came from, and what it is not #

The number is old, it is a standard rather than a theorem, and the difference is worth keeping. The proportions that go with it — one module of addendum, 1.25 of dedendum — are in exactly the same category, and the gears library says so at the point where they are typed in, because every number downstream of them inherits the convention.

What can be said from inside the geometry is narrower and firmer. Twenty degrees clears a contact ratio of 1.6 on a medium pair, allows a seventeen-tooth pinion without shift, leaves two thirds of a module of tooth at the tip, and sits about sixteen degrees below the pointing wall. Every one of those is comfortable and none of them is extreme, which is the signature of a compromise rather than of an optimum.

Twenty-five degrees is the angle for a drive that needs small pinions and can afford a 1.45 contact ratio, and it is used where packaging is tight. Fourteen and a half is the angle for a drive that wants quiet running and has room for large pinions. Both are correct answers to different questions, and a catalogue carrying both is not indecision — it is two standards because there are two questions.

What the angle does not do #

Three things are worth separating out, because each is regularly attributed to the pressure angle and none of them is its doing.

It does not change the ratio. Not by a part in 10¹⁵. The ratio is $z_2/z_1$ and it comes from counting, which is the exactness that survives measurement.

It does not change the constancy of the ratio. That is the law of gearing, satisfied by every involute pair at every pressure angle, and the site checks it on the drawn flank rather than on the parameterisation that produced it.

It does not change the centre distance the pair is cut for. That is $m(z_1+z_2)/2$, with no α in it. What α does control is how much the operating pressure angle moves when the pair is assembled at some other distance — and the fact that it can be assembled at another distance at all, with the ratio unchanged, is the involute’s own property and the reason it won.

The third deserves a number rather than a sentence, because “how much the operating angle moves” is precisely the sensitivity a tolerance budget needs.

Push a 20–32 pair’s shafts 0.05 modules further apart than the standard 26. The operating pressure angle rises from 20° to 20.300° — a rise of 0.300° for a fiftieth of a module. The same displacement on a pair cut at 14.5° raises its angle by 0.419°, and on a pair cut at 25° by 0.235°. A low pressure angle is more sensitive to where the shafts actually are, by nearly a factor of two across the standards in use.

That is a fourth item in the trade and it runs the same way as the contact ratio: the angle that buys quiet running and long contact also buys a drive that notices its housing more. It is the geometric half of what a centre-distance error costs, where the same 0.05 is priced in backlash instead of in degrees.

The last of those is the one that most rewards care. Two quantities here are called the pressure angle and they are not the same: the angle the gear was cut at, which is a property of the cutter and fixes the base circle for ever, and the angle the pair runs at, which depends on where the shafts actually are. Profile shift changes the second and never the first, which is why a shifted gear has the same involute cut at a different part of it.

What was measured, and against what #

Every number above comes from the same sweep, and it is worth saying what the sweep does and does not establish.

The floor is a closed form, $2/\sin^2\alpha$, and it is drawn rather than tabulated so that its shape is visible. The contact ratio is not a closed form in α: it is the usable length of the line of action divided by the base pitch, both computed from the two gears’ geometry, and the pair is rebuilt at every angle in the sweep rather than scaled from one result. The tooth thicknesses are read off the involute at two radii, at the base circle and at the tip, through the same function that prices backlash.

The claim the sweep supports is a claim about monotonicity, and it is required rather than eyeballed: the floor must fall, the contact ratio must fall, the thickness at the base circle must rise and the thickness at the tip must fall, across every step of the range. A single non-monotone step anywhere fails the figure, which is the form the statement “there is no optimum” has to take if it is to be checkable at all. A stationary point is exactly what a monotone series does not have.

What the sweep cannot support is the stronger claim that no combination of these quantities has an optimum. A weighted sum of a falling curve and a rising one certainly can, and a designer who prices root thickness against contact ratio is forming exactly such a sum. The argument here is narrower and survives that: the geometry supplies no weights, so any optimum is imported rather than found.

The floor and the ceiling are the same curve seen twice #

One tidy thing falls out of putting the two thresholds side by side, and it is the kind of coincidence worth checking rather than admiring.

The undercutting floor $2/\sin^2\alpha$ is about the rack — about a straight-sided cutter reaching past the base circle. The pointing angle is about the tooth — about two involutes converging above the pitch circle. They have no construction in common, and yet both are statements about how much involute there is between the base circle and the tip circle: undercutting is that interval being invaded from below, pointing is it running out from above.

That reading makes the trade a single statement rather than two. Raising the pressure angle lowers the base circle and shortens the usable flank at both ends, and everything in this essay is a consequence: fewer teeth can be cut cleanly because the base circle has dropped below the trouble, less contact is carried because the usable flank is shorter, the root is broader because the flank leans out, and the tip runs out because the flank has less height to work with.

Still open: what the angle costs in the other two directions #

Two consequences of the pressure angle are named here and not measured, and both belong further up this sequence rather than in this essay.

The first is sliding. The contact path’s position relative to the pitch point decides how much of the motion at a mesh is slip rather than roll, and a steeper line of action moves both ends of the path. That is a quantity per contact rather than a quantity per pair, it is antisymmetric between the two flanks, and it is measured in the essay that follows.

The second is what happens when one of the two curvatures is reversed. Everything above assumes an external pair, with the two flanks convex and the two centres on opposite sides of the line of action. An annulus puts them on the same side, and the undercutting threshold reappears there in a construction with no cutter in it at all — the same 17.10 teeth, arrived at from the other direction, which is the strongest evidence available that the number is about the involute rather than about the rack.