The cutter takes back the tooth

A gear with too few teeth comes out with a waist. The flank narrows near the root, the tooth looks as though it has been eaten, and the strength and the length of the working profile both suffer. The rule everybody carries is that it starts below seventeen teeth at a twenty-degree pressure angle, and the rule is right.

What the rule does not say is what happens, and the mechanism is more interesting than the threshold. Nothing is missing from the tooth. The tooth was made and then part of it was removed, by the same tool, later in the same cut.

Removal, not shape #

The generation gives a flank as a set of contacts: at each position of the rack, the meshing equation returns the point where the straight edge is touching, and that point is on the involute. Nothing about that calculation asks whether the point is still there later.

To find out, sweep. Take each generated point, put it back into the rack’s frame at every other position of the cut, and ask whether it is inside the cutter’s tooth. If it is, the metal at that place has been removed, and the point is a piece of involute that briefly existed and does not any more.

On a twelve-tooth wheel of module 4, 47 of 240 generated points are removed, and the deepest of them is $0.053$ mm inside the cutter. On a fourteen-tooth wheel, 26 of 224, deepest $0.0079$ mm. On sixteen teeth, 8 of 206, deepest $0.00019$. On seventeen, eighteen and twenty-four teeth: none at all.

The removed points are the ones nearest the base circle — the bottom of the working flank — and they are removed by the rack’s tip corner, which is the same corner whose path is the fillet. So the fillet and the undercut are not two features. They are one curve, and whether it is called a fillet or an undercut depends on whether it stays below the base circle or reaches above it.

What the sweep had to be careful about #

The removal test is a sweep, and two details of it decide whether it measures the geometry or its own settings.

The first is that a point is tested against the cutter at every position except the one that generated it. At its own generating instant a point is exactly on the cutter’s boundary — that is what being a contact means — so a test that included it would report every point as touching. The exclusion is a small window in the input angle, and the window has to be wide enough to skip the generating instant and narrow enough not to skip the removal, which happens shortly afterwards.

The second is that “inside the cutter” has to be a depth rather than a yes or no. Undercutting arrives continuously: as the tooth count rises the corner reaches the flank by less and less, and a boolean test flips at whatever tooth count the sampling happens to resolve. The sweep therefore keeps the deepest penetration in millimetres, which goes smoothly to zero as the threshold is approached — and even so it is not the quantity to bisect on, for the reason the next section gives.

The cutter itself is a polygon: the same straight flank that generates, its tip line, the mirrored flank and the root. It is the same object as the generating profile rather than a second description of it, which matters — a sweep against a slightly different rack would produce a plausible removal that belonged to no cut.

The sharp version of the question #

Counting removed sample points is a good picture and a poor measurement: the count depends on how the flank was sampled, and near the threshold the removal is so shallow that a coarser sampling misses it entirely. Bisecting on a count would measure the sampling.

The sharp version compares two points, both of which are measured.

The flank can only be generated while the contact is on the rack’s straight edge. The contact travels along the line of action — a locus fitted to the contacts themselves, straight to $10^{-14}$ mm — and the edge ends at the corner, which lies on the cutter’s tip line. So generation stops where the tip line crosses the line of action.

The other point is the foot of the perpendicular from the wheel’s centre onto the line of action, which is where the line touches the base circle and where the involute begins.

If the crossing is past the foot, the corner has gone beyond the point where the involute starts, and it is cutting flank that has already been made. If it is short of the foot, everything the straight edge generated survives.

That is a difference of two lengths along one line, both read off measured objects, and it passes through zero linearly — which makes it bisectable to any precision the arithmetic can carry.

Seventeen point zero nine seven #

Bisected on a non-integer tooth count until the crossing lands exactly on the foot:

The worst disagreement across the three is $2.6\times10^{-10}$ teeth.

Nothing in the measurement knows that formula, or any formula. It generates a flank by solving a dot product, fits a line to the contacts, projects a point onto that line, intersects the line with another line, and compares two lengths. The classical expression arrives as the answer to a question asked entirely in terms of measured geometry — which is the strongest form of agreement this site can produce, and the reason the seventeen-tooth rule is stated in this field as a consequence rather than as a rule.

Two things about the table are worth reading beyond the agreement. The threshold is not seventeen: it is 17.097, so a seventeen-tooth wheel is undercut, very slightly — measured as removal, at seventeen teeth the deepest bite is under a ten-thousandth of a millimetre, which is why the rule is quoted as seventeen and why nobody minds. And the pressure angle is the whole of the rest of it: at 14.5° the threshold is nearly thirty-two teeth, which is why the older standard produced so many wheels with waisted teeth, and at 25° it is eleven.

The other three cures #

Shift is the usual answer and it is not the only one. Each of the alternatives moves the same two points relative to each other, and it is worth seeing them as one family.

More teeth. The obvious cure, and the one the threshold is stated in. A larger wheel puts the foot of the perpendicular further out along the line of action than the tip line’s crossing.

A bigger pressure angle. At 25° the threshold falls to 11.2 teeth, because a steeper line of action reaches the base circle sooner. It costs a larger separating force and a noisier mesh, both of which need a load to talk about.

A shorter addendum on the cutter. The crossing is where the tip line meets the line of action, so a cutter whose teeth stand less proud crosses sooner. That is a “stub tooth” system, it cures undercut on small wheels, and it costs contact ratio directly — the teeth are shorter, so the arc over which a pair stays in mesh is shorter.

All three, and the shift, are the same manoeuvre: get the crossing point back inside the foot. Which one a designer reaches for is decided by what else is fixed — the centre distance, the ratio, the cutter in stock — and not by anything about tooth form.

Moving the rack out #

The cure is to hold the cutter further away from the blank’s centre while rolling it the same way. That is profile shift, and in this account it is not a modification of the tooth form at all — it is the same tool, the same rolling, and a different offset.

Moving the rack out by $x\,m$ moves its tip line out by the same amount, so the crossing point slides back along the line of action, and the undercut goes away when the crossing reaches the foot. The shift that does it can be bisected exactly as the threshold was:

to $3.6\times10^{-13}$ of a module. The seventeen-tooth entry is the interesting one: the shift needed is six thousandths of a module, which on a module of 4 is twenty-three microns. That is the size of the seventeen-tooth rule, expressed as a length.

Shift is not free, and the teeth field has measured what it costs: the tooth gets thicker at the root and thinner at the tip, the operating pressure angle changes, and a pair of shifted wheels no longer runs at the centre distance the tooth counts suggest. Those are the trades. What the shift buys is the whole of the involute back.

Reading the removal on the drawing #

There is a practical translation of all this, and it is what a gear drawing shows.

An undercut tooth’s flank is not a smooth curve with a waist. It is an involute from the tip down to a definite radius, then a trochoid that turns inward and comes back out to meet the root circle. The join between them is a point of tangency in the good case and a corner in the bad one, and its radius is the top of what the removal took: on a twelve-tooth wheel measured here, the highest removed point is at 22.815 mm against a base circle of 22.553 — so the material is missing from a quarter of a millimetre above the base circle downwards.

That number is the one to look at, rather than the count of removed samples. It says how much of the usable flank has gone in the units the mating wheel cares about, and it is the quantity that decides whether the pair still has enough profile to stay in contact.

Why it matters that the involute is missing #

An undercut tooth is weaker, and that is a force argument and not this site’s. Two of the consequences are geometric and are worth stating.

The working flank is shorter. The involute below the removal is gone, so the arc over which this tooth can be in contact with its mate is shorter than the tooth’s height suggests. That reduces the contact ratio, and a contact ratio below one is a drive that lets go between teeth.

The mating wheel’s tip may want the missing part. The contact between a large wheel and a small pinion runs down towards the pinion’s root, and the part of the pinion’s flank the large wheel’s tip needs is exactly the part the corner removed. That is why undercutting bites hardest on the pinion of a high-ratio pair — the small wheel is both the one at risk of undercut and the one whose root is used.

Two things this is not #

The word interference is used for several different failures in gearing, and the one measured here is only one of them. It is worth separating them, because the cures are different.

Cutter interference is what this essay is about: the tool removing flank it has generated. It happens during manufacture, it depends on the cutter’s proportions, and it leaves a wheel whose teeth are permanently short of profile. A wheel can be undercut sitting on a bench with nothing meshing with it.

Mating interference is when the tip of one wheel fouls the flank of the other below its base circle, where there is no involute for it to be conjugate to. It happens in service, it depends on both wheels and on the centre distance, and it is cured by tip relief, by fewer teeth on the large wheel or by a longer centre distance. A wheel can be perfectly cut and still interfere with a particular mate.

They are related — both are about the involute running out at the base circle — and they are not the same event. Confusing them produces the common suggestion that undercutting can be fixed by adjusting the mesh, which it cannot: the metal is already gone.

Why it is the corner and not the flank #

One more piece of the mechanism is worth making explicit, because it explains why undercutting is a small-wheel problem and nothing else.

The flank generates while the contact is on the straight edge; the corner cuts wherever it goes. The corner’s path in the wheel’s frame — the trochoid — is fixed by the rolling and by where the corner sits on the rack, and it dips towards the wheel’s centre by an amount that depends on the rolling radius. On a large wheel the rolling radius is large, the corner’s dip is shallow relative to the base circle, and the trochoid stays below it. On a small wheel the same corner, on the same rack, dips proportionally deeper — the rack has not changed, the wheel has — and the trochoid comes up past the base circle into the flank.

So nothing about the tooth form causes undercutting. It is caused by the ratio between a fixed feature of the cutter and a radius of the part, which is exactly why the threshold is a tooth count rather than a length: a tooth count is that ratio.

The general shape of the trap #

The thing worth carrying out of this essay is not the number.

The generation produces a correct curve. Every point on it satisfies the meshing equation, every point is on the involute, and the measurement that checks it against the involute passes at $10^{-14}$. And a third of that curve is not on the finished part, because the process that made it kept going.

A construction can be locally right everywhere and wrong as a whole. Contact is a statement about an instant; a solid is what survives a sweep. This site has met the same distinction from the other side, where a linkage assembles at every position it was checked at and cannot get between them, and where a synthesised mechanism reaches all its prescribed positions and passes through none of them in the right order.

The remedy is always the same and it is not cleverness: check the whole interval, not the instants.

There is a second, quieter lesson in how the threshold had to be measured. The first attempt bisected on the removal depth — a quantity that is genuinely zero above the threshold and genuinely positive below it, and therefore looks like exactly the right thing to bisect. It gave 16.94 against a formula value of 17.097, an error of one per cent that no amount of extra sampling removed, because the depth near the threshold falls away faster than linearly and the flank’s own sampling could not resolve it. A criterion that vanishes quadratically is a poor thing to bisect on; the difference of two lengths, which vanishes linearly, gives ten significant figures from the same geometry.

Reading undercutting as an event rather than as a shape has a practical dividend the shape reading cannot give: it says what to inspect. A tooth described as having a waist offers nothing measurable — waists are a matter of degree and every fillet blends into the flank somehow. A tooth described as having had material taken back offers a comparison: near the base circle, the actual flank against the involute it was supposed to be, and the discrepancy is the removal, in millimetres, at a stated radius. That is a measurement a gear inspection machine already makes — it traces the flank and reports its departure from a nominal involute — so the undercut is not a separate test but a reading of a chart everybody already produces. Which turns the seventeen-tooth rule from a design-stage classification into an acceptance criterion with a number attached. A tooth is undercut by so many microns at so many millimetres of radius; that number can be specified, measured and argued about, and it is available from the same trace that reports profile error and lead. None of that follows from the picture of a waist.