Moving the cutter out

A gear with fewer than about seventeen teeth is undercut: the rack cutter that generates it reaches below the point where the involute begins, and eats into the flank near the root. What it removes is exactly the part that would have done the work.

The fix is simple to state. Hold the cutter further out.

The shift, and the threshold #

The cutter’s offset is measured in fractions of the module and written x. Positive x means the cutter is further from the gear’s centre, so the tooth is cut from a part of the rack that is thicker, and the tooth comes out thicker.

The condition for no undercutting is

$$x \geq 1 - \frac{z \sin^2\!\alpha}{2}$$

where z is the tooth count and α the pressure angle. At x = 0 that reduces to z ≥ 2/sin²α, which is 17.097 at a 20° pressure angle — the familiar rule and the exact value it was rounded from. The two statements are one statement, and having them as separate rules in the library is how they came to disagree.

They did disagree, and the disagreement is worth recording because of where it was. The undercut test read “fewer teeth than the minimum and zero shift”, which says that a gear cut with a shift of any size at all is safe. A ten-tooth gear at x = 0.001 came back clean when the shift it actually needs is 0.415. It passed every existing check because those checks only ever asked about unshifted gears — the clause that was wrong was the one no test exercised.

For the figure’s eleven-tooth gear the threshold is 0.357. At 0.347 the gear still undercuts; at 0.357 it does not. Both sides are measured, because a rule with only one side is not a threshold.

What the cutter is doing #

The word “shift” describes the fix and hides the mechanism, so it is worth a paragraph on what physically happens.

A gear is generated by a rack — a straight-sided cutter — rolling against a blank. The rack’s pitch line rolls on the gear’s pitch circle without slipping, and the tooth space is the envelope of all the positions the rack passes through. That is why the involute appears: the flank is the envelope of a family of straight lines rolling on a circle, and that envelope is an involute of the base circle.

Undercutting happens when the rack’s tip dips below the point where the involute begins. Below the base circle there is no involute — the string has fully unwound — so any material the rack removes there is coming out of the region that carries the flank, and the tooth ends up waisted at the root.

Shifting moves the rack’s pitch line off the gear’s pitch circle: the rack is held x modules further out and rolls on a larger circle. The rack’s tip is therefore further from the centre and no longer reaches the interference point. Nothing about the rack changes, nothing about the involute changes, and the tooth is cut from the same family of straight lines with the family displaced.

That is why the technique costs nothing in ratio and why it is available on any hobbing machine: it is a setting, not a tool.

What it costs #

Profile shift is not free, and the cost is at the other end of the tooth.

Moving the cutter out makes the tooth thicker at the root and thinner at the tip, because the involute flanks converge as the radius grows. Shift far enough and the tooth comes to a point. For the eleven-tooth gear, the tip thickness falls from 0.606 of a module at zero shift to 0.240 at a shift of 0.507 — and a pointed tooth has no strength at its tip, chips in service, and cannot be finished.

So there is a floor set by undercutting and a ceiling set by tip thickness, and for a very small tooth count the two close on each other. That, rather than any rule about seventeen, is the real limit on how few teeth a gear can have.

What “the pitch circle” means after a shift #

There is a vocabulary problem here that causes more confusion than the geometry does, and it is worth clearing up before the pair.

A gear has a pitch circle of radius z m/2. That is a definition, it depends only on the tooth count and the module, and it does not move when the gear is shifted.

A pair of gears has an operating pitch circle each: the two circles that roll on each other without slipping at the centre distance the pair actually runs at. For an unshifted pair those coincide with the individual pitch circles. For a shifted pair they do not.

So after a shift a gear has two circles that might reasonably be called its pitch circle, at different radii, and the literature uses “pitch circle” for both. The rule that resolves it: the cut pitch circle is where the cutter’s pitch line was tangent when the tooth was generated, and it is a property of one gear; the operating pitch circle is where the pair rolls, and it is a property of two.

The same doubling happens to the pressure angle. The cut pressure angle is the rack’s, usually 20°, and it is a property of the tool. The operating pressure angle is the slant of the line of action in the assembled pair, and it is a property of the assembly. When a catalogue says “20° pressure angle” it means the first; when a load calculation says “pressure angle” it means the second, and for a shifted pair those differ by several degrees.

Nothing in the arithmetic is difficult. The confusion is entirely a matter of two things sharing a name, and it is worth the paragraph because a designer who substitutes one for the other in a force calculation gets an answer that is wrong by the cosine of the difference.

What it does to a pair #

Here is the part that is not obvious and is the reason the essay exists.

Shifting a gear does not move its base circle. The base circle is r cos α, and neither r nor α depends on the shift — the shift moves the cutter, not the pitch line. And the base circle is what the involute is generated from, so the tooth’s flank is a piece of the same involute as before, just a different piece of it: further out along the curve.

Two consequences follow immediately, and they point in opposite directions.

The ratio does not change. The velocity ratio of an involute pair is the ratio of the base radii, which is the ratio of the tooth counts, and shifting moves neither. This is the property the whole technique depends on: a designer can shift a gear to fix its strength and the transmission ratio is exactly what it was.

The centre distance does change. The teeth are thicker, so at the standard centre distance they would not fit — the pair would jam. They must run further apart, and running further apart means the line of action is steeper, so the operating pressure angle is larger than the one the gears were cut with.

Why the mesh is drawn from the flanks #

The measured route above deserves one more sentence about where its inputs come from, because it is the difference between a check and a restatement.

The tooth thickness at a radius is computed from the involute geometry of the flank that toothFlank actually generates — the same points that get drawn. So if the flank generator were wrong, the thickness would be wrong, the bisection would land somewhere else, and the disagreement with the involute equation would show it.

That is the property worth insisting on. A measurement taken from a formula that shares a derivation with the thing being checked measures nothing, and this site has a recorded instance: the conjugate-action test that parameterised both flanks by the same roll angle and announced the ratio was constant, having restated the definition of an involute rather than tested anything.

The two routes #

The relation between shift and operating pressure angle is classical:

$$\operatorname{inv}\alpha_w = \operatorname{inv}\alpha + \frac{2(x_1 + x_2)\tan\alpha}{z_1 + z_2}$$

where inv α = tan α − α is the involute function, and the operating centre distance follows from the base circles, which cannot change: a_w = a cos α / cos α_w.

That is one route, and it is an equation.

The other measures. Zero backlash means the two teeth together exactly fill the circular pitch on the circles they actually run on. So: pick a trial centre distance, compute the two operating pitch radii, compute each gear’s tooth thickness at its own radius from the drawn flank geometry, and see whether the two thicknesses plus nothing add up to the pitch. Bisect until they do.

Nothing in the second route knows the involute equation. It uses only tooth thicknesses, and it finds the centre distance at which the teeth fit.

They agree to 4 × 10⁻¹⁵. Two things had to be fixed before they did.

The two thicknesses are directly comparable and must not be scaled. The two operating pitch circles roll on each other without slipping, so r_w1/z₁ = r_w2/z₂ and one circular pitch serves both. The first version scaled one thickness by the ratio of the radii, as though the two lived on circles of different pitch, and put the compensated pair’s centre distance out by a part in fifty.

The bracket has to start at the base circles. Below r_b1 + r_b2 the operating pitch circles fall inside the base circles, where there is no involute and no tooth thickness to speak of, so the thickness function returns nothing and the bisection has nothing to bracket. A lower bound of 0.8 times the standard centre distance is inside that region for these tooth counts, and every measured centre distance came back undefined.

The compensated pair #

The last row of the figure is the one that gets used in practice, and it is a pleasant piece of arithmetic.

The operating pressure angle depends on x₁ + x₂ — the sum of the shifts, not either one separately. So shift the pinion out by 0.4 and the wheel in by 0.4, and the sum is zero: the operating pressure angle is 20°, the centre distance is exactly the standard one, and nothing about the installation changes.

Meanwhile the pinion, which was the part in trouble, has been shifted out by 0.4 and is no longer undercut. The wheel, which had teeth to spare, has been shifted in by the same amount and has lost a little strength it did not need.

That is how a weak pinion is fixed without moving a single bearing, and it is the reason profile shift is a routine tool rather than an exotic one. The figure checks it: the compensated row’s centre distance returns to the standard value to the last bit, and the ratio has moved by exactly zero across all four rows.

What else the shift moves #

Two more effects, both of which a designer has to look at and neither of which the ratio hints at.

The contact ratio falls. The contact ratio is the length of the path of contact divided by the base pitch, and it is how much of the cycle has more than one pair of teeth carrying the load. A larger operating pressure angle shortens the path of contact, so positive shift on both gears reduces the contact ratio — and if it falls below 1 there are moments with no teeth in contact at all, which is a mechanism that does not transmit anything. This is the constraint that usually decides how much combined shift is acceptable, and it is one more instance of a quantity that varies through the cycle being summarised into a single number — a contact ratio of 1.4 does not mean 1.4 teeth are always in contact, it means one pair for some of the cycle and two for the rest.

The tip may need relieving. A larger operating pressure angle moves the start of contact closer to the tip of the driven gear’s tooth, and if the tip circle reaches past the point where the mating flank’s involute begins, the two profiles interfere — the tip of one gouges the root of the other. This is a different failure from undercutting and it has a different fix, usually shortening the addendum. It is worth naming because it is the constraint that bites when both gears are shifted positively by a lot, and because it is the second way the mesh geometry can fail rather than merely perform badly.

The sliding changes. Along the path of contact the two flanks slide against each other, and the amount of sliding is not symmetric between the two gears. Shift redistributes it, and one of the classical reasons for shifting a pair even when neither gear is undercut is to balance the specific sliding between pinion and wheel so that they wear at the same rate.

Neither is a kinematic failure and neither is something this site computes past the geometry. Both are here because a change that leaves the ratio exactly alone is easy to mistake for a change that leaves everything alone, and it does not.

Shifting one gear, in an existing gearbox #

The compensated pair is the textbook answer. The case that comes up more often in practice is messier and is worth going through, because it shows the arithmetic being used rather than admired.

A gearbox exists. A pinion in it is failing at the root. The centre distance is fixed by the castings and cannot be changed, and the ratio is fixed by the application.

Shifting the pinion out strengthens it and demands a larger centre distance, which is not available. So the wheel must be shifted in by the same amount, and the sum of the shifts stays at zero. That is the compensated pair, and the constraint that forced it was the casting rather than any elegance.

What the designer has to check is the other end. The wheel’s tip is now thicker, not thinner — negative shift thickens the tip — so the tip-thickness limit is not the problem there. The wheel’s root is thinner, so the wheel has lost bending strength, and the question is whether it had enough to give. For a large tooth count it usually does, which is why the technique works: the pinion is the weak part in almost every pair, and negative shift on the wheel is cheap.

The number to watch is the contact ratio, and here it goes the other way and helps: with the shifts summing to zero the operating pressure angle is unchanged, so the contact ratio changes only through the altered tip radii, and the change is small. A compensated pair is very nearly a standard pair as far as the mesh is concerned, which is exactly what makes it a drop-in fix.

The shape of the fix #

There is a pattern worth naming, because it recurs.

Undercutting is a manufacturing problem: it is caused by the generating process, not by the gear. A gear form-milled with a cutter shaped like the tooth space is not undercut whatever its tooth count, because there is no rack sweeping past to interfere with. The rule about seventeen teeth is a fact about hobbing.

The fix is a manufacturing fix — move the tool — and it works entirely within the involute system, which is why it costs nothing in ratio. That is a consequence of the involute’s central property: an involute pair transmits a constant ratio at any centre distance, so the pair can be pulled apart to accommodate thicker teeth without the transmission being affected. A cycloidal tooth form, which was the main alternative in the nineteenth century, does not have that property — its ratio is sensitive to centre distance — and profile shift as a technique would be impossible with it.

So the whole of this essay rests on the one property the involute was chosen for, applied to a situation it was not chosen for. That is a reasonable summary of why the involute won.