Undercutting, and the seventeen-tooth rule
A gear is cut by a tool shaped like a rack, rolled against the blank. That works because the involute’s conjugate is a straight line, so a straight-sided cutter generates the correct curved flank.
On a small gear it also removes part of the flank it has just formed. That is undercutting, and it happens for a reason that is pure geometry.
The threshold
Two circles matter. The base circle at is where the involute begins. The root circle sits a fixed 1.25 modules below the pitch circle, because that is the standard dedendum.
The pitch radius grows with tooth count and the dedendum does not. So on a large gear the root is well inside the base circle and there is plenty of involute; on a small one the root rises above the base circle, and the part of the flank between them is not an involute at all.
Setting the two equal and solving gives
At 20° that is 17.097.
What the rounding hides
The rule is taught as “fewer than seventeen teeth will undercut”. The exact figure says something the rule does not: a seventeen-tooth gear does undercut, marginally, and eighteen is the smallest count that does not.
That is a small thing and it is the kind of small thing this site exists to notice. A threshold quoted as an integer has been rounded in some direction, and knowing which direction settles whether the boundary case is safe.
The threshold moves sharply with pressure angle, because it depends on :
| pressure angle | exact threshold | smallest clean count |
|---|---|---|
| 14½° | 31.90 | 32 |
| 20° | 17.10 | 18 |
| 25° | 11.20 | 12 |
That table is most of the argument for the twentieth century’s move from 14½° to 20°: it nearly halved the smallest usable pinion, which matters because a small pinion is what makes a large reduction possible in one stage.
What the cutter is doing
Undercutting is a consequence of the generating process rather than a property of the finished tooth, which is why it takes a moment to see.
A gear is cut by rolling a rack-shaped tool against the blank — the tool’s straight flanks generate the involute, because the involute’s conjugate is a straight line. The tool’s tip is the part that cuts the root region.
On a large gear the tip sweeps through material below the base circle, where there is no involute anyway, and leaves a fillet. On a small gear the pitch circle is tighter, so the same tool tip sweeps further round relative to the blank, and it reaches material that is part of the working flank. It cuts that away, and what remains is a tooth with a scooped root.
Nothing about the finished shape announces itself as a mistake; it looks like a tooth with a generous fillet. The defect is that the flank now starts higher up than it should, which is the second cost below.
Both costs, quantified
A weaker tooth. Bending stress in a gear tooth is highest at the root fillet, and undercutting removes material exactly there. The tooth is thinner where it is most loaded.
A shorter path of contact. The usable involute now begins above the base circle, so the segment of the line of action over which this tooth can carry load is shortened. That reduces the contact ratio, and a contact ratio below one means the drive is periodically not being driven.
The second is the one that decides designs. A weaker tooth can be compensated by a larger module; a contact ratio below one cannot be compensated at all.
The threshold derived
Two circles matter and both are fixed by convention rather than by nature.
The base circle sits at r·cos α, which follows from the pressure angle. The root circle sits 1.25 modules below the pitch circle, which is the standard dedendum and is a convention this site states wherever it is used, because every number downstream depends on it.
Setting them equal: r cos α = r − 1.25m, with r = mN/2. Rearranged, and using the standard addendum of one module rather than the dedendum for the interference condition, the threshold comes out as
Which is a formula about the pressure angle alone, and the reason a change of pressure angle moves the threshold so sharply.
What undercutting costs
Two things, and the second is worse.
The tooth is weaker, because material has been removed from the root where the bending stress is highest.
And the path of contact is shortened, because the usable flank is shorter. That reduces the contact ratio, and a contact ratio that drops below one means the drive is periodically not being driven.
Profile shift
The standard fix is to move the cutter outward from the blank by a fraction of a module — profile shift, or correction.
The tooth is then generated from a part of the involute further from the base circle, where there is more of it. The tooth comes out thicker at the root and thinner at the tip, and it does not undercut.
Nothing about the involute changes: the base circle, the pressure angle and the velocity ratio are untouched. What changes is which portion of the curve is used, which is why a shifted gear still meshes correctly with an unshifted one of the same module and pressure angle.
The cost is that the tooth tip gets pointed if the shift is too large, and that the pair no longer runs at the standard centre distance without further correction. Both are manageable and both are why gear design has more parameters than tooth count and module.
Asserting a threshold
A rule that says “below N it undercuts” is only meaningful if N − 1 and N behave differently, so this site checks both sides: a gear with one fewer tooth than the threshold must undercut, and one at the threshold must not.
That is the same discipline as everywhere else here — a check that only ever confirms is not a check — and it is cheap to write once the threshold is a formula rather than a remembered integer.
Why the standard moved to 20°
The table above is most of the argument, and it is worth drawing out because it is a case of a whole industry changing a convention for a computable reason.
At 14½° the smallest clean pinion has 32 teeth. At 20° it has 18. Since a reduction ratio is the ratio of tooth counts, and the large gear’s size is limited by the space available, the smallest usable pinion sets the largest reduction obtainable in one stage. Halving it nearly doubles what a single stage can do.
The cost is real and was accepted. A 20° tooth has a steeper line of action, so the force between teeth has a larger component pushing the shafts apart — bigger bearing loads, for the same transmitted torque. It also has a shorter path of contact and therefore a lower contact ratio, which is the constraint that stops the same argument being run again at 25°.
Twenty-five degrees is standard for some applications, gives a threshold of 12, and is not general because the contact ratio and the bearing loads both get worse. The convention settled at 20° because that is where the three effects balance for the majority of power transmission, which is a judgement rather than a theorem.
At 20° the contact ratio of the pair drawn earlier is 1.61, meaning two tooth pairs share the load for 61% of the cycle. At 25° that number falls, and the smallest usable pinion falls with it — which is the trade the standard settled.
Proving the threshold discriminates
A rule of the form “below N it undercuts” only means something if N − 1 and N behave differently, so this site checks both sides of it: seventeen teeth must undercut and eighteen must not.
That is the same discipline as everywhere here — a check that only ever confirms is not a check — and it is cheap once the threshold is a formula rather than a remembered integer. It also makes the rounding visible, which is how the discrepancy between 17 and 17.097 became something the essay could report rather than something it repeated.
Where the rule is broken deliberately
Seventeen is a limit for a standard, full-depth, unshifted tooth cut by a rack. Every one of those qualifications is a place production gearing departs from the rule.
Stub teeth. Reduce the addendum from one module to 0.8 and the interference limit falls with it, because the interference is between the cutter tip and the flank near the base circle. Fewer teeth become possible and the contact ratio falls, so the gain is paid for in smoothness.
Profile shift. Move the cutter outward and the tooth is generated from a part of the rack that does not reach below the base circle. Twelve teeth or fewer become routine, at the cost of a thinner tip and a changed operating pressure angle.
Higher pressure angle. The threshold is 2/sin²α, so raising α from 20° to 25° drops the limit from 17.1 to 11.2. That is why 25° gearing exists: it is bought for exactly this, and paid for with higher bearing loads and lower contact ratio.
Not being cut by a rack at all. Undercutting as described here is a generation phenomenon — the cutter removing material the tooth needed. A gear that is moulded, sintered, wire-cut or shaped by a pinion cutter has different interference conditions, and a moulded plastic pinion of eight teeth is not undercut in this sense at all. It may still interfere with its mate in service, which is a separate check.
So the rule is best read as: seventeen is where the cheapest process stops working. It is a manufacturing threshold that has been repeated until it sounds like a geometric one, and the geometric content — the 2/sin²α — is the part that survives translation to the other cases.
Why the exact threshold is not an integer
A recurring irritation with this rule is that the derivation gives 17.097 and the answer is 17, and the rounding is not neutral.
The threshold is the tooth count at which the interference exactly vanishes. Below it there is interference; above it there is none. Since 17.097 lies between 17 and 18, a seventeen-tooth pinion is below the threshold and is therefore undercut — slightly, and genuinely.
The convention that quotes seventeen as the minimum is accepting that slight undercut as harmless, which for most applications it is: the material removed at seventeen teeth is a small fraction of what is removed at fourteen, and the strength loss is within the margins the design already carries. The honest statement of the rule is “eighteen is clean and seventeen is close enough”, and the figure here plots the interference against tooth count so the size of “close enough” is visible rather than asserted.
That is the reason this site’s threshold assertion is written the way it is: it checks that the computed threshold sits strictly between the count that is undercut and the count that is not, and it checks that a gear on each side of it is classified correctly. A test that only compared a number to 17 would pass on a formula that was wrong by a tenth, and the tenth is the entire subject of this section.
What the rule is an instance of
The seventeen-tooth rule has a shape that recurs across engineering, and naming the shape makes it easier to handle the next one.
A continuous quantity — here the interference between the cutter tip and the flank — crosses zero at some value of a parameter. The parameter happens to be an integer in practice, so the threshold gets quoted as an integer. The rounding direction is then decided by convention rather than by the mathematics, and the convention gets repeated until the underlying continuous quantity disappears from the account entirely.
What is lost is the margin. Seventeen teeth is undercut and eighteen is not, and how badly seventeen is undercut — whether it is a rounding artefact or a real loss of root material — is a question the integer cannot answer. The continuous quantity can, and it says the interference at seventeen is small enough to ignore for most purposes and not zero.
The same shape appears elsewhere on this site. Grashof’s condition is an inequality between sums of link lengths, and a chain that satisfies it by a thousandth of a unit is classified identically to one that satisfies it by a factor of two — while behaving quite differently near its change point. The 40° transmission-angle rule is a threshold on a continuous quality measure with no discontinuity at the boundary at all.
In each case the useful thing to compute is not whether the threshold is met but by how much, and in each case this site plots the continuous quantity alongside the verdict for that reason. A figure that says “passes” has thrown away the information a designer needs; a figure that plots the margin lets the reader decide what margin is enough for their case.
That is the argument for measuring rather than classifying, made on a rule where the classification is correct and still incomplete.