Undercut and minimum teeth

Andrew Prestridge · 9 November 2026

Undercut is what a generating cutter does to a small pinion when the tip of the rack keeps cutting past the interference point. It slices away the base of the involute and leaves a thin neck above the root fillet. The tooth is weaker in bending, and the length of involute left for contact is shorter, so the real contact ratio is worse than the full-involute formula suggests.

For rack or hob generation, undercut is avoided when the profile shift satisfies:

x ≥ h_a* − (z / 2) sin²α

With no shift, the smallest tooth count that satisfies that is:

z_min = 2 h_a* / sin²α

For the ordinary full-depth addendum h_a* = 1 this is 17.10 teeth at 20°, 31.90 teeth at 14.5°, and 11.20 teeth at 25°. The first whole counts with a clean tooth at x = 0 are therefore 18, 32, and 12. Seventeen teeth at 20° is already a hair under the limit — x_min is just above zero — so a standard 20° pinion of 17 teeth is slightly undercut. A 12-tooth pinion at 20° is not a close call.

What the check assumes

The formula uses the straight flank of the basic rack, which for a full-depth ISO 53 style tool is h_a* modules tall. The extra depth between that and the dedendum (the usual 0.25·m) is clearance, and a radiused rack tip takes it up as fillet rather than as more straight flank. The designer follows that model when the root fillet is greater than zero. The default fillet, 0.38×module, is the familiar rack tip radius, so the check matches the textbook counts above.

Set the root fillet to zero and the tool treats the rack tip as sharp, reaching the full dedendum. A sharp 1.25·m rack undercuts more teeth than a radiused one. That is intentional. If you did not mean a sharp rack, put the fillet back.

Shop cures

More teeth, if the ratio allows it.
A larger pressure angle. Twenty-five degrees is much kinder to a 12-tooth pinion than 20°, at the cost of higher separating force.
Positive profile shift, which is the usual fix when the tooth count is stuck. The formula above is the minimum x. A little more is common so the involute starts cleanly above the fillet.
A stub addendum, a lower h_a*. That is a different tooth form. The mate has to be designed for it.

Undercut is a geometry problem. It is not the statement “this pinion is strong enough.” A tooth that clears the limit can still be the weak member in bending or wear. Those calculations need face width, material, hardness, and a load. That is the job of a gear-rating method — AGMA rating practice and ISO 6336 are the usual names — and of a calculator built for it. This page stops at the question of whether the involute is still there.

The drawing still shows a circular fillet when undercut is flagged. It does not pretend to be the hob trochoid. Read the banner, not just the picture. Eighteen teeth at 20° and x = 0 is the first clean full-depth pinion.