Gear undercut, minimum tooth count and profile shift

Below a certain number of teeth, the tool that generates an involute gear cuts away part of the tooth's own flank near the root. That undercut weakens the tooth exactly where bending stress peaks and shortens the usable involute. Profile shift is the standard fix.

Why do small gears undercut?

Involute teeth are generated by a rack-shaped cutter (or a hob, which behaves like one). The involute only exists outside the base circle. On a gear with few teeth the base circle is small, and the straight tip of the rack reaches below the point where the involute starts. Instead of stopping there it keeps cutting into the flank, leaving a notch at the root.

Printed and laser-cut gears are not generated by a cutter, but the same geometry still applies: if you model the tooth as the cutter would leave it, the root comes out undercut. GearStudio builds the root from the rack-cutter path, so its profiles show undercut where a real cutter would produce it.

What is the minimum number of teeth before undercut?

For a standard rack with addendum coefficient h_a* (1.0 for full-depth teeth), undercut starts below:

where α is the pressure angle. The tooth count has to be a whole number, so round up:

At 20°, a 17-tooth gear is right at the limit and its undercut is negligible, which is why 17 is often quoted. Raising the pressure angle or using stub teeth lowers the limit, at the cost of higher bearing loads or less contact.

How does profile shift prevent undercut?

Profile shift moves the cutter away from the gear centre by x · m while it generates the teeth, where x is the profile shift coefficient and m the module. The pitch circle does not change, but the teeth are cut further out on the involute. That moves the root clear of the region the cutter would undercut. The smallest shift that avoids undercut is:

What shift does to the tooth

For an external gear with shift x, the main dimensions become:

• Outside diameter: da = m · (z + 2ha* + 2x)
• Root diameter: df = m · (z − 2hf* + 2x), with hf* = 1.25 for full depth
• Tooth thickness at the pitch circle: s = m · (π/2 + 2x · tan α)

Positive shift makes the tooth thicker at the root and thinner at the tip. Too much and the tip comes to a point, so check tip thickness when shifting small gears heavily. An internal gear such as a planetary ring runs the opposite way: positive shift moves its teeth outward and thins them.

Can you use profile shift in a planetary gear set?

Shifting one gear changes its centre distance to every gear it meshes with. In a planetary set the sun–planet and planet–ring meshes share one centre distance, so shifts have to be chosen together. The simplest arrangement keeps the standard centre distance:

• shift the sun by +x to clear undercut,
• shift the planets by −x,
• shift the ring by the same amount as the planets, −x.

This only works if the planets have enough teeth to take a negative shift without undercutting themselves. When they do not, the alternative is to let the centre distance grow. That changes the operating pressure angle and needs the coaxial condition worked out with the shifted geometry. GearStudio lets you set shift per gear and override the centre distance, and reports undercut and contact ratio as you go.

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