Involute Gear Design: Undercut Limits and Contact Ratio Targets

Summary of design limits: For a standard involute spur gear cut at a 20° pressure angle, any tooth count below 17 results in undercut, in which the cutter removes material from the root involute, weakening the root section and lowering bending capacity. Designs should therefore begin at 17 teeth or above; for power transmission duty the usual working band is 17–25 teeth. Contact ratio — the mean count of tooth pairs sharing load at any moment — ought to be held above 1.2, with 1.3–1.6 representing the normal design window; once it falls under 1.1, the drive becomes rough, with noticeable noise and shock loading. If a lower tooth count is unavoidable, positive profile shift eliminates the undercut, with the required coefficient beginning near x = (17 − z)/17.

1. Why contact ratio governs running smoothness

Definition: contact ratio is the average quantity of tooth pairs carrying load at the same time while the gears rotate. Motion transfer in a spur gear pair stays continuous only while this value exceeds 1. Should it drop below 1, contact is momentarily lost, and the following engagement arrives as an impact.

For external spur gears at α = 20°, a widely used approximation is:

εα ≈ 1.88 − 3.2 × (1/z1 + 1/z2)

Taking two gears of z = 20 each: εα ≈ 1.88 − 3.2 × 0.1 = 1.56, so on average 1.56 pairs are meshed. Increasing the combined tooth number lifts εα, yet the improvement diminishes and flattens out close to 1.9. Practical design floors sit at 1.2, most handbooks keep the value within 1.2–1.4, and once it falls under 1.1 noise together with shock becomes evident.

Helical gearing contributes an axial term as well:

εβ = b·sinβ / (π·mn), giving a total εγ = εα + εβ

For parallel-shaft helical units, εγ ≥ 2 is the normal requirement. With module 2, a 40 mm face width and 10° helix angle, εβ ≈ 40 × 0.1736 / (3.1416 × 2) ≈ 1.1; combined with εα the total clears 2 without difficulty. This explains directly why helical teeth run more quietly than spur teeth.

2. Origin of the 17-tooth threshold

Undercut: when a rack-type cutter shapes the gear, its tip travels past the interference limit point and cuts away part of the involute adjacent to the root. The root section loses material, bending capacity falls, and the usable engagement length is shortened, which also penalises contact ratio.

The tooth count at which undercut begins is given by:

z_min = 2·h_a* / sin²α

Using h_a* = 1 and α = 20°, z_min = 2 × 1 / sin²20° ≈ 17.1, rounded to 17. Raising the pressure angle pushes this limit downward — roughly 13.5 teeth at α = 25°, compared with about 32 teeth under the older 14.5° standard. This behaviour is one reason 20° has become the mainstream selection.

Where z already sits below 17, positive profile shift restores the tooth form. The coefficient required is approximately x ≥ (17 − z)/17; for z = 14, x ≥ 0.176 is enough to clear the undercut.

3. Points to watch during design review

• A drawing that specifies fewer than 17 teeth without any profile shift called out should be flagged as an undercut risk immediately. Material removed from the root cannot be restored by any subsequent heat treatment or grinding operation, so the drawing itself must be revised.
• Planet gears and compound gear clusters operate in confined envelopes, and the smallest member is frequently reduced to z = 12–15. Positive shift is compulsory in that situation; begin at x = (17 − z)/17 and then verify that contact ratio remains above 1.2 rather than judging by root bending strength alone.
• With helical gears, helix angle can be used to purchase contact ratio: every additional degree of β raises εβ by about 7% (the relationship is linear in b·sinβ), while axial load rises by roughly 1.7% per degree (tanβ). Past 15° the bearing cost increases noticeably, so 8–15° is the customary range.
• Tip relief removes a small amount of effective contact ratio. On high-precision, high-speed gears that carry profile modification, keeping a margin of about 0.1 on εα is the safer choice. These calculation details are reviewed item by item during the process review stage of our precision gear machining service.

Resolving undercut and contact ratio questions while the design is still on paper is considerably cheaper than correcting them afterwards through heat treatment and grinding; the penalty appears as cost and lead time rather than as any precision figure.

4. Reference table: tooth count versus undercut and contact ratio

Holding centre distance fixed, a larger tooth count forces a smaller module, which shortens tooth height and reduces sliding, with bending capacity recovered through face width. A smaller tooth count permits a larger module, giving greater bending strength but a lower contact ratio. The majority of power gears end up in the 17–25 band, which is also where the batch-produced reducer gears in our gear product range are positioned.

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