title: “Undercut and Contact Ratio in Involute Gears: Why z Below 17 Costs You, and Why εα Should Sit Above 1.2” date: 2026-08-25 tags: [gear design, undercut, contact ratio, minimum tooth number, profile shift, technical learning] category: Gear Design
Undercut and Contact Ratio in Involute Gears: Why z Below 17 Costs You, and Why εα Should Sit Above 1.2
Bottom line: For a standard involute spur gear with a 20° pressure angle, fewer than 17 teeth means undercut — the cutter removes part of the root involute, thinning the tooth root and cutting bending strength. Design with 17 teeth or more first; power transmission commonly uses 17–25. Keep the contact ratio (the average number of tooth pairs in mesh at any instant) above 1.2, with 1.3–1.6 as the usual design band; below 1.1 the drive runs rough, with visible noise and shock. When tooth count is forced lower, positive profile shift fixes the undercut, starting at roughly x = (17 − z)/17.
1. Where the number 17 comes from
Undercut: when a gear is cut with a rack-type cutter, the cutter tip passes beyond the interference limit point and removes part of the involute near the tooth root. The root gets thinner, bending strength drops, and the effective length of engagement shortens — contact ratio suffers too.
The minimum tooth number to avoid undercut is:
z_min = 2·h_a* / sin²α
With h_a* = 1 and α = 20°, z_min = 2 × 1 / sin²20° ≈ 17.1, rounded to 17. A larger pressure angle lowers the limit — about 13.5 at α = 25°, versus about 32 at the old 14.5° standard. That is part of why 20° became the mainstream choice.
When z is already below 17, positive profile shift brings the tooth back. The shift coefficient needed is roughly x ≥ (17 − z)/17; a z = 14 gear needs x ≥ 0.176 to clear the undercut.
2. Contact ratio: the hard requirement for smooth transmission
Contact ratio: the average number of tooth pairs engaged simultaneously during meshing. A spur gear must have a contact ratio above 1 to transmit motion continuously — below 1, the teeth separate for an instant and re-engage with impact.
For external spur gears (α = 20°), the common approximation is:
εα ≈ 1.88 − 3.2 × (1/z1 + 1/z2)
A pair of z = 20 gears gives εα ≈ 1.88 − 3.2 × 0.1 = 1.56, meaning 1.56 tooth pairs in mesh on average. Larger tooth sums raise εα, but the gain slows down and levels off near 1.9. The recommended floor in practice is 1.2, and most design handbooks control 1.2–1.4; below 1.1, noise and shock become obvious.
Helical gears add an axial component:
εβ = b·sinβ / (π·mn), with total contact ratio εγ = εα + εβ
Parallel-shaft helical gears normally require εγ ≥ 2. With module 2, face width 40 mm and helix angle 10°, εβ ≈ 40 × 0.1736 / (3.1416 × 2) ≈ 1.1 — added to εα, the total passes 2 easily. That is the direct reason helical gears run smoother than spur gears.
3. Tooth count, undercut and contact ratio at a glance
| Teeth z (equal pair) | Undercut status | Approx. εα | Typical use |
|---|---|---|---|
| 10–13 | Clear undercut, positive shift required | ~1.2–1.35 | Space-limited pump gears, planet gears |
| 14–16 | Slight undercut, positive shift advised | ~1.4–1.5 | Small-module instruments, compact custom gears |
| 17–25 | No undercut | ~1.5–1.65 | General power transmission, reducer gears |
| 26–40 | No undercut | ~1.6–1.7 | Low-speed smooth running, noise-sensitive equipment |
At the same center distance, more teeth means a smaller module — shorter tooth height and lower sliding, with bending strength made up by face width. Fewer teeth means a larger module — stronger in bending but lower contact ratio. Most power gears settle in the 17–25 band, and that is also where batch-produced reducer gears in our gear product range sit.
4. Practice notes
💡 When a drawing shows fewer than 17 teeth with no profile shift marked, treat it as an undercut risk first. Once the root is cut away, no heat treatment or grinding step brings it back — the drawing has to change.
💡 Planet gears and multi-gear clusters are space-tight, and the small gear often gets pushed down to z = 12–15. Positive shift is mandatory there; start from x = (17 − z)/17, then check that contact ratio stays above 1.2 — do not look at root bending strength alone.
💡 For helical gears, use the helix angle to buy contact ratio: each degree of β adds roughly 7% to εβ (linear in b·sinβ), but axial force grows about 1.7% per degree (tanβ). Beyond 15° bearing cost climbs noticeably, so 8–15° is the common range.
💡 Tip relief eats a little effective contact ratio. On high-precision, high-speed gears with profile modification, leaving a margin of about 0.1 on εα is safer. These calculation details are checked item by item during the process review stage of our precision gear machining service.
Sorting out undercut and contact ratio at the design stage costs far less than patching things later with heat treatment and grinding — the difference shows up in cost and lead time, not in a precision number.
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