KEVOS® MAKEOS · Gears
Gear Tooth Top Land and Undercut Check (Involute Function, Pointed Radius, Profile Shift)
For a spur gear with few teeth or a profile shift, is the tooth undercut, how thick is the top land, and how far is the tip from becoming pointed?
Check of a spur gear tooth form: the pitch and base radii, the tooth thickness at the pitch circle with profile shift, the tip radius, the involute function at the pressure angle and at the tip (as the trigonometric functions tool), the tooth thickness at the tip (top land), the radius at which the tooth would come to a point found from the involute value (as the same tool, angle from an involute value), and the least profile shift that avoids undercut, against your least top land.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module | mm | Module of the gear. |
| Number of teeth | teeth | Number of teeth. |
| Pressure angle | ° | Standard pressure angle. |
| Profile shift coefficient | Profile shift coefficient (positive moves the tooth outwards, thickening it at the root). | |
| Addendum coefficient | Addendum as a multiple of the module (1 for standard teeth). | |
| Least top land (× module) | Least tooth thickness at the tip as a multiple of the module (often 0.25–0.4 for hardened gears). |
Method and formulas
- r = m z ÷ 2; r_b = r cos φ; tooth thickness s = m (π ÷ 2 + 2x tan φ); tip radius r_a = r + m (h_a + x).
- inv φ = tan φ − φ (as the trigonometric functions tool); φ_a = acos (r_b ÷ r_a); top land s_a = 2r_a (s ÷ 2r + inv φ − inv φ_a).
- Pointed when inv φ_p = s ÷ 2r + inv φ, solved for φ_p (as the same tool) at r_p = r_b ÷ cos φ_p; no undercut when x ≥ h_a − z sin² φ ÷ 2.
- Example: top land 0.87148 mm (0.43574 m); the tooth would point at 15.117 mm against a 14.6 mm tip.
Limitations
- Rack-generated involute teeth; shaper-cut and tip-relieved teeth differ.
- Undercut limit for a full rack without tip rounding.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a small pinion with profile shift
- Find the largest profile shift before teeth point
- Check a gear with long addendum
Who it is for
- Gear designers
- Design engineers
- Gear cutters
What you get
- Pitch, base and tip radii, tooth thickness
- Top land and pointed radius
- Undercut and top land checks
Interpreting the result
- Profile shift cures undercut on small pinions but thins the top land; the two limits set the useful range of shift.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 95–114, 2141–2160. Topic reference; no source tables or recommended settings are reproduced.
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