BIZOMAT 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). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Module
- What it is
- The module of the gear, which sets the tooth size: the pitch diameter divided by the number of teeth. The pitch radius, tooth thickness and tip radius all scale with it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- For an inch gear given as diametral pitch, module = 25.4 ÷ diametral pitch.
What this means: Number of teeth
- What it is
- The number of teeth on the gear being checked. With the module it sets the pitch radius, and it decides how much profile shift is needed to avoid undercut.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 4 teeth to 1000 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Fewer teeth need more profile shift to avoid undercut.
- Get it right
- Count the teeth on this gear, not its mate.
What this means: Pressure angle
- What it is
- The standard pressure angle of the gear’s basic rack, the angle between the tooth flank and a radial line at the pitch circle. It sets the base circle and the involute shape.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 10° to 35°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger pressure angle lowers the profile shift needed to avoid undercut.
- Get it right
- Use the standard pressure angle from the drawing, not the operating pressure angle of a shifted gear pair.
What this means: Profile shift coefficient
- What it is
- The profile shift coefficient: how far the cutting rack is moved away from the gear centre, as a multiple of the module. Positive shift thickens the root and thins the tip.
- Units and format
- Enter a plain number with no unit. The form accepts -1 to 1.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- More positive shift reduces undercut but raises the tip radius and thins the top land.
- Get it right
- Enter the coefficient, not the shift in mm: divide a shift in mm by the module. Keep the sign.
What this means: Addendum coefficient
- What it is
- The addendum, the height of the tooth above the pitch circle, as a multiple of the module before any profile shift is added.
- Units and format
- Enter a plain number with no unit. The form accepts 0.5 to 1.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger addendum raises the tip radius, thins the top land and raises the shift needed to avoid undercut.
- Get it right
- Take it from the gear’s tooth system on the drawing; stub teeth use a smaller value than full-depth teeth.
What this means: Least top land (× module)
- What it is
- The least tooth thickness you accept at the tip circle, as a multiple of the module, so the tip is not too thin and brittle.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 2. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.25 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter it as a multiple of the module, not in mm. Take the limit from your gear design standard or heat treatment supplier.
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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