Gear Undercut & Profile Shift
Determine the minimum teeth Zmin to avoid undercut and the minimum profile shift coefficient x₀ and shift amount Lₜ for low-tooth-count gears.
Inputs
Formula
Fundamentals
Undercut is a machining defect where the cutter digs into the tooth root, weakening the gear — it must be avoided or limited.
With a rack-type cutter, undercut occurs when tooth count is below $z_\text{min}=2h_a^*/\sin^2\alpha$ (for standard $\alpha=20^\circ$, $h_a^*=1$, $z_\text{min}=17$). Positive profile shift prevents it.
History
Undercut was known from early milling; to make small gears practical, profile-shifted (corrected) gears appeared in the late 19th c., systematised by Buckingham and others in the 20th c.
Engineering applications
Where space is tight, reducers and power tools need small tooth counts; positive shift avoids undercut and keeps root strength.
Glossary
| Min teeth $z_\text{min}$ | Smallest count without undercut; standard spur gear $z_\text{min}=17$. |
| Shift factor $x$ | Cutter offset from standard position; positive shift avoids undercut. |
| Pressure angle $\alpha$ | Angle between common normal and pitch-line velocity, usually $20^\circ$. |
How to use
- Fill in Teeth Z, Pressure angle α, Addendum coeff. ha*, Module m in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Min teeth Zmin, Min shift x₀, Shift amount Lt, Undercut (1=yes) in the results area.
Formula · Worked Example · Knowledge
Formula
Worked Example
Key Points
- Standard spur gears need Z≥17 to avoid undercut (α=20°).
- Profile shift rescues low-tooth-count strength and contact ratio.
- Undercut weakens root bending strength and raises noise.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Teeth Z | Z | 14 | |
| Pressure angle α (°) | alpha | ° | 20 |
| Addendum coeff. ha* | ha | 1 | |
| Module m (mm) | m | mm | 2 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Min teeth Zmin | Zmin | |
| Min shift x₀ | x0 | |
| Shift amount Lt | Lt | mm |
| Undercut (1=yes) | undercut |
Applications
- Common engineering use cases