Gear Undercut & Profile Shift
Calculate the minimum tooth count to avoid undercut and the required profile shift coefficient for spur gears
Inputs
Formula Interpretation
Minimum Teeth (Undercut Limit)
The minimum tooth count to avoid undercut with a standard full-depth cutter () at pressure angle . For α = 20°: theoretical 17, practical 14; for α = 14.5°: theoretical 32, practical 26.
Minimum Profile Shift Coefficient
When a gear has fewer teeth than the limit, the cutter must be shifted outward. The minimum positive shift coefficient prevents the cutter from cutting into the root fillet (undercutting the involute).
Profile Shift Amount
The physical shift distance of the cutter reference line from the gear pitch circle equals the shift coefficient multiplied by the module . A positive moves the cutter away from the gear centre.
Knowledge Points
What is Undercut?
Undercut occurs during hobbing or shaping when the cutting tool removes material from the root of the involute tooth profile. This weakens the tooth at its root and shortens the effective involute, reducing the contact ratio. Gears with fewer than the minimum safe tooth count require corrective action.
Pressure Angle and the Limit
A larger pressure angle (e.g. 20° vs 14.5°) allows a smaller minimum tooth count before undercut. At α = 20° the theoretical limit is 17 teeth (practical 14); at α = 14.5° it rises to 32 (practical 26). Modern standards prefer 20° to enable smaller, lighter gears.
Profile Shift (Addendum Modification)
Moving the cutter reference line outward by x·m introduces a positive profile shift. This avoids undercutting without changing module or tooth count, and also increases tooth root thickness and load capacity. The pair centre distance changes and the meshing geometry must be re-verified.
Worked Example
A standard hobbing cutter with module , pressure angle , is used to cut a gear with teeth. Find the minimum profile shift coefficient and the required shift amount.
Step 1 — Minimum shift coefficient (Formula ②)
Step 2 — Profile shift amount (Formula ③)
Result: x₀ ≈ 0.30 (rounded up from 0.298), Lₜ = 0.30 × 15 = 4.5 mm. Because Z = 12 < 17 (the theoretical limit for α = 20°), a positive shift of at least 4.5 mm is required to prevent undercutting.
Extended Knowledge
- •For peripheral (circular) machining such as hobbing, the limit Zmin = 2h₁/sin²α applies directly. In practice, the minimum is often set to 14 for α = 20° because the last few teeth in the theoretical band cause only very shallow undercut that does not significantly reduce load capacity.
- •A negative profile shift (x < 0) shifts the cutter inward. While this reduces tooth tip thickness and top-land width, it is sometimes used on the larger gear of a pair to equalise bending fatigue life between pinion and wheel.
- •When both gears of a meshing pair are profile-shifted, the sum x₁ + x₂ determines the change in centre distance. A zero-shift-sum pair (x₁ = −x₂) maintains the standard centre distance while still allowing undercut avoidance on the smaller gear.
Related Standards & Articles
Authoritative references for the formulas used in this calculator
Gear Design Symbol Reference (GB/T 2821 / ISO 701)
A complete notation reference for gear design calculations — geometric parameters, load factors, strength properties, and safety factors — per GB/T 2821-2003 and ISO 701:1998, with full LaTeX symbol rendering.
Gear Transmission Types: Selection Overview (Involute, Bevel, Worm & Planetary)
A comprehensive comparison of all major gear transmission types — involute cylindrical, bevel, worm, planetary, few-tooth-difference, and harmonic drives — covering key characteristics, transmission ratios, power capacities, speed ranges, and typical industrial applications for mechanical design selection.