How to use this calculator
- Enter module, pressure angle and teeth. The no-undercut minimum z_min = 2/sin²(α) and all involute functions follow from these.
- Check the minimum shift. If z < z_min, the calculator reports the x needed to clear undercut, with the customary practical tolerance shown as a warning band.
- Watch the tip. Large positive shifts thin the tip; keep s_a above ~0.4·m (0.25·m absolute floor).
- Run the pair. In pair mode the shift sum sets the working pressure angle and center distance; the tip-shortening coefficient keeps bottom clearance.
- Verify the cut gear. Confirm the manufactured shift with a span measurement — positive x widens W by 2·x·m·sin(α).
How it works
Cutting a gear with the rack shifted outward by x·m uses a
different stretch of the same involute. Three things follow, and this
calculator screens all of them. First, undercut: below
z_min = 2/sin²(α) teeth (≈17.1 at 20°) an unshifted rack
digs into the root flank; the minimum shift that prevents it is
x_min = (z_min − z) / z_min
Second, tooth thickness: the reference thickness grows to
s = m·(π/2 + 2x·tan α), but the tip thins — the calculator
evaluates the actual tip thickness
s_a = d_a·(s/d + inv α − inv α_a) against the customary
0.4·m / 0.25·m limits. Third, for a pair, the shift sum
re-prices the mesh:
inv(αw) = inv(α) + 2(x₁+x₂)·tan(α)/(z₁+z₂), a′ = a₀·cos(α)/cos(αw)
That is how gear pairs hit non-standard center distances without special tooling. Use it with the gear center distance calculator for the unshifted baseline, the contact ratio calculator to confirm the mesh still overlaps adequately (shift reduces contact ratio), and the span measurement calculator to verify the shift on the cut part.
Worked example
Verified against the live calculator
A 12-tooth module-2 pinion at 20° needs
x_min = (17.097 − 12)/17.097 = 0.298 to fully clear undercut
— so the default x₁ = 0.3 just clears it. Meshing with an
unshifted 36-tooth gear (x₂ = 0):
inv(αw) = inv(20°) + 2(0.3)(tan 20°)/48 = 0.019454 → αw = 21.787°
a′ = 48 × cos 20° / cos 21.787° = 48.575 mm
The center distance grows by 0.575 mm — less than x·m = 0.6 mm;
the difference is the tip-shortening coefficient
k* = 0.3 − 0.2875 = 0.0125, small enough that many designs
skip the trim. The pinion tip thickness comes out
s_a = 0.87 mm = 0.44·m, above the 0.4·m comfort line.
Push the shift to x = 0.6 on a 12-tooth gear and the tip
ratio collapses to about 0.20·m — the calculator flags it red.
Frequently asked questions
What is profile shift in gears?
Profile shift (addendum modification) moves the cutting rack outward (positive x) or inward (negative x) by x·m while keeping the same base circle. Positive shift thickens the root and avoids undercut on small pinions; it also changes the working center distance of a pair. The involute itself is unchanged — only which portion of it is used.
How do I calculate the minimum profile shift to avoid undercut?
x_min = (z_min − z)/z_min with z_min = 2/sin²(α) — about 17.1 teeth at 20°. A 12-tooth pinion therefore needs x ≥ (17.1 − 12)/17.1 ≈ 0.30. In practice light undercut is tolerated down to about 14 teeth at x = 0, which this calculator shows as a warning band.
How does profile shift change center distance?
Through the working pressure angle: inv(αw) = inv(α) + 2(x₁+x₂)·tan(α)/(z₁+z₂), then a′ = a₀·cos(α)/cos(αw). The center distance change is NOT simply (x₁+x₂)·m — it is always somewhat less, and the difference is the tip-shortening coefficient k* = (x₁+x₂) − y needed to preserve bottom clearance.
What is the downside of large positive shift?
Tip thinning. The tooth gets thicker at the root but the involute converges at the tip; customary minima are around 0.4·m for through-hardened gears and 0.25·m as the hard floor. The calculator computes the actual tip thickness s_a and flags both bands.
How should I split the shift between pinion and gear?
The center distance fixes only the SUM x₁+x₂. Designers usually give the pinion the larger share (it has fewer teeth and weaker roots) while watching its tip thickness. Balanced-strength or balanced-slip splits are refinements on the same idea.
Method & assumptions
- External involute spur gears with the standard full-depth basic rack (ha* = 1.0, αn = α). Stub racks, protuberance hobs and grinding stock change z_min, tip diameter and tip thickness.
- The theoretical z_min assumes the rack tip generates to the full addendum; the practical warning band (~0.18 below x_min, i.e. 14 teeth at x = 0, 20°) reflects customary tolerance of light undercut.
- Tip diameters use d_a = m(z + 2 + 2x) without tip shortening; apply k* when you adopt it.
- Backlash allowance, tooth strength (root bending, flank pressure) and contact ratio are separate checks.