Profile shift explained

Andrew Prestridge · 9 November 2026

Profile shift is a deliberate offset of the rack that generates the tooth. ISO 21771 calls the coefficient x. A positive x moves the rack out by x·m and leaves more metal on the tooth. A negative x sinks the rack into the blank and thins the tooth. On an inch drawing the same idea often shows up as a tooth-thickness callout, which is the thing a shop can measure.

At the reference pitch circle the circular tooth thickness is:

s = m (π/2 + 2 x tan α) − j

where j is circular backlash taken on that gear. With x = 0 and j = 0 the tooth and the space each take half of the circular pitch, so s = π m / 2. The designer removes j from the gear you are editing and leaves the mate at its own shift, with no extra backlash of its own.

What it is for

Three jobs cover almost every use:

• It is the usual cure for undercut on a small pinion. The minimum coefficient is x_min = h_a* − (z/2) sin²α.
• It thickens the root of a pinion that would otherwise be a skinny tooth.
• It lets a pair run at a center distance other than the sum of the reference pitch radii. The operating pressure angle changes with it.

Center distance and operating pressure angle

The operating pressure angle α_w satisfies:

inv α_w = inv α + 2 (x₁ + x₂) tan α / (z₁ + z₂)

and the center distance is:

a = m (z₁ + z₂) / 2 · cos α / cos α_w

Opening the centers uses up part of the shift. Write y = (a − a_ref) / m. The part that is left, k = x₁ + x₂ − y, is tip shortening. Each outside radius is reduced by k·m, but never increased, so the clearance at the mate’s root stays at the value the dedendum was written for. The mesh view and the contact ratio use those shortened tips. The single-gear outside diameter stays the unshortened, as-cut number, and the table shows both when they differ.

A useful check: at the operating pitch circles, the two tooth thicknesses sum to the operating circular pitch when backlash is zero. That is the zero-backlash condition the center-distance formula is built to satisfy. Add backlash and the centered picture shows a gap on both flanks.

The cost of a large positive shift

The tooth gets wider at the pitch circle and narrower at the tip. Far enough and the flanks meet below the nominal outside diameter. The tooth is pointed, the effective tip is the meeting point, and the designer says so. A practical tooth wants some land at the tip so it survives heat treat and ordinary handling. A large shift also raises the operating pressure angle, which raises separating force and bearing load.

Negative shift does the opposite: thinner tooth, more undercut risk, shorter center distance, lower operating pressure angle. It belongs on the larger gear when a pair has to fit a tight center and the pinion cannot give up any thickness.

Changing h_a* and h_f* is how you represent a stub tooth or a deeper whole depth without pretending the rack is a standard ISO 53 full-depth tool. The default here is h_a* = 1 and h_f* = 1.25, which is that full-depth rack, clearance 0.25·m included.

Related: the involute, minimum teeth.