3. Profile shift: rescuing a small pinion
A robot joint wants the biggest ratio it can get from one stage, which means the smallest pinion it can get away with. But below a certain tooth count the generating rack cuts into the involute near the root — undercut — and takes away exactly the material that carries the bending load.
The fix is a cutter setting, not a new cutter
Pull the rack outwards by x·m when generating and the tooth is formed further up the involute, where it is thicker and not undercut. The wheel keeps the same module and still meshes with everything else of that module. The cost is a thinner tip and a slightly lower contact ratio.
If the two shifts do not cancel, the pair no longer runs at the reference centre distance. The working pressure angle opens up, and it is found from the involute function:
There is no closed form for αw — the involute function has to be inverted numerically, which is what the calculator does.
Two ways to use it
- x₁ + x₂ = 0. Shift the pinion out and the wheel in by the same amount. The centre distance stays standard, and the strength moves from the wheel, which has it to spare, to the pinion, which does not. This is the common case.
- x₁ + x₂ ≠ 0. Used to hit a centre distance the standard module cannot reach, or to raise the contact strength by opening the pressure angle.
A 14-tooth pinion — undercut as standard — rescued with x₁ = +0.5 against a wheel at −0.5, so the centre distance stays where it was.