Gear Centre Distance

Two gears mesh when their pitch circles touch, so the distance between their shafts is the sum of the two pitch radii:

\[ C = \frac{d_1 + d_2}{2} = \frac{m(N_1 + N_2)}{2} \]

The second form is the useful one, because it shows what a designer is really up against: for a chosen module, only certain centre distances are reachable at all. The tooth counts are whole numbers, so $C$ moves in steps of $m/2$. A housing that wants exactly 112 mm between bores, at module 3, needs $N_1 + N_2 = 74.67$ — which does not exist.

Profile shift, the way out

The fix is profile shift (also called addendum modification or correction). Cutting the teeth with the rack displaced slightly outward or inward changes the tooth thickness at a given radius without changing the module or the tooth count. A pair cut this way runs at an operating centre distance different from the standard one, with a slightly altered pressure angle at the mesh. It is standard practice, it is how gearboxes hit awkward centre distances, and it does a second job: shifting a small pinion outward is the classical cure for undercutting below 17 teeth, thickening the root exactly where the tooth is weak.

Getting it wrong in both directions

Centre distance is one of the few gear dimensions that cannot be adjusted after the housing is bored, and the tolerance is tight. Too close and the backlash disappears: the teeth bind, the mesh loses its oil film, and the pair heats and seizes. Some backlash is not sloppiness, it is a requirement — it is the space that thermal expansion, deflection under load and the oil film all need, and gear standards specify a minimum. Too far apart and contact migrates toward the tooth tips, where the tooth is thinnest, where the sliding velocity is highest and where the load-sharing between successive teeth is worst.

Internal gears

For a pinion running inside a ring gear — planetary sets, and many hoist and winch drives — the geometry inverts and the centres come together rather than apart:

\[ C = \frac{d_2 - d_1}{2} \]

with $d_2$ the internal gear. That single sign change is the reason a planetary gearset packs so much reduction into so little space: the ring gear surrounds the whole mechanism rather than sitting beside it, and the input and output can be coaxial.