Involute spur gears without the guesswork
Every time I model a spur gear from scratch I end up looking up the same handful of formulas. This note is the page I wish I had pinned above the monitor: what the numbers mean, which circles matter, and what to check before you send the part to the printer or the machine shop.
Two numbers define the tooth
For a standard spur gear you choose the module m and the number of teeth z. The module is simply the pitch diameter divided by the number of teeth, in millimetres. Two gears mesh only if they share the same module and the same pressure angle, which is 20° in almost every catalogue today.
The four circles
| Circle | Formula | m = 2, z = 18 |
|---|---|---|
| Pitch diameter, d | m · z | 36.00 |
| Tip diameter, da | d + 2m | 40.00 |
| Root diameter, df | d − 2.5m | 31.00 |
| Base diameter, db | d · cos 20° | 33.83 |
The involute flank starts at the base circle. Below it, down to the root, the tooth is just a fillet. If the root circle ends up larger than the base circle — which happens with many teeth — the whole working flank is involute and the fillet is short.
Distance between two gears
For two standard gears with no profile shift, the centre distance is
a = m · (z1 + z2) / 2With m = 2, z1 = 18 and z2 = 36 that gives a = 54 mm. If the housing forces a different distance, that is a job for profile shift, not for "just move the shaft a little".
Things worth checking
- Undercut. With a 20° pressure angle, fewer than 17 teeth will be undercut by a standard rack cutter. Printed gears forgive it; cut steel gears lose strength.
- Backlash. A CAD model with perfect zero backlash will bind once printed. I thin each flank by about 0.05 mm for printed parts with m = 2.
- Face width. A common rule of thumb is 8 to 12 times the module. Narrower than that and the teeth start to bend.
- The bore. Leave at least about 2.5 m of material between the root circle and the keyway.
In the next note I will look at flange bolt circles, which have their own small set of traps.