6. Planetary trains
A planetary — sun, planets, ring, carrier — is how you get a large ratio in a short, coaxial package that shares the load between several meshes. It is the standard first stage of almost every industrial servo gearbox.
One equation covers every configuration
The Willis equation relates all three shaft speeds. It does not care which one you hold:
zs ωs + zr ωr = ( zs + zr ) ωcWillis
Fix one member and the familiar ratios drop out:
| Held | In → out | Ratio |
|---|---|---|
| ring | sun → carrier | 1 + zr/zs |
| carrier | sun → ring | −zr/zs |
| sun | carrier → ring | 1 + zs/zr |
The ring is not a free choice — geometry fixes it:
zr = zs + 2 zp
Two conditions that catch people out
- Assembly. Equally spaced planets only fit if (zs + zr) / np is a whole number. Otherwise the last planet arrives at a tooth instead of a gap and the thing will not go together.
- Neighbouring. Adjacent planets must not touch. More planets share the load better but leave less room, and three is the usual compromise.
The ceiling. A single planetary stage is practically limited to about
1:10. Beyond that the sun becomes too small to carry the torque. Two stages reach 1:100 at the cost
of length, efficiency and backlash — which is exactly the gap that harmonic and cycloidal drives
were invented to fill.