OK, let's rewind a little ! First, what are epicyclic gearbox and how do they achieve good reduction-ratio in small space ? The basic part for such a gearbox are :
- an input gear called the "sun"
- 2 or more gears revolving around the sun, called "planets"
- a ring containing the "solar system" formed by the aforementioned gears.
In the following picture, the sun is yellow, the planets are blue and the ring is grey.
Figure 1: Epicyclic gear-set.
For my application, viewing through the shaft is mandatory, so the sun is a hollow gear, but it doesn't have to be. This gearbox has a 21-tooth sun, 24-tooth planets, and, as a consequence, a 69-tooth ring. Yes, as the gears are connected to each other, they share the same tooth-size (aka modulus), so the number of teeth of the ring is linked to the number of teeth of both the sun and the planets:
Nring = Nsun + 2 * Nplanets
where Nxxx is the number of teeth of gear xxx.
The usual arrangement for an epicyclic reduction gearbox is to connect the planets together and create a shaft co-linear with the input shaft running the sun, the ring is held fixed and the sun is driven. In that case, the reduction ratio would be:
Reduction Ratio = 1 + Nring / Nsun
So, with Nring = 69 and Nsun = 21, we get 1+69/21=90/21=30/7, so around 4.2857. This formula is linked to the number of teeth the sum need to move a planet so it make a complete revolution. The planet size doesn't seem to come into play, but remember the number of teeth of the ring is connected to the number of teeth of both the sun and the planet. The reduction ratio could be written as:
Reduction Ratio = 1 + (Nsun + 2 * Nplanet) / Nsun
Obtaining the same ratio with a regular gear-set would require a 90-tooth gear driven by the 21-tooth sun, which takes more space and has the added default of radial load on the bearing.
Yet, the reduction ratio is still not that big and the order of magnitude or reduction I'm looking for would require cascading 3 or 4 of these gear-set. While doable (and done in real world application), this is not the way I went.
If you look at Figure 1, you see that the teeth of a planet don't move relative to the ring when touching the ring (non-slipping contact). What if we now add another planet on top of each input planet ? If it has the same number of teeth as the input planet, this will do nothing, but, if it has a different number of teeth (and a different diameter because we keep the same tooth-size for all gears), there will be a small motion relative to the input ring. Let's assume the difference in tooth-count is -1 or +1 (higher values are possible but will defeat the purpose of obtaining massive reduction ratio). Whenever the input planet has done a complete turn on itself (not to be confused with a complete revolution around the sun), the output planet has done exactly the same, but its outer side has moved one-more or one-less tooth relative to the input ring. And that's the trick: for every turn of a planet on itself, and external ring will move one tooth (forward or backward depending upon the sign of delta). So the reduction ratio is now the product of sun-to-planet ratio and planet-to-output-ring ratio.
Animation 1: split-ring epicyclic
In the above animation, the grey bottom ring is fixed, the yellow sun is the driving gear, rotating the hard-to-see blue bottom "input" planet, attached to the top red "output" planet (forming a compound-planet), driving the light-blue top ring. The compound-planet rotate around the sun is the same direction as the sun, but in the opposite direction around its own centre. With this configuration where the upper part of the compound has one more tooth than the lower part, the output ring (grey) goes in the opposite direction. This is the kind of trick also used with the Harmonic Drive system.
