Epicyclic Train (form 1) Mechanism Explained: How It Works, Parts, Willis Ratio and Uses

Reduce speed through the carrier

The gold sun is the input. Two blue planets spin on pins carried by the orange output arm. The fixed internal ring constrains their outer engagement. The carrier consequently turns more slowly than the sun, in the same direction.

The two gear meshes must have one common centre distance and module. Once sun and ring counts are chosen, planet teeth must equal half their difference. The calculator does not round a half-tooth result or stretch a planet to make it appear to fit.

Two opposed planets keep the engagements visible. Their tooth phases are synchronized with both the sun and the stationary ring.

Compare reductions from two tooth counts

The two controls select sun and ring teeth. The four results report reduction, carrier speed as a fraction of sun speed, ideal output torque multiplication and required planet count.

The torque ratio assumes lossless steady power transfer. It does not estimate a gearset’s allowable torque, motor size or bearing load.

For a calculator with an independently entered planet count and input speed, use the planetary gear-train calculator.

Speed reduction and ideal torque multiplication

The relative-speed relation is Ns(ωs−ωc)+Nr(ωr−ωc)=0. Fixing the ring sets ωr=0 and gives i=ωs/ωc=1+Nr/Ns. Carrier speed is therefore 100/i percent of sun speed.

With no losses, Tinωs=Toutωc, so Tout/Tin=i. The required planet count is Np=(Nr−Ns)/2 and must be a positive integer. If it is fractional or nonpositive, the assembled speed and torque results are withheld.

The animation uses that same speed relation. Its illustrative 25° stub tooth outlines have a 0.75-module addendum and simplified roots; they are not stock-gear or machining specifications.

20-tooth sun and 60-tooth ring

The required planet has (60−20)/2=20 teeth. The reduction is 1+60/20=4:1, so the carrier turns at 25% of the sun speed. Ideal output torque is four times input torque, before losses.

Changing the ring to 61 teeth would require 20.5-tooth planets. The animation instead shows separate parts and a clear assembly explanation. Selecting equal sun and ring counts would require a zero-tooth planet and is also invalid.

Keep kinematics separate from production design

Correct centre distances and speed ratios do not establish tooth strength, acceptable backlash, bearing life or manufacturability. The example uses two opposed planets and illustrative stub involutes; low-count cases outside its profile checks use pitch circles.

Operating limits, tolerances and wear depend on the specified gears and application. A detailed gearbox requires its own tooth geometry, interference, material, load and bearing calculations.

Form 1 questions

Why is the planet count calculated?

The selected sun and ring fix the radial space available at a common module.

Why not round fractional planet teeth?

Rounding would change the centre distance and prevent both meshes from fitting as entered.

Does four times torque mean a four-times-rated gearbox?

No. It is a lossless power ratio; tooth and bearing capacity require a separate calculation.

Does the carrier reverse?

No. In this fixed-ring, sun-input arrangement, it turns with the sun.