Planetary Gearset - 2D Lab

The gearset

A planetary (epicyclic) gearset has three members that turn about one axis: the sun gear (Zs teeth), the ring gear with internal teeth (Zr) and the carrier that holds the planet gears (Zp teeth each). The planets must fit between the sun and the ring, so Zr = Zs + 2 Zp, and equally spaced planets need (Zs + Zr) / n to be a whole number.

The Willis equation

Rolling without slipping at the sun and ring contacts gives, for any speeds, Zs w_s + Zr w_r = (Zs + Zr) w_c. The planet spins at w_p = w_c + (w_c - w_s) Zs / Zp. Hold one member, drive another, and the third is the output. For example, sun in with the ring held gives w_c / w_s = Zs / (Zs + Zr) (a reduction, 3.5:1 for 24/60); carrier in with the ring held gives w_s / w_c = (Zs + Zr) / Zs (an overdrive); holding the carrier gives w_r / w_s = -Zs / Zr (reverse). Locking any two members locks the whole set at 1:1. The three speeds always lie on one straight line (the lever analogy), with the carrier at Zr / (Zs + Zr) along the lever.

Torque and power

With no losses and no inertia the torques on the three members are in the ratio T_s : T_r : T_c = 1 : Zr/Zs : -(1 + Zr/Zs) and sum to zero, so the held member carries the reaction torque and T_s w_s + T_r w_r + T_c w_c = 0. Each mesh wastes a fraction 1 - eta_m of the power that flows through it relative to the carrier, P_rel = |T (w - w_c)|; this basic-efficiency method gives the total loss, so a locked set (no relative motion) has no gear loss at all.

What is simplified

Rigid gears, no backlash or shaft inertia in the speed calculation, constant mesh efficiency independent of speed and load, ideal brakes and clutches (the red caliper just marks the held member), equal load sharing between planets. Tooth profiles are drawn as simple trapezoids; the pitch circles, tooth counts and mesh phasing are exact.

• Start with the 1st-gear preset (sun in, ring held): note the 3.5:1 reduction. Press Overdrive: the same hardware now speeds up the output.
• Choose Reverse (carrier held): the ring turns the other way at Zs/Zr of the sun speed. The lever on the wall board flips.
• Press Lock-up: the three members turn together, the planets stop spinning relative to the carrier and the efficiency becomes 100 %.
• Change the sun teeth from 24 to 12 with the ring held: the reduction jumps to 5:1. Watch the number of planets that still satisfy the assembly rule.
• Lower the mesh efficiency to 92 % and compare the efficiency of a deep reduction against the direct drive.