Planetary (Epicyclic) Gear Ratio

The ratio of a planetary gearset (sun, planets, ring, carrier) - the thing gear-cascade cannot do because two members move and the ratio depends on which is HELD. Willis superposition, R0 = Nr/Ns: ring fixed with the sun driving the carrier gives 1 + R0 (reduction, same direction); a carrier-fixed set reverses (ratio -R0); driving the carrier gives an overdrive (ratio below 1). Sun 30, ring 72, 3,400 rpm in, ring held, sun to carrier: R0 2.4, ratio 3.4, output 1,000 rpm same direction, planet teeth Np = (72-30)/2 = 21; hold the carrier instead and it flips to -2.4, reversing to -1,417 rpm. Same gears, different held member, completely different drive - automatic transmissions, hub reductions, and gear reducers all run on it. Compound/multi-stage sets, torque split, and efficiency are separate; tooth strength is checked separately. A design aid; Machinery Handbook / Shigley and the gear maker govern.

R0 = Nr/Ns; Np = (Nr - Ns)/2. Held-member ratios: ring fixed sun->carrier 1+R0; sun fixed ring->carrier (R0+1)/R0; ring fixed carrier->sun 1/(1+R0); sun fixed carrier->ring R0/(1+R0); carrier fixed sun->ring -R0; carrier fixed ring->sun -1/R0.

The epicyclic (planetary) gear-train ratio by the Willis superposition method (Machinery's Handbook; Shigley, Mechanical Engineering Design), by name.

The Willis epicyclic ratio relations are standard published gear-train results; the tooth counts, input speed, and held-member configuration are the user's inputs.

Estimate. AHJ and licensed professional govern.

Field names used by the API: sun_teeth, ring_teeth, input_speed_rpm, configuration, gear_ratio, output_speed_rpm, planet_teeth

• Willis method ratio by superposition, R0 = Nr/Ns, six standard held-member configurationsMachinery's Handbook / Shigley
• Planet teeth Np = (Nr - Ns)/2 from Nr = Ns + 2 Np; Nr - Ns must be evenconcentric constraint
• Scope single stage; compound/multi-stage, torque split, efficiency, and assembly count are separatescope of this tile