Planetary Gear Set Calculator
Calculate planetary gear ratios, speeds, and torques from ring, sun, and planet gear teeth.
Find output speed for fixed ring, fixed carrier, or fixed sun configurations.
Planetary (Epicyclic) Gear Set A planetary gear set consists of: Sun gear (S): center gear, driven by input in many configurations Planet gears (P): mesh with both sun and ring; mounted on carrier Ring gear (R): outer internal gear; surrounds the planets Carrier (C): holds the planet gear axes; can be output or held fixed
Fundamental Equation (Willis Equation) (ω_R − ω_C) / (ω_S − ω_C) = −N_S / N_R Where ω = angular velocity (RPM), N = number of teeth. This single equation governs all possible configurations.
Three Common Configurations
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Fixed Ring (R locked): ω_R = 0 Gear ratio = ω_S / ω_C = 1 + N_R / N_S Output (carrier) is slower than input (sun). Used in: first gear of auto transmissions.
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Fixed Sun (S locked): ω_S = 0 Gear ratio = ω_R / ω_C = 1 + N_S / N_R Output (carrier) is slower than input (ring). Used in: reverse or overdrive in some transmissions.
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Fixed Carrier (C locked): ω_C = 0 Gear ratio = ω_S / ω_R = −N_R / N_S (negative = reverse rotation) Output (ring) is faster than input (sun) in opposite direction. Used in: planetary reducers.
Gear Constraints Planet teeth N_P: determined by geometry. N_R = N_S + 2 × N_P (mesh condition, internal gear) Assembly condition: (N_R + N_S) / number of planets = integer
Applications Automatic transmissions (multiple planetary sets in series for different gear ratios). Bicycle hub gears (Sturmey-Archer, Shimano Nexus). Wind turbine pitch drives. Helicopter gearboxes. Clock and watch escapements (historical use).