Planetary gears
An epicyclic gear train: a sun gear at the center, planet gears orbiting around it, a ring gear enclosing everything, and a carrier arm connecting the planets. Lock any one component and drive another — the gear ratio changes depending on which parts are fixed and which are free.
Ns + Nr = 2 · Np + Ns • Nr = Ns + 2Np • Willis equation: ωr−ωc / ωs−ωc = −Ns/Nr
Epicyclic gear trains
A planetary (epicyclic) gear set consists of a central sun gear, one or more planet gears that mesh with the sun and orbit around it, a ring gear (annulus) with internal teeth that mesh with the planets, and a carrier arm that holds the planet gears' axles. The planet gears simultaneously mesh with both the sun and ring gears.
The Willis equation
The fundamental relationship linking all three rotational speeds is the Willis equation: (ωr − ωc) / (ωs − ωc) = −Ns/Nr, where ω are angular velocities, N are tooth counts, and subscripts r, s, c denote ring, sun, and carrier. By fixing one component, you get a specific gear ratio between the other two.
Common configurations
Ring locked: Input on sun, output on carrier. Ratio = 1 + Nr/Ns.
This is the most common configuration, used in automatic transmissions.
Carrier locked: Input on sun, output on ring. Ratio = −Ns/Nr.
Reverses direction — used as a reverse gear.
Sun locked: Input on carrier, output on ring. Ratio = 1 + Ns/Nr.
Provides overdrive.
Applications
Planetary gear sets are found in automatic transmissions, bicycle hub gears, helicopter rotors, wind turbine gearboxes, power drills, and the Antikythera mechanism (c. 100 BCE). Their compact, coaxial design and multiple gear ratios from one assembly make them indispensable in mechanical engineering.