§1Why chain gears together
A single pair is limited: push the ratio much past about 6 : 1 and the gears become wildly mismatched in size, the large one dominating the housing. A train shares the total reduction across several modest stages.
Two questions define any train: what is the overall ratio, and which way does the output turn? Both are answered by tracking tooth counts through the mesh. Beyond the plain ratio, trains let a designer place input and output shafts where the machine needs them, split power down several paths, and — in the planetary form — pack a large ratio into a concentric, in-line package or switch ratios while running. This page works from the simple train to the planetary, the arrangement inside every automatic gearbox and most compact reducers.
Contents§2Simple trains and idlers
In a simple train each gear sits on its own shaft and meshes with the next. Only the first and last gears set the ratio; everything between is an idler.
An idler earns its place not by changing the ratio — its teeth appear once on top and once on the bottom of the chain of fractions and cancel — but by two other services: it spans a gap when the input and output shafts must sit far apart without a huge single gear, and it reverses the direction of rotation. Each mesh flips the sense of turn, so counting meshes (or idlers) tells you whether output turns with or against input: an even number of gears means opposite senses, an odd number the same.
Contents§3Compound trains
To make the ratio actually multiply, two gears are fixed on a shared intermediate shaft so they turn together — a compound train.
Stage one: a 20-tooth driver meshes a 60-tooth gear. On that gear’s shaft sits a 15-tooth gear driving a 45-tooth output. Overall i = (60 × 45)/(20 × 15) = 2700/300 = 9 : 1. A 1500 rev/min input therefore leaves at 1500/9 = 166.7 rev/min. The same 9 : 1 as two 3 : 1 stages keeps every individual gear pair sensibly matched, where a single 9 : 1 pair would pair a small pinion with a gear nine times its diameter.
§4Planetary trains
A planetary (epicyclic) train has three concentric members — a central sun, an outer internally-toothed ring (annulus), and planet gears that mesh both and ride on a rotating carrier.
What sets it apart is that the planet axes themselves move, orbiting the sun on the carrier, so the gears both spin and revolve. That gives it three shafts on one axis — sun, ring and carrier — any one of which can be input, output or held fixed, and each choice yields a different ratio from the same hardware. The load is shared among several planets, so a planetary carries more torque for its size than an ordinary train, in a package that is concentric and in-line. The cost is complexity: the members interact, so the ratio can no longer be read off by tracking one path — it needs the equation of §5.
Contents§5The epicyclic equation
Because the carrier moves, ratios are found by working relative to the carrier — imagining you ride on it, so the train looks momentarily like a simple one.
The minus sign records that, seen from the carrier, sun and ring turn in opposite directions (the ring is internally toothed). Fix any one member by setting its speed to zero and the equation gives the ratio between the other two. The tooth-count identity on the right — the ring equals the sun plus twice the planet — is the geometric constraint that the planets fit the annular space; it also fixes the planet size once sun and ring are chosen.
Contents§6Worked planetary
The commonest case — ring held fixed, sun driven, carrier out — gives a compact reduction.
Sun z = 24, ring z = 72; the planets are therefore (72 − 24)/2 = 24 teeth each. With the ring fixed (n_ring = 0), the equation reduces to n_sun = n_carrier(1 + z_ring/z_sun), so the reduction is 1 + 72/24 = 4 : 1. A sun turning at 2000 rev/min drives the carrier at 2000/4 = 500 rev/min, both turning the same way. Swap which member is fixed and the same gears give a different ratio — hold the sun instead and drive the ring, and the reduction changes, which is exactly how an automatic gearbox selects gears by clamping different members.
§7Quick reference
The working core of the page on one card rack.
Simple
i = z_last/z_first
idlers reverse only
Compound
i = Π driven / Π driver
Direction
even gears → opposite
odd gears → same
Planetary
(n_s−n_c)/(n_r−n_c) = −z_r/z_s
z_r = z_s + 2 z_p
Fixed ring
i = 1 + z_r/z_s
sun in, carrier out