Gear Trains: Ratio, Torque and Planetary Gears

Gear ratio and torque multiplication

Two meshing gears share a common pitch point, and the pitch-line velocities there must match. That single constraint is the whole of gear ratio theory: the speed ratio equals the driven gear's tooth count divided by the driver's, and — because power is conserved through an ideal, frictionless mesh — torque scales by exactly the inverse factor.

Speed ratio:  i = N₂/N₁ = ω₁/ω₂ = T₂/T₁
N₁, N₂ = tooth counts of driver, driven gear
External gears reverse rotation; internal gears rotate the same way
Pitch circle diameter: d = m·N  (m = module, mm)

If a gear pair steps speed down by a factor of four, torque goes up by the same factor of four — free force, paid for entirely in lost speed. Because meshing gears must share the same module (the ratio of pitch diameter to tooth count, essentially the tooth "size"), only gears cut to the same module will mesh cleanly, exactly like paper only fitting a matching envelope size.

The involute tooth profile

Gear teeth have to transmit uniform angular velocity, which the law of gearing shows requires the common normal at the contact point to always pass through the pitch point. The involute curve — the path traced by a point on a taut string unwinding from a circle — satisfies this exactly, and virtually every gear cut since the 19th century uses it. Its best property: nudging the centre distance slightly does not change the velocity ratio at all, which is why gearboxes tolerate manufacturing tolerances gracefully. Neighbouring teeth also need a contact ratio greater than 1 (typically 1.4–1.8) so a new tooth pair engages before the previous pair disengages, keeping the mesh smooth and quiet.

Compound trains and idler gears

A compound gear train stacks several meshing pairs on different shafts, and the overall ratio is simply the product of every stage: i_total = i₁ × i₂ × i₃. A small automotive first-gear stage might combine a 2:1 and a 4:1 reduction for an overall 8:1 — engine speed is eight times wheel speed, and torque at the wheels is eight times engine torque before the differential. An idler gear placed between driver and final gear meshes with both; it reverses rotation direction but its tooth count cancels out of the maths entirely, so it changes nothing about the overall ratio — only the spin direction.

Epicyclic (planetary) gears

A planetary gearset arranges a central sun gear, several planet gears riding on a carrier, and an outer ring gear that meshes with the planets from inside. All three — sun, carrier, ring — are related by the Willis equation:

(ω_ring − ω_carrier) / (ω_sun − ω_carrier) = −N_sun / N_ring

Hold any one of the three elements fixed and feed/take power from the other two, and the same physical hardware yields a different ratio — which is exactly how automatic transmissions pack several gears into one compact unit. The Toyota Prius takes this further: with no element locked, the engine drives the carrier, a generator (MG1) sits on the sun, and the wheels take the ring, giving continuously-variable behaviour with no discrete gear shifts at all.

Efficiency and where the losses go

Real gears are not free: friction between sliding tooth surfaces, rolling contact deformation, windage, and bearing drag all bleed off power. Spur and helical gears run around 97–99% efficient per stage; a worm gear, by contrast, can fall to 30–90% depending on its lead angle, and if that lead angle is small enough relative to the friction angle the worm becomes self-locking — the output cannot back-drive the input, which is deliberately exploited in lifts and winches.

Frequently asked questions

Try it live

Everything above runs in your browser — open Gear Train, build a 2–4 gear chain, set the tooth counts and input RPM, and watch the ratio, torque and direction update live.