Compound Gear Train Value

The train value of a compound gear train is the product of the driven tooth counts over the product of the drivers. For two stages:

\[ e = \frac{N_2 N_4}{N_1 N_3} \]

and it extends to any number of stages by continuing to multiply. The output turns $e$ times slower than the input and carries $e$ times the torque, less whatever each mesh loses — a good spur mesh runs 98–99% efficient, so a two-stage box keeps about 96–98% and the losses are small but not zero.

What makes it compound

The essential feature is that $N_2$ and $N_3$ share a shaft and turn together. That is what lets the second reduction multiply the first rather than merely follow it. Contrast an idler: a gear that receives on a shaft of its own and delivers straight onward. Run the algebra and the idler's tooth count appears once in a numerator and once in a denominator, cancelling exactly. An idler changes the direction of rotation and lets two shafts sit further apart. It changes the ratio by nothing at all, which surprises people every time.

Why compound rather than simple

Consider a 40:1 reduction. As a single mesh with a 20-tooth pinion, the wheel needs 800 teeth — at module 2 that is a gear 1.6 m across, which is absurd for a machine you can lift. Split it into two stages of about 6.3:1 each and the largest gear has about 126 teeth, or 252 mm. The box shrinks by a factor of six in every dimension and by rather more than that in mass.

How the split is chosen matters. Roughly equal stages give the smallest overall envelope. Where the stages must be unequal, put slightly MORE reduction in the first stage, where the torque is still low and the gears can be small, and less in the last, where the gear is already large and every extra tooth is expensive in both material and space. The old rule for a two-stage box is $N_2/N_1 \approx \sqrt{e}$ as the starting point, adjusted for whatever tooth counts are available.

Hunting teeth

A detail worth knowing: if the tooth counts share a common factor, the same pairs of teeth meet over and over, and any manufacturing error on one tooth wears its permanent partner. Choosing counts with no common factor — a "hunting" ratio, such as 21 and 62 rather than 20 and 60 — makes every tooth eventually meet every other tooth, spreading the wear and letting the pair run itself in. It costs nothing at the design stage and it is a real difference in service life.