Gears on parallel fixed shafts. Every external mesh reverses direction.
| Stage | Driver T | Driven T | Ratio | Direction flip |
|---|
Each stage is driven teeth ÷ driver teeth; the overall ratio is the product. A mesh trades speed for torque, not power: P = τω.
Each external mesh reverses direction: an even number of stages turns the output the same way as the input, an odd number reverses it.
Sun, planets and ring on one axis have two independent speeds, so it is not a ratio until you hold something. Hold a different member, get a different ratio.
Below 2/sin²α teeth (17.1 at 20°) a standard cutter gouges the tooth root, thinning the section that carries the load.
A simple train puts every gear on its own fixed shaft. A planetary set puts sun, planets and ring on one axis, with the planets on an arm: three usable connections instead of two. It is how an automatic gearbox changes ratio without moving a gear. A gear pair has one input and one output; a planetary set has three shafts and only two independent speeds, which is exactly what a differential is for.
Every external mesh reverses direction; an internal one does not, because the teeth face inward. Hold the ring and the sun and carrier turn the same way. Hold the carrier instead and the ring reverses: the one arrangement here that gives a negative ratio.
The planet must fill the gap exactly: its tooth count is (ring − sun) ÷ 2, so an odd difference asks for half a tooth. Separately, sun + ring must divide by the number of planets or the last lands where a tooth should be; three planets suit most counts, five often fails. Both are assembly constraints: the ratio looks fine either way.
The limit is 2/sin²α: 17.1 at 20° but 31.9 at 14.5° (why old imperial gearing looks so much coarser) and about 12 at 25°. On a small enough gear the rack tip sweeps inside the base circle and scoops the root. Profile shift can rescue a smaller pinion; a plain full-depth cutter cannot.
A ratio above 1:1 reduces speed and multiplies torque; below 1:1 does the reverse (a bicycle final drive, a wind-turbine generator stage). An idler cancels out of the ratio but adds one direction flip. Stacking stages on shared shafts (this tool’s model) multiplies ratios; several gears on one shaft meshing with a single larger gear is a different, true compound topology used to fit a big ratio into a small space. A single 50:1 spur stage needs a driven gear 50× the driver’s diameter, so splitting it keeps every gear a sane size.
Meshing gears must share module (metric) or diametral pitch (imperial), so an odd tooth count is not always drop-in. Sliding friction and churning bleed a few per cent per mesh: 96–99% for well-cut spur and helical gears, but worm meshes can be 50–90%. Output torque is an upper estimate.
Because the driven gear has more teeth meshing over the same distance, each tooth on the smaller driver has to push harder to keep the tangential mesh force consistent — more teeth sharing that force means more total leverage, which shows up as torque.
A gear inserted between driver and driven purely to reverse direction or bridge a gap — its own tooth count cancels out of the ratio math entirely (it appears once as a "driven" and once as a "driver"), but it does add one more direction flip.
Practical tooth counts and centre distances get unwieldy fast — a single 50:1 spur stage needs a driven gear 50× the driver's diameter. Splitting the same overall ratio across two or three stages keeps every gear a sane, manufacturable size.
Sliding friction between meshing teeth, plus some churning loss from lubricant, bleeds off a few percent of power at every mesh — well-cut spur and helical gears typically manage 96–99% per stage; worm gear meshes can be far lower, sometimes 50–90%.
Ratio = Ndriven ÷ Ndriver, the teeth on the driven gear over the teeth on the driver. A 20-tooth pinion driving a 60-tooth gear is 3:1: the output turns at a third of the speed with up to three times the torque, less friction losses. For several stages multiply the stage ratios, so 3:1 then 4:1 is 12:1.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.