Gear Trains, Ratios and Efficiency

Simple gear trains and idlers

In a simple train each gear is on its own shaft and meshes with the next. The overall ratio is the ratio of the last gear to the first, and the gears in between, idlers, only change the direction of rotation and the distance: they cancel out of the ratio, as the speed ratio of a pair is z_out ÷ z_in and the idler appears in both numerator and denominator. With an idler of 30 teeth between a 16-tooth gear and a 48-tooth gear the ratio is still 48 ÷ 16 = 3.0, but the output now turns in the same direction as the input. Each external mesh reverses the direction, so a train with an even number of meshes reverses overall, an odd number does not.

Compound and reverted trains

A compound train puts two gears on the same intermediate shaft: the second gear of one stage and the first gear of the next. The overall ratio is the product of the stage ratios. This reaches large ratios in a compact space with modest gear sizes, because the ratio of each stage stays moderate (below about 6:1). When the output shaft is in line with the input it is a reverted train, as in a clock or a coaxial reducer.

i = (z₂ ÷ z₁) × (z₄ ÷ z₃) × …    n_out = n_in ÷ i    T_out = T_in × i × η_total

Planetary (epicyclic) trains

A planetary set has three members: a central sun gear, several planets held by a carrier, and an outer ring (internal gear). Any one of the three can be the input, the output or held still, giving several ratios from one set. The planets share the load, so a planetary gearbox carries a lot of torque for its size, and the input and output shafts are in line. With the tooth counts of the sun z_s and the ring z_r (the planet z_p = (z_r − z_s) ÷ 2 follows from the geometry), the ratios are:

Arrangement	Input → output	Ratio i = input ÷ output	Direction
Ring fixed (most common)	Sun → carrier	1 + zr ÷ zs	Same direction
Sun fixed	Ring → carrier	1 + zs ÷ zr	Same direction (a small reduction)
Carrier fixed	Sun → ring	− zr ÷ zs	Opposite direction
Locked (two members joined)	Any	1	Direct drive

Choosing the numbers of teeth

• Split a big ratio evenly: for a two-stage reduction the stage ratios of √i each give the smallest gearbox and balance the loads. For a ratio of 7.2 the ideal stage ratio is √7.2 = 2.68.
• Use pinions of at least 17 teeth (see Gear Basics and Terminology), and 20 or more for quiet running; larger pinions in the final (high-torque) stage.
• Avoid common factors between the tooth counts of a pair: if z₁ and z₂ share a factor, the same teeth meet repeatedly and wear unevenly. A “hunting” tooth (a factor of one) spreads the wear. For example, 20 and 50 have a common factor of 10 (each tooth of the pinion meets only 5 gear teeth), while 18 and 49 have none.
• Match the center distance: for two stages on the same pair of shafts (a reverted train) m₁(z₁ + z₂) = m₂(z₃ + z₄).
• Check the ratio is exact enough. A ratio from small integers is exact; when the ratio must approximate a number (such as 7.2), search for tooth pairs: 17/51 × 18/49 … and compare the error.

Example: for the ratio 7.2 with two stages, try 51 ÷ 17 = 3.00, 49 ÷ 18 = 2.72: the product 8.17 is close to 7.2 (error 13.4 percent). The Gear Ratio Calculator helps to compare options.

Efficiency of a train

Each mesh and each pair of bearings loses a little. For a good spur or helical mesh the loss is 1 to 2 percent; the efficiencies multiply: η_total = η₁ × η₂ × … A three-stage gearbox at 98 percent per stage has η = 0.98³ = 94.1 percent, so 5.9 percent of the input power becomes heat. For a worm stage in the train (30 to 90 percent) the total efficiency is dominated by it (see Gear Types and Selection). Friction also depends on the lubricant, the speed and the load (Lubrication, Wear and Gear Failures).

Backlash, windup and stiffness in a train

Backlash accumulates through a train, referred to the output by the ratio of the stages that follow. In a speed-reducing train the backlash of the last stage matters most at the output; the first stage’s backlash is divided by the ratio of the rest. Torsional stiffness behaves the other way: the compliance of the early (high-speed, low-torque) stages is divided by the square of the ratio when referred to the output. So: make the last stage the most accurate and stiffest, and the earlier ones can be cheaper. See Mechanical Transmission for the effect on servo loops.

Example: a backlash of 0.06 mm at the pitch circle of a 40 mm gear (radius 20 mm) is 0.06 ÷ 20 = 3 mrad = 10.3 arc-minutes on that shaft. If that gear is in the first stage of a train whose second stage has a ratio of 2.0, it appears at the output as 10.3 ÷ 2.0 = 5.2 arc-minutes; the same backlash in the last stage appears in full (10.3 arc-minutes).

Reflected inertia and the choice of ratio

To the motor, the load inertia seems smaller by the square of the ratio: J_reflected = J_load ÷ N². A gear train therefore makes a heavy load easier for a small motor to accelerate and to control, which is why servo axes use gearheads (see Servo Systems and Tuning). With a load of 0.002 kg·m² and a motor rotor of 2.0 × 10^-5 kg·m², a 24:1 train reduces the reflected inertia to 3.47 × 10^-6 kg·m² (a ratio to the rotor of 0.17:1, compared with 100:1 direct).

Designing a reduction gearbox

• Define the duty: input power and speed, output speed, torque, shock, hours of use, ambient temperature and the arrangement of the shafts.
• Choose the layout: in-line (planetary or reverted), parallel shafts (helical), or right angle (bevel-helical, worm). Pick the number of stages from the ratio (about 6:1 per spur stage, 10:1 per planetary stage).
• Size the gears with the strength calculation (see Gear Strength and Design Calculations), starting from the output stage, where the torque is highest.
• Design the shafts, keys and bearings (see Shafts, Keys and Couplings and Bearings and Bearing Selection) with the forces of every mesh.
• Housing and lubrication: a stiff, well-aligned housing, oil level and a breather, and seals (see Lubrication, Wear and Gear Failures).
• Thermal check: the losses, the surface area and the oil temperature.
• Buy or build? For most uses a catalog gearbox (helical, planetary, worm) with a service factor is cheaper than a custom one.

Related

• Gears and Power Transmission: the overview of this guide.
• Gear Basics and Terminology, Gear Types and Selection and Gear Strength and Design Calculations: geometry, types and strength.
• Shafts, Keys and Couplings and Lubrication, Wear and Gear Failures: the shafts of a train, and its lubrication.
• Gear Ratio, Torque and Speed Generator and the Motor Torque and Inertia Matching Calculator.

FAQ

Why does an idler not change the ratio?

The idler is driven by the first gear and drives the last. Its tooth count cancels in the product of the ratios (z_idler ÷ z_in × z_out ÷ z_idler), so only the direction of rotation and the center distance change.

How do I get a very large reduction?

Use several stages (compound or planetary in series), a worm gear (up to about 100:1 in one stage), a harmonic or cycloidal drive (50 to 160:1), or a combination. Mind the efficiency and the backlash.

Why are planetary gearboxes so compact?

The load is shared between several planets and the input and output are in line, so the torque density is high for the size and the weight.

What is a hunting tooth?

A tooth count chosen so that the pinion and gear have no common factor: each tooth meets every tooth of the other gear in turn, spreading the wear evenly.

References

• Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design, McGraw-Hill: gear trains and planetary trains.
• Müller, H. W., Epicyclic Drive Trains, Wayne State University Press; Dudley, D. W., Handbook of Practical Gear Design.
• Manufacturer catalogs (Neugart, Apex, Bonfiglioli) for planetary and helical gearheads.

Last reviewed: October 2026. Efficiencies and ratios are typical; use the catalog of the gearbox for real data.