Gear Ratio Calculator
Gear Ratio Calculator
Overall ratio, output speed and output torque for one, two or three gear or belt stages, with the per-stage efficiency taken off, the speed and torque at every intermediate shaft, and the load inertia reflected back through the ratio — which falls with its square, and is the real reason a gearbox makes a load easier to accelerate.
gear ratio
two stages, 12:60 and 15:45 teeth, driven at 1,455 rpm and 49.22 N·m, 97% per stage, with 0.50 kg·m² on the output shaft and a 0.0045 kg·m² rotor
One shared pitch-line speed, and everything else
itotal = i₁ · i₂ · i₃ nout = nin / itotal
Tout = Tin · itotal · ηstages
Jreflected = Jload / i² — from ½Jω² with ωin = i·ωout
inertia-matched ratio: i = √(Jload / Jmotor)
- i
- reduction ratio, driven over driving. Greater than 1 means the output is slower and stronger than the input
- eta
- efficiency of ONE stage. It compounds: three stages at 97% give 91.3%, and the catalogue figure for a two-stage planetary is 94%, which is 0.97 squared
- J reflected
- the load’s inertia as the motor feels it. The square is the whole point: 15:1 makes an inertia look 225 times smaller
Worked example
two stages, 12:60 and 15:45 teeth, driven at 1,455 rpm and 49.22 N·m, 97% per stage, with 0.50 kg·m² on the output shaft and a 0.0045 kg·m² rotor
Stage 1 is 60 ÷ 12 = 5:1 and stage 2 is 45 ÷ 15 = 3:1, so overall 15:1
Output speed = 1,455 ÷ 15 = 97 rpm, with the intermediate shaft at 291 rpm
Ideal output torque = 49.22 × 15 = 738.3 N·m; at 97% per stage the train is 94.09% efficient, so 694.7 N·m arrives. That 94.09% is 0.97², which is exactly what a planetary gearbox catalogue quotes for a two-stage unit
Power in is 7.4995 kW and out 7.0563 kW — 443.2 W of heat in the gears and bearings
The 0.50 kg·m² on the output shaft comes back to the motor as 0.50 ÷ 15² = 0.002222 kg·m², which is 0.49 times the rotor inertia — comfortably inside the usual five-to-one guidance
If inertia matching were the only consideration the ratio would be √(0.50 ÷ 0.0045) = 10.5:1; at 15:1 you are a little under-geared for pure acceleration and probably right for everything else
What the square does to a load inertia
| Ratio | Speed from 1,455 rpm | Torque from 49.22 N·m at 97%/stage | 0.5 kg·m² reflected | As a multiple of a 0.0045 kg·m² rotor |
|---|---|---|---|---|
| 1:1 | 1,455 rpm | 49 N·m | 0.500000 kg·m² | 111.11× |
| 3:1 | 485 rpm | 143 N·m | 0.055556 kg·m² | 12.35× |
| 5:1 | 291 rpm | 239 N·m | 0.020000 kg·m² | 4.44× |
| 10:1 | 146 rpm | 463 N·m | 0.005000 kg·m² | 1.11× |
| 15:1 | 97 rpm | 695 N·m | 0.002222 kg·m² | 0.49× |
| 25:1 | 58 rpm | 1,158 N·m | 0.000800 kg·m² | 0.18× |
| 50:1 | 29 rpm | 2,246 N·m | 0.000200 kg·m² | 0.04× |
| 100:1 | 15 rpm | 4,492 N·m | 0.000050 kg·m² | 0.01× |
Efficiency per stage, and what it compounds to
| Drive | Per stage | Two stages | Three stages |
|---|---|---|---|
| Spur or helical gear pair, well lubricated | 98% | 96.0% | 94.1% |
| Planetary stage (catalogue figure) | 97% | 94.1% | 91.3% |
| Toothed belt | 97% | 94.1% | 91.3% |
| Roller chain, clean and tensioned | 96% | 92.2% | 88.5% |
| Bevel pair | 95% | 90.3% | 85.7% |
| Worm, high lead angle | 85% | 72.2% | 61.4% |
| Worm, low lead angle (self-locking) | 50% | 25.0% | 12.5% |
Speed down, torque up, inertia down by the square
Two gears in mesh share one thing: the speed of the point where the teeth touch. That gives the ratio immediately — a wheel with three times as many teeth turns at a third the speed — and it does not matter whether you count teeth or measure pitch diameters, because for a given tooth size those are proportional. The same argument works for a toothed belt, a chain and a friction drive, which is why the arithmetic on this page covers all of them.
Torque and power. Power is torque times angular speed, and a lossless gearbox passes power through, so torque must rise by exactly the factor the speed falls. Real gearing is not lossless: each mesh costs a few per cent, and the efficiencies multiply rather than add. Three stages at 97% is 91.3%. The loss is the part that heats the gearbox, and on a hard-working reducer it is enough to size the oil and the housing around — which is what ISO/TR 14179 is for.
The reflected inertia, which is the part people come for. Put an inertia J on the output shaft and ask what the motor feels. Equate the kinetic energies: ½Jωout² is the energy in the load, and the motor turns i times faster, so the equivalent inertia at its shaft is J/i². The square is the whole point. A 15:1 reduction makes a load inertia 225 times smaller as the motor sees it, while making the torque 15 times larger. On any axis that spends its life accelerating and decelerating, most of the torque goes into inertia rather than into the working load, and that asymmetry is why gearboxes turn up on servo axes that have no steady-state torque problem at all.
Inertia matching. Since reflected inertia falls as 1/i² and the torque demand rises as i, there is a best ratio for pure acceleration, and it is √(Jload/Jmotor) — the ratio at which the reflected load inertia equals the rotor inertia. That is a starting point rather than an answer: the top speed you need, the gearbox’s own inertia, and the steady-state torque all pull on it. The usual practical guidance, from motor manufacturers’ sizing guides, is to keep the reflected inertia within about five times the rotor inertia and to reach for a larger-rotor motor rather than a bigger gearbox beyond that.
Step-up is the same arithmetic with the sign of the surprise reversed. Drive a small wheel from a big one and the output is faster and weaker — and the reflected inertia is multiplied by the square, not divided by it. A flywheel overdriven 3:1 looks nine times heavier to the motor than it is, which is why step-up drives on inertial loads are so unpleasant.
Things this page does not do. It says nothing about whether the teeth will survive: gear rating is bending stress and contact stress, ISO 6336 or AGMA 2001, and it depends on face width, material, surface finish, lubrication and how many cycles you want. It says nothing about backlash, which is what actually limits a positioning axis, nor about torsional stiffness or the resonance that the gearbox and the load form together. And it assumes the efficiency you enter — a figure which on a worm drive can swing from 90% to below 50% with the lead angle, and which on any drive is worse at light load than at full load.
For the motor at the input end, the DC motor calculator gives torque from current and the motor power, torque and speed calculator relates any two of power, torque and speed in the same N·m and rpm the page uses here. For a stepper, where the ratio also multiplies the resolution, the stepper motor calculator.
Frequently asked questions
How do I calculate a gear ratio?
Divide the driven gear’s tooth count by the driving gear’s. Twelve teeth driving sixty is 5:1 — five turns of the input for one of the output. Pitch diameters give the same answer, because tooth size is common to both gears in a mesh. For several stages, multiply the stage ratios together.
How does a gear ratio affect torque and speed?
Speed divides by the ratio and torque multiplies by it, less the efficiency. A 15:1 reduction on 1,455 rpm and 49.22 N·m gives 97 rpm and, at 97% per stage over two stages, about 695 N·m rather than the ideal 738 N·m.
What is reflected inertia and why does it matter?
The load’s inertia as the motor feels it through the gearbox: J divided by the square of the ratio. It matters because most of a servo motor’s torque goes into accelerating inertia rather than into the working load, and dividing by the square is a far bigger effect than the torque multiplication. A 10:1 reduction makes a load inertia 100 times smaller and the available torque 10 times larger.
What inertia ratio should I aim for?
Reflected load inertia within about five times the rotor inertia is the usual industrial guidance; motor sizing guides suggest a motor with more rotor inertia when you exceed it. For pure acceleration the optimum is a ratio of √(J_load/J_motor), which makes the two equal, but the speed you need and the gearbox’s own inertia usually decide it first.
Is the gear ratio the same for belts and pulleys?
Yes. Pitch diameters, or tooth counts on a timing pulley, and the same torque and inertia relationships apply. Use pitch diameter and not the outside diameter of the flange, and keep the belt speed inside the manufacturer’s limit — this page prints it.
How efficient is a gearbox?
About 97–98% per spur, helical or planetary stage, and it compounds — a planetary catalogue will quote 97% for one stage and 94% for two, which is 0.97 squared. Bevel pairs are a little worse, chains and toothed belts similar, and a worm drive can be anywhere from 90% down to below 50% depending on its lead angle, with the low-efficiency ones being the self-locking ones.
Related calculators
References
- Kollmorgen, Application Sizing Guide — Servomotor Selection. Gives the reflected inertia as (JL + Jgearbox) divided by the square of the gear ratio, the output torque as input torque × ratio × gearbox efficiency, and the practical guidance that it is desirable to keep the reflected inertia at or below five times the motor inertia — with a higher-inertia motor recommended above that.
- Apex Dynamics AD-series high-precision planetary gearbox catalogue: efficiency ≥ 97% for the single-stage (L1) units and ≥ 94% for the two-stage (L2) units. 0.97² = 94.09%, which is the multiplicative per-stage model this page uses, checked against a real product rather than assumed.
- ISO/TR 14179-2:2001, Gears — Thermal capacity — Part 2: Thermal load-carrying capacity (ISO/TC 60/SC 2). The technical report covering gear-unit power loss and heat dissipation, and the reason a per-stage efficiency is a sizing approximation rather than a specification: real loss depends on load, speed, lubricant and sump temperature.
- ISO 6336-1:2019, Calculation of load capacity of spur and helical gears — Part 1: Basic principles, introduction and general influence factors — what actually decides whether a given ratio can be built in a given size, which this page does not attempt.