Engineering · Mechanical Engineering · Machine Elements
Gear Ratio Calculator
Calculates gear ratio, output speed, and output torque from input gear teeth, output gear teeth, input speed, and input torque.
Calculator
Formula
GR is the gear ratio (dimensionless). N_{out} is the number of teeth on the driven (output) gear. N_{in} is the number of teeth on the driver (input) gear. \omega_{in} is the input rotational speed (RPM). \omega_{out} is the output rotational speed (RPM). T_{in} is the input torque (N·m). T_{out} is the output torque (N·m), assuming 100% mechanical efficiency.
Source: Shigley's Mechanical Engineering Design, 10th Edition, McGraw-Hill — Chapter 13: Gears — General.
How it works
A gear ratio describes the relationship between the rotational speeds and torques of two meshing gears. When a small driver gear turns a larger driven gear, the output shaft rotates more slowly but delivers greater torque — a trade-off governed by the conservation of energy. Conversely, a large driver gear driving a small driven gear produces a speed increase at the cost of reduced torque. This principle underlies virtually every rotating machine, from bicycle derailleurs to jet engine accessory drives.
The gear ratio is calculated as the ratio of the number of teeth on the driven (output) gear to the number of teeth on the driver (input) gear: GR = N_out / N_in. Output speed follows as ω_out = ω_in / GR, and output torque is T_out = T_in × GR × η, where η is the mechanical efficiency expressed as a decimal. Real gear meshes always incur some friction loss, typically 1–5% per stage for well-lubricated spur and helical gears, which this calculator accounts for via the efficiency input. Input and output power are derived from the standard relation P = T × ω, where angular velocity is converted from RPM to rad/s by multiplying by 2π/60.
Practical applications span almost every engineering discipline. Automotive gearboxes use multi-stage gear trains to keep the engine operating in its efficient power band across a wide vehicle speed range. Industrial conveyors rely on worm gear reducers with ratios up to 100:1. Robotic joints use planetary gear sets for high torque density in compact packages. Wind turbine gearboxes step up the slow rotor speed (typically 10–20 RPM) to the 1,500 RPM required by standard generators. Understanding gear ratio allows engineers to match motor characteristics to load requirements, minimise motor size, and extend drivetrain life.
Worked example
Problem: A motor drives a gearbox with a 20-tooth driver gear meshing with a 60-tooth driven gear. The motor runs at 1,500 RPM and delivers 50 N·m of torque. The gear mesh efficiency is 98%. Find the gear ratio, output speed, output torque, and output power.
Step 1 — Gear Ratio:
GR = N_out / N_in = 60 / 20 = 3.000 : 1
The driven gear rotates three times slower than the driver gear.
Step 2 — Output Speed:
ω_out = ω_in / GR = 1,500 / 3 = 500 RPM
Step 3 — Output Torque:
T_out = T_in × GR × η = 50 × 3 × 0.98 = 147 N·m
Torque is tripled, reduced slightly by the 2% friction loss.
Step 4 — Input Power:
P_in = T_in × (ω_in × 2π / 60) = 50 × (1,500 × 2π / 60) = 50 × 157.08 = 7,854 W (≈ 7.85 kW)
Step 5 — Output Power:
P_out = P_in × η = 7,854 × 0.98 = 7,697 W (≈ 7.70 kW)
The 157 W difference represents heat generated by gear mesh friction.
Limitations & notes
This calculator models a single-stage, two-gear mesh and assumes constant efficiency across all load and speed conditions. In reality, gear efficiency varies with speed, load, lubrication viscosity, and temperature. For multi-stage gearboxes, the overall ratio is the product of individual stage ratios, and the total efficiency is the product of per-stage efficiencies. The formula does not account for backlash, gear deflection under load, dynamic load factors at high speeds, or bearing losses — all of which become significant in precision or high-speed applications. Worm gears have direction-dependent efficiency and may be non-back-drivable at low lead angles, which requires separate treatment. Planetary gear sets require different tooth-count rules (the fundamental planet equation) and cannot be analysed with this simple driver-driven model. Always verify gear selection against AGMA or ISO gear rating standards when designing for fatigue life and contact stress.
Frequently asked questions
What does a gear ratio of 3:1 mean?
A gear ratio of 3:1 means the input (driver) gear completes three full revolutions for every one revolution of the output (driven) gear. The output shaft rotates at one-third the input speed but produces three times the input torque (less efficiency losses). This is called a speed reduction or gear reduction, and it is the most common configuration in motors driving heavy loads.
How do I calculate gear ratio from tooth counts?
Divide the number of teeth on the driven (output) gear by the number of teeth on the driver (input) gear: GR = N_out / N_in. For example, a 15-tooth driver meshing with a 45-tooth driven gear gives GR = 45/15 = 3. This relationship holds because gears mesh tooth-by-tooth, so the gear with more teeth rotates more slowly by exactly the tooth-count ratio.
Does gear ratio affect torque or only speed?
Gear ratio affects both speed and torque simultaneously. A gear reduction (GR > 1) decreases output speed and increases output torque by the same factor, minus any efficiency losses. A gear increase (GR < 1) raises output speed and reduces torque. This is a direct consequence of power conservation: since power equals torque multiplied by angular velocity, if speed decreases, torque must increase proportionally to keep power constant.
What is a typical mechanical efficiency for gear pairs?
Well-lubricated spur and helical gear pairs typically achieve 97–99% efficiency per mesh stage. Bevel gears are slightly lower at 95–98%. Worm gears vary widely — from 50–90% — depending on the lead angle and sliding velocity. For a multi-stage gearbox, multiply the per-stage efficiencies together; a three-stage spur gearbox at 98% per stage delivers an overall efficiency of roughly 94%.
What is the difference between gear ratio and speed ratio?
In common engineering usage the terms are interchangeable for a simple two-gear mesh: the gear ratio equals the speed ratio, which equals N_out / N_in. Some texts define speed ratio as the inverse (N_in / N_out), expressing how many times faster the input turns relative to the output. Always confirm which convention is being used. In multi-stage or compound gear trains, the overall gear ratio is the product of the individual stage ratios, not the sum.
Last updated: 2025-01-15 · Formula verified against primary sources.