Calculate Gear Ratio

Calculator and formulas for calculating gear ratio with gears

Gear Ratio Calculator

Calculate gear transmission

Calculates the gear ratio, mechanical advantage, output speed and output torque of gear transmissions.

Enter number of teeth
teeth
teeth
Optional
rpm
Nm
Result
Gear ratio:
Mechanical advantage:
Output speed:
Output torque:

Example Calculation

Example: Bicycle transmission
Problem:

A bicycle transmission has a chainring (input gear) with 42 teeth and a sprocket (output gear) with 14 teeth. The rider pedals at 60 rpm. Calculate the gear ratio and rear wheel speed.

Given:
  • Chainring (input gear) n₁ = 42 teeth
  • Sprocket (output gear) n₂ = 14 teeth
  • Input speed ω₁ = 60 rpm
  • Find: Gear ratio i and output speed ω₂
Solution:
Gear ratio:
\[i = \frac{n_1}{n_2} = \frac{42}{14} = 3.0\]
Output speed:
\[\omega_2 = \omega_1 \times i = 60 \times 3.0 = 180 \text{ rpm}\]
Practical Applications
Mechanical engineering: Gearboxes, reducers, drive technology
Automotive engineering: Transmissions, differential, rear axle
Clock technology: Gear trains, time transmission, precision mechanics
Gear Types
  • Reduction: i > 1
  • Overdrive: i < 1
  • Direct: i = 1
  • Spur gear: parallel axes
  • Bevel gear: intersecting axes
  • Worm gear: skew axes

Formulas for gear ratio

The gear ratio describes the relationship between input and output gear. The result is the gear ratio and the mechanical advantage (reciprocal of the gear ratio).

Gear ratio

Basic formula for gear ratio with gears.

\[\displaystyle i = \frac{n_1}{n_2}\]
i = gear ratio, n₁ = teeth input gear, n₂ = teeth output gear
Mechanical advantage

Reciprocal of gear ratio, shows torque gain.

\[\displaystyle MA = \frac{1}{i} = \frac{n_2}{n_1}\]
MA = Mechanical Advantage
Output speed

Speed of output gear based on gear ratio.

\[\displaystyle \omega_2 = \omega_1 \times i = \omega_1 \times \frac{n_1}{n_2}\]
ω₁ = input speed, ω₂ = output speed
Output torque

Torque at output gear with mechanical advantage.

\[\displaystyle M_2 = M_1 \times MA = M_1 \times \frac{n_2}{n_1}\]
M₁ = input torque, M₂ = output torque
Important Notes
  • Reduction (i > 1): Speed decreased, torque increased
  • Overdrive (i < 1): Speed increased, torque decreased
  • The product of speed and torque remains constant (without losses)
  • Units are preserved: rpm → rpm, Nm → Nm

Detailed description of gear ratio

Physical Fundamentals

With this function, the gear ratio with gears can be calculated. The result is the gear ratio and the mechanical advantage (reciprocal of the gear ratio).

Usage Instructions

To calculate, enter the number of teeth for the input and output gear. Optionally, the transformed speed and torque can also be calculated. No fixed unit is specified here - the output matches what is entered.

Application Areas

Mechanical Engineering

Gearboxes, reducers, drive technology, industrial machines. Dimensioning of gear stages and drive chains.

Automotive Engineering

Transmissions, differential, rear axle ratio, bicycle chains. Adaptation of engine speed to driving speed.

Precision Mechanics

Clockwork, precision gears, robotics, measuring instruments. High-precision ratios for exact movements.

Understanding Gear Principles

Gears convert speed and torque, with power remaining (theoretically) constant:

Reduction (i > 1)

Input: High speed
Output: Low speed
Advantage: High torque
Example: Drill

Overdrive (i < 1)

Input: Low speed
Output: High speed
Advantage: High speed
Example: Bicycle large chainring

Direct drive (i = 1)

Input: = Output
Speed: Unchanged
Torque: Unchanged
Purpose: Direction change

Basic principle: P = ω × M = constant (without losses)
Higher speed → lower torque, and vice versa


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