# Gear Train Speed Ratio Calculator

Calculate the overall ratio and output speed of a multi-stage gear train. i_total = i₁·i₂·…·iₙ (each i = z₂/z₁). Enter tooth pairs per stage and the input speed to get the final output speed.

> Canonical page: https://elysiatools.com/en/tools/gear-train-speed-ratio

- **Category:** Math & Numbers

- **Keywords:** gear train, compound gear, transmission ratio, multi-stage gear, speed ratio, gearbox, mechanical engineering

## Overview

The Gear Train Speed Ratio Calculator helps mechanical engineers, students, and hobbyists quickly determine the overall transmission ratio and final output speed of multi-stage compound gear trains. By entering the driver and driven tooth counts for each stage along with the input shaft speed, you can instantly compute the cumulative reduction or overdrive ratio and the resulting RPM.

## Inputs

- **Tooth Pairs (one stage per line, 'driver,driven')** (textarea): Each line is one stage: driver teeth, then driven teeth, separated by a comma. Separate stages with new lines or semicolons.
- **Input Shaft Speed (rpm)** (number): Optional. If provided, the final output speed is computed.
- **Decimal Places** (number)

## When to use

- When designing custom gearboxes or transmission systems and needing to verify the final output speed based on motor RPM.
- When analyzing compound gear trains to determine the cumulative mechanical advantage and overall reduction ratio.
- When prototyping robotics projects or RC vehicles to select the correct gear tooth counts for target speed requirements.

## How it works

- Enter the tooth counts for each gear stage on a new line, formatted as driver teeth followed by driven teeth separated by a comma.
- Optionally input the rotational speed of the input shaft in revolutions per minute (RPM).
- Specify the desired decimal precision for the calculations.
- Click calculate to compute the individual stage ratios, the total compound ratio, and the final output speed.

## Use cases

- Calculating the output speed of a multi-stage reduction gearbox powered by an electric motor.
- Determining the overall gear ratio of a compound gear train in a mechanical clock or watch mechanism.
- Designing custom transmission setups for robotics drivetrains to match motor torque and speed specifications.

## Frequently asked questions

### How do I format the tooth pairs for multiple stages?

Enter each stage on a new line or separate them with semicolons, using the format 'driver,driven' (for example, '20,60').

### What formula is used to calculate the total gear ratio?

The total ratio is the product of the individual stage ratios, where each stage ratio is the number of driven teeth divided by the driver teeth.

### Is the input speed required to calculate the gear ratio?

No, the input speed is optional. If omitted, the calculator will still compute the overall transmission ratio.

### Can this calculator handle planetary or epicyclic gear systems?

No, this calculator is designed specifically for standard external or internal spur and helical gear trains in compound arrangements.

### How does the calculator handle speed reduction versus overdrive?

A total ratio greater than 1 indicates speed reduction (torque multiplication), while a ratio less than 1 indicates overdrive (speed multiplication).

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