# Gear Trains - Bicycle Gearing Calculator

A simple gear can change magnitude and line of action of an effort force.

The effort force is applied to the "driver" and the load is applied to the "follower" .

### Transmission - or Movement - Ratio

Transmission - or movement - ratio can be expressed as

_{ M }= n_{ D }/ n_{ F }= t_{ F }/ t_{ D }(1)

where

i_{ M }= movement ratio

n_{ D }= revolutions of driver (rpm)

n_{ F }= revolutions of follower (rpm)

t_{ F }= number of teeth on follower

t_{ D }= number of teeth on driver

When the same direction of rotation is required for the driver and the follower, an ** idler wheel ** is used.

### Gear Calculator

Calculate follower revolutions.

The ** movement ratio ** for a gear with an idler wheel can be expressed as

i_{ M }= n_{ D }/ n_{ F }= (t_{ I }/ t_{ D }) (t_{ F }/ t_{ I }) = t_{ F }/ t_{ D }(1b)

where

t_{ I }= number of teeth on idler

### Moment Ratio

The moment - or torque - ratio for a transmission can be expressed as

* i _{ T } = M _{ F } / M _{ L } = 1 / (i _{ M } μ) (2) *

* where *

* i _{ T } = moment ratio *

* μ = transmission efficiency *

* M _{ L } = output moment of load from follower (Nm, lb ft) *

* M _{ F } = input moment of force on driver (Nm, lb ft) *

### Transmission - Moment Ratio Calculator

Calculate follower output moment load.

The calculator is generic and can be used for SI and Imperial units - output units equals input units.

### Make 3D models of spurs and gears with the Engineering ToolBox SketchUp plugin

### Typical Gear Ratios

Typical gear ratios for different types of gear sets are indicated below.

Type of Gear set | Typical Gear Ratios | |
---|---|---|

Min | Max | |

Spur gear, external | 1 : 1 | 5 : 1 |

Spur gear, internal | 1.5 : 1 | 7 : 1 |

Helical gear, external | 1 : 1 | 10 : 1 |

Helical gear, internal | 1.5 : 1 | 10 : 1 |

Straight bevel gear | 1 : 1 | 8 : 1 |

Spiral bevel gear | 1 : 1 | 8 : 1 |

Epicyclic planetary gear | 3 : 1 | 12 : 1 |

Epicyclic star gear | 2 : 1 | 11 : 1 |

### Bicycle Gearing

The revolutions of a bicycle wheel when pedaling can be calculated by transforming * (2) * :

* n _{ F } = n _{ D } t _{ D } / t _{ F } (3) _{ } *

* where *

* n _{ F } = revolutions of the bicycle wheel (rpm) *

* n _{ D } = revolutions of the pedaling (rpm) *

* t _{ D } = number of teeth in the pedaling sprocket *

* t _{ F } = number of teeth in wheel sprocket *

The distance traveled by the wheel can be calculated by multiplying wheel revolutions with wheel circumference:

* l = c n _{ F } *

* = c n _{ D } t _{ D } / t _{ F } *

* = π d n _{ D } t _{ D } / t _{ F } (3b) *

* where *

* l = outer wheel traveled length or distance (m, in) *

* c = outer wheel circumference (m, in) *

* d = outer wheel diameter (m, in) *

### Example - Bicycle Gear

A mountain bike with * 26 inch * outer diameter wheels has a * 42/34/24T * chain-set and a 7-speed * 14-34 * cassette in the rear wheel.

The outer circumference of the wheel can be calculated as

* c = π (26 in) *

* = 81.7 in *

The distance traveled by the wheel in the lowest gear - for one pedaling revolution - using the smallest sprocket in the chainset * (24T) * and the largest sprocket in the cassette * (34T) * - can be calculated using * (3b) * as

* l = (1) (81.6 in) (24) / (34) *

* = 57.7 in *

The distance traveled by the wheel in the highest gear - for one pedaling revolution - using the largest sprocket in the chainset * (42T) * and the smallest sprocket in the cassette * (14T) * - can be calculated as

* l = (1) (81.6 in) (42) / (14) *

* = 244.8 in *

### Bicycle Gearing Calculator

Calculate distance traveled by wheel.

The calculator is generic and can be used for SI and Imperial units - output units equals input units.

* bicycle wheel outside diameter - d - (m, mm, in) *

* number of teeth in chain-set sprocket *

* number of teeth in cassette sprocket *

### Bicycle Gearing Calculator - Template

Make your own graphical Bicycle Gearing Calculator by using this Google Docs template!

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