# Gear Trains - Bicycle Gearing Calculator

## Gear train transmissions - bicycle gearing.

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

i_{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.

* n _{D} - revolutions of driver (rpm)*

* t _{D} - no. of teeth on driver *

* t _{F} - no. of teeth on follower*

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.

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

* n _{D} - revolutions driver (rpm)*

* n _{F} - revolutions follower (rpm)*

* μ - transmission efficiency*

### 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!