# Gear Trains

## A simple gear train is used to transmit rotary motion

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

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

### Transmission Ratio

Transmission or movement ratio can be expressed as

μ_{M}= n_{D}/ n_{F}= t_{F}/ t_{D}(1)

where

μ_{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 both the driver and the follower, an **idler wheel **is used.

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

μ_{M}= n_{D}/ n_{F}= (t_{I}/ t_{D}) (t_{F}/ t_{I}) = t_{F }/ t_{D}(2)

where

t_{I}= number of teeth on idler

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

Spurs and Gears - Use the awesome SketchUp to make 3D models of spurs and gears

### 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)* as:

*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 outer diameter *26 inch* wheels has a *42/34/24T* chainset 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

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