# Belt Transmissions - Speed and Length of Belts

## Calculate length and speed of belt and belt gearing.

Belts (or chains) are used to transfer power (or convert torque) through rotational motion from one shaft to an other. Belt driven fans are common in heating, ventilation, air-conditioning and cooling (HVAC) systems. Chain are common in many transmission systems like bicycles and other.

**friction belts**- transfer the power through friction between the pulley and the belt**synchronous belts**- transfer the power through a mechanical linkage between the teeth in the belt and the grooves in the pulley

### Belt Length

The length of the belt can be calculated as

l_{b}= (d_{m}π / 2) + (d_{f}π / 2) + (2 l_{fm}) + ((d_{f}- d_{m})^{2 }/ (4 l_{fm})) (1)

where

l_{b}= length of belt (mm, inches)

d_{f}= pulley (sheave) diameter fan (mm, inches)

d_{m}= pulley (sheave) diameter motor (mm, inches)

π = 3.14..

l_{fm}= center to center distance of fan and motor pulleys (mm, inches)

### Belt Velocity

The velocity at which a belt travels may be expressed as

v = π d_{m}n_{m}/ 12 (2)

where

v = velocity of belt (ft/min)

n_{m}= revolutions motor (rpm)

- or alternatively in metric units:

*v _{b} = π d_{m} n_{m} / 60 (2b)*

*where *

*v _{b} = velocity of belt (m/s)*

*d _{mb} = diameter motor pulley (m)*

Belt Velocity (ft/min) | |||||||
---|---|---|---|---|---|---|---|

Pitch Diameter of Pulley (in) | |||||||

Revolutions Pulley - n _{} -(rpm) | |||||||

850 | 1050 | 1075 | 1140 | 1550 | 1725 | 3450 | |

1 | 223 | 275 | 281 | 298 | 406 | 452 | 903 |

2 | 445 | 550 | 563 | 597 | 812 | 903 | 1806 |

3 | 668 | 825 | 844 | 895 | 1217 | 1355 | 2710 |

4 | 890 | 1100 | 1126 | 1194 | 1623 | 1806 | 3613 |

5 | 1113 | 1374 | 1407 | 1492 | 2029 | 2258 | 4516 |

6 | 1335 | 1649 | 1689 | 1791 | 2435 | 2710 | 5419 |

7 | 1558 | 1924 | 1970 | 2089 | 2841 | 3161 | 6322 |

8 | 1780 | 2199 | 2251 | 2388 | 3246 | 3613 | 7226 |

9 | 2003 | 2474 | 2533 | 2686 | 3652 | 4064 | 8129 |

10 | 2225 | 2749 | 2814 | 2985 | 4058 | 4516 | 9032 |

### Belt Gearing

The relationship between the rotational speed of the motor and the fan and the disc diameter can be expressed as

d_{f}n_{f}= d_{m}n_{m }(3)

or

n_{f}= n_{m }d_{m}/d_{f}

#### Belt Gearing Calculator

*n _{m} - speed driver (rpm)*

* d _{m} - diameter driver (mm, in)*

* d _{f} - diameter follower (mm, in)*

### Horsepower

If belt tension and belt velocity are known - horsepower transferred can be calculated as

P_{hp}=Fv_{b}_{b}/ 33000 (4)

where

P_{hp}= power (hp)

F_{b}= belt tension (lb_{f})

v_{b}= velocity of belt (ft/min)

If torque and revolution per minute are known - horsepower transferred can be calculated as

P_{hp}= T n / 5252 (4)

where

P_{hp}= power (hp)

T = torque (ft lb_{f})

n = revolutions per minute (rpm)

*Speed Ratio*

*Speed ratio can be calculated as *

SR = n_{f}/ n_{s}(5)

where

SR = speed ratio

n_{f}= revolutions per minute - fastest machine

n_{s}= revolutions per minute - slowest machine