# Belt friction

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Belt friction is a term describing the friction forces between a belt and a surface, such as a belt wrapped around a bollard. When one end of the belt is being pulled only part of this force is transmitted to the other end wrapped about a surface. The friction force increases with the amount of wrap about a surface and makes it so the tension in the belt can be different at both ends of the belt. Belt friction can be modeled by the Belt friction equation.[1]

In practice, the theoretical tension acting on the belt or rope calculated by the belt friction equation can be compared to the maximum tension the belt can support. This helps a designer of such a rig to know how many times the belt or rope must be wrapped around the pulley to prevent it from slipping. Mountain climbers and sailing crews demonstrate a standard knowledge of belt friction when accomplishing basic tasks.

## Equation

Main article: Capstan equation

The equation used to model belt friction is, assuming the belt has no mass and its material is a fixed composition:[2]

${\displaystyle T_{2}=T_{1}e^{\mu _{s}\beta }\,}$

where ${\displaystyle T_{2}}$ is the tension of the pulling side, ${\displaystyle T_{1}}$ is the tension of the resisting side, ${\displaystyle \mu _{s}}$ is the static friction coefficient, which has no units, and ${\displaystyle \beta }$ is the angle, in radians, formed by the first and last spots the belt touches the pulley, with the vertex at the center of the pulley.[3]

The tension on the pulling side of the belt and pulley has the ability to increase exponentially[1] if the magnitude of the belt angle increases (e.g. it is wrapped around the pulley segment numerous times).

## Friction coefficient

There are certain factors that help determine the value of the friction coefficient. These determining factors are:[4]

• Belting material used – The age of the material also plays a part, where worn out and older material may become more rough or smoother, changing the sliding friction.
• Construction of the drive-pulley system – This involves strength and stability of the material used, like the pulley, and how greatly it will oppose the motion of the belt or rope.
• Conditions under which the belt and pulleys are operating – The friction between the belt and pulley may decrease substantially if the belt happens to be muddy or wet, as it may act as a lubricant between the surfaces. This also applies to extremely dry or warm conditions which will evaporate any water naturally found in the belt, nominally making friction greater.
• Overall design of the setup – The setup involves the initial conditions of the construction, such as the angle which the belt is wrapped around and geometry of the belt and pulley system.

## Applications

An understanding of belt friction is essential for sailing crews and mountain climbers.[1] Their professions require being able to understand the amount of weight a rope with a certain tension capacity can hold versus the amount of wraps around a pulley. Too many revolutions around a pulley make it inefficient to retract or release rope, and too few may cause the rope to slip. Misjudging the ability of a rope and capstan system to maintain the proper frictional forces may lead to failure and injury.

## References

1. ^ a b c Attaway, Stephen W. (1999). The Mechanics of Friction in Rope Rescue (PDF). International Technical Rescue Symposium. Retrieved February 1, 2010.
2. ^ Mann, Herman (May 5, 2005). "Belt Friction". Ruhr-Universität. Retrieved 2010-02-01.
3. ^ Chandoo. "Couloumb Belt Friction". Missouri University of Science and Technology. Retrieved 2010-02-01.
4. ^ "Belt Tension Theory". CKIT – The Bulk Materials Handling Knowledge Base. Retrieved 2010-02-01.