# Belt friction

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

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).

## Generalization for a rope lying on an arbitrary orthotropic surface

If a rope is laying in equilibrium under tangential forces on a rough orthotropic surface then three following conditions (all of them) are satisfied:

1. No separation – normal reaction ${\displaystyle N}$ is positive for all points of the rope curve:

${\displaystyle N=-k_{n}T>0}$, where ${\displaystyle k_{n}}$ is a normal curvature of the rope curve.

2. Dragging coefficient of friction ${\displaystyle \mu _{g}}$ and angle ${\displaystyle \alpha }$ are satisfying the following criteria for all points of the curve

${\displaystyle -\mu _{g}<\tan \alpha <+\mu _{g}}$

3. Limit values of the tangential forces:

The forces at both ends of the rope ${\displaystyle T}$ and ${\displaystyle T_{0}}$ are satisfying the following inequality

${\displaystyle T_{0}e^{-\int _{s}\omega ds}\leq T\leq T_{0}e^{\int _{s}\omega ds}}$

with ${\displaystyle \omega =\mu _{\tau }{\sqrt {k_{n}^{2}-{\frac {k_{g}^{2}}{\mu _{g}^{2}}}}}=\mu _{\tau }k{\sqrt {\cos ^{2}\alpha -{\frac {\sin ^{2}\alpha }{\mu _{g}^{2}}}}}}$,

�where ${\displaystyle k_{g}}$is a geodesic curvature of the rope curve, ${\displaystyle k}$ is a curvature of a rope curve, ${\displaystyle \mu _{\tau }}$is a coefficient of friction in the tangential direction.

If ${\displaystyle \omega =const}$ then ${\displaystyle T_{0}e^{-\mu _{\tau }ks\,{\sqrt {\cos ^{2}\alpha -{\frac {\sin ^{2}\alpha }{\mu _{g}^{2}}}}}}\leq T\leq T_{0}e^{\mu _{\tau }ks\,{\sqrt {\cos ^{2}\alpha -{\frac {\sin ^{2}\alpha }{\mu _{g}^{2}}}}}}}$.

This generalization has been obtained by Konyukhov A.,[4][5]

## Friction coefficient

There are certain factors that help determine the value of the friction coefficient. These determining factors are:[6]

• 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.