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 a force applies a tension to one end of a belt or rope wrapped around a curved surface, the frictional force between the two surfaces increases with the amount of wrap about the curved surface, and only part of that force (or resultant belt tension) is transmitted to the other end of the belt or rope. Belt friction can be modeled by the Belt friction equation.
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 system determine how many times the belt or rope must be wrapped around a curved surface to prevent it from slipping. Mountain climbers and sailing crews demonstrate a working knowledge of belt friction when accomplishing tasks with ropes, pulleys, bollards and capstans.
Equation.
The equation used to model belt friction is, assuming the belt has no mass and its material is a fixed composition:
where formula_2 is the tension of the pulling side, formula_3 is the tension of the resisting side, formula_4 is the static friction coefficient, which has no units, and formula_5 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.
The tension on the pulling side of the belt and pulley has the ability to increase exponentially 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 formula_6 is positive for all points of the rope curve:
formula_7, where formula_8 is a normal curvature of the rope curve.
2. Dragging coefficient of friction formula_9 and angle formula_10 are satisfying the following criteria for all points of the curve
formula_11
3. Limit values of the tangential forces:
The forces at both ends of the rope formula_12 and formula_13 are satisfying the following inequality
formula_14
with formula_15,
�where formula_16is a geodesic curvature of the rope curve, formula_17 is a curvature of a rope curve, formula_18is a coefficient of friction in the tangential direction.
If formula_19 then formula_20.
This generalization has been obtained by Konyukhov A.,
Friction coefficient.
There are certain factors that help determine the value of the friction coefficient. These determining factors are:
Applications.
An understanding of belt friction is essential for sailing crews and mountain climbers. 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.