Belt friction is a term describing the friction forces between a
belt
Belt may refer to:
Apparel
* Belt (clothing), a leather or fabric band worn around the waist
* Championship belt, a type of trophy used primarily in combat sports
* Colored belts, such as a black belt or red belt, worn by martial arts practit ...
and a surface, such as a belt wrapped around a
bollard. When a force applies a
tension
Tension may refer to:
Science
* Psychological stress
* Tension (physics), a force related to the stretching of an object (the opposite of compression)
* Tension (geology), a stress which stretches rocks in two opposite directions
* Voltage or el ...
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
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different eleme ...
and its material is a fixed composition:
:
where
is the tension of the pulling side,
is the tension of the resisting side,
is the
static friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
coefficient, which has no units, and
is the angle, in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s, 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
Exponential may refer to any of several mathematical topics related to exponentiation, including:
*Exponential function, also:
**Matrix exponential, the matrix analogue to the above
*Exponential decay, decrease at a rate proportional to value
*Expo ...
[ 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 is positive for all points of the rope curve:
, where is a normal curvature of the rope curve.
2. Dragging coefficient of friction and angle are satisfying the following criteria for all points of the curve
3. Limit values of the tangential forces:
The forces at both ends of the rope and are satisfying the following inequality
with ,
where is a geodesic curvature of the rope curve, is a curvature of a rope curve, is a coefficient of friction in the tangential direction.
If then .
This generalization has been obtained by Konyukhov A.,
Friction coefficient
There are certain factors that help determine the value of the friction coefficient
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of t ...
. These determining factors are:
* 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.[ 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.
]
See also
* Capstan equation
*Frictional contact mechanics
Contact mechanics is the study of the deformation of solids that touch each other at one or more points. This can be divided into compressive and adhesive forces in the direction perpendicular to the interface, and frictional forces in the tangen ...
References
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Mechanics