Belt friction

The Euler Eytelwein formula, also called rope friction formula was developed by Leonhard Euler (1707-1783) and Johann Albert Eytelwein ( 1764-1848 ).

For example, if a bending ropes looped around one bollard and drawn at a wire end, the holding of the other end meets with a lower force in order to prevent slipping of the rope around the bollard. Because along the wetted perimeter bollards develop tangential friction forces that support the holding.

For the ratio of pulling force () and the holding force ( ) applies:

In which

  • Describes the angle of wrap (in radians), the rope is wrapped around the round object, and
  • The static friction coefficient.

When the rope slides on the round body, the static friction coefficient is replaced with the coefficient of sliding friction.

The formula can be derived from a local balance of forces in the radial direction at an infinitesimal piece of rope, with the relationships of adhesion or friction.

As can be easily seen, the forces increase very rapidly with the wrap. A steel cable which is placed over a bollard made ​​of steel, to keep a ship in a wrap only needs 40 % of the force to hold the force tending to cause movement. In three wraps already rich of 5.9 %.

Applications and Examples

In addition to the bollards where a ship with a few wraps can be safely moored, the Eytelwein equation provides a basis for estimating the moments that can transmit a low-speed belt drive as a function of its bias. With high-speed belt drives the centrifugal force must be considered to reduce the benefit. With V - grooved belt or the effect of friction is increased by the wedge effect. Already a loose end of the rope developed by gravity into a holding force when it is stuck at a certain length of a horizontally lying wrapped rod down or is rubbing on the ground in front of a vertical bollard. The stiffness of a steel cable, so its own counter-force against the forms of the noose around the bollard reduces the frictional effect more evident the lower the bollard diameter, so better rub soft bending ropes on bollards with larger diameters.

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