Euler-Eytelwein formula

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

Euler-Eytelwein: rope friction

If, for example, a flexible rope is wrapped around a bollard and pulled at one end of the rope, holding the other end with less force is sufficient to prevent the rope from sliding around the bollard. Because along the touched bollard circumference tangential static friction forces develop , which support holding.

The following applies to the ratio of pulling force ( ) and holding force ( ):

in which

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

When the rope slides on the round body, the coefficient of static friction must be replaced by the coefficient of sliding friction .

The formula can be derived from a local equilibrium of forces in the radial direction on an infinitesimal piece of rope, with the relationships for static and dynamic friction.

As can be easily seen, the forces increase very quickly with the angle of wrap. A steel cable, which is laid over a steel bollard to hold a ship, only needs 40% of the force to hold the force that wants to cause a movement when looped around . With three wraps, 5.9% is sufficient.

Applications and examples

literature

See also