# MATHEMATICAL APPROACH TO CURVE LINE OF FREE BENT ROPES

**Summary **

The rope curve line of a tensioned rope can be described by means of the catenary curve. Opposed to that, the curved line of a free bent rope cannot be described by an analytical function. Practical applications of free bending are for example at tail ropes at the bottom of shaft in rope drives with traction sheaves. The question whether the maximum diameter of rope loop is small enough for the diameter of the shaft is highly interesting. In [1] a method was presented to calculate the curved line of free bent ropes numerically by help of energy methods. An analytical description of rope curve line would be very helpful. Beginning with the structure of a rope curve line of tensioned rope (catenary curve) and considering the influence of bending stiffness, the structure of an analytical equation for the curve line of a free bent rope will be developed. The main focus of this paper is to develop and to describe the structure of such an analytical equation. To get a first idea about the values of the constants in that analytical equation a few test results were evaluated. But these equations consider the static rope behavior only. Due to dynamic effects in the rope while running through the loop at the bottom of a shaft, pendulousness of the tail rope occurs.

Rope Curvature, Catenary, Bending Stiffness, Tail Ropes, Free Bending

**Author: U. Briem**