Expressions for Strand and Yarn Paths in Braided Ropes

Abstract

This paper attempts to define a mathematical expression for the strand and yarn paths in a braided rope. The theoretical development is based on three assumptions as follows:
The projectional path of the center of a strand on a cross-sectional plane to the rope axis is an ellipse, namely the strand axis laid on the surface of an elliptic cylinder.
The relation between the length of a strand axis, l_s, and the corresponding eccentric angle of the ellipse, t, is given as d_sl/dt=c(constant).
The relation between the rotated angle of a yarn around the strand axis, φ, and the eccentric angle t is written as dφ/dt=ρ(constant).
Under the above assumptions, the positions of strand and yarn in the coordinate system with its center in the rope axis are introduced as mathematical expressions.
To experimentally confirm the applicability of the expressions to an actual rope, the positions of the strand and yarn in a thin cross section of a commercial polyethylene 8-strand braided rope solidified by a polyester resin were observed using a profile projector. The results of the observed positions are in good agreement with the calculated ones.