25—The Geometry of Multi-Ply Yarns

Abstract

The theory of the geometry of multi-ply yarns developed in this paper is based on the assumption that the individual filament in the ply has the form of a “doubly-wound” helix, i.e., of a helix wound about a helical axis. The analysis yields an expression for the length of filament in terms of the parameters defining the system in the form of an integral. The properties of this integral are examined in detail, and it is shown that it may be evaluated by introducing an approximation which is valid under all practical conditions. The relation between the single yarn twist (before cording) and the twist in the ply (after cording) is dealt with, and is shown to depend on the change in length of ply axis on cording ; an explicit expression for the latter quantity is derived. Properties such as yarn retraction, cord retraction, etc., are calculated as average properties for an assembly of coaxial helices, the filaments themselves being taken to be inextensible. Extensive numerical data calculated on the basis of the theory are presented for cord retraction and outside filament angles, as functions of single yarn twist and cord twist, in the case of 2-ply cord. Experimental data made available to the author and included in the paper show good agreement between calculated and observed yarn and cord retractions. Measurements of filament angles, however, show considerable discrepancies from the theoretical predictions.