On unwinding yarn from a cylindrical package

Abstract

The over-end unwinding of yarn from a stationary helically wound cylindrical package is considered. The motion of the yarn between the unwind point (where it first starts to slip across the package surface before flying into the unwinding balloon) and the guide eye located on the package axis is analysed. The motion is periodic as the unwind point moves backwards and forwards along the length of the package surface. In 1958 D. G. Padfield argued that, provided the helix angle is small, the time derivative terms in the equations of motion can be neglected and the problem can be reduced to a stationary (relative to rotating axes) balloon problem subject to a modified boundary condition at the unwind point. The problem of yarn slipping across the package surface has also been investigated by D. G. Padfield and by H. V. Booth. In the present paper a regular perturbation expansion is used to provide a theoretical framework for Padfield’s ideas and to remove the time dependence from the zero order equations of motion. To this order of approximation the time dependence appears in the ‘moving’ boundary condition at the unwind point. A new derivation of this boundary condition is given and a set of continuity conditions between the yarn slipping on the package and the yarn in the balloon is used to splice the two solutions together so that the package can be unwound through a complete period of the unwinding cycle.