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.

The Effect of Winding Angle on Unwinding Yarn

“…The winding angle is positive if during the unwinding the lift-off point moves in the direction of higher values of the coordinate z (this corresponds to the unwinding in the backward direction). The winding angle is negative if during the unwinding the lift-off point moves in the direction of lower values of z (the unwinding in the forward direction) [6][7][8][9]. We introduce a new auxiliary quantity V 1 , the velocity of the yarn at the lift-off point.…”

“…al. [6,7]. They showed that the entire time dependence can be shifted to moving boundary conditions.…”

“…Angle  is the winding angle of the yarn on the package. Its magnitude is the angle between the yarn at the unwinding point and the tangential direction at the package surface, while its sign determines whether the yarn is being withdrawn in the direction from the front end toward the rear end of the package ( > 0) or from the rear end toward the front end ( < 0) [6][7][8][9]. The yarn is parametrised with arc length s (s is therefore the length of yarn from the origin of the coordinate system to the given point on the yarn).…”

“…We have emphasized that coordinates of the point depend explicitly on both the time of observation t and on the arc length s, where the point is located at given time t. The vector t(s,t) is the vector tangential to the yarn at the given point; it equals the partial derivative of the radius vector r with respect to the arc-length s, that is t=r/s. The equation of motion of the yarn was derived in the previous paper [6,12]:…”

Matematical Model for Yarn Unwinding from Packages

“…Using elaborate numerical calculates they have shown that these two forces have only a small contribution to overall dynamics of the yarn. Recently Fraser et al have applied mathematical theory of perturbations to correctly eliminate the time dependance from equations of motion in stationary conditions [6]. They have show that the entire time dependance can be shifted to moving boundary conditions.…”