Effect of yarn non-uniformity on the stability of the ring-spinning balloon

This paper extends the results of W. B. Fraser and D. M. Stump on the stability of a ballooning yarn of uniform mass linear density to allow for non-uniformities in the yarn. A two-variable perturbation expansion procedure (two-timing ) is used to show that the effect of a slub (defined as a thick place or lump in a thread) on the stability of the ring-spinning balloon can be analysed as a sequence of quasistationary balloons for yarn of variable mass linear density as the slub travels slowly through the rapidly rotating balloon. The slub is modelled by the addition of a mass linear density term, in the shape of a gaussian or cosine distribution function, to the uniform mass linear density. Computational results are given for a ring spindle and yarn combination with representative geometry, air drag, and frictional parameters. These results show that even a slub that in practice would be considered quite small can have a significant effect on the stability of the yarn balloon. The results in this paper show how such effects can be quantified.