In the textile yarn manufacturing process of ring spinning, a loop of yarn rotates rapidly about a fixed axis. The surface generated by the rotating yarn loop is called a balloon. The solutions of the time-independent, nonlinear, yarn-balloon equations have been extensively investigated for a reference frame that rotates with constant angular velocity and are termed quasi-stationary solutions. A linear perturbation stability analysis of these solutions has shown that while single-loop balloons are stable, multiple-loop balloons are typically unstable. In this paper a numerical method for the calculation of transient solutions of the nonlinear time-dependent PDEs is developed, and the stability of representative quasi-stationary balloons subjected to a model velocity impulse is studied. The results of the linearized analysis are confirmed: Single-loop balloons remain stable while multiple-loop balloons typically collapse within only a few spindle revolutions.

1.
Ames, W. F., 1977, Numerical Methods for Partial Differential Equations, 2nd ed., Academic Press, New York.
2.
Batra
 
S. K.
,
Ghosh
 
T. K.
, and
Zeidman
 
M. I.
,
1989
a, “
An Integrated Approach to Dynamic Analysis of the Ring Spinning Process, Part I: Without Air Drag and Coriolis Acceleration
,”
Textile Research Journal
, Vol.
59
, pp.
309
317
.
3.
Batra
 
S. K.
,
Ghosh
 
T. K.
, and
Zeidman
 
M. I.
,
1989
b, “
An Integrated Approach to Dynamic Analysis of the Ring Spinning Process, Part II: Without Air Drag
,”
Textile Research Journal
, Vol.
59
, pp.
416
424
.
4.
de Barr, A. E., and Catling, H., 1965, The Principles and Theory of Ring Spinning, Manual of cotton spinning, Vol. 5, F. Charnley and P. W. Harrison, eds.), Butterworths Press, Manchester, UK.
5.
Fraser
 
W. B.
,
1993
a, “
On the Theory of Ring Spinning
,”
Philosophical Transactions of the Royal Society
, Vol.
A342
, pp.
439
468
.
6.
Fraser, W. B., 1993b, “The Dynamics of Ballooning Yarn in Ring Spinning and Two-for-one Twisting,” Proceedings of the IV-th Polish Conference on the Mechanics of Textile and Crane Machines, Politechnika Lodzka Fila w Bielsko-Biala, pp. 61–78.
7.
Fraser, W. B., Farnell, L. L., and Stump, D. M., 1994, “The Effects of Yarn Nonuniformity on the Stability of the Ring-Spinning Balloon,” Proceedings of the Royal Society, in press.
8.
Gray, B. F., and Roberts, M. J., 1988, “A Method for the Complete Qualitative Analysis of Two Coupled Ordinary Differential Equations Dependent on Three Parameters,” Proceedings of the Royal Society, Vol. A416, pp. 361–389.
9.
Klein, W., 1987, A Practical Guide to Ring Spinning, Manual of Textile Technology: Short-Staple Spinning Series, Vol. 4, ed. H. Sadler, ed., The Textile Institute, Manchester, U.K.
10.
Lisini
 
G. G.
,
Toni
 
P.
,
Quilghini
 
D.
, and
Campedelli
 
V. L. Di Gorgi
,
1992
, “
A Comparison of Stationary and Non-Stationary Mathematical Models for the Ring-Spinning Process
,”
Journal of the Textile Institute
, Vol.
83
, pp.
550
559
.
11.
Padfield
 
D. G.
,
1958
, “
The Motion and Tension in an Unwinding Balloon
,”
Proceedings of the Royal Society
, Vol.
A245
, pp.
382
407
.
12.
Press, W. H., Flannery, B. P., Teukolvsky, S. A., and Vetterling, W. T., 1986, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, U.K., pp. 588–600.
13.
Stump
 
D. M.
, and
Fraser
 
W. B.
,
1995
, “
Dynamic Bifurcations of the Ring-Spinning Balloons
,”
Mathematical Engineering for Industry
, Vol.
5
, pp.
161
185
.
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