Transient Solutions of the Ring-Spinning Balloon Equations

Abstract

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.