Limit cycle prediction for ballooning strings

This paper uses the Hopf bifurcation and center manifold theorems to predic t the limit cycles of ballooning strings observed in many textile manufactu ring processes. The steady state and linearized solutions of the non-linear governing equations are reviewed. The non-linear dynamic equations are dis cretized and projected to a finite-dimensional linear normal mode space, re sulting in a set of non-linearly coupled but linearly uncoupled ordinary di fferential equations. The first two equations, corresponding to the lowest normal mode, are analyzed using the Hopf bifurcation theorem. The first Lya punov coefficient is calculated to prove the existence of stable limit cycl es fbr double-loop balloons with small string length. Analysis of the first two modes using the center manifold method verifies the effectiveness of t he two-dimensional approximation. The bifurcation theorem, however, fails t o apply to the large string length Hopf bifurcation point because the first Lyapunov coefficient is indeterminable. Numerical simulation of the non-li near two-dimensional equations agrees with experimental results quantitativ ely for small string length and qualitatively for large string length. (C) 1999 Elsevier Science Ltd. All rights reserved.