Limit cycle prediction for ballooning strings

This paper uses the Hopf bifurcation and center manifold theorems to predict the limit cycles of ballooning strings observed in many textile manufacturing processes. The steady state and linearized solutions of the non-linear governing equations are reviewed. The non-linear dynamic equations are discretized and projected to a finite-dimensional linear normal mode space, resulting in a set of non-linearly coupled but linearly uncoupled ordinary differential equations. The first two equations, corresponding to the lowest normal mode, are analyzed using the Hopf bifurcation theorem. The first Lyapunov coefficient is calculated to prove the existence of stable limit cycles for double-loop balloons with small string length. Analysis of the first two modes using the center manifold method verifies the effectiveness of the two-dimensional approximation. The bifurcation theorem, however, fails to apply to the large string length Hopf bifurcation point because the first Lyapunov coefficient is indeterminable. Numerical simulation of the non-linear two-dimensional equations agrees with experimental results quantitatively for small string length and qualitatively for large string length.