Phase Plane Analysis in Bundle Flow

Abstract

In a roll draft mechanism “draft wave” caused by irregular behavior of floating fibers has been known to occur, which influences directly the thickness variation in the output fiber bundle due to the process resonance in bundle flow. However, in spite of such an important notion of the draft wave, there have scarcely been studies that could explain the existence or the phenomenon since the first conceptual suggestion by W. L. Balls in 1928. In this paper, based on the governing equations (nonlinear PDEs system) for bundle flow, we tried to analyze the dynamic characteristics of the bundle flow in a roll draft system by applying both the linear stability and phase plane analysis, and thus to verify the occurrence of the draft wave. Therefore, the steady state stability and the eigenvalue problem were treated to find the stability diagram, and the solutions of the equations in transient state were obtained by developing a numerical program to illustrate the trajectory of output thickness in a phase plane. Results show that the linear stability can be applied to analyze the bundle flow dynamics. Nonlinearity of the bundle flow reveals that the system oscillates at the critical draw ratio harmonically, indicating the onset of a Hopf bifurcation. For draft ratios above a critical value, the fluctuation in amplitude is amplified, and a sustained oscillation can be achieved. We could confirm that the roller drafting operation has a “bifurcation” property with which the topological structure of phase portrait changes as the draft ratio and the model parameters are varied. When the process variables or model parameters exceed some critical values, fixed points are destroyed and oscillations with the limit cycle occur, which can be called “draft wave”. Particularly, as the value of a draft ratio increases, the phase portrait converges to a shell-shaped curve.