Hopf Bifurcation of a Magnetic Bearing System with Time Delay

[+] Author and Article Information
J. C. Ji

School of Mechanical Engineering, The University of Adelaide, SA 5005, Australiajcji@mecheng.adelaide.edu.au

Colin H. Hansen

School of Mechanical Engineering, The University of Adelaide, SA 5005, Australiachansen@mecheng.adelaide.edu.au

J. Vib. Acoust 127(4), 362-369 (Dec 16, 2004) (8 pages) doi:10.1115/1.1924644 History: Received March 14, 2004; Revised December 16, 2004

This paper is concerned with a study of the influence of a time delay occurring in a PD feedback control on the dynamic stability of a rotor suspended by magnetic bearings. In the presence of geometric coordinate coupling and time delay, the equations of motion governing the response of the rotor are a set of two-degree-of-freedom nonlinear differential equations with time delay coupling in nonlinear terms. It is found that as the time delay increases beyond a critical value, the equilibrium position of the rotor motion becomes unstable and may bifurcate into two qualitatively different kinds of periodic motion. The resultant Hopf bifurcation is associated with two coincident pairs of complex conjugate eigenvalues crossing the imaginary axis. Based on the reduction of the infinite dimensional problem to the flow on a four-dimensional center manifold, the bifurcating periodic solutions are investigated using a perturbation method.

Copyright © 2005 by American Society of Mechanical Engineers
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Grahic Jump Location
Figure 1

Amplitude r1 of the possible periodic solutions as a function of the small perturbed parameters under p=2.0, q=0.5, α=0.25, (a) with α2=0.002 and (b) with α1=0.002

Grahic Jump Location
Figure 2

Variation of amplitude r1 of the possible periodic solutions with the small perturbed parameters under p=2.1, q=0.6, α=0.35, (a) with α2=0.002 and (b) with α1=0.002.

Grahic Jump Location
Figure 3

The time histories and phase portraits of the solutions of Eq. 5 in the neighborhood of the Hopf bifurcation of multiplicity two for the system parameters p=2.1, q=0.6, α=0.35, α1=0.005, α2=−0.005, (a) at τ=0.5 before Hopf bifurcation and (b) at τ=0.515 after Hopf bifurcation.



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