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TECHNICAL PAPERS

# Hopf Bifurcation of a Magnetic Bearing System with Time Delay

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

Colin H. Hansen

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

## Abstract

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.

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## Figures

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

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.

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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