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Research Papers

# Modeling and Nonlinear Vibration Analysis of a Rigid Rotor System Supported by the Magnetic Bearing (Effects of Delays of Both Electric Current and Magnetic Flux)

[+] Author and Article Information
Tsuyoshi Inoue

Department of Mechanical Science and Engineering, School of Engineering, Nagoya University, Nagoya 464-8603, Japaninoue@nuem.nagoya-u.ac.jp

Yasuhiko Sugawara

Daihatsu Motor Co., Ltd., 1-1 Daihatsu-cho, Ikeda, Osaka 563-8651, Japan

Motoki Sugiyama

Tokai Rika Co., Ltd., 3-260 Toyota, Oguchi-cho, Niwa-gun, Aichi 480-0195, Japan

J. Appl. Mech 77(1), 011005 (Sep 24, 2009) (10 pages) doi:10.1115/1.3172139 History: Received September 08, 2008; Revised April 28, 2009; Published September 24, 2009

## Abstract

Active magnetic bearing (AMB) becomes widely used in various kinds of rotating machinery. However, as the magnetic force is nonlinear, nonlinear phenomena may occur when the rotating speed becomes high and delays of electric current or magnetic flux in the AMB relatively increase. In this paper, the magnetic force in the AMB is modeled by considering both the second-order delay of the electric current and the first-order delay of the magnetic flux. The magnetic flux in the AMB is represented by a power series function of the electric current and shaft displacement, and its appropriate representation for AMB is discussed. Furthermore, by using them, the nonlinear theoretical analysis of the rigid rotor system supported by the AMB is demonstrated. The effects of the delays and other AMB parameters on the nonlinear phenomena are clarified theoretically, and they are confirmed experimentally.

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## Figures

Figure 1

Model of the AMB and the rotor system

Figure 2

Block diagram of the amplifier

Figure 3

Magnetic flux density Bx1 (piecewise function expression)

Figure 4

Comparison between two expressions (Bxp): (a) piecewise model and (b) seventh-order power series model

Figure 5

Comparison between two expressions (Bxm): (a) piecewise model and (b) seventh-order power series model

Figure 6

Time histories at ω=1.18 (close to the critical speed). The parameters correspond to those of Fig. 7.

Figure 7

Resonance curve: solid line: stable solution and open circle: steady state oscillation. The dimensional values of parameters kp, kd, and I0 are shown in Table 2.

Figure 8

Influence of feedback coefficient kp: solid line: stable solution, dashed line: unstable solution for the case with a positive real eigenvalue, dashed-dotted line: unstable solution for the case with a pair of conjugate complex eigenvalues with a positive real part (due to Hopf Bifurcation), solid circle: maximum and minimum amplitudes of almost periodic motion, and open circle: steady state oscillation. The dimensional values of parameters kp, kd, and I0 are shown in Table 2.

Figure 9

Time histories at point C

Figure 10

Influence of feedback coefficient kd: solid line: stable solution, dashed line: unstable solution for the case with a positive real eigenvalue, dashed-dotted line: unstable solution for the case with a pair of conjugate complex eigenvalues with a positive real part (due to Hopf Bifurcation), solid circle: maximum and minimum amplitudes almost periodic motion, and open circle: amplitude of steady state oscillation. The dimensional values of parameters kp, kd, and I0 are shown in Table 2.

Figure 11

Influence of bias current I0: solid line: stable solution, dashed line: unstable solution, solid circle: maximum and minimum amplitudes almost periodic motion, and open circle: steady state oscillation. The dimensional values of parameters kp, kd, and I0 are shown in Table 2.

Figure 12

Influence of the electric current delay: (a) influence of α1 and (b) influence of α2: solid line: stable solution, dashed line: unstable solution for the case with a positive real eigenvalue, dashed-dotted line: unstable solution for the case with a pair of conjugate complex eigenvalues with a positive real part (due to Hopf Bifurcation), solid circle: maximum and minimum amplitudes almost periodic motion, and open circle: steady state oscillation. The values α1=3.180 and α2=0.161 are the dimensionless value of the dimensional value shown in Table 1. The dimensional values of parameters kp, kd, and I0 are shown in Table 2.

Figure 13

Influence of the Magnetic Flux Delay β: solid line: stable solution, dashed line: unstable solution for the case with a positive real eigenvalue, dashed-dotted line: unstable solution for the case with a pair of conjugate complex eigenvalues with a positive real part (due to Hopf Bifurcation), solid circle: maximum and minimum amplitudes almost periodic motion, and open circle: steady state oscillation. The value β0=0.307 is the dimensionless value of the dimensional value β0=3.66×10−3 s shown in Table 1. The dimensional values of parameters kp, kd, and I0 are shown in Table 2.

Figure 14

Experimental model

Figure 15

Experimental results (influence of feedback coefficient kp)

Figure 16

Experimental results (influence of feedback coefficient kd)

Figure 17

Experimental results (influence of bias current I0)

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