Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations
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ORIGINAL PAPER
Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations N. A. Saeed
. Emad Mahrous
. Jan Awrejcewicz
Received: 18 April 2020 / Accepted: 20 August 2020 Ó Springer Nature B.V. 2020
Abstract The nonlinear dynamics of the six-pole rotor active magnetic bearings system is studied in this article for the first time. Two control configurations based on a proportional-derivative feedback current controller are proposed to mitigate lateral vibrations of the considered system. The first configuration is designed in such a way that only four electromagnetic poles are responsible for the system vibration control in the Y-direction, while all six poles control the lateral vibration in the X-direction. A second configuration is proposed in which the same four poles of the first configuration control the system vibration in the Ydirection, while the other two poles are responsible for controlling the system vibration in the X-direction. According to the suggested control methods, a mathematical model is derived that simulates lateral oscillations of the system. Using the perturbation
N. A. Saeed Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf 32952, Egypt E. Mahrous (&) Raytheon Chair for Systems Engineering (RCSE Chair), Advanced Manufacturing Institute, King Saud University, Riyadh, Saudi Arabia e-mail: [email protected] J. Awrejcewicz Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90924 Lodz, Poland
analysis, four autonomous and coupled first-order nonlinear differential equations that govern the system oscillation amplitudes and the corresponding phase angles in both X and Y-directions are extracted. Various bifurcation diagrams are obtained using rotor spinning-speed and disk eccentricity as a bifurcation control parameter. The conditions at which the system can whirl either forward or backward are investigated. Numerical validations for the obtained bifurcation diagrams are introduced which illustrate excellent agreement with the analytical results. Based on the acquired analytical and numerical results, it is found that the first control configuration has the best dynamical behavior for controlling the vibration of such systems. Keywords Six-pole rotor-AMB system Forward whirling motion Backward whirling motion Bifurcation diagram Quasi-periodic motion Chaotic motion Frequency spectrum Poincare´ map List of symbols x; x;_ x€
y; y;_ y€
Six-pole rotor active magnetic bearing system displacement, velocity, and acceleration in the X-direction Six-pole rotor active magnetic bearing system displacement, velocity, and acceleration in the Y-direction
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l1 ; l2
x1 ; x2
a1j ; j ¼ 1; 2; . . .; 8
a2j ; j ¼ 1; 2; . . .; 8
f
X
Six-pole rotor active magnetic bearing system linear damping coefficients in the X and Ydirection, re
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