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Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations

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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 \( Y \)-direction, 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 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.

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Abbreviations

\( x, \dot{x}, \ddot{x} \) :

Six-pole rotor active magnetic bearing system displacement, velocity, and acceleration in the \( X \)-direction

\( y, \dot{y}, \ddot{y} \) :

Six-pole rotor active magnetic bearing system displacement, velocity, and acceleration in the \( Y \)-direction

\( \mu_{1} , \mu_{2} \) :

Six-pole rotor active magnetic bearing system linear damping coefficients in the \( X \) and \( Y \)-direction, respectively

\( \omega_{1} , \omega_{2} \) :

Six-pole rotor active magnetic bearing system linear natural frequencies in the \( X \) and \( Y \)-direction, respectively

\( \alpha_{1j} , j = 1,2, \ldots ,8 \) :

Cubic nonlinearity coefficients of the six-pole rotor active magnetic bearing system in the \( X \)-direction

\( \alpha_{2j} , j = 1,2, \ldots ,8 \) :

Cubic nonlinearity coefficients of the six-pole rotor active magnetic bearing system in the \( Y \)-direction

\( f \) :

Dimensionless disk eccentricity of the six-pole rotor active magnetic bearing system

\( \varOmega \) :

Dimensionless disk spinning-speed of the six-pole rotor active magnetic bearing system

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Acknowledgements

This work has been supported by the National Science Centre, Poland, under the grant OPUS 14 No. 2017/27/B/ST8/01330. The authors are grateful to the Raytheon Chair for Systems Engineering for funding.

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Authors and Affiliations

  1. Department of Physics and Engineering Mathematics, Faculty of Electronic Engineering, Menoufia University, Menouf, 32952, Egypt

    N. A. Saeed

  2. Raytheon Chair for Systems Engineering (RCSE Chair), Advanced Manufacturing Institute, King Saud University, Riyadh, Saudi Arabia

    Emad Mahrous

  3. Department of Automation, Biomechanics, and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 90924, Lodz, Poland

    Jan Awrejcewicz

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  1. N. A. Saeed
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  2. Emad Mahrous
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Correspondence to Emad Mahrous.

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Saeed, N.A., Mahrous, E. & Awrejcewicz, J. Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations. Nonlinear Dyn 101, 2299–2323 (2020). https://doi.org/10.1007/s11071-020-05911-0

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