Performance enhancement of a 12-pole AMB model via robust control approach

Hany Samih Bauomy; Hany Samih Bauomy

doi:10.3934/math.2026502

AIMS Mathematics

2026, Volume 11, Issue 4: 12233-12286. doi: 10.3934/math.2026502

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Performance enhancement of a 12-pole AMB model via robust control approach

Hany Samih Bauomy ^,

Department of Mathematics, College of Science and Humanities in Alkharj, Prince Sattam Bin Abdulaziz University, Alkharj 11942, Saudi Arabia

Received: 06 February 2026 Revised: 21 March 2026 Accepted: 13 April 2026 Published: 30 April 2026
MSC : 70K20, 74H45

In this study, I investigated the energetic behavior of a twelve-pole active magnetic bearing (AMB) framework with a nonlinear proportional derivative cubic velocity feedback (NPDCVF) controller in the presence of mixed excitations and primary resonance ($ \varOmega \cong \, {\omega _1}, \, \, \varOmega \cong \, {\omega _2} $). The controller combines classic proportional-derivative (PD) control with nonlinear cubic velocity feedback to improve stability, reduce rotor oscillations, and increase robustness. A detailed mathematical framework of the 12-pole AMB system was developed, accounting for magnetic force nonlinearity, dynamic interactions between poles, and the effects of rotor eccentricity. The motion equations (ME) were investigated using the multiple time scales approach (MTSA), and the approximation solutions (AS) were numerically validated with the fourth-order Runge-Kutta (4RK) method. Simulations were performed to compare five controllers: PD, integral resonant controller (IRC), positive position feedback (PPF), nonlinear integral positive position feedback (NIPPF), and the proposed NPDCVF scheme. MATLAB 18.2 numerical simulations (4RK) were employed to analyze time-history responses, the effects of system parameters, and the performance of controllers. The time-domain results showed that the NPDCVF controller delivers the quickest vibration reduction, the least overshoot, and increased robustness to disturbances and parameter changes. Time-domain, frequency-response, and phase-plane analyses confirmed wider stability margins and increased damping effectiveness. Additional nonlinear dynamical assessments, such as bifurcation charts, frequency response curves, and stable and unstable zones, showed that nonlinear oscillations have been successfully reduced and the AMB system has stabilized reliably.
- vibration suppression,
- rotor dynamics,
- AMB,
- perturbation,
- stability enhancement
Citation: Hany Samih Bauomy. Performance enhancement of a 12-pole AMB model via robust control approach[J]. AIMS Mathematics, 2026, 11(4): 12233-12286. doi: 10.3934/math.2026502

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Abstract

In this study, I investigated the energetic behavior of a twelve-pole active magnetic bearing (AMB) framework with a nonlinear proportional derivative cubic velocity feedback (NPDCVF) controller in the presence of mixed excitations and primary resonance ($ \varOmega \cong \, {\omega _1}, \, \, \varOmega \cong \, {\omega _2} $). The controller combines classic proportional-derivative (PD) control with nonlinear cubic velocity feedback to improve stability, reduce rotor oscillations, and increase robustness. A detailed mathematical framework of the 12-pole AMB system was developed, accounting for magnetic force nonlinearity, dynamic interactions between poles, and the effects of rotor eccentricity. The motion equations (ME) were investigated using the multiple time scales approach (MTSA), and the approximation solutions (AS) were numerically validated with the fourth-order Runge-Kutta (4RK) method. Simulations were performed to compare five controllers: PD, integral resonant controller (IRC), positive position feedback (PPF), nonlinear integral positive position feedback (NIPPF), and the proposed NPDCVF scheme. MATLAB 18.2 numerical simulations (4RK) were employed to analyze time-history responses, the effects of system parameters, and the performance of controllers. The time-domain results showed that the NPDCVF controller delivers the quickest vibration reduction, the least overshoot, and increased robustness to disturbances and parameter changes. Time-domain, frequency-response, and phase-plane analyses confirmed wider stability margins and increased damping effectiveness. Additional nonlinear dynamical assessments, such as bifurcation charts, frequency response curves, and stable and unstable zones, showed that nonlinear oscillations have been successfully reduced and the AMB system has stabilized reliably.

References

[1]	G. Schweitzer, E. H. Maslen, Magnetic bearings: Theory, design, and application to rotating machinery, Berlin: Springer, 2009.https://doi.org/10.1007/978-3-642-00497-1
[2]	G. Schweitzer, H. Bleuler, A. Traxler, Active magnetic bearings: Basics, properties, and applications of active magnetic bearings, Zurich: vdf Hochschulverlag AG an der ETH, 1994.
[3]	H. Lv, H. Geng, J. Zhou, T. Du, H. Li, Structure design and optimization of thrust magnetic for the high-Speed motor, 2017 IEEE International Conference on Mechatronics and Automation (ICMA), 2017,805–809.https://doi.org/10.1109/ICMA.2017.8015919
[4]	R. Q. Wu, W. Zhang, M. H. Yao, Nonlinear dynamics near resonances of a rotor-active magnetic bearings system with 16-pole legs and time varying stiffness, Mech. Syst. Signal Process., 100 (2018), 113–134.https://doi.org/10.1016/j.ymssp.2017.07.033 doi: 10.1016/j.ymssp.2017.07.033
[5]	R. Bakri, R. Nabergoj, F. Tondl, F. Verhulst, Parametric excitation in non-linear dynamics, Int. J. Non-Linear Mech., 39 (2004), 311–329.https://doi.org/10.1016/S0020-7462(02)00190-7 doi: 10.1016/S0020-7462(02)00190-7
[6]	H. Abdelhafez, Resonance of a nonlinear forced system with two-frequency parametric and self-excitations, Math. Comput. Simul., 66 (2004), 69–83.https://doi.org/10.1016/j.matcom.2004.03.002 doi: 10.1016/j.matcom.2004.03.002
[7]	M. Kamel, H. S. Bauomy, Nonlinear behavior of a rotor-AMB system under multi-parametric excitations, Meccanica, 45 (2010), 7–22.https://doi.org/10.1007/s11012-009-9213-3 doi: 10.1007/s11012-009-9213-3
[8]	N. A. Saeed, W. A. El-Ganaini, Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system, Appl. Math. Model., 44 (2017), 523–539.https://doi.org/10.1016/j.apm.2017.02.019 doi: 10.1016/j.apm.2017.02.019
[9]	K. Akash, K. P. Lijesh, V. Chittlangia, H. Hirani, Design and implementation of adaptive PID controller for active magnetic bearings, International Conference on Advances in Tribology, ICAT14, 2014.
[10]	N. A. Saeed, M. S. Mohamed, S. K. Elagan, J. Awrejcewicz, Integral resonant controller to suppress the nonlinear oscillations of a two-degree-of-freedom rotor active magnetic bearing system, Processes, 10 (2022), 271.https://doi.org/10.3390/pr10020271 doi: 10.3390/pr10020271
[11]	A. Kandil, Y. S. Hamed, Tuned positive position feedback control of an active magnetic bearings system with 16 poles and constant stiffness, IEEE Access, 9 (2021), 73857–73872.https://doi.org/10.1109/ACCESS.2021.3080457 doi: 10.1109/ACCESS.2021.3080457
[12]	A. T. El-Sayed, H. S. Bauomy, NIPPF versus ANIPPF controller outcomes on semi-direct drive cutting transmission system in a shearer, Chaos Solitons Fractals, 156 (2022), 111778.https://doi.org/10.1016/j.chaos.2021.111778 doi: 10.1016/j.chaos.2021.111778
[13]	Y. Ren, W. Ma, Dynamic analysis and PD control in a 12-pole active magnetic bearing system, Mathematics, 12 (2024), 2331.https://doi.org/10.3390/math12152331 doi: 10.3390/math12152331
[14]	I. I. H. Jawaid, Bifurcations in the response of a rigid rotor supported by load sharing between magnetic and auxiliary bearings, Meccanica, 46 (2011), 1341–1351.https://doi.org/10.1007/s11012-010-9395-8 doi: 10.1007/s11012-010-9395-8
[15]	J. C. Ji, A. Y. T. Leung, Non-linear oscillations of a rotor-magnetic bearing system under super harmonic resonance conditions, Int. J. Non-Linear Mech., 38 (2003), 829–835. https://doi.org/10.1016/S0020-7462(01)00136-6 doi: 10.1016/S0020-7462(01)00136-6
[16]	T. Inoue, Y. Sugawara, Nonlinear vibration analysis of a rigid rotating shaft supported by the magnetic bearing (influence of the integral feedback in the PID control of the vertical shaft), J. Syst. Des. Dyn., 4 (2010), 471–483. https://doi.org/10.1299/jsdd.4.471 doi: 10.1299/jsdd.4.471
[17]	17. H. C. Sung, J. B. Park, Y. H. Joo, Robust fuzzy controller for active magnetic bearing system with 6-DOF, J. Korean Inst. Intell. Syst., 22 (2012), 267–272.https://doi.org/10.5391/JKIIS.2012.22.3.267 doi: 10.5391/JKIIS.2012.22.3.267
[18]	G. Zhang, G. Xi, Vibration control of a time-delayed rotor-active magnetic bearing system by time-varying stiffness, Int. J. Appl. Mech., 14 (2022), 2250007.https://doi.org/10.1142/S1758825122500077 doi: 10.1142/S1758825122500077
[19]	H. C. Wu, L. Zhang, J. Zhou, Y. F. Hu, Dynamic analysis and vibration control of a rotor-active magnetic bearings system with base motion, J. Vib. Control, 30 (2023), 2697–2708.https://doi.org/10.1177/10775463231183190 doi: 10.1177/10775463231183190
[20]	X. Xu, Y. Liu, Q. Han, A universal dynamic model and solution scheme for the electrical rotor system with wide range of eccentricity, Int. J. Non-Linear Mech., 152 (2023), 104402.https://doi.org/10.1016/j.ijnonlinmec.2023.104402 doi: 10.1016/j.ijnonlinmec.2023.104402
[21]	C. W. Chang, L. M. Chu, T. C. Chen, H. T. Yau, Nonlinear dynamic of turbulent bearing-rotor system under quadratic damping with HSFD and active control, J. Braz. Soc. Mech. Sci. Eng., 46 (2024), 123. https://doi.org/10.1007/s40430-024-04691-7 doi: 10.1007/s40430-024-04691-7
[22]	C. Wang, C. C. Liu, S. K. Cao, L. Sun, Nonlinear vibration control of magnetic bearing system considering positive and negative stiffness, J. Vib. Control, 31 (2024), 796–806. https://doi.org/10.1177/10775463241233616 doi: 10.1177/10775463241233616
[23]	M. Eissa, M. Kamel, H. S. Bauomy, Dynamics of an AMB-rotor with time varying stiffness and mixed excitations, Meccanica, 47 (2012), 585–601.https://doi.org/10.1007/s11012-011-9469-2 doi: 10.1007/s11012-011-9469-2
[24]	Y. A. Amer, U. H. Hegazy, Resonance behavior of a rotor active magnetic bearing with time-varying stiffness, Chaos Solitons Fractals, 34 (2007), 1328–1345.https://doi.org/10.1016/j.chaos.2006.04.040 doi: 10.1016/j.chaos.2006.04.040
[25]	M. Kamel, H. S. Bauomy, Nonlinear oscillation of a rotor-AMB system with time-varying stiffness and multi-external excitations, J. Vib. Acoust., 131 (2009), 031009.https://doi.org/10.1115/1.3085884 doi: 10.1115/1.3085884
[26]	W. Zhang, J. W. Zu, F. X. Wang, Global bifurcations and chaos for a rotor-active magnetic bearing system with time varying stiffness, Chaos Solitons Fractals, 35 (2008), 586–608.https://doi.org/10.1016/j.chaos.2006.05.095 doi: 10.1016/j.chaos.2006.05.095
[27]	W. Zhang, J. W. Zu, Transient and steady nonlinear responses for a rotor-active magnetic bearings system with time-varying stiffness, Chaos Solitons Fractals, 38 (2008), 1152–1167.https://doi.org/10.1016/j.chaos.2007.02.002 doi: 10.1016/j.chaos.2007.02.002
[28]	X. D. Yang, H. Z. An, Y. J. Qian, W. Zhang, M. H. Yao, Elliptic motions and control of rotors suspending in active magnetic bearings, J. Comput. Nonlinear Dyn., 11 (2016), 054503.https://doi.org/10.1115/1.4033659 doi: 10.1115/1.4033659
[29]	W. Zhang, M. H. Yao, X. P. Zhan, Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time varying stiffness, Chaos Solitons Fractals, 27 (2006), 175–186.https://doi.org/10.1016/j.chaos.2005.04.003 doi: 10.1016/j.chaos.2005.04.003
[30]	J. Li, Y. Tian, W. Zhang, S. F. Miao, Bifurcation of multiple limit cycles for a rotor-active magnetic bearings system with time-varying stiffness, Int. J. Bifurcation Chaos, 18 (2008), 755–778.https://doi.org/10.1142/S021812740802063X doi: 10.1142/S021812740802063X
[31]	J. Li, Y. Tian, W. Zhang, Investigation of relation between singular points and number of limit cycles for a rotor- AMBs system, Chaos Solitons Fractals, 39 (2009), 1627–1640.https://doi.org/10.1016/j.chaos.2007.06.044 doi: 10.1016/j.chaos.2007.06.044
[32]	J. Awrejcewicz, L. P. Dzyubak, Chaos caused by hysteresis and saturation phenomenon in 2-DOF vibrations of the rotor supported by the magneto- hydrodynamic bearing, Int. J. Bifurcation Chaos, 21 (2011), 2801–2823.https://doi.org/10.1142/S0218127411030155 doi: 10.1142/S0218127411030155
[33]	N. A. Saeed, M. Eissa, W. A. El-Ganaini, Nonlinear oscillations of rotor active magnetic bearing systems, Nonlinear Dyn., 74 (2013), 1–20.https://doi.org/10.1007/s11071-013-0967-8 doi: 10.1007/s11071-013-0967-8
[34]	J. I. Inayat-Hussain, Geometric coupling effects on the bifurcations of a flexible rotor response in active magnetic bearings flexible rotor response in active magnetic bearings, Chaos Solitons Fractals, 41 (2009), 2664–2671.https://doi.org/10.1016/j.chaos.2008.09.041 doi: 10.1016/j.chaos.2008.09.041
[35]	N. A. Saeed, W. A. El-Ganaini, Time-delayed control to suppress the nonlinear vibrations of a horizontally suspended Jeffcott-rotor system, Appl. Math. Modell., 44 (2017), 523–539.https://doi.org/10.1016/j.apm.2017.02.019 doi: 10.1016/j.apm.2017.02.019
[36]	M. R. Ghazavi, Q. Sun, Bifurcation onset delay in magnetic bearing systems by time varying stiffness, Mech. Syst. Signal Process., 90 (2017), 97–109.https://doi.org/10.1016/j.ymssp.2016.12.016 doi: 10.1016/j.ymssp.2016.12.016
[37]	R. Ebrahimi, M. Ghayour, H. M. Khanlo, Effects of some design parameters on bifurcation behavior of a magnetically supported coaxial rotor in auxiliary bearings, Eng. Comput., 34 (2017), 2379–2395.https://doi.org/10.1108/EC-04-2017-0141 doi: 10.1108/EC-04-2017-0141
[38]	A. Kandil, Nonlinear dynamic behavior, impact suppression, and stability control of rotor active magnetic bearing systems: A comparative study of fixed and adjustable surplus current strategies, Eur. J. Mech. A Solids, 116 (2026), 105928.https://doi.org/10.1016/j.euromechsol.2025.105928 doi: 10.1016/j.euromechsol.2025.105928
[39]	H. S. Bauomy, A. T. El-Sayed, T. S. Amer, M. K. Abohamer, Negative derivative feedback control and bifurcation in a two-degree-of-freedom coupled dynamical system, Chaos Solitons Fractals, 193 (2025), 116138.https://doi.org/10.1016/j.chaos.2025.116138 doi: 10.1016/j.chaos.2025.116138
[40]	M. K. Abohamer, T. S. Amer, A. A. Galal, M. A. Darweesh, A. Arab, T. A. Bahnasy, Nonlinear oscillations of a lumped system with series spring, piezoelectric device, and feedback controller, Sci. Rep., 15 (2025), 14642.https://doi.org/10.1038/s41598-025-97173-2 doi: 10.1038/s41598-025-97173-2
[41]	T. S. Amer, G. M. Moatimid, S. K. Zakria, A. A. Galal, Vibrational and stability analysis of planar double pendulum dynamics near resonance, Nonlinear Dyn., 112 (2024), 21667–21699.https://doi.org/10.1007/s11071-024-10169-x doi: 10.1007/s11071-024-10169-x
[42]	S. Ghanem, T. S. Amer, W. S. Amer, S. Elnaggar, A. A. Galal, Analyzing the motion of a forced oscillating system on the verge of resonance, J. Low Freq. Noise Vib. Act. Control, 42 (2023), 563–578.https://doi.org/10.1177/14613484221142182 doi: 10.1177/14613484221142182
[43]	A. H. Nayfeh, Introduction to perturbation techniques, New York: Wiley, 1993.
[44]	A. H. Nayfeh, Perturbation methods, New York: Wiley, 1973.
[45]	A. H. Nayfeh, D. T. Mook, Nonlinear oscillations, New York: Wiley, 1979.
[46]	A H. Nayfeh, Resolving Controversies in the Application of the Method of Multiple Scales and the Generalized Method of Averaging, Nonlinear Dyn., 40 (2005), 61–102.https://doi.org/10.1007/s11071-005-3937-y doi: 10.1007/s11071-005-3937-y

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