This paper analyzes the motion stability of a 16-pole rotor-active magnetic bearings (rotor-AMB) system and investigates the complex vibrations under a proportional-derivative (PD) controller. First, electromagnetic theory and Newton’s second law are applied to derive the two-degree-of-freedom differential governing equations for the 16-pole rotor-AMB system, incorporating the PD control terms. The resulting differential equations include parametrically excited, quadratic nonlinear, and cubic nonlinear terms. Subsequently, the multiple time scales perturbation analysis method is performed on the obtained governing equations, yielding four-dimensional averaged equations in both Cartesian and polar coordinates. Finally, numerical simulations including the amplitude–frequency response characteristics, motion trajectories, energy–amplitude relationships, as well as bifurcation and chaotic motion of the system are studied. The results indicate that the PD controller affects the softening and hardening spring characteristics of the system and has significant control effects on the system’s amplitude, energy, and stability. Additionally, increasing the differential gain coefficient can change the system’s motion from chaotic to periodic.
AmerYAEissaMHHegazyUH, et al. (2006) Dynamic behavior of an AMB/supported rotor subject to parametric excitation. Journal of Vibration and Acoustics128(5): 646–652.
2.
El-ShourbagySMSaeedNAKamelM, et al. (2021) Control performance, stability conditions, and bifurcation analysis of the twelve-pole active magnetic bearings system. Applied Sciences11(22): 10839.
3.
ElashmaweyRASaeedNAElganiniWA, et al. (2024) An asymmetric rotor model under external, parametric, and mixed excitations: nonlinear bifurcation, active control, and rub-impact effect. Journal of Low Frequency Noise, Vibration and Active Control0(0): 1–34. DOI: 10.1177/14613484241259315.
4.
Inayat-HussainJI (2009) Geometric coupling effects on the bifurcations of a flexible rotor response in active magnetic bearings. Chaos, Solitons & Fractals41(5): 2664–2671.
5.
JiJC (2003) Stability and hopf bifurcation of a magnetic bearing system with time delays. Journal of Sound and Vibration259(4): 845–856.
6.
JiJCHansenCH (2001) Non-linear oscillations of a rotor in active magnetic bearings. Journal of Sound and Vibration240(4): 599–612.
7.
JiJCHansenCH (2004) Analytical approximation of the primary resonance response of a periodically excited piecewise non-linear–linear oscillator. Journal of Sound and Vibration278(1-2): 327–342.
8.
JiJCLeungAYT (2003) Non-linear oscillations of a rotor-magnetic bearing system under superharmonic resonance conditions. International Journal of Non-linear Mechanics38(6): 829–835.
9.
JiJCYuLLeungAYT (2000) Bifurcation behavior of a rotor supported by active magnetic bearings. Journal of Sound and Vibration235(1): 133–151.
10.
KamelMBauomyHS (2009) Nonlinear behavior of a rotor-AMB system under multi-parametric excitations. Meccanica45(1): 7–22.
11.
KandilA (2020) Investigation of the whirling motion and rub/impact occurrence in a 16-pole rotor active magnetic bearings system with constant stiffness. Nonlinear Dynamics102: 2247–2265.
12.
KandilASayedMSaeedNA (2020) On the nonlinear dynamics of constant stiffness coefficients 16-pole rotor active magnetic bearings system. European Journal of Mechanics - A: Solids84: 104051.
13.
KandilAHouLSharafM, et al. (2024) Configuration angle effect on the control process of an oscillatory rotor in 8-pole active magnetic bearings. AIMS Mathematics9(5): 12928–12963.
14.
KayaI (2003) A PI-PD controller design for control of unstable and integrating processes. ISA Transactions42(1): 111–121.
15.
LiJTianYZhangW, et al. (2008) Bifurcation of multiple limit cycles for a rotor-active magnetic bearings system with time-varying stiffness. International Journal of Bifurcation and Chaos18(3): 755–778.
16.
LuoHYWangYF (2011) Stability and bifurcation of a continuum rotor with transverse electromagnetic and bearing excitations. Applied Mechanics and Materials105-107: 439–443.
17.
LuoHYWangYF (2012) Nonlinear vibration of a continuum rotor with transverse electromagnetic and bearing excitations. Shock and Vibration19(6): 1297–1314.
18.
MaWSZhangWZhangYF (2021) Stability and multi-pulse jumping chaotic vibrations of a rotor-active magnetic bearing system with 16-pole legs under mechanical-electric-electromagnetic excitations. European Journal of Mechanics - A: Solids85: 104120.
19.
MaZLiuGLiuY, et al. (2022) Research of a six-pole active magnetic bearing system based on a fuzzy active controller. Electronics11: 1723.
20.
SaeedNAKandilA (2019) Lateral vibration control and stabilization of the quasiperiodic oscillations for rotor-active magnetic bearings system. Nonlinear Dynamics98(2): 1191–1218.
21.
SaeedNAKandilA (2021) Two different control strategies for 16-pole rotor active magnetic bearings system with constant stiffness coefficients. Applied Mathematical Modelling92: 1–22.
22.
SaeedNAEissaMEl-GaniniWA (2013) Nonlinear oscillations of rotor active magnetic bearings system. Nonlinear Dynamics74(1-2): 1–20.
23.
SaeedNAMahrousEAwrejcewiczJ (2020a) Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations. Nonlinear Dynamics101: 2299–2323.
24.
SaeedNAAwwadEMEl-MeligyMA, et al. (2020b) Radial versus cartesian control strategies to stabilize the nonlinear whirling motion of the six-pole rotor-AMBs. IEEE Access8: 138859–138883.
25.
SaeedNAMahrousEAbouel NasrE, et al. (2021) Nonlinear dynamics and motion bifurcations of the rotor active magnetic bearings system with a new control scheme and rub-impact force. Symmetry13(8): 1502.
26.
SaeedNAEl-ShourbagySMKamelM, et al. (2022a) Nonlinear dynamics and static bifurcations control of the 12-pole magnetic bearings system utilizing the integral resonant control strategy. Vibration and Active Control41(4): 1532–1560.
27.
SaeedNAEl-ShourbagySMKamelM, et al. (2022b) On the resonant vibrations control of the nonlinear rotor active magnetic bearing systems. Applied Sciences12(16): 8300.
28.
SaeedNAAwrejcewiczJHafezST, et al. (2023) Stability, bifurcation, and vibration control of a discontinuous nonlinear rotor model under rub-impact effect. Nonlinear Dynamics111: 20661–20697.
29.
SaeedNAAwrejcewiczJElashmaweyRA, et al. (2024) On $$\frac{1}{2}$$-DOF active dampers to suppress multistability vibration of a $$2$$-DOF rotor model subjected to simultaneous multiparametric and external harmonic excitations. Nonlinear Dynamics112: 12061–12094.
30.
WangHLiuJ (2005) Stability and bifurcation analysis in a magnetic bearing system with time delays. Chaos, Solitons & Fractals26(3): 813–825.
31.
WangSZhuHWuM, et al. (2020) Active disturbance rejection decoupling control for three-degree-of-freedom six-pole active magnetic bearing based on BP neural network. IEEE Transactions on Applied Superconductivity30(4): 1–5.
32.
WuRQZhangWYaoMH (2018) Nonlinear dynamics near resonances of a rotor-active magnetic bearings system with 16-pole legs and time varying stiffness. Mechanical Systems and Signal Processing100: 113–134.
33.
YangXDAnHZQianYJ, et al. (2016) Elliptic motions and control of rotors suspending in active magnetic bearings. Journal of Computational and Nonlinear Dynamics11(5): 054503.
34.
YehTChungYWuW (2001) Sliding control of magnetic bearing systems. Journal of Dynamic Systems, Measurement, and Control123(3): 353–362.
35.
ZhangWWuRQSirigulengB (2020) Nonlinear vibrations of a rotor-active magnetic bearings system with 16-pole legs and two degrees of freedom. Shock and Vibration10: 1–29.
