Analysis of Nonlinear Dynamics for a Rotor-Active Magnetic Bearing System With Time-Varying Stiffness: Part I — Formulation and Local Bifurcations

2003 ◽  
Author(s):  
Wei Zhang ◽  
Jean W. Zu
Keyword(s):  
Nonlinear Dynamics ◽  
Steady State ◽  
Transient State ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Time Varying ◽  
Quasiperiodic Solutions ◽  
Local Bifurcations ◽  
Averaged Equations ◽  
Amb System

In this two-part paper, we investigate nonlinear dynamics in the rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The model of parametrically excited two-degree-of-freedom nonlinear system with the quadratic and cubic nonlinearities is established to explore the periodic and quasiperiodic motions as well as the bifurcations and chaotic dynamics of the system. The method of multiple scales is used to obtain the averaged equations in the case of primary parameter resonance and 1/2 subharmonic resonance. In Part I of the companion paper, numerical approach is applied to the averaged equations to find the periodic, quasiperiodic solutions and local bifurcations. It is found that there exist 2-period, 3-period, 4-period, 5-period, multi-period and quasiperiodic solutions in the rotor-AMB system with 8-pole legs and the time-varying stiffness. The catastrophic phenomena for the amplitude of nonlinear oscillations are first observed in the rotor-AMB system with 8-pole legs and the time-varying stiffness. The procedures of motion from the transient state chaotic motion to the steady state periodic and quasiperiodic motions are also found. The results obtained here show that there exists the ability of autocontrolling transient state chaos to the steady state periodic and quasiperiodic motions in the rotor-AMB system with 8-pole legs and the time-varying stiffness.

Nonlinear Oscillation of a Rotor-AMB System With Time Varying Stiffness and Multi-External Excitations

2009 ◽  
Vol 131 (3) ◽  
Author(s):  
M. Kamel ◽  
H. S. Bauomy
Keyword(s):  
Steady State ◽  
Order Approximation ◽  
Nonlinear Differential Equations ◽  
Steady State Solution ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Multiple Time ◽  
Time Varying ◽  
The Stability ◽  
Amb System

The rotor-active magnetic bearing system subjected to a periodically time-varying stiffness having quadratic and cubic nonlinearities is studied and solved. The multiple time scale technique is applied to solve the nonlinear differential equations governing the system up to the second order approximation. All possible resonance cases are deduced at this approximation and some of them are confirmed by applying the Rung–Kutta method. The main attention is focused on the stability of the steady-state solution near the simultaneous principal resonance and the effects of different parameters on the steady-state response. A comparison is made with the available published work.

NONLINEAR BEHAVIOR OF TUNED ROTOR-AMB SYSTEM WITH TIME VARYING STIFFNESS

2011 ◽  
Vol 21 (01) ◽  
pp. 195-207 ◽  
Author(s):  
M. EISSA ◽  
M. KAMEL ◽  
H. S. BAUOMY
Keyword(s):  
Steady State ◽  
Multiple Scales ◽  
Primary Resonance ◽  
Nonlinear Behavior ◽  
Steady State Solution ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Practical Case ◽  
Time Varying ◽  
Amb System

A rotor-active magnetic bearing (AMB) system with a periodically time-varying stiffness subjected to tuned and external excitations is studied and solved. The tuned excitation represents an imposed noise on the external excitation to simulate the practical case. The method of multiple scales is applied to analyze the response of the system two modes near the simultaneous combined and primary resonance cases. The stability of the steady state solution near this resonance case is studied applying Lyapunov's first method. The system exhibits many typical nonlinear behaviors including multiple-valued solutions, jump phenomenon, softening nonlinearity and saturation. The presence of the tuned excitation increased the steady state amplitudes and produced a chaotic system. The effects of the different parameters on the steady state solutions are investigated and discussed. Comparison with previous work is reported.

Multi-Pulse Chaotic Motions of a Rotor-Active Magnetic Bearing With Time-Varying Stiffness

2005 ◽  
Author(s):  
Wei Zhang ◽  
Ming-Hui Yao ◽  
Xue-Ping Zhan ◽  
Li-Lai Bai
Keyword(s):  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Time Varying ◽  
Chaotic Motions ◽  
Dimensionless Equation ◽  
System A ◽  
Averaged Equations ◽  
Asymptotic Perturbation Method ◽  
Amb System ◽  
Asymptotic Perturbation

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.

scholarly journals Nonlinear Vibrations of a Rotor-Active Magnetic Bearing System with 16-Pole Legs and Two Degrees of Freedom

2020 ◽  
Vol 2020 ◽  
pp. 1-29 ◽  
Author(s):  
W. Zhang ◽  
R. Q. Wu ◽  
B. Siriguleng
Keyword(s):  
Perturbation Method ◽  
Electromagnetic Force ◽  
Nonlinear Vibrations ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Time Varying ◽  
Asymptotic Perturbation Method ◽  
Amb System ◽  
Asymptotic Perturbation ◽  
Quadratic And Cubic Nonlinearities

The asymptotic perturbation method is used to analyze the nonlinear vibrations and chaotic dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs and the time-varying stiffness. Based on the expressions of the electromagnetic force resultants, the influences of some parameters, such as the cross-sectional area Aα of one electromagnet and the number N of windings in each electromagnet coil, on the electromagnetic force resultants are considered for the rotor-AMB system with 16-pole legs. Based on the Newton law, the governing equation of motion for the rotor-AMB system with 16-pole legs is obtained and expressed as a two-degree-of-freedom system with the parametric excitation and the quadratic and cubic nonlinearities. According to the asymptotic perturbation method, the four-dimensional averaged equation of the rotor-AMB system is derived under the case of 1 : 1 internal resonance and 1 : 2 subharmonic resonances. Then, the frequency-response curves are employed to study the steady-state solutions of the modal amplitudes. From the analysis of the frequency responses, both the hardening-type nonlinearity and the softening-type nonlinearity are observed in the rotor-AMB system. Based on the numerical solutions of the averaged equation, the changed procedure of the nonlinear dynamic behaviors of the rotor-AMB system with the control parameter is described by the bifurcation diagram. From the numerical simulations, the periodic, quasiperiodic, and chaotic motions are observed in the rotor-active magnetic bearing (AMB) system with 16-pole legs, the time-varying stiffness, and the quadratic and cubic nonlinearities.

Analysis of Nonlinear Dynamics of a Rotor-Active Magnetic Bearing System With 16-Pole Legs

2017 ◽  
Author(s):  
Ruiqin Wu ◽  
Wei Zhang ◽  
Ming Hui Yao
Keyword(s):  
Nonlinear Dynamics ◽  
Perturbation Method ◽  
Primary Resonance ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Chaotic Motions ◽  
Dimensionless Equation ◽  
Asymptotic Perturbation Method ◽  
Amb System ◽  
Asymptotic Perturbation

In this paper, we use the asymptotic perturbation method to analyze the nonlinear dynamics of a rotor-active magnetic bearing (AMB) system with 16-pole legs. The motion governing equation is derived by using classical Newton law. The resulting dimensionless equation of motion for the system is expressed as a two-degree-of-freedom system including the parametric excitation, quadratic and cubic nonlinearities. The asymptotic perturbation method is used to obtain the averaged equation when the primary resonance and 1/2 sub-harmonic resonance are taken into consideration. From the averaged equations obtained, numerical simulations are presented to investigate the modulation of vibration amplitudes of the rotor-AMB system. Based on a specific set of parameters, it is found that there exist the periodic, quasi-periodic and chaotic motions in the modulated amplitude of the rotor in the system.

Switching-Type Self-Tuning Fuzzy PID Control of an Active Magnetic Bearing System

2013 ◽  
Vol 284-287 ◽  
pp. 2330-2336
Author(s):  
Kuan Yu Chen ◽  
Pi Cheng Tung ◽  
Yi Hua Fan
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Pid Control ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
The Other ◽  
Fuzzy Pid ◽  
Switching Type ◽  
Self Tuning ◽  
Amb System

This paper presents a new switching control scheme for an active magnetic bearing (AMB) system using self-tuning fuzzy proportional-integral-derivative (PID) control. The research process consists of three stages. First, four types of self-tuning fuzzy PID-type controllers (FPIDCs) consisting of two most commonly used fuzzy inference systems: Mamdani and Takagi-Sugeno types, and two efficient parameter adaptive methods: function tuner and relative rate observer, are used to control a highly nonlinear AMB system, respectively. Hence, there are two kinds of FPIDCs can be obtained by comparing experimental results of these tests: one has the fastest transient response and the other has the minimum steady-state error. Next, the switching-type self-tuning FPIDC is proposed by combining the two kinds of FPIDCs. Namely, the AMB system is dominated by the scheme with the fastest transient response when the rotor is at rest and by the one with the best steady-state performance when the rotor is in rotation. Finally, experimental results demonstrate that the proposed switching-type self-tuning FPIDC performs better overall performance than the other self-tuning FPIDCs, particularly when controlling an AMB system.

Nonlinear Vibration of a Rotor-Active Magnetic Bearing System With 16-Pole Legs

2017 ◽  
Author(s):  
Ruiqin Wu ◽  
Wei Zhang ◽  
Ming Hui Yao
Keyword(s):  
Parametric Excitation ◽  
Multiple Scales ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Dynamical Response ◽  
Time Varying ◽  
Chaotic Motions ◽  
Dimensionless Equation ◽  
Amb System ◽  
Bearing System

In this paper, the nonlinear dynamics of a rotor-active magnetic bearing system with 16-pole legs and the time varying stiffness is investigated. The magnetic forces are obtained through an electromagnetic theory. The motion governing equation is derived by using Newton law. The resulting dimensionless equation of motion for the rotor-AMB system with 16-pole legs and the time varying stiffness is presented with the two-degree-of-freedom system including parametric excitation, the quadratic and cubic nonlinearities. The averaged equations of the rotor-AMB system are obtained by using the method of multiple scales under the case of the primary parametric resonance and 1/2 sub-harmonic resonance. The numerical results show that there exist the periodic, quasi-periodic and chaotic motions in the rotor-active magnetic bearing system. Since the weight of the rotor effect the system, it is also found that there are the different shapes of motion on the two directions of the rotor-AMB system. The parametric excitation, or the time-varying stiffness produced by the PD controller has great impact on the system. Thus, the complicated dynamical response in the rotor-AMB system can be controlled through adjusting the parametric excitation.

Some Nonlinear Effects of Magnetic Bearings

1999 ◽  
Author(s):  
Norbert Steinschaden ◽  
Helmut Springer
Keyword(s):  
Equations Of Motion ◽  
Period Doubling ◽  
Path Following ◽  
Nonlinear Effects ◽  
Magnetic Bearing ◽  
Operating Conditions ◽  
Active Magnetic Bearing ◽  
Nonlinear Phenomena ◽  
Amb System ◽  
Large Rotor

Abstract In order to get a better understanding of the dynamics of active magnetic bearing (AMB) systems under extreme operating conditions a simple, nonlinear model for a radial AMB system is investigated. Instead of the common way of linearizing the magnetic forces at the center position of the rotor with respect to rotor displacement and coil current, the fully nonlinear force to displacement and the force to current characteristics are used. The AMB system is excited by unbalance forces of the rotor. Especially for the case of large rotor eccentricities, causing large rotor displacements, the behaviour of the system is discussed. A path-following analysis of the equations of motion shows that for some combinations of parameters well-known nonlinear phenomena may occur, as, for example, symmetry breaking, period doubling and even regions of global instability can be observed.

Dynamic Modeling and Analysis of Active Magnetic Bearings

2014 ◽  
Vol 494-495 ◽  
pp. 685-688
Author(s):  
Rong Gao ◽  
Gang Luo ◽  
Cong Xun Yan
Keyword(s):  
Dynamic Characteristic ◽  
Dynamic Modeling ◽  
Magnetic Bearing ◽  
Integrated System ◽  
Magnetic Bearings ◽  
Active Magnetic Bearing ◽  
Active Magnetic Bearings ◽  
Modeling And Analysis ◽  
Amb System ◽  
Mathematics Method

Active magnetic bearing (AMB) system is a complex integrated system including mechanics, electronic and magnetism. In order to research for the basic dynamic characteristic of rotor supported by AMB, it is necessary to present mathematics method. The dynamics formula of AMB is established using theory means of dynamics of rotator and mechanics of vibrations. At the same tine, the running stability of rotor is analyzed and the example is presented in detail.

scholarly journals Nonlinear Control of an Active Magnetic Bearing System Achieved Using a Fuzzy Control with Radial Basis Function Neural Network

2014 ◽  
Vol 2014 ◽  
pp. 1-18 ◽  
Author(s):  
Seng-Chi Chen ◽  
Van-Sum Nguyen ◽  
Dinh-Kha Le ◽  
Nguyen Thi Hoai Nam
Keyword(s):  
Neural Network ◽  
Radial Basis Function ◽  
Basis Function ◽  
Fuzzy Logic Controller ◽  
Magnetic Bearing ◽  
Operating Conditions ◽  
Active Magnetic Bearing ◽  
Highly Nonlinear ◽  
Radial Basis ◽  
Amb System

Studies on active magnetic bearing (AMB) systems are increasing in popularity and practical applications. Magnetic bearings cause less noise, friction, and vibration than the conventional mechanical bearings; however, the control of AMB systems requires further investigation. The magnetic force has a highly nonlinear relation to the control current and the air gap. This paper proposes an intelligent control method for positioning an AMB system that uses a neural fuzzy controller (NFC). The mathematical model of an AMB system comprises identification followed by collection of information from this system. A fuzzy logic controller (FLC), the parameters of which are adjusted using a radial basis function neural network (RBFNN), is applied to the unbalanced vibration in an AMB system. The AMB system exhibited a satisfactory control performance, with low overshoot, and produced improved transient and steady-state responses under various operating conditions. The NFC has been verified on a prototype AMB system. The proposed controller can be feasibly applied to AMB systems exposed to various external disturbances; demonstrating the effectiveness of the NFC with self-learning and self-improving capacities is proven.

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