BIFURCATION OF MULTIPLE LIMIT CYCLES FOR A ROTOR-ACTIVE MAGNETIC BEARINGS SYSTEM WITH TIME-VARYING STIFFNESS

2008 ◽  
Vol 18 (03) ◽  
pp. 755-778 ◽  
Author(s):  
J. LI ◽  
Y. TIAN ◽  
W. ZHANG ◽  
S. F. MIAO
Keyword(s):  
Nonlinear Equation ◽  
Limit Cycles ◽  
Multiple Scales ◽  
Equation Of Motion ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Detection Function ◽  
Single Degree Of Freedom ◽  
Amb System

The bifurcations of multiple limit cycles for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The governing nonlinear equation of motion is established for the rotor-AMB system with single-degree-of-freedom and parametric excitation. Using the method of multiple scales, the governing nonlinear equation of motion is first transformed to the averaged equation, which is in the form of a Z2-symmetric perturbed polynomial Hamiltonian system of degree 5. Then, the bifurcation theory of planar dynamical system and the method of detection function are utilized to analyze the bifurcations of multiple limit cycles of the averaged equation. Four groups of parametric controlling conditions are given to obtain the configurations of compound eyes. It is found that there exist respectively at least 17, 19, 21 and 22 limit cycles in the rotor-AMB system with the time-varying stiffness under the different controlling conditions.

A Time-Varying Stiffness Rotor Active Magnetic Bearings Under Combined Resonance

2008 ◽  
Vol 75 (1) ◽  
Author(s):  
U. H. Hegazy ◽  
M. H. Eissa ◽  
Y. A. Amer
Keyword(s):  
Multiple Scales ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Multiple Scale ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Gyroscopic Effects ◽  
Nonlinear Equations Of Motion ◽  
Combined Resonance

This paper is concerned with the nonlinear oscillations and dynamic behavior of a rigid disk-rotor supported by active magnetic bearings (AMB), without gyroscopic effects. The nonlinear equations of motion are derived considering a periodically time-varying stiffness. The method of multiple scales is applied to obtain four first-order differential equations that describe the modulation of the amplitudes and the phases of the vibrations in the horizontal and vertical directions. The stability and the steady-state response of the system at a combination resonance for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical nonlinear behaviors, including multiple-valued solutions, jump phenomenon, hardening, and softening nonlinearity. A numerical simulation using a fourth-order Runge-Kutta algorithm is carried out, where different effects of the system parameters on the nonlinear response of the rotor are reported and compared to the results from the multiple scale analysis. Results are compared to available published work.

A time-varying stiffness rotor-active magnetic bearings system under parametric excitation

2008 ◽  
Vol 222 (3) ◽  
pp. 447-458 ◽  
Author(s):  
U H Hegazy ◽  
Y A Amer
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Multiple Scales ◽  
Magnetic Bearings ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Principal Parametric Resonance ◽  
Non Linear ◽  
Response Function Method ◽  
Non Linear System

The method of multiple scales is applied to investigate the non-linear oscillations and dynamic behaviour of a rotor-active magnetic bearings (AMBs) system, with time-varying stiffness. The rotor-AMB model is a two-degree-of-freedom non-linear system with quadratic and cubic non-linearities and parametric excitation in the horizontal and vertical directions. The case of principal parametric resonance is considered and examined. The steady-state response and the stability of the system at the principal parametric resonance case for various parameters are studied numerically, applying the frequency response function method. It is shown that the system exhibits many typical non-linear behaviours including multiple-valued solutions, jump phenomenon, hardening and softening non-linearity. Different effects of the system parameters on the non-linear response of the rotor are also reported. Results are compared with available published work.

Homoclinic Tangencies for a Rotor-Active Magnetic Bearings System

2013 ◽  
Vol 275-277 ◽  
pp. 941-944 ◽  
Author(s):  
Feng Hong Yang ◽  
Hong Zhi Tong
Keyword(s):  
Numerical Results ◽  
Magnetic Bearings ◽  
Homoclinic Tangency ◽  
Time Varying ◽  
Active Magnetic Bearings ◽  
Order Zero ◽  
Parameter Perturbations ◽  
Amb System ◽  
Homoclinic Tangencies

The homoclinic tangency for a rotor-active magnetic bearings (AMB) system with the time-varying stiffness are considered in this paper. The zeros of Melnikov equation are paid more attentions and a 3-order zero was gained and some numerical results under the parameter perturbations were shown.

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.

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.

Chaotic Vibration Analysis of a Coaxial Rotor System in Active Magnetic Bearings and Contact With Auxiliary Bearings

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Reza Ebrahimi ◽  
Mostafa Ghayour ◽  
Heshmatallah Mohammad Khanlo
Keyword(s):  
Dynamic Behavior ◽  
Rotor System ◽  
Magnetic Bearings ◽  
Effective Control ◽  
Speed Ratio ◽  
Maximum Lyapunov Exponent ◽  
Active Magnetic Bearings ◽  
Coaxial Rotor ◽  
Speed Parameter ◽  
Amb System

In many cases of rotating systems, such as jet engines, two or more coaxial shafts are used for power transmission between a high/low-pressure turbine and a compressor. The major purpose of this study is to predict the nonlinear dynamic behavior of a coaxial rotor system supported by two active magnetic bearings (AMBs) and contact with two auxiliary bearings. The model of the system is formulated by ten degrees-of-freedom in two different planes. This model includes gyroscopic moments of disks and geometric coupling of the magnetic actuators. The nonlinear equations of motion are developed by the Lagrange's equations and solved using the Runge–Kutta method. The effects of speed parameter, speed ratio of shafts, and gravity parameter on the dynamic behavior of the coaxial rotor–AMB system are investigated by the dynamic trajectories, power spectra analysis, Poincaré maps, bifurcation diagrams, and the maximum Lyapunov exponent. Also, the contact forces between the inner shaft and auxiliary bearings are studied. The results indicate that the speed parameter, speed ratio of shafts, and gravity parameter have significant effects on the dynamic responses and can be used as effective control parameters for the coaxial rotor–AMB system. Also, the results of analysis reveal a variety of nonlinear dynamical behaviors such as periodic, quasi-periodic, period-4, and chaotic vibrations, as well as jump phenomena. The obtained results of this research can give some insight to engineers and researchers in designing and studying the coaxial rotor–AMB systems or some turbomachinery in the future.

VIBRATION INDUCED FAILURE ANALYSIS OF A HIGH SPEED ROTOR SUPPORTED BY ACTIVE MAGNETIC BEARINGS

2015 ◽  
Vol 39 (4) ◽  
pp. 855-866 ◽  
Author(s):  
Sarvat M. Ahmad ◽  
Osman A. Ahmed ◽  
Zaharuddin Mohamed
Keyword(s):  
Fault Diagnosis ◽  
High Speed ◽  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Flexible Rotor ◽  
Describing Function ◽  
Industrial Grade ◽  
Sinusoidal Input ◽  
Diagnosis Algorithm ◽  
Amb System

Active Magnetic Bearings (AMBs) are increasingly used in various industries and a quick re-levitation of AMBs supported high speed flexible rotor is necessary in case of vibration induced failure. A robust fault diagnosis algorithm is presented to detect suspected saturation type of nonlinearity associated with a power amplifier. A five degree-of-freedom AMB system consisting of four opposing pair of radial magnets and a pair of axial magnets is considered. In this paper failure of an industrial grade AMB system is investigated using Sinusoidal Input Describing Function (SIDF) method. SIDF predicts the gain and frequency at which failure occurs. It is demonstrated that the predicted frequency is in agreement with the frequency at which failure occurs.

scholarly journals A 35,000 RPM Test Rig for Magnetic, Hybrid and Back-Up Bearings

1999 ◽  
Author(s):  
Erik E. Swanson ◽  
James F. Walton ◽  
Hooshang Heshmat
Keyword(s):  
High Speed ◽  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Test Rig ◽  
Low Loss ◽  
Dynamic Motion ◽  
Speed Test ◽  
Bearing Loads ◽  
Amb System ◽  
Overload Capacity

Magnetic bearings have long offered the potential for significant turbomachinery system improvements due to their oil-free, non-contact, low loss nature and their ability to actively control shaft dynamic motion. However, end-users and many designers are hesitant to apply this technology. There are two basic stumbling blocks: active magnetic bearings (AMBs) have little overload capacity, and failure of any portion of the AMB system could result in catastrophic damage to the machine. To cope with both of these problems, a secondary back-up bearing must be included in the system. This paper describes a new full scale, high speed test rig which has the capability to test a variety of back-up bearings at speeds of up to 35,000 RPM, and bearing loads of up to 6.7 kN. Preliminary data for two novel back-up bearings are presented as a demonstration of the test rig’s capabilities.

Nonsmooth Bifurcations in a Cracked Rotor-Active Magnetic Bearings System

2014 ◽  
Vol 989-994 ◽  
pp. 2825-2828 ◽  
Author(s):  
Feng Hong Yang ◽  
Hong Zhi Tong
Keyword(s):  
Equations Of Motion ◽  
Magnetic Bearings ◽  
Active Magnetic Bearings ◽  
Periodic Motions ◽  
Cracked Rotor ◽  
Rotating Shaft ◽  
Chaotic Motions ◽  
Amb System ◽  
Nonlinear Equations Of Motion ◽  
Smooth System

A cracked rotor-active magnetic bearings (AMB) system with the time-varying stiffness is modeled by a piecewise smooth system due to the breath of crack in a rotating shaft. The governing nonlinear equations of motion for the nonsmooth system are established and solved with the numerical method. The simulation results show that a grazing bifurcation, period-double bifurcation and chaotic motions exist in the response. These nonsmooth bifurcations can give rise to jumps between periodic motions, quasi-periodic motions and chaos.

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.

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