Nonlinear Dynamics of a Magnetically Supported Rotor on Safety Auxiliary Bearings

1998 ◽  
Vol 120 (2) ◽  
pp. 596-606 ◽  
Author(s):  
X. Wang ◽  
S. Noah
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Nonlinear Dynamic System ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Fixed Point Algorithm ◽  
Friction Forces ◽  
Flexible Supports ◽  
Multiple Periodic Solutions ◽  
The Stability

This study concerns the dynamic response of a rotor landed on auxiliary (catcher) bearings in an Active Magnetic Bearing (AMB) supported rotor, following postulated loss of power or overload of the AMB. An analytical model involving a disk, a shaft and auxiliary bearings on damped flexible supports is constructed and appropriate equations of the nonlinear dynamic system are developed. The equations include a switch function to indicate contact/non-contact events and determine the existence of contact normal forces and tangential friction forces between the shaft and the bearings. Steady state solutions are obtained. An analytical method was formulated and used to yield solutions for cases with well balanced rotors, in absence of any side forces. The Fixed Point Algorithm (FPA) is used to obtain steady state periodic solutions of the unbalanced rotor for various parameters. The FPA is used to determine the stability of the periodic solutions and the type of bifurcation involved. Multiple periodic solutions, quasi-periodic and chaotic responses are detected and discussed. A set of preliminary guidelines for selection of the parameters of the catcher bearings is given.

Dynamic Response of a Rotor Landed on Auxiliary Bearings

1995 ◽  
Author(s):  
Xinchao Wang ◽  
Sherif Noah
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Periodic Solutions ◽  
Nonlinear Dynamic System ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Fixed Point Algorithm ◽  
Friction Forces ◽  
Flexible Supports ◽  
Multiple Periodic Solutions

Abstract This study concerns the dynamic response of a rotor landed on auxiliary (catcher) bearings in an Active Magnetic Bearing (AMB) supported rotor, following postulated loss of power or overload of the AMB. An analytical model involving disk, shaft and auxiliary bearings on damped flexible supports is constructed and appropriate equations of the nonlinear dynamic system are developed. The equations include a switch function to indicate contact/noncontact events and determine the existence of contact normal forces and tangential friction forces between the shaft and the bearings. Steady state solutions are obtained. An analytical method was formulated and used to yield solutions for cases with well balanced rotors, in absence of any side forces. The Fixed Point Algorithm (FPA) is used to obtain steady state periodic solutions for various parameters. The FPA is used to determine the stability of the periodic solution and the type of bifurcations involved. Multiple periodic solutions, quasiperiodic and chaotic responses are detected and discussed. A set of preliminary guidelines for selection of the parameters of the catcher bearings are given.

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.

Mode-Locking and Chaos in a Jeffcott Rotor With Bearing Clearances

1994 ◽  
Vol 61 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Sang-Kyu Choi ◽  
Sherif T. Noah
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Mode Locking ◽  
Rotor System ◽  
Winding Number ◽  
Fixed Point Algorithm ◽  
Jeffcott Rotor ◽  
Multiple Periodic Solutions ◽  
The Stability

A complex mode-locking (or entrainment) structure underlying the nonlinear whirling phenomenon of a horizontal Jeffcott rotor with a discontinuous nonlinearity (bearing clearance) was identified. A winding number is introduced as a measure of the ratio between two frequencies involved in the aperiodic whirling motions of the rotor system considered. Utilizing the winding number map, it was revealed that the alternating periodic and quasi-periodic responses take place according to the Farey number tree. The winding number varies in the form of the so-called “Devil’s staircase” as a certain system parameter varies. From the mode-locking pattern in the parameter space of the forcing amplitude and frequency, it was observed that as the forcing amplitude increases, the size of each locking interval increases so that its growth takes place in the form of “Arnol’d tongues,” where the winding number remains a rational number. Moreover, inside each locking zone, i.e., each “Arnol’d tongue,” there exist many smaller tongues similar to the main tongue, in which a sequence of period-doubling bifurcations leading to chaos occurred. The boundaries of each locking zone was obtained using a fixed-point algorithm along with the Floquet theory for checking the stability of the periodic solutions. The winding numbers were estimated utilizing a fixed-point algorithm modified to obtain quasi-periodic responses. A jump phenomenon was also observed by tracking multiple periodic solutions for several parameters of the rotor system.

An Evaluation of Stability Indices Using Sensitivity Functions for Active Magnetic Bearing Supported High-Speed Rotor

2006 ◽  
Vol 129 (2) ◽  
pp. 230-238 ◽  
Author(s):  
Naohiko Takahashi ◽  
Hiroyuki Fujiwara ◽  
Osami Matsushita ◽  
Makoto Ito ◽  
Yasuo Fukushima
Keyword(s):  
High Speed ◽  
Transfer Functions ◽  
Mimo System ◽  
Magnetic Bearing ◽  
Singular Value ◽  
Active Magnetic Bearing ◽  
Sensitivity Functions ◽  
Phase Lags ◽  
The Stability ◽  
Stability Indices

In active magnetic bearing (AMB) systems, stability is the most important factor for reliable operation. Rotor positions in radial direction are regulated by four-axis control in AMB, i.e., a radial system is to be treated as a multi-input multioutput (MIMO) system. One of the general indices representing the stability of a MIMO system is “maximum singular value” of a sensitivity function matrix, which needs full matrix elements for calculation. On the other hand, ISO 14839-3 employs “maximum gain” of the diagonal elements. In this concept, each control axis is considered as an independent single-input single-output (SISO) system and thus the stability indices can be determined with just four sensitivity functions. This paper discusses the stability indices using sensitivity functions as SISO systems with parallel/conical mode treatment and/or side-by-side treatment, and as a MIMO system with using maximum singular value; the paper also highlights the differences among these approaches. In addition, a conversion from usual x∕y axis form to forward/backward form is proposed, and the stability is evaluated in its converted form. For experimental demonstration, a test rig diverted from a high-speed compressor was used. The transfer functions were measured by exciting the control circuits with swept signals at rotor standstill and at its 30,000 revolutions/min rotational speed. For stability limit evaluation, the control loop gains were increased in one case, and in another case phase lags were inserted in the controller to lead the system close to unstable intentionally. In this experiment, the side-by-side assessment, which conforms to the ISO standard, indicates the least sensitive results, but the difference from the other assessments are not so great as to lead to inadequate evaluations. Converting the transfer functions to the forward/backward form decouples the mixed peaks due to gyroscopic effect in bode plot at rotation and gives much closer assessment to maximum singular value assessment. If large phase lags are inserted into the controller, the second bending mode is destabilized, but the sensitivity functions do not catch this instability. The ISO standard can be used practically in determining the stability of the AMB system, nevertheless it must be borne in mind that the sensitivity functions do not always highlight the instability in bending modes.

Robust Active Chatter Control in Milling Processes With Variable Pitch Cutters

2018 ◽  
Vol 140 (10) ◽  
Author(s):  
Tao Huang ◽  
Lijun Zhu ◽  
Shengli Du ◽  
Zhiyong Chen ◽  
Han Ding
Keyword(s):  
Nonlinear Differential Equations ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Linear Matrix ◽  
Variable Pitch ◽  
Stability Lobe ◽  
Milling Operations ◽  
Controller Gain ◽  
Chatter Control ◽  
The Stability

Milling chatters caused by the regenerative effect is one of the major limitations in increasing the machining efficiency and accuracy of milling operations. This paper studies robust active chatter control for milling processes with variable pitch cutters whose dynamics are governed by multidelay nonlinear differential equations. We propose a state feedback controller based on linear matrix inequality (LMI) approach that can enlarge multiple stability domains in the stability lobe diagram (SLD) while the controller gain is minimized. Numerical simulations of active magnetic bearing systems demonstrate the effectiveness of the proposed method.

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.

Harmonic disturbance attenuation in a three-pole active magnetic bearing test rig using a modified notch filter

2016 ◽  
Vol 23 (5) ◽  
pp. 770-781 ◽  
Author(s):  
S Mahdi Darbandi ◽  
Mehdi Behzad ◽  
Hassan Salarieh ◽  
Hamid Mehdigholi
Keyword(s):  
Notch Filter ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Disturbance Attenuation ◽  
Test Rig ◽  
Stability Margins ◽  
Periodic Disturbances ◽  
Mass Unbalance ◽  
Gain Margin ◽  
The Stability

This study is concerned with the problem of harmonic disturbance rejection in active magnetic bearing systems. A modified notch filter is presented to identify both constant and harmonic disturbances caused by sensor runout and mass unbalance. The proposed method can attenuate harmonic displacement and currents at the synchronous frequency and its integer multiples. The reduction of stability is a common problem in adaptive techniques because they alter the original closed-loop system. The main advantage of the proposed method is that it is possible to determine the stability margins of the system by few parameters. The negative phase shift of the modified notch filter can be tuned to achieve a desired phase margin, while the gain margin can also be adjusted separately. It is shown that the modified notch filter can be designed to suppress multiple harmonics at the same time. It is implemented on a three-pole magnetic bearing test rig to evaluate its performance. Simulation and experimental results indicate that the presented method can be successfully applied to compensate the periodic disturbances such as sensor runout and mass unbalance in active magnetic bearing systems.

A Numerical Study on Effect of Electromagnetic Actuator on Rigid Rotor Supported on Gas Foil Bearing

2017 ◽  
Author(s):  
Kamal Kumar Basumatary ◽  
Gaurav Kumar ◽  
Karuna Kalita ◽  
Sashindra Kumar Kakoty
Keyword(s):  
Numerical Study ◽  
Load Capacity ◽  
Current Trend ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Normal Operation ◽  
Electromagnetic Actuator ◽  
Rigid Rotor ◽  
Magnetic Actuator ◽  
The Stability

Generally, Gas Foil Bearings (GFBs) are used in high speed machineries which are quite prone to instability or wear and tear. The current trend is to develop hybrid bearings which has conventional bearing (GFB) along with active magnetic bearing as an electromagnetic actuator (EMA). The GFBs are used for normal operation and the magnetic actuator can be used for the improvement of the stability and the load capacity of the bearing. In the present work a numerical study has been carried out to study the effects of magnetic actuator on the stability of bump type GFB supported rigid rotor. A rigid rotor supported on two identical GFBs with and without EMA has been investigated. The electromagnetic forces are incorporated in the equation of motion to provide the active control. A PD controller has been used as a controller for the magnetic actuator. It has been observed that the incorporation of EMA to the GFB reduces the sub synchronous vibrations and hence increases the stability.

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.

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