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 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.

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.

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 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.

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.

Computational Cost Reduction of MIMO Controllers for Active Magnetic Bearing Systems

2020 ◽  
Author(s):  
Alican Sahinkaya ◽  
Larry Hawkins ◽  
Jerzy T. Sawicki
Keyword(s):  
Dynamic Response ◽  
Cost Reduction ◽  
High Speed ◽  
Computational Cost ◽  
Order Reduction ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Robust Performance ◽  
Limited Bandwidth ◽  
Novel Method

Abstract Generalized MIMO controllers such as H∞ and μ have not been widely adopted in the magnetic bearing industry, partially due to high computational cost relative to simpler SISO schemes. Computational cost is important to industrial magnetic bearing vendors as their controller hardware is based on embedded processors that have limited bandwidth. Studies to mitigate the problem of high order controllers show the limit of the existing methods in order reduction while still maintaining satisfying robust performance. A novel method is proposed to reduce the computational cost of robust controllers by identifying the bounds in their dynamic response, such that an implementation of a controller within those bounds results in the robust performance. The bounds are used to develop two computational cost reduction schemes for controller implementation, i.e., 1) identifying a dual-rate implementation of a single-rate controller which uniformly reduces the computational cost via interlacing technique, and 2) redesign of a controller by identifying its negligible dynamics based on the identified bounds in the controllers’ dynamic response. The results of both approaches are demonstrated on two active magnetic bearing (AMB) systems, a model of a 300 kW turbine generator with permanent magnet biased AMBs and an experimental high-speed AMB machining spindle. μ-synthesis controllers are designed for both systems and the proposed method and schemes are applied accordingly. The comparison of standard implementation of the synthesized controllers and the proposed new implementations is presented. The results demonstrate considerable reduction in the computational cost in terms of required number of multiply-accumulate (MAC) operations.

Dynamic Response of Eccentric Face Seals to Synchronous Shaft Whirl

2004 ◽  
Vol 126 (2) ◽  
pp. 301-309 ◽  
Author(s):  
J. Wileman
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Solution Technique ◽  
Steady State Response ◽  
Analytical Technique ◽  
State Response ◽  
Flexible Supports ◽  
Simple Matrix ◽  
The Stability ◽  
Degenerate Cases

This work provides an analytical technique for computing the seal face misalignment which results from synchronous whirl of the shaft. The eccentric dynamic response is obtained for seals in which both mating faces are mounted on flexible supports. Responses for seals with a single flexibly mounted stator or rotor are also obtained as degenerate cases of the more general result. Synchronous shaft whirl is shown to have a significant effect on the steady-state response of all these seals, while not affecting the stability threshold. The steady-state response is obtained by solution of a simple matrix equation for the general case, and can be obtained in closed form for the degenerate cases of the flexibly mounted stator or flexibly mounted rotor. A numerical example of the solution technique is presented, and the influence of speed is examined. Extension of the method to shaft motions other than synchronous whirl is briefly discussed.

Computational Cost Reduction of MIMO Controllers for Active Magnetic Bearing Systems

2020 ◽  
Vol 142 (11) ◽  
Author(s):  
Alican Sahinkaya ◽  
Larry Hawkins ◽  
Jerzy T. Sawicki
Keyword(s):  
Dynamic Response ◽  
Cost Reduction ◽  
High Speed ◽  
Multiple Input Multiple Output ◽  
Computational Cost ◽  
Order Reduction ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Robust Performance ◽  
Single Input Single Output

Abstract Generalized multiple-input-multiple-output (MIMO) controllers such as H∞ and μ have not been widely adopted in the magnetic bearing industry, partially due to high computational cost relative to simpler single-input-single-output schemes. Computational cost is important to industrial magnetic bearing vendors as their controller hardware is based on embedded processors that have limited bandwidth. Studies to mitigate the problem of high-order controllers show the limit of the existing methods in order reduction while still maintaining satisfying robust performance. A novel method is proposed to reduce the computational cost of robust controllers by identifying bounds in their dynamic response, such that an implementation of a controller within those bounds results in the robust performance. The bounds are used to develop two computational cost reduction schemes for controller implementation, i.e., (1) identifying a dual-rate implementation of a single-rate controller which uniformly reduces the computational cost via interlacing technique, and (2) redesign of a controller by identifying its negligible dynamics based on the identified bounds in the controllers' dynamic response. The results of both approaches are demonstrated on two active magnetic bearing (AMB) systems, a model of a 300 kW turbine generator with permanent magnet biased AMBs, and an experimental high-speed AMB machining spindle. μ-synthesis controllers are designed for both systems, and the proposed method and schemes are applied accordingly. The comparison of standard implementations of the synthesized controllers and the proposed new implementations is presented. The results demonstrate considerable reduction in the computational cost in terms of required number of multiply accumulate (MAC) operations.

A New Approach for Health Monitoring and Detection of Shaft Crack Using an Active Magnetic Actuator During Steady-State Rotor Operation

2007 ◽  
Author(s):  
M. Kasarda ◽  
T. Bash ◽  
D. Quinn ◽  
G. Mani ◽  
D. Inman ◽  
...  
Keyword(s):  
Steady State ◽  
Natural Frequency ◽  
Health Monitoring ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Vibration Monitoring ◽  
Rotating Shaft ◽  
Rotating System ◽  
Rotating Machine ◽  
Input Signals

This work demonstrates the capability of an Active Magnetic Bearing (AMB) to be used as an actuator for interrogating a system by applying multiple forces to a rotating shaft in order to monitor and evaluate the associated responses to these inputs. Similar to modal analysis techniques which apply input signals to static structures in order to monitor responses to those inputs, this approach allows for the measurement of both input and output response in a rotating system for evaluation. However, unlike these techniques, the procedure developed here allows for multiple forms of force inputs to be applied to a rotating structure. This procedure facilitates the development of new improved techniques for diagnosing subtle changes in machinery health or for identifying faults that would potentially go undetected by conventional methods before failure. Although it is expected that this approach can be used in rotors supported in AMBs, the technique developed here uses an AMB on the rotor in conjunction with conventional support bearings. Therefore, this approach has the potential to be used on any rotating machine that can be designed or retrofitted with a single AMB actuator. To demonstrate this approach experimentally, a notched shaft was chosen to represent a shaft crack for identification purposes. Three cases were examined, including a healthy (unnotched) shaft, and three cases of a shaft with a mid-span notch extending to a depth of 10%, 25%, and 40% of shaft diameter, respectively. During testing, excitations up to 1000 Hz were applied via one axis of the AMB actuator to the four rotor cases while the rotor was operating at a steady-state speed of 2400 rpm, and corresponding responses were recorded at the proximity probes. No changes in the 1st or 2nd natural frequencies were detected, but distinct shifts in the 3rd natural frequency were detected from the Frequency Response Function (FRF) data. Since the vast majority of rotating machinery are designed to operate below the 3rd natural frequency, the effect of the notch on the 3rd natural frequency would not have been identified without the application of excitation forces through the AMB actuator. This paper represents an introduction to the new health monitoring approach and results presented here demonstrate the viability of the technique for detecting shaft cracks that might otherwise go undetected in typical steady-state vibration monitoring approaches.

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