Active Magnetic Bearing Rotor Research Articles

Tuned Positive Position Feedback Control of an Active Magnetic Bearings System With 16-Poles and Constant Stiffness

This paper presents a tuned positive position feedback controller (TPPF) to overcome the dual high peaks of the classical PPF. This control signal is merged with a proportional-derivative (PD) control signal to suppress the vibrations of a constant-stiffness 16-poles rotor active magnetic bearings (AMBs) system. Another benefit of the applied TPPF is turning the rotor’s quasiperiodic unstable motion into a periodic stable motion by eliminating both saddle-node and Hopf points. The whole group equations of motion are derived and their approximate solutions are sought with the aid of the multiple scales method. Different response curves are plotted to clarify the difference between the rotor’s vibratory behaviors before and after PPF and also with TPPF.

Rotor Active Magnetic Bearings System Control via a Tuned Nonlinear Saturation Oscillator

The proposed work focuses on improving the performance of the traditional nonlinear saturation controller (NSC) algorithm. In this work, a tuning mechanism can be implemented by providing the NSC control unit with the measured speed $\Omega $ of the rotor from a shaft encoder device in order to introduce a tuned NSC (TNSC). In other words, we are going to tune the NSC’s natural frequency ${\omega _{c}}$ at one half of $\Omega $ such that $\Omega =2{\omega _{c}}$ . This TNSC is adopted to reduce the vibratory amplitudes of a 16-pole constant-stiffness rotor-Active magnetic bearings AMBs) system for a wide range of speeds and eccentricities. The whole controlled system is studied mathematically to seek its approximate solutions via the multiple scales technique. Different relations between the rotor’s amplitudes and its parameters are plotted so as to verify the proposed tuning mechanism and its crucial role in improving the NSC’s performance.

Finite element modelling, analysis and identification using novel trial misalignment approach in an unbalanced and misaligned flexible rotor system levitated by active magnetic bearings

This paper investigates the finite element modelling, analysis and identification of a fully-levitated flexible rotor with faults using active magnetic bearings (AMBs). For simultaneous estimation of the rotor residual unbalance and AMB residual misalignment (combination of parallel and angular), a novel trial misalignment approach has been proposed. Additionally, the force–displacement and force-current stiffness coefficients of AMBs are identified. This novel work is similar to the idea of trial unbalance in the rotor balancing. An unbalanced flexible shaft with multiple discs levitated by multiple misaligned AMBs has been mathematically modelled utilizing finite element method based on Timoshenko beam theory. To overcome the practical difficulty in measuring responses, the dynamic reduction scheme has been implemented to eliminate the transverse rotational degrees of freedom from the system model, which are further utilized in development of the identification algorithm. To numerically illustrate the reliability and robustness of the proposed algorithm, the algorithm is tested against the measurement noise in responses and rotor modelling errors.

Investigation of the whirling motion and rub/impact occurrence in a 16-pole rotor active magnetic bearings system with constant stiffness

In this paper, the weight of the rotor is considered in a 16-pole rotor active magnetic bearing system with constant stiffness. The equations of motion are derived to show the asymmetry between the rotor’s horizontal and vertical displacements. Accordingly, the rotor may exhibit forward, backward, intermediate, or hybrid whirls. The possibility to overcome the backward whirl and to symmetrize the rotor’s motion again is discussed. Also, the rotor may rub/impact with the stator legs depending on the values of the adopted system parameters. The multiple-scales method is utilized to extract the approximate solutions of the studied model and to analyze its nonlinear dynamics and the aforementioned whirls. The discussion is enhanced by different analytical plots such as the rotor’s responses to its eccentricity $$f$$ and rotation speed $$\varOmega $$ . Numerical validation is carried out to demonstrate how these analytical plots describe precisely the nonlinear dynamical behavior of the whole system. Finally, whirl orbit maps are plotted to simulate the real-life motion of the rotor at different whirls and at rub/impact occurrence.

Two different control strategies for 16-pole rotor active magnetic bearings system with constant stiffness coefficients

Rotor active magnetic bearing is one of the preferable supporting techniques for the high-speed rotating machinery. This article proposes a comparison between two different control strategies for the constant stiffness coefficients 16-pole rotor active magnetic bearings system (16-pole Rotor-AMBs). The first strategy is the Cartesian control in which the applied currents on the poles depend on the horizontal or vertical displacements of the rotor. The second strategy is the radial control in which the applied currents depend on the radial displacement of the rotor. Both strategies are based on the proportional-derivative (PD) control algorithm to make the rotor's vibrations track a zero reference point. The major novelty here is that we compare the two strategies to explore their advantages and disadvantages in controlling the 16-pole Rotor-AMBs. The analytical and numerical approximate predictions of the rotor's vibrations are extracted to show and verify the nonlinear dynamical behavior of the rotor based on the two methods. The local stability of the extracted solutions is analyzed via Hartman–Grobman theorem to conclude the stable and unstable solutions regions and the boundary between them. Various responses are included to show, in most conditions, that the radial control has a privilege over the Cartesian control in avoiding either the multi-stable solutions or the rub/impact forces occurrence between the rotating disk and the stator pole legs.

Stable Control of Nutation and Precession for the Radial Four-Degree-of-Freedom AMB-Rotor System Considering Strong Gyroscopic Effects

The stable control of the active magnetic bearing rotor (AMB-rotor) system is the key technology for a magnetically suspended control moment gyroscope, which is used in the spacecraft attitude control system. The nutation and precession instability caused by strong gyroscopic effects is the principal issue of the AMB-rotor system. In order to solve the gyroscopic effects problem, in this article, a state-space model of the radial four-degree-of-freedom AMB-rotor system is established, and the strong gyroscopic effects term varying with the rotor speed is regarded as a bounded perturbation. The regional pole assignment method of an uncertain system is used to design a robust controller to maintain the system stability over the full rotor speed range. However, the controller designed by the regional pole assignment method does not consider the influence of closed-loop stiffness of the magnetic bearing, which will lead to the start-up oscillation and structural mode self-excited oscillation problems. Therefore, on the basis of the regional pole assignment, the system stiffness is further constrained and a robust controller is solved. Experiments show that the designed controller can not only deal with the start-up oscillation and the structural modal vibration problems, but also can achieve the stable control of the nutation and precession.

Robust control of active magnetic bearing systems with an add-on controller to cancel gyroscopic effects: Is it worth it?

One of the performance-limiting factors in the design of robust controllers for active magnetic bearing systems is the fact that the controller needs to be robust to the gyroscopic effects, that is rotational speed-dependent dynamics of the system. Studies in the literature show that better performance and stability can be achieved when gyroscopic effects are explicitly handled by a cross-feedback control for rigid rotor-active magnetic bearing systems. For flexible rotor-active magnetic bearing systems, gyroscopic effects are mainly dealt by defining the rotational speed as an uncertain parameter of the model or with linear parameter-varying controllers. This study explores the novel idea of compensating gyroscopic effects of a flexible rotor-active magnetic bearing system with an add-on controller and investigates its effects on the achieved performance of μ-controllers. The study is carried out on an experimental active magnetic bearing test rig with relatively high gyroscopic effects. An add-on controller is designed to compensate the gyroscopic effects of the first and second flexible modes of the rotor. Two μ-controllers are designed for the system: (1) benchmark controller that is designed using a standard control approach for active magnetic bearing systems and (2) controller designed with a modified model of the system in which the gyroscopic effects for the first and second flexible modes are reduced because of the presence of the add-on controller. Both controllers are implemented, and their performances are compared for initial levitation, run-up test from 0 to 10,000 r/min, orbit sizes at various speeds, and the computational cost of implementing each controller. The results suggest that better performance is potentially possible at the cost of significant increase in the computational complexity of the controller.

Stability and multi-pulse jumping chaotic vibrations of a rotor-active magnetic bearing system with 16-pole legs under mechanical-electric-electromagnetic excitations

The stability and Shilnikov-type multi-pulse jumping chaotic vibrations are investigated for a nonlinear rotor-active magnetic bearing (AMB) system with the time varying stiffness and 16-pole legs under the mechanical-electric-electromagnetic excitations. The ordinary differential governing equation of motion for the rotor-AMB system is given by a two-degree-of-freedom nonlinear dynamical system including the parametric excitation, quadratic and cubic nonlinearities. The averaged equations of the rotor-AMB system are obtained by using the method of multiple scales under the cases of 1:1 internal resonance, primary parametric resonance and 1/2 subharmonic resonance. Some coordinate transformations are employed to find the type and number of the equilibrium points for the averaged equations. Using the global perturbation method developed by Kavacic and Wiggins, the explicit sufficient conditions near the resonance are obtained for the existence of the Shilnikov-type multi-pulse jumping homoclinic orbits and chaotic vibrations. This implies that the Shilnikov-type multi-pulse jumping chaotic vibrations may occur for the rotor-AMB system in the sense of Smale horseshoes. Numerical simulations are presented to verify the analytical predictions by using the fourth-order Runge-Kutta method. The Shilnikov-type multi-pulse jumping chaotic vibrations can exist in the rotor-AMB system with the time varying stiffness and 16-pole legs under the mechanical-electric-magnetic excitations.

Experimental Study of Macro Fiber Composite-Magnet Energy Harvester for Self-Powered Active Magnetic Bearing Rotor Vibration Sensor

The paper presents the design, fabrication, and characterization of an energy harvester for an active magnetic bearing (AMB) rotor vibration using a macro fiber composite (MFC) with magnetic coupling. The MFC cantilevers configuration, together with neodymium magnets, is used for the contact-free rotor radial vibration self-powered sensor. The permanent magnets attached to the rotor and to the four MFC element beams ensure the mechanical energy transfer and the MFC cantilever vibration excitation. In the proposed prototype, the MFC transducer output voltage depends on the air-gap between two magnets. This paper investigates the optimum conditions to harvest as much as possible electric energy at different clearances and rotational speeds. Furthermore, to assess the rotor vibration sensitivity, the experimental results of the MFC-magnet self-powered sensor are compared with measurements obtained using a fiber optic sensor. The maximal obtained harvesting power equals 673.47 µW for the rotor speed of 3150 rpm. Moreover, the MFC cantilever was proposed as the rotor vibration sensor. The MFC-magnet self-powered vibration sensor output was compared with the fiber optic laser sensor. The mismatched vibration amplitude for both sensors does not exceed 1 µm.

Nonlinear dynamics of the six-pole rotor-AMB system under two different control configurations

The nonlinear dynamics of the six-pole rotor active magnetic bearings system is studied in this article for the first time. Two control configurations based on a proportional-derivative feedback current controller are proposed to mitigate lateral vibrations of the considered system. The first configuration is designed in such a way that only four electromagnetic poles are responsible for the system vibration control in the $$ Y $$ -direction, while all six poles control the lateral vibration in the $$ X $$ -direction. A second configuration is proposed in which the same four poles of the first configuration control the system vibration in the $$ Y $$ -direction, while the other two poles are responsible for controlling the system vibration in the $$ X $$ -direction. According to the suggested control methods, a mathematical model is derived that simulates lateral oscillations of the system. Using the perturbation analysis, four autonomous and coupled first-order nonlinear differential equations that govern the system oscillation amplitudes and the corresponding phase angles in both $$ X $$ and $$ Y $$ -directions are extracted. Various bifurcation diagrams are obtained using rotor spinning-speed and disk eccentricity as a bifurcation control parameter. The conditions at which the system can whirl either forward or backward are investigated. Numerical validations for the obtained bifurcation diagrams are introduced which illustrate excellent agreement with the analytical results. Based on the acquired analytical and numerical results, it is found that the first control configuration has the best dynamical behavior for controlling the vibration of such systems.

Synchronous Vibration Control for Magnetically Suspended Rotor System Using a Variable Angle Compensation Algorithm

In active magnetic bearings (AMBs) supported rigid rotor systems, the unbalance compensation strategy based on a notch filter was often used to suppress the synchronous unbalance vibration when the rotor operates in the steady-state rotational speed. Whereas the rotor operates in a rotational-speed range including the critical rotational speeds, conventional notch filters should be improved to suppress the unbalance vibration. In most cases, improving the structure of the notch filter, such as a cross-feedback notch filter, phase-shift notch filter, adding the polarity switch or two-stage switch, was considered to make the notch filter work in the rotational-speed range. However, the improved notch filter may affect the stability of the AMBs rotor systems and result in large vibrations when the polarity switch or two-stage switch shifts. To avoid the shortcomings of these methods, a method using a variable angle compensation algorithm was proposed to suppress the synchronous unbalance vibration in finite iterative seeking processes, directly realizing unbalance compensation according to real-time rotor position. Finally, the effectiveness of the method proposed is verified by simulation and experiment results.

On the nonlinear dynamics of constant stiffness coefficients 16-pole rotor active magnetic bearings system

The present study is devoted to investigate the oscillatory behaviors of the 16-pole rotor active magnetic bearing system. A controllable electromagnetic force generated via a conventional proportional-derivative controller is utilized to stabilize the system lateral oscillations that excited by the rotating disk eccentricity when the spinning speed (Ω) is close to or equal the system linear natural frequency (ω). The nonlinear dynamical equations governing the controlled system lateral vibrations at constant stiffness coefficients are derived in this article for the first time. Then, four nonlinear autonomous first-order differential equations to describe the considered system oscillation amplitudes and the corresponding phase angles are obtained applying the asymptotic analysis. Bifurcation behavior of the system periodic motions under varying the different control parameters is explored. The main acquired results confirm that the 16-pole rotor-AMB system at constant stiffness coefficients can exhibit one of three oscillatory motions that are periodic, quasiperiodic, or chaotic motions depending on the derivative gain coefficient. Moreover, the system may respond with one-stable solution, bi-stable solutions, tri-stable solutions, or quadri-stable solutions depending on the proportional gain coefficient. Numerical simulations for different system motions are validated via the system time response, Poincare map, orbit plot, and frequency spectrum that are showed an excellent agreement with the obtained analytical results.

Dynamic behaviour analysis of coupled rotor active magnetic bearing system in the supercritical frequency range

In this article, an attempt has been made to model and estimate speed-dependent characteristic parameters of coupling misalignment and AMB, in an AMB integrated coupled flexible rotor system running in the supercritical frequency range. For estimation algorithm, desired current and displacement responses in time series are acquired from Simulink. The practical limitation of FEM modelling is the number of sensors and availability of measurement locations that have been overcome by applying a novel gyroscopic high-frequency condensation (GHFC) scheme. The transformation matrix obtained from standard high-frequency condensation (SHFC) is modified by adding a gyroscopic effect into the transformation matrix to achieve the novel GHFC. The numerical analysis has been performed on a coupled rotor train system along with AMB. The estimations are presented and compared for SHFC and GHFC. The GHFC is found effective over SHFC techniques. The effect of noisy response and modelling error on the estimation algorithm is analysed and found competent. The experimental responses obtained for a different level of misalignment conditions have been compared with the numerically simulated response and found that the application of AMB suppresses the excessive vibration.

Modeling and experimental validation of transient response of magnetic suspension rotor contacting with auto-reducing clearance auxiliary bearing

The major purpose of this study is to indicate the transient responses of an active magnetic bearing (AMB) rotor contacting with auto-reducing clearance auxiliary bearing (ACAB). Considering the radial vibration of the single support, the real-time clearances between the outer ring (OR) and each support, and the swing of the supports, the dynamic models of the rotor and the ACAB are formulated. The dynamic behavior of the rotor-ACAB system is investigated by the dynamic trajectories of the rotor and the OR, the movement of each support, and the dynamic forces generated in the ball bearing and between the OR and supports. Furthermore, the AMB-rotor dropping tests, in which a strain gauge based measurement system and a high speed camera equipment are adopted to separately survey the interaction force between the support and the OR and record the movements of the OR and the supports, are performed. The results show that, after the rotor drops on ACAB, the swing supports rapidly reduce the protective clearance and then stably maintain the rotor to operate around the center of ACAB. Finally, the ball bearing only suffers the centrifugal force of the rotor without impact forces, while big interaction forces always exist between the OR and each support.

Whirl Mode Suppression for AMB-Rotor Systems in Control Moment Gyros Considering Significant Gyroscopic Effects

For a magnetically suspended control moment gyroscope with strong gyroscopic effects, the whirl mode (nutation and precession) stability control of the active magnetic bearing (AMB) rotor system has always been a challenging problem. This article proposes a robust control method combined with regional pole assignment to achieve the stability control of the whirl mode in the AMB-rotor system. First, the physical model of multi-degree-of-freedom magnetic bearing system is analyzed in detail, and the state-space model of the system is established. Then, the gyroscopic effects varying with the rotor speed are considered as a bounded parameter perturbation of the system, and the constraints on the controller to achieve the system stability are obtained. Furthermore, in order to increase the system stability margin and the whirl mode damping, the additional constraints on the controller are derived through the regional pole assignment method. Finally, all constraints are converted into the form of linear matrix inequalities (LMIs), and a controller satisfying all the constraints can be obtained by solving the LMIs. Experimental results demonstrate that the designed robust controller can achieve the stability control of nutation and precession as well as keeping a satisfactory system performance.

Design of Interval Type-2 Fuzzy Controllers for Active Magnetic Bearing Systems

This article proposes an active levitation control system composed of an active magnetic bearing (AMB) rotor and an interval type-2 (IT2) model-based fuzzy logic controller (FLC). The controller aims to achieve fast and stable levitation by compensating for system uncertainties via proper design of IT2 membership functions. In particular, the complex nonlinear dynamics of the AMB are modeled as a set of local linear systems around multiple operating points, which can be stabilized by a parallel distributed compensation scheme. Sufficient conditions are derived to guarantee the asymptotical stability of the closed-loop system based on the linear matrix inequality approach and the Lyapunov stability theory. Moreover, input constraints are applied in an enhanced version of the controller. Experiments on a real AMB platform were conducted to demonstrate the effectiveness and superiority of the proposed IT2 FLC.

Stability of Rotating Machinery Supported on Active Magnetic Bearings Subjected to Base Excitation

Abstract The stability of rotating machinery is a major challenge for the floating production storage and offloading (FPSO) units such as steam turbines or centrifugal compressors. The use of active magnetic bearings (AMBs) in turbomachines enables high operating speeds, active mechatronic system for the diagnostics, and the control and enables downsizing of the whole installation footprint. In case of strong base motions, the rotor can contact its touchdown bearings (TDBs) which are used as emergency and landing bearings. The aim of this study is to assess the stability of a rotating machine supported on AMBs during severe foundation excitation. The combined effect of unbalance forces, base motion excitation, and contact non-linearity on a rotor–AMB system response is analyzed focusing on the capacity of an augmented proportional-integral-derivative controller to maintain the system stable. An academic scale test rig was used for the experimental investigations. The controller was efficient and able to maintain the system stable during and after the application of the excitation, but the dynamic capacity of the AMBs was largely oversized with respect to the studied system. In order to check the capacity of the AMBs, when they are designed as a function of the rotor weight and expected excitation, numerical simulations were carried out (downsized). A finite element (FE) model was developed to model the on-board rotor–AMB system. Predicted and measured responses due to impulse excitation applied on the foundations were compared. The capacity of the controller to maintain the system stability is then discussed.

Unbalance Suppression for AMB Rotor System Using APF-SRF Algorithm

With the great capability in levitating rotors, active magnetic bearings (AMB) have been increasingly applied in high-speed rotating machinery applications. Typically, for the practical application of AMB rotor systems, synchronous unbalance vibration has long been a critical issue. Thus, exploring a facile yet effective approach to address this issue has attracted tremendous interest from both industrial and academic perspectives. Herein, in this work, we propose a novel strategy to suppress the unbalance vibration through a compensation control algorithm method. The compensation control algorithm which is established based on the combination of first-order all-pass filter (APF) and synchronous rotating frame (SRF) algorithm demonstrates to be efficient in greatly reducing the unbalance vibration and resulting in excellent system stability according to both simulation and experimental results. This work not only confirms the applicability of APF-SRF algorithm for the AMB rotor systems but also opens up a new paradigm to modulate and minimize the unbalance vibration of AMB rotor systems.

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

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.

A novel iterative learning control method and control system design for active magnetic bearing rotor imbalance of primary helium circulator in high-temperature gas-cooled reactor

As one of the key technologies of high-temperature gas-cooled reactor, primary helium circulator–equipped active magnetic bearing provides driving force for primary helium cooling system. However, repetitive periodic vibration produced by rotor imbalance may introduce risks to primary helium circulator (even for high-temperature gas-cooled reactors). First, this article analyzes a periodic component extraction algorithm which is widely used in active magnetic bearing rotor unbalance control methods and points out the problem that the periodic component extraction algorithm occupies numerous computing resources which cannot satisfy the real-time request of active magnetic bearing control system. Then, a novel iterative learning control algorithm based on the iteration before last iteration of system information (iterative learning control-2) and a plug-in parallel control mechanism based on the existing control system are put forward, meanwhile, an integrated independent distributed active magnetic bearing control system is designed to solve the problem. Finally, both the simulation and experiment are carried out, respectively. The corresponding results show that the control method and control system proposed in this article have significant suppression effect on the repetitive periodic vibration of the active magnetic bearing system without degrading the real-time requirement and can provide important technical support for the safe and stable operation of the primary helium circulator in high-temperature gas-cooled reactor.