Abstract
This study employed a variety of nonlinear dynamic analysis techniques to explore the complex phenomena associated with a nonlinear mathematical model of an active magnetic bearing (AMB) system. The aim was to develop a method with which to assume control over chaotic behavior. The bifurcation diagram comprehensively explicates rich nonlinear dynamics over a range of parameter values. In this study, we examined the complex nonlinear behaviors of AMB systems using phase portraits, Poincaré maps, and frequency spectra. Furthermore, estimates of the largest Lyapunov exponent based on the properties of synchronization confirmed the occurrence of chatter vibration indicative of chaotic motion. Thus, the proposed continuous feedback control approach based on synchronization characteristics eliminates chaotic oscillations. Finally, some simulation results demonstrated the feasibility and efficiency of the proposed control scheme.
1. Introduction
Active magnetic bearings (AMBs) are widely used in rotating machinery and have high potential for machining applications. These noncontact bearings use magnetic forces to firmly hold the rotor and maintain separation between the rotor and the stationary components. In such a linear case study, the electromagnetic force is linearized about the operating point. In AMB systems, the occurrence of large unbalanced forces in rotor bearings can cause nonlinear motions of high amplitude. However, a closed-loop control system is required to stabilize the system by eliminating vibrations caused by disturbing forces. AMBs are regarded as inherently nonlinear due to nonlinearities of the electromagnetic forces. Accurate control of the system requires that designers account for the effects of nonlinearities. Thus, these complex nonlinear dynamics must be considered for in the design of AMB systems. Accordingly, in the present author’s previous work [1], an experiment involving an unloaded symmetric rotor with a flexible coupler at one end and a bearing comprising two pairs of electromagnets at the other end (Figure 1) was conducted by applying a series of nonlinear electromagnetic forces to identify a nonlinear model of this system. Figure 2 illustrates the block diagram of the AMB control system along the horizontal direction. In the aforementioned study, we sought to predict these nonlinear dynamics by modifying the conventional identification technique based on the principle of harmonic balance to characterize the system with a higher degree of precision.
This model successfully captured the primary characteristics of the system by comparing the simulated frequency responses with those obtained experimentally [1]. However, theoretical analyses of this model to determine whether the nonlinear mathematical model derived experimentally can predict and characterize the dynamics of the physical system have not yet been undertaken. Various works have been carried out to study the chaotic motion in AMB systems [2–6]. But the chaotic motion that occurs when the rotor strikes the electromagnet that using the identification of a mathematical model for the AMB system has not been examined.
In this study, we used bifurcation diagrams, phase portraits, Poincaré maps, frequency spectra, and Lyapunov exponents to observe periodic orbits, bifurcation, and chaotic phenomena in an identified mathematical model of the AMB system. Across a broad range of parameters, the Lyapunov exponent offers the most powerful methods to measure the sensitivity of a dynamic system with regard to its initial conditions. This approach can be used to determine whether a system is susceptible to chaotic motion. The algorithms for computing Lyapunov exponents of smooth dynamic systems are well established [7–10]. However, a number of nonsmooth dynamic systems possess discontinuities, for example, discontinuities associated with dry friction, backlash, and saturation for which these algorithms cannot be applied directly. Numerous studies have developed procedures for the calculation of Lyapunov exponents associated with nonsmooth dynamic systems [11–13]. In this study, we adopted the method developed by Stefanski [13] for estimating the largest Lyapunov exponent for AMB systems with closed-loop control.
Several practical engineering problems involving chaos require control techniques to convert chaotic attractors into stable periodic orbits. Since the pioneering work of Ott et al. [14] on controlling chaos, various modified methods and other approaches have been proposed [15–22]. Improving the performance of a magnetic bearing system necessitates the transformation of chaotic motion into a steady-state periodic orbit by the state feedback control and the injection of dither signals [22]. Accordingly, a continuous feedback control method, based on the synchronization properties proposed by Kapitaniak [18] and Pyragas [19], was used. Such schemes convert chaotic motion into stable motion by using feedback combined with a periodic external force. Simulation results demonstrate the efficacy of the proposed approach mechanism.
This paper is organized as follows. Section 2 describes an AMB system model and describes the complex dynamic behavior of a power system, which was elucidated using numerical analysis methods, such as bifurcation diagrams, phase portraits, Poincaré maps, and frequency spectra. Section 3 presents the Lyapunov exponent used to determine whether the system exhibits chaotic motion. A synchronization control technique for controlling chaos in the AMB system is described in Section 4. Conclusions are outlined in Section 5.
2. Overall Characteristics of AMB System: Simulations and Discussion
The resulting nonlinear model of the AMB system [1] is as follows:wherewith the limitationswhere x1 is the displacement of the rotor around the equilibrium point, x2 is the velocity of the rotor, x3 and x4 are coil currents oscillating around the bias current, ec1 and ec2 are the outputs of the proportional-integral-derivative controller to the two coils, KP = −55, KD = −0.3, and KI = −50 are the control gains, KA(=2.4) and KH(=10000) represent the gains of the power amplifier and displacement sensor, and A0 sin Ωt is the forcing voltage generated by a programmable function generator. The procedures used to derive the other coefficients necessary for (1a)–(1d) are listed in Table 1.
Numerical simulations based on (1a)–(1d) were performed to clarify the characteristics of the proposed system. In (1a)–(1d), the amplitude of the input excitation, A0, equals 3.5 V. The commercial software package DIVPRK of IMSL [23] in Fortran subroutines for mathematics was used to solve these ordinary differential equations. The resulting bifurcation diagram is presented in Figure 3; this figure clearly demonstrates that the first period-doubling bifurcation occurred at approximately Ω = 19.5 Hz and chaotic motion appears at approximately Ω = 18.63 Hz. The period-doubling route to chaos has been observed in Figure 3. Additional details of the responses exhibited by the system are presented in Figures 4–7 , in which each type of response is characterized by time response, phase portrait, Poincaré map, and frequency spectrum. Figures 4(a)–4(d) prove that the -period includes a constant term and the fundamental components. From Figures 5 and 6, we inferred that a cascade of period-doubling bifurcations produced a series of subharmonic components, revealing bifurcations with new frequency components at 1, , , and so on. The essence of chaotic behavior can be described using Poincaré maps, which present an infinite set of points, referred to as a strange attractor. Chaotic motion also exhibits a broad continuous frequency spectrum. Thus, strange attractors and continuous-type Fourier spectra are generally regarded as strong indicators of chaos, as illustrated in Figures 7(a)–7(d).
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3. Analysis of Chaotic Phenomena in AMB
The largest Lyapunov exponent is a useful diagnostic index of the chaotic system. Every dynamic system possesses a spectrum of Lyapunov exponents (λ), which determine length, area, and volume changes in phase space. In simple terms, Lyapunov exponents measure the rate of divergence (or convergence) between two adjacent orbits. Chaos can be identified simply by calculating the largest Lyapunov exponent, thereby determining whether nearby trajectories generally diverge (λ > 0) or converge (λ < 0). Any bounded motion in a system of at least one positive Lyapunov exponent is defined as chaotic, whereas nonpositive Lyapunov exponents indicate periodic motion. Numerous well-established algorithms are available for computing the Lyapunov spectrum of smooth dynamic systems [7–10]. However, these algorithms cannot be directly applied to nonsmooth dynamic systems with discontinuities, such as dry friction, backlash, and saturation. In this study, we estimated the largest Lyapunov exponent in order to identify the onset of chaotic motion in AMB systems. Stefanski [13] proposed a simple method for estimating the largest Lyapunov exponent based on synchronization properties. Synchronization controls the response system by accessing the output of the drive system. Thus, the output of the response system asymptotically follows the output of the drive system. This method is described briefly in the following paragraph.
The dynamic system is decomposed into two subsystems, namely, a drive systemand a response system
Consider a dynamic system that is composed of two identical n-dimensional subsystems; a coupling coefficient d is applied only to the response system (5), while the drive system (4) remains unchanged. The first-order differential equations for such a system can be written as
The condition of synchronization is given by the following inequality:
The smallest value of the coupling coefficient d in the synchronization, , is assumed to equal the largest Lyapunov exponent, as follows:
Equation (6) provides an augmented system, based on (1a)–(1d), as follows:wherewith the following limitations:
In the next step, the largest Lyapunov exponent of the system under consideration was determined for the chosen parametric values, in an aforementioned manner. Figure 8 is a plot of the results of numerical calculations and shows the estimated largest Lyapunov exponents obtained using the synchronization method. All of the largest Lyapunov exponents were positive with regard to the forcing frequency (Ω ≤ 18.63 Hz), indicating that the system exhibited chaotic motion. These results help enhance our understanding of the chaos phenomenon in AMB systems under rotation.
4. Suppressing Chaos
Learning to predict the behaviors of a chaotic system provides a number of benefits, with the ultimate objective being to assume control over the system. Improving the performance of a dynamic system necessitates chaotic motion to be converted into a stable periodic orbit. Therefore, suitable control methods must be developed. Kapitaniak [18] and Pyragas [19] have proposed simple and effective time-continuous control methods based on synchronization characteristics capable of converting chaotic motion into periodic motion. This method involves the construction of a feedback mechanism in conjunction with a specific time-continuous perturbation. Figure 9 is a plot of the proposed feedback-controlled loop with an external periodic perturbation. The technique is explained briefly in the following paragraph.
Consider an n-dimensional dynamic system,where denotes the state vector and represents the input signal. System (14) without an input signal (F(t) = 0) is assumed to exhibit a strange attractor, whereas system (13) exhibits periodic motion. A periodic system is typically termed the drive system, whereas a chaotic system is called the response system. The strategy shown in Figure 9 is used to synchronize the two systems. The difference between signals q(t) and p(t) is utilized as a control signal:where K represents the feedback gain.
When Ω1 = 20 Hz is selected in the drive system, the following equations reveal period-one motion.wherewith the following limitations:
When Ω2 = 18.2 Hz is selected in the response system, the following equations reveal chaotic motion:wherewith the following limitations:
Control signal (15) was introduced into (19a)–(19d) as a feedback control to synchronize (16a)–(16d) and (19a)–(19d). Control signal (15) was incorporated into (19a)–(19d), yielding the following coupled system, to achieve synchronization.
Equation (22a)–(22d) exhibited chaotic motion when K = 0 and Ω2 = 18.2 Hz. The feedback gain K was adjusted to be between 0.0 and −3000 to convert the dynamics of system (22a)–(22d) from chaotic to periodic motion. Figure 10 presents the resulting bifurcation diagram, which comprehensively explains the dynamic behavior of the system over a range of feedback gains. Figure 10 clarifies that chaotic motion occurred in the region between K = 0 and K = −120. Stable periodic motion occurred when K decreased to below −120. When K < −1000, (22a)–(22d) display period-one motion. Accordingly, the synchronization was achieved in the AMB system by applying the control signal with K < −1000. When t = 10 s, the chaotic AMB system (19a)–(19d) can be synchronized by applying the control signal with K = −1300, as shown in Figure 11. Thus, this chaotic system can be managed using the control signal (K = −1300), such that chaotic motion is converted into period-one motion. Figure 11(a) presents the phase portrait of the controlled system, and Figure 11(b) plots the time response of displacement in which the synchronization control signal is injected after 10 s. Figure 12 presents the associated synchronization errors, e1 = q1 − p1 and e2 = q2 − p2. When the synchronization errors equal zeros, the synchronization is achieved. The control signal controlled the chaotic system, which converged to a stable equilibrium point. Therefore, the synchronization control method successfully suppressed chaotic motion and improved the chatter behavior of the AMB system. Thus, the performance of the AMB system could be improved considerably at high-speed rotations.
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5. Conclusions
This work used an identified nonlinear model to explore global bifurcations and chaos control problem in AMB systems. Dynamic behaviors over the entire range of parameter values could be observed in a bifurcation diagram, which revealed that AMB systems exhibit period-doubling bifurcations and chaotic motions. The largest Lyapunov exponent, derived using the properties of synchronization, provided the most powerful approach for analyzing chaotic motion in a system. Controlling chaotic motion is an effective method to prevent chatter vibration in AMB systems. We propose a continuous feedback control method based on synchronization properties to suppress chaotic motion in AMB systems. Our analysis revealed that the proposed nonlinear model could predict the occurrence of global bifurcations and chaos control in AMB systems. These results indicate that the proposed system has potential for application across a wide range of functions in the design of magnetic levitation systems. Furthering the development of magnetically levitated vehicles requires an understanding of their dynamic characteristics from the viewpoint of stability, safety, and ride quality at high speeds. We believe that studying the dynamics and controlling chaotic vibrations in the magnetically levitated systems will help to advance the development of magnetic transportation systems.
Data Availability
All data generated or analyzed during this study are included within this article.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the Ministry of Science and Technology in Taiwan, Republic of China, under project number MOST 108-2221-E-212-010-MY3.