Levitation Stability and Hopf Bifurcation of EMS Maglev Trains

Hu, Junxiong; Ma, Weihua; Chen, Xiaohao; Luo, Shihui

doi:https://doi.org/10.1155/2020/2936838

Mathematical Problems in Engineering

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Abstract Introduction Numerical Simulation Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 2936838 | https://doi.org/10.1155/2020/2936838

Levitation Stability and Hopf Bifurcation of EMS Maglev Trains

Junxiong Hu,^1,2Weihua Ma ,¹Xiaohao Chen,¹and Shihui Luo¹

Academic Editor: Xiao-Qiao He

Received28 Jul 2019

Accepted28 Feb 2020

Published07 Apr 2020

Abstract

This paper analyzed the mechanical characteristics of single electromagnet system and elastic track beam of EMS maglev train and established a five-dimensional dynamics model of single electromagnet-track beam coupled system with classical PD control strategy adopted for its levitation system. Then, based on the Hurwitz criterion and the high-dimensional Hopf bifurcation theory, the stability of the coupled system is analyzed; the existence of the Hopf bifurcation is discussed and the bifurcation direction and the stability of the periodic solution are determined with levitation control feedback coefficient as the bifurcation parameter; and numerical simulation is conducted to verify the validity of the theoretical analysis results. The results show that the Hurwitz algebra criterion can directly determine the eigenvalues and symbols of the dynamics system to facilitate the analysis on the stability of the system and the Hopf bifurcation without the necessity of calculating the specific eigenvalues; supercritical Hopf bifurcation will occur under the given parameters, that is, when , the real-time system is asymptotically stable, yet Hopf bifurcation occurs as increases gradually beyond , with the stability of the system lost and a stable limit cycle branched.

1. Introduction

Maglev train is a new type of rail transit vehicle that has gradually emerged in recent years. It uses electromagnets, permanent magnets, etc., to interact with the track to generate magnetic force and levitate the train and also uses linear motor for traction so that the contactless operation between the train and the track is realized. From the perspective of the principle of levitation, currently there are two typical and mature types of maglev trains: electromagnetic suspension (EMS) and electrodynamic suspension (EDS). EMS maglev train generates a magnetic field by energizing the electromagnet attached to the train, and the electromagnet and the track attract each other to levitate the train. The levitation gap is generally 8∼12 mm, and form of the elevated bridge line is adopted. Due to the small levitation gap of the EMS maglev train and the nonlinear characteristics of the levitation system, there is a special coupled vibration problem between the train and the track beam, which may even lead to a levitation failure in severe cases [1]. This vibration problem is a high-dimensional and nonlinear complex problem involving bridge structure vibration, vehicle vibration, and stability control of levitation system. To solve this technical problem, many experts and scholars have conducted studies from different angles in the aim of finding the couple mechanism and solving the problem once and for all to ensure the levitation stability of the train.

As early as 1986, Nagai noticed the nonlinear self-excited vibration of the track beam due to the neglect of the elastic deformation of the track beam during the design process of the controller and designed a control algorithm so that the vehicle can be stabilized in the elastic track beam and is insensitive to elastic conditions [2]. Cai et al. considered the interaction between the elastic track beam and the moving train, simplified the “vehicle-guideway” system to the “moving load-simple beam” system, and studied the different dynamics of the beam when the moving load was applied to single- and double-span beams [3]. Lee et al. have conducted a lot of research studies on the train-track beam coupled vibration behavior of the low-speed maglev in Korea and have obtained a series of research results [4–6]. Based on the analysis of the structural characteristics of the EMS train system, Xie and Chang simulated and analyzed the effects of the damping of elastic track beam and acceleration feedback gain upon static levitation stability and dynamic characteristics [7]. Zhao and Zhai studied and analyzed the vertical dynamics behavior of German TR (Transrapid) high-speed maglev trains and China’s self-developed low-speed maglev trains [8]. Ma et al. established test bench for electromechanical coupled vibration of single-levitation frame for medium- and low-speed maglev trains and carried out simulation analysis and experimental comparison of vehicle-guideway coupled vibration under different combinations of boundary conditions such as track beam support stiffness, levitation mass, and control algorithm [9].

The levitation control system of EMS maglev train is a typical nonlinear dynamic system, and the change of system parameters will generate new dynamics behavior and even undermine the stability of the system. Zhou and Zheng established the ordinary differential equations of the vehicle-track beam coupled dynamics with periodically variable coefficients. The Lyapunov characteristic index was used to determine the dynamic stability of the maglev system, and the possibility of chaotic phenomena in the system under the nonlinear conditions of the secondary suspension was studied [10]. Shi et al. analyzed the Hopf bifurcation of nonlinear magnetic levitation system based on PID controller and explained the reasons for the vibration of the system from the perspective of control parameters and the stability of periodic solution [11]. Wang et al. used the center manifold reduction method and Poincaré canonical method to study the Hopf bifurcation and resonance problems of the levitation system under positional delay and track beam disturbance [12, 13]. Li et al. established a single electromagnet-track beam model to study the mechanism of self-excited vibration caused by the transfer of vibration energy from levitation system to track beam and compared the stability of the system under different PD control parameters [14]. Zhao and Zhai numerically simulated the dynamic stability of a single electromagnet system, pointing out that the difference between the secondary suspension frequency, the track beam fundamental frequency, and the control system frequency should be as large as possible to avoid resonance between the single electromagnet and the track beam, which may lead to system instability [15]. Based on the similarity theory, Wang and Shen established a small-scale model of single-electromagnetic vehicle-track beam coupled system, derived the similarity relationship in the dynamic system, and analyzed the similarity characteristics of the small-scale model [16]. They also studied the optimal control method on the elastic track beam and established a test rig of small-scale single electromagnet-elastic track beam for verification [17]. Zhang et al. studied the stability of the time-delay levitation system and the Hopf bifurcation under the sliding mode control strategy [18, 19]. Based on the Hurwitz algebraic criterion, Wu and Fang analyzed the conditions for Hopf bifurcation to occur without considering the influence of the elasticity of the track beam [20]. Sun et al. analyzed the relationship between the control parameters and the Hopf bifurcation of the maglev vehicle-guideway coupling system and designed a fuzzy adaptive controller to adjust the control parameters automatically to keep the closed-loop system away from the Hopf bifurcation point [21].

In this paper, analysis is conducted on the stability of the levitation system of the EMS maglev train on elastic track beam. The research object is a levitation control unit in the decentralized control point of the maglev train—a single electromagnet system. In the first part, a simplified dynamics model of “electromagnet-levitation control system-elastic track beam” is established; the second part discusses the equilibrium point of the canonical form of the model and its stability; the third part uses the control coefficient as the bifurcation parameter, introduces the method of directly determining the Hopf bifurcation of the levitation control system by using the Hurwitz criterion, and further discusses the direction of Hopf bifurcation and the stability of bifurcation periodic solution by means of the high-dimensional Hopf bifurcation theory. The fourth part uses the numerical simulation method to verify the theoretical analysis results.

2. Modeling

EMS maglev train adopts electromagnetic attraction type of levitation, that is, it relies on the attraction between the vehicle electromagnet and the F-shaped ferromagnetic track to achieve levitation. The active control method must be adopted to ensure the stable levitation of the train. At present, EMS maglev trains mostly adopt the technical scheme of multipoint decentralized control. The electromagnet controlled by one set of levitation controller is a single-magnet levitation system, many of which compose the levitation system of EMS maglev train. The simplified single electromagnet system is shown in Figure 1.

Under the assumption of neglecting the leakage of the electromagnet winding and the magnetic resistance of the electromagnet core and the guideway, the following equations for the system model shown in (1) can be obtained:where is levitation force, is vacuum permeability, is the turns of electromagnet coil, is the area of electromagnet pole, is sum of a single electromagnet’s own weight and load weight, is exciting current, is electromagnet resistance, is the voltage at electromagnet end, is absolute gap between the electromagnet and the rigid track, and is actual levitation gap of the electromagnet.

The voltage feedback control adopts the classic PD control. and are the gap offset and the gap differential feedback coefficients, respectively. Therefore,where is the initial voltage at equilibrium position and is the desired levitation gap.

Assume that the track beam is a simply supported and evenly elastic beam with equal cross section; then, the following can be obtained by the Bernoulli–Euler beam equation:where is Young’s modulus of elasticity, is beam section inertia, is equivalent damping coefficient of the beam, is beam density, is area of beam section, and is levitation load intensity acting on the beam.

Using the modal superposition method, the displacement of the beam in equation (3) can be expressed as the linear superposition of a regular mode function and a generalized coordinate :

Integrating equation (4) into equation (3) gives the vibration equation corresponding to the i^th-order mode of the beamwhere is the i^th-order natural frequency of the beam, is the i^th-order damping ratio of the beam, and is the i^th-order generalized force.

For the simply supported single-span beam with the length of in Figure 1, the natural frequency and the regular mode function of the beam are, respectively, given by

While conducting studies on the coupled vibration of the EMS maglev train, Shi et al. found that the high-order vibration mode can only occur with exceptionally high energy excitation [11]. From the latest test results from Li and others on the Changsha maglev line, it can be seen that for a definite track beam structure, the vibration frequency corresponding to the maximum amplitude of track beam with maglev train running on its upper part is distributed within a range, wherein the first-order frequency of the track beam has the greatest impact [22]. Therefore, only the influence of the first-order bending mode of the track beam is taken into account in the analysis of this paper. Since the length of the electromagnet is much smaller than the span of the track beam, the electromagnetic force F can be considered as a concentrated load acting at position on the track beam when seeking the limit through the generalized force expression. The first-order vibration equation of the generalized coordinates of the track can be rewritten aswhere , and .

Introducing the deformation of the track beam into the vibration equation (8), where is a constant, we can obtain

In the midspan position where the amplitude of the simply supported beam is the largest, we have , and then .

Therefore, the coupled system model under the influence of the first-order vibration mode of the track can be defined by

3. Equilibrium Points and Their Stability

Let and levitation gap ; then, the canonical form of the coupled system model (10) is determined as follows:

It can be seen from equations (11) that the EMS maglev train-track beam coupled system is a typical nonlinear system, which also brings certain difficulties for solving. The core work of this paper is to study the local stability of the coupled system at the equilibrium point and the Hopf bifurcation of the control parameters.

Let and we have at the desired levitation gap and ; then, the equilibrium point can be given by

Shift the equilibrium point of the system (11) to the original point through the transformation , and the new system (13) is expressed as

The Jacobian matrix of system (13) at the original point is given by

The corresponding characteristic equation is written as

Model parameters are shown in Table 1.

Substituting the parameters, the coefficients of characteristic equation (15) can be obtained:

According to Liénard–Chipart criterion, the necessary and sufficient conditions for all eigenvalues of the characteristic equation (15) to have a negative real part are that the characteristic equation coefficients are all greater than 0, and half of the Hurwitz determinant is greater than 0, namely,

Therefore, the levitation control feedback coefficient and when the system is stable can be obtained, and the value range is shown in Figure 2.

4. Existence and Direction of Hopf Bifurcation

4.1. Existence of Hopf Bifurcation

For the classical Hopf bifurcation evaluation method, it is necessary to solve the eigenvalues of the Jacobian matrix whenever each parameter changes and judge whether the real part is zero. For the five-order coupled vibration model to be studied in this paper, the calculation is cumbersome and not conducive to the development of research work. Thus, the Hopf bifurcation of parameter is taken as an example. The algebraic criterion and calculation method of Hopf bifurcation occurring when the equilibrium point loses stability proposed in reference [23] are adopted, and the existence of the Hopf bifurcation point is directly determined based on the Hurwitz algebraic criterion, with three theorems first introduced as follows.

Theorem 1. The necessary and sufficient conditions for the characteristic equation (15) to have a pair of pure imaginary eigenvalue and the remaining n − 2 eigenvalues to have negative real parts arewhere is the Hurwitz determinant of the characteristic equation (15).

Theorem 2. If the Hurwitz determinant of the characteristic equation (15) satisfiesthen the characteristic equation (15) has a pair of pure imaginary eigenvalues , and the remaining n − 2 eigenvalues have negative real parts, and

Theorem 3. If the eigenvalues of characteristic equation (15) at all have negative real parts, and the Hurwitz determinant satisfies the following conditions:(i) where , then the characteristic equation (15) has a pair of pure imaginary eigenvalues at , and the other eigenvalues all have negative real parts. Let U and V be the left and right eigenvectors of the matrix corresponding to the eigenvalues , respectively, UV = 1.(ii)Re(UBV) ≠ 0, where .Then, the Hopf bifurcation occurs at , namely, there is periodic motion in the system near the parameters .

Applying the aforementioned theorems to the problem in this paper, according to Theorem 1, the characteristic equation (15) has a pair of pure imaginary eigenvalues, and the remaining n − 2 eigenvalues have negative real parts; then, we can obtain

The range of control parameter when the characteristic equation (15) has a pair of pure imaginary eigenvalues and the remaining n − 2 eigenvalues all have negative real parts is

According to Theorem 2, let , and discuss the bifurcation parameter . Get the critical value according to the equation and substitute it in other parameter expressions. The calculation results are shown in Table 2.

It can be seen that and hold. Directly calculating the pair of pure imaginary eigenvalues of characteristic equation (15) of the system, we have , and thus .

According to Theorem 3, substitute the obtained pure imaginary eigenvalues into the system equation, and we can obtain left and right eigenvectors corresponding to , satisfying UV = 1:

Let , and we have .

Therefore, there is Re (UBV) = 0.0049 ≠ 0, which satisfies the transversal condition. Then, when it is set that , is a Hopf bifurcation point of the system.

4.2. Direction and Stability of Hopf Bifurcating Periodic Solutions

Hassard et al. used the center manifold theory and the normal form approach to derive the formula for determining the Hopf bifurcation direction, periodic solution stability, and periodic variation of high-dimensional dynamic systems [24, 25] and further analyze the stability of periodic solutions near the aforementioned control parameters and Hopf bifurcation direction. Suppose there is a transformation matrix P that transforms the system Jacobian matrix into the following Jordan standard form:where the matrix , in which is the eigenvector corresponding to complex eigenvalue and is the real part of while is the imaginary part of ; D is the diagonal matrix composed of real eigenvalues , , and ; , , and are the eigenvectors of the corresponding eigenvalues; and P and are given by

By substituting , the original system is transformed into standardized form , which can be expressed as

At this time, the equilibrium point of the new system (26) is the original point, is the Jacobian matrix of system (26), and is a high-order term. Then, the following values are obtained:

Based on the above calculated values, letwhere and .

Theorem 4. When the bifurcation parameter of system (13) is increased beyond the critical value , the Hopf bifurcation occurs at the equilibrium point , and the following characteristics exist:(a)When , the Hopf bifurcation is a supercritical (subcritical) bifurcation, and a bifurcation periodic solution for exists(b)When , the bifurcation periodic solution is asymptotically stable (unstable)(c)When , the period of the periodic solution increases (decreases)Substitute the system parameters to discuss the bifurcation direction when and . The calculation results of the above values are shown in Table 3.

Substituting the values yields , , and .

It is known from that the Hopf bifurcation of system (13) at is a supercritical bifurcation, that is, when , the equilibrium point of the dynamical system is stable, yet Hopf bifurcation occurs as is gradually increased to . indicates that the periodic solution of the system bifurcation is a stable limit cycle, and the period of the periodic solution increases as the bifurcation parameter increases because .

5. Numerical Simulation

In this part, numerical simulation is used to verify the results of the above theoretical analysis. The dynamic characteristics of the system when and are analyzed through simulation, while other system parameters in Table 1 remain unchanged.

Give the system a small external disturbance, and the local Hopf bifurcation diagram of system (13) with as the bifurcation parameter is shown in Figure 3. It can be seen that the system has a supercritical Hopf bifurcation, that is, the system equilibrium point is in a stable state when and Hopf bifurcation occurs when is increased to , while a stable limit cycle appears when .

Figures 4–7 show the time history curves and phase before and after the bifurcation occurs when takes different values. It can be seen from the figure that the equilibrium point of the system is stable when , yet the stability of the system equilibrium point is lost and a set of stable limit cycles is branched when is increased beyond .

It can also be seen from Figures 6 and 7 that each periodic solution is stable because , and indicates that the period of the periodic solution increases as the bifurcation parameter increases so that the period of the periodic solution when is greater than that when .

6. Conclusions

(1)The dynamics model of the single electromagnet-track beam coupled system of the EMS maglev train is established, with the classical PD control method adopted and the influence of the first-order vibration mode of the track beam taken into consideration.(2)It is convenient to determine the stability of the system motion and the Hopf bifurcation, from the composition and symbol of the eigenvalues of the system based on the Hurwitz algebraic criterion instead of solving the specific eigenvalues.(3)Take as the bifurcation parameter. When , the critical Hopf bifurcation value is calculated as , and it is categorized as supercritical Hopf bifurcation, that is, when , the system is asymptotically stable, yet Hopf bifurcation occurs as increases gradually beyond , with the stability of the system lost and a stable limit cycle branched.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (grant no. 51875483), National Key R&D Program of China (grant nos. 2016YFB1200601-A03 and 2016YFB1200602-13), and Henan Engineering Research Center of Rail Transit Intelligent Security Open Fund Project 2019 (grant no. 2019KFJJ002).

References

D. F. Zhou, C. H. Hansen, J. Li et al., “Review of coupled vibration problems in EMS maglev vehicles,” International Journal of Acoustics and Vibration, vol. 15, no. 1, pp. 10–23, 2010.
View at: Publisher Site | Google Scholar
M. Nagai, “The control of magnetic suspension to suppress the self-excited vibration of a flexible guideway,” Transactions of the Japan Society of Mechanical Engineers Series C, vol. 52, no. 478, pp. 1780–1786, 1986.
View at: Publisher Site | Google Scholar
Y. Cai, S. S. Chen, D. M. Rote, and H. T. Coffey, “Vehicle/guideway interaction for high speed vehicles on a flexible guideway,” Journal of Sound and Vibration, vol. 175, no. 5, pp. 625–646, 1994.
View at: Publisher Site | Google Scholar
J.-S. Lee, S.-D. Kwon, M.-Y. Kim, and I. H. Yeo, “A parametric study on the dynamics of urban transit maglev vehicle running on flexible guideway bridges,” Journal of Sound and Vibration, vol. 328, no. 3, pp. 301–317, 2009.
View at: Publisher Site | Google Scholar
K.-J. Kim, J.-B. Han, H.-S. Han, and S.-J. Yang, “Coupled vibration analysis of maglev vehicle-guideway while standing still or moving at low speeds,” Vehicle System Dynamics, vol. 53, no. 4, pp. 587–601, 2015.
View at: Publisher Site | Google Scholar
J.-B. Han, H.-S. Han, S.-S. Kim, S.-J. Yang, and K.-J. Kim, “Design and validation of a slender guideway for maglev vehicle by simulation and experiment,” Vehicle System Dynamics, vol. 54, no. 3, pp. 370–385, 2016.
View at: Publisher Site | Google Scholar
Y. D. Xie and W. S. Chang, “Modelling and simulation of electromagnetic suspension vehicle systems (EMS) in vertical direction,” Journal of the China Railway Society, vol. 18, no. 4, pp. 47–54, 1996.
View at: Google Scholar
C. F. Zhao and W. M. Zhai, “Maglev vehicle/guideway vertical random response and ride quality,” Vehicle System Dynamics, vol. 38, no. 3, pp. 185–210, 2002.
View at: Publisher Site | Google Scholar
W. H. Ma, R. R. Song, J. Q. Xu, Y. Zang, and S. Luo, “A coupling vibration test bench and the simulation research of a maglev vehicle,” Shock and Vibration, vol. 2015, Article ID 586910, 14 pages, 2015.
View at: Publisher Site | Google Scholar
Y. H. Zhou, J. J. Wu, X. J. Zheng et al., “Analysis of dynamic stability for magnetic levitation vehicles by liapunov characteristic number,” Acta Mechanica Sinica, vol. 32, no. 1, pp. 42–51, 2000.
View at: Google Scholar
X. H. Shi, L. H. She, and W. S. Chang, “The bifurcation analysis of the EMS maglev vehicle-coupled-guideway system,” Acta Mechanica Sinica, vol. 36, no. 5, pp. 634–640, 2004.
View at: Publisher Site | Google Scholar
H.-P. Wang, J. Li, and K. Zhang, “Stability and hopf bifurcation of the maglev system with delayed speed feedback control,” Acta Automatica Sinica, vol. 33, no. 8, pp. 829–834, 2007.
View at: Publisher Site | Google Scholar
H. Wang, J. Li, and K. Zhang, “Non-resonant response, bifurcation and oscillation suppression of a non-autonomous system with delayed position feedback control,” Nonlinear Dynamics, vol. 51, no. 3, pp. 447–464, 2008.
View at: Publisher Site | Google Scholar
J.-H. Li, J. Li, D.-F. Zhou, and P.-C. Yu, “Self-excited vibration problems of maglev vehicle-bridge interaction system,” Journal of Central South University, vol. 21, no. 11, pp. 4184–4192, 2014.
View at: Publisher Site | Google Scholar
C. F. Zhao and W. M. Zhai, “Dynamic characteristics of electromagnet levitation system,” Journal of Southwest Jiaotong University, vol. 39, no. 4, pp. 464–468, 2004.
View at: Google Scholar
H. Wang and G. Shen, “Small-scale similarity model of maglev-guideway coupling vibration,” Journal of Traffic and Transportation Engineering, vol. 14, no. 1, pp. 49–56, 2014.
View at: Google Scholar
H. Wang, X. B. Zhong, and G. Shen, “Levitation control strategy for maglev on elastic track based on kalman filter,” Journal of Central South University, vol. 45, no. 3, pp. 965–972, 2014.
View at: Google Scholar
L. L. Zhang, L. H. Huang, and Z. Z. Zhang, “Stability and hopf bifurcation of the maglev system with delayed position and speed feedback control,” Nonlinear Dynamics, vol. 57, no. 1-2, pp. 197–207, 2009.
View at: Publisher Site | Google Scholar
L. L. Zhang, Z. Z. Zhang, Z. Q. Long, and A. M. Hao, “Sliding mode control with auto-tuning law for maglev system,” Engineering, vol. 2, no. 2, pp. 107–112, 2010.
View at: Publisher Site | Google Scholar
H. C. Wu and H. R. Fang, “Analysis on stability of suspension control system based on Hurwitz criterion,” Journal of the China Railway Society, vol. 34, no. 12, pp. 40–45, 2012.
View at: Google Scholar
Y. Sun, J. Xu, H. Qiang, W. Wang, and G. Lin, “Hopf bifurcation analysis of maglev vehicle-guideway interaction vibration system and stability control based on fuzzy adaptive theory,” Computers in Industry, vol. 108, pp. 197–209, 2019.
View at: Publisher Site | Google Scholar
X. Li, D. Wang, D. Liu, L. Xin, and X. Zhang, “Dynamic analysis of the interactions between a low-to-medium-speed maglev train and a bridge: field test results of two typical bridges,” Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, vol. 232, no. 7, pp. 2039–2059, 2018.
View at: Publisher Site | Google Scholar
J. Y. Zhang, Y. R. Yang, and J. Zeng, “An algorithm criterion for Hopf bifurcation and its applications in vehicle dynamics,” Acta Mechanica Sinica, vol. 32, no. 5, pp. 596–605, 2000.
View at: Google Scholar
B. Hassard, “Bifurcation of periodic solutions of the Hodgkin-Huxley model for the squid giant axon,” Journal of Theoretical Biology, vol. 71, no. 3, pp. 401–420, 1978.
View at: Publisher Site | Google Scholar
B. Hassard and Y. H. Wan, “Bifurcation formulae derived from center manifold theory,” Journal of Mathematical Analysis and Applications, vol. 63, no. 1, pp. 297–312, 1978.
View at: Publisher Site | Google Scholar

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Copyright © 2020 Junxiong Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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