Levitation Stability and Hopf Bifurcation of EMS Maglev Trains

-is paper analyzed the mechanical characteristics of single electromagnet system and elastic track beam of EMSmaglev 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.-en, 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 kp as the bifurcation parameter; and numerical simulation is conducted to verify the validity of the theoretical analysis results. -e 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 kp < kp0, the real-time system is asymptotically stable, yet Hopf bifurcation occurs as kp increases gradually beyond kp0, with the stability of the system lost and a stable limit cycle branched.


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. e 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].
is 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][5][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 selfdeveloped low-speed maglev trains [8]. Ma et al. established test bench for electromechanical coupled vibration of singlelevitation 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]. e 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 vehicletrack beam coupled dynamics with periodically variable coefficients. e 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 selfexcited 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 smallscale 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]. ey 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. e 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 k p 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. e fourth part uses the numerical simulation method to verify the theoretical analysis results.

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. e 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. e 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. e 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 F is levitation force, μ 0 is vacuum permeability, N is the turns of electromagnet coil, A m is the area of electromagnet pole, m is sum of a single electromagnet's own weight and load weight, I is exciting current, R is electromagnet resistance, u is the voltage at electromagnet end, z is absolute gap between the electromagnet and the rigid track, and δ is actual levitation gap of the electromagnet. e voltage feedback control adopts the classic PD control. k p and k d are the gap offset and the gap differential feedback coefficients, respectively. erefore, where u 0 is the initial voltage at equilibrium position and δ 0 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 E is Young's modulus of elasticity, I g is beam section inertia, C is equivalent damping coefficient of the beam, ρ is beam density, A g is area of beam section, and F(x , t) 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 ∅ i (x) and a generalized coordinate q i (t): Integrating equation (4) into equation (3) gives the vibration equation corresponding to the i th -order mode of the beam where ω i is the i th -order natural frequency of the beam, η i is the i th -order damping ratio of the beam, and Q i is the i thorder generalized force. For the simply supported single-span beam with the length of L in Figure 1, the natural frequency ω i and the regular mode function ∅ i (x) 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]. erefore, 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 x 0 on the track beam when seeking the limit through the generalized force expression. e first-order vibration equation of the generalized coordinates of the track can be rewritten as where ∅ 1 � ���� � 2/m G · sin(πx 0 /L), ω 1 � (π 2 /L 2 ) ��������� (EI g /ρA g ), and 2η 1 ω 1 � (C/ρ).
Introducing the deformation of the track beam z G � ∅ 1 q 1 into the vibration equation (8), where ∅ 1 is a constant, we can obtain In the midspan position where the amplitude of the simply supported beam is the largest, we have x 0 � (L/2), and then ∅ 1 � ������ (2/m G ). erefore, the coupled system model under the influence of the first-order vibration mode of the track can be defined by

Equilibrium Points and Their Stability
Let X � x 1 x 2 x 3 x 4 x 5 T � z _ z δ _ δ I T and levitation gap δ � z − z G ; then, the canonical form of the coupled system model (10) is determined as follows:

Mathematical Problems in Engineering
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. e 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 T � 0 and we have mg � k(I 0 /δ 0 ) 2 at the desired levitation gap and u 0 � I 0 R; then, the equilibrium point can be given by Shift the equilibrium point X 0 of the system (11) to the original point o 0 0 0 0 0 through the transformation X � X + X 0 , and the new system (13) is expressed as e Jacobian matrix of system (13) at the original point o 0 0 0 0 0 is given by 4 Mathematical Problems in Engineering e corresponding characteristic equation is written as Model parameters are shown in Table 1.
Substituting the parameters, the coefficients of characteristic equation (15) 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, erefore, the levitation control feedback coefficient k p and k d when the system is stable can be obtained, and the value range is shown in Figure 2

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. us, the Hopf bifurcation of parameter k p is taken as an example. e 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 2. If the Hurwitz determinant of the characteristic equation (15) satisfies
then the characteristic equation (15) has a pair of pure imaginary eigenvalues ± ωi, and the remaining n − 2 eigenvalues have negative real parts, and Theorem 3. If the eigenvalues of characteristic equation (15) at μ � μ 0 all have negative real parts, and the Hurwitz determinant satisfies the following conditions: where μ c � min μ |μ − μ 0 |: Δ n− 1 (μ) � 0 , then the characteristic equation (15) has a pair of pure imaginary eigenvalues at μ � μ c , and the other eigenvalues all have negative real parts. Let U and V be the left and right eigenvectors of the matrix A(μ c ) corresponding to the eigenvalues iω c , respectively, UV � 1.

Mathematical Problems in Engineering
en, the Hopf bifurcation occurs at μ � μ c , namely, there is periodic motion in the system near the parameters μ � μ c .
Applying the aforementioned theorems to the problem in this paper, according to eorem 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 a i > 0, (i � 1, 2, . . . , n), According to eorem 2, let k d � 100, and discuss the bifurcation parameter k p . Get the critical value k p 0 � 1690.681 according to the equation Δ 4 � 0 and substitute it in other parameter expressions. e calculation results are shown in Table 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 k p 0 and Hopf bifurcation direction. Suppose there is a transformation matrix P that transforms the system Jacobian matrix A(k p 0 ) into the following Jordan standard form: where the matrix P � Re(v 1 ) − Im(v 1 ) v 3 v 4 v 5 , in which v 1 is the eigenvector corresponding to complex eigenvalue λ 1 (k p 0 ) � ω 0 i � 20.044i and Re(v 1 ) is the real part of v 1 while − Im(v 1 ) is the imaginary part of v 1 ; D is the diagonal matrix composed of real eigenvalues λ 3 (k p 0 ), λ 4 (k p 0 ), and λ 5 (k p 0 ); v 3 , v 4 , and v 5 are the eigenvectors of the corresponding eigenvalues; and P and P − 1 are given by By substituting Y � P − 1 X, the original system , which can be expressed as _ y 1 � − 20.044y 2 + F 1 y 1 , y 2 , y 3 , y 4 , y 5 , _ y 2 � 20.044y 1 + F 2 y 1 , y 2 , y 3 , y 4 , y 5 , _ y 3 � (− 1.379 + 60.098i)y 3 + F 3 y 1 , y 2 , y 3 , y 4 , y 5 , _ y 4 � (− 1.379 − 60.098i)y 3 + F 4 y 1 , y 2 , y 3 , y 4 , y 5 , _ y 5 � − 2.389y 5 + F 5 y 1 , y 2 , y 3 , y 4 , y 5 . Amplitude of δ (m)

Conclusions
(1) e dynamics model of the single electromagnettrack 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 k p as the bifurcation parameter. When k d � 100, the critical Hopf bifurcation value is calculated as k p 0 � 1690.681, and it is categorized as supercritical Hopf bifurcation, that is, when k p < k p 0 , the system is asymptotically stable, yet Hopf bifurcation occurs as k p increases gradually beyond k p 0 , with the stability of the system lost and a stable limit cycle branched.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.