# Levitation Stability and Hopf Bifurcation of EMS Maglev Trains

Authors: Junxiong Hu; Weihua Ma; Xiaohao Chen; Shihui Luo
Journal: Mathematical Problems in Engineering
 (2020)

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## 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 coefficientkp 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 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.


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## Body

## 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 coefficientkp 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 theF-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.Figure 1
Simplified single electromagnet-track beam coupled model.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:(1)electromagnet force equation:F=kIδ2,k=μ0N2Am4,dynamic equation:mz¨=mg−F,electrical equation:u=RI+2kδI˙−2kIδ2δ˙,where F is levitation force, μ0 is vacuum permeability, N is the turns of electromagnet coil, Am 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.The voltage feedback control adopts the classic PD control.kp and kd are the gap offset and the gap differential feedback coefficients, respectively. Therefore,(2)uc=u0+kpδ−δ0+kdδ˙,where u0 is the initial voltage at equilibrium position and δ0 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:(3)EIg∂4zG∂4x+ρAg∂2zG∂2t+C∂zG∂t=Fx,t,where E is Young’s modulus of elasticity, Ig is beam section inertia, C is equivalent damping coefficient of the beam, ρ is beam density, Ag is area of beam section, and Fx,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 ∅ix and a generalized coordinate qit:(4)zGx,t=∑i=1n∅ix⋅qit.Integrating equation (4) into equation (3) gives the vibration equation corresponding to the ith-order mode of the beam(5)q¨it+2ηiωiq˙it+ωi2qit=Qi,where ωi is the ith-order natural frequency of the beam, ηi is the ith-order damping ratio of the beam, and Qi is the ith-order generalized force.For the simply supported single-span beam with the length ofL in Figure 1, the natural frequency ωi and the regular mode function ∅ix of the beam are, respectively, given by(6)ωi=iπL2EIρAg,(7)∅ix=2mG⋅siniπxL,0≤x≤L.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 x0 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 as(8)q¨1+2η1ω1q˙1=∅1F,where ∅1=2/mG⋅sinπx0/L, ω1=π2/L2EIg/ρAg, and 2η1ω1=C/ρ.Introducing the deformation of the track beamzG=∅1q1 into the vibration equation (8), where ∅1 is a constant, we can obtain(9)z¨G+2η1ω1z˙G+ω12zG=∅12F.In the midspan position where the amplitude of the simply supported beam is the largest, we havex0=L/2, and then ∅1=2/mG.Therefore, the coupled system model under the influence of the first-order vibration mode of the track can be defined by(10)mz¨=mg−F,I˙=Iδδ˙+δ2ku0+kpδ−δ0+kdδ˙−RI2kδ,z¨G+2η1ω1z˙G+ω12zG=∅12F.


## 3. Equilibrium Points and Their Stability

LetX=x1x2x3x4x5T=zz˙δδ˙IT and levitation gap δ=z−zG; then, the canonical form of the coupled system model (10) is determined as follows:(11)x˙1=x2,x˙2=g−kmx5x32,x˙3=x4,x˙4=g−kmx5x32−−2η1ω1x2−x4−ω12x1−x3+∅12kx5x32,x˙5=x5x3x4+x32ku0+kpx3−δ0+kdx4−R2kx3x5.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.LetX˙=x˙1x˙2x˙3x˙4x˙5T=0 and we have mg=kI0/δ02 at the desired levitation gap and u0=I0R; then, the equilibrium point can be given by(12)X0=x10x20x30x40x50T=δ0+mg∅12ω120δ00δ0mgkT.Shift the equilibrium pointX0 of the system (11) to the original point o00000 through the transformation X¯=X+X0, and the new system (13) is expressed as(13)x˙¯1=x¯2,x˙¯2=g−kmx¯5+x50x¯3+x302,x˙¯3=x¯4,x˙¯4=g−kmx¯5+x50x¯3+x302−−2η1ω1x¯2−x¯4−ω12x¯1+mg∅12ω12−x¯3+∅12kx¯5+x50x¯3+x302,x˙¯5=x¯5+x50x¯3+x30x¯4+x¯3+x302ku0+kpx¯3+kdx¯4−R2kx¯3+x30x¯5+x50.The Jacobian matrix of system (13) at the original point o00000 is given by(14)A=01000002gδ00−2kmδ0mgk00010ω122η1ω12g1+m∅12δ0−ω12−2η1ω1−2k1+m∅12mδ0mgk00kpδ02kkdδ02k+mgk−Rδ02k.The corresponding characteristic equation is written as(15)fλ=λ5+a1λ4+a2λ3+a3λ2+a4λ+a5.Model parameters are shown in Table1.Table 1
Model parameter values.
μ
0
M (kg)
R (Ω)
N
δ
0 (m)
4πe − 7
750
0.48
320
0.008
A
m (m2)
m
G (kg)
ω
1 (rad/s)
η
1
0.025
5e4
60
0.023Substituting the parameters, the coefficients of characteristic equation (15) can be obtained:(16)a1=5.147,a2=4.154kd+3600.589,a3=11.131kd+4.154kp+2563.807,a4=14518.156kd+11.131kp−16159.558,a5=14518.156kp−21077684.888.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,(17)ai>0,i=1∼5,Δ2=a11a3a2=a1a2−a3>0,Δ4=a1100a3a2a11a5a4a3a200a5a4=−a12a42−a1a22a5+a1a2a3a4+2a1a4a5+a2a3a5−a32a4−a52>0.Therefore, the levitation control feedback coefficientkp and kd when the system is stable can be obtained, and the value range is shown in Figure 2.Figure 2
Relationship between system stability region and levitation control feedback coefficients.


## 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 parameterkp 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 are(18)ai>0,i=1,2,…,n,Δi>0,i=n−3,n−5,…,Δn−1=0,where Δi is the Hurwitz determinant of the characteristic equation (15).Theorem 2.
If the Hurwitz determinant of the characteristic equation (15) satisfies(19)ai>0,i=1,2,…,n,Δi>0,i=n−3,n−5,…,Δn−1=0,then the characteristic equation (15) has a pair of pure imaginary eigenvalues ±ωi, and the remaining n − 2 eigenvalues have negative real parts, and(20)ω2=Δn−3Δn−2an.Theorem 3.
If the eigenvalues of characteristic equation (15) at μ=μ0 all have negative real parts, and the Hurwitz determinant satisfies the following conditions:(i)
Δn−3μc>0whereμc=minμμ−μ0:Δn−1μ=0, then the characteristic equation (15) has a pair of pure imaginary eigenvalues at μ=μc, and the other eigenvalues all have negative real parts. Let U and V be the left and right eigenvectors of the matrix Aμc corresponding to the eigenvalues iωc, respectively, UV = 1.(ii)
Re(UBV) ≠ 0, where B=dAμ/dμμ=μc.
Then, the Hopf bifurcation occurs atμ=μc, namely, there is periodic motion in the system near the parameters μ=μc.Applying the aforementioned theorems to the problem in this paper, according to Theorem1, 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(21)ai>0,i=1,2,…,n,Δ2=a11a3a2=10.250kd−4.154kp+16000.476>0,Δ4=a1100a3a2a11a5a4a3a200a5a4=1.656e6kd3−192kp3−6.7e5kd2kp−40.69kdkp2−1.722e9kd2+9.281e6kp2+1.109e9kdkp−3.655e11kd−5.468e11kp+7.749e14=0.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(22)1451.8<kp<2.468kd+3851.824.According to Theorem2, let kd=100, and discuss the bifurcation parameter kp. Get the critical value kp0=1690.681 according to the equation Δ4=0 and substitute it in other parameter expressions. The calculation results are shown in Table 2.It can be seen thatai>0 and Δ2>0 hold. Directly calculating the pair of pure imaginary eigenvalues of characteristic equation (15) of the system, we have ω02=Δ2/Δ3a5=401.766, and thus λ1,2kp0=±ω0i=±20.044i.According to Theorem3, substitute the obtained pure imaginary eigenvalues into the system equation, and we can obtain left and right eigenvectors corresponding to λ1kp0=±ω0i=±20.044i, satisfying UV = 1:(23)U=−1.973e3−33.863i,0.177+98.472i,1.246e4+1.427e3i,−0.189−10.987i,−3.476−0.415i,V=−2.850e−4+1.601e−8i,−3.209e−7−5.712e−3i,−2.839e−4−2.623e−9i,5.255e−8−5.691e−3i,−1T.LetB=dAkp0/dkp0=00000000000000000000004.97400, and we have UBV=0.0049+0.0006i.Therefore, there is Re (UBV) = 0.0049 ≠ 0, which satisfies the transversal condition. Then, when it is set that kd=100, kp0=1690.681 is a Hopf bifurcation point of the system.Table 2
Calculation results whenkd=100 and kp0=1690.681.
a
2
a
3
a
4
a
5
Δ
2
Δ
3
Δ
2
/
Δ
3
4.022e3
1.070e4
1.455e6
3.468e6
1.000e4
8.634e7
1.159e − 4


### 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 kp0 and Hopf bifurcation direction. Suppose there is a transformation matrix P that transforms the system Jacobian matrix Akp0 into the following Jordan standard form:(24)P−1Akp0P=0−ω0ω0000D=Jkp0,where the matrix P=Rev1−Imv1v3v4v5, in which v1 is the eigenvector corresponding to complex eigenvalue λ1kp0=ω0i=20.044i and Rev1 is the real part of v1 while −Imv1 is the imaginary part of v1; D is the diagonal matrix composed of real eigenvalues λ3kp0, λ4kp0, and λ5kp0; v3, v4, and v5 are the eigenvectors of the corresponding eigenvalues; and P and P−1 are given by(25)P=−2.850e−4−1.601e−8−3.165e−5+1.454e−6i−3.165e−5−1.454e−6i−3.314e−4−3.209e−75.712e−3−4.373e−5−1.904e−3i−4.373e−5+1.904e−3i7.915e−4−2.839e−42.623e−9−2.839e−4−8.870e−10i−2.839e−4+8.870e−10i−3.314e−45.255e−85.691e−33.912e−4−1.706e−2i3.912e−4+1.706e−2i7.916e−4−10−1,P−1=−3.947e30.3532.493e4−0.377−6.95367.7261.969e22.853e3−21.974−0.8301.974e3−56.591i−0.181−32.843i−1.928e3+57.127i0.189+32.967i−0.015−7.937e−5i1.974e3+56.591i−0.181+32.843i−1.928e3−57.127i0.189−32.967i−0.015+7.937e−5i−2.312e−29.660e−3−2.107e41.534e−55.983.By substitutingY=P−1X¯, the original system X˙¯=fX¯,kp0 is transformed into standardized form Y˙=P−1fPY,kp0, which can be expressed as(26)y˙1=−20.044y2+F1y1,y2,y3,y4,y5,y˙2=20.044y1+F2y1,y2,y3,y4,y5,y˙3=−1.379+60.098iy3+F3y1,y2,y3,y4,y5,y˙4=−1.379−60.098iy3+F4y1,y2,y3,y4,y5,y˙5=−2.389y5+F5y1,y2,y3,y4,y5.At this time, the equilibrium pointY0 of the new system (26) is the original point, Jkp0 is the Jacobian matrix of system (26), and Fiy1,y2,y3,y4,y5 is a high-order term. Then, the following values are obtained:(27)g20=14∂2F1∂y12−∂2F1∂y22+2∂2F2∂y1y2+i∂2F2∂y12−∂2F2∂y22−2∂2F2∂y1∂y2,g11=14∂2F1∂y12+∂2F2∂y22+i∂2F2∂y12+∂2F2∂y22,g02=14∂2F1∂y12−∂2F1∂y22−2∂2F2∂y1∂y2+i∂2F2∂y12−∂2F2∂y22+2∂2F2∂y1∂y2,G21=18∂3F1∂y13+∂3F1∂y1∂y22+∂3F2∂y12∂y2+∂3F2∂y23+i∂3F2∂y13+∂3F2∂y1∂y22−∂3F1∂y12∂y2−∂3F1∂y23,G110k−2=12∂2F1∂y1∂yk+∂2F2∂y2∂yk+i∂2F2∂y1∂yk−∂2F1∂y2∂ykk=3,4,5G101k−2=12∂2F1∂y1∂yk−∂2F2∂y2∂yk+i∂2F2∂y1∂yk+∂2F1∂y2∂ykk=3,4,5,w11k−2=−14λk∂2Fk∂y12+∂2Fk∂y22k=3,4,5,w20k−2=142ω0i−λk∂2Fk∂y12−∂2Fk∂y22−2i∂2Fk∂y1∂y2k=3,4,5,g21=G21+∑k=352G110k−2w11k−2+G101k−2w20k−2.Based on the above calculated values, let(28)C10=i2ω0g20g21−2g112−13g022+12g21,μ2=−ReC10α′0,τ2=−ImC10+μ2ω′0ω0,β2=2ReC10,where α′0=Reλ′kpkp=kp0 and ω′0=Imλ′kpkp=kp0.Theorem 4.
When the bifurcation parameterkp of system (13) is increased beyond the critical value kp0, the Hopf bifurcation occurs at the equilibrium point o00000, and the following characteristics exist:(a)
Whenμ2>0μ2<0, the Hopf bifurcation is a supercritical (subcritical) bifurcation, and a bifurcation periodic solution for kp>kp0kp<kp0 exists(b)
Whenβ2<0β2>0, the bifurcation periodic solution is asymptotically stable (unstable)(c)
Whenτ2>0τ2<0, the period of the periodic solution increases (decreases)
Substitute the system parameters to discuss the bifurcation direction whenkd=100 and kp0=1690.681. The calculation results of the above values are shown in Table 3.Substituting the values yieldsμ2=15.510>0, β2=−0.152<0, and τ2=6.682e−4>0.It is known fromμ2>0 that the Hopf bifurcation of system (13) at kp0 is a supercritical bifurcation, that is, when kp<kp0, the equilibrium point of the dynamical system is stable, yet Hopf bifurcation occurs as kp is gradually increased to kp0. β2<0 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 kp increases because τ2>0.Table 3
Calculation results of the relevant values in the bifurcation direction.
g
20
0.696–4.17e − 2i
G
110
3
1.171
w
11
3
−7.41e − 2
g
11
0.655
G
101
1
0.945–1.172i
w
20
1
−1.369e − 4 + 1.054e − 4i
g
02
0.613 + 4.17e − 2i
G
101
2
0.945 + 1.172i
w
20
2
−1.255e − 5–8.619e − 6i
G
21
2.27e − 2
G
101
3
1.184
w
20
3
−3e − 4 + 7.3e − 3i
G
110
1
1.727–1.172i
w
11
1
−2.304e − 5 + 7.558e − 6i
g
21
−0.151 + 8.9e − 3i
G
110
2
1.727 + 1.172i
w
11
2
−2.304e − 5–7.558e − 6i
C
1
0
−0.076–2.27e − 2i


## 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 parameterkp 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 are(18)ai>0,i=1,2,…,n,Δi>0,i=n−3,n−5,…,Δn−1=0,where Δi is the Hurwitz determinant of the characteristic equation (15).Theorem 2.
If the Hurwitz determinant of the characteristic equation (15) satisfies(19)ai>0,i=1,2,…,n,Δi>0,i=n−3,n−5,…,Δn−1=0,then the characteristic equation (15) has a pair of pure imaginary eigenvalues ±ωi, and the remaining n − 2 eigenvalues have negative real parts, and(20)ω2=Δn−3Δn−2an.Theorem 3.
If the eigenvalues of characteristic equation (15) at μ=μ0 all have negative real parts, and the Hurwitz determinant satisfies the following conditions:(i)
Δn−3μc>0whereμc=minμμ−μ0:Δn−1μ=0, then the characteristic equation (15) has a pair of pure imaginary eigenvalues at μ=μc, and the other eigenvalues all have negative real parts. Let U and V be the left and right eigenvectors of the matrix Aμc corresponding to the eigenvalues iωc, respectively, UV = 1.(ii)
Re(UBV) ≠ 0, where B=dAμ/dμμ=μc.
Then, the Hopf bifurcation occurs atμ=μc, namely, there is periodic motion in the system near the parameters μ=μc.Applying the aforementioned theorems to the problem in this paper, according to Theorem1, 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(21)ai>0,i=1,2,…,n,Δ2=a11a3a2=10.250kd−4.154kp+16000.476>0,Δ4=a1100a3a2a11a5a4a3a200a5a4=1.656e6kd3−192kp3−6.7e5kd2kp−40.69kdkp2−1.722e9kd2+9.281e6kp2+1.109e9kdkp−3.655e11kd−5.468e11kp+7.749e14=0.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(22)1451.8<kp<2.468kd+3851.824.According to Theorem2, let kd=100, and discuss the bifurcation parameter kp. Get the critical value kp0=1690.681 according to the equation Δ4=0 and substitute it in other parameter expressions. The calculation results are shown in Table 2.It can be seen thatai>0 and Δ2>0 hold. Directly calculating the pair of pure imaginary eigenvalues of characteristic equation (15) of the system, we have ω02=Δ2/Δ3a5=401.766, and thus λ1,2kp0=±ω0i=±20.044i.According to Theorem3, substitute the obtained pure imaginary eigenvalues into the system equation, and we can obtain left and right eigenvectors corresponding to λ1kp0=±ω0i=±20.044i, satisfying UV = 1:(23)U=−1.973e3−33.863i,0.177+98.472i,1.246e4+1.427e3i,−0.189−10.987i,−3.476−0.415i,V=−2.850e−4+1.601e−8i,−3.209e−7−5.712e−3i,−2.839e−4−2.623e−9i,5.255e−8−5.691e−3i,−1T.LetB=dAkp0/dkp0=00000000000000000000004.97400, and we have UBV=0.0049+0.0006i.Therefore, there is Re (UBV) = 0.0049 ≠ 0, which satisfies the transversal condition. Then, when it is set that kd=100, kp0=1690.681 is a Hopf bifurcation point of the system.Table 2
Calculation results whenkd=100 and kp0=1690.681.
a
2
a
3
a
4
a
5
Δ
2
Δ
3
Δ
2
/
Δ
3
4.022e3
1.070e4
1.455e6
3.468e6
1.000e4
8.634e7
1.159e − 4


## 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 kp0 and Hopf bifurcation direction. Suppose there is a transformation matrix P that transforms the system Jacobian matrix Akp0 into the following Jordan standard form:(24)P−1Akp0P=0−ω0ω0000D=Jkp0,where the matrix P=Rev1−Imv1v3v4v5, in which v1 is the eigenvector corresponding to complex eigenvalue λ1kp0=ω0i=20.044i and Rev1 is the real part of v1 while −Imv1 is the imaginary part of v1; D is the diagonal matrix composed of real eigenvalues λ3kp0, λ4kp0, and λ5kp0; v3, v4, and v5 are the eigenvectors of the corresponding eigenvalues; and P and P−1 are given by(25)P=−2.850e−4−1.601e−8−3.165e−5+1.454e−6i−3.165e−5−1.454e−6i−3.314e−4−3.209e−75.712e−3−4.373e−5−1.904e−3i−4.373e−5+1.904e−3i7.915e−4−2.839e−42.623e−9−2.839e−4−8.870e−10i−2.839e−4+8.870e−10i−3.314e−45.255e−85.691e−33.912e−4−1.706e−2i3.912e−4+1.706e−2i7.916e−4−10−1,P−1=−3.947e30.3532.493e4−0.377−6.95367.7261.969e22.853e3−21.974−0.8301.974e3−56.591i−0.181−32.843i−1.928e3+57.127i0.189+32.967i−0.015−7.937e−5i1.974e3+56.591i−0.181+32.843i−1.928e3−57.127i0.189−32.967i−0.015+7.937e−5i−2.312e−29.660e−3−2.107e41.534e−55.983.By substitutingY=P−1X¯, the original system X˙¯=fX¯,kp0 is transformed into standardized form Y˙=P−1fPY,kp0, which can be expressed as(26)y˙1=−20.044y2+F1y1,y2,y3,y4,y5,y˙2=20.044y1+F2y1,y2,y3,y4,y5,y˙3=−1.379+60.098iy3+F3y1,y2,y3,y4,y5,y˙4=−1.379−60.098iy3+F4y1,y2,y3,y4,y5,y˙5=−2.389y5+F5y1,y2,y3,y4,y5.At this time, the equilibrium pointY0 of the new system (26) is the original point, Jkp0 is the Jacobian matrix of system (26), and Fiy1,y2,y3,y4,y5 is a high-order term. Then, the following values are obtained:(27)g20=14∂2F1∂y12−∂2F1∂y22+2∂2F2∂y1y2+i∂2F2∂y12−∂2F2∂y22−2∂2F2∂y1∂y2,g11=14∂2F1∂y12+∂2F2∂y22+i∂2F2∂y12+∂2F2∂y22,g02=14∂2F1∂y12−∂2F1∂y22−2∂2F2∂y1∂y2+i∂2F2∂y12−∂2F2∂y22+2∂2F2∂y1∂y2,G21=18∂3F1∂y13+∂3F1∂y1∂y22+∂3F2∂y12∂y2+∂3F2∂y23+i∂3F2∂y13+∂3F2∂y1∂y22−∂3F1∂y12∂y2−∂3F1∂y23,G110k−2=12∂2F1∂y1∂yk+∂2F2∂y2∂yk+i∂2F2∂y1∂yk−∂2F1∂y2∂ykk=3,4,5G101k−2=12∂2F1∂y1∂yk−∂2F2∂y2∂yk+i∂2F2∂y1∂yk+∂2F1∂y2∂ykk=3,4,5,w11k−2=−14λk∂2Fk∂y12+∂2Fk∂y22k=3,4,5,w20k−2=142ω0i−λk∂2Fk∂y12−∂2Fk∂y22−2i∂2Fk∂y1∂y2k=3,4,5,g21=G21+∑k=352G110k−2w11k−2+G101k−2w20k−2.Based on the above calculated values, let(28)C10=i2ω0g20g21−2g112−13g022+12g21,μ2=−ReC10α′0,τ2=−ImC10+μ2ω′0ω0,β2=2ReC10,where α′0=Reλ′kpkp=kp0 and ω′0=Imλ′kpkp=kp0.Theorem 4.
When the bifurcation parameterkp of system (13) is increased beyond the critical value kp0, the Hopf bifurcation occurs at the equilibrium point o00000, and the following characteristics exist:(a)
Whenμ2>0μ2<0, the Hopf bifurcation is a supercritical (subcritical) bifurcation, and a bifurcation periodic solution for kp>kp0kp<kp0 exists(b)
Whenβ2<0β2>0, the bifurcation periodic solution is asymptotically stable (unstable)(c)
Whenτ2>0τ2<0, the period of the periodic solution increases (decreases)
Substitute the system parameters to discuss the bifurcation direction whenkd=100 and kp0=1690.681. The calculation results of the above values are shown in Table 3.Substituting the values yieldsμ2=15.510>0, β2=−0.152<0, and τ2=6.682e−4>0.It is known fromμ2>0 that the Hopf bifurcation of system (13) at kp0 is a supercritical bifurcation, that is, when kp<kp0, the equilibrium point of the dynamical system is stable, yet Hopf bifurcation occurs as kp is gradually increased to kp0. β2<0 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 kp increases because τ2>0.Table 3
Calculation results of the relevant values in the bifurcation direction.
g
20
0.696–4.17e − 2i
G
110
3
1.171
w
11
3
−7.41e − 2
g
11
0.655
G
101
1
0.945–1.172i
w
20
1
−1.369e − 4 + 1.054e − 4i
g
02
0.613 + 4.17e − 2i
G
101
2
0.945 + 1.172i
w
20
2
−1.255e − 5–8.619e − 6i
G
21
2.27e − 2
G
101
3
1.184
w
20
3
−3e − 4 + 7.3e − 3i
G
110
1
1.727–1.172i
w
11
1
−2.304e − 5 + 7.558e − 6i
g
21
−0.151 + 8.9e − 3i
G
110
2
1.727 + 1.172i
w
11
2
−2.304e − 5–7.558e − 6i
C
1
0
−0.076–2.27e − 2i


## 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 whenkd=100 and kp0=1690.681 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 kp 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 kp<kp0 and Hopf bifurcation occurs when kp is increased to kp0, while a stable limit cycle appears when kp>kp0.Figure 3
Local Hopf bifurcation diagram of system.Figures4–7 show the time history curves and phase before and after the bifurcation occurs when kp takes different values. It can be seen from the figure that the equilibrium point of the system is stable when kp<kp0, yet the stability of the system equilibrium point is lost and a set of stable limit cycles is branched when kp is increased beyond kp0.Figure 4
The response time domain and phase diagram whenkp=1650.Figure 5
The response time domain and phase diagram whenkp=1680.Figure 6
The response time domain and phase diagram whenkp=1700.Figure 7
The response time domain and phase diagram whenkp=1800.It can also be seen from Figures6 and 7 that each periodic solution is stable because β2<0, and τ2>0 indicates that the period of the periodic solution increases as the bifurcation parameter kp increases so that the period of the periodic solution when kp=1800 is greater than that when kp=1700.


## 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)
Takekp as the bifurcation parameter. When kd=100, the critical Hopf bifurcation value is calculated as kp0=1690.681, and it is categorized as supercritical Hopf bifurcation, that is, when kp<kp0, the 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.



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*Source: 2936838-2020-04-07.xml*