# Vertical Dynamic Response of the 2-DOF Maglev System considering Suspension Nonlinearity

Authors: XuDi Gu; GuoBin. Lin; Yuan Li; MeiQi Wang; PengFei Liu
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In order to study the dynamic characteristics of the nonlinear system of the maglev train, a vibration model of the two-degree-of-freedom nonlinear suspension system of the maglev train is established in this paper. Based on the linearized model of the suspension control module, the suspension system is controlled by the state feedback method. Compared with the results of the Runge–Kutta method, the accuracy of the settlement results is verified. This paper analyzes the influence of system parameters and feedback control parameters on the system. The results show that the linear stiffness mainly affects the left-right migration of the resonance peak and the difference between the maximum and minimum of the resonance region of the suspension frame; the nonlinear stiffness mainly affects the slope of the resonance region of the car body and the suspension frame; displacement feedback control parameters can reduce the amplitude of the system, velocity feedback control parameters can make the state change of the suspension frame more smoothly, and acceleration feedback control parameters make the car body and the suspension frame have a strong coupling effect. According to the Hurwitz criterion of Hopf bifurcation, this paper deduces the conditions that the control parameters should satisfy when the equilibrium point is stable, and the instability produces periodic vibration under PID control. Through numerical simulation, it is found that the system has complex dynamic behavior, the results show that the system works under some conditions, and there are multiple stable and unstable limit cycles simultaneously. Therefore, the system will appear alternately between multiperiod and chaotic motion, which will affect the stability of the system.


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## Body

## 1. Introduction

Maglev train has the advantages of good safety, low noise, strong climbing ability, high running speed, smooth, and comfortable [1–5]. There have been great development in recent years, and commercial routes using EMS technology already exist in Japan, China, and Korea [6]. The development and continuation of the maglev railway as an ordinary wheel-rail railway is expected to replace the traditional low-speed wheel-rail transportation system [7]. Maglev train is a kind of emerging railway technology, and many key structures have strong nonlinear characteristics, such as suspension control, metal rubber, and air springs. It is necessary to analyze and calculate it. The EMS type maglev train uses electromagnetic suction to overcome the weight of the train, and the magnitude of the electromagnetic force is proportional to the square of the current and inversely proportional to the square of the suspension gap. Therefore, the mathematical model of the maglev system has open-loop instability [8, 9]. When the maglev train is running, it is necessary to maintain a certain suspension gap, and only good stability can ensure the safety and comfort of the train running.Due to the nonlinear dynamic characteristics of the maglev train suspension system, many scholars have proposed and improved different suspension control schemes. Wang et al. [10] established a system model in which a joint structure replaces a single magnetic levitation unit, and a control strategy based on the linear-quadratic optimal control method is proposed. In the case of external disturbances, it has a good control performance. Li et al. [11] considered inherent nonlinearity, module coupling, coupling, and misalignment of sensors and actuators, and a feedback linear controller based on the mathematical model of the maglev module is proposed, which reduces the dynamic coupling between the two points. At present, the suspension control model of the maglev train mostly adopts the single electromagnet suspension system model, and the linearized model is applied to the classical control theory—PID or PD controller for control. In the suspension control of the maglev train, it is divided into voltage [12, 13] and current [14–18] as the control target.The main methods for nonlinear vibration analysis are the perturbation method, multiscale method, incremental harmonic balance method [19–21], and average method [22]. Wang et al. [18] established a simplified coupling model of the single suspension point and the elastic beam, and the Hopf bifurcation phenomenon caused by time delay and the influence of control parameters on forced vibration in periodic solutions of subharmonic resonance is discussed by using the multiscale method. Wu and Zeng [19] established a three-degree-of-freedom model of the maglev train body, maglev frame, and electromagnet, and the numerical solution is in good agreement with the incremental harmonic balance method. Zhang et al. [22] studied the nonlinear vehicle suspension system with a single degree of freedom using the average method. By comparing our understanding of different approaches, the IHB method can solve both the weak nonlinear system and the strong nonlinear system, which is a more comprehensive solution method, so this paper uses this method to study suspension control and nonlinear suspension.Bao et al. [23] established a 2-DOF nonlinear dynamics model, by fitting the experimental data of the nonlinear components. It is found that the stiffness expression of the two-system suspension is a unary quartic polynomial, and good fitting results are obtained under different preloads. Zheng et al. [24] used polynomials to describe the nonlinear stiffness of two-system suspension. The influence of nonlinear characteristics on system stability is studied, and it is found that the nonlinear spring system is chaotic. Han and Jing [25] through theoretical analysis and formula derivation of air spring, the obtained formula shows that the elastic restoring force of the air spring is a quartic polynomial with one variable about the spring shape. The above literature shows the feasibility of using polynomial to represent the maglev train two-system suspension model with air spring. The maglev system model in this paper adopts cubic polynomial to represent the two-system nonlinear suspension, and the dynamic behavior of the system is analyzed theoretically.To sum up, the existing literature mostly focuses on the suspension control of the single iron suspension model, and there are few maglev system models with two-system nonlinear suspension. To solve this problem, the nonlinear suspension model equation of the maglev system with two degrees of freedom is established in this paper, and after applying sinusoidal excitation to the suspension frame, the system response of the suspension frame and the vehicle body is studied. Taking velocity feedback control parameters as an example, the condition of Hopf bifurcation in the system is obtained. Finally, the bifurcation and chaos phenomena of the maglev system are analyzed by numerical simulation, and it is found that the system is in the alternating state of multiperiod and chaos in some working conditions.


## 2. Model

### 2.1. Vehicle-Rail Coupling Model

The nonlinear suspension vibration model of the maglev train with two degrees of freedom is shown in Figure1.Figure 1
Two-DOF nonlinear suspension vibration model of the maglev train.In Figure1, m1 and m2 are the mass of the suspension frame and the body, respectively, z1 and z2 are the displacement of the suspension frame and the body, respectively, h0 is the expected value of suspension clearance, the relation between the restoring force and displacement of the nonlinear spring is Fk = k1(–2 − z1) + k3(–2 − z1)3, k1 and k3 are linear and nonlinear stiffness coefficients of the suspension frame, respectively, c is the linear damping coefficient of the suspension frame, and F0 and ω represent the amplitude and frequency of the excitation force, respectively. The relationship between electromagnetic attraction F [26, 27] and suspension clearance is(1)Fi,c=μ0N2S4itct2.μ0 is the air permeability, N is the number of turns of electromagnet winding, S is the magnetic pole area of the electromagnet, i(t) is the electromagnet current at time t, and c(t) = –0 − z1. It can be seen from equation (1) that the function relation between electromagnetic attraction and suspension clearance is inverse square and nonlinear, and it is proved that the maglev train suspension system is unstable and needs to be controlled.The Taylor series expansion of equation (1) at the equilibrium point (c0,i0,h0) is obtained as(2)Fi,c=Fi0,c00+ΔF=F+Fii0,c0,Δit+Fci0,c0,Δct+αΔi,Δc.By ignoring higher-order terms, the linearized form of equilibrium neighborhood can be obtained as(3)Fi,c=Fi0,c0+kiΔit−kcΔct.In the previous formula,c0 and i0 are the suspension gap and current at the equilibrium point, respectively, and F(i0, c0) is the electromagnetic force generated by the electromagnet at the equilibrium point, and maglev train gravity, and other large reverse. The coefficients ki and kc are, respectively, written as follows:(4a)ki=Fii0,c0=∂F∂i|i0,c0=μ0N2Si02c02,(4b)kc=Fci0,c0=∂F∂i|i0,c0=μ0N2Si022c03.Using the state feedback method to control the current, with displacement-speed-acceleration as feedback variables, the control current is expressed as [14–18](5)Δit=kpΔct+kvΔz˙t+kaΔz¨t.The performance of the state feedback closed-loop system is determined bykp, ky, and ka.According to Newton’s second law, we get(6)m1z¨1−cz˙2−z˙1−k1z2−z1−k3z2−z13=Fcosωt−Fi,c+mg,m2z¨2+cz˙2−z˙1+k1z2−z1+k3z2−z13=0.Letx = z1 and y = –2 − z1; then, equation (6) can be rewritten as(7)m1+kikax¨+kikvx˙+kikp−kcx−cy˙−k1y−k3y3=Fcosωt,m2x¨+y¨+cy˙+k1y+k3y3=0.Formula (7) is normalized as follows.We makem0=m1+kika, ω1=kikp−kc/m0, ω2=k1/m2, μ=m2/m0, ξ=c/m2, ξ1=kikv/m0, γ=k3/m2, and f=F/m2. Then, equation (7) can be converted into(8)x¨+ξ1x˙+ω12x−μξy˙−μω22y−μγy3=fcosωt,x¨+y¨+ξy˙+ω22y+γy3=0.Let us write it in the matrix form(9)M¯X¨+C¯X˙+K1¯+K3¯X=P¯.In the previous formula, we get(10)X=xy,M¯=1011,C¯=ξ1−μξ0ξ,K1¯=ω12−μω220ω22,K3¯=0−μγy20γy2,P¯=fcosωt0.


### 2.2. Incremental Harmonic Balance Calculation

Letτ=ωtX′=dx/dτ, and X″=d2X/dτ2; then, formula (9) can be written as(11)ω2M¯X″+ωC¯X′+K1¯+K3¯X=P¯.LetX0 be the solution at a certain moment under the disturbance of fixed frequencyω0, then its adjacent state can be expressed in incremental form as(12)X=X0+ΔX.In the previous formula,X0=x0y0T and ΔX=ΔxΔyT.Substituting equation (12) into equation (11), using Taylor expansion and ignoring the higher-order small quantities in it, we get(13)ω2M¯X″+ωC¯X′+K1¯+K3¯X=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯+K3¯X0+ΔX=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯X0+ΔX+K3X0+ΔX3=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯X0+ΔX+K3X03+3K3X02ΔX=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯X0+ΔX+K3¯X0+3K3¯ΔX=P¯,ω2M¯ΔX′′+ωC¯ΔX′+K1¯+3K3¯ΔX=P¯−ω2M¯X0′′+ωC¯X0′+K1¯+K3¯X0.Fourier expansion of exact solutionX0 and incremental ΔX is as follows:(14)xj0=aj0+∑i=1Najkcoskτ+bjksinkτ,Δxj=Δaj0+∑i=1NΔajkcoskτ+Δbjksinkτ.We define(15)Aj=aj0,aj1,aj2…ajN,bj0,bj1,bj2…bjNT,ΔAj=Δaj0,Δaj1,Δaj2…ΔajN,Δbj0,Δbj1,Δbj2…ΔbjNT,Cs=1,cosτ,cos2τ,…cosNτ,sinτ,sin2τ,…sinNτT.We makeS=diagCsCs, A=A1A2T, ΔA=ΔA1ΔA2T, X0=SA, and ΔX0=SΔA. Applying the Galerkin averaging process to integrate equation (13), taking τ as the integration variable and [0,2π] as the integration interval, we can obtain(16)12π∫02πδΔXTω2M¯ΔX″+ωC¯ΔX′+K1¯+3K3¯ΔXdτ=12π∫02πδΔXTP¯−ω2M¯X0″+ωC¯ΔX0′+K1¯+K3¯X0dτ.We tidy up(17)12π∫02πSTω2M¯S″+ωC¯ΔS′+K1¯+3K3¯SdτΔA=12π∫02πSTP¯−ω2M¯S″+ωC¯S′+K1¯+K3¯SAdτ.We get the linearized equation forX0 and increment ΔX:(18)KmcΔA=R.Among them,(19)Kmc=12π∫02πSTω2M¯S″+ωC¯ΔS′+K1¯+3K3¯Sdτ,R=12π∫02πSTP¯−ω2M¯S″+ωC¯S′+K1¯+K3¯SAdτ.


## 2.1. Vehicle-Rail Coupling Model

The nonlinear suspension vibration model of the maglev train with two degrees of freedom is shown in Figure1.Figure 1
Two-DOF nonlinear suspension vibration model of the maglev train.In Figure1, m1 and m2 are the mass of the suspension frame and the body, respectively, z1 and z2 are the displacement of the suspension frame and the body, respectively, h0 is the expected value of suspension clearance, the relation between the restoring force and displacement of the nonlinear spring is Fk = k1(–2 − z1) + k3(–2 − z1)3, k1 and k3 are linear and nonlinear stiffness coefficients of the suspension frame, respectively, c is the linear damping coefficient of the suspension frame, and F0 and ω represent the amplitude and frequency of the excitation force, respectively. The relationship between electromagnetic attraction F [26, 27] and suspension clearance is(1)Fi,c=μ0N2S4itct2.μ0 is the air permeability, N is the number of turns of electromagnet winding, S is the magnetic pole area of the electromagnet, i(t) is the electromagnet current at time t, and c(t) = –0 − z1. It can be seen from equation (1) that the function relation between electromagnetic attraction and suspension clearance is inverse square and nonlinear, and it is proved that the maglev train suspension system is unstable and needs to be controlled.The Taylor series expansion of equation (1) at the equilibrium point (c0,i0,h0) is obtained as(2)Fi,c=Fi0,c00+ΔF=F+Fii0,c0,Δit+Fci0,c0,Δct+αΔi,Δc.By ignoring higher-order terms, the linearized form of equilibrium neighborhood can be obtained as(3)Fi,c=Fi0,c0+kiΔit−kcΔct.In the previous formula,c0 and i0 are the suspension gap and current at the equilibrium point, respectively, and F(i0, c0) is the electromagnetic force generated by the electromagnet at the equilibrium point, and maglev train gravity, and other large reverse. The coefficients ki and kc are, respectively, written as follows:(4a)ki=Fii0,c0=∂F∂i|i0,c0=μ0N2Si02c02,(4b)kc=Fci0,c0=∂F∂i|i0,c0=μ0N2Si022c03.Using the state feedback method to control the current, with displacement-speed-acceleration as feedback variables, the control current is expressed as [14–18](5)Δit=kpΔct+kvΔz˙t+kaΔz¨t.The performance of the state feedback closed-loop system is determined bykp, ky, and ka.According to Newton’s second law, we get(6)m1z¨1−cz˙2−z˙1−k1z2−z1−k3z2−z13=Fcosωt−Fi,c+mg,m2z¨2+cz˙2−z˙1+k1z2−z1+k3z2−z13=0.Letx = z1 and y = –2 − z1; then, equation (6) can be rewritten as(7)m1+kikax¨+kikvx˙+kikp−kcx−cy˙−k1y−k3y3=Fcosωt,m2x¨+y¨+cy˙+k1y+k3y3=0.Formula (7) is normalized as follows.We makem0=m1+kika, ω1=kikp−kc/m0, ω2=k1/m2, μ=m2/m0, ξ=c/m2, ξ1=kikv/m0, γ=k3/m2, and f=F/m2. Then, equation (7) can be converted into(8)x¨+ξ1x˙+ω12x−μξy˙−μω22y−μγy3=fcosωt,x¨+y¨+ξy˙+ω22y+γy3=0.Let us write it in the matrix form(9)M¯X¨+C¯X˙+K1¯+K3¯X=P¯.In the previous formula, we get(10)X=xy,M¯=1011,C¯=ξ1−μξ0ξ,K1¯=ω12−μω220ω22,K3¯=0−μγy20γy2,P¯=fcosωt0.


## 2.2. Incremental Harmonic Balance Calculation

Letτ=ωtX′=dx/dτ, and X″=d2X/dτ2; then, formula (9) can be written as(11)ω2M¯X″+ωC¯X′+K1¯+K3¯X=P¯.LetX0 be the solution at a certain moment under the disturbance of fixed frequencyω0, then its adjacent state can be expressed in incremental form as(12)X=X0+ΔX.In the previous formula,X0=x0y0T and ΔX=ΔxΔyT.Substituting equation (12) into equation (11), using Taylor expansion and ignoring the higher-order small quantities in it, we get(13)ω2M¯X″+ωC¯X′+K1¯+K3¯X=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯+K3¯X0+ΔX=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯X0+ΔX+K3X0+ΔX3=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯X0+ΔX+K3X03+3K3X02ΔX=P¯,ω2M¯X0+ΔX′′+ωC¯X0+ΔX′+K1¯X0+ΔX+K3¯X0+3K3¯ΔX=P¯,ω2M¯ΔX′′+ωC¯ΔX′+K1¯+3K3¯ΔX=P¯−ω2M¯X0′′+ωC¯X0′+K1¯+K3¯X0.Fourier expansion of exact solutionX0 and incremental ΔX is as follows:(14)xj0=aj0+∑i=1Najkcoskτ+bjksinkτ,Δxj=Δaj0+∑i=1NΔajkcoskτ+Δbjksinkτ.We define(15)Aj=aj0,aj1,aj2…ajN,bj0,bj1,bj2…bjNT,ΔAj=Δaj0,Δaj1,Δaj2…ΔajN,Δbj0,Δbj1,Δbj2…ΔbjNT,Cs=1,cosτ,cos2τ,…cosNτ,sinτ,sin2τ,…sinNτT.We makeS=diagCsCs, A=A1A2T, ΔA=ΔA1ΔA2T, X0=SA, and ΔX0=SΔA. Applying the Galerkin averaging process to integrate equation (13), taking τ as the integration variable and [0,2π] as the integration interval, we can obtain(16)12π∫02πδΔXTω2M¯ΔX″+ωC¯ΔX′+K1¯+3K3¯ΔXdτ=12π∫02πδΔXTP¯−ω2M¯X0″+ωC¯ΔX0′+K1¯+K3¯X0dτ.We tidy up(17)12π∫02πSTω2M¯S″+ωC¯ΔS′+K1¯+3K3¯SdτΔA=12π∫02πSTP¯−ω2M¯S″+ωC¯S′+K1¯+K3¯SAdτ.We get the linearized equation forX0 and increment ΔX:(18)KmcΔA=R.Among them,(19)Kmc=12π∫02πSTω2M¯S″+ωC¯ΔS′+K1¯+3K3¯Sdτ,R=12π∫02πSTP¯−ω2M¯S″+ωC¯S′+K1¯+K3¯SAdτ.


## 3. Stability Analysis

We definex=x1,x2,x3,x4T=x,x˙,y,y˙T, and the dynamic equation of the system can be written as the equation of the state as follows:(20)x=x2,x˙=fcosωt−ξ1x2−ω12x1+μω22y1−x1+μξy2−x2+μγy2−x23,y=x4,y˙=−ω22y1−x1−ξy2−x2−γy2−x23.At the equilibrium point,x0=0,0,0,0 linearizes to (21), and we get(21)x˙1=x2,x˙2=−ξ1x2−ω12x1+μω22y1−x1+μξy2−x2,x˙3=x4,x˙4=−ω22y1−x1−ξy2−x2.We write equation (21) as x˙=Ax, where A=0100−ω12−μω22−ξ1−μξμω22μξ0001ω22ξ−ω22ξ. The corresponding characteristic polynomial is(22)Jλ=λ4+a1λ3+a2λ2+a3λ+a4.Among them,a1=−ξ+ξ1+μξ,a2=ω12+ω22−2μξ2+μω22−ξξ1, a3=−ξω12+ξ1ω22−2ξμω22, and a4=ω12ω22.The stability interval of control parameters can be obtained as(23)kp≥7494.8,kv≥0,ka≥0.


## 4. Simulation Analysis of Stability and Bifurcation

### 4.1. Parameter Setting and Solution Results

The parameters used in the calculation and analysis in this paper are set as follows [28]: m1=8000kg, m2=39000kg, k1=2.05×105N/m, k3=6×108N/m3, c=10000N⋅s/m, R=0.61Ω, μ0=4π×10−7, z0=0.01m, N=290, and S=0.311m2.


### 4.2. Influence of Track Irregularity Amplitude

After the system is stable, a period of roughness amplitude excitation is applied to suspension to detect the stability of the system. The roughness amplitude excitation is shown in Figure2, and the dynamic response of the vehicle under the roughness amplitude is shown in Figure 3. The feedback control parameters are kp=7800, kv=50, and ka=5.Figure 2
Roughness amplitude excitation.Figure 3
Roughness amplitude excited dynamic response. (a) Car body. (b) Suspension.
(a)(b)


### 4.3. Influence of the Force Shock

When the system suddenly accelerates or brakes, the excitation is not periodic. Therefore, a shock force excitation of 1000 kN is applied to the suspension. The dynamic response of the system is shown in Figure4, where feedback control parameters are the same as in Section 4.2.Figure 4
Dynamic response of the system under impact excitation. (a) Car body. (b) Suspension.
(a)(b)


### 4.4. Solution Result Comparison

Using the Runge–Kutta numerical method to solve equation (5), compared with the approximate analytical solution of the system calculated by the IHB method. The calculation results are shown in Figure 5. It can be seen that the calculation results of the two methods have high consistency.Figure 5
Comparison of numerical and analytical solutions of the amplitude-frequency curve. (a) Car body. (b) Suspension.
(a)(b)


### 4.5. Influence of System Parameters on System Motion Characteristics

Figure6 shows that the resonance peaks of the amplitude-frequency curves of the vehicle body and suspension gradually shift to the right with the increase of stiffness k1. The increase of k1 makes the amplitude of the car body tend to increase slightly, the impact on the vibration response of the suspension frame is obvious, with the increase of k1, and the difference between the maximum value and the minimum value of the suspension frame increases with a relatively obvious trend. When the excitation frequency of the external excitation sweeps this frequency range, then the vibration of the suspension frame may be aggravated, which is not conducive to the stability of the suspension frame.Figure 6
Effects of different linear stiffnesses on the system. (a) Car body. (b) Suspension.
(a)(b)Figure7 shows that the highest point of the resonance peak of the amplitude-frequency curves of the vehicle body and suspension gradually shifts to the right with the increase of stiffness k3. Comparing Figures 6(a) and 7(a), it can be found that, with an increase in k1, the frequency corresponding to the formant peak of the peak of the car body formant is increased, and increasing k3 gradually increases the slope of the resonance peak. Comparing Figures 6(b) and 7(b), it is found that, k3 makes the difference between the maximum value and the minimum value of the resonance peak decrease with the increase of k3, and the amplitude-frequency response curve becomes smoother as the frequency changes, which is helpful for the stability of suspension.Figure 7
Effects of different nonlinear stiffnesses on the system (a) Car body. (b) Suspension.
(a)(b)


### 4.6. Influence of Feedback Control Parameters on System Motion Characteristics

Figures8–10 show the response results of the car body and the suspension frame under different feedback control parameters. The image shows that, with the gradual increase of kp, the vibration amplitude of the car body and the suspension frame has a relatively obvious decrease, the dynamic response of ky to the suspension frame is more obvious, and the maximum and minimum values of the formant are reduced and makes the curve drop at the formant smoother. The increase of ka increases the peak value of the resonance area of the car body and the suspension frame. However, as can be seen from Figure 10(b), the increase of ka makes the suspension amplitude decrease the fastest when the excitation frequency ω is greater than 5.Figure 8
Effects of different displacement control parameters on the system. (a) Car body. (b) Suspension.
(a)(b)Figure 9
Effects of different speed control parameters on the system. (a) Car body. (b) Suspension.
(a)(b)Figure 10
Effects of different acceleration control parameters on the system. (a) Car body. (b) Suspension. (c) Car body partial magnification.
(a)(b)(c)


### 4.7. Hopf Bifurcation

Theorem 1 (see [29, 30]).
We set the coefficient algebraic equation as(24)λn+a1λn−1+a2λn−2+…+an−1λ+an=0.The necessary and sufficient conditions for having a pair of pure imaginary eigenvalues and the remainingn−2 eigenvalues have negative real parts which are(25)ai>0i=1,2…n,Δi>0i=n−3,n−5…,Δn−1=0.From the coefficients of equation (25), the following Hurwitz determinant can be constructed:(26)Δi=a1100…0a3a2a11…0a5a4a3a2…0⋮⋮⋮⋮⋮a2i−1a2i−2a2i−3a2i−4…aim=1,2,…,n,where if i > n, then ai = 0.Applying Theorem1 to equation (23), the condition for the existence of a pair of pure imaginary eigenvalues and the remaining eigenvalues with negative real parts is(27)ai>0i=1,2,3,4,5,Δ3=a110a3a2a10a4a3,=−a4a12+a1a2a3−a32=0.So, the critical control parameter of system (21) is(28)kp=7494.7,kv=0,ka=0.Taking theky bifurcation parameter as an example, the motion characteristics of the system change as shown in Figures 11 and 12. As can be seen from Figures 11(a) and 12(a), when the speed feedback control parameter kd < 0, the system is unstable, the amplitudes of suspension and the car body do not converge. Figures 11(b) and 12(b) show that, when the speed feedback control parameter reaches the critical value kd = 0, Hopf bifurcation occurs in the system and periodic vibration occurs; Figures 11 and 12(c) show that, when the speed feedback control parameter kd > 0, the maglev system tends to be stable, and the amplitude of suspension and the car body tends to zero. It means that the suspension gap of the suspension frame converges to the desired suspension gap of 0.01 m.Figure 11
Hopf bifurcation of the car body. (a)ky=−0.1, x0 = 0.01. (b) ky=0, x0 = 0.01. (c) ky=−0.1, x0 = 0.01.
(a)(b)(c)Figure 12
Hopf bifurcation of suspension. (a)ky=−0.1, x0 = 0.01. (b) ky=0, x0 = 0.02. (c) ky=−0.1, x0 = 0.01.
(a)(b)(c)


### 4.8. Analysis of Bifurcation and Chaotic Motion

In order to analyze the influence of feedback control parameters on the dynamic response of the system, withkp as the variable parameter, using the numerical solution method to solve equation (8), the motion characteristic map of the system under parameter excitation is obtained, as shown in Figure 13. The main parameters of the system are as follows: F = 11000 kN, ky  = −0.9, ka = 0.5, and ω = 3.4.Figure 13
Bifurcation diagram of the system and its partial magnification.
(a)(b)With the change of the feedback control parameterkp, the system will produce complex dynamic behavior. To this end, we study the phase plane and Poincare section of the system.Whenkp = 7620, the system is a single-cycle motion, as shown in Figure 14. When kp = 7600, the system is multiperiod motion, as shown in Figure 15. When kp = 7600.18, the system is in chaotic motion, as shown in Figure 16.Figure 14
Phase diagram and Poincare cross section of the system whenkp = 7620.
(a)(b)Figure 15
Phase diagram and Poincare cross section of the system whenkp = 7600.
(a)(b)Figure 16
Phase diagram and Poincare cross section of the system whenkp = 7600.18.
(a)(b)


## 4.1. Parameter Setting and Solution Results

The parameters used in the calculation and analysis in this paper are set as follows [28]: m1=8000kg, m2=39000kg, k1=2.05×105N/m, k3=6×108N/m3, c=10000N⋅s/m, R=0.61Ω, μ0=4π×10−7, z0=0.01m, N=290, and S=0.311m2.


## 4.2. Influence of Track Irregularity Amplitude

After the system is stable, a period of roughness amplitude excitation is applied to suspension to detect the stability of the system. The roughness amplitude excitation is shown in Figure2, and the dynamic response of the vehicle under the roughness amplitude is shown in Figure 3. The feedback control parameters are kp=7800, kv=50, and ka=5.Figure 2
Roughness amplitude excitation.Figure 3
Roughness amplitude excited dynamic response. (a) Car body. (b) Suspension.
(a)(b)


## 4.3. Influence of the Force Shock

When the system suddenly accelerates or brakes, the excitation is not periodic. Therefore, a shock force excitation of 1000 kN is applied to the suspension. The dynamic response of the system is shown in Figure4, where feedback control parameters are the same as in Section 4.2.Figure 4
Dynamic response of the system under impact excitation. (a) Car body. (b) Suspension.
(a)(b)


## 4.4. Solution Result Comparison

Using the Runge–Kutta numerical method to solve equation (5), compared with the approximate analytical solution of the system calculated by the IHB method. The calculation results are shown in Figure 5. It can be seen that the calculation results of the two methods have high consistency.Figure 5
Comparison of numerical and analytical solutions of the amplitude-frequency curve. (a) Car body. (b) Suspension.
(a)(b)


## 4.5. Influence of System Parameters on System Motion Characteristics

Figure6 shows that the resonance peaks of the amplitude-frequency curves of the vehicle body and suspension gradually shift to the right with the increase of stiffness k1. The increase of k1 makes the amplitude of the car body tend to increase slightly, the impact on the vibration response of the suspension frame is obvious, with the increase of k1, and the difference between the maximum value and the minimum value of the suspension frame increases with a relatively obvious trend. When the excitation frequency of the external excitation sweeps this frequency range, then the vibration of the suspension frame may be aggravated, which is not conducive to the stability of the suspension frame.Figure 6
Effects of different linear stiffnesses on the system. (a) Car body. (b) Suspension.
(a)(b)Figure7 shows that the highest point of the resonance peak of the amplitude-frequency curves of the vehicle body and suspension gradually shifts to the right with the increase of stiffness k3. Comparing Figures 6(a) and 7(a), it can be found that, with an increase in k1, the frequency corresponding to the formant peak of the peak of the car body formant is increased, and increasing k3 gradually increases the slope of the resonance peak. Comparing Figures 6(b) and 7(b), it is found that, k3 makes the difference between the maximum value and the minimum value of the resonance peak decrease with the increase of k3, and the amplitude-frequency response curve becomes smoother as the frequency changes, which is helpful for the stability of suspension.Figure 7
Effects of different nonlinear stiffnesses on the system (a) Car body. (b) Suspension.
(a)(b)


## 4.6. Influence of Feedback Control Parameters on System Motion Characteristics

Figures8–10 show the response results of the car body and the suspension frame under different feedback control parameters. The image shows that, with the gradual increase of kp, the vibration amplitude of the car body and the suspension frame has a relatively obvious decrease, the dynamic response of ky to the suspension frame is more obvious, and the maximum and minimum values of the formant are reduced and makes the curve drop at the formant smoother. The increase of ka increases the peak value of the resonance area of the car body and the suspension frame. However, as can be seen from Figure 10(b), the increase of ka makes the suspension amplitude decrease the fastest when the excitation frequency ω is greater than 5.Figure 8
Effects of different displacement control parameters on the system. (a) Car body. (b) Suspension.
(a)(b)Figure 9
Effects of different speed control parameters on the system. (a) Car body. (b) Suspension.
(a)(b)Figure 10
Effects of different acceleration control parameters on the system. (a) Car body. (b) Suspension. (c) Car body partial magnification.
(a)(b)(c)


## 4.7. Hopf Bifurcation

Theorem 1 (see [29, 30]).
We set the coefficient algebraic equation as(24)λn+a1λn−1+a2λn−2+…+an−1λ+an=0.The necessary and sufficient conditions for having a pair of pure imaginary eigenvalues and the remainingn−2 eigenvalues have negative real parts which are(25)ai>0i=1,2…n,Δi>0i=n−3,n−5…,Δn−1=0.From the coefficients of equation (25), the following Hurwitz determinant can be constructed:(26)Δi=a1100…0a3a2a11…0a5a4a3a2…0⋮⋮⋮⋮⋮a2i−1a2i−2a2i−3a2i−4…aim=1,2,…,n,where if i > n, then ai = 0.Applying Theorem1 to equation (23), the condition for the existence of a pair of pure imaginary eigenvalues and the remaining eigenvalues with negative real parts is(27)ai>0i=1,2,3,4,5,Δ3=a110a3a2a10a4a3,=−a4a12+a1a2a3−a32=0.So, the critical control parameter of system (21) is(28)kp=7494.7,kv=0,ka=0.Taking theky bifurcation parameter as an example, the motion characteristics of the system change as shown in Figures 11 and 12. As can be seen from Figures 11(a) and 12(a), when the speed feedback control parameter kd < 0, the system is unstable, the amplitudes of suspension and the car body do not converge. Figures 11(b) and 12(b) show that, when the speed feedback control parameter reaches the critical value kd = 0, Hopf bifurcation occurs in the system and periodic vibration occurs; Figures 11 and 12(c) show that, when the speed feedback control parameter kd > 0, the maglev system tends to be stable, and the amplitude of suspension and the car body tends to zero. It means that the suspension gap of the suspension frame converges to the desired suspension gap of 0.01 m.Figure 11
Hopf bifurcation of the car body. (a)ky=−0.1, x0 = 0.01. (b) ky=0, x0 = 0.01. (c) ky=−0.1, x0 = 0.01.
(a)(b)(c)Figure 12
Hopf bifurcation of suspension. (a)ky=−0.1, x0 = 0.01. (b) ky=0, x0 = 0.02. (c) ky=−0.1, x0 = 0.01.
(a)(b)(c)


## 4.8. Analysis of Bifurcation and Chaotic Motion

In order to analyze the influence of feedback control parameters on the dynamic response of the system, withkp as the variable parameter, using the numerical solution method to solve equation (8), the motion characteristic map of the system under parameter excitation is obtained, as shown in Figure 13. The main parameters of the system are as follows: F = 11000 kN, ky  = −0.9, ka = 0.5, and ω = 3.4.Figure 13
Bifurcation diagram of the system and its partial magnification.
(a)(b)With the change of the feedback control parameterkp, the system will produce complex dynamic behavior. To this end, we study the phase plane and Poincare section of the system.Whenkp = 7620, the system is a single-cycle motion, as shown in Figure 14. When kp = 7600, the system is multiperiod motion, as shown in Figure 15. When kp = 7600.18, the system is in chaotic motion, as shown in Figure 16.Figure 14
Phase diagram and Poincare cross section of the system whenkp = 7620.
(a)(b)Figure 15
Phase diagram and Poincare cross section of the system whenkp = 7600.
(a)(b)Figure 16
Phase diagram and Poincare cross section of the system whenkp = 7600.18.
(a)(b)


## 5. Conclusion

In this paper, a two-degree-of-freedom nonlinear suspension dynamic model of a maglev train is established. According to the Hurwitz stability criterion, the boundary conditions for the stability of the suspension system are obtained, under sinusoidal excitation applied by suspension. The nonlinear response characteristics of the system are analyzed using the incremental harmonic balance method, and the following conclusions are drawn:(1)
The linear stiffness of the nonlinear suspension is gradually increased, the formants of the body and suspension will shift to the right, the amplitude of the car body formant increases slightly, and the maximum and minimum values at the resonance peak of the suspension frame tend to increase, which will cause the suspension frame to vibrate more violently under the external excitation in this frequency range.(2)
The nonlinear stiffness of the nonlinear suspension is gradually increased and will increase the slope of the body resonance peak. In the process of external excitation from small to large, the amplitude of the car body will have a huge drop in a short period of time. At this time, it has poor riding comfort; the difference between the maximum value and the minimum value at the resonance peak of the suspension frame decreases with the increase of nonlinear stiffness and reduces the vibration intensity of the suspension frame. It is helpful to the vibration stability of the suspension frame.(3)
If the PID controller is used, sinusoidal excitation is applied to the suspension frame.kp can effectively weaken the amplitude of the resonance peak of the car body and suspension and has a large positive effect on the stability of the suspension system. As ky increases, the amplitude of the resonance peak of the car body is only slightly reduced. The amplitude-frequency curve of the suspension frame has better smoothness, weakens the curve bulge at the maximum value of the formant, and makes the vibration response change of the maglev system smoother. Moreover, ky has a certain reduction effect on the overall vibration amplitude of the suspension frame. The increase of ka, the peaks of the resonance area of the car body, and the suspension frame are increased. In addition, it can be found that the increase of ka can effectively suppress the vibration of the suspension frame when the excitation frequency ω is greater than 5.(4)
This paper also analyzes the bifurcation and chaotic motion of the system, the bifurcation diagram of the system is obtained, from the phase diagram and Poincare cross section diagram. It can be found that, as the system parameters change, there will be an alternation of multiperiod motion and chaotic motion, and it is not conducive to the stability of the suspension system.



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*Source: 7802248-2022-08-03.xml*