# Random Vibration Analysis of Coupled Three-Dimensional Maglev Vehicle-Bridge System

Authors: Wei Liu; Wenhua Guo
Journal: Advances in Civil Engineering
 (2019)

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## Abstract

This paper presents a framework for the linear random vibration analysis of the coupled three-dimensional (3D) maglev vehicle-bridge system. Except for assembling the equation of motion of vehicle only via the principle of virtual work, the fully computerized approach is further expanded to assemble the governing equation of fluctuating current via the equilibrium relation. A state-space equation couples the equation of motion of the vehicle and the governing equation of fluctuating current. The equation of motion of a real three-span space continuous girder bridge is established by using finite element methods. A separated iteration method based on the precise integration method and the Newmark method is introduced to solve the state-space equation for the maglev vehicle and the equation of motion for the bridge. Moreover, a new scheme to application of the pseudoexcitation method (PEM) in random vibration analysis is proposed to maximize the computational efficiency of the random vibration analysis of the maglev vehicle-bridge system. Finally, the numerical simulation demonstrates that the proposed framework can efficiently obtain the mean value, root mean square (RMS), standard deviation (SD), and power spectral density (PSD) of dynamic response for the coupled 3D maglev vehicle-bridge system.


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## Body

## 1. Introduction

The guideway of the maglev vehicle mostly adopts continuous elevated girder bridges to save land and reduce foundation settlement [1]. The reason of the interaction between maglev vehicle and bridge is that flexibility of the bridge causes the change of the air gap between the suspension bogie and the bridge, so the control voltage and control current must correspondingly change with the air gap to provide adequate electromagnetic force supporting the moving maglev vehicle. As the construction deviation and rough track profile for the guideway surface, the stochastic factors exist in the guideway deformation, which leads to random vibration of the maglev vehicle-bridge system. As a result, conducting research efforts on random vibration analysis of the maglev vehicle-bridge system should be a very meaningful work.In the past decades, many investigators have conducted a large amount of research on the interaction between maglev vehicle and bridge. Popp and Schiehlen previously established a 2-degrees-of-freedom (2-DOFs) maglev vehicle-guideway model [2]. Assuming the car body as a rigid supported by different number of magnets (or bogies) with linear spring-dashpot systems, Cai and Chen conducted a comprehensive study on vertical dynamic responses of the maglev vehicle traveling on a single-span or double-span beam [3–5]. Yau investigated the resonance phenomenon and vibration control of the neuro-PI controller for a series of simplified 2-DOFs maglev vehicles moving on a flexible guideway [6]. He also studied the dynamic behaviors of the simple maglev vehicle-guideway system subjected to environmental excitations [7, 8]. Based on the urban transit maglev (UTM) in Korea, Min et al. simulated the EMS-type vehicle as a detailed 25-DOFs model with a rigid cabin supported by four suspension bogies to investigate the spatial vibration characteristics of the maglev vehicle traveling along the simple supported beam [9]. Xu et al. presented a framework for the dynamic analysis of high-speed maglev trains running on elastic transitional viaducts [10]. Deng et al. used a simplified levitation force model from the experiment to couple a 6-DOFs HTS maglev vehicle with a double-span continuous girder bridge [11, 12]. These literatures above treated the guideway irregularity as a certain input, and the response results cannot reflect the random properties of the vehicle-guideway system. Then, Zhao and Zhai developed a 10-DOFs model of the maglev vehicle traveling along three types of guideways to observe the vertical random response and ride quality [13]. Shi et al. studied the dynamic response of the high-speed EMS maglev vehicle-guideway system with random irregularity [14]. Certainly, these two literatures indicate that the reaction characteristics of the maglev vehicle-guideway system to different frequencies can be obtained by using the discrete fast Fourier transform from dynamic responses. However, to date, relatively little research attention seems to have provided completely random vibration analysis of the maglev vehicle-guideway system, especially concerning vertical and lateral directions.In this study, the proposed framework is that the random vibration model of the maglev vehicle-bridge system is decomposed into two independent parts: (i) the vehicle model with the state-space equation and (ii) the bridge model with the equation of motion. These two parts are coupled through linearized electromagnetic forces. In the vehicle model, except for assembling the equation of motion of vehicle only via the principle of virtual work, the fully computerized approach is further expanded to assemble the governing equation of fluctuating current via the equilibrium relation. In the bridge model, the equation of motion of a three-span continuous girder bridge is established using finite element methods. Based on the precise integration method and the Newmark method, a separated iteration method is introduced to solve the state-space equation for the maglev vehicle and the equation of motion for the bridge. To maximize the computational efficiency of the random vibration analysis of the maglev vehicle-bridge system, a new scheme to application of the pseudoexcitation method (PEM) in random vibration analysis is presented. Numerical simulation will indicate that the proposed framework is appropriate and efficient for the linear random vibration analysis of the 3D maglev vehicle-bridge system.


## 2. State-Space Equation of the 3D Maglev Vehicle

The fully computerized approach to assemble the matrices of equation of motion of the coupled vehicle-bridge system is presented by Guo and Xu [15]. In this approach, the single equation of motion of the bridge is expanded to the overall equation of motion of the coupled vehicle-bridge system, and the additional elements generated from the vehicle are automatically assembled into the corresponding position in the overall equation of motion based on the principle of virtual work. However, except for the general equations of motion from the maglev vehicle and the bridge, the coupled maglev vehicle-bridge system contains electromagnetic forces and control systems. In this study, the fully computerized approach will be further used to assemble the coefficient matrices of the electromagnetic forces. Moreover, unlike the common way of the fully computerized approach to assemble the mass matrix, damping matrix and spring matrix via the total virtual work done by the inertia forces, damping forces and spring forces, and the coefficient matrices of the governing equation of fluctuating current will also be automatically assembled row by row via the equilibrium relation. Here, the fully computerized approach will be further expanded to assemble coefficient matrices of any type of equations.


### 2.1. Equation of Motion of the Maglev Vehicle

As shown in Figure1, a detailed 3D model of the EMS-type maglev vehicle with the middle-low speed consists of a cabin body and 10 levitation bogies [16]. Each bogie has 4 electromagnets and there are 40 electromagnets in all. Secondary suspensions connecting the cabin with bogies are simulated by using spring-dashpot systems. Interaction ways between left and right bogies are also simplified as spring-dashpot systems. The linking modes between secondary suspensions and the cabin have fixed sliding table and lateral sliding table. The fixed sliding table is a mechanical device of the hinge joint that can transmit translational loads, while the lateral sliding stable only transmits vertical load. The main parameters of the maglev vehicle are listed in Table 1.Figure 1
3D middle-low speed maglev vehicle. (a) Side view. (b) Front view. (c) Plane view.
(a)
(b)
(c)Table 1
Main parameters of the maglev vehicle.
Parameter
Unit
Value
Cabin body mass (mc)
kg
23292
Bogie mass (mb)
kg
846
Roll inertia of cabin body (Jcx)
kg·m2
66800
Pitch inertia of cabin body (Jcy)
kg·m2
210000
Yaw inertia of cabin body (Jcz)
kg·m2
193000
Roll inertia of bogie (Jbx)
kg·m2
963
Pitch inertia of bogie (Jby)
kg·m2
1800
Yaw inertia of bogie (Jbz)
kg·m2
268
Longitudinal stiffness of secondary suspension (kcx)
N/m
900000
Vertical stiffness of secondary suspension (kcy)
N/m
140000
Lateral stiffness of secondary suspension (kcz)
N/m
6000
Longitudinal damping of secondary suspension (ccx)
N·s/m
50000
Vertical damping of secondary suspension (ccy)
N·s/m
70000
Lateral damping of secondary suspension (ccz)
N·s/m
4000
Stiffness of spring dampers connecting the left and right bogies (kby, kbz)
N/m
4
×
10
8, 4300
Damping of spring dampers connecting the left and right bogies (cby, cbz)
N·s/m
1
×
10
6, 2100
Resistance (R0)
Ω
1.6
Air gap at balanced point (h0)
mm
10
Pole area (A0)
m2
0.383
Coil turn (N0)
—
660
Vacuum permeability (μ0)
H/m
4
π
×
10
−
7
Width of electromagnet (d0)
m
0.164
Length of cabin
m
16
Width of cabin
m
2.62
Height of cabin
m
2.9The cabin and bogies in the vehicle are assumed to be rigid bodies. Each rigid body has 5-DOFs in which two translational and three rotation displacements at the geometry center of the rigid body only except for the running direction. Total DOFs of all rigid bodies in the maglev vehicle is 55. Thus, the displacement vector of the maglev vehicle is written as follows:(1)Xm=yc,zc,Φc,φc,θc,yb11,zb11,Φb11,φb11,θb11,…,ybpq,zbpq,Φbpq,φbpq,θbpqT,where y, z, Φ, φ, and θ are the vertical displacement, lateral displacement, rolling rotation, pitching rotation, and yawing rotation, respectively, the subscripts c and b represent the cabin body and levitation bogie, the value of the subscript p is selected as one for left bogies or two for right bogies, and the subscript q (q = 1, 2, 3, 4, 5) represents the qth bogie. For instance, Φb24 is the rotation by x-axis at the 4th bogie on the right. The distribution of loads for the cabin and the first bogie is shown in Figure 2, in which the spring-damping force represents the spring force plus the damping force; Fcxpl, Fcypk, and Fczpl are, respectively, the longitudinal, vertical, and lateral spring-damping forces between the cabin and bogies, Fbyg and Fbzg are, respectively, the vertical and lateral spring-damping forces connecting the left bogie with the right left bogie, and ΔFeypr and Fezpr are, respectively, the vertical and lateral electromagnetic forces, where the subscript e represents the electromagnet. Here, we know that 1≤l≤4, 1≤k≤10, 1≤g≤20, and 1≤r≤20.Figure 2
Distribution of loads for the cabin and the first bogie. (a) Distribution of spring-damping forces in the cabin. (b) Distribution of electromagnetic forces and spring-damping forces in the first bogie.
(a)
(b)The equation of motion of the maglev vehicle is usually described by(2)MmX¨m+CmX˙m+KmXm=ΔFm,in which Mm, Cm, and Km are the global mass matrix, damping matrix, and stiffness matrix of the vehicle, respectively, and ΔFm is the force vector. The total virtual work done by the 55 inertial forces, 56 damping forces, and 56 spring forces in the maglev vehicle can help to assemble the mass matrix, damping matrix, and stiffness matrix in equation (2) by using the fully computerized approach [15].


### 2.2. Expression of Fluctuating Electromagnetic Force

The calculation of the nonlinear electromagnetic force mainly relates to two variables of current and air gap, of which expression is [17](3)Fey=μ0A0N024ih2,where μ0 is the vacuum permeability, A0 is the effective areas of the magnetic poles, N0 is the number of turns of the windings, h is the vertical air gap between the electromagnet on the bogie and the guideway in the vertical direction, i is the control current in the circuit, and Fey is the vertical electromagnetic force. By applying the Taylor series expansion at the balanced point for equation (3) and ignoring the effect of the high order, the nonlinear electromagnetic force can be described by [17](4)Fey=G0+kiΔi−khΔh,with ki=μ0A0N02i0/2h02 and kh=μ0A0N02i02/2h03, where G0 is the equivalent gravity of the maglev vehicle, i0 and h0 are desired values of control current and air gap when the maglev vehicle maintains static at the balanced point, Δi is the fluctuating control current, Δh is the fluctuating air gap in the vertical direction, and ki and kh are the equivalent stiffness.In the middle-low speed maglev vehicle, the lateral guide force does not have the independent control system. The lateral guide forceFezpu can be simplified as the following expression [9]:(5)Fezpr=−kezΔyezpr,with kez=2G0/πd0, kez is the equivalent stiffness, Δyezpr is the relative displacement between the bogie and guideway in the lateral direction, and d0 is the width of the electromagnet.For the independent vehicle model, the influence of the guideway on the vehicle is considered as the external input. From equations (4) and (5), the calculation of external forces acting on the maglev vehicle is related to the following parameters: fluctuating currents, vertical and lateral displacements of bogies, and vertical and lateral displacements of the guideway. Hence, the force vector in equation (2) can be described in the following matrix-vector form:(6)ΔFm=AmXm+BmΔIm+EmU+QmV,(7a)ΔIm=Δi1,Δi2,Δi3,…,Δin−1,ΔinT,(7b)U=u11,u12,u13,…,uprT,(7c)V=v11,v12,v13,…,vprT,where ΔIm is the vector of fluctuating current, U and V are the guideway displacement vectors in vertical and lateral directions (including bridge displacement and guideway irregularity), and Δin is the nth control current (1 ≤ n ≤ 40). Am, Bm, Em, and Qm are, respectively, the 55 × 55, 55 × 20, 55 × 40, and 55 × 40 coefficient matrices of the fluctuating electromagnetic force. It has been pointed out that one bogie has four electromagnets as shown in Figure 1(c). The uniformly distributed forces between electromagnets on the bogies and ferromagnetic rails on the parallel track girders are assumed as concentrated forces. This assumption meets the requirement of computational accuracy, in which the quantity of forces between one bogie and the guideway is at least four [2].The total virtual work of fluctuating electromagnetic forces and guide forces acting on the maglev vehicle can be computed by the following expressions:(8a)δWeypr=∑p=12∑r=2n20ΔFeyprδΔheypr+∑p=12∑r=2n−120ΔFeyprδΔheypr=∑p=12∑r=2n20kiΔin−khΔheyprδΔheypr+∑p=12∑r=2n−120kiΔin−khΔheyprδΔheypr,(8b)δWezpr=∑p=12∑r=120FezprδΔzezpr=∑p=12∑r=120−kezΔzezprδΔzezpr,where δWeypr is the total virtual work of vertical electromagnetic forces, δWezpr is the total virtual work of guide forces, Feypr is the vertically fluctuating electromagnetic force, and Δheypr is the vertically relative displacement. After equations (8a) and (8b) are expanded, they will produce many terms and each term falls into one of the four categories: δyγAAyη, δyγABiμ, δyγAEuχ, and δyγAQvχ, where 1 ≤ η ≤ 55, 1 ≤ γ ≤ 55, 1 ≤ μ≤ 20, and 1 ≤ χ ≤ 40. Here, yη is the ηth element of Xm, δyγ is the γth element of δXm, iμ is the μth element of ΔIm, and uχ and uχ are the χth elements of U and V, respectively. All these coefficients AA, AB, AE, and AQ are added to the corresponding positions of the coefficient matrices Am, Bm, Em, and Qm under the help of the guide vectors, respectively. For instance, the coefficient AA will be added in the position of the matrix Am at the γth row with the ηth column.


### 2.3. Governing Equation of Fluctuating Current

The EMS-type maglev vehicle cannot remain stable without the control law [18]. The control law is very important in the ride quality of the maglev vehicle traveling along the guideway. The fluctuation of current in equation (4) mainly depends on control voltage. The relationship between current and voltage can be described as [19](9)Δi˙n=i0h0Δh˙en−R0L0Δin+1L0ΔUen,(10)L0=μ0N02A02h0,where L0 is the inductance of the magnet winding, Δh˙en is the first-order derivative of the vertical air gap related to the nth control current, and ΔUen is the nth control voltage. In order to maintain the maglev vehicle moving over the guideway stably, the linear negative feedback control is used to get the reliable voltage. The control voltage expression is obtained by(11)ΔUen=−kay¨en−kvΔh˙en−kdΔhen,where ka, kv, and kd are the feedback gain coefficients of acceleration, velocity, and displacement, respectively, and y¨en and Δhen are the vertical acceleration and air gap related to the nth control current, respectively. For instance, Δhe1 is equal to Δhey11+Δhey12/2. Here, the important function of the feedback control law is to compensate deviation against the target value. For the maglev vehicle-vehicle system in this paper, the control voltage is adjusted by using the feedback control law to ensure that the maglev vehicle fluctuates around the equilibrium point. If the air gap between the bogie and the guideway is less than 10 mm, the larger control voltage from the feedback control law produces the larger control current to obtain sufficient electromagnetic force. Thus, the bogie can move toward the equilibrium point. In practice, each variable of motion can be measured by using sensors on the bogies.Hence, we know that the first-order derivative of control current concerns with the vertical displacement, velocity and acceleration, and the control current of the vehicle, as well as the displacement and velocity of the guideway from equations (9) and (11). The general expression of the governing equation of fluctuating current can be formulated as the following matrix form:(12)ΔI˙m=DmX¨m+JmX˙m+OmXm+GmΔIm+HmU+SmU˙,where Dm, Jm, Om, Gm, Hm, and Sm are the 20 × 55, 20 × 55, 20 × 55, 20 × 20, 20 × 40, and 20 × 40 coefficient matrices of the governing equation of the fluctuating current, respectively. Substituting equation (11) into equation (9), the governing equation of the nth fluctuating current can be described by(13)Δin=−kaL0y¨en+i0h0−kvL0Δh˙en−kdL0Δhen−R0L0Δin.Expanding the expression above, we know that each item should belong to one of following six categories:ADy¨η, AJy˙η, AOyη,AGΔiμ, AHuχ, and ASvχ. Finally, all these coefficients from six types of items are added to the corresponding positions in the nth row of Dm, Jm, Om, Gm, Hm, and Sm, respectively.Although the maglev vehicle is a complicated multirigid-body dynamic system with 55-DOFs and 20 control currents, the coefficient matrices for the equation of motion of the vehicle and the governing equation of fluctuating current in the computer program can be assembled conveniently by using the fully computerized approach. Apart from these familiar parameters such as dynamic properties, positions of all the rigid bodies and springs and damping devices, and control parameters of electromagnetic forces, the required input data about the vehicle to the computer program also include the guide matrices, which help the generating elements get into corresponding positions of the coefficient matrices. For different types of the maglev vehicle, the fully computerized approach to assemble the matrix can quickly form the equation of motion and the governing equation of fluctuating current after the required input data.


### 2.4. State-Space Equation of the Maglev Vehicle

The equation of motion of the vehicle is a second-order differential equation, while the governing equation of fluctuating current is a first-order differential equation. The coupled relationships exist between these two equations. In this study, a state-space equation of the vehicle is derived to solve these two equations. The 55-DOFs vehicle has 130 state variables including the 55 displacements and 55 velocities of the vehicle and 20 fluctuating currents, which can be expressed as(14)X=Xm,X˙m,ΔImT.The dynamic behaviors of the 3D maglev vehicle system containing the maglev vehicle model, control electromagnetic forces, and control currents are coupled and described in the linear state form as follows:(15)X˙=AX+BU+CU˙+DV,(16)A=010Mm−1Am−Mm−1Km−Mm−1CmMm−1BmOm−DmMm−1Km+DmMm−1AmJm−DmMm−1CmGm+DmMm−1Bm,(17)B=0Mm−1EmHm+DmMm−1Em,(18)C=00Sm,(19)D=0Mm−1QmDmMm−1Qm.


## 2.1. Equation of Motion of the Maglev Vehicle

As shown in Figure1, a detailed 3D model of the EMS-type maglev vehicle with the middle-low speed consists of a cabin body and 10 levitation bogies [16]. Each bogie has 4 electromagnets and there are 40 electromagnets in all. Secondary suspensions connecting the cabin with bogies are simulated by using spring-dashpot systems. Interaction ways between left and right bogies are also simplified as spring-dashpot systems. The linking modes between secondary suspensions and the cabin have fixed sliding table and lateral sliding table. The fixed sliding table is a mechanical device of the hinge joint that can transmit translational loads, while the lateral sliding stable only transmits vertical load. The main parameters of the maglev vehicle are listed in Table 1.Figure 1
3D middle-low speed maglev vehicle. (a) Side view. (b) Front view. (c) Plane view.
(a)
(b)
(c)Table 1
Main parameters of the maglev vehicle.
Parameter
Unit
Value
Cabin body mass (mc)
kg
23292
Bogie mass (mb)
kg
846
Roll inertia of cabin body (Jcx)
kg·m2
66800
Pitch inertia of cabin body (Jcy)
kg·m2
210000
Yaw inertia of cabin body (Jcz)
kg·m2
193000
Roll inertia of bogie (Jbx)
kg·m2
963
Pitch inertia of bogie (Jby)
kg·m2
1800
Yaw inertia of bogie (Jbz)
kg·m2
268
Longitudinal stiffness of secondary suspension (kcx)
N/m
900000
Vertical stiffness of secondary suspension (kcy)
N/m
140000
Lateral stiffness of secondary suspension (kcz)
N/m
6000
Longitudinal damping of secondary suspension (ccx)
N·s/m
50000
Vertical damping of secondary suspension (ccy)
N·s/m
70000
Lateral damping of secondary suspension (ccz)
N·s/m
4000
Stiffness of spring dampers connecting the left and right bogies (kby, kbz)
N/m
4
×
10
8, 4300
Damping of spring dampers connecting the left and right bogies (cby, cbz)
N·s/m
1
×
10
6, 2100
Resistance (R0)
Ω
1.6
Air gap at balanced point (h0)
mm
10
Pole area (A0)
m2
0.383
Coil turn (N0)
—
660
Vacuum permeability (μ0)
H/m
4
π
×
10
−
7
Width of electromagnet (d0)
m
0.164
Length of cabin
m
16
Width of cabin
m
2.62
Height of cabin
m
2.9The cabin and bogies in the vehicle are assumed to be rigid bodies. Each rigid body has 5-DOFs in which two translational and three rotation displacements at the geometry center of the rigid body only except for the running direction. Total DOFs of all rigid bodies in the maglev vehicle is 55. Thus, the displacement vector of the maglev vehicle is written as follows:(1)Xm=yc,zc,Φc,φc,θc,yb11,zb11,Φb11,φb11,θb11,…,ybpq,zbpq,Φbpq,φbpq,θbpqT,where y, z, Φ, φ, and θ are the vertical displacement, lateral displacement, rolling rotation, pitching rotation, and yawing rotation, respectively, the subscripts c and b represent the cabin body and levitation bogie, the value of the subscript p is selected as one for left bogies or two for right bogies, and the subscript q (q = 1, 2, 3, 4, 5) represents the qth bogie. For instance, Φb24 is the rotation by x-axis at the 4th bogie on the right. The distribution of loads for the cabin and the first bogie is shown in Figure 2, in which the spring-damping force represents the spring force plus the damping force; Fcxpl, Fcypk, and Fczpl are, respectively, the longitudinal, vertical, and lateral spring-damping forces between the cabin and bogies, Fbyg and Fbzg are, respectively, the vertical and lateral spring-damping forces connecting the left bogie with the right left bogie, and ΔFeypr and Fezpr are, respectively, the vertical and lateral electromagnetic forces, where the subscript e represents the electromagnet. Here, we know that 1≤l≤4, 1≤k≤10, 1≤g≤20, and 1≤r≤20.Figure 2
Distribution of loads for the cabin and the first bogie. (a) Distribution of spring-damping forces in the cabin. (b) Distribution of electromagnetic forces and spring-damping forces in the first bogie.
(a)
(b)The equation of motion of the maglev vehicle is usually described by(2)MmX¨m+CmX˙m+KmXm=ΔFm,in which Mm, Cm, and Km are the global mass matrix, damping matrix, and stiffness matrix of the vehicle, respectively, and ΔFm is the force vector. The total virtual work done by the 55 inertial forces, 56 damping forces, and 56 spring forces in the maglev vehicle can help to assemble the mass matrix, damping matrix, and stiffness matrix in equation (2) by using the fully computerized approach [15].


## 2.2. Expression of Fluctuating Electromagnetic Force

The calculation of the nonlinear electromagnetic force mainly relates to two variables of current and air gap, of which expression is [17](3)Fey=μ0A0N024ih2,where μ0 is the vacuum permeability, A0 is the effective areas of the magnetic poles, N0 is the number of turns of the windings, h is the vertical air gap between the electromagnet on the bogie and the guideway in the vertical direction, i is the control current in the circuit, and Fey is the vertical electromagnetic force. By applying the Taylor series expansion at the balanced point for equation (3) and ignoring the effect of the high order, the nonlinear electromagnetic force can be described by [17](4)Fey=G0+kiΔi−khΔh,with ki=μ0A0N02i0/2h02 and kh=μ0A0N02i02/2h03, where G0 is the equivalent gravity of the maglev vehicle, i0 and h0 are desired values of control current and air gap when the maglev vehicle maintains static at the balanced point, Δi is the fluctuating control current, Δh is the fluctuating air gap in the vertical direction, and ki and kh are the equivalent stiffness.In the middle-low speed maglev vehicle, the lateral guide force does not have the independent control system. The lateral guide forceFezpu can be simplified as the following expression [9]:(5)Fezpr=−kezΔyezpr,with kez=2G0/πd0, kez is the equivalent stiffness, Δyezpr is the relative displacement between the bogie and guideway in the lateral direction, and d0 is the width of the electromagnet.For the independent vehicle model, the influence of the guideway on the vehicle is considered as the external input. From equations (4) and (5), the calculation of external forces acting on the maglev vehicle is related to the following parameters: fluctuating currents, vertical and lateral displacements of bogies, and vertical and lateral displacements of the guideway. Hence, the force vector in equation (2) can be described in the following matrix-vector form:(6)ΔFm=AmXm+BmΔIm+EmU+QmV,(7a)ΔIm=Δi1,Δi2,Δi3,…,Δin−1,ΔinT,(7b)U=u11,u12,u13,…,uprT,(7c)V=v11,v12,v13,…,vprT,where ΔIm is the vector of fluctuating current, U and V are the guideway displacement vectors in vertical and lateral directions (including bridge displacement and guideway irregularity), and Δin is the nth control current (1 ≤ n ≤ 40). Am, Bm, Em, and Qm are, respectively, the 55 × 55, 55 × 20, 55 × 40, and 55 × 40 coefficient matrices of the fluctuating electromagnetic force. It has been pointed out that one bogie has four electromagnets as shown in Figure 1(c). The uniformly distributed forces between electromagnets on the bogies and ferromagnetic rails on the parallel track girders are assumed as concentrated forces. This assumption meets the requirement of computational accuracy, in which the quantity of forces between one bogie and the guideway is at least four [2].The total virtual work of fluctuating electromagnetic forces and guide forces acting on the maglev vehicle can be computed by the following expressions:(8a)δWeypr=∑p=12∑r=2n20ΔFeyprδΔheypr+∑p=12∑r=2n−120ΔFeyprδΔheypr=∑p=12∑r=2n20kiΔin−khΔheyprδΔheypr+∑p=12∑r=2n−120kiΔin−khΔheyprδΔheypr,(8b)δWezpr=∑p=12∑r=120FezprδΔzezpr=∑p=12∑r=120−kezΔzezprδΔzezpr,where δWeypr is the total virtual work of vertical electromagnetic forces, δWezpr is the total virtual work of guide forces, Feypr is the vertically fluctuating electromagnetic force, and Δheypr is the vertically relative displacement. After equations (8a) and (8b) are expanded, they will produce many terms and each term falls into one of the four categories: δyγAAyη, δyγABiμ, δyγAEuχ, and δyγAQvχ, where 1 ≤ η ≤ 55, 1 ≤ γ ≤ 55, 1 ≤ μ≤ 20, and 1 ≤ χ ≤ 40. Here, yη is the ηth element of Xm, δyγ is the γth element of δXm, iμ is the μth element of ΔIm, and uχ and uχ are the χth elements of U and V, respectively. All these coefficients AA, AB, AE, and AQ are added to the corresponding positions of the coefficient matrices Am, Bm, Em, and Qm under the help of the guide vectors, respectively. For instance, the coefficient AA will be added in the position of the matrix Am at the γth row with the ηth column.


## 2.3. Governing Equation of Fluctuating Current

The EMS-type maglev vehicle cannot remain stable without the control law [18]. The control law is very important in the ride quality of the maglev vehicle traveling along the guideway. The fluctuation of current in equation (4) mainly depends on control voltage. The relationship between current and voltage can be described as [19](9)Δi˙n=i0h0Δh˙en−R0L0Δin+1L0ΔUen,(10)L0=μ0N02A02h0,where L0 is the inductance of the magnet winding, Δh˙en is the first-order derivative of the vertical air gap related to the nth control current, and ΔUen is the nth control voltage. In order to maintain the maglev vehicle moving over the guideway stably, the linear negative feedback control is used to get the reliable voltage. The control voltage expression is obtained by(11)ΔUen=−kay¨en−kvΔh˙en−kdΔhen,where ka, kv, and kd are the feedback gain coefficients of acceleration, velocity, and displacement, respectively, and y¨en and Δhen are the vertical acceleration and air gap related to the nth control current, respectively. For instance, Δhe1 is equal to Δhey11+Δhey12/2. Here, the important function of the feedback control law is to compensate deviation against the target value. For the maglev vehicle-vehicle system in this paper, the control voltage is adjusted by using the feedback control law to ensure that the maglev vehicle fluctuates around the equilibrium point. If the air gap between the bogie and the guideway is less than 10 mm, the larger control voltage from the feedback control law produces the larger control current to obtain sufficient electromagnetic force. Thus, the bogie can move toward the equilibrium point. In practice, each variable of motion can be measured by using sensors on the bogies.Hence, we know that the first-order derivative of control current concerns with the vertical displacement, velocity and acceleration, and the control current of the vehicle, as well as the displacement and velocity of the guideway from equations (9) and (11). The general expression of the governing equation of fluctuating current can be formulated as the following matrix form:(12)ΔI˙m=DmX¨m+JmX˙m+OmXm+GmΔIm+HmU+SmU˙,where Dm, Jm, Om, Gm, Hm, and Sm are the 20 × 55, 20 × 55, 20 × 55, 20 × 20, 20 × 40, and 20 × 40 coefficient matrices of the governing equation of the fluctuating current, respectively. Substituting equation (11) into equation (9), the governing equation of the nth fluctuating current can be described by(13)Δin=−kaL0y¨en+i0h0−kvL0Δh˙en−kdL0Δhen−R0L0Δin.Expanding the expression above, we know that each item should belong to one of following six categories:ADy¨η, AJy˙η, AOyη,AGΔiμ, AHuχ, and ASvχ. Finally, all these coefficients from six types of items are added to the corresponding positions in the nth row of Dm, Jm, Om, Gm, Hm, and Sm, respectively.Although the maglev vehicle is a complicated multirigid-body dynamic system with 55-DOFs and 20 control currents, the coefficient matrices for the equation of motion of the vehicle and the governing equation of fluctuating current in the computer program can be assembled conveniently by using the fully computerized approach. Apart from these familiar parameters such as dynamic properties, positions of all the rigid bodies and springs and damping devices, and control parameters of electromagnetic forces, the required input data about the vehicle to the computer program also include the guide matrices, which help the generating elements get into corresponding positions of the coefficient matrices. For different types of the maglev vehicle, the fully computerized approach to assemble the matrix can quickly form the equation of motion and the governing equation of fluctuating current after the required input data.


## 2.4. State-Space Equation of the Maglev Vehicle

The equation of motion of the vehicle is a second-order differential equation, while the governing equation of fluctuating current is a first-order differential equation. The coupled relationships exist between these two equations. In this study, a state-space equation of the vehicle is derived to solve these two equations. The 55-DOFs vehicle has 130 state variables including the 55 displacements and 55 velocities of the vehicle and 20 fluctuating currents, which can be expressed as(14)X=Xm,X˙m,ΔImT.The dynamic behaviors of the 3D maglev vehicle system containing the maglev vehicle model, control electromagnetic forces, and control currents are coupled and described in the linear state form as follows:(15)X˙=AX+BU+CU˙+DV,(16)A=010Mm−1Am−Mm−1Km−Mm−1CmMm−1BmOm−DmMm−1Km+DmMm−1AmJm−DmMm−1CmGm+DmMm−1Bm,(17)B=0Mm−1EmHm+DmMm−1Em,(18)C=00Sm,(19)D=0Mm−1QmDmMm−1Qm.


## 3. Equation of Motion of the Three-Span Space Continuous Girder Bridge

The guideway model takes a real three-span space continuous girder bridge as shown in Figure3. The continuous girder bridge has a total length of 280 m with the main span measured at 110 m and the two side spans of 85 m. It is also a single-box and single-cell concrete bridge. The midsection of the midspan is illustrated in Figure 4, and the detailed size of the cross section can be seen in Reference [20]. Young’s modulus and Poisson’s ratio of the bridge are 3.55e10 Pa and 0.3, respectively. A 3D dynamic finite element model is established by using 3D Euler–Bernoulli beam elements [21]. In the model, track girders on the bridge are considered, and F-shape rails used to support the vehicle are neglected because of their slight stiffness compared to the continuous girder bridge. Therefore, the equation of motion of the bridge is described as(20)MbrY¨+CbrY˙+KbrY=Fbr,where Y is the displacement vector of the bridge, Mbr and Kbr is the mass and stiffness matrices of the bridge, Cbr is the Rayleigh damping matrix, and the damping ratio of the bridge is taken as 0.03, and Fbr is the time-dependent vector from the electromagnetic forces acting on the bridge:(21)Fbr=N1Gbr+kiN3X+khN4X−khU+kezN2N5X−V,where N1 and N2 are the function matrices to solve the node forces of the bridge from the vertical electromagnetic forces and the lateral guide forces, N3 is the shape function matrix to solve the fluctuating currents, and N4 and N5 are the function matrices to solve the vertical and lateral displacements of the electromagnets. Gbr is the vector of the equivalent gravity and its value of each element is equal to G0.Figure 3
Three-span continuous girder bridge.Figure 4
Midsection of the midspan in the continuous girder bridge.


## 4. Pseudoexcitation Method

### 4.1. Proposed Scheme for the PEM

The nonlinear maglev vehicle-guideway system has been changed into a linear system to conduct random vibration analysis using the PEM. From the expression of electromagnetic force in equation (4), the maglev vehicle-bridge system contains the random input from the guideway irregularity and the deterministic input from the gravity of the maglev vehicle. Here, the response results for the deterministic input are treated as the mean values of stochastic behaviors by adopting time-history analysis [22]. For the random input of the dynamic system, its self-power spectral density Sxxw needs to be converted into the pseudoexcitation, which can be described as [22](22)x˜t,w=Sxxweiwvt,where i=−1 and eiwvt is a series of harmonic waves at given discrete frequencies. It should be emphasized that w is the spatial angular frequency of guideway irregularity and v is the traveling speed of the maglev vehicle. The whole computation of the random vibration using the PEM is carried out in the complex field. Then, by using the response result multiplied by its own conjugate complex, the self-power spectral density of the desired system response is expressed as the following form [23]:(23)y˜t,w⋅y˜∗t,w=HwSxxweiwvt⋅e−iwvtSxxwH∗w=Hw2Sxxw=Syyt,w.From equations (22) and (23), the dynamic system has two major variables of time and frequency during computation. The conventional scheme adopted by much more research is to conduct a complete time-history analysis under each one of discrete frequencies [24–28]. The total number of independent computations for this way in the dynamic system reaches Nw×Nt, where Nw and Nt are the numbers of discrete frequency and time, respectively. This leads to numerous repetitive calculations, especially for the required huge time cost to solve inverse matrices at each time step. Thus, the pseudoexcitation of the maglev vehicle-bridge system should be expressed as(24)X˜t=SxxwNweiwNwvt−D11SxxwNweiwNwvt−D12⋮SxxwNweiwNwvt−Dpr,where Dpr is the longitudinal distance from the first electromagnetic force to the rth electromagnetic force. In this study, a new scheme that all discrete frequencies are computed simultaneously under each time step is proposed to further manifest the advantage of the computational efficiency of the PEM. The proposed scheme can be achieved because there is no interaction relationship between any two discrete frequencies. Thus, any repetitive calculations cannot exist in the computer program, so the computational efficiency of the PEM is maximized by using the proposed idea. The advantage of the proposed scheme will be illustrated in the following numerical simulation. Here, for the maglev vehicle-bridge system, the pseudoexcitation cannot be a vector as shown in equation (24) but a matrix expressed as(25)X˜t=Sxxw1eiw1vt−D11Sxxw2eiw2vt−D11⋯SxxwNweiwNwvt−D11Sxxw1eiw1vt−D12Sxxw2eiw2vt−D12⋯SxxwNweiwNwvt−D12⋮⋮⋱⋮Sxxw1eiw1vt−DprSxxw2eiw2vt−Dpr⋯SxxwNweiwNwvt−Dpr.


### 4.2. PSD of Guideway Irregularity

Many researches have shown that the guideway irregularity is one of the main reasons that induce random vibration of the vehicle and guideway [29]. Frequency characteristic of guideway irregularity can be expressed by the PSD function. However, to date the PSD function specialized for the maglev vehicle guideway irregularity has not been measured. As a result, the PSD function proposed by the Federal Railroad Administration is adopted to describe the spatial irregularity characteristics of two parallel F-shape rails, which have four types of track irregularities: elevation irregularity, superelevation irregularity, alignment irregularity, and gauge irregularity [30, 31]. They are expressed in the following forms:(26)S1w=A1Ω12w2+Ω22w2+Ω12,(27)S2w=A1bh−2Ω12w2w2+Ω12w2+Ω22w2+Ω32,(28)S3w=A2Ω12w2+Ω22w2+Ω12,(29)S4w=A3Ω12w2w2+Ω12w2+Ω22w2+Ω32,where Ω1=0.8246rad/m, Ω2=0.0206rad/m, and Ω3=0.438rad/m are the cutoff frequencies, A1 = 1.5× 10−7 m2·rad/m, A2 = 0.6 × 10−7 m2·rad/m, and A3 = 14× 10−8 m2·rad/m are the roughness constants, bh is the half of the distance between two parallel F-shape rails, and the value of w ranges from 0.314 rad/m to 6.28 rad/m. Based on the elevation irregularity, superelevation irregularity, alignment irregularity, and gauge irregularity, the pseudoexcitations of vertical and lateral space guideway irregularities can be described as(30a)R˜up=X˜1t+−1pbhX˜2t,(30b)R˜vp=X˜3t−−1p2X˜4t,where R˜up and R˜up are the vertical and lateral pseudoexcitations of guideway irregularities. For instance, p=1 indicates that the vertical pseudoexcitation R˜up is on the left. We know that R˜u=R˜u1,R˜u2T and R˜v=R˜v1,R˜v2T.The vectors ofU and V are the vertical and lateral displacements, which can be written as the following expressions:(31a)U=N6Y+R˜u,(31b)V=N7Y+R˜v,where N6 and N7 are the vertical and lateral shape function matrices of the bridge. The first derivative of U in equation (15) is computed by(32)U˙=v∂N6∂xY+N6Y˙+iwvR˜u.


## 4.1. Proposed Scheme for the PEM

The nonlinear maglev vehicle-guideway system has been changed into a linear system to conduct random vibration analysis using the PEM. From the expression of electromagnetic force in equation (4), the maglev vehicle-bridge system contains the random input from the guideway irregularity and the deterministic input from the gravity of the maglev vehicle. Here, the response results for the deterministic input are treated as the mean values of stochastic behaviors by adopting time-history analysis [22]. For the random input of the dynamic system, its self-power spectral density Sxxw needs to be converted into the pseudoexcitation, which can be described as [22](22)x˜t,w=Sxxweiwvt,where i=−1 and eiwvt is a series of harmonic waves at given discrete frequencies. It should be emphasized that w is the spatial angular frequency of guideway irregularity and v is the traveling speed of the maglev vehicle. The whole computation of the random vibration using the PEM is carried out in the complex field. Then, by using the response result multiplied by its own conjugate complex, the self-power spectral density of the desired system response is expressed as the following form [23]:(23)y˜t,w⋅y˜∗t,w=HwSxxweiwvt⋅e−iwvtSxxwH∗w=Hw2Sxxw=Syyt,w.From equations (22) and (23), the dynamic system has two major variables of time and frequency during computation. The conventional scheme adopted by much more research is to conduct a complete time-history analysis under each one of discrete frequencies [24–28]. The total number of independent computations for this way in the dynamic system reaches Nw×Nt, where Nw and Nt are the numbers of discrete frequency and time, respectively. This leads to numerous repetitive calculations, especially for the required huge time cost to solve inverse matrices at each time step. Thus, the pseudoexcitation of the maglev vehicle-bridge system should be expressed as(24)X˜t=SxxwNweiwNwvt−D11SxxwNweiwNwvt−D12⋮SxxwNweiwNwvt−Dpr,where Dpr is the longitudinal distance from the first electromagnetic force to the rth electromagnetic force. In this study, a new scheme that all discrete frequencies are computed simultaneously under each time step is proposed to further manifest the advantage of the computational efficiency of the PEM. The proposed scheme can be achieved because there is no interaction relationship between any two discrete frequencies. Thus, any repetitive calculations cannot exist in the computer program, so the computational efficiency of the PEM is maximized by using the proposed idea. The advantage of the proposed scheme will be illustrated in the following numerical simulation. Here, for the maglev vehicle-bridge system, the pseudoexcitation cannot be a vector as shown in equation (24) but a matrix expressed as(25)X˜t=Sxxw1eiw1vt−D11Sxxw2eiw2vt−D11⋯SxxwNweiwNwvt−D11Sxxw1eiw1vt−D12Sxxw2eiw2vt−D12⋯SxxwNweiwNwvt−D12⋮⋮⋱⋮Sxxw1eiw1vt−DprSxxw2eiw2vt−Dpr⋯SxxwNweiwNwvt−Dpr.


## 4.2. PSD of Guideway Irregularity

Many researches have shown that the guideway irregularity is one of the main reasons that induce random vibration of the vehicle and guideway [29]. Frequency characteristic of guideway irregularity can be expressed by the PSD function. However, to date the PSD function specialized for the maglev vehicle guideway irregularity has not been measured. As a result, the PSD function proposed by the Federal Railroad Administration is adopted to describe the spatial irregularity characteristics of two parallel F-shape rails, which have four types of track irregularities: elevation irregularity, superelevation irregularity, alignment irregularity, and gauge irregularity [30, 31]. They are expressed in the following forms:(26)S1w=A1Ω12w2+Ω22w2+Ω12,(27)S2w=A1bh−2Ω12w2w2+Ω12w2+Ω22w2+Ω32,(28)S3w=A2Ω12w2+Ω22w2+Ω12,(29)S4w=A3Ω12w2w2+Ω12w2+Ω22w2+Ω32,where Ω1=0.8246rad/m, Ω2=0.0206rad/m, and Ω3=0.438rad/m are the cutoff frequencies, A1 = 1.5× 10−7 m2·rad/m, A2 = 0.6 × 10−7 m2·rad/m, and A3 = 14× 10−8 m2·rad/m are the roughness constants, bh is the half of the distance between two parallel F-shape rails, and the value of w ranges from 0.314 rad/m to 6.28 rad/m. Based on the elevation irregularity, superelevation irregularity, alignment irregularity, and gauge irregularity, the pseudoexcitations of vertical and lateral space guideway irregularities can be described as(30a)R˜up=X˜1t+−1pbhX˜2t,(30b)R˜vp=X˜3t−−1p2X˜4t,where R˜up and R˜up are the vertical and lateral pseudoexcitations of guideway irregularities. For instance, p=1 indicates that the vertical pseudoexcitation R˜up is on the left. We know that R˜u=R˜u1,R˜u2T and R˜v=R˜v1,R˜v2T.The vectors ofU and V are the vertical and lateral displacements, which can be written as the following expressions:(31a)U=N6Y+R˜u,(31b)V=N7Y+R˜v,where N6 and N7 are the vertical and lateral shape function matrices of the bridge. The first derivative of U in equation (15) is computed by(32)U˙=v∂N6∂xY+N6Y˙+iwvR˜u.


## 5. Application of the Separated Iteration Method

The state-space equation of the maglev vehicle for equation (15) is a first-order differential equation, while the equation of motion of the bridge for equation (20) is a second-order differential equation. The equation of motion of the bridge can be turned into an equivalent first-order differential equation to be solved, but it will reduce the computing speed of the program because the matrix dimensions of the equation of motion for the bridge are doubled. In this study, the state-space equation of the maglev vehicle and the equation of motion of the bridge are solved using the precise integration method and the Newmark method [32, 33], respectively, and the separated iteration method is applied in the numerical simulation. Hence, these two equations for the time from t to t+Δt with the iteration step from j to j+1 can be expressed as follows:(33a)Xjt+Δt=expAΔtXt+∫tΔtexpAt−τF¯jm,t+Δtτdτ,(33b)Yj+1t+Δt=K¯−1Fjbr,t+Δt+F¯br,t,where F¯jm,t+Δt is the external input of the state-space equation and denoted by(34)F¯jm,t+Δt=BN6,t+ΔtYjt+Δt+R˜u,t+Δt+Cv∂N6,t+Δt∂xYjt+Δt+N6,t+ΔtYjt+Δt+iwvR˜u,t+Δt+DN7,t+ΔtYjt+Δt+R˜v,t+Δt.Equivalent forces in equation (33b) yield(35)Fjbr,t+Δt=N1,t+ΔtGbr+kiN3Xjt+Δt+khN4Xjt+Δt−khN6,t+ΔtYjt+Δt−khR˜u,t+Δt+kezN2,t+ΔtN5Xjt+Δt−N7,t+ΔtYjt+Δt−R˜v,t+Δt,(36)F¯br,t=Cbr2YtΔt+Y˙t+Mbr4YtΔt2+4Y˙tΔt+Y¨t,where Yt, Y˙t, and Y¨t are the computing results at time t and their values are Yj+1t, Y˙j+1t, and Y¨j+1t, respectively. The equivalent stiffness in equation (33b) is given as(37)K¯=Kbr+2CbrΔt+4MbrΔt2.During the separated iteration, the initial iterative displacement of the bridgeY1t+Δt is selected as Yt, and the first-order derivative of Yjt+Δt in equation (34) is described by(38)Y˙jt+Δt=2Yjt+Δt−YtΔt−Y˙t.After the iteration steps terminate at each time step, the acceleration of the bridge can be calculated as(39)Y¨j+1t+Δt=M−1Fj+1br,t+Δt−CbrY˙j+1t+Δt−KbrYj+1t+Δt.When the bridge displacements between the current iteration step and the previous iteration step almost are the same, the iterations at the time step will be ended. The convergence condition is(40)maxmaxYj+1t+Δt−Yjt+Δt>10−10.When the pseudoexcitation of the guideway irregularitiesR˜u,t+Δt and R˜v,t+Δt in equations (34) and (35) is set as zero vectors, equations (33a) and (33b) can be used to solve the mean values of random responses only considering the influence of the gravity on the maglev vehicle-bridge system by conducting time-history analysis. Moreover, when the deterministic input Gbr is selected as the zero matrix, equations (33a) and (33b) are also used to solve dynamic responses of the maglev vehicle-bridge system subjected to random input of guideway irregularity by using the PEM. As a result, only the time-history analysis seems to appear in these two types of numerical simulations. The computer program is very convenient for both the time-history analysis and the random vibration analysis, and the required change in the program is very small when either of these two analyses is conducted. It can also maximize the computational efficiency of the PEM in the random vibration analysis of the maglev vehicle-bridge system.


## 6. Numerical Simulations

### 6.1. Computational Efficiency of the Proposed Scheme

To verify the computational efficiency and correctness of the proposed scheme, the solution to the RMS of the random vibration response for a maglev vehicle with one car traveling along the continuous girder bridge is conducted by using three methods, i.e., the proposed PEM-based scheme, the conventional PEM-based scheme, and the Monte Carlo (MC) method [34]. Major computational procedures for the PEM using the proposed scheme or the conventional scheme are illustrated in the previous Sections 4 and 5. The MC method is to statistically analyze response results from substantial random samples. The accuracy of the MC method for random vibration analysis increases with the number of samples. In this study, all calculations are conducted on a personal computer with Intel (R) Core (TM) i7-8700 processors. Figures 5(a) and 5(b) show the time-history RMSs of the vertical displacements for the cabin and the bridge midpoint. The proposed scheme, the conventional scheme, and the MC method with the guideway irregularity samples of 500 consume 1,120 s, 9,487 s, and 15,707 s for computing the RMSs of dynamic responses of the vehicle-bridge system, respectively. Also, the proposed scheme and the conventional scheme have the same computational results, in which the stable RMS curves have higher precision than the fluctuating curves of the MC method with 500 samples. Consequently, the PEM using the proposed scheme for the linear random analysis of the maglev vehicle-bridge system is very efficient compared with the conventional scheme and the MC method.Figure 5
RMSs of vertical displacements for (a) the cabin and (b) the bridge midpoint when the maglev vehicle with one car moves on the bridge at the speed of 100 km/h.
(a)
(b)


### 6.2. Mean Value

Figures6(a) and 6(b) show the mean values of vertical accelerations for the second cabin and the bridge midpoint. It should be noted that the random vibration analysis of the maglev vehicle in this paper is mainly conducted under the case of the maximum commercial operating speed of 100 km/h. Here, the mean values of lateral accelerations for the vehicle and bridge are not given because their lateral responses are smaller without specific excitations. Major interaction between the maglev vehicle and three-span continuous girder bridge is the vertical motions when the guideway surface is smooth. When t = 2.39 s, the second car in the maglev vehicle begins to move on the bridge. The significant response of the second cabin focuses on the case of the vehicle moving on the bridge, especially for the vehicle on the bridge midpoints of side spans and main span, while the effects on the three-span continuous girder bridge mainly concentrate on the main span. The maximum values of vertical accelerations for the second cabin and bridge midpoint are 0.16 m2/s and 0.09 m2/s. Besides, the negative feedback control in the maglev vehicle is a reasonable design because the oscillation in the vertical acceleration curve of the cabin can reduce to zero rapidly after the second car leaves the bridge at t = 13.05 s.Figure 6
Mean values of vertical accelerations for (a) the second cabin and (b) the bridge midpoint.
(a)
(b)


### 6.3. 3σ Rule

With the assumption that the random process of guideway irregularity is a Gaussian distribution, the 3σ rule can be used to estimate upper or lower boundaries of response, and its accuracy rate has 99.73%. Figures 7(a) and 7(b) show the estimated boundaries for vertical and lateral accelerations of the cabin, and response results of any arbitrary two irregularity sample samples stay within the scope of the boundaries except for very few points. Thus, the extreme value of response that is of most interest in practical engineering can be picked from its corresponding boundary. When the vehicle moves on the bridge, the maximum vertical and lateral accelerations of the cabin can get to 0.32 m2/s and 0.03 m2/s, respectively. Moreover, the curve of the lateral acceleration of the cabin shown in Figure 7(b) appears to be low fluctuation, and this is a clear demonstration that lateral responses of the maglev vehicle mainly come from the influence of the guideway irregularity. By comparing with accelerations of the second cabin in Figures 6(a) and 7(a), it can also be found that the random irregularity still has great influence on vertical motion of the vehicle. Certainly, response accelerations of the bridge midpoint for any two irregularity sample curves also almost stay within boundaries estimated using the 3σ rule as shown in Figures 8(a) and 8(b). Maximum vertical and lateral accelerations of the bridge midpoint are 0.11 m2/s and 0.03 m2/s, respectively.Figure 7
Comparison of time histories of the cabin acceleration between the estimated boundary and the conventional calculation for (a) vertical and (b) lateral directions.
(a)
(b)Figure 8
Comparison of time histories of the bridge midpoint acceleration between the estimated boundary and the conventional calculation for (a) vertical and (b) lateral directions.
(a)
(b)


### 6.4. Time-Dependent PSD of Dynamic Response

Time-dependent PSDs of vertical and lateral accelerations for the second cabin are given in Figures9(a) and 9(b). When the maglev vehicle begins to move on the ground, PSD of the cabin rapidly enlarge around zero frequency. There are a series of peaks distributed in the frequency range of 0–2.2 rad/m for the PSD of the cabin vertical acceleration (see Figure 9(a)) and then they slowly diminish with the increasing frequency. Here, the maximum peak appears at the frequency 0.16 rad/m. On the contrary, the influence of frequency on the lateral acceleration of the cabin is concentrated in the range of 0–0.52 rad/m.Figure 9
Time-dependent PSDs of the second cabin for (a) vertical and (b) lateral directions.
(a)
(b)Displayed in Figures10(a) and 10(b) are the time-dependent PSDs of bridge midpoint accelerations for vertical and lateral directions. Before the maglev vehicle reaches the bridge, the vertical and lateral PSDs of the bridge midpoint acceleration are zeros. With the increase of moving time of the vehicle on the bridge, the PSD of the vertical acceleration for the bridge midpoint appears some peaks and valleys in the frequency range of 0.2–0.8 rad/m, in which major frequencies concentrate on 0.24 and 0.66 rad/m. For the lateral acceleration of the bridge midpoint, the effect of frequency mainly focuses on 0.38 rad/m (see Figure 10(b)).Figure 10
Time-dependent PSDs of the bridge midpoint acceleration for (a) vertical and (b) lateral directions.
(a)
(b)


## 6.1. Computational Efficiency of the Proposed Scheme

To verify the computational efficiency and correctness of the proposed scheme, the solution to the RMS of the random vibration response for a maglev vehicle with one car traveling along the continuous girder bridge is conducted by using three methods, i.e., the proposed PEM-based scheme, the conventional PEM-based scheme, and the Monte Carlo (MC) method [34]. Major computational procedures for the PEM using the proposed scheme or the conventional scheme are illustrated in the previous Sections 4 and 5. The MC method is to statistically analyze response results from substantial random samples. The accuracy of the MC method for random vibration analysis increases with the number of samples. In this study, all calculations are conducted on a personal computer with Intel (R) Core (TM) i7-8700 processors. Figures 5(a) and 5(b) show the time-history RMSs of the vertical displacements for the cabin and the bridge midpoint. The proposed scheme, the conventional scheme, and the MC method with the guideway irregularity samples of 500 consume 1,120 s, 9,487 s, and 15,707 s for computing the RMSs of dynamic responses of the vehicle-bridge system, respectively. Also, the proposed scheme and the conventional scheme have the same computational results, in which the stable RMS curves have higher precision than the fluctuating curves of the MC method with 500 samples. Consequently, the PEM using the proposed scheme for the linear random analysis of the maglev vehicle-bridge system is very efficient compared with the conventional scheme and the MC method.Figure 5
RMSs of vertical displacements for (a) the cabin and (b) the bridge midpoint when the maglev vehicle with one car moves on the bridge at the speed of 100 km/h.
(a)
(b)


## 6.2. Mean Value

Figures6(a) and 6(b) show the mean values of vertical accelerations for the second cabin and the bridge midpoint. It should be noted that the random vibration analysis of the maglev vehicle in this paper is mainly conducted under the case of the maximum commercial operating speed of 100 km/h. Here, the mean values of lateral accelerations for the vehicle and bridge are not given because their lateral responses are smaller without specific excitations. Major interaction between the maglev vehicle and three-span continuous girder bridge is the vertical motions when the guideway surface is smooth. When t = 2.39 s, the second car in the maglev vehicle begins to move on the bridge. The significant response of the second cabin focuses on the case of the vehicle moving on the bridge, especially for the vehicle on the bridge midpoints of side spans and main span, while the effects on the three-span continuous girder bridge mainly concentrate on the main span. The maximum values of vertical accelerations for the second cabin and bridge midpoint are 0.16 m2/s and 0.09 m2/s. Besides, the negative feedback control in the maglev vehicle is a reasonable design because the oscillation in the vertical acceleration curve of the cabin can reduce to zero rapidly after the second car leaves the bridge at t = 13.05 s.Figure 6
Mean values of vertical accelerations for (a) the second cabin and (b) the bridge midpoint.
(a)
(b)


## 6.3. 3σ Rule

With the assumption that the random process of guideway irregularity is a Gaussian distribution, the 3σ rule can be used to estimate upper or lower boundaries of response, and its accuracy rate has 99.73%. Figures 7(a) and 7(b) show the estimated boundaries for vertical and lateral accelerations of the cabin, and response results of any arbitrary two irregularity sample samples stay within the scope of the boundaries except for very few points. Thus, the extreme value of response that is of most interest in practical engineering can be picked from its corresponding boundary. When the vehicle moves on the bridge, the maximum vertical and lateral accelerations of the cabin can get to 0.32 m2/s and 0.03 m2/s, respectively. Moreover, the curve of the lateral acceleration of the cabin shown in Figure 7(b) appears to be low fluctuation, and this is a clear demonstration that lateral responses of the maglev vehicle mainly come from the influence of the guideway irregularity. By comparing with accelerations of the second cabin in Figures 6(a) and 7(a), it can also be found that the random irregularity still has great influence on vertical motion of the vehicle. Certainly, response accelerations of the bridge midpoint for any two irregularity sample curves also almost stay within boundaries estimated using the 3σ rule as shown in Figures 8(a) and 8(b). Maximum vertical and lateral accelerations of the bridge midpoint are 0.11 m2/s and 0.03 m2/s, respectively.Figure 7
Comparison of time histories of the cabin acceleration between the estimated boundary and the conventional calculation for (a) vertical and (b) lateral directions.
(a)
(b)Figure 8
Comparison of time histories of the bridge midpoint acceleration between the estimated boundary and the conventional calculation for (a) vertical and (b) lateral directions.
(a)
(b)


## 6.4. Time-Dependent PSD of Dynamic Response

Time-dependent PSDs of vertical and lateral accelerations for the second cabin are given in Figures9(a) and 9(b). When the maglev vehicle begins to move on the ground, PSD of the cabin rapidly enlarge around zero frequency. There are a series of peaks distributed in the frequency range of 0–2.2 rad/m for the PSD of the cabin vertical acceleration (see Figure 9(a)) and then they slowly diminish with the increasing frequency. Here, the maximum peak appears at the frequency 0.16 rad/m. On the contrary, the influence of frequency on the lateral acceleration of the cabin is concentrated in the range of 0–0.52 rad/m.Figure 9
Time-dependent PSDs of the second cabin for (a) vertical and (b) lateral directions.
(a)
(b)Displayed in Figures10(a) and 10(b) are the time-dependent PSDs of bridge midpoint accelerations for vertical and lateral directions. Before the maglev vehicle reaches the bridge, the vertical and lateral PSDs of the bridge midpoint acceleration are zeros. With the increase of moving time of the vehicle on the bridge, the PSD of the vertical acceleration for the bridge midpoint appears some peaks and valleys in the frequency range of 0.2–0.8 rad/m, in which major frequencies concentrate on 0.24 and 0.66 rad/m. For the lateral acceleration of the bridge midpoint, the effect of frequency mainly focuses on 0.38 rad/m (see Figure 10(b)).Figure 10
Time-dependent PSDs of the bridge midpoint acceleration for (a) vertical and (b) lateral directions.
(a)
(b)


## 7. Conclusions

This study has been presented a framework for the linear random vibration analysis of a coupled 3D maglev vehicle-bridge system. The framework coupled the motion of the maglev vehicle, the control system, the dynamic behavior of the bridge, and the guideway irregularity. The proposed scheme has maximized the advantage of the PEM in the computational efficiency of the random vibration analysis. In addition, any type of matrices in the maglev vehicle-bridge system can be assembled automatically by using the fully computerized approach. It is very useful for solving the dynamic problem of various maglev vehicle-bridge models.



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*Source: 4920659-2019-12-12.xml*