Research Article  Open Access
Numerical Simulation of Vertical Random Vibration of TrainSlab TrackBridge Interaction System by PEM
Abstract
The paper describes the numerical simulation of the vertical random vibration of trainslab trackbridge interaction system by means of finite element method and pseudoexcitation method. Each vehicle is modeled as fourwheelset massspringdamper system with twolayer suspension systems. The rail, slab, and bridge girder are modeled by threelayer elastic BernoulliEuler beams connected with each other by spring and damper elements. The equations of motion for the entire system are derived according to energy principle. By regarding rail irregularity as a series of multipoint, differentphase random excitations, the random load vectors of the equations of motion are obtained by pseudoexcitation method. Taking a ninespan simply supported beam bridge traveled by a train consisting of 8 vehicles as an example, the vertical random vibration responses of the system are investigated. Firstly, the suitable number of discrete frequencies of rail irregularity is obtained by numerical experimentations. Secondly, the reliability and efficiency of pseudoexcitation method are verified through comparison with Monte Carlo method. Thirdly, the random vibration characteristics of trainslab trackbridge interaction system are analyzed by pseudoexcitation method. Finally, applying the 3σ rule for Gaussian stochastic process, the maximum responses of trainslab trackbridge interaction system with respect to various train speeds are studied.
1. Introduction
The dynamic analysis of railway track or bridge subjected to a moving train has long been a subject of great interest in the field of civil engineering [1–10]. As is well known [11–19], rail irregularity, with random nature, is one of the most important factors that can amplify the vibration responses of traintrack/bridge interaction system. The dynamic responses in most of the previous researches are usually calculated by using only one timehistory sample of rail irregularity. From the point of view of random vibration, those results can merely represent one of a series of possible random samples due to the fact that each timehistory sample generated from the PSD function of rail irregularity is random.
With the increasing speed and axle load of train, the random vibration of traintrack/bridge interaction system has received more and more attention and is usually analyzed by means of Monte Carlo method (MCM) [12, 16]. In this approach, the timehistory samples of dynamic responses are calculated by using a series of samples of rail irregularity firstly, and then the statistical characteristics of responses are obtained according to the maximum values of each sample of dynamic responses. In order to ensure the reliability of the simulation, tens, hundreds, or even thousands of samples must be calculated, which may result in intolerable computer time, especially for the sophisticated trainslab trackbridge interaction system.
Pseudoexcitation method (PEM), originally developed from the field of bridge seismic resistance [20], has been applied to analyze efficiently and accurately the linear timeindependent or timedependent systems with stationary or nonstationary random excitations [21–25]. Lu et al. [23] studied the stationary random vibration responses of vehicleballasted track interaction system by PEM in which a timeindependent vehicletrack interaction model was adopted. By adopting a trainbridge interaction element with the track structure omitted [26], PEM was applied to research the nonstationary random vibration of timedependent trainbridge interaction system [24].
With the increasing use of ballastless track on bridge in modern highspeed railway, this paper investigates the vertical random vibration responses of slab ballastless track and bridge subjected to a moving train based on finite element method (FEM) and PEM. The train, slab track, and bridge are regarded as an integrated system. Each moving vehicle is modeled as fourwheelset massspringdamper system with twolayer suspension systems possessing 6 degrees of freedom (DOFs) [17]. The rail, slab, and bridge girder are modeled by threelayer elastic BernoulliEuler beams [17, 19, 27]. The elasticity and damping properties of the fastener and bridge bearing are represented by discrete springs and dampers. Simultaneously, the elasticity and damping properties of the concrete asphalt layer (CA layer) are represented by continuous springs and dampers [27, 28]. The equations of motion of finite element form for the entire system are derived by means of energy principle [28]. The effects of rail irregularity are regarded as a series of multipoint, differentphase random excitations by taking time lags between the wheels into account. The random excitations between rail and wheels caused by rail irregularity are then transformed into a series of deterministic pseudoharmonic excitation vectors by PEM, so that the random vibration responses of train, slab track, and bridge can be obtained by stepbystep integration method such as Newmarkβ method [29] or Wilson method [30]. Taking a ninespan simply supported beam bridge traveled by a train consisting of 2 motor cars and 6 trailer cars as an example, the suitable number of discrete frequencies of rail irregularity for calculating the random vibration responses of the system is obtained by numerical experimentations, the reliability and efficiency of the random vibration responses obtained by PEM are verified through comparison with those by MCM; the random vibration characteristics of trainslab trackbridge interaction system are analyzed by PEM, and finally the maximum responses of trainslab trackbridge interaction system with respect to various train moving speeds are studied by applying the 3σ rule for Gaussian stochastic process.
2. Models of Train, Slab Track, and Bridge
2.1. Model of Train
Figure 1 shows a train consisting of a series of fourwheelset vehicles moving with constant speed on a slab track structure resting on a series of simply supported beams to model a railway bridge and the two approach subgrades. The train consists of the front and rear motor cars numbered 1 and 2, respectively, and trailer cars numbered from right to left. It is assumed that the downward vertical displacements and clockwise direction rotation of train are taken as positive and that they are measured with reference to their respective static equilibrium positions before moving upon the track concerned.
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(b)
Each trailer car in the train is modeled as a massspringdamper system consisting of one carbody, two bogies, four wheelsets, and twostage suspensions. As shown in Figure 1, the carbody rests on the front and rear bogies, each of which in turn is supported by two wheelsets. The carbody is modeled as a rigid body with a mass and a moment of inertia about the transverse horizontal axis through its center of gravity. Similarly, each bogie is considered as a rigid body having a mass and a moment of inertia about the transverse horizontal axis through its center of gravity. Each wheelset has a mass . The spring and shock absorber in the primary suspension for each wheelset are characterized by spring stiffness and damping coefficient , respectively. Likewise, the secondary suspension between carbody and each bogie is characterized by spring stiffness and damping coefficient . As the carbody is assumed to be rigid, the motions of the th trailer carbody may be described by the vertical displacement and rotation at its center of gravity, where the subscript denotes the trailer car number. Similarly, the motions of the front bogie of the th trailer car may be described by the vertical displacement and rotation at its center of gravity; the motions of the rear bogie of the th trailer car may be described by the vertical displacement and rotation at its center of gravity. The motions of the four wheelsets from right to left of the th trailer car may be described by the vertical displacements , , , and , respectively. Therefore, the total number of DOFs for each trailer car is ten. However, it is assumed that each wheelset of all vehicles is always in full contact with the rail in this paper; that is, the vertical displacement of each wheelset is constrained by the displacement of the rail. Consequently, the independent DOFs for each trailer car become six.
Each motor car in the train is also modeled as a massspringdamper system consisting of one carbody, two bogies, four wheelsets, and twostage suspensions. The carbody has a mass and a moment of inertia . Each bogie has a mass and a moment of inertia . Each wheelset has a mass . The primary suspension is characterized by spring stiffness and damping coefficient . Likewise, the secondary suspension is characterized by spring stiffness and damping coefficient . The independent DOFs for each motor car include, too, the carbody vertical displacement and rotation , the front bogie vertical displacement and rotation , and the rear bogie vertical displacement and rotation , where the subscript denotes the motor car number.
2.2. Models of Slab Track and Bridge
As shown in Figure 1, the rail, slab, and bridge girder are all modeled as elastic BernoulliEuler beam. The two rails are effectively treated as one in the subsequent analysis. On the basis of FEM, the rail, slab, and bridge girder are all divided into a series of beam elements of equal length . The elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness and dampers with damping coefficient . The elasticity and damping properties of the CA layer underlying the slab are represented by continuous massless springs with stiffness and dampers with damping coefficient . Simultaneously, the elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness and dampers with damping coefficient . It is assumed that the damping of the rail and slab is neglected [19, 31], and the bridge girder has linear viscous damping [15, 32]. In addition, by neglecting axial deformations of the rail, slab, and bridge girder, each node of the rail, slab, and bridge girder has two DOFs, that is, vertical displacement and rotation about an axis normal to the plane of paper. The cubic Hermitian interpolation polynomials are used as shape functions of the rail, slab, and bridge girder elements.
It is assumed also that the downward deflections of rail, slab, and bridge are taken as positive and that they are measured with reference to their respective vertical static equilibrium positions. Let denote the initial vertical irregularities of rail and be measured with reference to smooth profile of rail; that is, , if the top surface of rail is smooth. Likewise, it is considered positive in the downward direction.
3. Equations of Motion for a TrainSlab TrackBridge Interaction System
By using the energy principle, such as the principle of a stationary value of total potential energy of a dynamic system [28], one can derive the equations of motion written in submatrix for the trainslab trackbridge interaction system as where the subscripts “,” “,” “,” and “” denote the train, rail, slab, and bridge girder, respectively. The displacement vector, the matrices of mass, stiffness, and damping, as well as the load vector for the train, rail, slab, and bridge girder, are explained as follows.
3.1. Displacement Vectors
The total train displacement vector with order can be written as where the superscript “” denotes the transpose of the vector and () and () are the displacement vectors of the th trailer car and the th motor car, respectively, which can be expressed as
The displacement vector of rail with order can be written as where denotes the total number of DOFs of rail.
The displacement vector with order for a series of continuously supported beams to model slab can be written as where () denotes the displacement vector of the th slab, denotes the total number of slab, and denotes the total number of DOFs of all slab. with order and can be expressed as
The displacement vector with order for a series of simply supported beams to model the bridge can be written as where () denotes the displacement vector of the th bridge girder, denotes the total number of bridge girder, and denotes the total number of DOFs of all bridge girder. with order and can be expressed as
3.2. Matrices for Train
The matrices of train are marked with the subscript “.” The mass matrix of train, with order () × (), can be written as where and with order denote the mass matrices of the th trailer car and th motor car, respectively, and can be expressed as
The stiffness matrix of train, with order () × (), can be written as where and with order , denoting the stiffness matrices of the th trailer car and th motor car, respectively, can be expressed as in which denotes half of longitudinal distance between the centers of gravity of trailer car’s front and rear bogies, denotes half of longitudinal distance between the centers of gravity of motor car’s front and rear bogies, denotes half of trailer car’s bogie axle base, and denotes half of motor car’s bogie axle base.
The damping matrix of train with order () × () can be obtained by simply replacing in the corresponding stiffness matrix by .
3.3. Matrices for Rail, Slab, and Bridge
The matrices of rail are marked with the subscript “.” The mass matrix of rail, with order , can be written as with where , with order , represents the overall mass matrix of the rail itself, represents rail mass per unit length, represents the total number of rail elements, represents the local coordinate measured from the left node of a beam element, and , with order , represents the shape function matrix for the th rail element. It should be noted that each element is zero in except for those corresponding to the four DOFs of the two nodes of the th rail element. , with order , represents the overall mass matrix induced by all the wheel masses of trailer cars; , with order , represents the shape function matrix for the rail element, evaluated at the position of the th wheelset of the th trailer car; , , , and , respectively, represent the distances between the 1st, 2nd, 3rd, and 4th wheelsets of the th trailer car and the left node of the rail element upon which the wheelset is acting. It should be noted that each element is zero in except those corresponding to the four DOFs of the two nodes of the rail element upon which the th wheelset of the th trailer car is acting. is time dependent as the th wheelset of the th trailer car moves from one position to another within one rail element. As the th wheelset of the th trailer car moves to the next rail element, will shift in position corresponding to the DOFs of the rail element where the th wheelset of the th trailer car is positioned. Similarly, , with order , represents the overall mass matrix induced by all the wheel masses of motor cars; , with order , represents the shape function matrix for the rail element, evaluated at the position of the th wheelset of the th motor car; , , , and , respectively, represent the distances between each wheelset of the th motor car and the left node of the corresponding rail element upon which the wheelset is acting. It should be noted too for that, apart from those elements corresponding to the four DOFs of the two nodes of the rail element upon which the th wheelset of the th motor car is acting, all other elements are zero. is also time dependent.
The stiffness matrix of rail, with order , can be similarly expressed in terms of the overall stiffness matrix of the rail itself, overall stiffness matrix induced by all trailer cars, overall stiffness matrix induced by all motor cars, and overall stiffness matrix induced by the stiffness of all fasteners as with where denotes Young’s modulus of the rail and denotes the constant moment of inertia of the rail cross section and the prime differentiation with respect to local coordinate . In the formulation of , the shape function matrix () of order for the rail element is evaluated at the position of the th fastener; denotes the distance between the th fastener and the left node of the rail element containing the th fastener, and denotes the total number of fastener. It should be noted for that, apart from those elements corresponding to four DOFs of the two nodes of the rail element containing the th fastener, all other elements are zero.
With omission of the damping of the rail itself, the damping matrix of rail, with order , can be written in terms of the overall damping matrix induced by all trailer cars, overall damping matrix induced by all motor cars, and overall damping matrix induced by the damping of all fasteners as with
The matrices of slab, with order , are marked with the subscript “” and can be written in terms of the matrices of the slab itself, the matrices induced by fastener on the slab, and the matrices induced by CA layer underlying the slab. The matrices for bridge, with order , are marked with the subscript “” and can be written in terms of the matrices of the bridge girder itself, the matrices induced by CA layer lying on the bridge girder, and the matrices induced by bearing at the support of bridge girder. The matrices of slab and bridge are not derived in detail here but can be obtained by following the similar procedure for derivation of rail matrices.
3.4. Matrices for TrainRailSlabBridge Interaction
The matrices for trainrail interaction are marked with subscript “” or “.” The stiffness matrices of order , of order , of order , and of order induced by the interaction between the train and rail can be written, respectively, as with in which and represent the stiffness matrices induced by the interaction between the th wheelset of the th trailer car and rail, and and are the corresponding damping matrices. , , , and consist of zero row vectors except for those corresponding to the two DOFs of front bogie of the th trailer car, while , , , and consist of similar column vectors. Accordingly, , , , , , , , and are formed by row or column vectors where the only nonzero elements correspond to the two DOFs of the rear bogie.
The matrices induced by the interaction between motor car and rail can be worked out similarly.
The matrices for railslabbridge interaction can be also derived by following the similar procedure for derivation of trainrail interaction matrices. The matrices for railslab interaction, induced by the matrices of fastener between rail and slab, are marked with subscript “” or “.” The matrices for slabbridge interaction, induced by the matrices of CA layer between slab and bridge girder, are marked with subscript “” or “.”
3.5. Load Vector for Train, Rail, Slab, and Bridge
The load vector of train with order can be written as where the load vector of the th trailer car and and the load vector of the th motor car and with order can be written, respectively, as in which denotes the first derivative of rail irregularity .
The load vector of rail with order can be written as with where denotes the second derivative of rail irregularity and , , , and , respectively, denote the load vectors of the th wheelset of the th trailer car acting upon the rail caused by gravity force of trailer car, rail irregularity, velocity of rail irregularity, and acceleration of rail irregularity. , , , and are the load vectors of the th wheelset of the th motor car acting upon the rail caused by gravity force of motor car, rail irregularity, velocity of rail irregularity, and acceleration of rail irregularity, respectively.
In addition, each element for load vector of slab with order and load vector of bridge with order is zero.
Now let
Then, the load vector of the total trainslab trackbridge interaction system can be expressed by the deterministic load vector induced by the train gravity force and the random load vector induced by the rail irregularity can be expressed as
4. Random Vibration Analysis of TrainSlab TrackBridge Interaction System by PEM
4.1. The Pseudoexcitation for the System
Let be the distance from the lefthand starting point of the track model to the wheel/rail contact point at time ; there exists a transformation from the PSD function of in space domain to the PSD function of in time domain, according to : where , , and denote, respectively, the time frequency (rad/s), the spatial frequency (rad/m), and train speed (m/s). Obviously, .
According to (21)–(26), the pseudoexcitation induced by the rail irregularity can be obtained by PEM: with
4.2. The Calculation of Random Responses of the System
Based on (1) and (26), the equations of motion for a trainslab trackbridge interaction system can be expressed as in which , , and denote mass, damping, and stiffness matrices of the total trainslab trackbridge interaction system, respectively; , , and denote, respectively, displacement, velocity and acceleration vectors of the system.
By assuming the rail irregularity to be a zero mean valued Gaussian random process, the mean value (MV) of arbitrary response of the system is only caused by the train gravity force and can be calculated easily by solving (31) in time domain using a stepbystep integration method. Consider
The arbitrary pseudoresponse of the system caused by can be obtained by solving (32) in frequency domain and time domain. Consider
According to PEM and random vibration theory, the timedependent power spectral density (PSD) and standard deviation (SD) of arbitrary response of the system can be written as where the superscript “” denotes complex conjugate, denotes the total number of discrete frequencies of rail irregularity, and denotes the th frequency interval of rail irregularity.
4.3. The Solution Procedures
The flowchart for efficiently analyzing the vertical random vibration of trainslab trackbridge interaction system under the action of rail irregularity is shown in Figure 2 by following the above explanation of random vibration analysis by PEM.
5. Case Study
5.1. The Properties of TrainSlab TrackBridge Interaction System
In order to reduce the boundary effect of the subgrade as far as possible, a ninespan simply supported beam highspeed railway bridge with the span length of 32 m is considered as shown in Figure 1. The central part of railway slab track is supported on bridge, while the left and right parts of the track are supported on subgrades adjacent to the bridge. The track is assumed to be continuous throughout, while the lengths of left and right parts of track considered are, respectively, 260 m and 210 m, in order that the train moving from left to right can not only reach the steadystate response [15] before it runs upon the track on bridge but also move fully out of the right boundary of bridge. The lengths of rail, slab, and bridge girder element are all equal to fastener spacing of 0.65 m. The train comprises front and rear motor cars with the same properties and six identical trailer cars moving with constant velocity . The physical parameters of train, track, and bridge are listed, respectively, in Tables 1 and 2. The Germany highspeed track vertical profile irregularity PSD function [13, 33] is adopted; that is, where rad/m, rad/m, and m·rad.


To solve the equations of motion for the trainslab trackbridge interaction system, the Wilson method is used with and moving length of 0.1 m of vehicles along track for each time step. In addition, the spatial frequency of the PSD ranges from to rad/m; that is to say, the wavelength of rail irregularity ranges from 1 to 250 m. It should be noted that a series of wavelengths of unequal intervals are selected by the following expression; that is, where denotes the wavelength of rail irregularity () and denotes a constant related to the maximum frequency points .
5.2. Selection of the Suitable Number of Discrete Frequencies of Rail Irregularity
From the aforementioned solution procedures for random analysis of trainslab trackbridge system, one can find that the larger the number of discrete frequencies is selected, the more calculation steps are. In order to save computer time as much as possible and, meanwhile, ensure a high accuracy of solution, a sequence of numerical experimentations is carried out to obtain the suitable number of discrete frequencies of rail irregularity with train running at speed of 69.44 m/s (250 km/h). As increases from 50 to 250 with increment 20, the corresponding is listed in Table 3, based on which the solutions are plotted in Figures 3, 4, 5, 6, 7, and 8. The variation ratio VR_{1} of the responses of train, track, and bridge and the CPU times for different cases are shown in Table 4. Herein VR_{1} is defined as (36), in which denote the maximum dynamic response SD of train, track, and bridge with the total number of discrete frequencies being equal to :

