# Numerical Simulation of Vertical Random Vibration of Train-Slab Track-Bridge Interaction System by PEM

Authors: Zhi-ping Zeng; Zhi-wu Yu; Yan-gang Zhao; Wen-tao Xu; Ling-kun Chen; Ping Lou
Journal: Shock and Vibration
 (2014)

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## Abstract

The paper describes the numerical simulation of the vertical random vibration of train-slab track-bridge interaction system by means of finite element method and pseudoexcitation method. Each vehicle is modeled as four-wheelset mass-spring-damper system with two-layer suspension systems. The rail, slab, and bridge girder are modeled by three-layer elastic Bernoulli-Euler 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, different-phase random excitations, the random load vectors of the equations of motion are obtained by pseudoexcitation method. Taking a nine-span 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 train-slab track-bridge interaction system are analyzed by pseudoexcitation method. Finally, applying the 3σ rule for Gaussian stochastic process, the maximum responses of train-slab track-bridge interaction system with respect to various train speeds are studied.


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## Body

## 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 train-track/bridge interaction system. The dynamic responses in most of the previous researches are usually calculated by using only one time-history 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 time-history sample generated from the PSD function of rail irregularity is random.With the increasing speed and axle load of train, the random vibration of train-track/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 time-history 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 train-slab track-bridge 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 time-independent or time-dependent systems with stationary or nonstationary random excitations [21–25]. Lu et al. [23] studied the stationary random vibration responses of vehicle-ballasted track interaction system by PEM in which a time-independent vehicle-track interaction model was adopted. By adopting a train-bridge interaction element with the track structure omitted [26], PEM was applied to research the nonstationary random vibration of time-dependent train-bridge interaction system [24].With the increasing use of ballastless track on bridge in modern high-speed 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 four-wheelset mass-spring-damper system with two-layer suspension systems possessing 6 degrees of freedom (DOFs) [17]. The rail, slab, and bridge girder are modeled by three-layer elastic Bernoulli-Euler 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, different-phase 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 step-by-step integration method such as Newmark-β method [29] or Wilson-θ method [30]. Taking a nine-span 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 train-slab track-bridge interaction system are analyzed by PEM, and finally the maximum responses of train-slab track-bridge 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

Figure1 shows a train consisting of a series of four-wheelset vehicles moving with constant speed v 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 Nv trailer cars numbered 1,2,…,Nv 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.Model of train-slab track-bridge interaction system. (a) A train moving upon a slab track resting on a multispan simply supported beam bridge. (b) Thejth trailer car moving upon the ith span simply supported beam.
(a)
(b)Each trailer car in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. As shown in Figure1, 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 mc and a moment of inertia Jc about the transverse horizontal axis through its center of gravity. Similarly, each bogie is considered as a rigid body having a mass mt and a moment of inertia Jt about the transverse horizontal axis through its center of gravity. Each wheelset has a mass mw. The spring and shock absorber in the primary suspension for each wheelset are characterized by spring stiffness kp and damping coefficient cp, respectively. Likewise, the secondary suspension between carbody and each bogie is characterized by spring stiffness ks and damping coefficient cs. As the carbody is assumed to be rigid, the motions of the jth trailer carbody may be described by the vertical displacement ycj and rotation θcj at its center of gravity, where the subscript j denotes the trailer car number. Similarly, the motions of the front bogie of the jth trailer car may be described by the vertical displacement yt1j and rotation θt1j at its center of gravity; the motions of the rear bogie of the jth trailer car may be described by the vertical displacement yt2j and rotation θt2j at its center of gravity. The motions of the four wheelsets from right to left of the jth trailer car may be described by the vertical displacements yw1j, yw2j, yw3j, and yw4j, 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 mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. The carbody has a massmJc and a moment of inertia JJc. Each bogie has a mass mJt and a moment of inertia JJt. Each wheelset has a mass mJw. The primary suspension is characterized by spring stiffness kJp and damping coefficient cJp. Likewise, the secondary suspension is characterized by spring stiffness kJs and damping coefficient cJs. The independent DOFs for each motor car include, too, the carbody vertical displacement yJci and rotation θJci, the front bogie vertical displacement yJt1i and rotation θJt1i, and the rear bogie vertical displacement yJt2i and rotation θJt2i, where the subscript i denotes the motor car number.


### 2.2. Models of Slab Track and Bridge

As shown in Figure1, the rail, slab, and bridge girder are all modeled as elastic Bernoulli-Euler 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 l. The elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness krs and dampers with damping coefficient crs. The elasticity and damping properties of the CA layer underlying the slab are represented by continuous massless springs with stiffness k-sb and dampers with damping coefficient c-sb. Simultaneously, the elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness kbb and dampers with damping coefficient cbb. 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. Letr(x) denote the initial vertical irregularities of rail and be measured with reference to smooth profile of rail; that is, r(x)=0, if the top surface of rail is smooth. Likewise, it is considered positive in the downward direction.


## 2.1. Model of Train

Figure1 shows a train consisting of a series of four-wheelset vehicles moving with constant speed v 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 Nv trailer cars numbered 1,2,…,Nv 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.Model of train-slab track-bridge interaction system. (a) A train moving upon a slab track resting on a multispan simply supported beam bridge. (b) Thejth trailer car moving upon the ith span simply supported beam.
(a)
(b)Each trailer car in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. As shown in Figure1, 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 mc and a moment of inertia Jc about the transverse horizontal axis through its center of gravity. Similarly, each bogie is considered as a rigid body having a mass mt and a moment of inertia Jt about the transverse horizontal axis through its center of gravity. Each wheelset has a mass mw. The spring and shock absorber in the primary suspension for each wheelset are characterized by spring stiffness kp and damping coefficient cp, respectively. Likewise, the secondary suspension between carbody and each bogie is characterized by spring stiffness ks and damping coefficient cs. As the carbody is assumed to be rigid, the motions of the jth trailer carbody may be described by the vertical displacement ycj and rotation θcj at its center of gravity, where the subscript j denotes the trailer car number. Similarly, the motions of the front bogie of the jth trailer car may be described by the vertical displacement yt1j and rotation θt1j at its center of gravity; the motions of the rear bogie of the jth trailer car may be described by the vertical displacement yt2j and rotation θt2j at its center of gravity. The motions of the four wheelsets from right to left of the jth trailer car may be described by the vertical displacements yw1j, yw2j, yw3j, and yw4j, 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 mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. The carbody has a massmJc and a moment of inertia JJc. Each bogie has a mass mJt and a moment of inertia JJt. Each wheelset has a mass mJw. The primary suspension is characterized by spring stiffness kJp and damping coefficient cJp. Likewise, the secondary suspension is characterized by spring stiffness kJs and damping coefficient cJs. The independent DOFs for each motor car include, too, the carbody vertical displacement yJci and rotation θJci, the front bogie vertical displacement yJt1i and rotation θJt1i, and the rear bogie vertical displacement yJt2i and rotation θJt2i, where the subscript i denotes the motor car number.


## 2.2. Models of Slab Track and Bridge

As shown in Figure1, the rail, slab, and bridge girder are all modeled as elastic Bernoulli-Euler 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 l. The elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness krs and dampers with damping coefficient crs. The elasticity and damping properties of the CA layer underlying the slab are represented by continuous massless springs with stiffness k-sb and dampers with damping coefficient c-sb. Simultaneously, the elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness kbb and dampers with damping coefficient cbb. 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. Letr(x) denote the initial vertical irregularities of rail and be measured with reference to smooth profile of rail; that is, r(x)=0, if the top surface of rail is smooth. Likewise, it is considered positive in the downward direction.


## 3. Equations of Motion for a Train-Slab Track-Bridge 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 train-slab track-bridge interaction system as
(1)[Mtt0000Mrr0000Mss0000Mbb]{X¨tX¨rX¨sX¨b}+[CttCtr00CrtCrrCsr00CrsCssCsb00CbsCbb]{X˙tX˙rX˙sX˙b}+[KttKtr00KrtKrrKrs00KsrKssKsb00KbsKbb]{XtXrXsXb}={FtFrFsFb},
where the subscripts “t,” “r,” “s,” and “b” 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 vectorXt with order Tdof  (Tdof=(6×Nv)+6×2) can be written as
(2)Xt=[XJ1Xv1Xv2⋯XvNvXJ2]T,
where the superscript “T” denotes the transpose of the vector and Xvj (j=1,2,…,Nv) and XJi (i=1,2) are the displacement vectors of the jth trailer car and the ith motor car, respectively, which can be expressed as
(3)Xvj=[ycjθvjyt1jθt1jyt2jθt2j],XJi=[yJciθJciyJt1iθJt1iyJt2iθJt2i].The displacement vector of railXr with order Nr×1 can be written as
(4)Xr=[qr1qr2⋯qrNr]T,
where Nr denotes the total number of DOFs of rail.The displacement vectorXs with order N-s×1 for a series of continuously supported beams to model slab can be written as
(5)Xs=[Xs1Xs2⋯XsNs]T,
where Xsi (i=1,2,…,Ns) denotes the displacement vector of the ith slab, Ns denotes the total number of slab, and N-s denotes the total number of DOFs of all slab. Xsi with order 1×n-si and N-s can be expressed as
(6)Xsi=[qs1qs2⋯qsn-si],N-s=∑i=1Nsn-si.The displacement vectorXb with order N-b×1 for a series of simply supported beams to model the bridge can be written as
(7)Xb=[Xb1Xb2⋯XbNb]T,
where Xbi (i=1,2,…,Nb) denotes the displacement vector of the ith bridge girder, Nb denotes the total number of bridge girder, and N-b denotes the total number of DOFs of all bridge girder. Xbi with order 1×n-bi and N-b can be expressed as
(8)Xbi=[qb1qb2⋯qsn-bi],N-b=∑i=1Nbn-bi.


### 3.2. Matrices for Train

The matrices of train are marked with the subscript “tt.” The mass matrix Mtt of train, with order (6×Nv+6×2) × (6×Nv+6×2), can be written as
(9)Mtt=diag⁡[MJ1Mv1Mv2⋯MvNvMJ2],
where Mvj and MJi with order 6×6 denote the mass matrices of the jth trailer car and ith motor car, respectively, and can be expressed as
(10)Mvj=diag⁡[mcJcmtJtmtJt],MJi=diag⁡[mJcJJcmJtJJtmJtJJt].The stiffness matrixKtt of train, with order (6×Nv+6×2) × (6×Nv+6×2), can be written as
(11)Ktt=diag⁡[KJ1Kv1Kv2⋯KvNvKJ2],
where Kvj and KJi with order 6×6, denoting the stiffness matrices of the jth trailer car and ith motor car, respectively, can be expressed as
(12)Kvi=[2ks0-ks0-ks02ksLc2ksLc0-ksLc0ks+2kp0002kpLt200symmks+2kp02kpLt2],KJi=[2kJs0-kJs0-kJs02kJsLJc2kJsLJc0-kJsLJc0kJs+2kJp0002kJpLJt200symmkJs+2kJp02kJpLJt2]
in which Lc denotes half of longitudinal distance between the centers of gravity of trailer car’s front and rear bogies, LJc denotes half of longitudinal distance between the centers of gravity of motor car’s front and rear bogies, Lt denotes half of trailer car’s bogie axle base, and LJt denotes half of motor car’s bogie axle base.The damping matrixCtt of train with order (6×Nv+6×2) × (6×Nv+6×2) can be obtained by simply replacing k in the corresponding stiffness matrix Ktt by c.


### 3.3. Matrices for Rail, Slab, and Bridge

The matrices of rail are marked with the subscript “rr.” The mass matrix Mrr of rail, with order Nr×Nr, can be written as
(13)Mrr=Mrr1+Mrr2+Mrr3
with
(14)Mrr1=∑g=1nr∫0lm-rNr,egTNr,egdξ,Mrr2=∑j=1Nv∑h=14mwNjhTNjh,Mrr3=∑i=12∑h=14mJwNJihTNJih,Nr,eg=[00⋯0Nr1Nr2Nr3Nr40⋯00],Njh=[00⋯0Nr1Nr2Nr3Nr40⋯00]ξ=ξjh,NJih=[00⋯0Nr1Nr2Nr3Nr40⋯00]ξ=ξJih,Nr1=1-3(ξl)2+2(ξl)3,Nr2=ξ[1-2(ξl)+(ξl)2],Nr3=3(ξl)2-2(ξl)3,Nr4=ξ[(ξl)2-(ξl)],
where Mrr1, with order Nr×Nr, represents the overall mass matrix of the rail itself, m-r represents rail mass per unit length, nr represents the total number of rail elements, ξ represents the local coordinate measured from the left node of a beam element, and Nr,eg, with order 1×Nr, represents the shape function matrix for the gth rail element. It should be noted that each element is zero in Nr,eg except for those corresponding to the four DOFs of the two nodes of the gth rail element. Mrr2, with order Nr×Nr, represents the overall mass matrix induced by all the wheel masses of trailer cars; Njh, with order 1×Nr, represents the shape function matrix for the rail element, evaluated at the position of the hth wheelset of the jth trailer car; ξj1, ξj2, ξj3, and ξj4, respectively, represent the distances between the 1st, 2nd, 3rd, and 4th wheelsets of the jth 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 Njh except those corresponding to the four DOFs of the two nodes of the rail element upon which the hth wheelset of the jth trailer car is acting. Njh is time dependent as the hth wheelset of the jth trailer car moves from one position to another within one rail element. As the hth wheelset of the jth trailer car moves to the next rail element, Njh will shift in position corresponding to the DOFs of the rail element where the hth wheelset of the jth trailer car is positioned. Similarly, Mrr3, with order Nr×Nr, represents the overall mass matrix induced by all the wheel masses of motor cars; NJih, with order 1×Nr, represents the shape function matrix for the rail element, evaluated at the position of the hth wheelset of the ith motor car; ξJi1, ξJi2, ξJi3, and ξJi4, respectively, represent the distances between each wheelset of the ith motor car and the left node of the corresponding rail element upon which the wheelset is acting. It should be noted too for NJih that, apart from those elements corresponding to the four DOFs of the two nodes of the rail element upon which the hth wheelset of the ith motor car is acting, all other elements are zero. NJih is also time dependent.The stiffness matrixKrr of rail, with order Nr×Nr, can be similarly expressed in terms of the overall stiffness matrix Krr1 of the rail itself, overall stiffness matrix Krr2 induced by all trailer cars, overall stiffness matrix Krr3 induced by all motor cars, and overall stiffness matrix Krr4 induced by the stiffness of all fasteners as
(15)Krr=Krr1+Krr2+Krr3+Krr4
with
(16)Krr1=∑g=1nr∫0lErIrNr,eg′′TNr,eg′′dξ,Krr2=∑j=1Nv∑h=14kpNjhTNjh,Krr3=∑i=12∑h=14kJpNJihTNJih,Krr4=∑p=1npkrsNr,pTNr,p,Nr,p=[00⋯0Nr1Nr2Nr3Nr40⋯00]ξ=ξr,p,
where Er denotes Young’s modulus of the rail and Ir denotes the constant moment of inertia of the rail cross section and the prime differentiation with respect to local coordinate ξ. In the formulation of Krr4, the shape function matrix Nr,p (p=1,2,…,np) of order 1×Nr for the rail element is evaluated at the position of the pth fastener; ξr,p denotes the distance between the pth fastener and the left node of the rail element containing the pth fastener, and np denotes the total number of fastener. It should be noted for Nr,p that, apart from those elements corresponding to four DOFs of the two nodes of the rail element containing the pth fastener, all other elements are zero.With omission of the damping of the rail itself, the damping matrixCrr of rail, with order Nr×Nr, can be written in terms of the overall damping matrix Crr2 induced by all trailer cars, overall damping matrix Crr3 induced by all motor cars, and overall damping matrix Crr4 induced by the damping of all fasteners as
(17)Crr=Crr2+Crr3+Crr4
with
(18)Crr2=∑j=1Nv∑h=14cpNjhTNjh,Crr3=∑i=12∑h=14cJpNJihTNJih,Crr4=∑p=1npcrsNr,pTNr,p.The matrices of slab, with orderN-s×N-s, are marked with the subscript “ss” 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 N-b×N-b, are marked with the subscript “bb” 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 Train-Rail-Slab-Bridge Interaction

The matrices for train-rail interaction are marked with subscript “tr” or “rt.” The stiffness matrices Ktr of order Tdof×Nr, Krt of order Nr×Tdof, Ctr of order Tdof×Nr, and Crt of order Nr×Tdof induced by the interaction between the train and rail can be written, respectively, as
(19)Ktr=∑j=1Nv∑h=14Kvj-rh+∑i=12∑h=14KJi-rh,Krt=KtrT,Ctr=∑j=1Nv∑h=14Cvj-rh+∑i=12∑h=14CJi-rh,Crt=CtrT,
with
(20)Kvj-r1=[00⋮-kpNj1-kpLtNj1⋮00]Tdof×Nr,Kvj-r2=[00⋮-kpNj2kpLtNj2⋮00]Tdof×Nr,Kvj-r3=[00⋮-kpNj3-kpLtNj3⋮00]Tdof×Nr,Kvj-r4=[00⋮-kpNj4kpLtNj4⋮00]Tdof×Nr,Cvj-r1=[00⋮-cpNj1-cpLtNj1⋮00]Tdof×Nr,Cvj-r2=[00⋮-cpNj2cpLtNj2⋮00]Tdof×Nr,Cvj-r3=[00⋮-cpNj3-cpLtNj3⋮00]Tdof×Nr,Cvj-r4=[00⋮-cpNj4cpLtNj4⋮00]Tdof×Nr
in which Kvj-rk and Krk-vj represent the stiffness matrices induced by the interaction between the kth wheelset of the jth trailer car and rail, and Cvj-rk and Crk-vj are the corresponding damping matrices. Kvj-r1, Kvj-r2, Cvj-r1, and Cvj-r2 consist of zero row vectors except for those corresponding to the two DOFs of front bogie of the jth trailer car, while Kr1-vj, Kr2-vj, Cr1-vj, and Cr2-vj consist of similar column vectors. Accordingly, Kvj-r3, Kvj-r4, Cvj-r3, Cvj-r4, Kr3-vj, Kr4-vj, Cr3-vj, and Cr4-vj 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 rail-slab-bridge interaction can be also derived by following the similar procedure for derivation of train-rail interaction matrices. The matrices for rail-slab interaction, induced by the matrices of fastener between rail and slab, are marked with subscript “rs” or “sr.” The matrices for slab-bridge interaction, induced by the matrices of CA layer between slab and bridge girder, are marked with subscript “sb” or “bs.”


### 3.5. Load Vector for Train, Rail, Slab, and Bridge

The load vectorFt of train with order Vdof×1 can be written as
(21)Ft=[Ft1Ft2]T,Ft1=[FJ11Fv11Fv21⋯FvNv1FJ21],Ft2=[FJ12Fv12Fv22⋯FvNv2FJ22],
where the load vector of the jth trailer car Fvj1 and Fvj2 and the load vector of the ith motor car FJi1 and FJi2 with order 6×1 can be written, respectively, as
(22)Fvj1={00kpr(xj1)+kpr(xj2)kpLtr(xj1)-kpLtr(xj2)kpr(xj3)+kpr(xj4)kpLtr(xj3)-kpLtr(xj4)}T=∑h=14kpr(xjh)Rvh,Fvj2={00cpr˙(xj1)+cpr˙(xj2)cpLtr˙(xj1)-cpLtr˙(xj2)cpr˙(xj3)+cpr˙(xj4)cpLtr˙(xj3)-cpLtr˙(xj4)}T=∑h=14cpr˙(xjh)Rvh,FJi1={00kJpr(xi1)+kJpr(xi2)kJpLJtr(xi1)-kJpLJtr(xi2)kJpr(xi3)+kJpr(xi4)kJpLJtr(xi3)-kJpLJtr(xi4)}T=∑h=14kJpr(xih)RJh,FJi2={00cJpr˙(xi1)+cJpr˙(xi2)cJpLJtr˙(xi1)-cJpLJtr˙(xi2)cJpr˙(xi3)+cJpr˙(xi4)cJpLJtr˙(xi3)-cJpLJtr˙(xi4)}T=∑h=14cJpr˙(xih)RJh,Rv1=[001Lt00],Rv2=[001-Lt00],Rv3=[00001Lt],Rv4=[00001-Lt],RJ1=[001LJt00],RJ2=[001-LJt00],RJ3=[00001LJt],RJ4=[00001-LJt]
in which r˙(x) denotes the first derivative of rail irregularity r(x).The load vector of railFr with order Nr×1 can be written as
(23)Fr=Fr0+Fr1+Fr2+Fr3
with
(24)Fr0=∑j=1Nv∑h=14Frjh0+∑i=12∑h=14FJrih0,Fr1=∑j=1Nv∑h=14Frjh1+∑i=12∑h=14FJrih1,Fr2=∑j=1Nv∑h=14Frjh2+∑i=12∑h=14FJrih2,Fr3=∑j=1Nv∑h=14Frjh3+∑i=12∑h=14FJrih3,Frjh0=g(mw+12mt+14mc)NjhT,Frjh1=-kpr(xjh)NjhT,Frjh2=-cpr˙(xjh)NjhT,Frjh3=-mwr¨(xjh)NjhT,FJrih0=g(mJw+12mJt+14mJc)NJihT,FJrih1=-kJpr(xih)NJihT,FJrih2=-cJpr˙(xih)NJihT,FJrih3=-mJpr¨(xih)NJihT,
where r¨(x) denotes the second derivative of rail irregularity r(x) and Frjh0, Frjh1, Frjh2, and Frjh3, respectively, denote the load vectors of the hth wheelset of the jth trailer car acting upon the rail caused by gravity force of trailer car, rail irregularity, velocity of rail irregularity, and acceleration of rail irregularity. FJrih0, FJrih1, FJrih2, and FJrih3 are the load vectors of the hth wheelset of the ith 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 slabFs with order N-s×1 and load vector of bridge Fb with order N-b×1 is zero.Now let(25)FG(t)=[0Vdof×1Fr00N-s×10N-b×1],FR1(t)=[Ft1Fr10N-s×10N-b×1],FR2(t)=[Ft2Fr20N-s×10N-b×1],FR3(t)=[0Vdof×1Fr30N-s×10N-b×1].Then, the load vector of the total train-slab track-bridge interaction systemF(t) can be expressed by the deterministic load vector FG(t) induced by the train gravity force and the random load vector FR(t) induced by the rail irregularity can be expressed as
(26)F(t)=[FtFrFsFb]=FG(t)+FR(t)=FG(t)+FR1(t)+FR2(t)+FR3(t).


## 3.1. Displacement Vectors

The total train displacement vectorXt with order Tdof  (Tdof=(6×Nv)+6×2) can be written as
(2)Xt=[XJ1Xv1Xv2⋯XvNvXJ2]T,
where the superscript “T” denotes the transpose of the vector and Xvj (j=1,2,…,Nv) and XJi (i=1,2) are the displacement vectors of the jth trailer car and the ith motor car, respectively, which can be expressed as
(3)Xvj=[ycjθvjyt1jθt1jyt2jθt2j],XJi=[yJciθJciyJt1iθJt1iyJt2iθJt2i].The displacement vector of railXr with order Nr×1 can be written as
(4)Xr=[qr1qr2⋯qrNr]T,
where Nr denotes the total number of DOFs of rail.The displacement vectorXs with order N-s×1 for a series of continuously supported beams to model slab can be written as
(5)Xs=[Xs1Xs2⋯XsNs]T,
where Xsi (i=1,2,…,Ns) denotes the displacement vector of the ith slab, Ns denotes the total number of slab, and N-s denotes the total number of DOFs of all slab. Xsi with order 1×n-si and N-s can be expressed as
(6)Xsi=[qs1qs2⋯qsn-si],N-s=∑i=1Nsn-si.The displacement vectorXb with order N-b×1 for a series of simply supported beams to model the bridge can be written as
(7)Xb=[Xb1Xb2⋯XbNb]T,
where Xbi (i=1,2,…,Nb) denotes the displacement vector of the ith bridge girder, Nb denotes the total number of bridge girder, and N-b denotes the total number of DOFs of all bridge girder. Xbi with order 1×n-bi and N-b can be expressed as
(8)Xbi=[qb1qb2⋯qsn-bi],N-b=∑i=1Nbn-bi.


## 3.2. Matrices for Train

The matrices of train are marked with the subscript “tt.” The mass matrix Mtt of train, with order (6×Nv+6×2) × (6×Nv+6×2), can be written as
(9)Mtt=diag⁡[MJ1Mv1Mv2⋯MvNvMJ2],
where Mvj and MJi with order 6×6 denote the mass matrices of the jth trailer car and ith motor car, respectively, and can be expressed as
(10)Mvj=diag⁡[mcJcmtJtmtJt],MJi=diag⁡[mJcJJcmJtJJtmJtJJt].The stiffness matrixKtt of train, with order (6×Nv+6×2) × (6×Nv+6×2), can be written as
(11)Ktt=diag⁡[KJ1Kv1Kv2⋯KvNvKJ2],
where Kvj and KJi with order 6×6, denoting the stiffness matrices of the jth trailer car and ith motor car, respectively, can be expressed as
(12)Kvi=[2ks0-ks0-ks02ksLc2ksLc0-ksLc0ks+2kp0002kpLt200symmks+2kp02kpLt2],KJi=[2kJs0-kJs0-kJs02kJsLJc2kJsLJc0-kJsLJc0kJs+2kJp0002kJpLJt200symmkJs+2kJp02kJpLJt2]
in which Lc denotes half of longitudinal distance between the centers of gravity of trailer car’s front and rear bogies, LJc denotes half of longitudinal distance between the centers of gravity of motor car’s front and rear bogies, Lt denotes half of trailer car’s bogie axle base, and LJt denotes half of motor car’s bogie axle base.The damping matrixCtt of train with order (6×Nv+6×2) × (6×Nv+6×2) can be obtained by simply replacing k in the corresponding stiffness matrix Ktt by c.


## 3.3. Matrices for Rail, Slab, and Bridge

The matrices of rail are marked with the subscript “rr.” The mass matrix Mrr of rail, with order Nr×Nr, can be written as
(13)Mrr=Mrr1+Mrr2+Mrr3
with
(14)Mrr1=∑g=1nr∫0lm-rNr,egTNr,egdξ,Mrr2=∑j=1Nv∑h=14mwNjhTNjh,Mrr3=∑i=12∑h=14mJwNJihTNJih,Nr,eg=[00⋯0Nr1Nr2Nr3Nr40⋯00],Njh=[00⋯0Nr1Nr2Nr3Nr40⋯00]ξ=ξjh,NJih=[00⋯0Nr1Nr2Nr3Nr40⋯00]ξ=ξJih,Nr1=1-3(ξl)2+2(ξl)3,Nr2=ξ[1-2(ξl)+(ξl)2],Nr3=3(ξl)2-2(ξl)3,Nr4=ξ[(ξl)2-(ξl)],
where Mrr1, with order Nr×Nr, represents the overall mass matrix of the rail itself, m-r represents rail mass per unit length, nr represents the total number of rail elements, ξ represents the local coordinate measured from the left node of a beam element, and Nr,eg, with order 1×Nr, represents the shape function matrix for the gth rail element. It should be noted that each element is zero in Nr,eg except for those corresponding to the four DOFs of the two nodes of the gth rail element. Mrr2, with order Nr×Nr, represents the overall mass matrix induced by all the wheel masses of trailer cars; Njh, with order 1×Nr, represents the shape function matrix for the rail element, evaluated at the position of the hth wheelset of the jth trailer car; ξj1, ξj2, ξj3, and ξj4, respectively, represent the distances between the 1st, 2nd, 3rd, and 4th wheelsets of the jth 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 Njh except those corresponding to the four DOFs of the two nodes of the rail element upon which the hth wheelset of the jth trailer car is acting. Njh is time dependent as the hth wheelset of the jth trailer car moves from one position to another within one rail element. As the hth wheelset of the jth trailer car moves to the next rail element, Njh will shift in position corresponding to the DOFs of the rail element where the hth wheelset of the jth trailer car is positioned. Similarly, Mrr3, with order Nr×Nr, represents the overall mass matrix induced by all the wheel masses of motor cars; NJih, with order 1×Nr, represents the shape function matrix for the rail element, evaluated at the position of the hth wheelset of the ith motor car; ξJi1, ξJi2, ξJi3, and ξJi4, respectively, represent the distances between each wheelset of the ith motor car and the left node of the corresponding rail element upon which the wheelset is acting. It should be noted too for NJih that, apart from those elements corresponding to the four DOFs of the two nodes of the rail element upon which the hth wheelset of the ith motor car is acting, all other elements are zero. NJih is also time dependent.The stiffness matrixKrr of rail, with order Nr×Nr, can be similarly expressed in terms of the overall stiffness matrix Krr1 of the rail itself, overall stiffness matrix Krr2 induced by all trailer cars, overall stiffness matrix Krr3 induced by all motor cars, and overall stiffness matrix Krr4 induced by the stiffness of all fasteners as
(15)Krr=Krr1+Krr2+Krr3+Krr4
with
(16)Krr1=∑g=1nr∫0lErIrNr,eg′′TNr,eg′′dξ,Krr2=∑j=1Nv∑h=14kpNjhTNjh,Krr3=∑i=12∑h=14kJpNJihTNJih,Krr4=∑p=1npkrsNr,pTNr,p,Nr,p=[00⋯0Nr1Nr2Nr3Nr40⋯00]ξ=ξr,p,
where Er denotes Young’s modulus of the rail and Ir denotes the constant moment of inertia of the rail cross section and the prime differentiation with respect to local coordinate ξ. In the formulation of Krr4, the shape function matrix Nr,p (p=1,2,…,np) of order 1×Nr for the rail element is evaluated at the position of the pth fastener; ξr,p denotes the distance between the pth fastener and the left node of the rail element containing the pth fastener, and np denotes the total number of fastener. It should be noted for Nr,p that, apart from those elements corresponding to four DOFs of the two nodes of the rail element containing the pth fastener, all other elements are zero.With omission of the damping of the rail itself, the damping matrixCrr of rail, with order Nr×Nr, can be written in terms of the overall damping matrix Crr2 induced by all trailer cars, overall damping matrix Crr3 induced by all motor cars, and overall damping matrix Crr4 induced by the damping of all fasteners as
(17)Crr=Crr2+Crr3+Crr4
with
(18)Crr2=∑j=1Nv∑h=14cpNjhTNjh,Crr3=∑i=12∑h=14cJpNJihTNJih,Crr4=∑p=1npcrsNr,pTNr,p.The matrices of slab, with orderN-s×N-s, are marked with the subscript “ss” 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 N-b×N-b, are marked with the subscript “bb” 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 Train-Rail-Slab-Bridge Interaction

The matrices for train-rail interaction are marked with subscript “tr” or “rt.” The stiffness matrices Ktr of order Tdof×Nr, Krt of order Nr×Tdof, Ctr of order Tdof×Nr, and Crt of order Nr×Tdof induced by the interaction between the train and rail can be written, respectively, as
(19)Ktr=∑j=1Nv∑h=14Kvj-rh+∑i=12∑h=14KJi-rh,Krt=KtrT,Ctr=∑j=1Nv∑h=14Cvj-rh+∑i=12∑h=14CJi-rh,Crt=CtrT,
with
(20)Kvj-r1=[00⋮-kpNj1-kpLtNj1⋮00]Tdof×Nr,Kvj-r2=[00⋮-kpNj2kpLtNj2⋮00]Tdof×Nr,Kvj-r3=[00⋮-kpNj3-kpLtNj3⋮00]Tdof×Nr,Kvj-r4=[00⋮-kpNj4kpLtNj4⋮00]Tdof×Nr,Cvj-r1=[00⋮-cpNj1-cpLtNj1⋮00]Tdof×Nr,Cvj-r2=[00⋮-cpNj2cpLtNj2⋮00]Tdof×Nr,Cvj-r3=[00⋮-cpNj3-cpLtNj3⋮00]Tdof×Nr,Cvj-r4=[00⋮-cpNj4cpLtNj4⋮00]Tdof×Nr
in which Kvj-rk and Krk-vj represent the stiffness matrices induced by the interaction between the kth wheelset of the jth trailer car and rail, and Cvj-rk and Crk-vj are the corresponding damping matrices. Kvj-r1, Kvj-r2, Cvj-r1, and Cvj-r2 consist of zero row vectors except for those corresponding to the two DOFs of front bogie of the jth trailer car, while Kr1-vj, Kr2-vj, Cr1-vj, and Cr2-vj consist of similar column vectors. Accordingly, Kvj-r3, Kvj-r4, Cvj-r3, Cvj-r4, Kr3-vj, Kr4-vj, Cr3-vj, and Cr4-vj 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 rail-slab-bridge interaction can be also derived by following the similar procedure for derivation of train-rail interaction matrices. The matrices for rail-slab interaction, induced by the matrices of fastener between rail and slab, are marked with subscript “rs” or “sr.” The matrices for slab-bridge interaction, induced by the matrices of CA layer between slab and bridge girder, are marked with subscript “sb” or “bs.”


## 3.5. Load Vector for Train, Rail, Slab, and Bridge

The load vectorFt of train with order Vdof×1 can be written as
(21)Ft=[Ft1Ft2]T,Ft1=[FJ11Fv11Fv21⋯FvNv1FJ21],Ft2=[FJ12Fv12Fv22⋯FvNv2FJ22],
where the load vector of the jth trailer car Fvj1 and Fvj2 and the load vector of the ith motor car FJi1 and FJi2 with order 6×1 can be written, respectively, as
(22)Fvj1={00kpr(xj1)+kpr(xj2)kpLtr(xj1)-kpLtr(xj2)kpr(xj3)+kpr(xj4)kpLtr(xj3)-kpLtr(xj4)}T=∑h=14kpr(xjh)Rvh,Fvj2={00cpr˙(xj1)+cpr˙(xj2)cpLtr˙(xj1)-cpLtr˙(xj2)cpr˙(xj3)+cpr˙(xj4)cpLtr˙(xj3)-cpLtr˙(xj4)}T=∑h=14cpr˙(xjh)Rvh,FJi1={00kJpr(xi1)+kJpr(xi2)kJpLJtr(xi1)-kJpLJtr(xi2)kJpr(xi3)+kJpr(xi4)kJpLJtr(xi3)-kJpLJtr(xi4)}T=∑h=14kJpr(xih)RJh,FJi2={00cJpr˙(xi1)+cJpr˙(xi2)cJpLJtr˙(xi1)-cJpLJtr˙(xi2)cJpr˙(xi3)+cJpr˙(xi4)cJpLJtr˙(xi3)-cJpLJtr˙(xi4)}T=∑h=14cJpr˙(xih)RJh,Rv1=[001Lt00],Rv2=[001-Lt00],Rv3=[00001Lt],Rv4=[00001-Lt],RJ1=[001LJt00],RJ2=[001-LJt00],RJ3=[00001LJt],RJ4=[00001-LJt]
in which r˙(x) denotes the first derivative of rail irregularity r(x).The load vector of railFr with order Nr×1 can be written as
(23)Fr=Fr0+Fr1+Fr2+Fr3
with
(24)Fr0=∑j=1Nv∑h=14Frjh0+∑i=12∑h=14FJrih0,Fr1=∑j=1Nv∑h=14Frjh1+∑i=12∑h=14FJrih1,Fr2=∑j=1Nv∑h=14Frjh2+∑i=12∑h=14FJrih2,Fr3=∑j=1Nv∑h=14Frjh3+∑i=12∑h=14FJrih3,Frjh0=g(mw+12mt+14mc)NjhT,Frjh1=-kpr(xjh)NjhT,Frjh2=-cpr˙(xjh)NjhT,Frjh3=-mwr¨(xjh)NjhT,FJrih0=g(mJw+12mJt+14mJc)NJihT,FJrih1=-kJpr(xih)NJihT,FJrih2=-cJpr˙(xih)NJihT,FJrih3=-mJpr¨(xih)NJihT,
where r¨(x) denotes the second derivative of rail irregularity r(x) and Frjh0, Frjh1, Frjh2, and Frjh3, respectively, denote the load vectors of the hth wheelset of the jth trailer car acting upon the rail caused by gravity force of trailer car, rail irregularity, velocity of rail irregularity, and acceleration of rail irregularity. FJrih0, FJrih1, FJrih2, and FJrih3 are the load vectors of the hth wheelset of the ith 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 slabFs with order N-s×1 and load vector of bridge Fb with order N-b×1 is zero.Now let(25)FG(t)=[0Vdof×1Fr00N-s×10N-b×1],FR1(t)=[Ft1Fr10N-s×10N-b×1],FR2(t)=[Ft2Fr20N-s×10N-b×1],FR3(t)=[0Vdof×1Fr30N-s×10N-b×1].Then, the load vector of the total train-slab track-bridge interaction systemF(t) can be expressed by the deterministic load vector FG(t) induced by the train gravity force and the random load vector FR(t) induced by the rail irregularity can be expressed as
(26)F(t)=[FtFrFsFb]=FG(t)+FR(t)=FG(t)+FR1(t)+FR2(t)+FR3(t).


## 4. Random Vibration Analysis of Train-Slab Track-Bridge Interaction System by PEM

### 4.1. The Pseudoexcitation for the System

Letx be the distance from the left-hand starting point of the track model to the wheel/rail contact point at time t; there exists a transformation from the PSD function Srr(Ω) of r(x) in space domain to the PSD function Srr(ω) of r(t) in time domain, according to x=vt:
(27)Srr(ω)=Srr(Ω)v,
where ω, Ω, and v denote, respectively, the time frequency (rad/s), the spatial frequency (rad/m), and train speed (m/s). Obviously, ω=Ωv.According to (21)–(26), the pseudoexcitation F~R(ω,t) induced by the rail irregularity can be obtained by PEM:
(28)F~R(ω,t)=F~R1(ω,t)+F~R2(ω,t)+F~R3(ω,t)
with
(29)F~R1(ω,t)=[F~t1(ω,t)F~r1(ω,t)0N-s×10N-b×1],F~R2(ω,t)=[F~t2(ω,t)F~r2(ω,t)0N-s×10N-b×1],F~R3(ω,t)=[0Vdof×1F~r3(ω,t)0N-s×10N-b×1],F~t1(ω,t)=[F~J11F~v11F~v21⋯F~vNv1F~J21]T,Ft2(ω,t)=[FJ12Fv12Fv22⋯FvNv2FJ22]T,F~vj1(ω,t)=∑h=14kpRvhSrr(ω)eiω(t-tjh),F~Ji1(ω,t)=∑h=14kJpRJhSrr(ω)eiω(t-tJh),F~vj2(ω,t)=iω∑h=14cpRvhSrr(ω)eiω(t-tjh),F~Ji2(ω,t)=iω∑h=14cJpRJhSrr(ω)eiω(t-tJh),F~r1(ω,t)=∑j=1Nv∑h=14F~rjh1+∑i=12∑h=14F~Jrih1,F~r2(ω,t)=∑j=1Nv∑h=14F~rjh2+∑i=12∑h=14F~Jrih2,F~r3(ω,t)=∑j=1Nv∑h=14F~rjh3+∑i=12∑h=14F~Jrih3,F~rjh1(ω,t)=-kpNjhTSrr(ω)eiω(t-tjh),F~rjh2(ω,t)=-iωcpNjhTSrr(ω)eiω(t-tjh),F~Jrjh3(ω,t)=ω2mwNjhTSrr(ω)eiω(t-tjh),F~Jrih1(ω,t)=-kJpNJihTSrr(ω)eiω(t-tJih),F~Jrjh2(ω,t)=-iωcJpNJihTSrr(ω)eiω(t-tJih),F~Jrjh3(ω,t)=ω2mJwNJihTSrr(ω)eiω(t-tJih).


### 4.2. The Calculation of Random Responses of the System

Based on (1) and (26), the equations of motion for a train-slab track-bridge interaction system can be expressed as
(30)M(t)U¨+C(t)U˙+C(t)U=FG(t)+FR(t)
in which M(t), C(t), and K(t) denote mass, damping, and stiffness matrices of the total train-slab track-bridge interaction system, respectively; U, U˙, and U¨ 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)Y-out(t) 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 step-by-step integration method. Consider
(31)M(t)U¨-+C(t)U˙-+C(t)U-=FG(t).The arbitrary pseudoresponseY~out(ω,t) of the system caused by F~R(ω,t) can be obtained by solving (32) in frequency domain and time domain. Consider
(32)M(t)U¨~+C(t)U˙~+C(t)U~=F~R(ω,t).According to PEM and random vibration theory, the time-dependent power spectral density (PSD) and standard deviation (SD) of arbitrary responseY~out(ω,t) of the system can be written as
(33)SYout(ω,t)=Y~out(ω,t)Y~out*(ω,t),σYout2(t)=∑j=1mSYout(ωj,t)Δωj,
where the superscript “*” denotes complex conjugate, m denotes the total number of discrete frequencies of rail irregularity, and Δωj denotes the jth frequency interval of rail irregularity.


### 4.3. The Solution Procedures

The flowchart for efficiently analyzing the vertical random vibration of train-slab track-bridge interaction system under the action of rail irregularity is shown in Figure2 by following the above explanation of random vibration analysis by PEM.Figure 2
Flowchart for the vertical random vibration analysis of train-slab track-bridge interaction system.


## 4.1. The Pseudoexcitation for the System

Letx be the distance from the left-hand starting point of the track model to the wheel/rail contact point at time t; there exists a transformation from the PSD function Srr(Ω) of r(x) in space domain to the PSD function Srr(ω) of r(t) in time domain, according to x=vt:
(27)Srr(ω)=Srr(Ω)v,
where ω, Ω, and v denote, respectively, the time frequency (rad/s), the spatial frequency (rad/m), and train speed (m/s). Obviously, ω=Ωv.According to (21)–(26), the pseudoexcitation F~R(ω,t) induced by the rail irregularity can be obtained by PEM:
(28)F~R(ω,t)=F~R1(ω,t)+F~R2(ω,t)+F~R3(ω,t)
with
(29)F~R1(ω,t)=[F~t1(ω,t)F~r1(ω,t)0N-s×10N-b×1],F~R2(ω,t)=[F~t2(ω,t)F~r2(ω,t)0N-s×10N-b×1],F~R3(ω,t)=[0Vdof×1F~r3(ω,t)0N-s×10N-b×1],F~t1(ω,t)=[F~J11F~v11F~v21⋯F~vNv1F~J21]T,Ft2(ω,t)=[FJ12Fv12Fv22⋯FvNv2FJ22]T,F~vj1(ω,t)=∑h=14kpRvhSrr(ω)eiω(t-tjh),F~Ji1(ω,t)=∑h=14kJpRJhSrr(ω)eiω(t-tJh),F~vj2(ω,t)=iω∑h=14cpRvhSrr(ω)eiω(t-tjh),F~Ji2(ω,t)=iω∑h=14cJpRJhSrr(ω)eiω(t-tJh),F~r1(ω,t)=∑j=1Nv∑h=14F~rjh1+∑i=12∑h=14F~Jrih1,F~r2(ω,t)=∑j=1Nv∑h=14F~rjh2+∑i=12∑h=14F~Jrih2,F~r3(ω,t)=∑j=1Nv∑h=14F~rjh3+∑i=12∑h=14F~Jrih3,F~rjh1(ω,t)=-kpNjhTSrr(ω)eiω(t-tjh),F~rjh2(ω,t)=-iωcpNjhTSrr(ω)eiω(t-tjh),F~Jrjh3(ω,t)=ω2mwNjhTSrr(ω)eiω(t-tjh),F~Jrih1(ω,t)=-kJpNJihTSrr(ω)eiω(t-tJih),F~Jrjh2(ω,t)=-iωcJpNJihTSrr(ω)eiω(t-tJih),F~Jrjh3(ω,t)=ω2mJwNJihTSrr(ω)eiω(t-tJih).


## 4.2. The Calculation of Random Responses of the System

Based on (1) and (26), the equations of motion for a train-slab track-bridge interaction system can be expressed as
(30)M(t)U¨+C(t)U˙+C(t)U=FG(t)+FR(t)
in which M(t), C(t), and K(t) denote mass, damping, and stiffness matrices of the total train-slab track-bridge interaction system, respectively; U, U˙, and U¨ 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)Y-out(t) 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 step-by-step integration method. Consider
(31)M(t)U¨-+C(t)U˙-+C(t)U-=FG(t).The arbitrary pseudoresponseY~out(ω,t) of the system caused by F~R(ω,t) can be obtained by solving (32) in frequency domain and time domain. Consider
(32)M(t)U¨~+C(t)U˙~+C(t)U~=F~R(ω,t).According to PEM and random vibration theory, the time-dependent power spectral density (PSD) and standard deviation (SD) of arbitrary responseY~out(ω,t) of the system can be written as
(33)SYout(ω,t)=Y~out(ω,t)Y~out*(ω,t),σYout2(t)=∑j=1mSYout(ωj,t)Δωj,
where the superscript “*” denotes complex conjugate, m denotes the total number of discrete frequencies of rail irregularity, and Δωj denotes the jth frequency interval of rail irregularity.


## 4.3. The Solution Procedures

The flowchart for efficiently analyzing the vertical random vibration of train-slab track-bridge interaction system under the action of rail irregularity is shown in Figure2 by following the above explanation of random vibration analysis by PEM.Figure 2
Flowchart for the vertical random vibration analysis of train-slab track-bridge interaction system.


## 5. Case Study

### 5.1. The Properties of Train-Slab Track-Bridge Interaction System

In order to reduce the boundary effect of the subgrade as far as possible, a nine-span simply supported beam high-speed railway bridge with the span length of 32 m is considered as shown in Figure1. 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 steady-state 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 v. The physical parameters of train, track, and bridge are listed, respectively, in Tables 1 and 2. The Germany high-speed track vertical profile irregularity PSD function [13, 33] is adopted; that is,
(34)Srr(Ω)=AvΩc2(Ω2+Ωr2)(Ω2+Ωc2),
where Ωc=0.8246 rad/m, Ωr=0.0206 rad/m, and Av=4.032×10-7 m·rad.Table 1
Parameters for trailer car and motor car.
Notation
Item
Unit
Value
Trailer car
Motor car
m
c/mJc
Mass of carbody
kg
4.4 × 104
4.8 × 104
J
c/JJc
Mass moment of inertia of carbody
kg·m2
2.7 × 106
2.7 × 106
m
t/mJt
Mass of bogie
kg
2.4 × 103
3.2 × 103
J
t/JJt
Mass moment of inertia of bogie
kg·m2
2.2 × 103
7.2 × 103
m
w/mJw
Mass of wheelset
kg
2.4 × 103
2.4 × 103
k
s/kJs
Spring stiffness of the second suspension system
N/m
6.0 × 105
8.0 × 105
c
s/cJs
Damping coefficient of the second suspension system
N·s/m
8.0 × 104
9.0 × 104
k
p/kJp
Spring stiffness of the primary suspension system
N/m
1.4 × 106
2.08 × 106
c
p/cJp
Damping coefficient of the primary suspension system
N·s/m
8.0 × 104
6.0 × 104
L
v/LJv
Longitudinal distance between the center of bogie and the nearest side of carbody
m
3.45
3.45
L
c/LJc
Half of longitudinal distance between the centers of front and rear bogies
m
8.6875
8.6875
L
t/LJt
Half of bogie axle base
m
1.25
1.25
ω
c
1/ωJc1
The fundamental frequency for vertical motion of carbody
Hz
0.75
0.84
ω
c
2/ωJc2
The fundamental frequency for pitching motion of carbody
Hz
0.84
0.97
ω
t
1/ωJt1
The fundamental frequency for vertical motion of bogie
Hz
6.00
4.78
ω
t
2/ωJt2
The fundamental frequency for pitching motion of bogie
Hz
7.10
6.28Table 2
Parameters for slab track and bridge.
Notation
Item
Unit
Value
Slab track
E
r
Young’s modulus of rail
N/m2
2.06 × 1011
υ
r
Poisson’s ratio of rail
—
0.3
I
r
Mass moment of inertia of rail cross section
m4
2 × 3.217 × 10−5
m
-
r
Mass per unit length of rail
kg/m
2 × 60.64
E
s
Young’s modulus of slab
N/m2
3.6 × 1010
υ
s
Poisson’s ratio of slab
—
0.2
I
s
Mass moment of inertia of slab cross section
m4
1.6 × 10−3
m
-
s
Mass per unit length of slab
kg/m
1.2 × 103
k
rs
Spring stiffness of fastener
N/m
2 × 5.0 × 107
c
rs
Damping coefficient of fastener
N·s/m
2 × 3.625 × 104
k
-
sb
Spring stiffness of CA layer per unit length
N/m2
2 × 1.5 × 109
c
-
sb
Damping coefficient of CA layer per unit length
N·s/m2
2 × 8.3 × 104
Bridge
E
b
Young’s modulus of girder
N/m2
3.45 × 1010
υ
b
Poisson’s ratio of girder
—
0.2
I
b
Mass moment of inertia of girder cross section
m4
12.744
m
-
b
Mass per unit length of bridge girder
kg/m
2.972 × 104
ζ
b
Damping ratio of girder
—
0.05
ω
b
1
The fundamental frequency of girder
Hz
5.72
ω
b
2
The second natural frequency of girder
Hz
22.89
k
bb
Spring stiffness of bearing
N/m
6.0 × 109
c
bb
Damping coefficient of bearing
N·s/m
2.0 × 106To solve the equations of motion for the train-slab track-bridge interaction system, the Wilson-θ method is used with θ=1.4 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 0.004×2π to 1×2π 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,
(35)WLen(j)=CLene(j-1)(j=1,2,3,…,m),
where WLen denotes the wavelength of rail irregularity (m) and CLen denotes a constant related to the maximum frequency points m.


### 5.2. Selection of the Suitable Number of Discrete Frequencies of Rail Irregularity

From the aforementioned solution procedures for random analysis of train-slab track-bridge system, one can find that the larger the number of discrete frequenciesm 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 m increases from 50 to 250 with increment 20, the corresponding CLen is listed in Table 3, based on which the solutions are plotted in Figures 3, 4, 5, 6, 7, and 8. The variation ratio VR1 of the responses of train, track, and bridge and the CPU times for different cases are shown in Table 4. Herein VR1 is defined as (36), in which RPEM(i) denote the maximum dynamic response SD of train, track, and bridge with the total number of discrete frequencies being equal to i:
(36)VR1=RPEM(i)-RPEM(250)RPEM(250)×100%(i=50~250).Table 3
Relationship between the number of discrete frequenciesm and the constant CLen.
m
50
70
90
110
130
150
170
190
210
230
250
C
Len
1.11929
1.08331
1.06342
1.05197
1.04374
1.03776
1.03322
1.02965
1.02677
1.02441
1.02243Table 4
The variation ratioVR1 of the responses (%) and CPU times.
Item
Total number of discrete frequenciesm
50
70
90
110
130
150
170
190
210
230
250
Vertical acceleration of carbody
Motor car
-
1.99
-
1.29
-
0.88
-
0.62
-
0.47
-
0.34
-
0.23
-
0.16
-
0.08
-
0.03
0.00
Trailer car
-
1.97
-
1.30
-
0.86
-
0.63
-
0.44
-
0.35
-
0.25
-
0.16
-
0.10
-
0.03
0.00
Vertical acceleration of rear bogie
Motor car
-
2.02
-
1.27
-
0.90
-
0.56
-
0.45
-
0.35
-
0.28
-
0.17
-
0.10
-
0.05
0.00
Trailer car
-
2.33
-
1.45
-
0.98
-
0.62
-
0.56
-
0.35
-
0.28
-
0.17
-
0.10
-
0.04
0.00
Wheel/rail vertical force
Motor car
-
1.20
-
2.99
-
0.49
-
0.31
-
0.56
-
0.38
-
0.27
-
0.21
-
0.15
-
0.10
0.00
Trailer car
-
1.06
-
1.97
-
0.24
-
0.13
-
0.60
-
0.32
-
0.15
-
0.06
-
0.23
-
0.14
0.00
Vertical displacement of bridge midpoint
Rail
0.98
0.57
0.41
0.33
0.24
0.16
0.08
0.08
0.08
0.00
0.00
Girder
-
2.99
-
1.12
-
0.93
-
0.56
-
0.37
-
0.37
-
0.19
-
0.19
0.00
0.00
0.00
Vertical acceleration of bridge midpoint
Rail
9.40
6.21
4.33
3.08
2.26
1.63
1.15
0.77
0.47
0.22
0.00
Slab
12.26
9.67
6.60
4.28
3.30
2.27
1.59
1.05
0.62
0.32
0.00
Girder
-
0.38
-
1.76
11.42
3.14
1.38
-
0.69
-
0.19
-
0.69
0.00
0.06
0.00
Vertical fastener pressure
1.04
0.67
0.47
0.33
0.24
0.18
0.13
0.08
0.04
0.02
0.00
CPU times (h)
0.60
0.80
0.96
1.13
1.31
1.43
1.64
1.84
2.03
2.23
2.41Figure 3
Maximum SD of vertical acceleration of carbody.Figure 4
Maximum SD of vertical acceleration of rear bogie.Figure 5
Maximum SD of wheel/rail vertical force.Figure 6
Maximum SD of vertical displacement of bridge midpoint.Figure 7
Maximum SD of vertical acceleration of bridge girder midpoint.Figure 8
Maximum SD of vertical fastener pressure above bridge midpoint.According to Yang et al. [15], the dynamic response of the last vehicle of a train should be given much more attention concerning the running safety and riding comfort of the train moving over a bridge because the last vehicle tends to vibrate more violently than the ones ahead. Therefore, “motor car” and “trailer car” considered in the following examples, respectively, mean the rear motor car and the Nvth trailer car connecting with the rear motor car (Figure 1). In addition, “bridge midpoint” and “wheel/rail vertical force,” respectively, are the midpoints of the fifth span for the nine-span bridge and the vertical contact force between the 3rd wheelset of the rear motor car or of the Nvth trailer car and rail.From Figures3–8 and Table 4, it is observed that, as the total number of discrete frequencies increases, each of the maximum SD of train, track, and bridge dynamic responses tends to approach a limit value. Comparing the results obtained by 250 discrete frequencies with those by 170 discrete frequencies, the variation ratios VR1 are −0.23% and −0.25% for the vertical acceleration SD of the motor and trailer carbody, respectively, and 1.15%, 1.59%, and −0.19% for the vertical acceleration SD of rail, slab, and bridge girder, respectively. Therefore, one can conclude that the suitable number of discrete frequencies of rail irregularity can be equal to 170.


### 5.3. Comparing PEM with MCM

The running safety and riding comfort of train have been of great concern in railway engineering for a long time, particularly due to the development of high-speed railway. Several mechanisms that result in the derailment or discomfort of a running train have been identified through analytical and experimental investigations, based on which some indices have been proposed for evaluating the running safety and riding comfort of train [34], such as carbody vertical acceleration, wheel load decrement ratio, Sperling’s ride index, and acceleration of bridge girder. In practice, the maximum dynamic responses are usually of the most interest for us. As is mentioned above, the conventional method of estimation to pick the maximum values from a set of response samples is extremely time consuming. However, the upper and lower boundaries of the responses by PEM can be estimated easily by the MV ± 3 times SD according to the 3σ rule for Gaussian stochastic process [24], since the corresponding MV and SD can be obtained efficiently and accurately according to the aforementioned solution procedures.For the purpose of comparison, the random responses of the train-slab track-bridge interaction system calculated by PEM, with total discrete frequenciesm=170, are compared with those by MCM, with 1000 of rail irregularity samples. Herein, the method proposed by Chen and Zhai [35] is implemented to generate the rail irregularity sample r(x) from the PSD function described as (34), and the train is also assumed to pass through the bridge with a constant speed 69.44 m/s. Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 show the estimated upper and lower boundaries of the responses by PEM and the corresponding extreme values of each time-history response obtained by MCM. The variation ratio VR2 of the responses of train, track, and bridge girder, determined according to (37), is shown in Table 5. Consider
(37)VR2=max⁡|RMCME(i)|-min⁡|RMCME(i)|min⁡|RMCME(i)|×100%(i=1~1000),
where RMCME denote the extreme values of each time-history response obtained by MCM and i denotes the number of rail irregularity samples.Table 5
The variation ratioVR1 of the responses (%) and CPU times.
Item
Total number of rail irregularity samples
100
200
400
600
800
1000
Vertical acceleration of carbody
Motor car
152.25
193.00
228.30
228.30
280.37
280.37
Trailer car
150.17
150.17
158.91
160.20
184.66
184.66
Vertical acceleration of rear bogie
Motor car
61.94
74.82
79.93
98.72
98.72
98.72
Trailer car
62.92
69.90
83.86
112.07
112.07
112.07
Wheel load decrement ratio
Motor car
68.31
77.28
96.66
96.66
96.96
96.96
Trailer car
71.72
71.72
78.99
94.03
120.08
120.08
Sperling’s ride index
Motor car
23.96
29.76
34.09
34.09
39.00
39.00
Trailer car
16.24
21.69
25.36
25.36
27.40
27.40
Vertical displacement of rail above bridge midpoint
49.74
60.51
61.94
61.94
68.28
68.28
Vertical fastener pressure above bridge midpoint
49.70
60.25
62.48
62.48
68.12
68.12
Vertical displacement of bridge girder midpoint
8.98
8.98
8.98
10.90
10.90
10.90
Vertical acceleration of bridge girder midpoint
111.92
115.58
160.73
160.73
160.73
170.05
CPU times (h)
13.12
26.24
52.48
78.72
91.84
131.20Figure 9
Comparison of vertical acceleration of motor carbody.Figure 10
Comparison of vertical acceleration of trailer carbody.Figure 11
Comparison of rear bogie vertical acceleration of motor car.Figure 12
Comparison of rear bogie vertical acceleration of trailer car.Figure 13
Comparison of wheel load decrement ratio of motor car.Figure 14
Comparison of wheel load decrement ratio of trailer car.Figure 15
Comparison of Sperling’s ride index of motor car.Figure 16
Comparison of Sperling’s ride index of trailer car.Figure 17
Comparison of vertical displacement of rail above bridge midpoint.Figure 18
Comparison of vertical pressure of fastener above bridge midpoint.Figure 19
Comparison of vertical displacement of bridge girder midpoint.Figure 20
Comparison of vertical acceleration of bridge girder midpoint.From Figures9–20 and Table 5, the following observations can be made easily. (1) There exist dramatic variations for the extreme values of each time-history response by MCM, except for that of bridge girder vertical displacement. For example, variation ratio VR2 of vertical acceleration of motor carbody, wheel load decrement ratio of trailer car, Sperling’s ride index of motor car, and vertical acceleration of bridge girder midpoint among the 1000 samples are 280.37%, 120.08%, 39.00%, and 170.05%, respectively. In general, a larger number of rail irregularity samples tend to enlarge the variation ratio VR2. (2) All of extreme values of responses obtained by MCM fall with the corresponding limits of upper and lower boundaries of those by PEM. (3) Compared with MCM, the PEM helps to save computer time drastically. For example, the total CPU times for PEM with 170 discrete frequencies and MCM with just 100 samples are 1.64 hours and 13.12 hours, respectively, on a 2.8 GHz personal computer. The latter equals about eight times the former. That is to say, PEM, in dealing with random vibration of train-slab track-bridge interaction system or evaluating the running safety and riding comfort of train, is not only more reliable but also more efficient than the conventional MCM.


### 5.4. The Random Vibration Characteristics of Train-Slab Track-Bridge Interaction System

The dominant vibration frequency of train and bridge can be obtained easily according to the PSDs of dynamic responses calculated by PEM. To investigate the random vibration characteristics of train-slab track-bridge interaction system, the solutions obtained by PEM, with 170 discrete frequencies of rail irregularity and three train speeds, are employed.Figures21 and 22 exhibit the PSDs of carbody vertical acceleration. As can be seen, the PSDs vary only slightly with time which indicates that the influence of bridge vibration on carbody vibration is insignificant because of the comparatively high mass and flexural rigidity of bridge girder. On the other hand, the PSDs change violently with rail irregularity frequency, implying the great influence of rail irregularity on the dynamic response of carbody. Of interest is that, no matter how the train speed changes, the peak values of PSDs of carbody vertical acceleration appear always approximately, respectively, to be around 0.77~0.86 and 0.69~0.77 Hz for motor car and trailer car, which show good agreement with their own fundamental frequencies of 0.84 and 0.75 Hz.PSD of vertical acceleration of motor carbody.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)PSD of vertical acceleration of trailer carbody.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)The similar phenomenon for bogie vertical acceleration can be also observed from Figures23 and 24. As can be seen, the bogie also vibrates periodically, implying three dominant frequencies of 4.22, 16.79, and 33.49 Hz for v=41.67 m/s, 6.31, 26.99, and 55.82 Hz for v=69.44 m/s and 6.37 (motor car), or 7.92 (trailer car), 39.18, and 78.15 Hz for v=97.22 m/s. It is found that the vertical vibration frequency ft of bogie, the half of bogie axle base Lt, and the train speeds v satisfy the relation ft=v/(n×Lt), n=1,2,3,…,∞, where relatively large contributions are made by harmonic components with n=1, 2, or 8~15.PSD of vertical acceleration of motor car rear bogie.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)PSD of vertical acceleration of trailer car rear bogie.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)The PSDs of vertical acceleration of bridge girder midpoint to the passage of the train are plotted in Figure25. Similar with the cases of bogie vertical acceleration, there exist in Figure 25 three dominant frequency ranges of 4.7~5.4, 14~18, and 32~37 Hz for v=41.67 m/s, 4.4~6.3, 25~33, and 53~58 Hz for v=69.44 m/s and 4.4~6.8, 35~43, and 72~81 Hz for v=97.22 m/s. Obviously, the first dominant frequency ranges of bridge girder vertical acceleration vary slightly around the fundamental frequency of bridge girder ωb1=5.72 Hz under various train speeds. However, the second and the third dominant frequency ranges increase as the train speed increases, both of whose trends quite coincide with those of bogie vertical acceleration (Figures 23 and 24). Furthermore, the contribution of the latter two ranges of dominant frequency to bridge girder vertical acceleration show an obvious trend of increase for higher train speeds.PSD of vertical acceleration of bridge girder midpoint.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)From the point of view of structural dynamics, so-called resonance may occur if the wavelengths or frequencies implied by rail irregularity are close to the vehicle or bridge girder frequencies. Therefore, it is especially important to control strictly the rail irregularity with wavelengths ofv/ωc1 and v/ωb1 in high-speed railway track maintenance, where ωc1 and ωb1 denote the fundamental frequencies of vertical motion of carbody and bridge girder, respectively. In addition, it is also significant to control the rail irregularity with wavelengths of Lt and 2Lt to reduce the vibration of bogie and bridge girder as much as possible for high-speed railway.


### 5.5. Influence of Train Speeds on the Dynamic Responses of Train-Slab Track-Bridge Interaction System

In reality, the train can move over the bridge at various speeds. There exists a need to investigate the dynamic responses of train-track-bridge system under various train speeds, as they may be quite different. The effect of train speeds on the dynamics responses of train-slab track-bridge interaction system is now investigated by PEM with speeds from 0 to 420 km/h at 10 km/h intervals and 170 discrete frequencies of rail irregularity considered. The estimated maximum responses of train, track, and bridge with respect to train speeds are shown in Figures26, 27, 28, 29, and 30.Figure 26
Maximum carbody vertical acceleration under various train speeds.Figure 27
Maximum Sperling’s ride index under various train speeds.Figure 28
Maximum wheel load decrement ratio under various train speeds.Figure 29
Maximum vertical impact factor of bridge midpoint under various train speeds (rail/girder).Figure 30
Maximum vertical acceleration of bridge girder midpoint under various train speeds.The maximum vertical accelerations of carbody under various train speeds are plotted in Figure26. In general, the maximum response of the train moving over irregular track increases as train speeds increase, with a moderate growth for train speeds over 260 km/h. The reason for existence of such trend is that the exciting frequencies induced by rail irregularity at higher speeds are filtered out by the suspension system of each vehicle in transmission to the carbody. Moreover, both the maximum vertical accelerations of motor and trailer carbody are below the allowable limit of 1.3 m/s2 imposed by China’s provision [33] for speeds up to 420 km/h.Figure27 shows Sperling’s ride index Wz computed for the train under various speeds. Similar with the case of carbody vertical acceleration, Sperling’s ride index of both motor car and trailer car increase comparatively slightly for train moving at moderate to high speeds (200–420 km/h), with the former a mite higher than the latter. Compared with the allowable limit values set in China’s provision for different comfort levels [33], Sperling’s ride indices of motor car and trailer car fall with the limit of “tolerate (2.75<Wz<3.0)” and “good (2.5<Wz<2.75),” respectively, under the track conditions specified for speeds up to 420 km/h.Maximum wheel load decrement ratio with respect to train speeds are presented in Figure28, which shows a trend of increase for higher train speeds. Of interest is the fact that the wheel load decrement ratio of trailer car is always greater than that of motor car. At the meantime, the former increases faster than the later for train speeds over 260 km/h. As a result, the wheel load decrement ratio of trailer car exceeds the limit of allowable 0.65 set in China’s provision for the running safety of train [33] for speeds up to 370 km/h. In contrast, the wheel load decrement ratio of motor car remains to be lower than the allowable limit of 0.65 for speeds up to 420 km/h. From this point of view, the larger axle load is beneficial to ensure the running safety of train.The impact responses of track and bridge girder traveled by train moving at speeds 0–420 km/h are plotted in Figures29 and 30, in which the impact factor I is defined as I=[Rd(x)-Rs(x)]/Rs(x), where Rd(x) and Rs(x), respectively, denote the maximum dynamic and static vertical responses of track or bridge girder at section x due to the action of train. As can be seen, the impact factor and vertical acceleration of bridge girder increase generally as the train speed increases, but no phenomenon of train-track-bridge resonance is observed, indicating that the fundamental frequency of bridge girder is suitable under the train properties specified for speeds over 420 km/h. In addition, the impact factor for rail vertical displacement is generally higher than that for bridge girder vertical displacement. The implication is that the extent of amplification for rail vertical displacement is larger than that for bridge girder vertical displacement, because of the comparatively small mass of rail and the effects of elasticity and damping of fastener and CA layer. Of interest is that the bridge girder vertical acceleration is far below the allowable limit of 3.5 m/s2 set in China’s provision under the track conditions specified for speed up to 420 km/h.


## 5.1. The Properties of Train-Slab Track-Bridge Interaction System

In order to reduce the boundary effect of the subgrade as far as possible, a nine-span simply supported beam high-speed railway bridge with the span length of 32 m is considered as shown in Figure1. 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 steady-state 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 v. The physical parameters of train, track, and bridge are listed, respectively, in Tables 1 and 2. The Germany high-speed track vertical profile irregularity PSD function [13, 33] is adopted; that is,
(34)Srr(Ω)=AvΩc2(Ω2+Ωr2)(Ω2+Ωc2),
where Ωc=0.8246 rad/m, Ωr=0.0206 rad/m, and Av=4.032×10-7 m·rad.Table 1
Parameters for trailer car and motor car.
Notation
Item
Unit
Value
Trailer car
Motor car
m
c/mJc
Mass of carbody
kg
4.4 × 104
4.8 × 104
J
c/JJc
Mass moment of inertia of carbody
kg·m2
2.7 × 106
2.7 × 106
m
t/mJt
Mass of bogie
kg
2.4 × 103
3.2 × 103
J
t/JJt
Mass moment of inertia of bogie
kg·m2
2.2 × 103
7.2 × 103
m
w/mJw
Mass of wheelset
kg
2.4 × 103
2.4 × 103
k
s/kJs
Spring stiffness of the second suspension system
N/m
6.0 × 105
8.0 × 105
c
s/cJs
Damping coefficient of the second suspension system
N·s/m
8.0 × 104
9.0 × 104
k
p/kJp
Spring stiffness of the primary suspension system
N/m
1.4 × 106
2.08 × 106
c
p/cJp
Damping coefficient of the primary suspension system
N·s/m
8.0 × 104
6.0 × 104
L
v/LJv
Longitudinal distance between the center of bogie and the nearest side of carbody
m
3.45
3.45
L
c/LJc
Half of longitudinal distance between the centers of front and rear bogies
m
8.6875
8.6875
L
t/LJt
Half of bogie axle base
m
1.25
1.25
ω
c
1/ωJc1
The fundamental frequency for vertical motion of carbody
Hz
0.75
0.84
ω
c
2/ωJc2
The fundamental frequency for pitching motion of carbody
Hz
0.84
0.97
ω
t
1/ωJt1
The fundamental frequency for vertical motion of bogie
Hz
6.00
4.78
ω
t
2/ωJt2
The fundamental frequency for pitching motion of bogie
Hz
7.10
6.28Table 2
Parameters for slab track and bridge.
Notation
Item
Unit
Value
Slab track
E
r
Young’s modulus of rail
N/m2
2.06 × 1011
υ
r
Poisson’s ratio of rail
—
0.3
I
r
Mass moment of inertia of rail cross section
m4
2 × 3.217 × 10−5
m
-
r
Mass per unit length of rail
kg/m
2 × 60.64
E
s
Young’s modulus of slab
N/m2
3.6 × 1010
υ
s
Poisson’s ratio of slab
—
0.2
I
s
Mass moment of inertia of slab cross section
m4
1.6 × 10−3
m
-
s
Mass per unit length of slab
kg/m
1.2 × 103
k
rs
Spring stiffness of fastener
N/m
2 × 5.0 × 107
c
rs
Damping coefficient of fastener
N·s/m
2 × 3.625 × 104
k
-
sb
Spring stiffness of CA layer per unit length
N/m2
2 × 1.5 × 109
c
-
sb
Damping coefficient of CA layer per unit length
N·s/m2
2 × 8.3 × 104
Bridge
E
b
Young’s modulus of girder
N/m2
3.45 × 1010
υ
b
Poisson’s ratio of girder
—
0.2
I
b
Mass moment of inertia of girder cross section
m4
12.744
m
-
b
Mass per unit length of bridge girder
kg/m
2.972 × 104
ζ
b
Damping ratio of girder
—
0.05
ω
b
1
The fundamental frequency of girder
Hz
5.72
ω
b
2
The second natural frequency of girder
Hz
22.89
k
bb
Spring stiffness of bearing
N/m
6.0 × 109
c
bb
Damping coefficient of bearing
N·s/m
2.0 × 106To solve the equations of motion for the train-slab track-bridge interaction system, the Wilson-θ method is used with θ=1.4 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 0.004×2π to 1×2π 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,
(35)WLen(j)=CLene(j-1)(j=1,2,3,…,m),
where WLen denotes the wavelength of rail irregularity (m) and CLen denotes a constant related to the maximum frequency points m.


## 5.2. Selection of the Suitable Number of Discrete Frequencies of Rail Irregularity

From the aforementioned solution procedures for random analysis of train-slab track-bridge system, one can find that the larger the number of discrete frequenciesm 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 m increases from 50 to 250 with increment 20, the corresponding CLen is listed in Table 3, based on which the solutions are plotted in Figures 3, 4, 5, 6, 7, and 8. The variation ratio VR1 of the responses of train, track, and bridge and the CPU times for different cases are shown in Table 4. Herein VR1 is defined as (36), in which RPEM(i) denote the maximum dynamic response SD of train, track, and bridge with the total number of discrete frequencies being equal to i:
(36)VR1=RPEM(i)-RPEM(250)RPEM(250)×100%(i=50~250).Table 3
Relationship between the number of discrete frequenciesm and the constant CLen.
m
50
70
90
110
130
150
170
190
210
230
250
C
Len
1.11929
1.08331
1.06342
1.05197
1.04374
1.03776
1.03322
1.02965
1.02677
1.02441
1.02243Table 4
The variation ratioVR1 of the responses (%) and CPU times.
Item
Total number of discrete frequenciesm
50
70
90
110
130
150
170
190
210
230
250
Vertical acceleration of carbody
Motor car
-
1.99
-
1.29
-
0.88
-
0.62
-
0.47
-
0.34
-
0.23
-
0.16
-
0.08
-
0.03
0.00
Trailer car
-
1.97
-
1.30
-
0.86
-
0.63
-
0.44
-
0.35
-
0.25
-
0.16
-
0.10
-
0.03
0.00
Vertical acceleration of rear bogie
Motor car
-
2.02
-
1.27
-
0.90
-
0.56
-
0.45
-
0.35
-
0.28
-
0.17
-
0.10
-
0.05
0.00
Trailer car
-
2.33
-
1.45
-
0.98
-
0.62
-
0.56
-
0.35
-
0.28
-
0.17
-
0.10
-
0.04
0.00
Wheel/rail vertical force
Motor car
-
1.20
-
2.99
-
0.49
-
0.31
-
0.56
-
0.38
-
0.27
-
0.21
-
0.15
-
0.10
0.00
Trailer car
-
1.06
-
1.97
-
0.24
-
0.13
-
0.60
-
0.32
-
0.15
-
0.06
-
0.23
-
0.14
0.00
Vertical displacement of bridge midpoint
Rail
0.98
0.57
0.41
0.33
0.24
0.16
0.08
0.08
0.08
0.00
0.00
Girder
-
2.99
-
1.12
-
0.93
-
0.56
-
0.37
-
0.37
-
0.19
-
0.19
0.00
0.00
0.00
Vertical acceleration of bridge midpoint
Rail
9.40
6.21
4.33
3.08
2.26
1.63
1.15
0.77
0.47
0.22
0.00
Slab
12.26
9.67
6.60
4.28
3.30
2.27
1.59
1.05
0.62
0.32
0.00
Girder
-
0.38
-
1.76
11.42
3.14
1.38
-
0.69
-
0.19
-
0.69
0.00
0.06
0.00
Vertical fastener pressure
1.04
0.67
0.47
0.33
0.24
0.18
0.13
0.08
0.04
0.02
0.00
CPU times (h)
0.60
0.80
0.96
1.13
1.31
1.43
1.64
1.84
2.03
2.23
2.41Figure 3
Maximum SD of vertical acceleration of carbody.Figure 4
Maximum SD of vertical acceleration of rear bogie.Figure 5
Maximum SD of wheel/rail vertical force.Figure 6
Maximum SD of vertical displacement of bridge midpoint.Figure 7
Maximum SD of vertical acceleration of bridge girder midpoint.Figure 8
Maximum SD of vertical fastener pressure above bridge midpoint.According to Yang et al. [15], the dynamic response of the last vehicle of a train should be given much more attention concerning the running safety and riding comfort of the train moving over a bridge because the last vehicle tends to vibrate more violently than the ones ahead. Therefore, “motor car” and “trailer car” considered in the following examples, respectively, mean the rear motor car and the Nvth trailer car connecting with the rear motor car (Figure 1). In addition, “bridge midpoint” and “wheel/rail vertical force,” respectively, are the midpoints of the fifth span for the nine-span bridge and the vertical contact force between the 3rd wheelset of the rear motor car or of the Nvth trailer car and rail.From Figures3–8 and Table 4, it is observed that, as the total number of discrete frequencies increases, each of the maximum SD of train, track, and bridge dynamic responses tends to approach a limit value. Comparing the results obtained by 250 discrete frequencies with those by 170 discrete frequencies, the variation ratios VR1 are −0.23% and −0.25% for the vertical acceleration SD of the motor and trailer carbody, respectively, and 1.15%, 1.59%, and −0.19% for the vertical acceleration SD of rail, slab, and bridge girder, respectively. Therefore, one can conclude that the suitable number of discrete frequencies of rail irregularity can be equal to 170.


## 5.3. Comparing PEM with MCM

The running safety and riding comfort of train have been of great concern in railway engineering for a long time, particularly due to the development of high-speed railway. Several mechanisms that result in the derailment or discomfort of a running train have been identified through analytical and experimental investigations, based on which some indices have been proposed for evaluating the running safety and riding comfort of train [34], such as carbody vertical acceleration, wheel load decrement ratio, Sperling’s ride index, and acceleration of bridge girder. In practice, the maximum dynamic responses are usually of the most interest for us. As is mentioned above, the conventional method of estimation to pick the maximum values from a set of response samples is extremely time consuming. However, the upper and lower boundaries of the responses by PEM can be estimated easily by the MV ± 3 times SD according to the 3σ rule for Gaussian stochastic process [24], since the corresponding MV and SD can be obtained efficiently and accurately according to the aforementioned solution procedures.For the purpose of comparison, the random responses of the train-slab track-bridge interaction system calculated by PEM, with total discrete frequenciesm=170, are compared with those by MCM, with 1000 of rail irregularity samples. Herein, the method proposed by Chen and Zhai [35] is implemented to generate the rail irregularity sample r(x) from the PSD function described as (34), and the train is also assumed to pass through the bridge with a constant speed 69.44 m/s. Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, and 20 show the estimated upper and lower boundaries of the responses by PEM and the corresponding extreme values of each time-history response obtained by MCM. The variation ratio VR2 of the responses of train, track, and bridge girder, determined according to (37), is shown in Table 5. Consider
(37)VR2=max⁡|RMCME(i)|-min⁡|RMCME(i)|min⁡|RMCME(i)|×100%(i=1~1000),
where RMCME denote the extreme values of each time-history response obtained by MCM and i denotes the number of rail irregularity samples.Table 5
The variation ratioVR1 of the responses (%) and CPU times.
Item
Total number of rail irregularity samples
100
200
400
600
800
1000
Vertical acceleration of carbody
Motor car
152.25
193.00
228.30
228.30
280.37
280.37
Trailer car
150.17
150.17
158.91
160.20
184.66
184.66
Vertical acceleration of rear bogie
Motor car
61.94
74.82
79.93
98.72
98.72
98.72
Trailer car
62.92
69.90
83.86
112.07
112.07
112.07
Wheel load decrement ratio
Motor car
68.31
77.28
96.66
96.66
96.96
96.96
Trailer car
71.72
71.72
78.99
94.03
120.08
120.08
Sperling’s ride index
Motor car
23.96
29.76
34.09
34.09
39.00
39.00
Trailer car
16.24
21.69
25.36
25.36
27.40
27.40
Vertical displacement of rail above bridge midpoint
49.74
60.51
61.94
61.94
68.28
68.28
Vertical fastener pressure above bridge midpoint
49.70
60.25
62.48
62.48
68.12
68.12
Vertical displacement of bridge girder midpoint
8.98
8.98
8.98
10.90
10.90
10.90
Vertical acceleration of bridge girder midpoint
111.92
115.58
160.73
160.73
160.73
170.05
CPU times (h)
13.12
26.24
52.48
78.72
91.84
131.20Figure 9
Comparison of vertical acceleration of motor carbody.Figure 10
Comparison of vertical acceleration of trailer carbody.Figure 11
Comparison of rear bogie vertical acceleration of motor car.Figure 12
Comparison of rear bogie vertical acceleration of trailer car.Figure 13
Comparison of wheel load decrement ratio of motor car.Figure 14
Comparison of wheel load decrement ratio of trailer car.Figure 15
Comparison of Sperling’s ride index of motor car.Figure 16
Comparison of Sperling’s ride index of trailer car.Figure 17
Comparison of vertical displacement of rail above bridge midpoint.Figure 18
Comparison of vertical pressure of fastener above bridge midpoint.Figure 19
Comparison of vertical displacement of bridge girder midpoint.Figure 20
Comparison of vertical acceleration of bridge girder midpoint.From Figures9–20 and Table 5, the following observations can be made easily. (1) There exist dramatic variations for the extreme values of each time-history response by MCM, except for that of bridge girder vertical displacement. For example, variation ratio VR2 of vertical acceleration of motor carbody, wheel load decrement ratio of trailer car, Sperling’s ride index of motor car, and vertical acceleration of bridge girder midpoint among the 1000 samples are 280.37%, 120.08%, 39.00%, and 170.05%, respectively. In general, a larger number of rail irregularity samples tend to enlarge the variation ratio VR2. (2) All of extreme values of responses obtained by MCM fall with the corresponding limits of upper and lower boundaries of those by PEM. (3) Compared with MCM, the PEM helps to save computer time drastically. For example, the total CPU times for PEM with 170 discrete frequencies and MCM with just 100 samples are 1.64 hours and 13.12 hours, respectively, on a 2.8 GHz personal computer. The latter equals about eight times the former. That is to say, PEM, in dealing with random vibration of train-slab track-bridge interaction system or evaluating the running safety and riding comfort of train, is not only more reliable but also more efficient than the conventional MCM.


## 5.4. The Random Vibration Characteristics of Train-Slab Track-Bridge Interaction System

The dominant vibration frequency of train and bridge can be obtained easily according to the PSDs of dynamic responses calculated by PEM. To investigate the random vibration characteristics of train-slab track-bridge interaction system, the solutions obtained by PEM, with 170 discrete frequencies of rail irregularity and three train speeds, are employed.Figures21 and 22 exhibit the PSDs of carbody vertical acceleration. As can be seen, the PSDs vary only slightly with time which indicates that the influence of bridge vibration on carbody vibration is insignificant because of the comparatively high mass and flexural rigidity of bridge girder. On the other hand, the PSDs change violently with rail irregularity frequency, implying the great influence of rail irregularity on the dynamic response of carbody. Of interest is that, no matter how the train speed changes, the peak values of PSDs of carbody vertical acceleration appear always approximately, respectively, to be around 0.77~0.86 and 0.69~0.77 Hz for motor car and trailer car, which show good agreement with their own fundamental frequencies of 0.84 and 0.75 Hz.PSD of vertical acceleration of motor carbody.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)PSD of vertical acceleration of trailer carbody.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)The similar phenomenon for bogie vertical acceleration can be also observed from Figures23 and 24. As can be seen, the bogie also vibrates periodically, implying three dominant frequencies of 4.22, 16.79, and 33.49 Hz for v=41.67 m/s, 6.31, 26.99, and 55.82 Hz for v=69.44 m/s and 6.37 (motor car), or 7.92 (trailer car), 39.18, and 78.15 Hz for v=97.22 m/s. It is found that the vertical vibration frequency ft of bogie, the half of bogie axle base Lt, and the train speeds v satisfy the relation ft=v/(n×Lt), n=1,2,3,…,∞, where relatively large contributions are made by harmonic components with n=1, 2, or 8~15.PSD of vertical acceleration of motor car rear bogie.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)PSD of vertical acceleration of trailer car rear bogie.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)The PSDs of vertical acceleration of bridge girder midpoint to the passage of the train are plotted in Figure25. Similar with the cases of bogie vertical acceleration, there exist in Figure 25 three dominant frequency ranges of 4.7~5.4, 14~18, and 32~37 Hz for v=41.67 m/s, 4.4~6.3, 25~33, and 53~58 Hz for v=69.44 m/s and 4.4~6.8, 35~43, and 72~81 Hz for v=97.22 m/s. Obviously, the first dominant frequency ranges of bridge girder vertical acceleration vary slightly around the fundamental frequency of bridge girder ωb1=5.72 Hz under various train speeds. However, the second and the third dominant frequency ranges increase as the train speed increases, both of whose trends quite coincide with those of bogie vertical acceleration (Figures 23 and 24). Furthermore, the contribution of the latter two ranges of dominant frequency to bridge girder vertical acceleration show an obvious trend of increase for higher train speeds.PSD of vertical acceleration of bridge girder midpoint.
(a)
v
=
41.67 m/s (150 km/h)
(b)
v
=
69.44 m/s (250 km/h)
(c)
v
=
97.22 m/s (350 km/h)From the point of view of structural dynamics, so-called resonance may occur if the wavelengths or frequencies implied by rail irregularity are close to the vehicle or bridge girder frequencies. Therefore, it is especially important to control strictly the rail irregularity with wavelengths ofv/ωc1 and v/ωb1 in high-speed railway track maintenance, where ωc1 and ωb1 denote the fundamental frequencies of vertical motion of carbody and bridge girder, respectively. In addition, it is also significant to control the rail irregularity with wavelengths of Lt and 2Lt to reduce the vibration of bogie and bridge girder as much as possible for high-speed railway.


## 5.5. Influence of Train Speeds on the Dynamic Responses of Train-Slab Track-Bridge Interaction System

In reality, the train can move over the bridge at various speeds. There exists a need to investigate the dynamic responses of train-track-bridge system under various train speeds, as they may be quite different. The effect of train speeds on the dynamics responses of train-slab track-bridge interaction system is now investigated by PEM with speeds from 0 to 420 km/h at 10 km/h intervals and 170 discrete frequencies of rail irregularity considered. The estimated maximum responses of train, track, and bridge with respect to train speeds are shown in Figures26, 27, 28, 29, and 30.Figure 26
Maximum carbody vertical acceleration under various train speeds.Figure 27
Maximum Sperling’s ride index under various train speeds.Figure 28
Maximum wheel load decrement ratio under various train speeds.Figure 29
Maximum vertical impact factor of bridge midpoint under various train speeds (rail/girder).Figure 30
Maximum vertical acceleration of bridge girder midpoint under various train speeds.The maximum vertical accelerations of carbody under various train speeds are plotted in Figure26. In general, the maximum response of the train moving over irregular track increases as train speeds increase, with a moderate growth for train speeds over 260 km/h. The reason for existence of such trend is that the exciting frequencies induced by rail irregularity at higher speeds are filtered out by the suspension system of each vehicle in transmission to the carbody. Moreover, both the maximum vertical accelerations of motor and trailer carbody are below the allowable limit of 1.3 m/s2 imposed by China’s provision [33] for speeds up to 420 km/h.Figure27 shows Sperling’s ride index Wz computed for the train under various speeds. Similar with the case of carbody vertical acceleration, Sperling’s ride index of both motor car and trailer car increase comparatively slightly for train moving at moderate to high speeds (200–420 km/h), with the former a mite higher than the latter. Compared with the allowable limit values set in China’s provision for different comfort levels [33], Sperling’s ride indices of motor car and trailer car fall with the limit of “tolerate (2.75<Wz<3.0)” and “good (2.5<Wz<2.75),” respectively, under the track conditions specified for speeds up to 420 km/h.Maximum wheel load decrement ratio with respect to train speeds are presented in Figure28, which shows a trend of increase for higher train speeds. Of interest is the fact that the wheel load decrement ratio of trailer car is always greater than that of motor car. At the meantime, the former increases faster than the later for train speeds over 260 km/h. As a result, the wheel load decrement ratio of trailer car exceeds the limit of allowable 0.65 set in China’s provision for the running safety of train [33] for speeds up to 370 km/h. In contrast, the wheel load decrement ratio of motor car remains to be lower than the allowable limit of 0.65 for speeds up to 420 km/h. From this point of view, the larger axle load is beneficial to ensure the running safety of train.The impact responses of track and bridge girder traveled by train moving at speeds 0–420 km/h are plotted in Figures29 and 30, in which the impact factor I is defined as I=[Rd(x)-Rs(x)]/Rs(x), where Rd(x) and Rs(x), respectively, denote the maximum dynamic and static vertical responses of track or bridge girder at section x due to the action of train. As can be seen, the impact factor and vertical acceleration of bridge girder increase generally as the train speed increases, but no phenomenon of train-track-bridge resonance is observed, indicating that the fundamental frequency of bridge girder is suitable under the train properties specified for speeds over 420 km/h. In addition, the impact factor for rail vertical displacement is generally higher than that for bridge girder vertical displacement. The implication is that the extent of amplification for rail vertical displacement is larger than that for bridge girder vertical displacement, because of the comparatively small mass of rail and the effects of elasticity and damping of fastener and CA layer. Of interest is that the bridge girder vertical acceleration is far below the allowable limit of 3.5 m/s2 set in China’s provision under the track conditions specified for speed up to 420 km/h.


## 6. Summary

In this study, the vertical vibration model of train-slab track-bridge interaction system is firstly established by FEM; the equations of motion for the entire system are then derived by means of energy principle, and simultaneously the excitations between rail and wheels caused by random rail irregularity are transformed into a series of deterministic pseudoharmonic excitation vectors by PEM, and the random vibration responses of train, slab track, and bridge are finally obtained by step-by-step integration method. From the numerical results obtained in this work, the following conclusions can be drawn.(1)
In order to save computer time and ensure accuracy of solution as far as possible, some numerical experimentation should be done firstly to obtain the suitable number of discrete frequencies of rail irregularity in the process of random analysis of train-slab track-bridge interaction system.(2)
The PEM can be applied not only to study the vertical random vibration characteristics of train-slab track-bridge interaction system but also to evaluate the running safety and riding comfort of train moving over bridge with various speeds, which is more reliable and efficient than the conventional MCM.(3)
The PEM may provide a new way to either optimize the design of dynamic parameters of train-slab track-bridge interaction system or provide guidance on the maintenance of track structure according to certain evaluation standard for railway.(4)
The equations of motion of vertical interaction system with other types of train, track, and bridge and the corresponding random vibration responses can be also obtained by the method proposed in this paper.(5)
In this work, only the vertical random vibration characteristics of train-slab track-bridge interaction system are investigated. In the future work, we plan on further developing the numerical method and extending its range of applicability to model the spatial random vibration characteristics of train-slab track-bridge interaction system.



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*Source: 304219-2014-08-12.xml*