Construction of a dynamic model for the interaction between the versatile tracks and a vehicle

Engineering Structures 206 (2020) 110067

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Construction of a dynamic model for the interaction between the versatile tracks and a vehicle

T

Lei Xua,b, Lifeng Xinc, , Zhiwu Yua,b, Zhihui Zhua,b ⁎

a

School of Civil Engineering, Central South University, Changsha 410075, China National Engineering Laboratory for High-Speed Railway construction, Central South University, Changsha 410075, China c Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, 610031, PR China b

ARTICLE INFO

ABSTRACT

Keywords: Railway engineering Vehicle-track interactions Versatile track model Finite element method Wheel-rail contacts

Take hypothesis of wheel-rail rigid contact, this work presents a new method to characterize the vehicle-track interaction in the three-dimensional (3-D) space. The vehicle model is considered as a multi-rigid-body system with one car body, two bogie frames and four wheelsets that are connected by two-stage suspension systems. The track model is constructed by a versatile approach, where the finite elemental size and type of track components can be arbitrarily chosen and combined. The key matrices in depicting vehicle-track interaction are established with satisfaction of wheel-rail displacement complementarity by geometrical constraint equations and the wheel-rail force equilibrium by energy variational principle. Besides the functionality of depicting wheel-rail separations has been solved and compiled in the program. This model possesses the superiority of the previous models in computational stability. Besides the vehicle-track interaction at a 3-D space, which considers the wheel-rail contact geometries/creepages and infinite length calculation, are further formulated in the vehicletrack coupling matrices. Comparisons with general solutions built by well-known wheel-rail contact theories are conducted to show the engineering practicality of this model; moreover the differences of modelling results and the advancements of this model are also illustrated.

1. Introduction Dating back to since 1960s, characterizations of the vehicle-track interaction in physics, mechanics and mathematics have been a hot topic for worldwide researchers [1–4]. At early stages, this work mainly concentrates on railway vehicle dynamics, and the vehicle and the tracks are modelled and considered separately. For example, the tracks are generally regarded as a rigid support in railway vehicle dynamics [5–9]. Later, railway researchers gradually look into the necessities of considering the coupled interactions between the vehicle and the tracks [9–14]. The dynamic models have also been gradually improved from considering vertical vibrations only to fully incorporating the threedimensional (3-D) vibrations [15–18]. Until now, the modelling of vehicle-track interaction is greatly developed, and the coupling interaction between multi-body systems can be compiled and depicted by a computer program, even considering the complex elastic-plastic wheel-rail contact. The 3-D elastic-plastic dynamic models apply accurate wheel-rail contact approaches such as CONTACT by Kalker [19], 3-D wheel-rail rolling contacts [20], modified FASTSIM-based [21] non-elliptic contacts [22–25], etc. These

⁎

methods are mainly implemented to solve local contact problems with small scales and high frequency vibrations. However these models are generally of low computational efficiency and can be only solved by explicit numerical methods. To clarify the dynamic performance, e.g., curve negotiation, of an entire vehicle or a train (not just moving loads or mass-stiffness-damping suspensions), the modelling method, which possess high accurateness and computational efficiency in vehicle-track multi-body dynamics due to the large scale analysis, is the main point of this study. Generally, the vehicle is regarded as multi-rigid-body system and the tracks are modelled as an entire elastic system or rigid-elastic hybrids in vehicle-track dynamical analysis,. The algorithm to numerically describe the wheel-rail contact in geometries and forces is the key in model construction. In this period, non-linear/linear Hertzian theory, Kalker′s linear creep theory and saturated nonlinear modifications are widely applied in vehicle-track interaction models. For instance, Zhai et al. [26] presented a 3-D nonlinear vehicle-track coupled model, where wheel-rail contact geometries/creepages were accurately obtained by a wheel-rail coupling model [27], besides the normal and tangential wheel-rail forces were respectively calculated by theories of

Corresponding author. E-mail addresses: [email protected] (L. Xu), [email protected] (L. Xin), [email protected] (Z. Yu), [email protected] (Z. Zhu).

https://doi.org/10.1016/j.engstruct.2019.110067 Received 15 May 2019; Received in revised form 6 December 2019; Accepted 7 December 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature

the half of the width of the rail bottom the lateral distance between the left-side of the track slab and the contact point to the rail pad (k rty ,k rtz ) the lateral and vertical stiffness coefficient of the rail pad respectively m w, j the mass of the jth wheelset Iwx the moment of inertia of the wheelset around X-axis (k zp ,czp ) the vertical stiffness and damping coefficient of the primary suspension lH the half of the lateral distance between the left- and rightside wheel-rail contact points lV the vertical distance between the wheel/rail contact point and the rail centroid W1y the half of the lateral distance between primary suspensions in a bogie frame kgX the equivalent angular stiffness of wheel/rail contact kgY the equivalent stiffness of gravity ( f11, f22 ) the longitudinal and lateral creep coefficient at the wheel/ rail interfaces the nominal rolling speed V Wj the axle-load of the jth wheelset

br b

the spacing interval between two adjacent rail pads the length and the width of a track slab respectively the Young′s modulus of the rail the rail sectional area the moment of inertia of the rail around Z- and Y-axis respectively k rt the rail torsional stiffness m ¯r the rail mass per unit length Wr the polar moment of inertia Es the Young′s modulus of the track slab Isz the moment of inertia of the track slab around Z-axis Hs the height of the track slab µ the Poisson ratio m ¯s the mass of the track slab per unit volume (k tsy ,k tsz ) the lateral and vertical stiffness coefficient between the track slab and the subgrade respectively (ctsy ,ctsz ) the lateral and vertical damping coefficient between the track slab and the subgrade respectively ar the vertical distance between the rail neural axis and the rail bottom

Lr (L s ,L w ) Er Ar (Irz ,Iry )

Hertzian, Kalker, nonlinear modifications. With similar work of Ref. [26], a series of studies using almost the same wheel-rail contact approaches were further conducted by Xu and Zhai [28], Montenergro et al. [29], Wang et al. [30], Kouroussis et al. [31], Dinh et al. [32], Yu and Mao [33], etc. In these related works, finite element theory, iterative schemes, substructure and ground motions, etc., were further introduced in analyzing vehicle-track interaction and broadening the usage of classical wheel-rail contact theories to different structural, mechanical and physical conditions. As another alternative, it is assumed that there is no elastic compression between the wheel and the rail and consequently, the wheelrail normal contact is no longer modelled by the Hertzian theory. Instead it is replaced by wheel-rail geometry constraint equations in vertical and lateral directions at instantaneous time t , that is,

Xw = Xr + XI + Xwr

to comply with the displacement complementarity between the wheel and the rail, unless the adoption of iterative procedures or a very small time integral step. By energy principle, such as the work presented in [37,38], vehicle-track (bridge) interaction has been depicted by matrix formulations, in which the vehicle and its substructures have been united as an entire system, thus possessing high superiority in computational stability and efficiency. But in these models, the moving vehicles are assumed to keep contact with the rail. Besides the refinement and nonlinear wheel-rail contact geometry, creepage and force in a 3-D space are roughly considered as two-dimensional (2-D) contact and linear creep scenario. Recently an efficient moving element method was developed by Dai et al. [39], in which the global stiffness and damping matrices in one period is prepared for easier retrieval in the subsequent time-domain dynamic analysis. This method has also been applied to study the dynamics performance of vehicle-track systems subject to unsupported sleepers [40] and abrupt braking of partially filled freight train [41]. In this paper, a coupled approach to construct the vehicle-track interaction model is specifically proposed by further improving the models built upon the hypothesis of wheel-rail rigid contact. Moreover, the matrices of mass, stiffness, damping and load vectors are fully integrated into a coupled system. Using this constructed model, the vehicle and the tracks are united as one system, where the wheel-rail contact surfaces are no longer the system boundaries and accordingly, no iterative procedures are needed. Mostly important, arbitrary time step sizes can be used. This approach is based on the finite element method (FEM) and draws essence from wheel-rail coupling model [27] to accurately characterize the wheel-rail contact geometries/creepages. Another technical issue in FEM modelling is the requirement of elemental node conjunctions. For example, in the well-known ANSYS®, different elements must be directly connected by elemental nodes, which brings the modelling inconvenience. Regarding this, a versatile method for constructing the track systems will be put forward, in which the elemental size can be arbitrarily chosen without one-to-one match between nodes. Correspondingly, an extended approach, through which only a section of tracks is required to achieve infinite length computation, is further presented. The following part of this paper will be organised by

(1)

in Eq. (1), the subscript ‘ w’, ‘r’ and ‘I’ denote the wheel, the rail and the track irregularity respectively; X denotes the displacement and Xwr denotes the wheel-rail relative displacement. In most of the abovementioned work, Xwr is dealt with a nonlinear relation to wheel-rail normal force following Hertzian assumption. However, with hypothesis of no wheel-rail penetration, Xwr can be treated as zeros in no-jump running environment. Following this hypothesis, Wu et al. [34], Dimitrakopoulos et al. [35], Chen et al. [36], etc., constructed vehicletrack (bridge) interaction models, where the vehicle and the track were entirely coupled as an entire system in time-dependent matrix representations, and the system responses of the vehicle and the tracks could be simultaneously obtained by time integrations without iterative procedures. But in their work, the wheel-rail relative motion and creepages were ignored. It is obviously an deficiency in accurately revealing the lateral dynamic performance of vehicle-track systems. In this period the multi-body dynamics for vehicle-track interaction is gradually perfected and governed by conventional wheel-rail contact theories. For example, when using uncoupling approaches to establish the dynamic model, the vehicle and the tracks are separately constructed and connected by wheel-rail force equilibrium equations explicitly, in which the wheel-rail elastic compression is calculated based on Hertzian contact theory. However, as mentioned above, the Hertz- or Kalker-based models are basically explicit in formulating wheel-rail interaction forces. Essentially speaking this class of models only satisfies the force equilibrium on wheel/rail contact interfaces, but failing

(1) In Section 2, the matrix representations for constructing the versatile track model are presented; 2

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L. Xu, et al.

(2) In Section 3, the modelling of vehicle-track interaction is given, and an extension for infinite length calculation is also achieved. (3) In Section 4, numerical examples are implemented to show the effectiveness and advancement of the proposed methods. (4) Finally in Section 5, some conclusions are drawn accordingly.

n i=1

2.1. Energy variation principle

2.2.1. The track model In this present study, the track model is constructed by FEM, in which the rail is modelled as Bernoulli-Euler beam, the track slab is modelled as thin-plate element, the rail pad is modelled as discrete supports of spring and dashpot, and the track slab and the subgrade are connected by continuous plane springs and dashpots. It is rather different from the conventional track elements presented in Refs. [28,36,37–38,45–46] where the rail, track and subgrade are coupled as an entire element with equal length. In this versatile model, the scale and meshing of the finite elements can be arbitrarily selected with unequal length of the rail and the track slab in 3-D space, and the nodes connecting the rail and the tracks are non-essential to match with each other in coordinates, as shown in Fig. 1.

(2)

( uT u ¨ )d , Vc = ( uTc u)d , VF = uTFsign (u), with Vm = Vp = uTp(t ) , Vg = uTG ,where Ui is the strain energy of the elastic system; Vm is the negative numerical value of work done by the inertial force; Vc is the negative numerical value of work by the viscous damping force; VF , Vp and Vg are respectively the negative numerical value of work done by the Coulomb frictional force, the applied loading and the potential energy of the gravitational force; is the density parameter; c is the damping coefficent; F is the Coulomb frictional force vector; p(t ) is the external loading force vector; G is the gravitational ¨ denote respectively the displacement, velocity and force; u , u and u acceleration vector. Besides, Eq. (2) must satisfy [43] d

=0

(4)

2.2. A versatile track model

Based on principles of virtual work and D′ Alembert, and considering the damping force, the total potential energy of an elastic system can be expressed by [42]

= Ui + Vm + Vc + VF + Vp + Vg

ui = 0

In Eq. (4), ui 0 , thus u d = 0 and it represents the i th dynamic i equilibrium equation, which consists of displacement vector uj , j = 1, 2, ...,n . In the matrices derived, the serial number uj and ui respectively indicate the position of the coefficient at the j th column and the i th row of the matrix, which has been technically developed as a ‘set-in-right-position’ rule in [44].

2. Construction of a versatile track model

d

d

ui

2.2.2. Establishment of the track matrices by energy variational method Firstly it is assumed that the longitudinal distance between two rail pads is denoted by Lr , and the length and the width of a track slab are denoted by Ls and L w , respectively. The symbols cr , cs and c w respectively denote the scale coefficients, which means that the real size of the rail beam element along Z-direction, the track slab element in X- and Ydirection are respectively cr Lr , cs L s and c w L w . To establish the track model, there are mainly three matrices required: the rail matrices, the track slab-subgrade matrices and the interaction matrices between the rail and the track slab. In the next, the matrices will be respectively derived through the elastic strain energy, deformation energy or the work.

(3)

where is the variational symbol; the subscript “ ” indicates that the variation of d aims only at the elastic strain and displacement vectors to guarantee the nature of virtual work principle. Eq. (3) is a corollary of the D′Alembert′s principle when fixing the time t transiently and accordingly, it can be viewed that d is only the function of displacement and strain vector u for an elastic dynamic system. Obviously, for a dynamical system with n independent displacement coordinates ui (i = 1, 2, ...,n ), it can be deduced as

Fig. 1. The versatile track model by FEM (a. side view; b. end view). 3

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L. Xu, et al.

(1) Establishment of the rail matrices by

The variation of the elastic strain energy of the rail can be expressed

with 2

Nr

Urx = l= 1 e= 1 2 Nr

= l= 1 e= 1 2

Nr

Urz = l= 1 e = 1 2 Nr

= l= 1 e = 1

{ {q} { {q}

eT r ·

(E r A r

0

eT r ·

(Er Irz

0

{ {q} { {q}

eT r ·

(Er Iry

0

eT r ·

(krt

cr L r

cr L r

cr L r

cr L r 0

)·{q}re},

[Nrx]T [Nrx] [N

T ry ] [N ry ]d

[Nrz]T [Nrz]d

Ury

)·{q}er },

)·{q}er },

Urt

Ns e=1

Nw

Vm,rx =

\{ q\} l= 1 e = 1

2

Nr

Um,ry = l=1 e=1 2

Nr

Um,rz = l= 1 e = 1

{ \{ q\}

eT r ·

{ \{ q\}

eT r ·

m ¯ + r Wr) Ar cr L r

(m¯r

0

(m¯r

0

cr L r

0

[Nrx]T [Nrx]d

)·\{ q¨\}

)·\{ q¨\}

[Nrz]T [Nrz]d

)·\{ q¨\} er}

Uts =

tsy

Ns e=1

e r

e=1

Nw

{ Ns

Usz = l= 1 e = 1

csL s

{

{q}seT ·

(

0

[N

cw L w 0

T sy ] [N sy ]d

csL s 0

[Nsz]T [Nsz]d d

q es } )·{¨}

eT s ·

(ktsy

{ \{ q\}

eT s ·

csL s

[Nsy]T [Nsy]d

0

(ktsz

cw L w 0

csL s

)·\{ q\} es} )·\{ q\} se}

[Nsz]T [Nsz]d d

0

(10)

Vc,ts = Vc,tsy + Vc,tsz with Ns

Vc,tsy = e=1 Nw

{ \{ q\} Ns

Vc,tsz = l=1 e=1

eT s ·

{ \{ q\}

(ctsy eT s ·

csL s 0

(ctsz

[Nsy]T [Nsy]d cw L w

0

csL s 0

)·\{ q\} se}

[Nsz]T [Nsz]d d

)·\{ q\} es}

where ctsy and ctsz denote the lateral and vertical damping coefficient between the track slab and the subgrade, respectively. (3) Establishment of the interaction matrices between the rail and the track slab The variation of the elastic deformation energy of the rail pad can be expressed by (11)

Urt = Urt,y + Urt,z

)·{q}es }

BTDBd d

csL s 0

where k tsy and k tsz denote the lateral and vertical stiffness coefficient between the track slab and the subgrade respectively. The variation of the work done by the damping forces between the rail and the track slab can be expressed by

e r

with

(

Ns

l=1 e=1

(7)

{q}eT s · Es Isz

cw L w 0

(9)

{ \{ q\}

Utsz =

The track slab-subgrade coupling matrices consist of the track slab matrices derived by the elastic strain energy and the work done by the inertial force and the track slab-subgrade interaction matrices derived by the elastic deformation energy and the work done by the damping forces. The variation of elastic strain energy of the track slab can be expressed by

Ns

(c w m¯sHs L w

+ Utsz

Utsy =

(2) Establishment of the track slab-subgrade coupling matrices

Usy =

q es } )·{¨}

[Nsy]T [Nsy]d

with

¯ r is the rail mass per unit length; Wr is the polar moment of where m inertia; \{ q¨\} er denotes the acceleration vector of the rail element.

Us = Usy + Usz

{ {q}

csL s 0

¯ s is the mass of the track slab per unit volume; \{ q¨\} es is the where m acceleration vector of the track slab. The variation of the elastic deformation energy of plane spring between the rail and the track slab can be expressed by

}

[Nry ]T [Nry ]d

(m¯sHs eT s ·

l=1 e=1

Nw eT ¯r r ·((m

Ns

Vm,sz =

with cr L r

{ {q}

eT s ·

Vm,sy =

(6)

Vm,r = Vm,rx + Vm,ry + Vm,rz + Vm,rt

Nr

(8)

with

where Er is the Young′s modulus of the rail; Ar is the rail sectional area; Irz and Iry is the moment of inertia around Z- and Y-axis, respectively; Nr is the number of the rail beam elements; l = 1, 2 denote the left- and right-side of the rail respectively; the subscript ‘r’ denotes the rail; the superscript ‘T’ denotes the transpose of the matrix or the vector; \{ q\} er denotes the displacement of a rail beam element; k rt is the rail torsional stiffness; Nrx (linear), Nry (nonlinear), Nrz (nonlinear) denote the shape functions of the rail beam element for the motions along X-, Y- and Zaxis respectively; the superscript ‘‘’’ (single quotation) and ‘’’’ (double quotation) denote the 1st and 2nd derivatives with respect to load coordinate respectively. The variation of the negative value of the work done by the inertial force can be expressed by

2

1 µ 0 Es Hs3 µ 1 0 , 2 12(1 µ ) 0 0 (1 µ )/2

,D=

Vm,s = Vm,sy + Vm,sz

)·{q}er },

[N rx]T [N rx]d

T 2N sz

2N sz 2

where Es is the Young′s modulus of the track slab; Isz is the moment of inertia of the track slab around Z-axis; is the height of the track slab; µ is the Poisson ratio; Ns and Nw denote the number of the track slab element along the X - and Y -direction respectively; \{ q\} es is the assemblage of the displacement vector of the Ns and Nw track slab elements; Nsy is the shape function of the track slab in Y-direction; Nsz is the shape function of the track slab in Z-direction, as shown in Appendix A. The variation of the work done by the inertial force of the track slab can be expressed by

(5)

Ur = Urx + Ury + Urz + Urt

2N sz 2

B=

with 2

)·{q}es }

Nr

Urt,y = l= 1 e = 1

4

\{ q\}

eT rt ·(krty

( [Nry] = rl

( [Nry ] = rl ar [Nrx] =

ar [Nrx] = rl

rl

[Nsy]

[Nsy] = sl ) )·\{

= sl q\} ert

)T

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L. Xu, et al.

with Vc,rt,y =

Urt,z 2

\{ q\}

Nr

l=1 e=1

( [Nrz] = rl

(

1) l + 1b

cr L r ([cr (e

1)]),

rl

= L r (e

1)

sl

= L wl

( 1) lb

\{ q\}

e rt

1

= {\{ q\}

( 1) l+ 1br [Nrx] =

rl

e r

c w L w [(L wl

1

r [Nrx ] = rl

sl

= L r (e

[Nsz] =

rl

[Nsz] = 1)

sl, = sl

sl, = sl

)T

) )·\{ q\}

cs Ls ([cs (e

e rt

1)]),

( 1) lb)/(c w L w )]

\{ q\} es}

where [·] denotes an operator rounding the reality in it to the nearest integer towards zero; ar denotes the vertical distance between the rail neural axis and the rail bottom; br denotes the half of the width of the rail bottom; b is the lateral distance between the left-side of the track slab and the contact point to the rail pad; e denotes the elemental number; k rty and k rtz denote the lateral and vertical stiffness coefficient of the rail pad, respectively. The variation of the work done by the damping force of the rail pad can be expressed by

Vc,rt = Vc,rt,y + Vc,rt,z

Urt,y , Vc,rt,z =

crtz krtz

Urt,z ,where crty and crtz denote the

lateral and vertical damping coefficient of the rail pad, respectively. The establishment of the interaction matrices between the rail and the track slab is a demonstration of the versatile modelling approach. Because the nodes of the spring-dashpot element of the rail pad have not directly connected to the elemental nodes of the rail and the track slab, as shown in Fig. 1, the parameters, e.g., rl , sl , sl , Urt,y and Urt,z , are therefore critical in characterizing the interactions between the rail subsystem and the track slab subsystem. Summarily the mass matrix of the track is assembled by variation of the work by the inertial force, that is,

eT rt ·

(krtz ( [Nrz] =

=

crty krty

Vr + Vs = \{ q\} tTMtt \{ q¨\}

(13)

t

The damping matrix of the track is assembled by variation of the work by the damping force, that is,

Vc,rt + Vc,ts = \{ q\} tTC tt \{ q\}

(14)

t

The stiffness matrix of the track is assembled by variation of the elastic strain/deformation energy, that is,

Ur + Us + Uts + Urt = \{ q\} tTKtt \{ q\}

(12)

Fig. 2. Three-dimensional vehicle-slab track coupled model (a. side view; b. end view). 5

t

(15)

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L. Xu, et al.

3. Modelling of a vehicle-track interaction model

all time-dependent.

In the vehicle-track interaction model, as shown in Fig. 2, the vehicle is modelled as a multi-rigid-body system consisting of a car body, two bogie frame, four wheelset and the two-stage suspension systems. The coupling of the vehicle and the tracks is mainly characterized by the wheel-rail interactions. Without loss of generality, the dynamic equations of motion for vehicle-track interactions can be expressed by matrix form, that is,

3.2.2. Vehicle-track coupling mass matrices The coupling mass matrices of Vehicle-track system can be ex¯ vt , with order of n vr × n vr , n vr = 86 . M ¯ vt can be derived by pressed by M the work done by the inertial force of the wheelset, that is,

¨v X ¨t X

Mvv Mvt Mtv Mtt

+

C vv C vt Ctv C tt

Xv Xt

+

K vv K vt Ktv Ktt

Xv Xt

=

4 j=1

Nm, j =

where M, C , K and F respectively denote the mass, damping, stiffness matrix and the load vector; the subscript ‘v ’ and ‘t ’ denote the vehicle ¨ denote the vectors of and the track subsystem respectively; X , X and X displacement, velocity and acceleration, respectively. In Eq. (16), the dynamic matrices of the vehicle and the wheel-rail interactions should be further compiled except for the track sub-matrices presented in Section 2.

1 2

1 2

1 2

1 2

|

m, j

},

m, j

= [ L v, j Rv, j Lz , j Rz , j ],

3.2.3. Vehicle-track coupling stiffness matrices ¯ vt,p The stiffness matrices regarding the Vehicle-track interactions K can be derived by the elastic deformation energy of the primary suspension, that is,

The matrices for characterizing the interactions of vehicle components have been elaborated in literatures [28,36], which can be consulted for references, here not presented for brevity.

4

2

T k zpNk,j,u Nk,j,u ){q}vt

Uvt,p = {q}vt ( with

Nk, j, u = { [1

The method for coupling the interaction between the vehicle and track has already been depicted in [28]. Here a brief presentation about the displacement sub-vector, and the stiffness, damping, mass sub-matrices of the vehicle-track interaction and the force sub-vector will be elaborated as follows.

k , j, u

1/2

1/2 ( 1)uW1y

1/2

= [ zfb | rb L v, j Lz , j Rv, j Rz , j

fb | rb

4

(17)

2

with

4

z w, j

k , j, u}

w, j ]

T k gY Ngy,j,u Ngy,j,u){q}vt

j=1

(20)

with

[ xk,j yk,j zk,j

k,j

k

T k,j ]

Ngx , j, u = { [( 1)u + 1lH , j, i 1

k =c,fb,rb,wj j = 1

=

4

k gX NTgx,j,u Ngx,j,u +

VF,vt,g = {q}vt ( j=1 u =1

4

fb

Lt 0 0]|

is derived by the work done by the vehicle gravity, that is,

V r

1/2

whereu = 1, 2 respectively denotes the left- and right-side wheel/rail contact by symbol ‘l ’ and ‘r ’; k zp denotes the vertical stiffness coefficient of the primary suspension; lH , j denotes the half of the lateral distance between the left- and right-side wheel-rail contact points; W1y denotes the half of the lateral distance between primary suspensions in a bogie frame. ¯ vt,g The gravity stiffness derived by the wheel tread characteristics K

3.2.1. Displacement vector The displacement vector is built up upon the wheel-rail contact positions, in which the track irregularities are also regarded as virtual displacement vectors for modelling convenience, that is,

=

(19)

j=1 u =1

3.2. The vehicle-track interaction matrices

r,j

{

where m w, j is the mass of the jth wheelset; Iwx, j is the moment of inertia around X-axis; the subscript ‘m’ denotes the shape function and displacement vector related to the construction of mass matrices; N·, j is the shape function (cubic Hermitian interpolation); is the assemblage of degrees of freedom (DOFs) with respect to the interaction matrices.

3.1. The matrices of the vehicle

v

(18)

with

Fv Ft (16)

{q} Vr =

T ¯ vt {q¨} Vr (mw,jNm,j Nm,j)){¨} q Vr = {q} Vr M

Vm,vt = {q} Vr (

gx , j

[ L v,j R v,j L z,j R z,j Lh,j Rh,j L y,j R y,j L n,j Rn,j L x,j Rx,j ]T

=[

w, j

Ngy, j, u = {[1

j=1

where the superscript ‘T’ denotes transpose of a vector or a matrix; the subscript ‘k ’ denotes different vehicle bodies, e.g., car body, bogie frame and wheelset; ‘ j ’ denotes the j th wheelset; L v and Rv denote the vertical irregularity at the left- and right-side rail, respectively; Lh and Rh denote the lateral irregularities at the left- and right-side rail, respectively; Lz and Rz denote the rail vertical displacement at the leftand right-side, respectively; L y and Ry denote the rail lateral displacement at the left- and right-side, respectively; L x and Rx denote the rail longitudinal displacement at the left and right side respectively. The above parameters are corresponding to j th wheel/rail contact position of the i th vehicle. It should be noted that there is no direct DOF of r,j with respect to the dynamic matrices in Eq. (16), where L v, j , Rv, j , Lh, j and Rh, j represent the track irregularities at the j th wheel-rail contact point, and L x, j , Rx, j , L y, j , Ry, j , Lz, j and Rz , j are the interpolation of the nodal DOFs. They are

gy, j

x w, j L x , j ]

u=1

1/2

1]| ,

1/2

gx , j

gx , j}

=[

1/2

w, j

x w, j Rx , j ]

1/2]|

u=2

;

gy, j }

= [ yw, j L y, j Ry, j Lh, j Rh, j ]

where kgX is the equivalent angular stiffness of wheel/rail contact; kgY is the equivalent stiffness of gravity [36]. ¯ vt can be therefore The stiffness matrix of the train-rail interactions K obtained by

¯ vt {q}vt Uvt,p + Vvt,g = {q}vt K

(21)

3.2.4. Vehicle-track coupling damping matrices ¯ vt , with order The Vehicle-track coupling damping matrices C n vr × n vr , derived by assemblage of the variation of the work done by the damping force of the primary suspension Vc,vt,p and the variation of the work done by the wheel/rail creep forces Vc,vt,g , that is, Vc,vt,p can be expressed by 6

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2

VF,W + VF,I = {q}vt F¯vt

T czp Nk,j,i,u Nk,j,i,u){q}vt

Vc,vt,p = {q}vt (

(22)

j=1 u=1

where Vc,vt,p has almost the same expression as Uvt,p , only needing to replaced k zp by czp . Vc,vt,g can be expressed by 4

2

j,i,u j,i,u T T (f11 /V· Ncx,j,u Ncx,j,u + f 22 /V· Ncy,j,u Ncy,j,u)){q}vt

Vc,vt,g = {q}vt ( j=1 u=1

(23) with

Ncx, i, j, u = { [1 ( 1)u + 1lH , j, i = [ x w, j

cx , j

Ncy, j, u = [1 cy, j

1]|

L x , j ]u = 1 ,

w, j

1

cx , j

cx , j }

= [ x w, j

w, j

Rx , j ]u = 2 ,

3.3. Infinite length calculation

1 lV , j ]

= [ yw, j Lh, j L y, j Ln, j ]u = 1 ,

cy, j

To a vehicle-track dynamic system, it is a necessity to be informed of the system performance at a large scale, especially in high speed scenario. But in a numerical model built by finite element method, it will lower the computational consumption and occupy too much computer memory if the finite element model has high DOFs. Xu and Zhai [49] have developed a cyclic calculation method (CCM) to solve this problem, which can be introduced and further modified in this work. In CCM, two coordinate systems are defined:

= [ yw, j Rh, j Ry, j Rn, j ]u = 2 ,

where and denote the longitudinal and lateral creep coefficient at the wheel/rail interfaces, which can be derived by the wheelrail 3-D coupling model [27]; V is the nominal rolling speed; lV , j, i is the vertical distance between the wheel/rail contact point and the rail centroid. Finally the vehicle-track coupling damping matrix can be expressed by

f11j, i, u

f22j, i, u

¯ vt {q}vt Vc,vt,p + Vc,vt,g = {q}vt C

• Global coordinate system, denoted by X O Y , which represent the actual coordinates the vehicle running on the track. • Local coordinate system, X O Y , which represents the com-

(24)

puter coordinates the dynamic equations of motion solved in the numerical integrations.

3.2.5. Vehicle-track load vectors The vehicle-track load vector F¯vt , with order 1 × nVr , is derived by the work done by the axle load VF,W and the work done by the structural geometrical excitation VF,I . VF,W can be obtained by

VF,W = {q}vt

Wj 2

Ng,j

No matter what kind of approaches applied, the ultimate goal is to achieve the solution of the dynamic equations, i.e., Eq. (16), in X O Y without track boundary constraint. Simultaneously, the system responses, i.e., displacement, velocity and acceleration, must follow the corresponding realities in global coordinate X O Y , and guaranteeing the continuity of solutions towards the running time. Because the tracks are established by finite element method, the serial number of track elements where the wheel and the rail contacts actually indicate the location of the vehicle wheel on the track in both coordinates of X O Y and X O Y . Since the nodal responses applied in the numerical integral scheme follow the realities in X O Y , while in the cyclic procedures, the matrix representations in dynamic equation are intercepted from the track model in X O Y coordinate. Thus the key work in CCM is building the mapping relationship of the track elemental number in these two coordinates. Fig. 3 shows a graphic illustration of CCM, in which ls denotes the pre-stationary distance due to the falling of vehicle due to gravity before running into the X O Y coordinate system, lccm denotes the length for circulatory calculation, and l 0 denotes the length of track truncated from the front of the 1st wheelset and the rear of the 4th wheelset for promoting computational efficiency. lc and lt respectively

(25)

with

Ng , j = { [1 1]|

g,j},

g,j

= [ Lz , j Rz , j ],

where Wj is the axle load. can be obtained by

¨ vt,j + C vt,jX vt,j + Kvt,jX vt,j) NI,j VF,I = {q}vt (Mvt,jX

(26)

with

NI , j = { [1 1 1 1]|

I , j },

I ,j

(27)

¯ vt , K ¯ vt and the ¯ vt , C Finally the vehicle-track interaction matrices M load vector F¯vt will be partitioned into the dynamic matrices in Eq. (4) according to the “set-in-right-position” rule stated in Section 2.1. The equations of motion for vehicle-track interactions have now been completely constructed. Because the displacement complementarity and force equilibrium between the wheel and the rail are satisfied in the coupled Eqs. (18)–(27), the iterative procedures can be therefore avoided. The system responses of the vehicle and the tracks are simultaneously obtained by implicit numerical integration method such as wilson method [47]. The equivalent shape functions with respect to the modified matrices regarding wheel-rail separations are presented in Ref. [48].

= [ Lz , j Rz, j L y, j Ry, j ],

¨ vt, j , X vt, j and X vt, j respectively denote the acceleration, velocity where X and displacement vector of track irregularities, which correspond to the realities of L v, j , Rv, j , Lh, j and Rh, j . The detail formula for obtaining wheelrail normal force has been shown in Appendix B. Thus the vehicle-track load vector F¯vt can be assembly obtained by

Fig. 3. Illustration of cyclic calculation method (CCM) (the vehicle remarked by red denotes the vehicle at the original position, and the black ones denote the moving vehicle at different positions). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 7

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represent the half-length between centres of front and rear of bogie frames and half-length of axle bases in one bogie frame. When the vehicle moves over ls length in X O Y coordinates, the cyclic procedures are put into practice in X O Y coordinates. By preliminary calculation, it is obtained that nccm and nl are corresponding to the elemental number with respect to lengths of lccm and l 0 respectively. If the elemental numbers of the rail at the 1st and 4th wheel-rail contact positions in X O Y coordinates are n1 and n4 , O Y coordinates can be carried out then the serial numbers in X by

ns = n1 + nl

([lccm /l ccm]

1) n ccm

(28)

n e = n4

([lccm / lccm]

1) nccm

(29)

nl

4.1. Example 1. Comparisons with general solutions In this example, the validity of the proposed model will be revealed by comparing with other models or solutions. 4.1.1. Comparison with a track model constructed by track segment element The conventional methods to build track model have been shown in Refs. [36,40–41], in which the track components between two adjacent rail pads are combined as one track segment element using finite element method, and then the whole track structures are established by these longitudinally united track segment elements. However, the track model in this study is established in a more versatile way where the two main track components, i.e., the rail and the track slab, can be arbitrarily combined with different elemental size. It is assumed that the vehicle runs with a constant velocity of 300 km per hour on the slab tracks, and the lateral and vertical track irregularity excitations at the left- and right-side are set to be the same, as shown in Fig. 4. The time step size t is 0.002 sec, unless otherwise stated. The time step size is determined by the sampling interval of track irregularity, denoted by s , and the speed of the moving vehicle, denoted by V , and then the time step size can be chosen by satisfying: t s / V , in which s = 0.25m, V = 83.33m. Fig. 5 plots the vertical displacements of the rail and the track slab respectively derived by the versatile track model and the conventional track segment element model. It should be noted the rail beam element and track-slab plate element have the same longitudinal length in these two models. It can be observed from Fig. 5 that results obtained by this versatile track model coincide well with conventional track segment element model absolutely, which are almost the same in response amplitudes and distributions. It therefore indicates that these two track model construction methods have the same capability in characterizing the track structural interactions, while with the same element size.

where [·] indicates the operator for rounding towards zero direction, lccm denotes the distance the vehicle has moved over ls . And the rail serial number at front of the 1st wheel set and rear of the 4th wheel set in X O Y coordinates are respectively expressed by

ns = n1 + nl

(30)

ne = n4

(31)

nl

Eqs. (30) and (31) are key equations for cyclic calculation, through which a railway vehicle can be guaranteed to run circularly within a finite length lccm + 2l 0 + 2(lc + lt ) . Obviously the rail DOFs in local and global coordinates are respectively represented by

Dof Xr O Y = Nr (n e

1) + 1: Nr (ns + 1)

Dof Xr O Y = Nr (ne

1) + 1: Nr (ns + 1)

(32)

where Nr is the half number of DOFs of the left- and right-side rail beam element; the symbol ‘:’ denotes increments from the integer at its leftside to the integer at the right-side with an interval of 1. Correspondingly the track-slab DOFs in local and global coordinates with respect to the rail DOFs are respectively represented by

Dof Xt O Y = N¯r + Nt (ne,t Dof t = N¯ r + Nt (ne,t XOY

4.1.2. Comparison with a vehicle-track coupled model

1) + 1: Nt (ns,t + 1) 1) + 1: Nt (ns,t + 1)

(1) Vehicle-track system responses (33)

Zhai et al. [26] developed a 3-D nonlinear vehicle-track coupled model, and established a theory of vehicle-track coupled dynamics [50]. In their work, a series of advanced work has been condensed in the vehicle-track coupled model, e.g., Hertz nonlinear contact theory, Kalker’s wheel-rail contact theory and nonlinear modifications [51,52]. The dynamic interactions between the vehicle and the tracks are coupled by the wheel-rail forces and predicted by an explicit integration method [53]. Fig. 6 shows the result comparisons of dynamic indices, i.e., wheelrail forces and car body accelerations, between this model and the model of Zhai et al. [26]. It can be observed from Fig. 6 that the system responses derived by this model coincide well with those derived by the model of Zhai et al. in response curves and amplitudes, however, it should be noticed that there inevitably exist differences between the results of these two model

where N¯r is the total number of rail DOFs in the track finite element model; N¯ r is the total number of rail DOFs pre-allocated in the response vector; Nt is the half number of DOFs of a track-slab element; ns,t and ne,t are respectively the track-slab elemental number regarding the locations that are at the front and the rear of the 1st wheelset and the 4th wheelset with a length of l 0 in X O Y , while ns,t and ne,t respectively denote the elemental number in X O Y against ns,t and ne,t . 4. Numerical examples Three examples are presented to illustrate the accurateness and application of this model in evaluating the dynamic behaviours of vehicle-track systems. The main parameters of the vehicle and the track have been listed in Appendix C.

Fig. 4. Track irregularity excitation (a. vertical irregularity of the left- and right- side rail; b. lateral irregularity of the left- and right- side rail). 8

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Fig. 5. Comparisons on track vertical displacement (a. rail vertical displacement; b. track-slab vertical displacement).

due to the absolutely different strategies on characterizing wheel-rail normal interactions, construction and solution methods for dynamic equations of motion, etc. Moreover wheel-rail separations, where the wheel-rail forces become zero, can be also characterized by this model. Through amplifying the irregularity amplitudes, wheel-rail separations can be triggered accordingly. Fig. 7 plots the comparisons on wheel-rail vertical forces between this model and the well-known nonlinear Hertzian contact theory in which wheel-rail separations are represented by wheel-rail vertical forces. It can be clearly observed from Fig. 7 that the wheel-rail separations mainly occur at locations with short wavelength irregularities. The response curve of wheel-rail vertical forces of this model agrees well with that derived by Hertzian contact theory, which indicates that this model is capable of investigating the wheel jumping phenomena. But one should notice that the differences on force amplitudes are also significant, where the wheel-rail vertical force derived by this model is generally smaller than that derived by Hertzian theory. The reason for this difference is originated from the differences of system excitation. In the Hertzian contact model, the wheel-rail force is induced by the wheel-rail elastic compression, namely it mainly relates to the displacement of track irregularities. While the excitation of this model, as shown in Eq. (26), include not only the elastic force induced by the displacement of track irregularities, but also the damping and inertia force induced by track irregularities’ first- and second-derivative of time respectively. Therefore this model requires smoothness of high-

Fig. 7. Characterization of wheel-rail separations.

Fig. 6. Comparisons between this model and vehicle-track coupled model by Zhai et al. (a. wheel-rail lateral force; b. wheel-rail vertical force; c. rail lateral displacement; d. vertical acceleration of the car body). 9

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order derivatives of track irregularities. In Fig. 8, the power spectral densities (PSD) of wheel-rail vertical force derived by this model and Hertzian contact theory are presented. One can see from Fig. 8 that the PSDs of vertical forces derived by these two models coincide well with each other at frequencies below 80 Hz. However there exist obvious deviations. Based on sampling theorem, it is known that the minimum effective wavelength of track irregularities is 0.5 m, thus the effective frequency is below 166.67 Hz. The frequency responses above 166.67 Hz are actually not caused by track irregularities, which are regarded as “false frequency”. In numerical simulation, track irregularities are discretely sampled with specific interval. Generally the sampling interval is larger than the time-domain integration interval, and accordingly, interpolations are required. In the interpolations, virtual track irregularities will be introduced, which will inevitably bring unreliable excitation, especially unsmooth time derivatives. The influence of the interpolations on vertical forces can be seen from Fig. 8 that the PSD derived by this model is relatively high at frequencies above 166.67 Hz comparing to those of Hertzian contact model, due to the unstable high-order derivatives. In the frequency range of 80 Hz–166.67 Hz, the PSD derived by the Hertzian contact theory is relatively higher than those of this model. Summarily the comparisons shown in Fig. 8 indicate that this model shows very different sensitivity to track irregularity excitations at the middle-high frequencies. Thus we should pay special attention on wheel-rail interactions when applying different wheel-rail contact hypothesis.

pressure and the range of the contact area. Besides the wheel-rail contact points can be also figured out in this model. As presented in Figs. 12 and 13, the trace of the wheel-rail contact points has been recorded at the rail profiles of the left-side and right-side. Comparing the contact traces of this model and the force equilibrium model, one can clearly observe that they coincide well with each other in distribution. From Fig. 9(b) and (c), it can be cognised that these two kinds of models are solved with totally different mode, thus the deviations are inevitable. From above comparisons, it can be seen that this model has been provided with the capability of describing wheel-rail 3-D contacts in force and geometry. 4.2. Example 2. Dynamic influence of the size of track finite elements In this example, a trail will be conducted to clarify the dynamic influence of the element size of the rail and the track slab on system performance. The element size is determined by the original element size L and its scale coefficient c , namely the elemental size in the track model is L /c . Three cases represented by C1, C2 and C3 respectively indicate the cases of altering the width of the track slab element (L w = 2.4m ), the longitudinal length of the track slab element (Ls = 0.63m ) and the rail (Lr = 0.63m ). The subscripts ‘w’, ‘s’ and ‘r’ correspond to the notations about the width size of the track slab, the longitudinal length size of the track slab and the rail respectively. In each case, only the elemental size is changed. Fig. 14 shows the influence of elemental sizes on displacement responses of the tracks, from which one can notice that the elemental sizes of track structures have significant effects on track vibrations. Comparing to conventional element with the same length, where track sections between two adjacent rail pads are contained in one track segment element and the track slab has not been segmented in width, namely c = 1 for all elements, one can noticed that the rail lateral displacement is generally unchanged when c is larger than one; while for the rail vertical displacement, c should be 3 for reaching a completely steady state, but comparing to c = 1, only a sight deviation of 0.23%, 0.97% and 0.22% existed for cases of C1, C2 and C3 , respectively. This is obviously allowable in engineering practices. Similarly, it can be seen from Fig. 14(d) that the track-slab lateral displacement is little influenced by the size of the track-slab element 1/2 . However this elemental size width with scale coefficient c w shows significant influence on the vertical displacement of the track slab, as shown in Fig. 14(c), it is better to set cr 4 , otherwise a maximum deviation of 12.02% might exist against c w = 1. Besides, the elemental size of the rail and the track slab in length, i.e., cr and cs should also be technically investigated. To obtain more accurate track-slab vertical vibrations, cr 4 and cs 4 should be satisfied accordingly, otherwise there are deviations of 2.19% and 5.72% when comparing to the results of c = 1. As to the track slab lateral vibrations, cr and cs might be better to be larger than 3, or the deviations will reach 6.57% and 14.91% respectively. Fig. 15 illustrates the car body accelerations with respect to different cases, from which it can be observed that the track elemental size

(2) Wheel-rail contact from aspects of pressure and geometry Apart from assessing the dynamic performance of the vehicle and the tracks, this present model has also compiled the advanced wheelrail coupling model in [27] and the semi-Hertzian method in [54,55] into the program. Accordingly this model also possesses the capability of quantifying the wheel-rail contact geometry and pressure. As illustrated in Fig. 9, the wheel and the rail are contacted at a 3-D space. The rail and the wheelset both possess motions of linear displacement along X-, Y- and Z-axis and angular displacement around X-, Y- and Z-axis. Besides, the profiles of the rail and the wheel are both considered. In this model, the wheel-rail vertical contact is depicted by displacement complementarity conditions, and the wheel-rail lateral contact is equivalently characterized by the time-dependent stiffness and damping coefficients, as shown in Fig. 9(b). Some other work, such as the models in [26,28], deploys force equilibrium method, as shown in Fig. 9(c). To illustrate the accuracy of this model in estimating the wheel-rail contact pressure, Figs. 10 and 11 respectively show the average contact pressure at the left- and right-side wheel profile. As observed in Fig. 10, the maximum contact pressures at the left-side of the wheel are respectively 0.94 GPa and 0.97 GPa for this model and the force equilibrium model in [28], that is, the relative error is about 3.2% between these two models. For this model, the contact patch is within the range of [−0.763, −0.743] m and [−0.0061, 0.0062] m respectively for the Y-coordinate of the wheel profile and the longitudinal contact range. For the force equilibrium model, the contact patch is within the range of [−0.762, −0.742] m and [−0.0081, 0.0073] m respectively for the Y-coordinate of the wheel profile and the longitudinal contact range. As seen from Fig. 11, the maximum contact pressures at the rightside wheel profile are respectively 0.92 GPa and 0.89 GPa with a deviation of 3.37%. Besides the contact patch is within the range of [0.736, 0.764] m towards the Y –coordinate of the wheel profile for the force equilibrium model, and within the range of [0.735, 0.765] m for this model. In the longitudinal contact range, the amplitude range is respectively [−0.009, 0.0091] m and [−0.011, 0.009] m for this model and the force equilibrium model. Obviously, this model has obtained approachable results comparing to those derived by a model of force equilibrium method, on both the amplitude of the contact

Fig. 8. PSD of wheel-rail vertical force. 10

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Fig. 9. General view of wheel-rail contacts (a. wheel-rail contact model; b. wheel-rail rigid contact; c. force equilibrium model).

Fig. 10. The average contact pressure at the left-side of the wheel (a. force equilibrium model in [28]; b. this model). 11

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Fig. 11. The average contact pressure at the right-side of the wheel (a. force equilibrium model in [28]; b. this model).

Fig. 12. Time-dependent contact points at the rail profile for the force equilibrium model (a. left-side of the rail; b. right-side of the rail).

Fig. 13. Time-dependent contact points at the rail profile for this model (a. rail left-side; b. rail right-side). 12

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Fig. 15. Influence of elemental size on car body acceleration (a. vertical acceleration; b. lateral acceleration).

4.3. Example 3. Advancements of this model In this example, we will further show the advancements and deficiencies of this model to make a clearer presentation about this developed model. 4.3.1. Infinite length calculation In Section 2, the new versatile method (in this paper) has been successfully used to construct a track model with a total length of 100 m. To further validate the advantages of this versatile model, a

Fig. 14. Influence of elemental size on maximum track vibration (a. rail vertical displacement; b. rail lateral displacement; c. track-slab vertical displacement at the centre; d. track-slab lateral displacement at the centre).

can be actually roughly chosen as c 1/2 , especially for the car body vertical acceleration. To the car body lateral acceleration, the maximum deviation is just 1.02% if c 1/2 . Fig. 16 further plots results of wheel-rail interaction forces. It can be observed from Fig. 16 the wheel-rail lateral force is more sensitive to the elemental size of the tracks comparing to the wheel-rail vertical force. However, if c 1, the fluctuation of the wheel-rail forces is actually very small. Though the wheel-rail lateral force seems to be enlarged when c w and cr are larger than 3, the maximum fluctuation is smaller than 0.2 kN, which is rather small.

Fig. 16. Influence of elemental size on wheel-rail force (a. wheel-rail lateral force; b. wheel-rail vertical force). 13

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computational length of 4.16 km is conducted to analyse the vehicletrack interactions, where the vehicle runs on the tracks to simulate relative motions. As a comparison to CCM, fix-point excitation (FEP), as shown in Fig. 17, can be adopted to achieve the long-length calculation in [11,12]. In FEP method, the vehicle is set to be static, but the track irregularities are regarded as an irregular strip pulled with a backward vehicle speed. FEP method is incapable of considering the relative motions between the vehicle and the tracks. A comparison between the CCM and FEP is implemented. Fig. 18 plots the dynamic results of car body accelerations derived from these two methods. From Fig. 18, the results derived by these two methods coincide well with each other. For further comparing the relative errors of these two results, the Root-Mean-Square (RMS) of the dynamic results is listed in Table 1. Obviously there exist slight and inevitable deviations between the RMS of these two methods due to the totally different modelling construction mechanisms in the wheel-rail coupling analysis, the track model and the integration method, etc. 4.3.2. Adjustable time step size without iterative procedures Another advance of this dynamic model is its computational stability in large time step without iterative procedures required, which will greatly promote the computational efficiency. In this example, the time steps, 0.002 sec, 0.001 sec and 0.0005 sec are respectively selected in the numerical computation. It is known that different time steps mean different frequency of analysis, the larger of the time step, the lower of the maximum frequency. In the numerical integration of this model, the selected time steps can all get stable dynamic results. To elaborate the practicability of this model in characterizing the frequency characteristics of responses with respect to various time steps, Fig. 19 shows the power spectral density (PSD) comparisons on wheel-rail forces and car body accelerations, from which it can be observed that the PSD values at the overlapping frequency parts of different time steps approach to each other, indicating that the dynamic responses will not change its amplitude-frequency characteristics even if the time steps are changed. From the view point of frequency analysis, it can be concluded that the time step in this model can be selected according to the characteristic frequency of dynamic indices cared.

Fig. 18. Comparisons on car body acceleration between CCM and FPE (a. car body lateral acceleration; b. car body vertical acceleration). Table 1 RMS of dynamic indices. Dynamic indices

Car body lateral acceleration (g) Car body vertical acceleration (g) Wheel-rail vertical force (kN) Wheel-rail lateral force (kN)

5. Conclusions In this paper, a fully matrix-represented model for characterizing vehicle-track interactions is proposed by a simple energy variational principle. This model is based on rough hypothesis of wheel-rail noelastic-compression contacts, and it shows significant advancements in computational stability, as time-varying characteristics can be guaranteed without iterative procedures. Besides, improvements on versatile track construction and infinite length calculation are also achieved in this work. Through numerical examples, the engineering practicability of the proposed model is validated, and other conclusions can be drawn accordingly,

RMS

Absolute error

CCM

FPE

0.081 0.059 71.05 1.98

0.079 0.062 69.12 1.82

2.47% 5.08% 2.79% 8.72%

rail pads in length; the track slab should also be segmented into three elements in width. (2) The model built by a wheel-rail no-elastic-compression hypothesis can obtain approachable results in response curves compared to the 3-D nonlinear dynamic model with consideration of wheel-rail elastic compressions by Hertzian theory. But there exist amplitude differences on response amplitudes, especially in wheel-rail interaction forces. The wheel-rail normal forces derived by a no-elasticcompression model are generally smaller than those derived by an elastic-compression model. (3) One of the most remarkable deficiencies in no-elastic-compression models, i.e., characterization of wheel-rail separation, has been properly solved in this model by comparing to the Hertzian nonlinear contacts.

(1) The track elemental size shows significant influence on vehicletrack dynamic behaviours, especially for the track vibrations. In this study, the rail and the track-slab elemental size should be smaller than the three times of the longitudinal distance between adjacent

Fig. 17. Fix-point-excitation (FEP) method. 14

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Fig. 19. PSD comparisons on dynamic indices regarding various time steps (a. car body lateral acceleration; b. wheel-rail lateral force; c. car body vertical acceleration; d. wheel-rail vertical force).

(4) The integral time step is adjustable according to the characteristic frequency interested in this model.

2017YFB1201204). Declaration of Competing Interest

Ethical statement

The authors declare that there is no conflict of interest.

This article was conducted according to ethical standards.

Acknowledgement

Funding body

This work was supported by the National Natural Science Foundation of China (No.: 51820105014; 51708558; 51578549; 51678576); the National Key R&D Program of China (No.: 2017YFB1201204)

This work was supported by the National Natural Science Foundation of China (No.: 51820105014; 51708558; 51578549; 51678576); the National Key R&D Program of China (No.: Appendix A

Here a “Lateral Finite Strip and Slab Segment” method [51] is applied to describe the shape function of the track slab. For arbitrary point o within the track slab element, as shown in Fig. A.1, its vertical displacement can be expressed as

Nsz = NV

(A.1)

ij

with

NVo = [1 - 3(s yo/l)2 + 2(syo/l)3 - (syo/l - 2(s yo/l)2 + (syo/l)3) l 3(s yo/l)2 ij

= [W,i

Xi ,

W,j

T Yj ] ,

2(s yo/l)3 ((s yo/l)3

(syo/l) 2) l]

l = cl L s ,

where Wi and Xi denotes vertical displacement and angular around X -axis for point i , and Wj and o; l is the length of a track slab element along Y -axis. Meanwhile, the displacement or angular in ij can be further expressed as

Wi = NVi [Wa,

Ya ,

Wb,

T Yb] ,

Xi

= NTi [

Xa,

T Xb] ,

Wj = NVj [Wc,

Yc ,

Wd,

T Yd ] ,

Xj

= NTj [

Xc ,

Xj

for point j ; syo is the local Y -coordinate of point

T Xd ] ,

with

NVi = [1 - 3(sxi/w ) 2 + 2(sxi/w )3 - (sxi/w - 2(sxi/w )2 + (sxi /w )3) w 3(sxi/w ) 2

2(sxi/w )3 ((sxi/w )3

(sxi /w )2) w]

NTi = [1 - sxi/w sxi /w]

Fig. A1. Top view of the track slab element. 15

(A.2)

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NVj = [1 - 3(sxj /w ) 2 + 2(sxj /w )3 - (sxj /w - 2(sxj /w )2 + (sxj /w )3) w 3(sxj/w ) 2

2(sxj /w )3 ((sxj /w )3

(sxj /w )2) w]

NTj = [1 - sxj/w sxj /w]

w = c w Ws where sxi , sxj are respectively the local X -coordinate for points i and j ; w is the width of TSE along X -coordinate; Y is the angular displacement around Y -axis. Accordingly, by combining Eqs. (A.1) and (A.2), the interpolation function for vertical displacement of o can be written by

Nv,o (sxo, s yo ) = [NV (1) NV , NV (2) NT , NV (3) NV , NV (4) NT]sxi = sxj= sxo

(A.3)

Appendix B The formula of wheel-rail normal force for the j th wheelset at the th side ( = 1, left-side of the wheelset; be expressed by

Nj, = Fw, j, /cos(

j,

+ ( 1)

+1

= 2 , right side of the wheelset) can

w, j )

(B.1)

with [56]

Fw, j, = F , j, + Fz , j, + FG, j, F , j, = ( 1) mb y¨b, j 4b0

Fz , j, =

FG, j, =

+1 [

(

+ ( 1)

(Htw + Rw ) +

mb z¨b, j

mc z¨c 8

W , 2

mc y¨c 8b0

4

(j ) =

¨ (j ) + 1 Icz c 8lc b0

Ibx ¨ b, j 4b0

+ ( 1)

) (H

cb

+ ( 1) j + 1

¨

I (j) + 1 cy c 8lc

+ Hbt + Htw + Rw ) +

Ibz ¨ b, j Htw + Rw 2lt 2b0

+ ( 1) j + 1

+

Icx ¨c 8b0

+

Iwx ¨w ] 2b0

Iby ¨b, j

m wj z¨wj

4lt

2

1 j = 1,2 , 2 j = 3,4

where F , Fz and FG respectively denote the wheel-rail interaction force due to the vehicle rolling, bounce and force of gravity; when j = 1,2 , y¨b, j , ¨b, j , ¨ b, j and ¨ respectively denote the transverse, yaw, rolling and pitch acceleration of the front bogie frame, and j = 3,4 denote the responses of rear bogie b, j

frame; the subscripts ‘c’ and ‘w’ denote the car body and the wheelset respectively; b0 denotes the half of lateral distance between the wheel-rail left- and right-side contact points; j, is the wheel contact angle at the side of the j th wheelset; Rw denotes the wheel rolling radius; Hcb denotes the vertical distance between the centre of the car body and the upper plane of the secondary suspension; Hbt denotes the vertical distance between the centroid of a bogie frame and the bottom plane; Htw is the vertical distance between the centroid of a bogie frame and the wheel centre; W is the static axle weight. Appendix C (See Tables 2 and 3).

Table 2 Main parameters of the railway vehicles used in the simulation. Notation

Parameter

Value

mc mt mw Icx Icy Icz Itx Ity Itz Iwx Iwy Iwz kpy kzp ksy ksz Czp csy csz lc lt R0

Car body mass (kg) Bogie mass (kg) Wheelset mass (kg) Mass moment of the inertia of the car body about the X-axis (kg·m2) Mass moment of the inertia of the car body about the Y-axis (kg·m2) Mass moment of the inertia of the car body about the Z-axis (kg·m2) Mass moment of the inertia of the bogie about the X-axis (kg·m2) Mass moment of the inertia of the bogie about the Y-axis (kg·m2) Mass moment of the inertia of the bogie about the Z-axis (kg·m2) Mass moment of the inertia of the wheelset about the X-axis (kg·m2) Mass moment of the inertia of the wheelset about the Y-axis (kg·m2) Mass moment of the inertia of the wheelset about the Z-axis (kg·m2) Stiffness coefficient of the primary suspension along the Y-axis (MN/m) Stiffness coefficient of the primary suspension along the Z-axis (MN/m) Stiffness coefficient of the secondary suspension along Y-axis (MN/m) Stiffness coefficient of the secondary suspension along Z-axis (MN/m) Damping coefficient of the primary suspension along Z-axis (kN·s/m) Damping coefficient of the secondary suspension along Y-axis (kN·s/m) Damping coefficient of the secondary suspension along Z-axis (kN·s/m) Semi-longitudinal distance between bogies (m) Semi-longitudinal distance between wheelsets in a bogie (m) Wheel radius (m)

46,280 3300 1780 149,970 2,267,765 2,139,900 2673 1807 3300 949 118 967 6.47 1.176. 0.167 0.323 9.8 39.2 9.8 8.75 1.25 0.43

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Table 3 Main parameters of the railway tracks used in the simulation. Notation Er Ar m ¯r Iry Irz

m ¯s Es Isz µ Hs Lw ktsy ktsz

krty krtz

Parameter

Value 2

Elastic modulus of the rail (N/m ) Rail cross-sectional area (m2) Mass of the rail per unit length (kg/m) Second moment of inertia of the rail around Y-axis (m4) Second moment of inertia of the rail around Z-axis (m4) Mass of the track slab per unit volume (kg/m3) Elastic modulus of the track slab (N/m2) Second moment of inertia of the track slab around Zaxis (m4) Poisson ratio of the track slab Thickness of the track slab (m) Width of the track slab (m) The lateral stiffness coefficient between the track slab and the subgrade (N/m) The vertical stiffness coefficient between the track slab and the subgrade (N/m) The lateral stiffness coefficient of the rail pad (N/m) The vertical stiffness coefficient of the rail pad (N/m)

2.059 × 1011 0.0077 60.64 3.215 × 10−5 5.24 × 10-6 4623 3.6 × 1010 0.004 0.25 0.19 2.4 1 × 108 6 × 107 3.5 × 107 4 × 107

Appendix D. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engstruct.2019.110067.

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