1

Railway Dynamics with Flexible Tracks

Extended Abstract

João Pedro Marçal Neves Costa Instituto Superior Técnico – Universidade de Lisboa

Abstract

When studying the dynamic behaviour of the railway vehicle, a meticulous description of the vehicle is

made using a multibody system formulation whilst the track is modelled, generally, as a rigid structure

comprising only the rails. The problem with railway dynamic analyses, from a mechanical point of view, is

that railway track exhibits in fact some flexibility. This fact has implications on the vehicle performance

and should not be disregarded. When studying the dynamic behaviour of the track the vehicle loading is

generally taken as a constant moving load. However, the loading from the vehicles is neither constant neither

with a vertical component only.

In response to the necessity of more realistic railway models a finite element generation tool for tracks

is presented here. The tracks are built using two sets of inputs: a MATLAB structure, containing the spatial

geometry of the track and an excel file, containing the material and geometric properties necessary for the

finite element methodology. For rigid tracks, used in conventional railway dynamic analysis, the

geometrical description is enough to generate the complete track geometry. For flexible tracks, used in

conventional railway track dynamics, the geometric properties are enough to describe straight tracks. The

innovation of this work is the ability to model general flexible railway tracks for railway dynamics. A case

study is presented, with the purpose to compare the dynamic behaviour of the track with two loading

scenarios. The first loading scenario considers vertical prescribed moving loads, as a standard procedure,

while the second scenario considers realistic vehicle loads extracted from the wheel-rail contact mechanics

using a multibody system simulation. It is concluded that, for realistic tracks, the multibody wheel-rail

interaction loading should always be used instead of the unrealistic prescribed loading, being the issue of

major importance in the presence of curves.

Keywords

Railway Dynamic, Track Dynamics, Finite Element Methodology, Multibody System, Track Modelling.

1 Introduction

When considering the vehicle-track interaction problem, there are two approaches for railway dynamic

analysis. The dynamic analysis of railway vehicles is usually made considering the track as a rigid structure.

The track is composed only by the rails, which may include deviations from the nominal geometry known

as imperfections. This model is widely used in analysis for technological developments and yields good

results. However, this simplification is not realistic, for specific types of analysis, because the track exhibits,

in fact, some flexibility. The flexibility of the track is not only due to the rail being a deformable structure,

but also because of the underlying structure, comprising the rail pads, sleepers and foundation. On the other

hand, the approach used in the dynamic analysis of the track, is to model the whole track, considering all

the elements and their respective flexibilities. As the vehicle is typically not included in this type of analysis,

prescribed moving loads are used to represent the vehicle loading. Although this approach is not as realistic

as it can be with current knowledge, it is representative and used as a research standard. Recent

developments using these two approaches are overviewed here.

The dynamic analysis of a multibody system involves the study of its motion and forces transmitted

over a time period, as functions of the initial conditions, external forces and/or prescribed motions [1]. A

multibody system consists in a set of rigid and/or deformable components interconnected by joints and/or

force elements. The joints, also known as kinematic constraints, define the relative motions between the

bodies. The force elements can be organised in two types: passive force elements, such as springs and

2

dampers, and active force elements, like actuators. Generally, a multibody model of a railway vehicle

comprises one carbody, two bogie frames, four wheelsets and one axle box for each wheel. Both the primary

suspensions, connecting the wheelsets to the bogie frames, and the secondary suspensions connecting the

bogie frames to the carbody are modelled by sets of force elements while the motion between the pairs

wheelsets/axle boxes and carbody/bogies is imposed by kinematic constraints [2].

There are several works related with the track dynamic analysis, especially in the field of civil

engineering. Zhai presents in his work [3] a two-dimensional numerical model that simulates the vertical

dynamic interaction between the vehicle and the track. It consists on a model where the carbody and bogies

are considered lumped masses and the four wheelset are unsprung masses. The wheels are connected to the

bogie, which in turn is connected to the carbody, by springs and dampers representing the primary and

secondary suspensions, respectively. The rail is modelled as an infinitely long beam discretely supported at

rail/sleepers junctions by a series of springs, dampers and masses representing the elasticity of the rail pads,

ballast and subgrade. The interaction between the wheelsets and rail is accomplished by using non-linear

Hertzian theory. More recently, Varandas [4] has developed a more complex model for two dimensional

analysis of the railway track. In this model the rail is assumed to follow the Euler-Bernoulli beam theory on

a Winkler foundation, the stiffness of the sleeper is considered non-linear and increases with the increasing

deformation of the sleeper contacting the ballast, the sleepers may be hanging allowing contact to be lost/re-

established during dynamic loading. In the work of Almeida [5], a methodology to find the equivalent

properties between a foundation modelled as three-dimensional solid finite elements and a foundation using

two-dimensional spring/dampers finite elements is presented.

Regarding the use of multibody formulation and finite element modelling Shabana proposes a non-

linear finite element formulation for modelling the rail and track flexibility [6, 7]. This method can be used

to systemically couple finite element computer programs with flexible multibody system codes, allowing

the development of track models that include rail, tie, and fastener flexibility as well as soil characteristics.

More recently, Pombo and Almeida proposed a tool [8, 9] to automatically generate straight tracks, based

on the FEM, that includes rails, rail pads, sleepers and foundation as flexible elements.

The objective of this work is to develop a tool to build FEM models of railway tracks, with arbitrary

geometry, that can be used with realistic railway vehicle models. The FEM model includes the rail, with or

without imperfections, and a simplification of the underlying structure of the track. This new approach

builds on the dynamic analysis of railway vehicles, by borrowing elements from the dynamic analysis of

railway tracks, allowing more realistic railway structural design and track degradation evaluation.

2 Railway Track Geometry and Structure

The railway track is composed by the rails, rail pads, sleepers and foundation, as outlined in Figure 1. The

rails are supported, through the rail pads, by the sleepers, which in turn are supported by the foundation.

The term foundation is used to refer to the ballast and substructure. The ballast is formed by a layer of coarse

granular material while the substructure is a prepared earthworks comprising the sub-ballast, the form layer

and the subgrade. Since the performance of railway vehicles is strongly dependent on the track conditions

an accurate characterization of the track is essential. The nominal railway track model is composed by

several segments that define the profile of the track centreline. Each segment is characterised by: length,

horizontal curvature, cant and grade. While curving, railway vehicles are subjected to centrifugal forces,

acting away from the centre of the curve, tending to overturn the vehicle. The sum of the vehicle weight and

centrifugal forces produces a resultant force directed toward the outer rail. To counteract this force the outer

rail is raised [10-12]. The height difference between the two rails is called cant, ht. The track grade is the

vertical profile of the track and represents the ascending or descending slope of the track. Changes in

curvature, cant and grade are accomplished by transition segments in order to guarantee continuity of the

variables without geometric singularities.

3

Figure 1: Railway track

For modelling purposes this description of the track is insufficient and a geometric reconstruction of

the track is necessary. The goal of the geometric reconstruction is to transform the values of curvature and

grade into points with coordinates xyz, along the track centreline, and associate to each of these points the

respective cant angle. For the plane reconstruction of the track centreline the method presented by Iverson

[13] is used. The reconstruction of the left and right rail is achieved using the method proposed by Pombo

[1].

3 Finite Element Railway Track Modelling

The track model is achieved by using linear finite elements. The rails and sleepers are modelled using

3D beam elements, based on Euler-Bernoulli theory, with six DOF per node, allowing shear/axial forces

and bending/torsion moments. For the 3D Euler-Bernoulli beam proportional damping is used [14].

Damping assumes a crucial role in the structural dynamics of the track due to the implications of the rail

dynamic response on the wheel-track interaction forces. The rail pads and the foundation are modelled as

massless spring-damper elements that connect to the nodes of the finite elements representing rails, sleepers

and foundations. The foundation is modelled considering two types of massless spring-dampers, below

sleepers and between sleepers, as depicted in Figure 2 and Figure 3.

Figure 2: Cross-section view of the track model

Figure 3: Longitudinal view of the track model

The vertical foundation, depicted in Figure 2, is represented by the vertical elements below de sleepers

and accounts for the flexibility of the foundations directly below the sleeper. The horizontal foundation,

depicted in Figure 3, is represented by the horizontal elements connecting the sleepers and accounts for the

fact that the track supporting layers should behave as continuum. A displacement of a sleeper’s node should

also cause a displacement field in the vicinity of that node, and consequently the surrounding sleepers. To

Flexibility of

the Foundation

Flexibility of

Rail Pad

Sleeper

Elements

Rail

Element

Fixed

FixedFlexibility of the

Foundation

Flexibility of Rail

PadSleeper

Element

Rail

Elements

Flexibility of the

foundation between

sleepers

4

avoid the elastic wave reflection in the end of the rail track, massless spring-damper elements are added in

the beginning and in the end of the track, and constrained in the opposite ends.

To model correctly a 3D realistic track a considerable amount of information is required to define the

position of the nodes, and consequently, the elements. The spatial geometry of the track centreline and rails

can be fully characterized by the arc length, s, the Cartesian coordinates, (x,y,z), and the moving frame,

(t,n,b) of each of these components, as shown in Figure 4. The information necessary for the centreline and

both rails is stored in a MATLAB structure in order to be used in the mesh generation tool. Aside from the

spatial geometry of the track, the geometric and material properties of the track components, i.e., sleepers,

rails, pads and foundation, are also necessary to complete de data for the FEM formulation. This information

is provided in a pre-formatted Excel file.

Figure 4: Parametric representation of a railway track with rails

The whole track is generated based in track centreline arc length. The left and right rail positions and

moving frames’ are interpolated over the track centreline, leading to the data structure specified in Table 1.

With Table 1, and the refinement level of the mesh and the distance between sleepers, provided by the Excel

input file, the computational tool assesses the track arc length placing the nodes in the correct position. To

achieve this positioning there are two options, if the arc length corresponds to a sleeper, nodes for the rails,

sleepers and foundations are created, if the arc length does not correspond to a sleeper only rail nodes are

created.

sc xc tc nc bc xLr tLr nLr bLr xRr tRr nRr bRr

0.0 … … … … … … … … … … … …

1.0 … … … … … … … … … … … …

… … … … … … … … … … … … …

Table 1: Geometric data necessary to automatically generate the mesh

Besides the rail coordinates, connectivity and properties tables necessary for a FEM formulation, the

mesh generation tool builds an auxiliary table for the rail-wheel contact force model used in the dynamic

analysis. This table, shown as Table 2: Table for wheel-rail contact in the dynamic analysis, comprises, for

each rail, the element numbers, ne, nodes i and j of the element, and the arc length of nodes i and j, si and sj,

respectively.

ne i j si sj

0.0 … … … …

1.0 … … … …

… … … … …

Table 2: Table for wheel-rail contact in the dynamic analysis

Rn

Rt Rb

LL RL

z

y

x

Ln

Lt

Lb

Lg

Rg

5

A mesh generated using this tool, serving only as a demonstrative example of the computational tool,

is shown in Figure 2. The track consist in two straights segments of 100 m, connected by a semi-circular

segment with a 50 m radius.

Figure 5: Example of a mesh generated using the tool

4 Case Study with Realistic Operation Scenarios

4.1 Vehicle Multibody Model

Generally, a multibody model of a railway vehicle comprises one carbody, two bogie frames, four

wheelsets and one axle box for each wheel, as shown in Figure 6. Both the primary suspensions, connecting

the wheelsets to the bogie frames, and the secondary suspensions connecting the bogie frames to the carbody

are modelled by a set of springs and dampers placed in parallel, as depicted in Figure 6.The constraints used

between the pairs wheelset/axle box and carbody/bogie are revolute joints and cylindrical joints with

bushing, respectively. In this work, the vehicle multibody models have their dynamics analysed for given

railway tracks. The interaction forces are retrieved and used, later on, as loading forces for the flexible track.

Figure 6: General multibody model of a railway vehicle

4.2 Track of a Suburban Railway Line

For the dynamic analysis of the track, two approaches are explored. In the first, the wheel-rail contact forces

are retrieved from DAP-3D dynamic analysis with the railway track to be analysed here. This loading case

referred to as realistic loading. The second approach considers a loading condition where eight prescribed

vertical loads represent the vehicle’s weight moving along the rails, deemed here as prescribed loading. The

wheel-rail contact forces are extracted from the MBS simulation considering the vehicle moving at a

Attaching Points of the:

Carbody

Bogie Frame

Wheelset

Primary Suspension

Secondary SuspensionBogie Frame

Wheelset

2CG

3CG

3

332

2

2

x

y

z

1

1

11CG

Carbody

6

constant velocity of 67.5 km/h. There are two areas of interest depicted as dots in Figure 7. For each dot,

control nodes are placed, in both the left and right rail, between two consecutive sleepers. The vertical and

lateral displacements of the track resulting from the dynamic analysis are presented as well as their velocities

and accelerations. The lateral component of the displacement is defined as the component perpendicular to

both, the rail longitudinal tangent and the vertical normal direction.

Figure 7: Top view of the track

4.3 Track Dynamic Analysis

The vertical displacement of the control rail nodes in the second curve is presented in Figure 8. The vertical

displacement of the left rail is similar for both loading scenarios with the realistic loading being responsible

for a slightly larger deformation upon passage of the last wheelset. For the right rail the shape of the vertical

displacement is similar for the realistic and prescribed loading, with the latter being responsible in general

for larger deformations. The difference between the vertical displacement of the left and right rail, for each

loading case, is due to the cant angle of the curve. In the case of the lateral displacement there is no similarity

between the loading cases for both rails. For the realistic loading, the left rail deforms outward with the first

wheelset causing larger deformations than the first. The behaviour of the deformations is the same for both

bogies. In the case of the right rail node, for each bogie, the deformation caused by the first wheelset is

outward followed by deformation inward upon passage of the second wheelset. The displacement of the

prescribed forces shows the same behaviour for both rails, deforming them in the positive direction of the

axis, i.e., toward the centre of the curve.

(a) (b)

Figure 8: Displacement of the control rail nodes in the 2nd curve: (a) vertical displacement; (b) Lateral

displacement

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

98.50 99.50 100.50 101.50

Dis

pla

ce

me

nt [

10

-5m

]

Time [s]

LR Realistic uz

LR Prescribed uz

RR Realistic uz

RR Prescribed uz

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

98.50 99.50 100.50 101.50

Dis

pla

ce

me

nt [

10

-5m

]

Time [s]

LR Realistic ul

LR Prescribed ul

RR Realistic ul

RR Prescribed ul

7

The vertical velocity and acceleration of the right rail node in the second curve are presented in Figure

9. The shape of the vertical velocities induced by the prescribed loading is in agreement with the velocities

induced by the realistic loading. The velocity profile due to the prescribed loading shows a smooth behavior

while the profile due to the realistic loading presents an oscillatory behavior. The oscillatory behaviour

increases upon passage of the first wheelset and decreases upon the passage of the second.

There is no resemblance between the shapes of the accelerations induced by both loading scenarios.

The acceleration due the realistic loading presents an oscillatory behavior, reaching higher values, while the

acceleration due the prescribed loading presents a smooth behavior.

(a) (b)

Figure 9: Kinematics of the control node in the 2nd curve for the right rail: (a) vertical velocity; (b) vertical

acceleration

The lateral velocity and acceleration of the right rail node in the second curve are presented in Figure

10. There is no similarity between the shape of the lateral velocities induced by the realistic and the

prescribed loading. The realistic loading produces higher velocities and presents an oscillatory behaviour

that increases before the passage of the first wheelset and decreases after the passage of the second wheelset.

Figure 10 (b) shows how poorly the prescribed loading simulates the actual accelerations felt by the

rail. The acceleration due to the prescribed loading presents a smooth behaviour and is almost negligible

when compared with the highly oscillatory behaviour of the accelerations due to the realistic loading.

(a) (b)

Figure 10: Kinematics of the control node in the right rail: (a) lateral velocity; (b) lateral acceleration

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

98.50 99.50 100.50 101.50

Veclo

city [

10

-3m

/s]

Time [s]

Realistic vz

Prescribed vz

-40.00

-30.00

-20.00

-10.00

0.00

10.00

20.00

30.00

40.00

98.50 99.50 100.50 101.50

Accele

ration [

10

-3g]

Time [s]

Realistic az

Prescribed az

-4.00

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

98.50 99.50 100.50 101.50

Velo

city [

10

-3m

/s]

Time [s]

Realistic vl

Prescribed vl-210.00

-160.00

-110.00

-60.00

-10.00

40.00

90.00

140.00

98.50 99.50 100.50 101.50

Accele

ration [

10

-3g]

Time [s]

Realistic al

Prescribed al

8

5 Conclusion and Future Work

The computational finite element model initialization and dynamic analysis tool developed in this work

comes as a response to the demand of more realistic railway track analysis. Track flexibility has implications

in the vehicle performance and should not be disregarded. The development of this tool allows to build any

FEM track model, as long as the geometry and properties are provided. The automatic mesh generation tool

allows creating tracks, with different segments and/or properties. For instance, a straight segment with twin-

block sleepers over a ballasted foundation, followed by a curved segment with mono-block sleepers on a

concrete foundation can be built. The development of this tools is also done taking into account that the

FEM models built are to be used in a co-simulation procedure, which is left for future work.

Vertical prescribed forces are used as an industry standard for homologation purposes and structural

design of tracks. Both cases studies presented here demonstrate this to be a conservative approach, leading

to slightly larger displacements and velocities than considering realistic loads, only if the track is considered

straight, which is unrealistic. For curved segments, either with a high or small curvature, the displacement,

velocities and accelerations of the rail nodes, caused by the prescribed loading do not agree with the ones

caused by the realistic loading. While traveling in a straight track, considering that there are no

imperfections, the lateral forces, which are responsible for keeping the vehicle on rails, are not significant

and the standard approach is viable. On the other hand, while negotiating a curve the lateral forces become

more significant and the standard approach is not enough to predict the dynamic behaviour of the track. The

prescribed loading also fails to describe the oscillating behaviour of the velocity and acceleration, especially

present in the curve’s outer rail.

In this work, the dynamic analyses of the vehicle and track are decoupled. The dynamic analysis of

the vehicle is performed, using MBS formulation, and considering only rigid bodies. Then, the dynamic

analysis of the track is performed, using FEM formulation, and considering the dynamic loads extracted

from the MBS simulation. By decoupling the dynamic analyses an important aspect of real phenomena is

left out of the analysis. The flexibility of the track should influence the performance of the vehicle, and vice

versa. To realistically model the interaction between the track and the vehicle a co-simulation procedure

should be developed, allowing the two codes to exchange information. This is not a trivial task since the co-

simulation procedure requires the wheel-rail contact to be evaluated by the finite element dynamics.

For the foundation and rails pads, modelled as massless spring-dampers, a linear behaviour is

considered. However, some authors [3, 4] consider the foundation as non-linear elements. Considering the

foundation springs/dampers as non-linear element would allow studying other interesting cases, for instance

the influence of hanging sleepers on the vehicle’s performance, or stiffness transitions.

Since the method presented in this work is not a standard, for either vehicle or track analyses, there

is little information available about the elastic and damping properties of the foundation. Therefore, some

of the properties of the foundation used in this work are arbitrary. Experimental data about the displacement

and velocity fields would allow to validate the tool, find more accurate values of these properties and

consequently, more reliable results.

The track includes rails that form curves and straight sections which are continuous and smooth.

Modelling the rails with two noded linear beam elements does not allow the correct representation of the

geometric smoothness. Therefore, the use of curved beam elements with three or more nodes must be

explored for the models of the rails, in particular when a co-simulation of the vehicle track interaction is

implemented.

The realistic loading of the track is achieved, in this work, by using the wheel-rail interaction loads

obtained with the vehicle analysis tool. Appropriate filters for these forces, which fulfil the applicable

norms, can be explored for application to the force loading histories.

References

[1] J. Pombo, "A Multibody Methodology for Railway Dynamics Applications," PhD Dissertation,

IDMEC/Department of Mechanical Engineering, Instituto Superior Técnico, Lisbon, Portugal,

2004.

9

[2] H. Magalhães, J. F. A. Madeira, J. Ambrósio, and J. Pombo, "Railway Vehicle Performance

Optimization Using Virtual Homologation," Vehicle System Dynamics, 2015.

[3] W. Zhai and Z. Cai, "Dynamic Interaction Between a Lumped Mass Vehicle and a Discretely

Supported Continuous Rail Track," Computers and Structures, vol. 63, pp. 987-997, 1997.

[4] J. Varandas, P. Hölscher, and M. Silva, "Dynamic Behaviour of Railway Tracks on Transitions

Zones," Computers and Structures, vol. 89, pp. pp. 1468-1479, 2011.

[5] T. Almeida, "Modelação Numérica de Entrada em Ponte Ferroviária," Master Thesis, Universidade

Nova de Lisboa, Lisbon, Portugal, 2015.

[6] A. A. Shabana, R. Chamorro, and C. Rathod, "A Multibody System Approach for Finite Element

Modeling of Rail Flexibility in Railroad Vehicle Applications," Institution of Mechanical

Engineers, Part K: J. Multi-body Dynamics, vol. 222, pp. pp. 1-15, 2008.

[7] A. A. Shabana and G. Sanborn, "An Alternative Simple Multibody System Approach for Modelling

Rail Flexibility in Railroad Vehicle Dynamics," Institution of Mechanical Engineers, Part K: J.

Multi-body Dynamics, vol. 223, pp. pp. 107-120, 2009.

[8] T. Almeida, "Development of Flexible Track Models for Railway Dynamics Applications," Master

Thesis, Department of Mechanical Engineering, Instituto Superior Técnico, Lisbon, Portugal, 2013.

[9] J. Pombo, T. Almeida, H. Magalhães, P. Antunes, and J. Ambrósio, "Development of a Finite

Element Methodology for Flexible Track Models in Railway Dynamics Applications," in 3rd

International Conference on Recent Advances in Railway Engineering (ICRARE-2013), Tehran,

Iran, 2013.

[10] V. K. Garg and R. V. Dukkipati, Dynamics of Railway Vehicle Systems. New York, New York:

Academic Press, 1984.

[11] E. Andersson, M. Berg, and S. Stichel, Rail Vehicle Dynamics, Fundamentals and Guidelines.

Stockholm, Sweden: Royal Institute of Technology (KTH), 1998.

[12] T. Wayne, Roller Coaster Physics - An Educational Guide to Roller Coaster Design and Analysis

for Teachers and Students. Charlottesville, Virginia, 1998.

[13] W. Iverson. (1974, May/June). Analysis of the Reconstruction of Rail Geometry from Curvature

Data. IA-10.

[14] K. J. Bathe, Finite Element Procedures. New Jersey: Prentice-Hall, 1996.