Development of Flexible Track Models for
Railway Dynamics Applications
Tiago Miguel Candeias de Almeida
Thesis to obtain the Master of Science Degree in
Mechanical Engineering
Examination Committee
Chairperson:
Prof. Luís Manuel Varejão de Oliveira Faria
Supervisor:
Prof. Jorge Alberto Cadete Ambrósio
Co-Supervisor:
Prof. João Carlos Elói de Jesus Pombo
External Supervisor:
Prof. Manuel Frederico Oom de Seabra Pereira
October 2013
Acknowledgements
I want to thank Professor Manuel Seabra Pereira for introducing me to the field of
dynamic simulations that spiked my interest in the field, and to presenting me with the chance
to study what became my favourite subject.
To my advisor, Professor Jorge Ambrósio, I thank him for his fair advice and help with
the problems along the way in a fast and effective manner.
My deepest thanks to my co-advisor, Professor João Pombo, who took so much time to
help me develop my work and advising me on how to improve it and this thesis.
To Professor José Varandas, I want to thank for his explanation on the integration
methods and help with the correct modelling of the railway track.
Thanks to Professor Virgínia Infante I was able to progress with the development of the
post-processor and had the chance to apply it to real cases.
For my friends and colleagues, Pedro Antunes and Hugo Magalhães, I thank them for
their support and opinions on my work and how their own work helped in the development of
this thesis.
My final thanks go to all those who intervened in my life, from family to friends and
teachers, for pushing me and supporting me in my decisions which lead me to be able to
produce this thesis.
The work reported here has been developed in the course of several national projects
funded by FCT (Portuguese Foundation for Science and Technology): SMARTRACK
(contract no. PTDC/EME-PME/101419/2008), WEARWHEEL (contract no. PTDC/EMEPME/115491/2009) and the industrial project VOUGA.
i
ii
Resumo
A análise dinâmica de veículos ferroviários envolve três modelos independentes: o modelo do
veículo, o modelo da via e o modelo de contacto roda-carril. Neste trabalho, uma formulação
de multicorpo é utilizada para descrever a estrutura cinemática de corpos rígidos e juntas que
constituem o modelo do veículo. Também é proposta uma metodologia que cria modelos de
vias tridimensionais, que incluem a flexibilidade dos carris e da subestrutura. A metodologia
proposta modela os carris como vigas suportadas discretamente por elementos molaamortecedor que representam a flexibilidade das palmilhas, das travessas, do balastro e da
subestrutura. A inclusão de modelos flexíveis de via é muito importante para o estudo realista
do comportamento dinâmico de veículos ferroviários, especialmente para analisar as
consequências das operações ferroviárias na infraestrutura e os danos nos veículos
provocados pelas condições da via. Este tópico tem um impacto económico significativo na
manutenção de veículos e de vias ferroviárias. A formulação do contato roda-carril aqui
introduzida permite obter, durante a análise dinâmica, a localização dos pontos de contacto,
para qualquer movimento tridimensional. A metodologia proposta para a construção de
modelos de via flexíveis é validado através da comparação dos resultados obtidos com os
obtidos em ANSYS, mostrando que a metodologia proposta é adequada para aplicações
ferroviárias. Neste trabalho é ainda desenvolvido uma ferramenta de pós-processamento para
avaliar se um dado veículo está em conformidade com as normas e regulamentos ferroviários
e permite analisar quantitativamente o desempenho de veículos em diferentes vias tendo em
vista a sua aprovação para serviço.
Palavras-Chave
Dinâmica Ferroviária;
Modelos de Via Férreas Flexíveis;
Operações Ferroviárias Realísticas;
Interação Veículo-Via;
Homologação de Veículos.
iii
Abstract
The dynamic analysis of railway vehicles involves the construction of three independent
models: the vehicle model; the track model; and the wheel-rail contact model. In this work, a
multibody formulation is used to describe the kinematic structure of the rigid bodies and
joints that constitute the vehicle model. A methodology is also proposed in order to create
detailed three-dimensional track models, which include the flexibility of the rails and of the
substructure. This approach uses a finite element methodology to represent the rails as beams
supported in a discrete manner by spring-damper elements that represent the flexibility of the
pads, sleepers, ballast and substructure. The inclusion of flexible track models is very
important to study realistically the dynamic behaviour of railway vehicles, especially the
impact of train operations on the infrastructure and the damages on vehicles provoked by the
track conditions. This topic has a significant economic impact on the vehicles maintenance
and life cycle costs of tracks. The wheel-rail contact formulation proposed here allows
obtaining, online during the dynamic analysis, the contact points location, for any threedimensional motion. The methodology proposed to build flexible track models is validated by
comparing the results with the ones from ANSYS, showing that the proposed methodology is
appropriate to railway applications. In this work a post-processing tool is also developed to
assess if a given vehicle is conform to the norms and regulations in practice and allows
assessing quantitatively the dynamic behaviour of the vehicle in different operation
conditions, being used for vehicle approval.
Keywords
Railway Dynamics;
Flexible Railway Track Models;
Realistic Railway Operations;
Vehicle-Track Interaction;
Vehicle Approval.
iv
Contents
Acknowledgements ..............................................................................................................................i
Resumo ............................................................................................................................................ iii
Palavras-Chave ................................................................................................................................. iii
Abstract ............................................................................................................................................. iv
Keywords .......................................................................................................................................... iv
Contents ............................................................................................................................................. v
List of Figures .................................................................................................................................. vii
List of Tables ...................................................................................................................................... x
1 Introduction ................................................................................................................................... 1
1.1 Motivation ............................................................................................................................ 1
1.2 Literature Review ................................................................................................................. 5
2 Railway Vehicle Models .............................................................................................................. 11
2.1 Railway Vehicles ................................................................................................................ 11
2.2 Description of the Vehicle Multibody Model ....................................................................... 14
3 Development of Advanced Track Models ..................................................................................... 16
3.1 Track Description ............................................................................................................... 16
3.2 The Finite Element Method on the Track System ................................................................ 17
3.2.1
Dynamic Analysis of Railway Tracks Using Linear FEM .................................... 19
3.2.2
Time Integration .................................................................................................. 21
3.3 Automatic Finite Element Method Mesh Generation ........................................................... 26
3.4 Vehicle-Track Interaction ................................................................................................... 28
3.5 Case Studies of the Flexible Track ...................................................................................... 31
3.5.1
Simple Flexible Track and Static Validation ........................................................ 31
3.5.2
Realistic Flexible Track with Moving Loads and Vehicle Forces ......................... 34
4 Definition of the Post-Processing Tool ......................................................................................... 39
4.1 Data to be Measured and Simulated .................................................................................... 40
4.2 Limit Values ....................................................................................................................... 42
4.2.1
Limit Values of Running Safely .......................................................................... 42
4.2.2
Track Loading Limit Values ................................................................................ 46
4.3 Experimental Tests ............................................................................................................. 48
4.3.1
Recording the Measuring Signals ........................................................................ 49
4.3.2
Processing the Measuring Signals ........................................................................ 49
5 Post-Processor Application .......................................................................................................... 57
5.1 Case Study 1 ....................................................................................................................... 57
5.1.1
Measured Raw Data ............................................................................................ 57
5.1.2
Filtered Data ....................................................................................................... 58
v
5.1.3
Classification Method.......................................................................................... 61
5.1.4
Characteristic Values for Track Sections.............................................................. 61
5.1.5
Characteristic Values for Test Zones ................................................................... 66
5.1.6
Discussion ........................................................................................................... 67
5.2 Case Study 2 ....................................................................................................................... 69
5.2.1
Measured Raw Data ............................................................................................ 69
5.2.2
Characteristic Values for Test Zones ................................................................... 69
5.2.3
Discussion ........................................................................................................... 72
5.3 Discussion of the Case Studies ............................................................................................ 74
6 Conclusions and Future Development .......................................................................................... 75
References ........................................................................................................................................ 77
Annex A: Flexible Track Properties .................................................................................................. 83
Annex B: Communication between Multibody and FE Codes ........................................................... 87
Annex C: Case Study Properties........................................................................................................ 88
vi
List of Figures
Figure 1.1: Schematic representation of the methodology used for the dynamic analysis and postprocessing of railway systems ......................................................................................... 3
Figure 2.1: Generic multibody system ............................................................................................... 11
Figure 2.2: Rigid frame vehicle with carbody suspended on two wheelsets ........................................ 12
Figure 2.3: Bogie vehicle with two-axle bogies ................................................................................. 13
Figure 2.4: Relative rigid body motions of a carbody ........................................................................ 14
Figure 2.5: Schematic representation of the Alfa Pendular trainset..................................................... 14
Figure 2.6: Alfa Pendular multibody model ....................................................................................... 15
Figure 2.7: Subsystems of multibody model: (a) Track and Infrastructure; (b) Carbody; .................... 15
Figure 3.1: Main components of the railway track (Longitudinal view) ............................................. 17
Figure 3.2: Main components of the railway track (Cross-section view) ............................................ 17
Figure 3.3: Main components of the track model (Cross-section view) .............................................. 18
Figure 3.4: Main components of the track model (Longitudinal view) ............................................... 18
Figure 3.5: Schematic representation of the Pre-Processing Tool ....................................................... 19
Figure 3.6: Stability of Newmark’s parameters γ and ζ ...................................................................... 22
Figure 3.7: Newmark’s (a) Explicit and (b) Implicit Time Integration Methods ................................. 25
Figure 3.8: Representation of (a) the curvature of a straight track and (b) the finite element mesh of a
straight track using the Pre-Processing Tool .................................................................. 27
Figure 3.9: Representation of (a) the curvature of a realistic curved track and (b) the finite element
mesh of a realistic track using the Pre-Processing Tool .................................................. 28
Figure 3.10: Representation of (a) the transversal view of a rail element and contact forces and (b) the
location of a contact point relative to the rail finite element ........................................... 29
Figure 3.11: Representation of the three cases for the different values of id where (a) 0  id  ij (b)
id  0 and (c) id  ij ................................................................................................ 30
Figure 3.12: Representation of the potential point of contact on the rail element ................................ 31
Figure 3.13: Simple flexible track model: (a) Finite element mesh; (b) External loads applied ........... 32
Figure 3.14: Perspective view of the track deformation (deformation scaled 100): (a) Computational
tool; (b) ANSYS ........................................................................................................... 32
Figure 3.15: Lateral view of the track deformation (deformation scaled 100): (a) Computational tool;
(b) ANSYS ................................................................................................................... 32
Figure 3.16: Relative error for the track vertical deformation. ............................................................ 33
Figure 3.17: Relative error on the nodes in the vicinity of the applied loads: (a) Nodes on the rail; (b)
Nodes on the sleeper ..................................................................................................... 33
vii
Figure 3.18: Realistic Flexible track model ....................................................................................... 34
Figure 3.19: Comparison of (a) the moving load method and (b) realistic vehicle forces .................... 34
Figure 3.20: Comparison of the moving load method (Case 1) and realistic vehicle forces (Case 2) for
the front wheels of the vehicle for (a) the vertical loads and (b) the transversal loads ..... 35
Figure 3.21: Results of the Dynamic Analysis before the first transition curve: (a) vertical deformation
of the rails with moving loads (Case 1) and realistic vehicle forces (Case 2) and (b) their
comparison in percentage .............................................................................................. 35
Figure 3.22: Results of the Dynamic Analysis on the first transition curve: (a) vertical deformation of
the rails with moving loads (Case 1) and realistic vehicle forces (Case 2) and (b) their
comparison in percentage .............................................................................................. 35
Figure 3.23: Results of the Dynamic Analysis on the curve: (a) vertical deformation of the rails with
moving loads (Case 1) and realistic vehicle forces (Case 2) and (b) their comparison in
percentage ..................................................................................................................... 36
Figure 3.24: Results of the Dynamic Analysis on the second transition curve: (a) vertical deformation
of the rails with moving loads (Case 1) and realistic vehicle forces (Case 2) and (b) their
comparison in percentage .............................................................................................. 36
Figure 3.25: Results of the Dynamic Analysis before the first transition curve: (a) transversal
deformation of the rails with moving loads (Case 1) and realistic vehicle forces (Case 2)
and (b) their comparison in percentage .......................................................................... 36
Figure 3.26: Results of the Dynamic Analysis on the first transition curve: (a) transversal deformation
of the rails with moving loads (Case 1) and realistic vehicle forces (Case 2) and (b) their
comparison in percentage .............................................................................................. 37
Figure 3.27: Results of the Dynamic Analysis on the curve: (a) vertical deformation of the rails with
moving loads (Case 1) and realistic vehicle forces (Case 2) and (b) their comparison in
percentage ..................................................................................................................... 37
Figure 3.28: Results of the Dynamic Analysis on the second transition curve: (a) transversal
deformation of the rails with moving loads (Case 1) and realistic vehicle forces (Case 2)
and (b) their comparison in percentage .......................................................................... 37
Figure 4.1: Schematic representation of the Post-Processing Tool...................................................... 39
Figure 4.2: Representation of the Wheel’s Guiding Force (Y) and Wheel Force (Q) ........................... 40
Figure 4.3: Representation of the Sum of Guiding Forces (ΣY) and Vertical Wheel Forces (Q), in (a)
straight line and (b) curved track ................................................................................... 41
Figure 4.4: Representation of a Wheelset .......................................................................................... 41
Figure 4.5: Relative rigid body accelerations ..................................................................................... 41
Figure 4.6: Bode Plot of low-pass Butterworth filters with a cut-off frequency of 1 Hz and variable
order (from 1 to 5) ......................................................................................................... 52
viii
Figure 4.7: Filtered data and some of the windows used in the calculation of the Sliding Mean
(window length: 4 m and step length: 1 m) vs. displacement graph ................................ 53
Figure 4.8: Filtered data and its Sliding Mean (window length: 4 m and step length: 1 m) vs.
displacement graph ....................................................................................................... 53
Figure 4.9: Filtered data and some of the windows used in the calculation of the Sliding RMS (window
length: 4 m and step length: 1 m) vs. displacement graph ............................................... 54
Figure 4.10: Filtered data and its Sliding RMS (window length: 4 m and step length: 1 m) vs.
displacement graph ....................................................................................................... 54
Figure 4.11: Filtered data vs. displacement graph .............................................................................. 55
Figure 4.12: Reordered absolute data vs. position graph .................................................................... 55
Figure 4.13: Cumulative percentile curve graph ................................................................................ 55
Figure 5.1: Raw ÿ+ vs. time graph ..................................................................................................... 58
Figure 5.2: Raw ÿ* vs. time graph...................................................................................................... 58
Figure 5.3: Raw
*
vs. time graph ..................................................................................................... 58
Figure 5.4: Filtered ÿ+S vs. time graph ............................................................................................... 59
Figure 5.5: Filtered ÿ*S vs. time graph................................................................................................ 59
Figure 5.6: Filtered
*
S
vs. time graph ............................................................................................... 60
Figure 5.7: Instability Criterion ÿ+S vs. time graph ............................................................................. 60
Figure 5.8: Filtered ÿ*qst vs. time graph .............................................................................................. 60
Figure 5.9: Filtered ÿ*q vs. time graph................................................................................................ 61
*
Figure 5.10: Filtered
q
vs. time graph ............................................................................................. 61
Figure 5.11: Reordered absolute ÿ+S data vs. position graph for the first section ................................. 62
Figure 5.12: Cumulative curve of ÿ+S vs. position graph for the first section ...................................... 62
Figure 5.13: Characteristic values of ÿ+S for each section of a zone .................................................... 63
Figure 5.14: Characteristic values of ÿ*S for each section of a zone .................................................... 63
Figure 5.15: Characteristic values of
Figure 5.16: RMS
ÿ+S
*
S
for each section of a zone .................................................... 63
vs. time graph ................................................................................................. 64
Figure 5.17: Characteristic values of ÿ*qst for each section of a zone................................................... 64
Figure 5.18: Characteristic values of ÿ*q for each section of a zone .................................................... 65
Figure 5.19: Characteristic values of
*
q
for each section of a zone .................................................... 65
Figure 5.20: Characteristic values of the sÿ*q for each section of a zone ............................................. 65
Figure 5.21: Characteristic values of the s
*
q
for each section of a zone ............................................. 66
Figure 5.22: Raw ÿ+ vs. time graph ................................................................................................... 70
Figure 5.23: Raw ÿ* vs. time graph .................................................................................................... 70
Figure 5.24: Raw
*
vs. time graph.................................................................................................... 70
Figure A.1: Sleeper general geometry ............................................................................................... 85
ix
List of Tables
Table 2.1: Definitions of relative motions.......................................................................................... 13
Table 3.1: Number of elements and their element type used to model each component of the railway
track model ................................................................................................................... 27
Table 4.1: Limit values for Maximum Accelerations in the Vehicle Body ......................................... 45
Table 4.2: Limit values for Track Loading ........................................................................................ 47
Table 4.3: Limit values for Ride Characteristics ................................................................................ 48
Table 4.4: Conditions for the processing of the measuring signals from EN 14363 [11] ..................... 50
Table 4.5: Conditions for the processing of the measuring signals from UIC 518 [12] ....................... 51
Table 5.1: Conditions for the processing of the measuring signals for the Simplified Method from UIC
518 [12] ........................................................................................................................ 57
Table 5.2: Case Study 1 safety approval table.................................................................................... 68
Table 5.3: Case Study 1 ride characteristics approval table ................................................................ 69
Table 5.4: case Study 2 safety approval table .................................................................................... 73
Table 5.5: Case Study 2 ride characteristics approval table ................................................................ 73
Table 5.6: Characteristic Values for the analysed case studies ........................................................... 74
Table A.1: Rail geometry data........................................................................................................... 83
Table A.2: Track segments data ........................................................................................................ 83
Table A.3: Track segment components data ...................................................................................... 83
Table A.4: Rail geometry data........................................................................................................... 84
Table A.5: Sleeper properties data ..................................................................................................... 84
Table A.6: Foundation properties data ............................................................................................... 85
Table A.7: Sleeper geometry data...................................................................................................... 86
Table A.8: Track model constants and output parameters .................................................................. 86
Table C.1: Track segments data for the case study ............................................................................. 88
Table C.2: Rail data for the case study .............................................................................................. 88
Table C.3: Foundation properties for the case study .......................................................................... 88
Table C.4: Sleeper data for the case study ......................................................................................... 89
Table C.5: Sleeper geometry for the case study ................................................................................. 89
x
1 Introduction
1.1 Motivation
The railway system is increasingly becoming a key-player in worldwide transport policies.
This results from the rising oil prices and from the urgency for reduction of CO2 emissions.
For short and medium distances, modern high speed trains are able to compete with air
transportation, having the advantage of presenting better energy efficiency and causing less
pollution. For longer distances the railway system is still the most economical mean for
transportation of goods and starts to have a competitive edge in the transport of passengers.
One of the main disadvantages of railway transport is the high costs of construction and
maintenance, when compared to other means of transportation. Furthermore, the increase of
speed, axle loads and traffic has led to higher-rates of degradation of the ballasted railway
tracks [1,2]. Hence, a considerable effort is necessary for maintenance of the tracks, with a
corresponding increase in costs for the infrastructure managers. The main cause for the
degradation of the track is the deformation and densification of the ballast layer, representing
75% of the total track position maintenance [3-5].
The use of profiled-flanged steel wheels running on steel tracks in order to provide
simultaneously support, guidance and traction was a brilliant concept in the early days of this
industry. Nevertheless, the simplicity of the concept masked the complexity of the contact
phenomenon. In fact, the complex contact forces that develop in the wheel-rail interface
strongly influence the dynamic behaviour of a rail guided vehicle. Also the characteristics of
the vehicle suspensions, the masses and inertias of the structural elements, and the geometry
and irregularities of the tracks play a dominant role in this regard.
Despite the complexity of the physical phenomena involved, the demands of increasing
speeds, better comfort and greater load capacity do not stop increasing in order to improve the
competitiveness and attractiveness of railway networks. Therefore, the increasing demands
for network capacity, either by increasing the traffic speed or the axle loads, put pressure on
the existing infrastructures and the effects of these changes have to be carefully considered.
Such requirements bring new problems to control the wheel-rail wear and to maintain the
vehicle stability and reliability in the different operation conditions.
Future developments are directed towards studies involving the influence of the track
settlement conditions on vehicles performance and analyses associated to railway
infrastructure degradation resulting from trainsets operation. The European Strategic Rail
1
Research Agenda [6] and the European Commission White Paper for Transports [7] have
identified key scientific and technological priorities for rail transport over the next 20 years.
One of the points emphasized is the need to reduce the cost of approval for new vehicles and
infrastructure products with the introduction of virtual certification.
The development of computer resources led simulations to be an essential part of the design
process of railway systems. Furthermore, the use of advanced computational tools during the
design phase of new trains allows carrying out several simulations, under various scenarios, in
order to improve its dynamic performance and reach an optimized design. In this way, studies to
evaluate the impact of design changes or failure mode risks can be performed in a much faster and
less costly way than the physical implementation and test of those changes in real prototypes.
Usually the track imperfections are measured by the infrastructure managers, and these can
be included in the track model when performing the computing simulations. Such feature allows
assessing the consequences of the track conditions on the vehicles performance, namely noise and
vibration. It can then help scheduling the track maintenance procedures by identifying the levels
of track irregularities that promote the increase of wear and/or vehicle-track interaction forces.
Due to their multidisciplinary, all issues involving railway systems are complex.
Therefore, the use of computational tools that represent the state of the art and that are able to
characterise the modern designs and predict the vehicles’ performance by using validated
mathematical models is essential. The use of a valid model of the railway track is necessary to
correctly determine the dynamic behaviour of the railway vehicle, which implies using flexible
track models in order to account for the deformation, wear, sags and maintenance state of the
track. While a complete model for the entire track layer would be desired, it would greatly
increase the computational cost of the simulations. Thus a simplified and computationally
friendly spring-damper model that simulates the track conditions is preferable.
In this work, the dynamic behaviour of the railway vehicle is studied using a multibody
formulation [8-10] where the main structural elements are treated as rigid bodies connected with
flexible links that represent the suspension elements. The relative motions between the bodies of
the system are restrained by using appropriate kinematic constraints. Recent computer codes for
railway applications use specific methodologies that, in general, only allow studying each
particular phenomenon at a time. By analysing such phenomena independently, it is not possible
to capture all the dynamics of the complete railway system and relevant coupling effects.
Developing innovative and more complex methodologies, each requiring different mathematical
formulations and numerical procedures, in a co-simulation environment allows, not only to
integrate all physical phenomena, but also to assess the cross influence between them.
2
Track Model
Vehicle Model
Wheel-Rail
Contact Model
Railway Dynamic
Analysis
Post-Processing Tool
Raw Data
Filtering and
Data Processing
Characteristic
Values
Check Acceptance
Criteria
Assess Dynamic
Performance
Figure 1.1: Schematic representation of the methodology used for the dynamic analysis and
post-processing of railway systems
The main innovation of this work lies in the development of computational tools that are
able to model with detail the vehicle, the track and the subgrade. Instead of using the traditional
approach, in which these systems are handled independently, here they are integrated in a
common and reliable tool, where the interaction among them is considered. Then, the results
from the dynamic analysis are post-processed in order to determine if a given vehicle would be
accepted to operate on the track according to the international norms and regulations [11,12].
The methodology used in this work is represented schematically on Figure 1.1.
The track flexibility is included in the formulation by using finite element models [13,14] to
represent the rails, which are supported by discrete elastic elements, representing the flexibility of
the sleepers, pads, ballast (or slab) and subgrade. Another advantage of this methodology is that it
allows building realistic track models by considering the track irregularities in the formulation [15].
The finite element formulation proposed here to build flexible track models is based on an
analogous formulation used by Ambrósio et al. [16,17] to study the pantograph-catenary interaction.
3
The pre-processor tool that was developed in this work to build the flexible track model
allows dynamic analysis for much longer distances than the traditional approach, since it has a
much lower computational cost due to not using solid 3D finite elements to model the
foundation, instead opting to use discrete spring-dampers. The track model pre-processor and
the numerical implementation of the finite element methodology are validated in this work by
comparing the results with the ones obtained from ANSYS in a static analysis where the track
is loaded with wheelset loads. Also a comparative analysis between the use of moving loads
and realistic vehicle forces is performed in order to analyse the validity of the State of the Art,
where moving loads that represent the vehicle is more often considered.
A generic wheel-rail contact detection formulation [18,19] is introduced here to
determine, online during the dynamic analysis, the contact points location, without need to
use pre-computed lookup tables. This computational efficient methodology uses an elastic
force model that allows computing the normal contact forces in the wheel-rail interface,
accounting for the energy loss during contact [20,21]. The tangential wheel-rail contact forces
can be calculated using one of the creep force models, namely the Kalker linear theory [22],
Heuristic nonlinear method [23] and the Polach formulation [24].
It is important to assess the accuracy and suitability of the proposed methodologies
through the comparison of the dynamic analysis results against those obtained by experimental
testing. For this purpose, a partnership between this research group and the Portuguese railroad
company has been established in order to validate the developed methodologies using real data.
When studying the dynamic performance of railway vehicles, it is also necessary to assess
of the vehicle fulfils the requirements defined by international regulations such as EN 14636
[11] and UIC 518 [12] and to compare the performance of different vehicles in several railway
tracks at different speeds. A computational post-processing tool that handles all filtering and
data analysis as required by regulations was developed in this work, handling the acceptance
criteria that a vehicle must pass in order to operate on a given track at a given velocity.
The methodologies described in this work are meant to be applied in the study of the
dynamic behaviour of the Alfa Pendular railway vehicle, which is operated by the Portuguese
Railway company in the intercity service for passenger transportation in Portugal. It is a
trainset with an active tilting system which allows it to curve at speeds higher than the
balanced speed [25] and keeping the non-compensated acceleration within admissible values
for passenger comfort [26]. But the work presented here is flexible and generic enough,
allowing its use to model any vehicle/track combination, study its dynamic behaviour and
assessing its performance according to the international standards.
4
1.2 Literature Review
The computational simulation of a railway vehicle requires the implementation of a
mathematical model that describes it. From this perspective, the use of multibody dynamics
methodologies is the most flexible approach to create such models [8,10,27-32]. Due to its
simplicity and computational implementation easiness, Cartesian coordinates [8,27,28] are
used. Considering the high level of complexity of these equations, analytical solutions are
impractical to obtain and, therefore, numerical algorithms must be applied.
The numerical solution of the Differential Algebraic Equations (DAE), and their
consequent integration in time, introduces several problems, namely the existence and
uniqueness of solution and the numerical instabilities of the solutions [28]. An alternative
approach for the solution of the equations of motion transforms the set of DAE in a set of
Ordinary Differential Equations (ODE). With such approach, the solution is obtained by
integrating in time the ODE using direct integration algorithms [8,33-40]. In a constrained
multibody system, the equations of motion are solved by appending the constrained acceleration
equations to the formulation [8]. The approach used by Gonçalves and Ambrósio [39,41]
involves the successful use of a sparse matrix solver in rigid and flexible multibody systems. A
sparse matrix solver is also used in this work for the solution of the equations of motion.
Modelling and simulation in the field of railway dynamics is a complex
interdisciplinary topic [25,42-47]. The theoretical basis of the methodologies implemented is
now mature and the programs, originally written by research institutes [9,48-50], have
evolved into powerful, reliable and user friendly packages [51-75]. Shabana et al. [76] use an
analytical track description defined by a three-step procedure. In this approach, a few nodal
points that define the space curves of the left and right rails are obtained and stored in a preprocessor, which is subsequently used by the dynamic simulation code. During the dynamic
analysis, the rail space curves are obtained by means of the absolute nodal coordinate
formulation [10], leading to an isoparametric beam element that can be conveniently used to
describe curved rigid and flexible rails. The method considers each rail as a separate body in
order to account for their relative motion.
Pombo and Ambrósio [77-80] suggest an appropriate methodology for the accurate
description of the track centerline, in the general case of a fully three-dimensional track
geometry, and demonstrate the approach with roller coaster applications. For railway
applications, the geometry of the left and right rails has to be described, which leads Pombo
and Ambrósio to extend the methodology used in roller coaster applications [81,82].
5
The complete characterization of a railway requires not only the description of its
design geometry, but also the description of the track irregularities that arise from its
construction, usage and change on the foundations [25,43,83,84]. In ADAMS/Rail 9.1.1 [54],
the analytical description of the track centerline is completely de-coupled from the description
of the track irregularities. Using this formulation, the track irregularities are given as a
correction of the wheelset position, and the geometrical contact parameters on the left and right
wheels are obtained according to the corrected location. In the track model proposed by Pombo
and Ambrósio [81,82], the left and right rails are considered as separate entities in order to
account for their relative misalignment due to the track irregularities. According to this
approach, the irregularity parameters are expressed with respect to each rail and not with
respect to the track centerline.
The wheel-rail interaction plays a dominant role in the study of the dynamic behaviour of
railway vehicles. The first step for the solution of the wheel-rail contact problem is the
determination of the contact geometry. The second step is related with the contact kinematics and
consists in the calculation of the creepages, or normalized relative velocities, at the point of contact.
The third and last step for the solution of the contact problem is the accurate characterization of the
contact mechanics that consists in the calculation of the wheel-rail contact forces.
Since the wheel and rail have profiled surfaces, the prediction of the contact point location
online during dynamic analysis is not a trivial problem. A common method used in many
existing computer algorithms consists in finding the location of the contact point, and the values
of some related parameters, by interpolating a set of pre-calculated table entries. Despite the fact
that this interpolating method does not represent a rigorous procedure for predicting the location
of the contact points, it is a fast and robust technique that is commonly used in railway
applications [51,54,65]. Many researchers use this approach in their studies [74,75,85,86].
In addition to the approximations due to the interpolation process, the pre-computed
contact table method fails to capture all possible three-dimensional configurations of the
wheelset with respect to the track, and does not account for possible simulation scenarios, such
as the lead and lag contact. Two alternative and conceptually different methods can be found in
the literature for solving the wheel-rail contact problem. The first is the constraint approach
that involves the definition of nonlinear kinematic contact constraints, leading to a model in
which the wheel has five degrees of freedom (DOF) with respect to the rail. The second method
is the elastic approach in which the wheel is assumed to have six DOF with respect to the rail.
One of the initial works using the constraint approach was presented by De Pater [87],
describing of the motion of a rigid wheelset on a pair of straight rigid rails, using four DOF.
6
The method was later implemented in a more general nonlinear way by Fisette and Samin
[71,72], who derived a new multibody wheel-rail contact model assuming a rigid independent
wheel on a rigid straight rail. More recently, Shabana et al. [88-91] implemented the
constraint method using an augmented Lagrangian formulation. Using this methodology, no
simplifications are required in the description of the surfaces geometry or in the kinematics of
the wheels and rails. Moreover, the most general motion of the wheelset can be considered
and the rotation and/or translation of the rails can be easily accounted for.
The method based on the elastic approach is also often used to solve the wheel-rail
contact problem. In this method, the wheel is assumed to have six DOF with respect to the rail
and the normal contact forces are defined in terms of the indentation between the surfaces and
using the Hertz contact theory. One of the initial works in this area was presented by Kik and
Steinborn [92]. In general, the main problem encountered when using the elastic approach is the
accurate determination of the contact points. Difficulties in predicting the location of the contact
points rise when dynamic analyses on curved tracks are considered. The accurate prediction of
the rail arc length travelled by each wheel over the track is, therefore, crucial. Shabana et al.
[90,91] address this problem and propose a methodology for the accurate prediction of the
contact points coordinates when the elastic approach is used. In their formulation, four surface
parameters are used to describe the geometry of the wheel and rail surfaces. In order to be able
to accurately determine the location of the contact points, a first order differential equation is
introduced. The solution gives the system generalized coordinates and velocities as well as the
rail arc length travelled by each wheel. This last parameter defines the rail cross section in
which the point of contact lies. Since the algorithm allows for an arbitrary number of contact
points, the methodology can be used to study multiple contacts between wheel and rail surfaces.
Kik and Piotrowski [93] suggest an approach to calculate the normal contact force between
wheel and rail. They propose a fast approximate method that allows estimating the area of
contact and the normal load for a prescribed penetration between the surfaces. Kik and
Piotrowski [93] show that the methodology predicts the normal load well when compared with
exact theories and with measured data.
Railway manufacturers fit their vehicles with conical wheels whose flanges are
fundamental to avoid derailment. Whenever the wheel moves laterally with respect to the rail, a
second point of contact between the wheel flange and the rail edge may occur. The contact
forces that result from the second point of contact influence the forces at the first contact point
and have a significant effect on the dynamic behaviour of the vehicle. The simulation of the two
points of contact scenario is one of the most difficult problems in the dynamic analysis of
7
railway systems. One of the initial works about the multi-contact problem was presented in
1982 by Piotrowski [94]. The basic assumption proposed is that the distance between the
contact patches is sufficient for the cross-influence of the normal and tangential stresses to be
neglected. Under this assumption, two distinct contact areas are considered for the wheel tread
and wheel flange and the rolling contact formulation can be stated independently. Several other
researchers also use this approach in their two-point wheel-rail contact studies [71,72,90,91,95].
Shabana and Sanborn [96] proposed a method allowing for general modelling of rail
and track flexibility and can be used to systematically couple finite element computer
programs with flexible multibody system codes, allowing for the development of detailed
track models that include rail, tie, and fastener flexibility as well as soil characteristics. In the
work reported by Dahlberg [97] the possibility to smooth out track stiffness variations is
discussed. It is demonstrated that by modifying the stiffness variations along the track, for
example by use of grouting or under-sleeper pads, the variations of the wheel/rail contact
force may be considerably reduced. Zhai et al. [4], established a five-parameter model for
analysis of the ballast vibration based upon the hypothesis that the load-transmission from a
sleeper to the ballast approximately coincides with the cone distribution. The concepts of shear
stiffness and shear damping of the ballast are introduced in the model in order to consider the
continuity of the interlocking ballast granules. A full-scale field experiment is carried out to
measure the ballast acceleration excited by moving trains and the theoretical simulation results
agreed well with the measured data.
The work developed by Ferreira [98] takes part of the development of a global
train/track dynamic model and carried out its validation with real experimental measurements
with the purpose to evaluate the influence of very high speeds in the induced track vibrations.
A quantification of the consequences of using different track design solutions was also
performed. The model was also developed to predict track differential settlements evolution
through the simulation of millions of train passages at high speeds.
Pombo et al. [15] propose a methodology that includes the track imperfections in the
definition of the track model. The methodology described in this work is applied to study the
influence of the track irregularities on the dynamic behaviour of the railway vehicle ML95.
For this purpose, a multibody formulation is used to build the vehicle model and a generic
wheel-rail contact formulation was applied in order to determine the contact points’ location
and the respective normal and tangential forces. The accuracy and suitability of the
methodology presented were demonstrated through the comparison of the dynamic analysis
results against those obtained by experimental testing.
8
In the work developed by Rauter [99], a general computational tool was developed for the
dynamic analysis of the pantograph-catenary interaction in nominal, operational and perturbed
conditions for high speed railway operation. The tool was characterised by a modular structure
with two independent codes running in a co-simulation environment. The pantograph dynamic
behaviour is analysed using a flexible multibody formulation and the catenary is modelled using
finite element method models. The contact force is modelled as a normal force using elastic
contact models with energy dissipation. In this work the multibody module, the contact module
which works as a co-simulation procedure and the pantograph module were developed.
In order to investigate the dynamic derailment of a railway vehicle Xiao et al. [100]
developed a coupled vehicle/track dynamics model, in which the vehicle is modelled as a
multibody system and the track is modelled as a 3-layer discrete elastic support model. Rails were
assumed to be Timoshenko beams supported by discrete sleepers, and the effects of vertical and
lateral motions and rolling of the rail on the wheel/rail creepages were taken into account. The
sleepers were treated as Euler beams on elastic foundation for the vertical vibration, while as
lumped masses in the lateral direction. A moving sleeper support model was developed to
simulate the effect of the periodical discrete sleepers on the vehicle/track interaction. The vehicle
and the track are coupled by wheel/rail contacts whereas the normal forces and the creep forces
are calculated using the Hertzian contact theory and the nonlinear creep theory by Shen et al. [23],
respectively. The equations of motion of the coupled vehicle/track system are solved by means of
an explicit integration method. The numerical results obtained indicate that track misalignment
and vehicle speed have a great influence on the whole vehicle running safely.
Feng [101] studied the influence of design parameters on dynamic response of the
railway track structure by implementing Finite Element Method (FEM). The rails and sleepers
have been modelled by Euler-Bernoulli beam elements, while springs and dashpots have been
used for the simulation of rail pads and the connection between the sleeper and ballast ground.
Dynamic explicit analysis has been used for the simulation of a moving load, and the train
speed effect has been studied. The displacement of the track bed has been evaluated and
compared to the measurement taken in Sweden in the static analysis [101].
The work presented by Antunes [102] refers to a computational tool and a modelling
methodology that handles the dynamics of pantograph-catenary interaction using a threedimensional methodology. To exploit the advantages of using a multibody formulation to model
the pantograph, a high-speed co-simulation procedure was setup in order to allow the
communication between the multibody code and the finite element catenary module. A contact
model, based on a penalty formulation, was selected to represent the interaction between the
9
two modelling procedures. Pombo and Ambrósio [103] present a fully three-dimensional
methodology for the computational analysis of the interaction between catenary and
pantographs. The Finite Element Method (FEM) is used to support the model of the catenary,
while a multibody (MB) dynamics methodology is applied to support the pantograph model.
The contact between the two subsystems is described using a penalty contact formulation. A
high-speed co-simulation procedure is proposed to ensure the communication between the two
methodologies. The numerical results were compared against experimental data and the results
show that the passage of the front pantograph excites the catenary, leading to the deterioration
of the contact conditions on the rear one.
Montalbán et al. [104] use the Finite Element Method to analyse the mechanical
behaviour of different track types by considering the different conditions to which they may be
subjected. The values of the stress and deformation as a function of depth are obtained at each
point of the cross-section of the considered track. These values allow quantification of the basic
design parameters for railway structures. The work developed by Varandas [105] describes the
research undertaken to model the dynamic response of the railway tracks, taking into account
the behaviour of ballast at railway transition zones, where the long-term settlements are
amplified by dynamical loading on the ballast due to the discontinuities. The numerical
simulations showed that the use of soft rail pads on the stiff side of the transition is beneficial,
provided the problem is mostly caused by stiffness variation of the track support. The slab track
solution was also tested and showed advantages over the ballasted track by displaying much
smaller differential rail displacements, for identical change of the track support stiffness.
In the work by Pombo et al. [106] a finite element methodology is used to create detailed
three-dimensional track models, which include the flexibility of the rails and of the substructure.
In the approach, the rails are modelled as beams supported in a discrete manner by springdamper systems that represent the flexibility of the pads, sleepers, ballast and subgrade. A
multibody formulation is used to describe the kinematic structure of the rigid bodies and joints
that constitute the vehicle model. The inclusion of flexible track models is very important to
study the dynamic behaviour of railway vehicles in realistic operation scenarios, especially
when studying the impact of train operations on the infrastructure and, conversely, the damages
on vehicles provoked by the track conditions. The wheel-rail contact formulation used here
allows obtaining, online during the dynamic analysis, the contact points location, even for the
most general three-dimensional motion of the wheelsets with respect to the track. The
methodology proposed to build flexible track models is validated here by comparing the results
obtained with this new approach with the ones obtained with ANSYS.
10
2 Railway Vehicle Models
The dynamic analysis of a multibody system involves the study of its motion and forces
transmitted during a given time period, as a function of the initial conditions, external applied
forces and/or prescribed motions. A multibody system can be defined as a collection of rigid
and/or flexible bodies interconnected by kinematic joints and/or force elements. The kinematic
joints control the relative motion between the bodies, while the force elements represent the
internal forces that develop among bodies due to their relative motion. The external forces may
be applied to the system components as a consequence of their interaction with the surrounding
environment. A generic multibody system is represented in Figure 2.1.
Flexible Body
Spring
Rigid Body
Rigid Body
Da mper
Revolute Joint
Spherica l Joint
Rigid Body
Spherical Joint
Force
Moment
Spring
Revolute Joint
Rigid Body
Revolute Joint
Ground
Figure 2.1: Generic multibody system
The dynamic analysis of a multibody system [8-10,27,30,107] involves the solution of a
system of second order differential equations, possibly mixed with algebraic equations [39,108].
Depending on the type of system modelled and/or on the type of coordinates used, the number of
coordinates may be larger than the number of DOF of the multibody system. Different sets of
coordinates may be chosen to describe the configuration of the bodies at any instant of time [109].
In the following, a multibody methodology is overviewed to present the formulation adopted
in the development of this work. In the formulation presented throughout this work only rigid
bodies are considered for the multibody formulation.
2.1 Railway Vehicles
Rail guided vehicles can be divided into different classes, depending on the field of
application [109]. The most common vehicles for different types of rail traffic systems are:
11
a) Railways: Often with different types of traffic like passenger, freight, suburban, mainline traffic, etc.
b) Subways or metros: Suburban rail systems fully or partly underground, separated from
other rail traffic and from road traffic.
c) Tramways: Suburban rail systems with light-weight vehicles fully or partly operating
in combination with the road traffic.
d) Roller-Coaster: Rail systems for roller-coaster applications.
Rail vehicles are also divided into two categories, depending if they are powered or not:
a) Tractive stock: The vehicles in this class are powered. Locomotives take no passengers
whereas motor coaches do. The vehicles can be powered electrically or by diesel fuel.
b) Rolling stock: The vehicles are not powered and can be divided into coaches, if they
take passengers, or wagons, if they don’t.
Rail guided vehicles consist of two main parts:
a) Running gear: It consists of wheels, axles, and suspension, which include the
components connecting these parts. A wheelset normally consists of two wheels and a
connecting axle. The running gear should support the carbody and guide, brake and,
for a tractive unit, drive the vehicle.
b) Carbody: This part of the vehicle carries the payload, i.e. passengers or goods, and/or
the traction equipment.
Depending on the running gear characteristics, there are two vehicle types:
a) Rigid frame vehicles: The running gear only consists of wheelsets and suspension
components, as the vehicle schematically shown in Figure 2.2.
b) Bogie vehicles: The running gear is a so called bogie. It consists of wheelsets, a
framework and suspension elements, as shown in Figure 2.3.
Wheelset
Carbody
Figure 2.2: Rigid frame vehicle with carbody suspended on two wheelsets
Rigid frame vehicles are the simplest, most inexpensive and lighter ones. However these
vehicles have limited payload capacity and its length is also restricted by the need to have an
acceptable curving performance. From the dynamic point of view, the rigid frame vehicles give
12
a rather shaky and uncomfortable ride since they have only one suspension level. A horizontally
stiffer wheelset-carbody connection increases the so-called critical speed, but gives a worse
curving performance [43]. In summary, rigid frame vehicles are most appropriate for rather
unqualified transports, for instance, light weight freight traffic with speeds up to 100–120 Km/h.
Bogie
frame
Carbody
Wheelset
Figure 2.3: Bogie vehicle with two-axle bogies
The bogie vehicles have two levels of suspension, the primary suspension, between the
wheelsets and the bogie frame, and the secondary suspension, between the bogie frame and
the carbody. Though the bogies increase the vehicle weight and costs, they provide isolation
for the high frequency contents of the motion due to the inertia of bogie frames. This
assemblage has also a geometric advantage since disturbances acting on one wheelset are, in
principle, halved at the bogie frame longitudinal midpoint, decreasing their transmission to
the carbody. The vehicles assembled with bogies have better curving performance and the
derailment risk is lower than for rigid frame vehicles. The carbody vibrations and the wheelrail contact forces are also reduced as a result of the two levels of suspension [43].
In rail-guided vehicle dynamics, the motion of the vehicle as a whole and the motion of
the particular vehicle parts are very important to quantify. In Table 2.1 the motions
corresponding to the six relative degrees of freedom of the rigid bodies that compose a rail
vehicle are defined and are represented in Figure 2.4. An example would be a vehicle composed
by a carbody that is supported by two bogies through a set of mechanical elements that constitute
the secondary suspension. The bogies are the subsystems that, through the wheelsets, are in contact
with the track and include another group of mechanical elements that constitute the primary
suspension. Further detail on this topic is outside the scope of this thesis; the interested reader is
referred to see the work developed by Pombo [109].
Relative motions
Translation in direction of travel
Translation in transverse direction, parallel to the track plane
Translation perpendicular to the track plane
Rotation about longitudinal axis
Rotation about a transverse axis, parallel to the track plane
Rotation about an axis perpendicular to the track plane
Symbol
Table 2.1: Definitions of relative motions
13
x
y
z



Notation
Longitudinal
Lateral
Vertical
Roll, Sway
Pitch
Yaw
z

y

x

Figure 2.4: Relative rigid body motions of a carbody
2.2 Description of the Vehicle Multibody Model
In this section, the Alfa Pendular trainset, a railway vehicle used for passenger transportation
in Portugal, is described. It is a trainset with an active tilting system which allows it to
negotiate curves at higher speeds, maintaining the passengers comfort within admissible
values [110]. This trainset is composed of six vehicles, being four motor units and two
trailers, as shown in Figure 2.5. In the following, all mechanical elements that are relevant to
build the multibody model, namely the structural and the suspension elements, are described.
Motor
Motor
Trailer
Trailer
Motor
Motor
Figure 2.5: Schematic representation of the Alfa Pendular trainset
Due to the trainset configuration, it is assumed that the dynamic behaviour of each vehicle
has a non-significant influence on the others and, therefore, each vehicle of the trainset can be
studied independently. In this way, the vehicle model introduced here is composed only by one
trailer unit of the trainset. It should be noted that the methodology described is generic and can
be applied to any railway vehicle. The Alfa Pendular trailer vehicle is composed by one
carbody, where the passengers travel. It is supported by two bogies through a set of mechanical
elements that constitute the secondary suspension. The main function of these elements is to
minimize the vibrations, resulting from the vehicle-track interaction, transmitted to the
passenger compartment, improving the comfort and reducing the problems associated to the
structural fatigue. Each bogie includes the wheelsets, which are in contact with the rails, and
another group of mechanical elements that constitute the primary suspension. These elements
are responsible mainly for the steering capabilities and stability behaviour of the whole group
and, ultimately, being responsible for the critical speed of the vehicle.
14
Figure 2.6: Alfa Pendular multibody model
The first step for modelling a railway vehicle using a multibody formulation is the division of
the group in several subsystems, which are simpler to handle. This strategy allows building each
subsystem independently, being the whole vehicle model built by assembling the subsystems. The
subsystems considered here to model the Alfa Pendular vehicle are shown in Figure 2.6.
The subsystem 0 is used to represent the track and the infrastructure, as shown in Figure
2.7 (a). The subsystem 1, depicted in Figure 2.7 (b), represents the carbody of the vehicle. The
subsystems 2 and 3, shown in Figure 2.7 (c), represent the front and the rear bogies. As these
last two are equal, it is only necessary to build one subsystem representing the bogie. Then,
when assembling the railway vehicle, this subsystem is used twice to represent both the front
and the rear bogies. The subsystem 1 is connected to subsystems 2 and 3 by attaching
elements, which represent the secondary suspension and the bogie-carbody connection
elements. The interaction between the rails (from subsystem 0) and the wheels (from
subsystems 2 and 3) is performed by using an appropriate wheel-rail contact model [18,19].
(a)
(b)
(c)
Figure 2.7: Subsystems of multibody model: (a) Track and Infrastructure; (b) Carbody;
(c) Front and Rear Bogies
For each subsystem it is necessary to provide the information about the rigid bodies,
kinematic joints and linear and/or nonlinear force elements. The relative motion between the
bodies is limited by kinematic joints [8], which restrain relative degrees-of-freedom between
the bodies connected by them. The suspension components, such as springs and dampers that
connect the rigid bodies, are modelled as force elements. These are responsible for
transmitting the internal forces that are developed in the system as function of the relative
motion between the bodies. Further detail on this topic is outside the scope of this thesis; the
interested reader is referred to see the work developed by Pombo et al. [106].
15
3 Development of Advanced Track Models
3.1 Track Description
The performance of railway vehicles is dependent on the track conditions. The loads induced
on the vehicle by the track and the corresponding forces transmitted to the track by the vehicle
also depend on the track geometry. Therefore, the accurate description of the track is essential
for the dynamic analysis of railway systems.
The description of a railway requires not only the characterization of its design
geometry, but also the description of the irregularities that are associated with the track. The
track irregularities represent the deviations of the track from its design geometry and result
from construction imperfections, usage operations and change on the foundations. The
realistic definition of a railway involves a combination between its design geometry and the
parameters that define the track irregularities. In the dynamic analysis of railway systems the
track irregularities must be considered, especially when studying the wheel-rail interaction
forces and the passenger ride comfort. Further detail on this topic is outside the scope of this
thesis; the interested reader is referred to see the work developed by Pombo [109].
A railway track is generally composed by an assembly of elements of distinct elasticity
responsible for gradually transmitting to the subsoil the dynamic loadings coming from the
trains’ passage, besides the important function of guiding the vehicles. These elements are the
rails, which are supported by the sleepers through the pads. The sleepers rest on an elastic bed
made up of supporting layers as ballast, sub-ballast, form layer and subsoil, as represented in
Figure 3.1 and Figure 3.2.
The most common railway track consists of steel rails supported on timber or prestressed concrete sleepers, which are laid on crushed stone ballast. A plastic or rubber pad is
usually placed between the rail and the concrete sleepers with the rail held down to the sleeper
with resilient fastenings.
The railway tracks are generally laid on a bed of stone ballast or track bed, which in turn
is supported by prepared earthworks known as the substructure. The substructure comprises the
subgrade, a layer of sand or stone dust (often sandwiched in impervious plastic), known as the
form layer, which restricts the upward migration of wet clay or silt and the sub-ballast, which
consists of smaller crushed stone than the ballast. This may also contain layers of waterproof
fabric to prevent water penetrating to the subgrade. The term foundation may be used to refer to
the ballast and substructure, i.e. all man-made structures below the tracks.
16
Rail
Sleeper
Rail Pad
Superstructure
Ballast
Track
Supporting
Layers
Substructure
Subsoil or Subgrade Soil
Sub-ballast
Form Layer
Figure 3.1: Main components of the railway track (Longitudinal view)
Rail
Sleeper
Rail Pad
Ballast
Subsoil or Subgrade Soil
Sub-ballast
Form Layer
Figure 3.2: Main components of the railway track (Cross-section view)
The track and ballast form the superstructure. The track ballast is customarily crushed
stone, and the purpose of this is to support the sleepers and allow some adjustment of their
position, while allowing free drainage.
3.2 The Finite Element Method on the Track System
Despite being considered as rigid by many authors and computational tools, the railway track
exhibits some flexibility that is characterised by small deformations and rotations, which,
besides other phenomena, originate track irregularities. Due to its nature and magnitude, these
deformations can be characterised as linear.
In this work the railway track system is modelled with linear finite elements, being the
wheel-rail contact forces included in the force vector of the finite element formulation. The
rails and sleepers are modelled by using Euler-Bernoulli beam elements [110], while the
foundations and rail pads are represented by spring-damper elements acting in the six degrees
of freedom, as shown in Figure 3.3 and Figure 3.4.
17
Rail
Element
Flexibility of
Rail Pad
Sleeper
Elements
Flexibility of
Ballast and
Substructure
Rigid Foundation
Figure 3.3: Main components of the track model (Cross-section view)
Flexibility of the
Sleeper Interaction
Rail
Elements
Flexibility of Rail Pad
Sleeper
Element
Flexibility of Ballast
and Substructure
Rigid Foundation
Figure 3.4: Main components of the track model (Longitudinal view)
A realistic track model requires a lot of variable information for it to approximate real
cases. It requires a detailed 3D geometry that includes irregularities, the different track zones
detailing the properties inherent to the elements within them and the transitions between the
different zones and sections where there are discontinuity of properties. The required data to
build this flexible track model is detailed in Annex A.
In this work, a pre-processor tool was developed in order to build detailed flexible track
models using a finite element formulation. The pre-processing tool builds a given track using
its 3D geometry as a pathway reference and places along that pathway the different track
segments on the intended order, each of them with their own elements, i.e., with specified rail,
pad, sleeper and foundation elements. The track geometry comes from the designed track
layout with the irregularities present, while the rails, pads, sleepers and foundations have their
own material and geometric properties that need to be defined. A schematic representation of
the pre-processor tool is presented on Figure 3.5.
The methodology used here to build the track model takes into account the influence of all
elements, namely the different properties of the foundation and its variation along the track,
allowing the presence of transitions. While other approaches also allow these features, few allow a
practical evaluation of a long track due to the computational cost of studying the soil properties, as it
requires many layers of 3D solid elements to be correctly modelled. Since the focus of this study is
on the influence of the track on the vehicle and of the vehicle on the track, this model uses a discrete
foundation that allows for dynamic analyses on longer tracks than the traditional approach.
18
Pad
Properties
Track Layout
Sleeper
Properties
Foundation
Properties
Track
Track
Segment
1
Track
Segment
2
Track
Segment
...
Segment n
Track Irregularities
Track
Geometry
Track Properties
Pre-Processing Tool
Rail
Properties
Track Model
Figure 3.5: Schematic representation of the Pre-Processing Tool
To capture the dynamic behaviour of the track, an integration algorithm was
implemented based on the implicit Newmark’s trapezoidal rule [111], taking into account the
modelling needs of the dynamics of the track model, due to the integration algorithm’s
accuracy, stability and other crucial aspects, especially considering the computational costs.
These aspects are discussed in the following.
3.2.1 Dynamic Analysis of Railway Tracks Using Linear FEM
The equilibrium equations of the finite element method for the railway track structural system
are assembled as [112]:
Ma Cv K d  f
(3.1)
where M, C and K are the finite element global mass, damping and stiffness matrices of the
finite element model of the railway track [110,112]. The accelerations, velocities and
displacements vectors are represented respectively as a, v and d while the sum of all external
applied forces is depicted by vector f. The 3D linear Euler-Bernoulli beam element Ke and the
local mass matrix Me are [110]:
19
 EA
 l

 0


 0


 0


 0


 0

Ke  
EA

 l

 0


 0


 0


 0


 0
12 EI z
l3
12 EI y
0
0
GJ
l
0

0

Symmetric
l3
6 EI y
4 EI y
0
l2
l
6 EI z
l2
0
0
0
4 EI z
l
0
0
0
0
0
EA
l
12 EI z
l3
0
0
0
6 EI z
l2
0
12 EI z
l3
0
0
0
0
0
0
0
0
0
0
6 EI
 2z
l
0

12 EI y
0

6 EI y
6 EI z
l2
1
3

0


0

0


0


0
e
M   lA 
1
6

0


0

0


0


 0

0
0
6 EI y
0
l3
l2
GJ
l
0
2 EI y
0
l2
0

l
0
2 EI z
l
0
12 EI y
l3
0
GJ
l
6 EI y
0
l2
0
4 EI y
l
0
0
13
35
0
13
35
0
0
Symmetric
Jx
3A
11l
210
0
l2
105
11l
210
0
0
0
l2
105
0
0
0
0
0
1
3
9
70
0
0
0
13l
420
0
13
35
0
9
70
0
13l
420
0
0
0
13
35
0
0
Jx
6A
0
0
0
0
0
Jx
3A
0
13l
420
0
3l 2
420
0
0
0
11l
210
0
13
35
0
0
0
0
13l

420

0
0


0
3l 2

420
11l
0 
210





























 (3.2)


4 EI z 
l 





























13 
35 
(3.3)
in which E is the Young modulus, G is the transversal modulus of rigidity, l is the element length,
A is the cross section area, ρ is the material density, and Iy, Iz and Jx are the second area moments
of inertia about the respective y, z, and x axis. The global stiffness and mass matrixes, K and M,
are built by assemblage of the matrices of the elements according to the railway track mesh.
20
In order to model the damping behaviour of the system, proportional damping, also
known as Rayleigh damping [112], is used. The global damping matrix C is obtained by
assembling the element damping matrices, Ce, for each element as:
Ce =  eΚ e +  eΜ e
(3.4)
where αe and βe are proportionality factors associated with each type of railway track element
e, such as sleeper, rail and others.
The force vector, f, containing the sum of all external applied loads, is evaluated at
each time step of the time integration. For a time t+Δt the force vector is calculated as:
f t  t  f g  ftc t
(3.5)
where the vector f g contains the gravitational forces of all element which remains constant.
The force vector f c represents the wheel contact forces being evaluated as:
ftct    B c fc i
(3.6)
i
where fc represents the equivalent forces and moments applied at appropriate nodes of the rail
element where a contact force, at time t+Δt, is to be applied. The matrix Bc means the Boolean
operation of assembling each contact force fc
i
in the global force vector. The contact force
value to be applied and its point of application are evaluated, at each integration time step, by
geometric interference and a proper contact modelling method, to be discussed on section 3.4.
3.2.2 Time Integration
To solve the time integration problem, Newmark [111] proposed that for a given time t and a
fixed time step Δt the solution of the equilibrium equation for a forthcoming time t+Δt is
represented as:
M at  t  C v t t  K dt t  ft t
(3.7)
Which would require the knowledge of how the d, v and a evolve over time and their
relation to each other. Admitting that the solution of the dynamic equilibrium equation is known
at time t, the direct use of Taylor’s series provides an approach to obtain these relations:
d t t  d t  v t t  a t
t 2 da t t 3

...
2
dt 6
21
(3.8)
da t  t 2
...
dt 2
v t t  v t  a t t 
(3.9)
Newmark truncated these equations in his methods assuming that the acceleration
would be linear within the time step:
dat at t  at

dt
t
(3.10)
Which leads to Newmark’s equations in the standard form, where the displacements and
velocities on time t+Δt can be obtained by:
 1
d t t  d t  v t t    
 2
 2

 at   at t  t


(3.11)
v t t  v t  1    at   at t  t
(3.12)
The parameters γ and ζ are determined in order to obtain integration accuracy and stability
[111]. The ζ controls the numerical dampening, where if ζ < 1/2 there is negative dampening and
introduce a self-excited vibration; similarly if ζ > 1/2 there is positive dampening and will
decrease the magnitude of the response even without real dampening. On the other hand γ controls
the convergence rate, where if γ < (1/2 + ζ)2/4 means that the results will not converge, and if γ >
(1/2 + ζ)2/4 they will converge, but will periodically introduce errors the smaller it is [111].
However when ζ = 1/2 and γ = 1/4, parameters known for the “trapezoidal rule”, and the
above stated assumptions are used implicitly to solve the equilibrium equation, results in a
particulate application of the Newmark method that is unconditionally stable. The stability of
ζ= ½
both γ and ζ on Newmark methods is shown on Figure 3.6.
γ
Unconditionally
Stable
3/2
1/2
Unstable
1
γ = (1/2 + ζ)2/4
Conditionally
Stable
1/4
1/2
1
3/2
2
ζ
Figure 3.6: Stability of Newmark’s parameters γ and ζ
22
The selection of a proper time integration numerical procedure to solve the governing
dynamic equilibrium equations of a system is usually decided by engineering judgement.
Such decision must take into account not only the stability and accuracy of the selected
algorithm, but also its computer processing effort.
3.2.2.1 Choice of Time Integration Method
An explicit method does not involve the solution of a set of linear equations at each time step.
Basically, these methods use the differential equation at a given time t to predict a solution for time
t+Δt. For most real structures a very small time step is required in order to obtain a stable solution,
since all explicit methods are conditionally stable with respect to the size of the time step.
For example, using Newmark’s Explicit Constant Average Acceleration Method [113],
the next time step’s displacement and velocity would be approximated by:
2
t
d t t  dt  t vt 
at
4
v tt  v t 
t
at
2
(3.13)
(3.14)
With those values, the next time step’s acceleration can now be calculated by solving:

t
t 2 
K  at t  ft t  Cv t t  Kd t t
M  C
2
4


(3.15)
And the approximated displacement and velocity would be connected by doing:
2
t
dt t  d t t 
at t
4
v tt  v t t 
t
a
2 t t
(3.16)
(3.17)
Then, the next time step’s acceleration is corrected by solving equation (3.15) with the
new values obtained from equations (3.16) and (3.17). This process is repeated until a given
stability value is reached, before proceeding to the next time step.
The explicit methods are corrective methods and as such only approximate the solution
of the problem at each time step, thus requiring a small Δt to prevent instability, which
increases the computational time.
23
Implicit methods attempt to satisfy the differential equation at time t+Δt after the
solution at time t is found. These methods require the solution of a set of linear equations at
each time step; however, larger time steps may be used in comparison with an explicit
method. Implicit methods can be conditionally or unconditionally stable [111].
For example, in the Newmark’s Implicit Constant Average Acceleration Method [113],
the equations (3.11) and (3.12) are rearranged respectively for at t and v t t in terms of d t t :
1
1
 1

d  dt  
v t    1 at
2  t t
t
t
 2 
(3.18)




t  
 dt t  dt     1 v t    2  a t t
t
2 



(3.19)
at t 
v t t 
Which applied to (3.7) and assuming ζ = 1/2 and γ = 1/4, result in the following
equation to be solved at each time step:
4
2 
4

 4

 2

 K  2 M  C  dt t  ft t  M  2 d t  v t  at   C  dt  vt  (3.20)
t
t 
t

 t

 t

And with the displacements at the next time step, it is possible to calculate the
remaining variables:
at t 
4
4
d  dt   vt
2  t t
t
t
v t t  v t 
t
t
a t  a t  t
2
2
(3.21)
(3.22)
Newmark’s Methods were chosen as examples because both assure the dynamic
equilibrium, which isn’t the case with other Explicit Methods such as the Central Differences
Method. With some alterations done to Newmark’s Methods, these allow different “numerical
dampening” and “period elongation”, for faster resolution for some specific problems [111].
A schematic comparison of these two methods is shown Figure 3.7, which shows the
required variables necessary to obtain a given value.
3.2.2.2 Implementation of the Time Integration Method
The time integration method embraced for this specific implementation is an implicit Newmark
family integration algorithm [112,114]. This particular method was chosen due to its
unconditional stability nature when used implicitly and its proven robustness in FEM applications.
24
While this method requires more computational power, it enforces the equilibrium between the
internal structure forces and the external applied loads at each time step, which an explicit
integration algorithm would not, allowing the time step to increase and solving the problem faster.
Newmark’s Explicit Method
vt
t
1
Newmark’s Implicit Method
at
dt
dt
d t t
v t  t
3
v t+Δt
t+Δt
v t+Δt
vt
dt+Δt
a t+Δt
2
at
a t+Δt
dt+Δt
a t+Δt
v t+Δt
dt+Δt
dt+Δt
a t+Δt
v t+Δt
t
(a)
(b)
Figure 3.7: Newmark’s (a) Explicit and (b) Implicit Time Integration Methods
To solve the implicit problem, the relations (3.18) and (3.19) are substituted into the
equilibrium equation (3.7) which than can then be solved for the displacements d t t as:
ˆ
ˆ
Kd
t+Δt = f t+Δt

LUd t+Δt = fˆt+Δt
(3.23)
ˆ K a MaC
K
0
1
(3.24)
fˆt + Δt  f t + Δt  M  a0 d t  a 2 v t  a3a t   C  a1d t  a 4 v t  a5 a t 
(3.25)
a0 
1
;
 t 2

a4   1;

a1 

;
t
a2 

t  
a5    2  ;
2 

1
;
t
a3 
a6  t 1    ;
1
 1;
2
(3.26)
a7    t
The notation LU is used in equation (3.23) to mean a factorization of the stiffness
matrix in the solution of the implied system of equations [115]. Afterwards the accelerations
and velocities can be calculated by using equations (3.21) and (3.22).
25
For the time integration of a linear system the matrix K̂ is constant unless the time step size
changes. An important computational advantage can be taken out of this predicament in integration
algorithms, because the largest computation cost that occurs at each integration time step is the
solution of the system of linear equations (3.23). More particularly when numerically solving this
system, a relevant part of the processing effort is strongly influenced by the numerical solver used
and its implicit matrix factorization algorithm [112,116]. In this case a LU decomposition is
selected. Taking the advantage on the fact that the effective stiffness matrix K̂ remains constant,
means that the factorization is done only once and the same products are used on the procedure at
every time step, resulting in a methodology that saves computational cost for the dynamic analysis.
Another aspect of the integration algorithm involves the calculation of the effective loads
vector fˆ . As the external loads vector f, expressed in (3.5), is not constant in time the effective
t + Δt
loads vector must be calculated at every integration time step. Moreover the calculation of the
wheel contact forces, as expressed in equation (3.6), depends on a close prediction of the node
displacements, d t  t , and velocities, v t t ,that would belong to the solution of the dynamic
equilibrium equations at time t+Δt. In order to be accurately close to this prediction, the
approximation of the displacements and velocities is evaluated iteratively within each time step of
the integration algorithm. On the first iteration the last time step displacements d t and v t are
considered a close enough prediction and used to form the effective loads vector and to evaluate
the dynamic equilibrium equations. The solution obtained is considered as the new displacements
and velocities prediction for the next iteration. This correction procedure is done iteratively until a
good enough convergence is reached where, d  d
  and v  v
  , being εd
t t
t t
d
t t
t t
d
and εv user defined tolerances. This iterative process is similar to the one used by Antunes [102],
but applied to the track instead of to the catenary.
3.3 Automatic Finite Element Method Mesh Generation
One of the key features of the Pre-Processing Tool is its ability to automatically generate a FEM
mesh using the information stored in a database, containing the rails geometry, and its use as a
guide for all the elements underneath, such as the rail pads, the sleepers and the foundation. The
rail geometry database includes all the characteristics of the rails, such as its position, length,
curvature and cant angle, as well as the irregularities. The method of construction of this
database was developed by Pombo [109] and is outside the scope of this thesis.
26
In addition to the definition of the rail position, the Pre-Processor also adds the
remaining elements that compose the track model using the data provided by the user. These
elements are equidistantly positioned at a selected distance in order to represent the flexibility
of the track components, namely the sleepers, pads and foundation. Further detail on the data
required to correctly represent this flexibility is presented in Annex A.
In this work, the construction of finite element models of railway track involves modelling, all
rail and sleeper elements with 3D beam elements based on Euler-Bernoulli beam theory [117]. This
3D beam element, which formulation is developed in [110], is assumed to be a straight beam of
uniform cross section capable of resisting axial forces, bending moments about the two principal
axes of its cross section and twisting moments about its centroid axis. The other element type used
here is the spring-damper element to represent the pad and foundation components, which are better
modelled as a spring-damper in all degrees of freedom due to their intrinsic properties.
The approach proposed here uses symmetric sleepers composed of six collinear
elements in order to account for the common transitions of section and thus a spring-damper
element must be placed below each node in order to accurately represent the sleeper support
system. There is no special requirement on the number of elements needed to model each rail
between sleepers as shown in Table 3.1.
Component
Foundation
Sleeper
Pad
Rail
Element Type
Spring-Damper Element
Euler-Bernoulli Beam
Spring-Damper Element
Euler-Bernoulli Beam
Number of Elements
7 below each sleeper
6 each
1 each
1 between pads (minimum)
Table 3.1: Number of elements and their element type used to model each component of the
railway track model
Two examples of generated meshes produced by the Pre-Processing Tool are shown in
Figure 3.8 and Figure 3.9. On the first case a straight track is shown, while the second case a
similar example was produced, but for a realistic curved track.
1
Curvature (1/R) [m-1]
0,8
0,6
0,4
0,2
0
-0,2
-0,4
-0,6
-0,8
-1
0
100
200
300
400
500
Length [m]
(a)
(b)
Figure 3.8: Representation of (a) the curvature of a straight track and (b) the finite element
mesh of a straight track using the Pre-Processing Tool
27
Curvature (1/R) [m-1]
0,0025
0,002
0,0015
0,001
0,0005
0
0
100
200
300
400
500
600
700
800
900
1000 1100
Length [m]
(a)
(b)
Figure 3.9: Representation of (a) the curvature of a realistic curved track and (b) the finite
element mesh of a realistic track using the Pre-Processing Tool
3.4 Vehicle-Track Interaction
The contact in the vehicle-track interaction involves the surface of the wheels and the top
surface of the rails, and other phenomena, the wear of the wheels and of the rails is deeply
influenced by the quality of this contact. This implies that the correct modelling of the contact
mechanics involved is crucial for accurate and efficient railway dynamic studies.
The contact problem can be treated either by a kinematic constraint between the wheel and
the rail or by a penalty formulation of the contact force. In the first procedure, the contact force is
simply the joint reaction force of the kinematic constraint [118,119]. With the second procedure,
the contact force is defined in function of the relative penetration between the wheel and the rail
[120,121]. The use of the kinematic constraint between the rail and the wheel forces these
elements to be in permanent contact, which is only valid if no contact loss exists; while the use of
the penalty formulation allows for the loss of contact and it is the method chosen in this work.
Since the vehicle is modelled by multibody formulation [8-10] and the track is modelled
by finite element formulation [13,14], there is a communication problem between the two
methods. This problem is solved by running the two simulations simultaneously and
performing a communication between them, where both exchange data with each other. This
communication procedure is detailed in Annex B.
In order to model the contact force using a penalty formulation it is necessary to
geometrically assess if there is contact and identify the contact point location on the wheel and on
the rail, in addition to the relative penetration of the contact. For this purpose a three step
procedure is implemented at every time step of the track integration algorithm and for each wheel
present on the dynamic analysis. In the first step a rail finite element is evaluated to be a candidate
for the contact solution. The first element that starts to be evaluated is the one used for the contact
on the last time step. If the element is not eligible for contact the procedure restarts for the next
28
element on the rail. On the second stage, it is assumed that there is contact and by geometric
interference and by using shape functions of the rail finite element, the potential points of contact
on the rail and the wheel are located. At the third step the relative penetration of the contact is
calculated and it is assessed if there is indeed contact or there is a contact loss.
To find the rail element candidate for contact, consider the representation of the
transversal view of a rail element and the contact forces present in it on Figure 3.10.
c
F1
j
c1
F2
c2
d
z
ûij
i
y
(a)
λid
(b)
Figure 3.10: Representation of (a) the transversal view of a rail element and contact forces
and (b) the location of a contact point relative to the rail finite element
The nodes i and j represent the rail element extremities and each node c represents a
contact point at the actual time. The node c represents the position on the rail element where
the contact force is located. The node d is located where a line, containing the point c and
perpendicular to the rail element, intersects the rail element. Since these are perpendicular:
rij rdc  0
(3.27)
where rij and rdc define the vectors from nodes i to j and from nodes d to c, respectively. The
vector rdc can also be defined by:
rdc  rc  rd
(3.28)
where rc and rd define the positions of nodes c and d respectively. But the position of node d
can also be obtained by:
rd  ri  id uˆ ij
(3.29)
where ri defines the position of node i, id is the norm from node i to d and uˆ ij is a versor of a
generic vector that goes from node i to node j. Then combining the equations (3.27) to (3.29),
it is possible to obtain:
rij  rc  ri  id uˆ ij   0
29
(3.30)
this equation can be further simplified, since:
(3.31)
ric  rc  ri
where ric defines the vector from node i to c. Substituting equation (3.31) into equation (3.30)
and solving in order to id , results in:
id 
rij ric
(3.32)
rij uˆ ij
Now, assuming that there is contact and the rail element is a rigid body, its potential
point d, can be calculated as presented in equation (3.29). Now three possible solutions exists
depending of ij . Which is the norm from node i to j, as represented in Figure 3.11.
d
j
j
j
ûij
d
ûij
i
i
d
ûij
λid
λid
λid
i
(a)
(b)
(c)
Figure 3.11: Representation of the three cases for the different values of id where (a)
0  id  ij (b) id  0 and (c) id  ij
If 0  id  ij , it means that the candidate for the contact is correct and the program can
advance to the second stage. If id  0 , it means that the contact is occurring in a previous rail
element and the program repeats the first step for the previous rail element. If id  ij , it
means that the contact is occurring ahead of the candidate rail element and the program
repeats the first step for the next element.
The correct position of node d on the rail element can be calculated as:
rd  rd0  N( )di , j
(3.33)
where, as presented on Figure 3.12, rd0 corresponds to the rail node d position without the
deformation accounted for; the vector di, j contains the displacements of the node i and j; and
the matrix N( ) contains the element shape functions [122] in order of  , which is the local
element relative position of the contact point in its longitudinal direction defined as:

id i0 d0

ij i0 j0
(3.34)
where i0 d0 and i0 j0 correspond to id and ij without the deformation, respectively.
30
j0
dj
d0
j
dd
i0
d
di
i
λid
λ ij
potential point of
contact on the ra il
Figure 3.12: Representation of the potential point of contact on the rail element
3.5 Case Studies of the Flexible Track
In the following three case studies are presented, including a static validation of a simple
flexible track and the dynamic analysis of realistic tracks with moving loads and realistic
vehicle forces. These results demonstrate the differences between using moving vertical loads
and realistic vehicle-track interaction forces.
3.5.1 Simple Flexible Track and Static Validation
The typical data required to build a finite element model representing a railway track is
presented in Annex C, using the data provided by the SMARTRACK partners. Using that data
and the curvature graph presented in Figure 3.8 (a), a finite element model of a generic railway
track is obtained, as shown in Figure 3.13. In order to validate the methodology proposed here
to define finite element flexible track models, a static analysis was performed and compared to a
similar case study on a commercial program. A realistic flexible track model was built and
subjected to wheelset loads of a railway vehicle, as depicted in Figure 3.13 (b). The results
obtained were compared against the ones provided by ANSYS 12. The data used to build the
flexible track model is given in Annex C. These forces represent the maximum wheelset load of
22.5 ton that a railway vehicle can have to be allowed to operate in the Portuguese railway
network. In ANSYS, the BEAM4 [123] element was used as an Euler-Bernoulli beam element
and the MATRIX27 [123] element was used as substitution for spring-damper elements, as it
allows to create user defined elements. All other parameters required to build the track model in
ANSYS match the ones used by the computational tool proposed here.
31
P
P
(a)
(b)
Figure 3.13: Simple flexible track model: (a) Finite element mesh; (b) External loads applied
A pair of static downward vertical forces P of 112.5 kN are applied, as depicted in
The deformations obtained with the two numerical tools are shown in Figure 3.14 and
Figure 3.15. As the deformations are very small when compared with the other dimensions of
the track, they are multiplied by a factor of 100 in these figures. The results obtained show
that the maximum vertical deformation of the track is 2.9 mm on the nodes where the loads
are applied. On the other hand, the maximum displacement in the lateral direction is 44.910-6
m, also on those nodes. The displacement in the longitudinal direction is negligible.
(b)
(a)
Figure 3.14: Perspective view of the track deformation (deformation scaled 100): (a)
Computational tool; (b) ANSYS
(a)
(b)
Figure 3.15: Lateral view of the track deformation (deformation scaled 100): (a)
Computational tool; (b) ANSYS
32
When comparing the results obtained with the methodology proposed here and with
ANSYS, it is observed that the maximum relative error for the track vertical deformation is
less than 0.04%, as shown in Figure 3.16, corresponding to a maximum absolute error of
57.410-9 m. Notice that the 0% error corresponds to the constrained nodes on the foundation.
0,045
0,04
Z Relative Error [%]
0,035
0,03
0,025
0,02
0,015
0,01
0,005
0
9600
9700
9800
9900
10000
10100
Node Number
10200
10300
10400
Figure 3.16: Relative error for the track vertical deformation.
Figure 3.17 (a) presents the relative errors on the rail nodes that are in the vicinity of those
where vertical wheelset forces were applied. The relative error of the vertical deformation on
the nodes of the sleeper subjected to the external loads is shown in Figure 3.17 (b).
0,0035
0,0021
0,0021
0,0030
Lef t Rail
Right Rail
0,0020
Z Relative Error [%]
Z Relative Error [%]
0,0021
0,0025
0,0021
0,0021
0,0021
0,0015
0,0021
0,0010
-10
-5
0
5
Node Distance to Load Location on Rail
0,0021
9992
10
9993
9994
9995
Node Number
9996
9997
9998
(a)
(b)
Figure 3.17: Relative error on the nodes in the vicinity of the applied loads: (a) Nodes on the
rail; (b) Nodes on the sleeper
Similar analyses were performed considering four and eight loads on the rail, representing the
bogies of an Alfa-Pendular vehicle. The results obtained are similar to the ones presented above.
These results demonstrate that the proposed finite element methodology represents the track
flexibility in an appropriate manner and it is quantitatively validated for static loads.
33
3.5.2 Realistic Flexible Track with Moving Loads and Vehicle Forces
As in the case study presented before, the typical data required to build a finite element model
representing a realistic curved railway track is also presented in Annex C. Using that data and
the curvature graph presented in Figure 3.9 (a), a finite element model of a realistic railway
track is automatically generated by the Pre-Processor, as shown in Figure 3.18.
Figure 3.18: Realistic Flexible track model
Since there was no opportunity to fully develop the communication procedure between
the finite element and the multibody modules, this case study was analysed using two
different approaches. In Case 1 constant moving loads are considered representative of the
total weight of the vehicle divided by the vehicle wheels, as depicted in Figure 3.19 (a). In
Case 2 realistic wheel-rail contact forces obtained from the vehicle-track interactions forces
on a realistic dynamic simulation are considered, as represented in Figure 3.19 (b). These
loads are 3D, including the normal contact forces and creep forces.
P
P
P
P
P
P
P
v
P
v
(a)
(b)
Figure 3.19: Comparison of (a) the moving load method and (b) realistic vehicle forces
Both cases are analysed considering that the vehicle is moving at v = 20 m/s (72 km/h)
on the realistic track represented in Figure 3.18. The vertical and transversal loads applied on
the track in both cases are depicted in Figure 3.20.
34
100000
Vertical Load on Rails [N]
80000
70000
60000
50000
40000
30000
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
20000
10000
0
0
200
400
600
800
Transversal Load on Rails [N]
40000
90000
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
30000
20000
10000
0
-10000
-20000
0
1000
200
400
600
800
1000
Track Length [m]
Track Length [m]
(a)
(b)
Figure 3.20: Comparison of the moving load method (Case 1) and realistic vehicle forces
(Case 2) for the front wheels of the vehicle for (a) the vertical loads and (b) the
transversal loads
The vertical displacement of the rails at four selected locations is shown in Figure 3.21
to Figure 3.24, which are located before the first transition curve, on the first transition curve,
on the curve and on the second transition curve; together with their corresponding comparison
in percentage. Here the maximum vertical deformation is 6.19 mm and is found on the left rail
at the second transition curve.
0,6
Z Cases Comparison [%]
0
Z Displacement [m]
-0,001
-0,002
-0,003
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
-0,004
-0,005
0
0,2
0,4
0,6
0,8
Left Rail
Right Rail
0,5
0,4
0,3
0,2
0,1
0
1
0
0,2
0,4
Time [s]
0,6
0,8
1
Time [s]
(a)
(b)
Figure 3.21: Results of the Dynamic Analysis before the first transition curve: (a) vertical
deformation of the rails with moving loads (Case 1) and realistic vehicle forces
(Case 2) and (b) their comparison in percentage
2,5
Z Cases Comparison [%]
0
Z Displacement [m]
-0,001
-0,002
-0,003
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
-0,004
-0,005
7
7,5
8
8,5
Left Rail
Right Rail
2
1,5
1
0,5
0
9
7
Time [s]
7,5
8
8,5
9
Time [s]
(a)
(b)
Figure 3.22: Results of the Dynamic Analysis on the first transition curve: (a) vertical
deformation of the rails with moving loads (Case 1) and realistic vehicle forces
(Case 2) and (b) their comparison in percentage
35
140
0
120
Z Cases Comparison [%]
Z Displacement [m]
0,001
-0,001
-0,002
-0,003
-0,004
Case 1
Case 1
Case 2
Case 2
-0,005
-0,006
26
26,5
27
Left Rail
Right Rail
Left Rail
Raight Rail
27,5
Left Rail
Right Rail
100
80
60
40
20
0
28
26
26,5
27
Time [s]
27,5
28
Time [s]
(a)
(b)
Figure 3.23: Results of the Dynamic Analysis on the curve: (a) vertical deformation of the
rails with moving loads (Case 1) and realistic vehicle forces (Case 2) and (b)
their comparison in percentage
1200
0,002
Z Cases Comparison [%]
Z Displacement [m]
0,001
0
-0,001
-0,002
-0,003
-0,004
Case 1
Case 1
Case 2
Case 2
-0,005
-0,006
-0,007
45
45,5
46
Left Rail
Right Rail
Left Rail
Raight Rail
46,5
Left Rail
Right Rail
1000
800
600
400
200
0
45
47
45,5
46
46,5
47
Time [s]
Time [s]
(a)
(b)
Figure 3.24: Results of the Dynamic Analysis on the second transition curve: (a) vertical
deformation of the rails with moving loads (Case 1) and realistic vehicle forces
(Case 2) and (b) their comparison in percentage
The transversal displacement of the rails for the previous four selected locations is
shown in Figure 3.25 to Figure 3.28. Here the maximum transversal deformation is 0.29 mm
and is found on the left rail at the curve. Comparably, the deformations on the longitudinal
direction are almost negligible, with a maximum inferior to 310-9 m.
0,00007
0,00005
Y Cases Comparison [%]
Y Displacement [m]
35
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
0,00006
0,00004
0,00003
0,00002
0,00001
0
-0,00001
Left Rail
Right Rail
30
25
20
15
10
5
0
0
0,2
0,4
0,6
0,8
1
0
Time [s]
0,2
0,4
0,6
0,8
1
Time [s]
(a)
(b)
Figure 3.25: Results of the Dynamic Analysis before the first transition curve: (a) transversal
deformation of the rails with moving loads (Case 1) and realistic vehicle forces
(Case 2) and (b) their comparison in percentage
36
14
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
0,00008
0,00006
0,00004
Y Cases Comparison [%]
Y Displacement [m]
0,00012
0,0001
0,00002
0
-0,00002
-0,00004
-0,00006
-0,00008
-0,0001
Left Rail
Right Rail
12
10
8
6
4
2
0
7
7,5
8
8,5
9
7
7,5
Time [s]
8
8,5
9
Time [s]
(a)
(b)
Figure 3.26: Results of the Dynamic Analysis on the first transition curve: (a) transversal
deformation of the rails with moving loads (Case 1) and realistic vehicle forces
(Case 2) and (b) their comparison in percentage
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
Y Displacement [m]
0,0002
0,0001
0
-0,0001
-0,0002
-0,0003
40
Y Cases Comparison [%]
0,0003
-0,0004
Left Rail
Right Rail
35
30
25
20
15
10
5
0
26
26,5
27
27,5
28
26
26,5
Time [s]
27
27,5
28
Time [s]
(a)
(b)
Figure 3.27: Results of the Dynamic Analysis on the curve: (a) vertical deformation of the
rails with moving loads (Case 1) and realistic vehicle forces (Case 2) and (b)
their comparison in percentage
Case 1 Left Rail
Case 1 Right Rail
Case 2 Left Rail
Case 2 Raight Rail
Y Displacement [m]
0,00006
0,00004
0,00002
0
-0,00002
-0,00004
-0,00006
-0,00008
100
Left Rail
Right Rail
90
Y Cases Comparison [%]
0,0001
0,00008
80
70
60
50
40
30
20
10
-0,0001
0
45
45,5
46
46,5
47
45
Time [s]
45,5
46
46,5
47
Time [s]
(a)
(b)
Figure 3.28: Results of the Dynamic Analysis on the second transition curve: (a) transversal
deformation of the rails with moving loads (Case 1) and realistic vehicle forces
(Case 2) and (b) their comparison in percentage
Similar results were obtained for the sleeper below the rail, where the maximum vertical
deformation is 5.56 mm also on the left side of the sleeper positioned at the second transition
curve. On the sleeper, the longitudinal deformation is even inferior to the one presented on the
rail, with a peak at 810-9 m, and the transversal deformation that peaks at 0.29 mm at the curve.
Although there are differences between the use of moving loads and realistic vehicle forces,
these are small, allowing the correct study of the track using either of them when the contact forces
are considered independent from the deformation of the rail. A future development of this work
37
consists on the development of a Co-Simulation Procedure, allowing for the correct calculation of
the track deformation and contact forces, since these quantities are dependent on each other.
In the future, with the completion of the Co-Simulation Procedure, the coupling
between the track deformation and the contact forces will be considered for future studies, as
it is fundamental for the understanding of realistic dynamic track deformation and how it
affects the vehicle performance.
38
4 Definition of the Post-Processing Tool
The results obtained from a railway dynamic analysis are kinematic data, such as positions,
velocities and accelerations from all vehicle components and kinetic data, such as the wheel-rail
interaction forces. When the flexibility of the track is considered, it is also possible to obtain the
realistic loading of the track and its deformation. Unfortunately, these results vary greatly, becoming
impossible to compare similar analysis, namely using other vehicles or tracks, and obtain significant
conclusions. In order to become comparable to other analyses, the results need to be filtered and
processed, to become possible to analyse the safety and comfort parameters of the case study.
This chapter describes the post-processing tool that was developed in this work to
perform all filtering and data processing that are required to assess the dynamic performance of
the railway vehicle according to the international regulations EN 14636 [11] and/or UIC 518
[12]. To comply with the requirements defined in these norms a set of values must be
determined. These characterise the behaviour of the vehicle on the different zones of the track,
being derived from measured or simulated data, namely accelerations and forces exerted on the
vehicle, which are filtered and processed. This process is represented in Figure 4.1.
Post-Processing Tool
Raw Data
Filtering and
Data Processing
Characteristic
Values
Check Acceptance
Criteria
Assess Dynamic
Performance
Figure 4.1: Schematic representation of the Post-Processing Tool
Two different methods can be used to approve vehicles: the Normal Method that uses all
assessment values, with the exception of the sum of the lateral axle box forces; while the
Simplified Method only uses accelerations and has the option to use the sum of the lateral axle
box forces as assessment values. The use of axle box forces and applicability is outside the
scope of this thesis; the interested reader is referred to see EN 14636 [11] and/or UIC 518 [12].
39
The industry specifies a series of limit values on the forces that the vehicle can apply on
the rails, the forces that the vehicle bogies or wheelsets can be subjected to and the
accelerations on the bogies or wheelsets and on the carbody. These limit values are then
divided into three categories: the safety limit values that ensure that the vehicle will not be in
any risk of derailment, the track loading limit values that ensure that the track will not be
damaged by the passage of the vehicle and the comfort limit values that ensure that the vehicle
is comfortable enough to be used for passengers transportation. To certificate the validity of a
given vehicle to travel on a given track, the vehicle must respect all these limit values.
4.1 Data to be Measured and Simulated
This section contains the information collected from EN 14636 [11] and UIC 518 [12] related
to the necessary data to assess the correct functioning of the vehicle. Since both norms are
similar in this matter, their contents will be condensed in a single section being the differences
between the two identified. Assessment values for running behaviour are either measured
directly or derived from other measured parameters. They are used in assessing the interaction
between vehicle and track and mainly describe the wheel/rail system or are closely related to
it. The following assessment values are generally used for the testing of running characteristics:
a) Forces between wheel and rail:
1) Guiding Force Y, lateral measuring direction, for each axle on bogies or
wheelset on non-bogie vehicles (Figure 4.2)
2) Wheel Force Q, vertical measuring direction, for each axle on bogies or
wheelset on non-bogie vehicles (Figure 4.2)
3) Sum of Guiding Forces ΣY of a wheelset, per axle on bogies or wheelset on
non-bogie vehicles (Figure 4.3)
4) Quotient Y/Q of Guiding Force/Wheel Force, for each axle on bogies or
wheelset on non-bogie vehicles (Figure 4.2)
Q
Y
Figure 4.2: Representation of the Wheel’s Guiding Force (Y) and Wheel Force (Q)
40
ΣY
ΣY
QR
QL
QL
QR
(a)
(b)
Figure 4.3: Representation of the Sum of Guiding Forces (ΣY) and Vertical Wheel Forces (Q),
in (a) straight line and (b) curved track
b) Forces at the bogie:
1) Sum of the lateral Axle Box Forces H, for each axle on instrumented bogies
or wheelset on non-bogie vehicles (Figure 4.4)
c) Accelerations:
1) Accelerations at axles, lateral measuring direction ÿ, on each wheelset for
non-bogie vehicles (Figure 4.5)
2) Accelerations at bogie, lateral measuring direction ÿ+, on bogie frame or on
each wheelset for non-bogie vehicles (Figure 4.5)
3) Accelerations in the vehicle body, lateral ÿ* and vertical
̈ ∗ measuring
directions, above the bogies or the wheelsets on non-bogie vehicles (Figure 4.5)
Axlebox
Axlebox
Axle
Wheel
flange
Wheel
tread
Figure 4.4: Representation of a Wheelset
̈
ÿ
̈
Figure 4.5: Relative rigid body accelerations
41
4.2 Limit Values
This section contains the information collected from EN 14636 [11] and UIC 518 [12] related
to the limit values that a vehicle may possess and still be able to operate in a given track.
4.2.1 Limit Values of Running Safely
The running safely limit values must be used restrictively. These limit values can only be changed
nationally and/or multi-nationally if the track and operating conditions differ from the basis
conditions used by UIC for the definition of limit values. Accelerations or H forces shall not be
considered for safety assessment of vehicles or vehicle parts on which Y and Q are measured.
4.2.1.1 Sum of Guiding Forces ΣYmax
The safety-critical limit for track shifting is:
2Q 0 

ΣYmax,lim  k 1  10 

3 

[k N ]
(4.1)
where ΣYmax,lim and 2Q0 (static axle-load) are expressed in kN and the factor k1:
k1 = 1.0
for locomotives, power cars, multiple units, passenger coaches and
track maintenance vehicles
k1 = 0.85
for freight wagons and special transport vehicles
The factor k1 takes into account the minimum Guiding Force values of a wheelset that a
track is still able to withstand without permanent lateral displacements. The limit ΣYmax,lim refers
to ballasted track; with timber sleepers, with a distance between sleepers inferior to 65 cm; and
rails with a weight greater than 46 kg/m; where the track bed has been recently tamped (see DT
66 and RP1 from ORE Committee C138 [124]). To take into account variations in geometrical
dimensions and the state of maintenance, a smaller factor k1 is assumed for freight wagons, but
exceptions are permissible for well-founded cases. For vehicles with short axle spacing the
influence of the adjacent axles increase the limit value ΣY a track can endure. It is allowed to
use extended calculation methods which take this fact into consideration.
4.2.1.2 Quotient of Guiding Force and Wheel Force (Y/Q)max
The safety-critical limit for the quotient of a leading wheel is:
Y / Q  max,lim  0.8
(4.2)
on curved tracks with radius of R ≥ 250 m. This recommended limit value, applicable for
dynamic on-line tests according to UIC 518 [12], was given by ORE C138 in RP9 [125]. To
42
assess safety against derailment at low speed on twisted track, the conditions quoted in ORE
B55/RP8 [126] have to be met. According to previous investigations, it was verified the limit
(Y/Q)max,lim for constant track curves (without transition curves and ramps) with radii R ≥ 300 m (see ERRI C138 [124]), in some loading conditions. Evidence of suitability for curves R <
300 m has not been provided. Until reliable results are available, it is recommended that the
limit value referred in equation (4.2) is also used for curves 250 m ≤ R < 300 m.
In transition curves it is recognized that higher values than 0.8 may be encountered. The
maximum limit value of 1.2 (for flange angle of 70º) applied for quasistatic testing according
to EN 14363 [11] section 4.1 shall be respected. Actually in transition curves no specific limit
can be specified, however it shall not exceed 1.2 and in the case where 0.8 is exceeded each
case shall be investigated and justified. These values are currently being revisited, on the basis
of test results from various vehicle types. Pending conclusions of these studies, when this
limit is exceeded it is allowed to recalculate the Y/Q estimated maximum value according to
the following process, considered by C138 [124] when setting at 0.8 the limit value:
1) Create an alternative test zone made up of all track sections with 300 m ≤ R ≤ 500 m.
2) For the statistical processing per section (see page 36 of UIC 518 [12]), use
xi(97.5 %) instead of xi(99.85%).
3) For the statistical processing per zone (see page 37 of UIC 518 [12]), use
Student coefficient t(N – 2; 95%) – (see table in page 78 of UIC 518 [12]) to
replace k = 3 (when using one-dimensional method) or Student coefficient t(N –
2; 99%) – (when using two-dimensional method).
4) Both results (before and after recalculation) shall be reported.
A recent UIC study, i.e. page 118 of UIC 518 [12], based on tests with empty freight
wagons equipped with Y25 bogies, showed that values up to 0.83 were evaluated as above on
lines comprising many sections with track quality above QN2. If the recalculation is made for
vehicles with axle load > 15 t there may be a track loading problem due to an unfavourable
angle of the force leading to failure of fastening on sharp curves. In this case operation may not
be accepted on some networks.
4.2.1.3 Overturning Criterion (Train Category IV) η
UIC 518 considers an additional criteria for category IV vehicles [12], where this limit value is:
η,lim  1
(4.3)
43
The definition of category IV vehicles and of the overturning criterion are outside the scope
of this thesis; the interested reader is referred to see UIC 518 [12].
4.2.1.4 Sum of Lateral Axle Box Forces Hmax
This limit value is only used in the simplified measuring method when measurement of lateral
axle box forces is carried out. The safety-critical limit is:
2Q 0 

H max,lim  k 2  10 

3 

(4.4)
[kN]
where Hmax,lim and 2Q0 (static axle load) are expressed in kN and the factor k2:
k2 = 0.90
for locomotives, power cars, multiple units and passenger coaches,
track-maintenance vehicles (and special vehicles in EN 14363 [11])
k2 = 0.75
for empty freight wagons (and special vehicles in UIC 518 [12])
k2 = 0.80
for loaded freight wagons (and special vehicles in UIC 518 [12])
The factor k2 takes into account the dynamic behaviour of the wheelset in the lateral
direction. To take account of greater variations in geometrical dimensions and of the state of
maintenance, smaller factor k2 is assumed for freight wagons. Exceptions are permissible in
well-founded individual cases.
4.2.1.5 Maximum Acceleration at the Bogie ÿ+max
The application of this assessment value is only used in the simplified measuring method
when measurement of lateral axle box forces is not carried out. Depending on the mass m+ of
the complete bogie (including wheelsets) the following limit values ÿ+max,lim are to be applied:
ÿ +max,lim  12 –
m+
5
[m / s 2 ]
(4.5)
where m+ is the mass in tonnes.
EN 14363 [11] allows for partial on-track tests with the simplified measuring method a
reduced limit value of ÿ+max,lim,simp shall be calculated at a third of the remaining margin
between the highest estimated maximum value of this assessment value and its limit value:
ÿ
with
+
max,lim,simp
(


 max Y  PA  max,normal 
)
,

ÿ + max,lim  max Y  PA  max,normal
3

[m / s 2 ] (4.6)
as the highest estimated maximum value of all test conditions
during the initial complete on-track test.
44
4.2.1.6 Maximum Accelerations in the Vehicle Body ÿ*Smax and ̈ *Smax
The limit value ÿ*Smax,lim is only used in the simplified measuring method when measuring of
lateral axle box forces is not carried out. The Table 4.1 shows the necessary limit values. This
table assumes that the static axle force 2Q0 is in kN and that ̈ *Smax,lim for empty freight
wagons is known to be a problem in test zone 1 and 2, it is currently being reviewed by UIC
so deviations from this limit value may be allowed.
EN 14363 [11] allows for partial on-track tests with the simplified measuring method a
reduced limit value of ÿ*Smax shall be calculated at a third of the remaining margin between the
highest estimated maximum value of this assessment value and its limit value:
ÿ
with
*
S,max,lim,simp
(


 max Y  PA max,normal 
)

ÿ* S,max,lim  max Y  PA  max,normal

3
[m / s 2 ] (4.7)
as the highest estimated maximum value of all test conditions
,
during the initial complete on-track test.
Limit values for (m/s²)
Vehicle, test conditions
ÿ*Smax,lim
Locomotives
Power Cars
Multiple Units
Passenger Coaches
̈ *Smax,lim
Single Suspension level or deflated air
spring condition
5
Double Suspension level
3
Test Zone 1 and 2
3
Test Zone 3
2.8
Test Zone 4
2.6
Freight Wagons (loaded) and Special Vehicles
5
Freight Wagons (empty)
5
Freight Wagons, Special Vehicles with Bogies
Freight Wagons
Special
Vehicles
Bogies
without
3
2Q0 < 60 kN
4
60 kN ≤ 2Q0 ≤ 200 kN
4.43 – 2Q0/140
2Q0 > 200 kN
3
Table 4.1: Limit values for Maximum Accelerations in the Vehicle Body
4.2.1.7 Instability Criterion
Depending on the applied measuring method and the vehicle type the following limit values
shall be used:
a) Normal measuring method: Sum of Guiding Forces,
45
ΣYrms,lim  ΣYmax,lim / 2
(4.8)
b) Simplified measuring method and measurement of lateral axle box forces: Sum
of lateral Axle Box Forces,
H rms,lim  H max,lim / 2
(4.9)
c) Simplified measuring method without measurement of lateral axle box forces on
non-bogie vehicles: Accelerations on Axle,
ÿrms,lim  5
[m / s 2 ]
(4.10)
d) Simplified measuring method without measurement of lateral axle box forces on
bogie vehicles: Accelerations at bogie frame,
ÿ+rms,lim  ÿ+max,lim / 2
(4.11)
4.2.2 Track Loading Limit Values
These values only apply for the Normal Measuring Method and are applicable to vehicles up
to maximum static wheel force of 112.5 kN. For operation of heavier vehicles on selected
tracks the limit values may be increased.
4.2.2.1 Quasi-static Guiding Force Yqst
EN 14363 [11] uses a constant limit value that is:
Yqst,lim  60
(4.12)
[kN]
for curved test zones with large, small and very small radius, excluding transition sections.
This limit value is known to be a problem for vehicles in curved test zones with very small
radius. It is currently being reviewed by UIC. Deviations from this value may be allowed. On
the other hand UIC 518 uses the following limit value is:

10500 
Yqst ,lim   30 

Rm 

[kN]
(4.13)
being Rm the mean radius of the track sections retained for the evaluation. When this limit
value is exceeded due to severe friction conditions, it is allowed to recalculate the estimated
value of Yqst on the zone after replacing the individual (Yqst)i values on the track sections “i”
where (Y/Q)ir (mean value of Y/Q ratio on the inner rail over the section) exceeds 0.40 by:
(Yqst)i – 50[(Y/Q)ir – 0.4]. Both results (before and after the recalculation) shall be reported.
46
4.2.2.2 Quasi-static Wheel Force Qqst
This limit value is:
Qqst,lim  145
(4.14)
[kN]
for curved test zones with large, small and very small radius, excluding transition sections.
UIC 518 [12] makes an exception for freight trains with Q0 > 112.5 kN and Vadm ≤ 100 km/h where it is:
Qqst ,lim  155
(4.15)
[kN]
4.2.2.3 Maximum Wheel Force Qmax
This limit value is:
Qmax,lim  90  Q0
[kN]
(4.16)
where Qmax,lim and Q0 are expressed in kN, Q0 being the static loading on each wheel and are
limited to the values present on Table 4.2 depending on the permissible maximum speed of
the vehicle Vadm.
Vadm* ≤ 100 km/h:
Vadm
≤ 160 km/h:
160 km/h <
Vadm
≤ 200 km/h:
200 km/h <
Vadm
≤ 250 km/h:
250 km/h <
Vadm
≤ 300 km/h:
Vadm
> 300 km/h:
*For freight trains with Q0 > 112.5 kN only according to UIC 518 [12]
Qmax,lim* ≤ 210 kN
Qmax,lim ≤ 200 kN
Qmax,lim ≤ 190 kN
Qmax,lim ≤ 180 kN
Qmax,lim ≤ 170 kN
Qmax,lim ≤ 160 kN
Table 4.2: Limit values for Track Loading
Where Vadm is the vehicle’s operating speed limit. The limiting value to be selected is
the smaller of the values obtained by applying the law of variation and the limitation due to
speed. The track loading limit values take into account rails with a weight ≥ 46 kg/m and the minimum values of rail strength of 700 N/mm2.
4.2.2.4 Quasi-static Track Loading Forces Bqst
UIC 518 [12] considers an additional criteria on curved test zones with large, small and very
small radius, excluding transition sections. The limit value is:
B 
qst lim
 185
[kN]
where:
47
(4.17)


10500  
Bqst  Yqst  0.83Qqst   a –  30 

Rm  


(4.18)
with a = 53.3 for small radius curves or a = 67.5 for very small curves.
In case of severe friction conditions it is allowed to use the recalculated estimated value
of Yqst on the test zone using (Yqst)i - 50[(Y/Q)ir – 0.4] for each test section “i” where (Y/Q)ir >
0.4. This limit value is based on the fatigue strength of the rail type 49 E1 (S 49) [127]. In
cases where (Bqst)lim is exceeded, a reduction of operating speed in curves may be considered.
4.2.2.5 Limit Values of the Ride Characteristics
For the assessment of the vehicle’s ride characteristics the following accelerations are used:
a) Quasistatic Accelerations in the Vehicle Body ÿ*qst
b) Maximum Accelerations in the Vehicle Body ÿ*max, ̈ *max
c) Root mean square of Accelerations in the Vehicle Body ÿ*rms, ̈ *rms
Assessment, Vehicle, Test Conditions
Limit Values for Accelerations in Vehicle Body (m/s²)
ÿ*qst,lim
ÿ*max,lim
z̈ *max,lim
ÿ*rms,lim
z̈ *rms,lim
Locomotives, Power Cars
1.5
2.5
2.5
0.5
1.0
Multiple Units, Passenger Coaches
1.5
2.5
2.5
0.5
0.75
Freight Wagon, Special Vehicles with Bogies
1.3
3.0
5.0
1.3
2.0
Freight Wagon, Special Vehicles without Bogies
1.3
4.0
5.0
1.5
2.0
Ride Characteristics
Table 4.3: Limit values for Ride Characteristics
Table 4.3 shows the values for good ride characteristics. If higher values occur, the influence
on passengers or loading safety and the strength of the vehicle and its mounted parts shall be
regarded. Number and duration of the incidents as well as the service concept shall be considered.
ÿ*qst,lim only is applicable in curved test zones. For degraded suspension conditions (see section
5.4.3.4 of EN 14363 [11]) running safely will be respected according to the limits in 4.2.
4.3 Experimental Tests
This section contains the information collected from EN 14636 [11] and UIC 518 [12] related
to the procedures required to analyse the data collected from experimental tests.
48
4.3.1 Recording the Measuring Signals
In principle, the measuring signals of all measured parameters and influencing parameters
intended for subsequent evaluation shall be recorded using machine-readable data carriers.
For the recording of the measuring signals, a low-pass filter shall be used. The cut-off value
of the frequency depends on the type of recording and of the type of parameter:
a) ≥ 40 Hz for data carriers
b) Graphical representation:
–
Lateral parameters: ≥ 10 Hz
–
Vertical parameters: ≥ 20 Hz
4.3.2 Processing the Measuring Signals
The filtering for recording and evaluation, method of classification and numerical values of
the accumulative curve are effective during the processing of measuring signals and affect the
characteristics values of frequency distribution and consequently all the dependent results.
Therefore, conditions once defined shall not be altered without good reason in order to
prevent systematic deviations and for comparability reasons.
The method of classification is taken to mean a specific method for the acquisition of
random vibrations. Applied methods of classification include the following:
1) Sampling Method: At specified intervals, the instantaneous value of the variable is
determined and counted according to classes.
2) Sliding Mean Method: First, the arithmetic mean is determined from a specific number
of instantaneous values over the window length. This mean shall be classified. A new
mean, displaced by the sampling step, shall be created and also classified.
3) Sliding RMS Method: The rms-value is calculated from a specific number of
instantaneous values (window length), a new rms-value shall be calculated displaced
by the sampling step length.
Table 4.4 and Table 4.5 give the conditions that apply to the processing of measuring
signals according to EN 14363 [11] and UIC 518 [12], respectively. The test zones are: 1)
strait track and curved tracks with very large radius, 2) curved track with a large radius, 3)
curved track with a small radius and 4) curved track with a very small radius.
49
Assessment
Symbol
Value
Running Safely
Sum
of
Guiding
ΣYmax
Forces
Wheelset 1, 2
Sum
of
Lateral Axle
Hmax
Box Forces
Wheelset 1, 2
Quotient
Leading
Wheelset
Acceleration
at Bogie
Wheelset 1, 2
Acceleration
in
Vehicle
Body
End I, II
Instability
Criterion
Filtering for
Evaluation
Low-pass
filter: 20 Hz
Method of
Classification
Low-pass
filter: 10 Hz
ÿ*smax
Low-pass
filter: 6 Hz
ÿrms
3
3
3
3
Per
end
group:
xj(h2) and
xj(h1)*(-1)
Random
Sampling
Method
Band-pass
filter: 0.4-4
Hz
Band-pass
filter:
f0 ± 2 Hz
k
Per wheelset group:
xj(h2) for left hand curves
xj(h1)*(-1) for right hand curves
For leading wheelset group:
x11(h2) for left hand curves
x12(h1)*(-1) for right hand
curves
h1 = 0.15%
h2 = 99.85%
ÿ+max
ΣYrms
Hrms
ÿ+rms
Grouping and Conversion
Test Zone 1
Test Zone 2, 3, 4
Per wheelset
group:
xj(h2) and
xj(h1)*(-1)
Sliding
Mean
Method:
Window length:
2.0 m
Step length: 0.5
m
(Y/Q)max
̈ *smax
Characteristic
Values
3
Per end group:
xj(h2) and xj(h1)*(-1)
Sliding
RMS
Method:
–
Window
length: 100 m
– Step length: 10
m
Max-Values
Per wheelset
Per wheelset
3
-
Track Loading
Guiding
Force
Wheelset 1, 2
Wheel Force
Wheels 11,
12, 21, 22
Yqst
Qqst
Low-pass
filter: 20 Hz
Random
Sampling
Method
h0 = 50.0%
h2 = 99.85%
Qmax
xjk(h2)
Per wheelset group:
xj1(h0) for left hand curves
xj2(h0)*(-1) for right hand
curves
Per bogie group:
xj1(h0) for left hand curves
xj2(h0) for right hand curves
Per bogie group:
xj1(h2) for left hand curves
xj2(h2) for right hand curves
0
0
2.2
Ride Characteristics
Acceleration
in
Vehicle
Body
End I, II
ÿ*qst
ÿ*max
̈ *max
ÿ*rms
̈ *rms
Influencing Parameters
Speed
V
Cant
cd
Defiency
Low-pass
filter: 20 Hz
Band-pass
filter:
0.4-10 Hz
Low-pass
filter: 4 Hz
h0 = 50.0%
Random
Sampling
Method
h1 = 0.15%
h2 = 99.85%
Per end group:
xj1(h0) for left hand curves
xj2(h0) for right hand curves
Per end group: xj(h2) and xj (h1)*(-1)
rms-values
Random
Sampling
Method
h0 = 50.0%
Table 4.4: Conditions for the processing of the measuring signals from EN 14363 [11]
50
2.2
2.2
2.2
0
0
Assessment
Value
Symbol
Running Safely
Sum
of
Guiding
Forces
(ΣY)2m
All
instrumented
wheelsets
Sum
of
Lateral Axle
(H)2m
Box Forces
Wheelset 1, 2
Quotient
Leading
(Y/Q)2m
Wheelset
Filtering
for
Evaluation
Method of
Classification
Low-pass
filter: 20 Hz
Sliding
Mean
Method
Window length:
2.0 m
Step length: 0.5
m
Overturning
Criterion
η
Low-pass
filter: 1.5 Hz
Acceleration
at Bogie
Outer
wheelsets
ÿ +s
Low-pass
filter: 10 Hz
Acceleration
in
Vehicle
Body
End I, II
ÿ* s
Low-pass
filter: 6 Hz
Instability
Criterion
All
instrumented
wheelsets
ΣY
H
ÿ +s
ÿ* s
ÿs
Track Loading
Guiding
Force
All wheels on
instrumented
wheelsets
Wheel Force
All wheels on
instrumented
wheelsets
̈ *s
Characteristic
Values
Grouping and Conversion
Test Zone 1
Per wheelset
group:
xj(h2)
and
|xj (h1)|
h1 = 0.15%
h2 = 99.85%
Random
Sampling
Method
Band-pass
filter:
f0 ± 2 Hz
Per
bogie
group:
xj(h2)
and
|xj (h1)|
Per wheelset
group:
xj(h2) and
|xj (h1)|
Per
end
group:
xj(h2) and
|xj (h1)|
For leading wheelset group:
x11(h2) for right hand curves
|x12(h1)| for left hand curves
3
Per bogie group:
xj (h2) for right hand curves
|xj(h1)| for left hand curves
3
Per wheelset group:
xj(h2) for right hand curves
|xj(h1)| for left hand curves
3
Per end group:
xj(h2) for right hand curves
|xj(h1)| for left hand curves
3
Per end group:
xj(h2) and |xj (h1)|
3
Per wheelset group:
xj1(h0) for right hand curves
|xj2(h0)| for left hand curves
Yqst
Qqst
3
Per wheelset group:
xj(h2) for right hand curves
|xj(h1)| for left hand curves
Sliding RMS Method:
– Window length: 100 m
– Step length: 10 m
Random
Sampling
Method
h0 = 50.0%
h2 = 99.85%
Q
k
3
Band-pass
filter: 0.4-4
Hz
Low-pass
filter: 20 Hz
Test Zone 2, 3, 4
xjk(h2)
Per bogie group:
xj1(h0) for right hand curves
xj2(h0) for left hand curves
Per bogie group:
xj1(h2) for right hand curves
xj2(h2) for left hand curves
-
0
0
2.2
Ride Characteristics
Acceleration
in
Vehicle
Body
End I, II
ÿ*qst
ÿ* q
̈ *q
sÿ*q
s ̈ *q
Low-pass
filter: 20 Hz
Band-pass
filter:
0.4-10 Hz
h0 = 50.0%
Random
Sampling
Method
Per end group:
xj1(h0) for right hand curves
xj2(h0) for left hand curves
0
h1 = 0.15%
h2 = 99.85%
Per end group: xj(h2) and |xj (h1)|
2.2
rms-values
Per end group: rms-values
2.2
Table 4.5: Conditions for the processing of the measuring signals from UIC 518 [12]
The sampling frequency of the Random Sampling Method should be at least 200 Hz.
The f0 is the instability frequency, defined as the dominant frequency in the case of unstable
behaviour and must be determined before evaluation of test results. Filter with cut-off
frequency at -3 dB, gradient ≥ 24 dB/octave, tolerance ± 0.5 dB up to the cut-off frequency, ±
1 dB beyond that value.
51
4.3.2.1 Filtering Raw Data
An ideal filter completely eliminates all frequencies outside of the passable frequency band
while passing those inside unchanged. The transition region present in practical filters does
not exist in an ideal filter. However, the ideal filter is impossible to realize without a signal of
infinite extent in time, and so generally needs to be approximated for real ongoing signals.
There are many different types of filter circuits, with different responses to changing
frequency. The frequency response of a filter is generally represented using a Bode plot, and
the filter is characterised by its cut-off frequency and rate of frequency roll-off. In all cases, at
the cut-off frequency, the filter attenuates the input power by half or 3 dB. In the case of the
Butterworth filter, at the cut-off frequency the filter always attenuates the input power by 3
dB, regardless of its order, unlike other filters.
The order of the filter then determines the amount of additional attenuation for
frequencies outside the passable frequency window. In general, the final rate of power roll-off
for an order-n all-pole filter is 6n dB/octave (i.e., 20n dB/decade). Here the poles define the
order of the filter. In Figure 4.6, a series of low-pass Butterworth filters with various orders
and their influence in the attenuation of the signal is presented. As shown, the signal suffers
almost no attenuation before the cut-off frequency and is much reduced after it, with the
greater the number of poles influencing that attenuation.
40
Attenuation [dB]
0
-40
-80
Order-1
Order-2
Order-3
Order-4
Order-5
-120
-160
-200
0,01
0,1
1
10
100
Frequency [Hz]
Figure 4.6: Bode Plot of low-pass Butterworth filters with a cut-off frequency of 1 Hz and
variable order (from 1 to 5)
4.3.2.2 The Sliding Mean Method
To use the Sliding Mean Method there are two required parameters, the Window Length and the
Step Length. With them, a series of windows are calculated, each with the length given by the
Window Length and displaced by the Step Length in relation to each other, so that the mean
52
value for those windows is possible to calculate. This results in the cut of the initial and end
data. Figure 4.7 presents some of the windows that will have their content averaged to a single
point, as an example of the process needed to determine the data set that to be averaged.
6
4
Data Value (y)
2
0
-2
-4
-6
-8
-10
-12
-14
0
10
20
30
40
50
60
Displacement [m]
Figure 4.7: Filtered data and some of the windows used in the calculation of the Sliding Mean
(window length: 4 m and step length: 1 m) vs. displacement graph
Figure 4.8 shows an example of a Sliding Mean with a window length of 4 m and a step
length of 1 m when applied to a set of data. As shown, the resulting signal has fewer peaks
and is much smoother.
4
2
Data Value (y)
0
-2
-4
-6
-8
Filtered Data
Sliding Mean Data
-10
-12
0
10
20
30
40
50
60
Displacement [m]
Figure 4.8: Filtered data and its Sliding Mean (window length: 4 m and step length: 1 m) vs.
displacement graph
4.3.2.3 The Sliding RMS Method
To use the Sliding Root Mean Square Method there are also two required parameters, the
Window Length and the Step Length. With them, a series of windows are calculated, each
with the length given by the Window Length and displaced by the Step Length in relation to
each other, so that the square root of the mean value of the squares of the data for those
53
windows is possible to calculate. This is also results in the cut of the initial and end data like
the Sliding Mean Method, but ignores the sign of the signal, seeing it as its absolute value.
Figure 4.9 presents some of the windows here the RMS method will be applied and
converted to a single point, as an example of the process needed to determine the data set
where the RMS method will be applied.
6
4
Data Value (y)
2
0
-2
-4
-6
-8
-10
-12
-14
0
10
20
30
40
50
60
Displacement [m]
Figure 4.9: Filtered data and some of the windows used in the calculation of the Sliding RMS
(window length: 4 m and step length: 1 m) vs. displacement graph
Figure 4.10 shows an example of a Sliding RMS with a window length of 4 m and a
step length of 1 m when applied to a set of data. The resulting signal has much less peaks and
is much smoother, using the absolute value of the raw data.
15
Data Value (y)
10
5
0
-5
-10
Filtered Data
Sliding RMS Data
-15
0
10
20
30
40
50
60
Displacement [m]
Figure 4.10: Filtered data and its Sliding RMS (window length: 4 m and step length: 1 m) vs.
displacement graph
4.3.2.4 Calculation of Characteristic Values for Track Sections
From the measuring signals, which were processed in accordance with the tables present in
4.3.2 the cumulative curve shall be determined by the sum of the absolute values y ordered from
smallest to greatest, for any parameter being processed. So the absolute values from the filtered
54
data, shown in Figure 4.11, are ordered from smallest to greatest, as represented by Figure 4.12.
Now the accumulative values are determined and then divided by the maximum accumulated
value as to obtain the percentile accumulative curve, as represented in Figure 4.13.
4
2
Data Value (y)
0
-2
-4
-6
-8
-10
-12
0
10
20
30
40
50
60
Displacement [m]
Figure 4.11: Filtered data vs. displacement graph
12,00
Data Value (y)
10,00
8,00
6,00
4,00
2,00
0,00
0
10
20
30
40
50
60
Position x(h)
Figure 4.12: Reordered absolute data vs. position graph
100
90
Frequency (h) [%]
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
Position x(h)
Figure 4.13: Cumulative percentile curve graph
55
60
From the cumulative curve (Figure 4.13), the frequency values x(hj) can be obtained, so that
it is possible to determine the value of y correspondent to the desired value of hj from Figure 4.12:
–
x(h1), frequency of cumulative curve h1 =
0.15%
–
x(h0), frequency of cumulative curve h0 =
50.00%
–
x(h2), frequency of cumulative curve h2 =
99.85%
4.3.2.5 Calculation of Characteristic Values for Test Zones
The values y, gathered for each section of a zone as described in 4.3.2.4, are then used to
determine the value for the zone. To do that, it is calculated the arithmetic mean
and the
standard deviation s for the quantities grouped as seen in the tables present in 4.3.2. These
statistical values serve to determine the estimated maximum value, using the equation ymax =
+ k ∙ s, where k is a factor that depends, among other things, on the level of confidence
selected (and is present in the tables in 4.3.2).
56
5 Post-Processor Application
The post-processor tool is applied here to two case studies using the Simplified Method and
only considers the accelerations measurements. Both cases were taken from the experimental
data gathered on the Vouga track, in Portugal, and studied under the Vouga Project. While the
first case is fully described, the second case only present the raw data, its derived characteristic
values and the conclusions; following the same procedures as the ones presented for the first
case. Table 5.1 shows the required data for a Simplified Method analysis.
Assessment
Symbol
Value
Running Safely
Acceleration
at Bogie
ÿ +s
Outer
wheelsets
Acceleration
in
Vehicle
Body
End I, II
ÿ
*
s
̈ *s
Instability
Criterion
All
ÿ +s
instrumented
wheelsets
Ride Characteristics
Acceleration
in
Vehicle
Body
End I, II
ÿ*qst
ÿ* q
̈ *q
sÿ*q
s ̈ *q
Filtering for
Evaluation
Method of
Classification
Characteristic
Values
Low-pass
filter: 10 Hz
Low-pass
filter: 6 Hz
Random
Sampling
Method
h1 = 0.15%
h2 = 99.85%
Band-pass
filter: 0.4-4
Hz
Band-pass
filter:
f0 ± 2 Hz
Low-pass
filter: 20 Hz
Band-pass
filter:
0.4-10 Hz
Grouping and Conversion
Test Zone 1
Test Zone 2, 3, 4
Per wheelset
group:
xj(h2) and
|xj(h1)|
Per
end
group:
xj(h2) and
|xj(h1)|
Per wheelset group:
xj(h2) for right hand curves
|xj(h1)| for left hand curves
3
Per end group:
xj(h2) for right hand curves
|xj(h1)| for left hand curves
3
Per end group:
xj(h2) and |xj(h1)|
3
Sliding RMS Method:
– Window length: 100 m
– Step length: 10 m
h0 = 50.0%
Random
Sampling
Method
h1 = 0.15%
h2 = 99.85%
rms-values
k
Per end group:
xj1(h0) for right hand curves
xj2(h0) for left hand curves
-
0
Per end group: xj(h2) and |xj(h1)|
2.2
Per end group: rms-values
2.2
Table 5.1: Conditions for the processing of the measuring signals for the Simplified Method
from UIC 518 [12]
5.1 Case Study 1
As it will be shown, this case study fails to respect all limit values, namely fails two safety
limit values and, therefore, this vehicle would not be allowed to operate on the analysed track
at the considered speed.
5.1.1 Measured Raw Data
Based on the information provided by Table 5.1, the raw data required would be ÿ+, ÿ* and ̈ *,
as presented in Figure 5.1, Figure 5.2 and Figure 5.3. This data was measured directly from
the vehicle and is unprocessed.
57
25
20
Accelaration [g]
15
10
5
0
-5
-10
-15
-20
0
5
10
Time [s]
+
15
20
10
Time [s]
15
20
15
20
Figure 5.1: Raw ÿ vs. time graph
1,5
Accelaration [g]
1
0,5
0
-0,5
-1
-1,5
-2
0
5
Figure 5.2: Raw ÿ* vs. time graph
4
3
Accelaration [g]
2
1
0
-1
-2
-3
-4
-5
0
5
10
Time [s]
Figure 5.3: Raw ̈ * vs. time graph
5.1.2 Filtered Data
A filter that fulfils all the criteria defined by EN 14636 [11] and UIC 518 [12] is a 4th-order
Butterworth filter. It will be the one used in this case study. Several filtering procedures need to be
applied to the data presented in Figure 5.1, Figure 5.2 and Figure 5.3. This will be explained in the
following, step by step.
58
5.1.2.1 Safety Parameters
Acceleration at Bogie ÿ+S: The lateral acceleration at the bogies ÿ+S is filtered with a lowpass filter at 10 Hz (see 4.3.2 after Table 4.5 for the remaining filter characteristics). The
results are shown in Figure 5.4.
2,5
2
Accelaration [g]
1,5
1
0,5
0
-0,5
-1
-1,5
-2
-2,5
-3
0
5
Figure 5.4:
10
Time [s]
Filtered ÿ+S vs.
15
20
time graph
Acceleration in Vehicle Body ÿ*S and ̈ *S: The lateral acceleration in the vehicle body ÿ*S is
filtered with a low-pass filter at 6 Hz and the vertical acceleration in the vehicle body ̈ *S is
filtered with a band-pass filter at 0.4-4 Hz (see 4.3.2 after Table 4.5 for the remaining filter
characteristics). The results are shown in Figure 5.5 and Figure 5.6, respectively.
0
-0,02
Accelaration [g]
-0,04
-0,06
-0,08
-0,1
-0,12
-0,14
-0,16
0
5
Figure 5.5:
10
Time [s]
Filtered ÿ*S vs.
15
20
time graph
Instability Criterion ÿ+S: The Instability Criterion defined by ÿ+S is filtered with a band-pass
filter at f0 ± 2 Hz (see 4.3.2 after Table 4.5 for the remaining filter characteristics). In this
instance the instability frequency f0 was determined to be at 89.57 Hz. The results are shown
in Figure 5.7.
59
0,3
Accelaration [g]
0,2
0,1
0
-0,1
-0,2
-0,3
0
5
Figure 5.6:
10
Time [s]
Filtered ̈ *S vs.
15
20
time graph
4
3
Accelaration [g]
2
1
0
-1
-2
-3
-4
0
5
10
Time [s]
15
20
Figure 5.7: Instability Criterion ÿ+S vs. time graph
5.1.2.2 Ride Characteristics
Acceleration in Vehicle Body ÿ*qst: The lateral acceleration in the vehicle body ÿ*qst is filtered
with a low-pass filter at 20 Hz (see 4.3.2 after Table 4.5 for the remaining filter characteristics).
The results are shown in Figure 5.8.
3
Accelaration [g]
2
1
0
-1
-2
-3
-4
0
5
Figure 5.8:
10
Time [s]
Filtered ÿ*qst vs.
60
15
time graph
20
Acceleration in Vehicle Body ÿ*q, ̈ *q, sÿ*q and s ̈ *q: The lateral and vertical accelerations in
the vehicle body ÿ*q, ̈ *q, sÿ*q, s ̈ *q are filtered with a band-pass filter at 0.4-4 Hz (see 4.3.2
after Table 4.5 for the remaining filter characteristics). The results are shown in Figure 5.9
and Figure 5.10, respectively.
0,15
Accelaration [g]
0,1
0,05
0
-0,05
-0,1
0
5
Figure 5.9:
10
Time [s]
Filtered ÿ*q vs.
15
20
time graph
0,4
0,3
Accelaration [g]
0,2
0,1
0
-0,1
-0,2
-0,3
-0,4
-0,5
0
5
Figure 5.10:
10
Time [s]
Filtered ̈ *q vs.
15
20
time graph
5.1.3 Classification Method
The direct use of the accelerations at the bogie and vehicle body is possible if they include
more than 200 measurements per second. As such, no action is required in this step for this
Simplified Method.
5.1.4 Characteristic Values for Track Sections
Each section of a given zone is characterised by values that define it. In this case study, it is
considered that each zone has 50 sections equally spaced, in order to accommodate the
61
smallest track sections. A step by step example will be given for the first section of the first
assessment value, with the remaining sections following similar methodologies.
5.1.4.1 Safety Parameters
Acceleration at Bogie ÿ+S: The lateral acceleration at the bogies ÿ+S is characterised by the
values at h1 = 0.15% and at h2 = 99.85% using the accumulative curve. The filtered data
presented in Figure 5.4 is divided in 50 equal sections and reordered using the method
presented in 4.3.2.4 and then used to build the cumulative curve. The reordered absolute data
from the first section is shown in Figure 5.11, while the cumulative curve for the first section
is shown in Figure 5.12.
0,40
Acceleration (y) [g]
0,35
0,30
0,25
0,20
0,15
0,10
0,05
0,00
0
50
100
150
200
Position x(h)
Figure 5.11: Reordered absolute ÿ+S data vs. position graph for the first section
100
90
Frequency (h) [%]
80
70
60
50
40
30
20
10
0
0
50
100
150
200
Position x(h)
Figure 5.12: Cumulative curve of ÿ+S vs. position graph for the first section
From the cumulative curve it is possible to acquire the position for a given h value and from
that its corresponding acceleration. Resulting in the first section having an y(h1) = 0.006985 g and
an y(h2) = 0.373682 g. The Figure 5.13 shows the y(hi) values for each section along the zone.
62
2,5
y(h1)
y(h2)
Acceleration (y) [g]
2
1,5
1
0,5
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.13: Characteristic values of ÿ+S for each section of a zone
Acceleration in Vehicle Body ÿ*S and ̈ *S: The lateral acceleration in the vehicle body ÿ*S is
characterised by values h1 = 0.15% and h2 = 99.85% using the accumulative curve. The
filtered data presented in Figure 5.5 is divided in 50 equal sections and reordered using the
method presented in 4.3.2.4 and then used to build the cumulative curve. Following a similar
procedure, Figure 5.14 shows the y(hi) values for each section along the zone. The same
method is applied to the vertical acceleration in the vehicle body ̈ *S, as shown in Figure 5.15.
0,18
y(h1)
0,16
y(h2)
Acceleration (y) [g]
0,14
0,12
0,1
0,08
0,06
0,04
0,02
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.14: Characteristic values of ÿ*S for each section of a zone
0,3
y(h1)
y(h2)
Acceleration (y) [g]
0,25
0,2
0,15
0,1
0,05
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.15: Characteristic values of ̈ *S for each section of a zone
63
Instability Criterion ÿ+S: The Instability Criterion defined by ÿ+S uses the RMS Method (with
the characteristics presented in Table 5.1). The resulting graph is represented by Figure 5.16.
0,7
0,6
Accelaration [g]
0,5
0,4
0,3
0,2
0,1
0
0
5
Figure 5.16:
10
Time [s]
RMS ÿ+S vs.
15
20
time graph
5.1.4.2 Ride Characteristics
Acceleration in Vehicle Body ÿ*qst: The lateral acceleration in the vehicle body ÿ*qst is characterised
by the values at h0 = 50% using the accumulative curve. The filtered data presented in Figure 5.8 is
divided in 50 equal sections and reordered using the method presented in 4.3.2.4 and then used to
build the cumulative curve. Figure 5.17 shows the y(h0) values for each section along the zone.
2,5
Acceleration (y) [g]
2
1,5
1
0,5
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.17: Characteristic values of ÿ*qst for each section of a zone
Acceleration in Vehicle Body ÿ*q, ̈ *q, sÿ*q and s ̈ *q: The lateral acceleration in the vehicle
body ÿ*q is characterised by the values at h1 = 0.15% and at h2 = 99.85% using the
accumulative curve. The filtered data presented in Figure 5.9 is divided in 50 equal sections
and reordered using the method presented in 4.3.2.4 and then used to build the cumulative
curve. Following a similar procedure, Figure 5.18 shows the y(hi) values for each section
64
along the zone. The same method is applied to the vertical acceleration in the vehicle body
̈ *q. The results are represented in Figure 5.19.
The lateral and vertical accelerations in the vehicle body sÿ*q and s ̈ *q use the rms value
for each section. The results are represented in Figure 5.20 and Figure 5.21, respectively.
0,12
y(h1)
y(h2)
Acceleration (y) [g]
0,1
0,08
0,06
0,04
0,02
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.18: Characteristic values of ÿ*q for each section of a zone
0,4
y(h1)
y(h2)
Acceleration (y) [g]
0,35
0,3
0,25
0,2
0,15
0,1
0,05
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.19: Characteristic values of ̈ *q for each section of a zone
0,05
0,045
Acceleration (y) [g]
0,04
0,035
0,03
0,025
0,02
0,015
0,01
0,005
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.20: Characteristic values of the sÿ*q for each section of a zone
65
0,2
0,18
Acceleration (y) [g]
0,16
0,14
0,12
0,1
0,08
0,06
0,04
0,02
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
Section
Figure 5.21: Characteristic values of the s ̈ *q for each section of a zone
5.1.5 Characteristic Values for Test Zones
After the analysis for each track section the characteristic value can be calculated for the
whole zone using the values of the track section.
5.1.5.1 Safety Parameters
Acceleration at Bogie ÿ+S: For straight track zones, the lateral acceleration at the bogies ÿ+S is
determined by the method explained in 4.3.2.5, using all the values gathered by all the
sections contained within the zone, i.e., the sum of the average plus three times the standard
deviation of the sections’ characteristic values for ÿ+S. For the analysed zone, ÿ+S = 2.1962 g.
For curved track zones, the lateral acceleration at the bogies ÿ+S is determined by the
same method, but using only some of the values required, as shown in Table 5.1. For the
analysed zone it was obtained ÿ+S = 2.5313 g.
Acceleration in Vehicle Body ÿ*S and ̈ *S: For straight track zones, the lateral acceleration in
the vehicle body ÿ*S is determined by the method explained in 4.3.2.5 using all the values
gathered by all the sections contained within the zone. For the analysed zone, ÿ*S = 0.1664 g.
For curved track zones, the lateral acceleration in the vehicle body ÿ*S is determined by the
same method, but using only some of the values required as shown by Table 5.1 and for the
analysed zone the computed value is ÿ*S = 0.1511 g. For the vertical acceleration in the vehicle
body ̈ *S is determined by the same method, but only applied to the curved test zones and for
the analysed zone it was obtained ̈ *S = 0.1883 g.
66
Instability Criterion ÿ+S: The Instability Criterion defined by ÿ+S is only valid for strait and
large radius curve test zone using the method explained in 4.3.2.5 for the maximum value in
the zone. For the present case, ÿ+S = 0.6475 g.
5.1.5.2 Ride Characteristics
Acceleration in Vehicle Body ÿ*qst: The lateral acceleration in the vehicle body ÿ*qst is only relevant
for curve test zones and is determined by the method explained in 4.3.2.5 using all the values
gathered by all the sections contained within the zone. For the analysed zone, ÿ*qst = 0.7312 g.
Acceleration in Vehicle Body ÿ*q, ̈ *q, sÿ*q and s ̈ *q: For straight and curved track zones, the
lateral acceleration in the vehicle body ÿ*q is determined by the method explained in 4.3.2.5
using all the values gathered by all the sections contained within the zone. For the analysed
zone, ÿ*q = 0.0849 g. For the vertical acceleration in the vehicle body ̈ *q is determined by the
same method and for the analysed zone ̈ *q = 0.2182 g. For the lateral acceleration in the
vehicle body sÿ*q is determined by the same method and for the analysed zone sÿ*q = 0.0408
g. For the vertical acceleration in the vehicle body s ̈ *q is determined by the same method and
for the analysed zone s ̈ *q = 0.1254 g.
5.1.6 Discussion
After determining the characteristic values for each track zone it is then possible to compare
them to the limit values imposed by EN 14636 [11] and UIC 518 [12] and previously
presented in section 4.2.
5.1.6.1 Safety Parameters
Acceleration at Bogie ÿ+S: In the present case the bogie with all its constituent parts and
wheelsets weights 4 tonnes, and the limit value is defined by equation (4.5), so we have that
the limit for the lateral acceleration at the bogie is:
(ÿ+S )lim  12 
mb
4
 12   11.2
5
5
[m / s 2 ]
(5.1)
or about 1.1417 g. This limit value is lower than the maximum encountered for the analysed
zone (2.5313 g) and so the operation is outside of the safety limits.
67
Acceleration in Vehicle Body ÿ*S and ̈ *S: The limit value for the lateral acceleration in the
vehicle body ÿ*S in straight tracks and curved tracks with large radius is 3 m/s2 or 0.3058 g. This
limit is much higher than the maximum encountered of 0.1664 g. The limit value becomes 2.8
m/s2 or 0.2854 g in curved tracks with small radius and 2.6 m/s2 or 0.2650 g in curved tracks
with very small radius, both of which are respected. For the vertical acceleration in the vehicle
body ̈ *S the limit value depends on the type of suspension that the vehicle has, with single
suspension level or deflated air spring condition as 5 m/s2 (0.5097 g) and double suspension
level 3 m/s2 (0.3058 g), both of which are much higher than the maximum found of 0.1883 g.
Instability Criterion ÿ+S: The limit value for the Instability Criterion defined by ÿ+S is half of
the safety limit value of the lateral acceleration at the bogie, defined in equation (5.1). In this
case it becomes 5.6 m/s2 or 0.5708 g. In this instance, the maximum value encountered (0.6475
g) also surpasses this limit and as such the vehicle instability surpasses the safety limits.
Safety Assessment: In the present case the lateral accelerations at the bogie ÿ+S exceed both
the safety limits values for such acceleration and the stability limits. As such the vehicle
cannot circulate at the analysed speed, but it is likely that at a lower speed these concerns
cease to exist and so further research is required. Table 5.2 presents the condensed results
concerning the safety of Case Study 1.
Parameter
ÿ+S
ÿ* S
̈ *S
Instability ÿ+S
Obtained Characteristic Value
2.5313 g
0.1664 g
0.1883 g
0.6475 g
Limit Value
1.1417 g
0.2650 g
0.3058 g
0.5708 g
Conclusion
Not Approved
Approved
Approved
Not Approved
Table 5.2: Case Study 1 safety approval table
5.1.6.2 Ride Characteristics
The limit values for the ride characteristics are greatly influenced by the type of vehicle it is.
In this case we will consider this vehicle as a Power Car and its correspondent limit values
can be found in Table 4.3.
Acceleration in Vehicle Body ÿ*qst: The limit for the lateral acceleration at the bogies ÿ*qst for
a Power Car is 1.5 m/s2 or 0.1529 g, which is much lower than the maximum encountered of
0.7312 g. As such the vehicle’s comfort is outside of the established comfort limit.
Acceleration in Vehicle Body ÿ*q, ̈ *q, sÿ*q and s ̈ *q: The limit for the lateral acceleration in the
vehicle body ÿ*q on a Power Car is 2.5 m/s2 or 0.2548 g, which is higher than the maximum
68
encountered of 0.0849 g. For the vertical acceleration in the vehicle body ̈ *q the limit on a Power
Car is 2.5 m/s2 or 0.2548 g, which is higher than the maximum encountered of 0.2182 g. For the
lateral acceleration in the vehicle body sÿ*q the limit on a Power Car is 0.5 m/s2 or 0.0510 g, which
is higher than the maximum encountered of 0.0405 g. For the vertical acceleration in the vehicle
body s ̈ *q the limit on a Power Car is 1.0 m/s2 or 0.1019 g, which is lower than the maximum
encountered of 0.1254 g. As such the vehicle’s comfort is outside of the established limit.
Ride Characteristics Assessment: In the present case the both lateral and vertical accelerations
in the vehicle body exceed the quality limits imposed for a comfortable ride. The vehicle can
still circulate at the analysed speed (if it didn’t exceed the safety limits, as shown in 0), but it is
likely that at a lower speed this problem is reduced or completely eliminated. Table 5.3 presents
the condensed results concerning the ride characteristics of Case Study 1.
Parameter
ÿ*qst
ÿ* q
̈ *q
sÿ*q
s ̈ *q
Obtained Characteristic Value
0.7312 g
0.0849 g
0.2182 g
0.0405 g
0.1254 g
Limit Value
0.1529 g
0.2548 g
0.2548 g
0.0510 g
0.1019 g
Conclusion
Not Approved
Approved
Approved
Approved
Not Approved
Table 5.3: Case Study 1 ride characteristics approval table
5.2 Case Study 2
This case study respects all limit values and, therefore, this vehicle would is able to operate on
the analysed track. The measured data is processed exactly the same way as shown for Case
Study 1. Hence, only the raw data and the results are shown.
5.2.1 Measured Raw Data
Based on the information provided by Table 5.1, the raw data required would be ÿ+, ÿ* and ̈ *,
as presented in Figure 5.22, Figure 5.23 and Figure 5.24 for Case Study 2.
5.2.2 Characteristic Values for Test Zones
After the analysis for each track section, the characteristic value can be calculated for the
whole zone using the values of the track section. As the steps required to go from the Raw
Data presented in 5.2.1 to the Characteristic Values for the Test Zone are the same as for Case
1, they are omitted here.
69
25,00
20,00
Accelaration [g]
15,00
10,00
5,00
0,00
-5,00
-10,00
-15,00
-20,00
0
10
20
30
Time [s]
+
40
30
Time [s]
*
40
30
Time [s]
*
40
50
60
50
60
50
60
Figure 5.22: Raw ÿ vs. time graph
1,00
0,80
Accelaration [g]
0,60
0,40
0,20
0,00
-0,20
-0,40
-0,60
-0,80
-1,00
0
10
20
Figure 5.23: Raw ÿ vs. time graph
6,00
Accelaration [g]
4,00
2,00
0,00
-2,00
-4,00
-6,00
-8,00
0
10
20
Figure 5.24: Raw ̈ vs. time graph
5.2.2.1 Safety Parameters
Acceleration at Bogie ÿ+S: For straight track zones, the lateral acceleration at the bogies ÿ+S is
determined by the method explained in 4.3.2.5 using all the values gathered by all the sections
contained within the zone. For the analysed zone, ÿ+S = 0.9728 g. For curved track zones, the
70
lateral acceleration at the bogies ÿ+S is determined by the same method, but using only some
of the values required, as shown by Table 5.1 and for the analysed zone ÿ+S = 1.0437 g.
Acceleration in Vehicle Body ÿ*S and ̈ *S: For straight track zones, the lateral acceleration in
the vehicle body ÿ*S is determined by the method explained in 4.3.2.5 using all the values
gathered by all the sections contained within the zone. For the analysed zone, ÿ*S = 0.1494 g.
For curved track zones, the lateral acceleration in the vehicle body ÿ*S is determined by the
same method, but using only some of the values required as shown by Table 5.1 and for the
analysed zone, ÿ*S = 0.1301 g. For the vertical acceleration in the vehicle body ̈ *S is
determined by the same method, but only applied to the curved test zones. For the analysed
zone, ̈ *S = 0.0751 g.
Instability Criterion ÿ+S: The Instability Criterion defined by ÿ+S is filtered with a band-pass
filter at f0 ± 2 Hz (and other characteristics see 4.3.2). The f0 frequency was determined to be
approximately 0 Hz, which results in an empty signal after filtering, meaning that there is no
instability.
5.2.2.2 Ride Characteristics
Acceleration in Vehicle Body ÿ*qst: The lateral acceleration in the vehicle body ÿ*qst is only relevant
for curve test zones and is determined by the method explained in 4.3.2.5 using all the values
gathered by all the sections contained within the zone. For the analysed zone, ÿ*qst = 0.0137 g.
Acceleration in Vehicle Body ÿ*q, ̈ *q, sÿ*q and s ̈ *q: For straight and curved track zones, the
lateral acceleration in the vehicle body ÿ*q is determined by the method explained in 4.3.2.5
using all the values gathered by all the sections contained within the zone. For the analysed
zone, ÿ*q = 0.0620 g. For the vertical acceleration in the vehicle body ̈ *q is determined by the
same method and for the analysed zone ̈ *q = 0.1341 g. For the lateral acceleration in the
vehicle body sÿ*q is determined by the same method and for the analysed zone, sÿ*q = 0.0220
g. For the vertical acceleration in the vehicle body s ̈ *q is determined by the same method and
for the analysed zone s ̈ *q = 0.0396 g.
71
5.2.3 Discussion
After determining the characteristic values for each track zone it is then possible to compare
them to the limit values imposed by EN 14636 [11] and UIC 518 [12] and previously
presented in section 4.2.
5.2.3.1 Safety Parameters
Acceleration at Bogie ÿ+S: In the present case the bogie with all its constituent parts and
wheelsets weights 4.3 tonnes, and the limit value is defined by equation (4.5), so we have that
the limit for the lateral acceleration at the bogie is:
(ÿ+S )lim  12 
mb
4
 12   11.2
5
5
[m / s 2 ]
(5.2)
or about 1.1417 g. This limit value is higher than the maximum encountered for the analysed
zone (1.0437 g) and so the operation is within the safety limits.
Acceleration in Vehicle Body ÿ*S and ̈ *S: The limit value for the lateral acceleration in the
vehicle body ÿ*S in straight tracks and large radius curve tracks is 3 m/s2 or 0.3058 g. This
limit is much higher than the maximum encountered of 0.1494 g. The limit value becomes 2.8
m/s2 or 0.2854 g in small radius curve tracks and 2.6 m/s2 or 0.2650 g in very small radius
curve tracks, both of which are respected. For the vertical acceleration in the vehicle body ̈ *S
the limit value depends on the type of suspension that the vehicle has, with single suspension
level or deflated air spring condition as 5 m/s2 (0.5097 g) and double suspension level 3 m/s2
(0.3058 g), both of which are much higher than the maximum found of 0.0751 g.
Instability Criterion ÿ+S; The limit value for the Instability Criterion defined by ÿ+S is half of
the safety limit value of the lateral acceleration at the bogie, in this case it becomes 5.6 m/s2 or
0.5708 g. In this instance, since there is no instability, the vehicle is within the safety limits.
Safety Assessment: In the present case all safety limit values are respected and as such the
vehicle can operate on the analysed track at the analysed speed. Further analyses are required
to determine the vehicle maximum operating speed within the analysed track. Table 5.4
presents the condensed results concerning the safety of Case Study 2.
72
Parameter
ÿ+S
ÿ* S
̈ *S
Instability ÿ+S
Obtained Characteristic Value
1.0437 g
0.1494 g
0.0751 g
0g
Limit Value
1.1417 g
0.2650 g
0.3058 g
0.5708 g
Conclusion
Approved
Approved
Approved
Approved
Table 5.4: case Study 2 safety approval table
5.2.3.2 Ride Characteristics
The limit values for the ride characteristics are greatly influenced by the type of vehicle it is.
In this case we will consider this vehicle as a Power Car and its correspondent limit values
can be found in Table 4.3 for the EN 14636 [11] for UIC 518 [12].
Acceleration in Vehicle Body ÿ*qst: The limit for the lateral acceleration at the bogies ÿ*qst for
a Power Car is 1.5 m/s2 or 0.1529 g, which is much higher than the maximum encountered of
0.0137 g.
Acceleration in Vehicle Body ÿ*q, ̈ *q, sÿ*q and s ̈ *q: The limit for the lateral acceleration in
the vehicle body ÿ*q on a Power Car is 2.5 m/s2 or 0.2548 g, which is higher than the maximum
encountered of 0.0620 g. For the vertical acceleration in the vehicle body ̈ *q the limit on a
Power Car is 2.5 m/s2 or 0.2548 g, which is higher than the maximum encountered of 0.1341 g.
For the lateral acceleration in the vehicle body sÿ*q the limit on a Power Car is 0.5 m/s2 or
0.0510 g, which is higher than the maximum encountered of 0.0220 g. For the vertical
acceleration in the vehicle body s ̈ *q the limit on a Power Car is 1.0 m/s2 or 0.1019 g, which is
higher than the maximum encountered of 0.0396 g.
Ride Characteristics Assessment: In the present case all ride characteristics limit values are
respected and as such the vehicle is validated for passenger transport on the analysed track at
the analysed speed. Further analyses are required to determine the vehicle maximum
operating speed within the analysed track. Table 5.5 presents the condensed results
concerning the ride characteristics of Case Study 2.
Parameter
ÿ*qst
ÿ* q
̈ *q
sÿ*q
s ̈ *q
Obtained Characteristic Value
0.0137 g
0.0620 g
0.1341 g
0.0220 g
0.0396 g
Limit Value
0.1529 g
0.2548 g
0.2548 g
0.0510 g
0.1019 g
Conclusion
Approved
Approved
Approved
Approved
Approved
Table 5.5: Case Study 2 ride characteristics approval table
73
5.3 Discussion of the Case Studies
Now that the data has been processed, these cases can be analysed and compared to the data
used by the Industry in order to assess the safety and comfort and compared with each other.
In the first case the lateral accelerations at the bogie exceed both the safety limits for such
acceleration and the stability limits and the both lateral and vertical accelerations in the
vehicle body exceed the quality limits imposed for a comfortable ride. As a result, the vehicle
cannot circulate at the analysed speed, but it is likely that at a lower speed these concerns
cease to exist and so further research is required.
In the second case all safety and ride characteristics limit values are respected and,
therefore, the vehicle can operate on the analysed track at the analysed speed. Further
analyses are required to determine the vehicle maximum operating speed on the track
considered here. This case also has overall lower characteristic values than the first analysed
case, but still retains a high lateral acceleration at the bogie. This high value can be caused by
many factors, such as a damaged or unbalanced bogie, bad track conditions, irregularities...
In Table 5.1 is presented a resume of the characteristic values for both case studies and
their comparison with each other and their limit values. Here, red indicates that the limit value
was exceeded, orange indicates that the values exceeds 75% of the limit value, yellow
indicates that the values exceed 50% of the limit value, green indicates that the values exceed
25% of the limit value and white indicate that the values are inferior to 25% of the limit value.
Assessment Values
Running Safely
Accelerations at Bogie
Accelerations in Vehicle Body
Instability Criteria
Ride Characteristics Values
Accelerations in Vehicle Body
Symbol
Case 1 [g]
Case 2 [g]
Limit Values [g]
ÿ+S
ÿ* S
̈ *S
+
ÿ S
2.5313
0.1664
0.1883
0.6475
1.0437
0.1494
0.0751
0
1.1417
0.2650
0.3058
0.5708
ÿ*qst
ÿ* q
̈ *q
sÿ*q
s ̈ *q
0.7312
0.0849
0.2182
0.0405
0.1254
0.0137
0.0620
0.1341
0.0220
0.0396
0.1529
0.2548
0.2548
0.0510
0.1019
Table 5.6: Characteristic Values for the analysed case studies
74
6 Conclusions and Future Development
The correct evaluation of the loads imposed to the railway infrastructure by trainsets and,
conversely, the damages on vehicles provoked by the track conditions has been attracting the
attention of railway industry in recent years. The raising interest on this subject has occurred
mainly due to the development of new high-speed railway lines and to the common drive to
upgrade the capacity of existing infrastructures. The increasing demands on railway transportation
require improvements of the network capacity, which can be achieved either by increasing the
speed of the traffic or by increasing the axle loads. However, both of these options place pressures
on the existing infrastructures and the effects of these changes have to be carefully considered.
The main goal of this work is to develop advanced computational tools for railway
dynamics, with innovative methodologies that are handled in a co-simulation environment, where
all physical phenomena can be integrated, as shown in Figure 1.1. This includes not only the
detailed representation of the vehicle, track and subgrade, but also the interaction among them.
Such tools can indicate solutions with technological relevance and give answer to the industry’s
most recent needs, contributing to improve the competitiveness of the railway transportation
system. The two main tools developed in this work are: a) the pre-processor that builds the
flexible track model from provided geometric and material properties for the track and its
elements, and b) the post-processor that computes the results of the dynamic analysis in order to
determine if a given vehicle is acceptable to operate on a given track at the proposed speed.
The mathematical models of the railway vehicles are created using a multibody
formulation. The kinematic constraints between the different system components are formulated
in terms of the set of generalized coordinates. On the other hand, the flexible track model uses the
finite element methodology. While the rails and the sleepers are modelled as beams, the pads and
foundations are modelled as spring-dampers to account for their intrinsic flexibility. Between the
multibody and the finite element codes lies the contact model, connecting the vehicle’s wheels
and the track’s rails using a co-simulation procedure. Although other procedures exist, the use of a
contact penalty formulation demonstrates to be enough to obtain all main contact features.
From the results obtained in the present work, it can be concluded that the proposed
numerical tools are appropriate for railway applications and that finite element methodology,
proposed here to represent the track flexibility, is suitable for railway studies and it is
quantitatively validated for static loads.
The developed post-processing tool intends to verify if the a given vehicle-track combination
is within the safety and comfort parameters defined by EN 14636 [11] and UIC 518 [12], which are
75
commonly used by the railway industry. This tool was developed and demonstrated in two case
studies. One failed to meet all required criteria and the other that complies requirements.
Although the co-simulation procedure required to perform the dynamic analysis of the
modelled railway vehicle running on the modelled flexible track isn’t complete, the results
taken from both the pre-processing and post-processing tools developed in this work are valid
and useful when integrated on a complete dynamic analysis.
The first future development to be realised is the conclusion of this co-simulation procedure
and the integration of the track, vehicle and contact models in a common tool, so that it can perform
complete dynamic analysis and evaluate the results according to the current industry requirements.
Other future development of this work is to perform comparative studies in order to
investigate the influence of track flexibility and of track conditions on vehicles performance. Also
studies involving the consequences of trainset operation on railway infrastructure degradation are
possible to develop. The establishment of partnerships with Portuguese railway operators and
infrastructure manager gives good perspectives for the industrial application of these studies.
Other aspect which needs further investigation is the identification of the railway track
damping parameters. However, it is recognized that the estimation of the structural damping
of structures is still a technological challenge. Rayleigh damping, also known as proportional
damping, was used to model the developed track model. Still, these damping parameters need
to be identified with further detail, either on tracks under current operation or in the design
phase. So it is important to find methodologies able to identify the track damping on existent
tracks with experimental testing and validation. Moreover, it is important to relate these
findings to tracks that are still in the design phase.
76
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Pereira, C., Ramalho, A. and Ambrósio, J., "A Critical Overview of Internal and External
Cylinder Contact Force Models", Nonlinear Dynamics, 63, No. 4, 681-697, 2011.
Flores, Paulo, "Kinematics and dynamics of multibody systems with imperfect joints : models
and case studies", Springer, Berlin, 2008.
Theodore, G. T. and Khozeimen, K., "Computer Analysis of Rigid Frames", Computers and
Structures, 1, 193-221, 1971.
ANSYS, Inc, "Theory Reference for the Mechanical APDL and Mechanical Applications",
Canonsburg, Pennsylvania, USA2009.
ERRI C138, "Permissible limit values for the y and q forces and derailment criteria", 1977.
ORE C138/RP9, "Permissible Maximum Values for the Y- and Q-Forces and Derailment
Criteria", 1986.
ORE B55/RP8, "Conditions for negotiating track twists - Recomended values for the track
twist and cant - Calculation and measurement of the relevant vehicle parameters - Vehicle
testing", 1983.
UNI EN 13674, "Rail 49 E1 - DIN S 49", 2011.
Ambrósio, J., Pombo, J., Rauter, F. and Pereira, M., "A Memory Based Communication in the
Co-Simulation of Multibody and Finite Element Codes for Pantograph-Catenary Interaction
Simulation", Multibody Dynamics, (Bottasso C.L., Ed.), Springer, Dordrecht, The
Netherlands, pp. 211-231, 2008.
Rauter, F., "Dynamic Analysis of the Pantograph-Catenary Interaction in Railway Systems",
PhD Dissertation, Instituto Superior Técnico, Lisbon, Portugal, 2011.
82
Annex A: Flexible Track Properties
In the following, the data required to define the flexible track model is described together with
the pre-processor developed to build its FE mesh.
In order to define a given railway track, it is necessary to provide information about the
geometry of each rail. This is done in 3D space by defining a set of control points that are
representative of the geometry of each rail. In addition, it is necessary to provide the Cartesian
components of the tangential t, normal n and binormal b vectors that define the rail referential
associated to each nodal point. These quantities are tabulated as function of the rail arc length, as
represented in Table A.1.
Rail arc
Length
<Num>
Xi
…
<Num>
Yi
Zi
Txi
Tyi
Tzi
Nxi
Nyi
Nzi
Bxi
Byi
Bzi
<Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num>
…
…
…
…
…
…
…
…
…
…
…
…
<Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num> <Num>
Table A.1: Rail geometry data
After defining the 3D geometry of each rail, it is necessary to provide information about
the number of track segments to be considered in the finite element mesh. For each segment,
it is necessary to define its name, length and the refinement level of the mesh, as represented
in Table A.2.
Number of
Track Types:
Track Type i
Track Type 1
Track Type 2
…
Track Type n
<Number>
Track Type Name
<Track Type 1 Name>
<Track Type 2 Name>
…
<Track Type n Name>
Length of Track Type i
<Number>
<Number>
…
<Number>
Refinement Level of Track Type i
<Number>
<Number>
…
<Number>
Table A.2: Track segments data
For each track segment defined in Table A.2, it is necessary to provide information about
the types of rails, sleepers and foundations that compose each one, as represented in Table A.3.
Track Type 1
Rail Data Type
Sleepers Data Type
Foundations Data Type
<Track Type 1 Name>
<Rail Data Type Name>
<Sleepers Data Type Name>
<Foundations Data Type Name>
Table A.3: Track segment components data
83
Then, for each rail, it is necessary to define the properties required for the EulerBernoulli beam elements formulation, as represented in Table A.4.
Rail Data Type
<Rail Data Type n Name>
UIC Rail Code:
<Code>
Young Modulus - E [Pa]:
<Number>
Poisson Coefficient:
<Number>
Cross Section Area [m²]:
<Number>
Effective Shear Section Area in yy Direction [m²]:
<Number>
Effective Shear Section Area in zz Direction [m²]:
<Number>
Second Moment of Area in xz Plane - Iyy [m⁴]:
<Number>
Second Moment of Area in xy Plane- Izz [m⁴]:
<Number>
Second Moment of Area in yz Plane - Ixx [m⁴]:
<Number>
Density [kg/m³]:
<Number>
Torsion Modulus - G [Pa]:
<Number>
Rayleigh Damping Parameter α:
<Number>
Rayleigh Damping Parameter β:
<Number>
Table A.4: Rail geometry data
After introducing the information about the rails, it is necessary to provide all properties
required to define the sleepers for each track segment, as represented in Table A.5.
Sleepers Data Type:
<Sleepers Data Type n Name>
Sleepers Distance [m]:
<Number>
Number of Nodes Between Sleepers:
<Number>
Sleeper Geometry:
<Sleeper Geometry Name>
Pad Longitudinal Stiffness Kx [N/m]:
<Number>
Pad Transversal Stiffness Ky [N/m]:
<Number>
Pad Vertical Stiffness Kz [N/m]:
<Number>
Pad Torsional Stiffness Kt [N/m]:
<Number>
Pad Vertical Rotation Stiffness Kry [N/m]:
<Number>
Pad Transversal Rotation Stiffness Krz [N/m]:
<Number>
Pad Longitudinal Damping Cx [N.s/m]:
<Number>
Pad Transversal Damping Cy [N.s/m]:
<Number>
Pad Vertical Damping Cz [N.s/m]:
<Number>
Pad Torsional Damping Ct [N.s/m]:
<Number>
Pad Vertical Rotation Damping Cry [N.s/m]:
<Number>
Pad Transversal Rotation Damping Crz [N.s/m]:
<Number>
Table A.5: Sleeper properties data
Besides the information about the rails and sleepers, the properties for the definition of
the foundations for each track segment are required, as represented in Table A.6.
84
Foundations Data Type:
<Foundations Data Type n Name>
Foundation Longitudinal Stiffness Kx [N/m]:
<Number>
Foundation Transversal Stiffness Ky [N/m]:
<Number>
Foundation Vertical Stiffness Kz [N/m]:
<Number>
Foundation Torsional Stiffness Kt [N/m]:
<Number>
Foundation Vertical Rotation Stiffness Kry [N/m]:
<Number>
Foundation Transversal Rotation Stiffness Krz [N/m]:
<Number>
Foundation Longitudinal Damping Cx [N.s/m]:
<Number>
Foundation Transversal Damping Cy [N.s/m]:
<Number>
Foundation Vertical Damping Cz [N.s/m]:
<Number>
Foundation Torsional Damping Ct [N.s/m]:
<Number>
Foundation Vertical Rotation Damping Cry [N.s/m]:
<Number>
Foundation Transversal Rotation Damping Crz [N.s/m]:
<Number>
Sleeper Interaction Longitudinal Stiffness Kx [N/m]:
<Number>
Sleeper Interaction Transversal Stiffness Ky [N/m]:
<Number>
Sleeper Interaction Vertical Stiffness Kz [N/m]:
<Number>
Sleeper Interaction Torsional Stiffness Kt [N/m]:
<Number>
Sleeper Interaction Vertical Rotation Stiffness Kry [N/m]:
<Number>
Sleeper Interaction Transversal Rotation Stiffness Krz [N/m]:
<Number>
Sleeper Interaction Longitudinal Damping Cx [N.s/m]:
<Number>
Sleeper Interaction Transversal Damping Cy [N.s/m]:
<Number>
Sleeper Interaction Vertical Damping Cz [N.s/m]:
<Number>
Sleeper Interaction Torsional Damping Ct [N.s/m]:
<Number>
Sleeper Interaction Vertical Rotation Damping Cry [N.s/m]:
<Number>
Sleeper Interaction Transversal Rotation Damping Crz [N.s/m]:
<Number>
Table A.6: Foundation properties data
As previously referred, the rails and sleepers are modelled by using Euler-Bernoulli
beam elements. The rail geometry data is provided in Table A.4. For the sleepers, with a
general geometry shown in Figure A.1, the data required to define their geometry is
represented in Table A.7.
C
B
A
Figure A.1: Sleeper general geometry
85
Sleeper Geometry:
<Sleeper Geometry n Name>
Sleeper Length (Parameter A) [m]:
<Number>
Rail-to-End Position (Parameter C) [m]:
<Number>
Rail-to-Start Position (Parameter B) [m]:
<Number>
Number of Additional Nodes on Half Sleeper:
<Number>
End Young Modulus - E [Pa]:
<Number>
End Poisson Coefficient:
<Number>
End Cross Section Area [m²]:
<Number>
End Effective Shear Section Area in yy Direction [m²]:
<Number>
End Effective Shear Section Area in zz Direction [m²]:
<Number>
End Second Moment of Area in xz Plane - Iyy [m⁴]:
<Number>
End Second Moment of Area in xy Plane - Izz [m⁴]:
<Number>
End Second Moment of Area in yz Plane - Ixx [m⁴]:
<Number>
End Density [kg/m³]:
<Number>
End Torsion Modulus - G [Pa]:
<Number>
End Rayleigh Damping Parameter α:
<Number>
End Rayleigh Damping Parameter β:
<Number>
Start Young Modulus - E [Pa]:
<Number>
Start Poisson Coefficient:
<Number>
Start Cross Section Area [m²]:
<Number>
Start Effective Shear Section Area in yy Direction [m²]:
<Number>
Start Effective Shear Section Area in zz Direction [m²]:
<Number>
Start Second Moment of Area in xz Plane - Iyy [m⁴]:
<Number>
Start Second Moment of Area in xy Plane - Izz [m⁴]:
<Number>
Start Second Moment of Area in yz Plane - Ixx [m⁴]:
<Number>
Start Density [kg/m³]:
<Number>
Start Torsion Modulus - G [Pa]:
<Number>
Start Rayleigh Damping Parameter α:
<Number>
Start Rayleigh Damping Parameter β:
<Number>
Middle Young Modulus - E [Pa]:
<Number>
Middle Poisson Coefficient:
<Number>
Middle Cross Section Area [m²]:
<Number>
Middle Effective Shear Section Area in yy Direction [m²]:
<Number>
Middle Effective Shear Section Area in zz Direction [m²]:
<Number>
Middle Second Moment of Area in xz Plane - Iyy [m⁴]:
<Number>
Middle Second Moment of Area in xy Plane - Izz [m⁴]:
<Number>
Middle Second Moment of Area in yz Plane - Ixx [m⁴]:
<Number>
Middle Density [kg/m³]:
<Number>
Middle Torsion Modulus - G [Pa]:
<Number>
Middle Rayleigh Damping Parameter α:
<Number>
Middle Rayleigh Damping Parameter β:
<Number>
Table A.7: Sleeper geometry data
Finally it is necessary to define the constants and output parameters for the track model.
These quantities are represented in Table A.8.
Track Constants
Output Parameters
Gravity Acceleration [m/s2]: <Number> Deformation Scalar Factor: <Number>
Table A.8: Track model constants and output parameters
86
Annex B: Communication between Multibody and FE Codes
In this work, a 3D methodology to study the interaction of a railway vehicle, described by a
multibody formulation, with a flexible track, represented by a finite element methodology, is
proposed. Instead of using the conventional approach where the vehicle and track dynamics
are handled independently, here an integrated strategy is used to handle the vehicle-track
coupled dynamics. For this purpose, a high-speed co-simulation procedure is established in
order to communicate between the multibody and the finite element codes. The vehicle-track
interaction forces are computed by using an appropriate wheel-rail contact formulation
[18,19].
For the dynamic analysis of the finite elements model, a Newmark family numerical
integrator [111,114] using a fixed time step is employed, while for the multibody vehicle
model the integration procedure is based on a predictor-corrector algorithm with variable time
step [128]. Each code handles independently their equations of motion of their referred subsystem and applies the contact forces on the contact points both shared between them.
The compatibility between the two integration algorithms imposes readily available
state variables of the two sub-systems during the integration procedure and that a prediction
of the contact forces is available at any given time step. There are occasions in which one of
the algorithms has to wait for the other and vice-versa. The developed communication
interface is composed of two stages. In the first stage, the codes exchange input data
necessary to their own initialization procedures. No contact at the track is implied or allowed
at the initial time step. In the second stage, data is shared between codes to perform dynamic
analysis, exchanging data as previously described.
One critical issue of using co-simulation procedures is the added computational cost due
to the data exchange between codes. The time spent on data exchange between applications
must be negligible compared to the computation time costs of the two analyses. In order to
reduce this computational cost, the data exchange methodology adopted will use virtual
memory sharing via memory mapped files [129]. Further details on this topic are outside the
scope of this thesis, the interested reader is referred to the work developed by Antunes [102],
where is it applied to the catenary instead of the track and to the pantograph instead of the
vehicle.
87
Annex C: Case Study Properties
The following track data was provided by the SMARTRACK partners from New University
of Lisbon.
Number of Track Types
1
Track Type i
Track Type Name Length of Track Type i Refinement Level of Track Type i
Track Type 1
Track1
500
1
Table C.1: Track segments data for the case study
Rail Data Type
UIC Rail Code
Young Modulus - E [Pa]
Poisson Coefficient
Cross Section Area [m2]
Second Moment of Area in xz Plane - Iyy [m4]
Second Moment of Area in xy Plane - Izz [m4]
Second Moment of Area in yz Plane - Ixx [m4]
Density [kg/m3]
Torsion Modulus - G [Pa]
Rayleigh Damping Parameter α
Rayleigh Damping Parameter β
UIC60
UIC60
210×109
0.3
7.6700×10-3
30.383×10-6
5.123×10-6
35.506×10-6
7.860×103
80.770×109
0
0
Table C.2: Rail data for the case study
Foundations Data Type
Foundation1
Foundation Longitudinal Stiffness Kx [N/m]:
3×106
Foundation Transversal Stiffness Ky [N/m]:
55×106
Foundation Vertical Stiffness Kz [N/m]:
55×106
Foundation Torsional Stiffness Kt [N/m]:
1×10-20
Foundation Vertical Rotation Stiffness Kry [N/m]:
1×10-20
Foundation Transversal Rotation Stiffness Krz [N/m]:
1×10-20
Foundation Longitudinal Damping Cx [N.s/m]:
31×103
Foundation Transversal Damping Cy [N.s/m]:
31×103
Foundation Vertical Damping Cz [N.s/m]:
31×103
Foundation Torsional Damping Ct [N.s/m]:
1×10-20
Foundation Vertical Rotation Damping Cry [N.s/m]:
1×10-20
Foundation Transversal Rotation Damping Crz [N.s/m]:
1×10-20
Sleeper Interaction Longitudinal Stiffness Kx [N/m]:
55×106
Sleeper Interaction Transversal Stiffness Ky [N/m]:
55×106
Sleeper Interaction Vertical Stiffness Kz [N/m]:
3×106
Sleeper Interaction Torsional Stiffness Kt [N/m]:
1×10-20
Sleeper Interaction Vertical Rotation Stiffness Kry [N/m]:
1×10-20
Sleeper Interaction Transversal Rotation Stiffness Krz [N/m]:
1×10-20
Sleeper Interaction Longitudinal Damping Cx [N.s/m]:
31×103
Sleeper Interaction Transversal Damping Cy [N.s/m]:
31×103
Sleeper Interaction Vertical Damping Cz [N.s/m]:
31×103
Sleeper Interaction Torsional Damping Ct [N.s/m]:
1×10-20
Sleeper Interaction Vertical Rotation Damping Cry [N.s/m]:
1×10-20
Sleeper Interaction Transversal Rotation Damping Crz [N.s/m]:
1×10-20
Table C.3: Foundation properties for the case study
88
Sleepers Data Type
Sleeper1
Sleepers Distance [m]:
0,6
Number of Nodes Between Sleepers:
5
Sleeper Geometry:
SleeperGeo1
Pad Longitudinal Stiffness Kx [N/m]:
260×106
Pad Longitudinal Stiffness Kx [N/m]:
260×106
Pad Transversal Stiffness Ky [N/m]:
65×106
Pad Vertical Stiffness Kz [N/m]:
68×106
Pad Torsional Stiffness Kt [N/m]:
1×10-20
Pad Vertical Rotation Stiffness Kry [N/m]:
1×10-20
Pad Transversal Rotation Stiffness Krz [N/m]:
1×10-20
Pad Longitudinal Damping Cx [N.s/m]:
75×103
Pad Transversal Damping Cy [N.s/m]:
19×103
Pad Vertical Damping Cz [N.s/m]:
19×103
Pad Torsional Damping Ct [N.s/m]:
1×10-20
Pad Vertical Rotation Damping Cry [N.s/m]:
1×10-20
Pad Transversal Rotation Damping Crz [N.s/m]:
1×10-20
Table C.4: Sleeper data for the case study
Sleeper Geometry
Sleeper Length (Parameter A) [m]:
Rail-to-End Position (Parameter C) [m]:
Rail-to-Start Position (Parameter B) [m]:
Number of Additional Nodes on Half Sleeper:
End Young Modulus - E [Pa]:
End Poisson Coefficient:
End Cross Section Area [m²]:
End Second Moment of Area in xz Plane - Iyy [m⁴]:
End Second Moment of Area in xy Plane - Izz [m⁴]:
End Second Moment of Area in yz Plane - Ixx [m⁴]:
End Density [kg/m³]:
End Torsion Modulus - G [Pa]:
End Rayleigh Damping Parameter a:
End Rayleigh Damping Parameter b:
Start Young Modulus - E [Pa]:
Start Poisson Coefficient:
Start Cross Section Area [m²]:
Start Second Moment of Area in xz Plane - Iyy [m⁴]:
Start Second Moment of Area in xy Plane - Izz [m⁴]:
Start Second Moment of Area in yz Plane - Ixx [m⁴]:
Start Density [kg/m³]:
Start Torsion Modulus - G [Pa]:
Start Rayleigh Damping Parameter a:
Start Rayleigh Damping Parameter b:
Middle Young Modulus - E [Pa]:
Middle Poisson Coefficient:
Middle Cross Section Area [m²]:
Middle Second Moment of Area in xz Plane - Iyy [m⁴]:
Middle Second Moment of Area in xy Plane - Izz [m⁴]:
Middle Second Moment of Area in yz Plane - Ixx [m⁴]:
Middle Density [kg/m³]:
Middle Torsion Modulus - G [Pa]:
Middle Rayleigh Damping Parameter a:
Middle Rayleigh Damping Parameter b:
A
2.6
450×10-3
425×10-3
0
37×109
0.2
50×10-3
260.42×10-6
166.70×10-6
427.12×10-6
2.5×103
15×109
0.04
0.96
37×109
0.2
50×10-3
260.42×10-6
166.70×10-6
427.12×10-6
2.5×103
15×109
0.04
0.96
37×109
0.2
50×10-3
260.42×10-6
166.70×10-6
427.12×10-6
2.5×103
15×109
0.04
0.96
Table C.5: Sleeper geometry for the case study
89