Development of Multibody Pantograph and Finite

Element Catenary Models for Application to High-speed

Railway Operations

Pedro Cabaço Antunes

Thesis to obtain the Master of Science Degree in

Mechanical Engineering

Examination Commitee

Chairperson: Prof. Luis Manuel Varejão de Oliveira Faria

Supervisor: Prof. Jorge Alberto Cadete Ambrósio

Co- Supervisor: Prof. João Carlos Elói de Jesus Pombo

Member of the Commitee: Prof. Manuel Frederico Oom Seabra Pereira

Outubro de 2012

i

Acknowledgements

I especially want to thank, Prof. Manuel Seabra Pereira, for his guidance during this

work. His energy and enthusiasm in research have motivated me as all his advices. In addition,

he was always accessible and willing to help, for that I wish to express my gratitude.

To my advisor, Prof. Jorge Ambrósio, a very special thanks. His work discipline and

commitment have inspired me. For his close attention to my thesis and all the teachings that

came along the way I leave my grateful acknowledgment.

My thanks to my co-adviser, Prof. João Pombo, for the given dedication and help on the

work here presented. I would like also to express my thanks for his given encouragement that

always left me with desire to do more.

For last but not the least on my personal acknowledgments, I leave my thanks to my

parents and brother who provided the support only family can give. Even on the toughest

moments they gave me the care and moral support needed to soldier on.

The work reported here has been developed in the course of several national and

international projects. Among these, two national projects funded by FCT (Portuguese

Foundation for Science and Technology); SMARTRACK (contract no. PTDC/EME-

PME/101419/2008) and WEARWHEEL (contract no. PTDC/EME-PME/115491/2009), and

also the European Project PANTOTRAIN, funded by the EC in the FP7 Program with the

contract SC8-GA-2009-234015 coordinated by UNIFE.

I also want to thank Frederico Rauter (Siemens Portugal), Stefano Bruni, Alan Facchinetti

and Andrea Colina (Politecnico di Milano) and Jean-Pierre Massat (SNCF), whose discussions

and recommendations were important to achieve some of the developments reported here.

ii

iii

Resumo

Nos comboios de alta velocidade a captação de energia eléctrica para os motores é

assegurada pelo sistema pantógrafo-catenária, o qual é alvo de apertados requisitos funcionais

para garantir a sua fiabilidade e controlo dos períodos de manutenção que necessitam de ser

aumentados. A necessidade de garantir a interoperabilidade entre pantógrafos e diferentes

sistemas de catenária, quer existentes quer em projecto, coloca um maior nível de exigência

sobre a capacidade de controlar os seus comportamentos dinâmicos. É também vital assegurar a

qualidade da captação de energia por forma a evitar perdas de contacto entre o pantógrafo e a

catenária, com consequente arqueamento eléctrico. Este factor não só limita a velocidade

máxima dos comboios como também conduz à degradação das condições funcionais dos

elementos mecânicos destes equipamentos. Para abordar estes aspectos, de importância para

projecto e análise do sistema de pantógrafo-catenária, é necessário desenvolver procedimentos

computacionais que sejam fiáveis, eficientes e precisos de forma a ser possível capturar todas as

características relevantes de seu comportamento dinâmico. Este trabalho apresenta uma

metodologia de modelação capaz de lidar com a dinâmica envolvida na interacção pantógrafo-

catenária utilizando uma abordagem totalmente tridimensional e a sua implementação numa

ferramenta computacional. Com o fim de explorar as vantagens da utilização de uma

formulação multicorpo para modelar o pantógrafo, um processo de co-simulação especializado

foi desenvolvido para permitir a comunicação entre o modelo multicorpo e um modelo de

elementos finitos da catenária. De forma a modelar a interacção entre o pantógrafo e a catenária

a força de contacto é descrita por um método de penalidades. Os métodos e metodologias aqui

desenvolvidos são usados em condições de operação realistas para comboios de alta velocidade.

Uma série de estudos sobre o comportamento dinâmico do sistema pantógrafo-catenária é

apresentada incluindo catenárias com dados geométricos adquiridos experimentalmente em

cenários de operação com pantógrafos múltiplos e secções de intersecção de catenária a fim de

aceder às condições que limitam o aumento da velocidade das composições.

Palavras-Chave

Dinâmica ferroviária;

Interacção pantógrafo-catenário;

Elementos finitos;

Sistemas multi-corpo;

Mecânica do contacto;

Co-simulação.

iv

v

Abstract

High-speed railway overhead systems are subjected to tight functional requirements to

deliver electrical energy to train’s engines, in order to ensure their reliability and to control their

maintenance periods. The quest for interoperability of different pantographs, in existing and

projected catenary systems, puts an extra demand on the ability to control their dynamic

behaviour. The quality of the current collection is of fundamental importance as the loss of

contact and consequent arching, not only limit the top velocity of high-speed trains but also

imply the deterioration of the functional conditions of these mechanical equipments. To address

such important aspects for the design and analysis of the pantograph-catenary system, it is

necessary to develop reliable, efficient and accurate computational procedures that allow

capturing all the relevant features of their dynamic behaviour. This work presents a

computational tool and a modelling methodology able to handle the dynamics of pantograph-

catenary interaction using a fully three-dimensional methodology. In order to exploit the

advantages of using a multibody formulation to model the pantograph, a high-speed co-

simulation procedure is setup in order to allow the communication between the multibody

model and the finite element catenary model. A contact model, based on a penalty formulation,

is selected to represent the interaction between the two modelling procedures. The methods and

approaches developed here are used in realistic operation conditions for high speed trains. A

series of studies of the dynamic behaviour of the pantograph-catenary system is presented,

including experimental catenary models with single and multiple pantograph operation

scenarios and overlap catenary sections, in order to access the conditions that limit the increase

of the trainset speed.

Keywords

Railway dynamics;

Pantograph-Catenary interaction;

Finite elements;

Multibody systems;

Contact mechanics;

Co-simulation.

vi

vii

Contents

Acknowledgements ........................................................................................................................ i 

Resumo ......................................................................................................................................... iii 

Palavras-Chave ............................................................................................................................. iii 

Abstract ......................................................................................................................................... v 

Keywords ...................................................................................................................................... v 

Contents....................................................................................................................................... vii 

List of Figures .............................................................................................................................. ix 

List of Tables .............................................................................................................................. xiii 

1  Introduction ............................................................................................................................ 1 

2  Catenary Dynamic Analysis and Modelling ........................................................................... 9 

2.1  Characteristics of Current High-Speed Catenaries ....................................................... 10 

2.2  Catenary Modelling ...................................................................................................... 13 

2.3  The Finite Element Method on the Pantograph-Catenary System ................................ 15 

2.3.1  Dynamic Analysis of Catenaries Using Linear FEM ........................................ 16 

2.3.2  FEM Catenary Models ...................................................................................... 19 

2.3.3  Lumped Mass Pantograph Dynamic Analysis in Finite Elements

Application ................................................................................................. 22 

2.3.4  Time integration method ................................................................................... 24 

3  General Pantograph Dynamic Analysis and Modelling ......................................................... 27 

3.1  Characteristics of High-Speed Pantographs ................................................................. 27 

3.2  Multibody Dynamic Analysis of Pantographs .............................................................. 27 

3.3  Multibody Pantograph Models ..................................................................................... 30 

3.4  Lumped Mass Pantograph Model ................................................................................. 33 

4  Pantograph-Catenary Interaction ........................................................................................... 35 

4.1  Contact Modelling ........................................................................................................ 35 

4.1.1  Geometric Conditions for Contact .................................................................... 36 

4.1.2  Continuous Contact Force Model ..................................................................... 38 

viii

4.2  Co-Simulation of Multibody and Finite Elements ........................................................ 39 

4.2.1  Co-Simulation Procedure .................................................................................. 39 

4.2.2  Communication Protocol .................................................................................. 41 

4.2.3  Data Exchange Methodology ............................................................................ 42 

5  Applications to Overhead Current Collecting Systems .......................................................... 45 

5.1  Pantograph-catenary pairs in current operation ............................................................ 45 

5.2  Analysis of existing pantograph-catenary national pairs .............................................. 51 

5.3  Multiple Pantograph Operation .................................................................................... 60 

5.4  Pantograph-Catenary performance with overlap sections ............................................ 66 

6  Conclusions and Future Development ................................................................................... 77 

References ................................................................................................................................... 80 

ix

List of Figures

Figure 1.1: Detail of the pantograph catenary contact, during the operation of a high-speed train,

with the occurrence of dropper slacking and arcing. ................................................. 1 

Figure 1.2: Different high-speed catenary types: The French Southeast LN1 (a), and the German

Re330 (b) are Stitch wire catenaries; The French Atlantic LN2 (c) is a Simple

catenary; Japanese Shinkansen (d) is a compound catenary. ..................................... 3 

Figure 1.3: Different high-speed railway vehicles roof pantographs: a) Faiveley CX; b)

Stemman DSA380; c) Stemman ASP400; d) Contact ATR95. ................................. 4 

Figure 2.1: General structural and functional elements in a high speed catenary. ........................ 9 

Figure 2.2: Representation of the functional elements of the catenary geometry ......................... 9 

Figure 2.3: LN1 catenary with stitch wire in use in the Southeast TGV line. ............................. 11 

Figure 2.4: LN2 catenary, without stitch wire, in use in the Atlantic TGV line. ........................ 11 

Figure 2.5: Re330 catenary, with stitch wire, in use in the German ICE lines. .......................... 12 

Figure 2.6: Japanese compound catenary for the Shinkansen lines. ........................................... 12 

Figure 2.7: Photo of a weights line tensioning system; Photo: Rainer Knäpper, Freenbsp;

Art License (http://artlibre.org/licence/lal/en) ......................................................... 14 

Figure 2.8 Dropper slacking for the same operation scenario: (a) pantograph-catenary contact is

maintained, (b) loss of contact with arcing .............................................................. 15 

Figure 2.9: Finite element model of a generic catenary with the sag highlighted. ...................... 21 

Figure 2.10: Representation of the constraints applied on a general catenary model ................. 21 

Figure 2.11: Representation of the pantograph lumped mass model .......................................... 22 

Figure 2.12: Flowchart of the integration algorithm steps as each time step .............................. 26 

Figure 3.1: Generic multibody system. ....................................................................................... 28 

Figure 3.2: Flowchart with the forward dynamic analysis of a multibody system ..................... 29 

Figure 3.3: (a) Pantograph model; (b) Roof mounted pantograph on the vehicle guided on the

track; (c) Pantograph with prescribed base motion ................................................. 30 

Figure 3.4: The pantograph system: (a) Complete pantograph; (b) Base; (c) Lower arm; (d)

Upper arm; (e) Lower link; (f) Upper link; (g) Head support; (h) Bow .................. 31 

Figure 3.5: Lumped mass pantograph model: (a) Laboratory parameter identification procedure;

(b) Three stage lumped mass model; (c) Parameter values ..................................... 33 

Figure 3.6: Test rig for the experimental identification of the lumped mass pantograph, courtesy

of Politecnico di Milano (PoliMi) ............................................................................ 34 

Figure 4.1: Pantograph-catenary contact: (a) Pantograph bow and catenary contact wire; (b)

Cross-section of the contact wire; (c) Cross-section of the collector strip .............. 35 

x

Figure 4.2: Representation of the top view of a catenary contact wire element and a pantograph

registration strip ....................................................................................................... 36 

Figure 4.3: Representation of the potential point of contact on the catenary contact wire element

................................................................................................................................. 38 

Figure 4.4: Co-simulation flowchart between a finite element and a multibody code ................ 40 

Figure 4.5: Initialization stage flowchart of the communication interface ................................. 41 

Figure 4.6: Dynamic analysis stage flowchart of the communication interface ......................... 42 

Figure 4.7: Representation of the data exchange procedure between applications using memory

mapped files ............................................................................................................. 43 

Figure 5.1: Representation of the finite element model of the LN2 catenary with the static

deformation accounted for ....................................................................................... 48 

Figure 5.2: Representation of the finite element model of the C270 catenary with the static

deformation accounted for ....................................................................................... 49 

Figure 5.3: Representation of the finite element model of the Re330 catenary with the static

deformation accounted for ....................................................................................... 50 

Figure 5.4: Pantograph-catenary contact force for the CX-LN2 pair with different lumped mass

pantograph formulation, finite element formulation (FEM) and multibody

formulation (MB). ................................................................................................... 52 

Figure 5.5: Statistical quantities of the pantograph-catenary contact force for the CX-LN2 pair

with different lumped mass pantograph formulation, finite element formulation

(FEM) and multibody formulation (MB). ............................................................... 52 

Figure 5.6: Pantograph-catenary contact force for the ATR95-C270 pair .................................. 54 

Figure 5.7: Statistical quantities of the pantograph-catenary contact force for the ATR95-C270

pair ........................................................................................................................... 54 

Figure 5.8: Pantograph-catenary contact force for the DSA800-Re330 pair .............................. 55 

Figure 5.9: Statistical quantities of the pantograph-catenary contact force for the DSA800-R330

pair ........................................................................................................................... 55 

Figure 5.10: Comparison of the statistical quantities of the pantograph-catenary contact force

between pairs .......................................................................................................... 56 

Figure 5.11: Effort and lower node displacement of the first dropper on the span, (a), and other

at middle span, (b), registered on the CX-LN2 pair. .............................................. 58 

Figure 5.12: Effort and lower node displacement of the first dropper on the span, (a), and other

at middle span, (b), registered on the CX-LN2 pair. .............................................. 58 

Figure 5.13: Effort and lower node displacement of the first dropper on the span, (a), and other

at middle span, (b), registered on the DSA800-Re330 pair. ................................... 59 

Figure 5.14: Multiple pantograph operations of high-speed trains with typical distances between

pantographs. ............................................................................................................ 60 

xi

Figure 5.15: Pantograph-catenary contact force for several pantograph separations in catenaries

with different proportional damping: (a) =0.00275; (b) =0.0275. ..................... 61 

Figure 5.16: Statistical quantities of the pantograph-catenary contact force in catenaries with

different proportional damping: (a) =0.00275; (b) =0.0275. ............................. 62 

Figure 5.17: Histograms of the pantograph-catenary contact force in catenaries with different

proportional damping: (a) =0.00275; (b) =0.0275. ............................................ 63 

Figure 5.18: Typical steady-arm uplift in catenaries with different proportional damping for two

different separations of pantographs: (a) =0.00275; (b) =0.0275. ..................... 64 

Figure 5.19: Typical mid-span dropper forces in catenaries with different proportional damping

for two different separations of pantographs: (a) =0.00275; (b) =0.0275. ........ 64 

Figure 5.20: Statistical quantities associated to the contact force for a catenary with low

damping (=0.00275): (a) Leading pantographs; (b) Trailing pantographs. .......... 65 

Figure 5.21: Statistical quantities associated to the contact force for a catenary with average

damping (=0.0275): (a) Leading pantographs; (b) Trailing pantographs. ............ 65 

Figure 5.22: Schematic of a LN2 overlap section arrangement with projected views ................ 67 

Figure 5.23: Representation of the finite element model of the LN2 catenary (TGV Atlantique

line) with the static deformation already accounted. .............................................. 68 

Figure 5.24: Pantograph-catenary contact force for a single pantograph and multiple

pantographs with several separations in a regular catenary section ....................... 69 

Figure 5.25: Statistical quantities of the pantograph-catenary contact force on a single

pantograph and multiple pantographs with several separations in a regular catenary

section ..................................................................................................................... 70 

Figure 5.26: Pantograph-catenary contact force for a single pantograph and multiple

pantographs with several separations in a catenary overlap section ....................... 71 

Figure 5.27: Statistical quantities of the pantograph-catenary contact force on a single

pantograph and multiple pantographs with several separations in a catenary overlap

section ..................................................................................................................... 72 

Figure 5.28: Statistical quantities associated to the contact force between an overlap and a

normal section of the catenary system for different pantograph separations. ......... 73 

Figure 5.29: Statistical quantities associated to the contact force between single, leading and

trailing pantographs on a normal catenary section for different pantograph

separations .............................................................................................................. 74 

Figure 5.30: Statistical quantities associated to the contact force between single, leading and

trailing pantographs on a catenary’s overlap section for different pantograph

separations .............................................................................................................. 75 

xii

Figure 5.31: Discretized contact forces on the catenary overlap section for the leading (a); and

trailing pantograph (b), with 31 meters separation. ................................................ 76 

xiii

List of Tables

Table 2.1: Summary of the characteristics of different catenary systems [4, 5, 10]. Data for the

French catenaries is courtesy and property of SNCF .................................................. 13 

Table 2.2: Number of elements and element type used to model each component of the catenary

model .......................................................................................................................... 19 

Table 2.3: Geometric and material properties of a generic simple catenary ............................... 20 

Table 3.1: Rigid body data of the pantograph multibody model ................................................. 31 

Table 3.2: Kinematic joints used in the pantograph multibody model ........................................ 32 

Table 3.3: Linear force elements data used in the pantograph multibody model ........................ 32 

Table 5.1: Identified lumped mass model parameters for the CX, ATR95 and DSA800

pantographs ................................................................................................................. 46 

Table 5.2: Geometric and material properties of the LN2 catenary ............................................ 46 

Table 5.3: Geometric and material properties of the C270 catenary ........................................... 47 

Table 5.4: Geometric and material properties of the Re330 catenary ......................................... 47 

Table 5.5: Description of the intervals of interest for the reports of the dynamic analysis used for

each catenary model. ................................................................................................... 51 

Table 5.6: Maximum steady arm uplifts registered on the CX-LN2 pair ................................... 53 

Table 5.7: Statistical quantities of the pantograph-catenary contact force for the CX-LN2 pair

for different lumped mass pantograph formulations and their absolute deviation ...... 53 

Table 5.8: Maximum steady arm uplifts registered on the ATR95-C270 pair ............................ 55 

Table 5.9: Maximum steady arm uplifts registered on the ATR95-C270 pair ............................ 56 

Table 5.10: Description of the intervals of interest used for each catenary section .................... 67 

Table 5.11: Geometric and material properties of the LN2 catenary (TGV Atlantique line) ..... 67 

xiv

1

1 Introduction

The high speed railway systems are becoming key-players in worldwide transport policies. This

results from the rising oil prices and from the urgency for reduction of CO2 emissions, among

others. To improve the competitiveness and attractiveness of railway networks, the trains have

to travel faster, with improved safety and comfort conditions and with lower life cycle costs.

Furthermore, the railway operators are demanding reductions in the overall operational costs.

They put particular emphasis on the railway vehicles maintenance costs and on the

aggressiveness of rolling stock on the infrastructures. The quest for interoperability has in the

compatibility of different pantographs with existing and projected catenary systems puts an

extra level of demand on the ability to control their interface.

Within respect to other means of transportation, 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.

As railway vehicles with electrical traction are, today, the most economical, ecological

and safe means of transportation the energy collection of the pantograph on the catenary is a

crucial element for their reliable running. A limitation on the velocity of high-speed trains

concerns the ability to supply the proper amount of energy required to run the engines, through

the catenary-pantograph interface [1]. Due to the loss of contact not only the energy supply is

interrupted but also arcing between the collector bow of the pantograph and the contact wire of

the catenary occurs, as depicted in Figure 1.1, leading to the deterioration of the functional

conditions of the two systems.

Figure 1.1: Detail of the pantograph catenary contact, during the operation of a high-speed train,

with the occurrence of dropper slacking and arcing.

2

The increase of the average contact force would improve the energy collecting

capabilities with less incidents of loss of contact but would also lead to higher wear of the

catenary contact wire and pantograph collector strip [2, 3]. A balance between contact force

characteristics and wear of the energy collection system is the objective of improving contact

quality. Even in normal operating conditions, a control on the catenary-pantograph contact force

is required to ensure longer maintenance cycles and a better reliability of the systems.

The quest for higher speeds for train operations in lines that were designed for lower

speed requires changes in the catenaries, pantographs and eventually in the vehicles. By

decreasing the weight per unit of length of the contact wire and increasing its tension higher

operation speeds are allowed in the same line due to the higher wave propagation speed. For

instance, in order to achieve the world record of 574.8 km/h by French high-speed train, the

contact wire used is made of copper with a cross-section of 150 mm2, the axial tension of the

contact wire increased from the current 26KN to 40N, being the voltage increased from 25 kV

to 31 kV [4]. In any case, the total tension in the overhead system must remain unchanged if the

changes that lead to a speed increase are to be implemented in an existing line [5, 6]. For

instance, to allow for the Shinkansen to operate at 360 km/h, in lines that were designed for

operations up to 240 km/h not only the tension of the contact wire is increased from 14.6 KN to

19.6 KN but the tension on the messenger and auxiliary wires also changed in order to maintain

the total tension at 53.9 KN [5]. Certainly the technological means of tensioning the catenary

wires and to maintain it regardless of temperature variations and material degradation is

important to guarantee the quality of the catenary-pantograph interface [7]. With standard

balance-weighted systems variations of 10% on the contact wire tension can be observed in just

few hours [8].

Another important aspect of the catenary design is to maintain the stiffness of the contact

wire to transversal loading by the pantograph as constant as possible [8, 9]. Different types of

catenaries exist, as seen in Figure 1.2, with alternative topological arrangements. The compound

catenary, commonly used in the Japanese Shinkansen, guarantees an almost uniform stiffness

while maintaining the contact wire at constant height, without requiring pre-sag, i.e., without

requiring the height of the contact wire to be lower at the middle of the span [10]. Catenaries,

such as the French LN1 and the German Re330, use the stitch wire to improve the uniformity of

stiffness around the steady-arms while in the simple catenaries, such as the French LN2 or

Italian C270, the stiffness is controlled via the dropper distance around the steady-arms. For

both stitch and simple catenary types there is a pre-sag of 1/1000 of the span to further improve

the uniformity of the stiffness [8, 10]. Due to less costly new designs that allow a better stiffness

uniformity simple catenaries are being preferred over compound and stitch wire catenaries in

new high speed line implementations [9, 11, 12].

3

a) b)

c) d)

Figure 1.2: Different high-speed catenary types: The French Southeast LN1 (a), and the German

Re330 (b) are Stitch wire catenaries; The French Atlantic LN2 (c) is a Simple catenary;

Japanese Shinkansen (d) is a compound catenary.

The catenary system dynamics exhibits small displacements about the static equilibrium

position. The only source of nonlinearity results from the slacking of the droppers. Therefore,

the linear finite element method has all features necessary to the modelling of this type of

systems, provided that the nonlinear effects are suitably modelled as nonlinear forces, in this

case, the dropper slacking can be handled by adding corrective terms to the system force vector.

The overhead catenary system is a very lightly damped structure in which the damping

characterization is important, in particular when the trains are equipped with multiple

pantographs [8]. The introduction of damping devices in the droppers of the catenary has been

attempted to better control the contact wire vibrations [11]. Different studies show that the

evaluation of the pantograph-catenary contact quality is highly dependent on the amount of

structural damping considered for the catenary structural elements [13]. However, it is also

recognized that the estimation of the structural damping of the catenary is still a technological

challenge.

The aerodynamic forces due to the direct effects of the wind on the overhead contact line

and on the pantograph components and indirect effect due to the additional motion of the

carbody imparted to the base of the pantograph influence the dynamics of the pantograph and

catenary [8, 14]. Apart from the consideration of a mean wind speed for the design of catenary

and its supporting structure, according to wind maps [15], investigations have been carried out

4

for the consideration of galloping instability of catenary wires motion, due to wind action in

particularly exposed areas. Mitigations of such phenomena, based on wind shield or increase

damping of catenary have been considered[16]. The problem of characterizing and controlling

the pantograph aerodynamics, although addressed in several technological projects, is still an

open issue for research [14, 17].

Different pantographs are currently used in train vehicles in the World, among which

some of the pantographs in operation in Europe are depicted in Figure 1.3. With the exception

of the Shinkansen 500 series telescopic pantograph, all other current collectors are of the two

stage type. The topology of a pantograph must address three ranges of its operation: lift the pan

head to the contact wire height and compensate for spans with lower catenary heights, generally

with frequencies of 1-2 Hz; handle the displacements with middle range frequencies associated

to steady-arms passage, up to 10 Hz; deal with the higher frequency but low amplitude events

[8]. Typically the pantograph head, with its suspension, is responsible for handling the high-

frequency excitations, up to 20 Hz, while the lower stage, including the pneumatic bellow, deal

with the low frequency excitations, below 5 Hz. Some other effects such as the pantograph bow

flexibility and aerodynamics may result in the contact force to exhibit frequency contents over

20 Hz [18].

a) b) c) d)

Figure 1.3: Different high-speed railway vehicles roof pantographs: a) Faiveley CX; b)

Stemman DSA380; c) Stemman ASP400; d) Contact ATR95.

A large majority of the pantographs in operation have been developed with particular

catenary systems in mind, forming national pantograph-catenary pairs such as the CX-LN2,

DSA380-Re330 and ATR95-C270 pairs. However, present trends for interoperability result in

new pantograph design requirements allowing operations in different catenary systems. It is

accepted that the improvement of the current collection capabilities, in the pantograph side,

require lighter bows, which in turn suggest the use of new materials and construction concepts

and that the head suspension is adjusted accordingly [19]. The need for the pantographs to have

low aerodynamic drag and noise emission and to be compatible for cross-border operation sets

some of the development directions for improved collectors [8]. Due to the range of motion of

5

the pantograph mechanical components and to the nonlinear elements present on its

construction, multibody methods are well suited to handle the pantograph dynamics [14].

Special models based on the use of lumped masses can still be used in the framework of linear

finite element methods [1], however, the use of multibody methods ensure that both lumped

mass and detailed nonlinear pantograph models still be used in the analysis of the pantograph-

catenary interaction problem. Hybrid methodologies in which the catenary dynamics is

evaluated using a linear finite element model and the pantograph dynamics is obtained using a

real prototype in a test bench are hardware-in-the-loop alternatives to fully computational

oriented approaches [20].

The interaction of the pantograph and catenary is achieved through the contact of the

pantograph collector strip on the catenary contact wire. The ability of collecting reliable data on

the contact forces to allow not only monitoring the operating conditions of the overhead system

but also to allow for the validation of numerical models is one of the important key issues of the

pantograph/catenary dynamics [8, 17]. Different experimental approaches have been proposed

and are now implemented by operators, manufacturers and infrastructure owners that allow for

the acquisition of reliable and complete data sets [21-23] . The norm EN50317 specifies the

requirements for the measurement systems to be used [24]. In face of the ability to correctly

measure the contact forces the use of the statistical occurrence of contact losses has been

replaced by the statistical parameters extracted from the contact force to the measure for the

contact quality. The norm EN50367 specifies the technical criteria for the interaction between

pantograph and overhead line [25]. The experimental data on the contact force allows obtaining

the most important parameters required to evaluate the quality of the contact, i.e., mean contact

force, standard deviation of the contact force, maximum contact force, minimum contact force,

maximum and minimum statistical contact forces, histogram of the forces, number of contact

losses, duration of the contact losses and uplifts of the registration arms. The norm EN50367

specifies the following thresholds for pantograph acceptance:

Mean contact force (Fm) Fm=0.00097 v2+70 N

Standard deviation (max) max<0.3 Fm

Maximum contact force(Fmax) Fmax<350 N

Maximum CW uplift at steady-arm (dup) dup120 mm

Maximum pantograph vertical amplitude(z) z80 mm

Percentage of real arcing (NQ) NQ0.2%

The complete study of design and operational alternatives for the mechanics of the

overhead electrical system require that the dynamics of the pantograph-catenary are properly

6

modelled and that software, used for analysis, design or maintenance support, is not only

accurate and efficient but also allows for the modelling of all relevant details to the train

overhead energy collector system. Most of the software tools used for the simulation of the

pantograph-catenary interaction is based in the finite element method and on multibody

dynamic procedures [26-29]. However, in what the catenary modelling is concerned, the use of

the finite difference method and of modal analysis is also reported [8]. The modelling of contact

between the pantograph collector strip and the catenary contact wire can be done using

unilateral kinematic constraints [28, 30], which does not require the estimation of any contact

law parameter but prevents any loss of contact to be detected. Alternatively, penalty

formulations can be used [19, 26] with no limitations on how contact may develop but

requiring that the penalty terms of the contact law are estimated. In any case, the use of different

methods to handle the dynamics of the catenary and pantograph requires that either a single

code in which both methods are implemented is developed or that a co-simulation strategy

between the two codes is implemented [29, 31] . The contact modelling plays a central role in

the establishment of the co-simulation strategies [31].

The development of computer resources led simulations to be an essential part of the

design process of railway systems. Moreover, 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. The European

Strategic Rail Research Agenda [32] and the European Commission White Paper for Transports

[33] 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. Also, an important

issue arising during the design phase of new trains is the improvement of its dynamic

performance. The concurrent use of different computational tools allows carrying out several

simulations, under various scenarios, in order to reach an optimized design. In this way, studies

to evaluate the impact of design changes or failure modes risks can be performed in a much

faster and less costly way than the physical implementation and test of those changes in real

prototypes. Due to their multidisciplinary, all the issues involving railway systems are complex.

Therefore, the use of computational tools that represent the state of the art and that are able to

characterize the modern designs and predict their dynamic behaviour by using validated

mathematical models is essential. 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. However, developing innovative and

more complex methodologies in a co-simulation environment allow, not only to integrate all

physical phenomena, but also to assess the cross influence between them.

7

The work presented here follows this trend and proposes a modelling approach and a

computational methodology gathered in a software tool that enables the dynamic analysis of

pantograph-catenary interaction. The finite element method is used for the dynamic analysis of

the catenary and a multibody dynamics approach is used for the dynamic analysis of the

pantographs, regardless of being lumped or multibody models. A co-simulation environment is

setup to run interference between the independent catenary and pantograph dynamic analyses.

The methods proposed in this work are demonstrated in the framework of the application of the

regulation EN50367 to three case studies. One of the case studies is composed by a comparison

between three pantograph-catenary pairs that are currently operating in Europe. On another case

it is presented the analysis of multiple pantograph operation in high-speed trains between

realistic catenary and a high speed pantograph models. This case addresses one of the limiting

factors in high-speed railway operation that is the need to use more than a single pantograph for

current collection. As the overlap section represents a critical zone on the contact quality,

having the responsibility to provide a smooth transition between subsequent catenaries, in the

final case study, an analysis on catenary overlap section is presented with comparison to a

normal section of the same catenary system

8

9

2 Catenary Dynamic Analysis and Modelling

High-speed railway catenaries are periodic structures that ensure the availability of electrical

energy for the train vehicles running under them. A typical construction, such as the one

presented in Figure 2.1, includes the masts (support, stay and console), serving as support for

the registration arms and messenger wire, the steady arms, which not only support the contact

wire but also ensure the correct stagger, the messenger wire, the droppers, the contact wire and,

eventually, the stitch wire. Furthermore the functionality of the catenaries impose that spans

have limited length, to allow for curve insertion and that the contact and messenger wires are

not longer than 1.5 Km, depending each particular network. As shown on Figure 2.2 the

catenary geometry requires overlaps between the starting and ending spans of different sections.

Figure 2.1: General structural and functional elements in a high speed catenary.

span overlap section

Figure 2.2: Representation of the functional elements of the catenary geometry

Stay

Messenger wire

Stitch wire

ConsoleSupport

Registration arm

Steady arm

Contact wire

Dropper

10

Depending on the catenary system installed in a particular high-speed railway all the

elements or only some of them may be implemented. However, in all cases both messenger and

contact wires are tensioned with high axial forces not only to ensure the correct geometry, i. e.,

to limit the sag, guarantee the appropriate smoothness of the pantograph contact and ensure the

stagger of the contact and messenger wires, but also to allow for the correct wave travelling

speed to develop.

2.1 Characteristics of Current High-Speed Catenaries

The types of high-speed catenaries existing today in the different countries are either simple,

stich wire or compound catenaries. The trend of new implementations of high-speed lines is to

use simple catenaries. In any case, the need for increasing the train operating velocities not only

forces the catenary designs to present a smoother geometry of the contact wire, while

maintaining a flexibility as constant as possible, but also to withstand higher tension forces on

the messenger and contact wires.

One of the critical parameters that limits the operational velocity of the trains is the wave

propagation velocity on the contact wire, C, which is given by [34]

2

2

EI F

CL

(2.1)

where F is the tension of the contact wire, ρ is the contact wire mass per length unit, EI is the

beam bending stiffness and L is the beam length. When the train speeds approach the wave

propagation velocity of the contact wire, called critical velocity, the contact between the

pantograph and the catenary is harder to maintain due to increase in the amplitude of the

catenary oscillations and bending effects. In order to avoid this deterioration of the contact

quality the train speed should not exceed 70-80% of the contact wire wave propagation speed

[1]. For safety the maximum train operating speed, V, is set to be V=0.7C.

The high-speed railway line catenaries in use in France are depicted in Figure 2.3, for the

Southeast TGV line, and Figure 2.4, for the Atlantique TGV. The Southeast TGV line catenary,

built in 1980, was the first high-speed line built in Europe. The catenary is of the stitch wire

type being the registration wires supported by droppers and pinned to the console. In order to

allow the increase of the operating speed and to overcome some problems experienced with the

LN1 catenary, the Atlantique TGV line catenary LN2, built in 1990, is a simple catenary, with

the registration arms fixed to the console, and has a higher contact wire tension than the LN1.

11

Figure 2.3: LN1 catenary with stitch wire in use in the Southeast TGV line.

Figure 2.4: LN2 catenary, without stitch wire, in use in the Atlantic TGV line.

The German high-speed line catenary Re330 is of the stitch wire type, as presented in

Figure 2.5. Due to the very high tension of the contact wire this catenary has the maximum

critical speed in the commercial catenaries.

F = 14 kN

F = 15 kN

F = 14 kN

F = 15 kN

½ stagger½ stagger

F = 14 kN

F = 20 kN

F = 14 kN

F = 20 kN

½ stagger ½ stagger

12

Figure 2.5: Re330 catenary, with stitch wire, in use in the German ICE lines.

The catenary used in the Japanese Shinkansen, depicted by Figure 2.6, is of the

compound type to provide a more constant flexibility for the pantograph contact. The catenary

suffered an evolution since it first started in operation, in 1964, until now to allow for an

increase of the contact wire tension. In the new Japanese high-speed lines the compound

catenary type is being abandoned being the simple type of catenary used instead.

Figure 2.6: Japanese compound catenary for the Shinkansen lines.

A summary of the main characteristics of the current catenaries is provided in Table 2.1.

It should be noticed that due to the research for higher critical speeds of the catenaries some of

the characteristics listed may change, especially in what the wire materials, cross-section and

tension is concerned.

F = 21 kN

F = 27 kN

F = 21 kN

F = 27 kN

½ stagger ½ stagger

F = 21.6 kN

F = 12.7 kN

F = 21.6 kN

F = 12.7 kN

F = 19.6 kN

F = 19.6 kN

½ stagger ½ stagger

13

France (LN6)

France (LN2)

Germany (Re330)

Italy (C270)

Japan (Shinkansen)

Catenary type Simple Simple Stitched Simple Compound Std. span (m) 54 54 65 57 50 Std. wire height from TOR (m)

5.08 5.08 5.3 5.3 5.0

Stagger (mm) ±200 ±200 ±300 ±200 ±100

Messenger wire

Section (mm2) 117 117 120 117 180 Mass (kg/m) 1.038 1.038 1.068 1.038 1.450 Tension (N) 20000 16000 21000 16000 21600

Stitch wire

Section (mm2) 35 150 Mass (kg/m) 0.312 1.375 Tension (N) 3500 12700

Contact wire

Section (mm2) 150 150 120 150 110 Mass (kg/m) 1.334 1.334 1.075 1.335 0.936 Tension (N) 26000 20000 27000 20000 19600

Wave velocity (km/h) 503 441 572 441 521 Max operat. speed (km/h) 350 350 350 350 Train speed (km/h) 320 300 330 300 330 Pre-sag 1/2000 1/1000 1/1000 1/1000 none

Table 2.1: Summary of the characteristics of different catenary systems [4, 5, 10]. Data for the

French catenaries is courtesy and property of SNCF

Many new high-speed lines are being planned and built around the World. China, Korea,

India, Brazil, United States of America, Morocco, Russia, and Portugal are some of the

countries that exemplify this trend. It is noticed that the majority of these lines are selecting

catenaries of the simple type for their networks, with some design variations. The software

analysis tools for the pantograph-catenary interaction must be able not only to allow the

implementation in the models of the relevant design characteristics of the catenaries but also be

fitted with numerical methods that allow obtaining the sensitivity of the dynamic response of

the system to design features.

2.2 Catenary Modelling

When modelling catenary systems, two main concerns need to be addressed, the line tensioning

and the dropper slacking. The line tension on a catenary is often employed by means of weights

or occasionally by hydraulic tensioners. In either case the method ensures that the line tension

on the catenary wires, respectively the contact wire and the messenger wire, are kept under a

constant tension as much as possible considering temperature changes. To apply the tension on

the wires by weights a pulley system is mounted on both end supports of the catenary where the

wires meet and a set of weights hangs as represented on Figure 2.7.

14

Figure 2.7: Photo of a weights line tensioning system; Photo: Rainer Knäpper, Freenbsp;

Art License (http://artlibre.org/licence/lal/en)

As already pointed out in section 2.1, the wires are kept under mechanical tension so that

the oscillations on the wire due to pantograph-catenary contact have a wave propagation speed

faster than the train. This avoids the increase in the amplitude of the catenary oscillations and

bending effects that would otherwise not only introduce more disturbances on the contact but

also cause excessive wear and possible wire breakage. Also the line tensioning has a direct

effect on the contact wire sag. With no tensioning the sag is much more accentuated and would

not represent the catenary accurately. Evidently the more sag is present in a contact wire the less

constant the contact force will be as more perturbation is introduced on the contact. Because of

the influence of line tension on the wave propagation speed and on the wire sag, it is than

understandable that to correctly capture the dynamic behaviour of the pantograph-catenary

interaction it is important to take the wire tension into account when constructing catenary

system models.

The dropper slacking is also an important nonlinear behaviour that is necessary to

consider in any catenary dynamic analysis tool. This phenomenon is shown in Figure 2.8 for a

normal operation on a high-speed train.

15

(a) (b)

Figure 2.8 Dropper slacking for the same operation scenario: (a) pantograph-catenary contact is

maintained, (b) loss of contact with arcing

When supporting the contact wire, the dropper is in a tension stress state. However when

the pantograph passes under the dropper, it is compressed and its tension vanishes. As the

dropper is a braided cable, which does not offer any compressive resistance, it slacks

transmitting no reaction forces to the rest of the catenary system. This occurrence represents a

nonlinearity when modelling the dynamic behaviour of the catenary. Nonlinear problems

usually require more complex numerical methodologies that lead to bring larger computational

cost. However this nonlinearity is well localized and its implied behaviour is known. So it is

possible that by adding corrective measures on the numerical solution of the problem a linear

methodology can still be used, thus avoiding the use of more complex methodologies and

saving valued computational time.

2.3 The Finite Element Method on the Pantograph-Catenary System

The motion of the catenary is characterized by small rotations and small deformations, in which

the only nonlinear effect is the slacking of the droppers. The axial tension on the contact, stitch

and messenger wire is constant and must be considered in the analysis. Therefore, the catenary

system is typically modelled with linear finite elements in which the dropper slacking

compensating forces, pantograph contact and gravitational forces are included in the force

vector.

To capture the dynamic behaviour of the catenary, an integration algorithm was

implemented based on the implicit Newmark’s trapezoidal rule but taking into account specific

modelling needs of the dynamics of the catenary model. Besides the integration algorithm

accuracy and stability a other aspect that is crucial to consider is its effectiveness, especially

when considering computational costs.

16

2.3.1 Dynamic Analysis of Catenaries Using Linear FEM

Using the finite element method, the dynamic equilibrium equations for the catenary structural

system are assembled as [35, 36],

M a C v K d f (2.2)

where M, C and K are the finite element global mass, damping and stiffness matrices of the

finite element model of the catenary. The accelerations, velocities and displacements vector are

represented respectively as a, v and d while the sum of all external applied forces is depicted by

vector f. In order to represent accurately the stress stiffening of the catenary structure due to the

tension stress state caused by the line tensioning with high axial tension forces the beam finite

element used for the messenger, stitch and contact wire, designated as element i, is written as

e e ei L GFK K K (2.3)

in which KeL is the linear Euler-Bernoulli beam element, F is the axial tension and Ke

G is the

element geometric matrix. The droppers and the registration and steady arms are also modelled

with the same beam element but disregarding the geometric stiffening. The 3D linear Euler-

Bernoulli beam element KeL, the element geometric matrix Ke

G plus the local mass matrix Me

are [35],

3

3

2

2

3 2 3

3 2 3

2 2

2

120

120 0

0 0 0

6 40 0 0

6 40 0 0 0

0 0 0 0 0

12 6 120 0 0 0 0

12 6 120 0 0 0 0 0

0 0 0 0 0 0 0 0

6 2 6 40 0 0 0 0 0 0

6 2 60 0 0 0 0

z

y

y y

z z

eL

z z z

y y y

y y y y

z z

EA

lEI

lEI

Symmetricl

GJ

lEI EI

llEI EI

llEA EA

l lEI EI EI

l l lEI EI EI

l l lGJ GJ

l lEI EI EI EI

l ll lEI EI EI

ll

K

2

40 0 0

z zEI

ll

(2.4)

17

0

60

56

0 05

0 0 0 0

1 20 0 0

10 151 2

0 0 0 010 15

0 0 0 0 0 0 0

6 1 60 0 0 0 0

5 10 56 1 6

0 0 0 0 0 05 10 5

0 0 0 0 0 0 0 0 0 0

1 1 20 0 0 0 0 0 0

10 30 10 151 1 2

0 0 0 0 0 0 0 010 30 10 15

eG

l

Symmetricl

l

l

l l

l l

l l

l l

K

(2.5)

2

2

2

2

1

313

035

130 0

35

0 0 03

110 0 0

210 105

110 0 0 0

210 1051 1

0 0 0 0 06 3

9 13 130 0 0 0 0

70 420 359 13 13

0 0 0 0 0 070 420 35

0 0 0 0 0 0 0 06 3

13 3 11 130 0 0 0 0 0 0

420 420 210 35

13 3 11 130 0 0 0 0 0 0 0

420 420 210 35

x

e

x x

Symmetric

J

A

l l

l l

lA

l

l

J J

A A

l l l

l l l

M

(2.6)

In which E, is the Young modulus, G, is the 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 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 catenary mesh. In order to model

the damping behaviour of the system, proportional damping, also known as Rayleigh damping

[36]. The global damping matrix C is evaluated as

18

C = Κ + Μ (2.7)

were parameters and are defined to represent an adequate damping response of the system

with the reasoning of the overall stiffness and mass characteristics of the system. It is also

possible to implement a more particular approach of this method by evaluating a proper

damping matrix Ce for each element as

e e e e eC = Κ + Μ (2.8)

where e and e are proportionality factors associated with each type of catenary element e,

such as dropper, messenger wire, stitch wire and so on. In this case the global C matrix is

obtained by assemblage of the elements damping matrices.

The force vector, f, containing the sum of all external applied loads, is evaluated at

every time step of the time integration. For a time t+Δt the force vector is calculated as

g t c dt t t t t tf f f f f (2.9)

where the vector gf contains the gravitational forces of all elements and vector tf is made of

forces responsible for line tensioning the wires, i. e. , they model the weights installed in the

endings of the wires. Both gf and tf remain constant. The force vector cf represents the

pantograph contact forces being evaluated as,

ct t c c i

i

f B f (2.10)

where cf represents the equivalent forces and moments applied at appropriate nodes of the

contact wire element where a contact force, at time t+Δt, is to be applied. The matrix cB means

the Boolean operation of assembling each contact force icf 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 4.1. The force vector d

t tf contains dropper slacking compensating terms which need to

be corrected at each time step whenever a dropper gets slack. Otherwise this vector is null.

Although the droppers perform as bar elements during extension, which is most of the time,

their stiffness during compression is either null or about 1/100th of the extension stiffness. This

extent occurs when contact forces are applied in the vicinity of the dropper. As the droppers

stiffness is included in the stiffness matrix K as a bar element, anytime one of them is

compressed its contribution to the catenary stiffness has to be removed, or modified to include

the dropper slacking. In order to keep the dynamic analysis linear, the strategy pursued here is to

19

compensate the contributions of any compressed dropper, i, to the stiffness matrix. This is done

by adding the vector force d

t tf , to the global vector f , equal to the bar compression force as

d et t d d t t i

i

f B K d (2.11)

where edK is the dropper i element stiffness matrix and

t td is a close prediction of the dropper

node displacements. The Boolean matrix dB simply maps the global nodal coordinates into the

coordinates of the dropper element, having the same meaning as for the contact forces.

2.3.2 FEM Catenary Models

In this work, for the construction of finite element models of catenary systems, all

catenary elements are modelled with a 3D beam element based on Euler Bernoulli beam theory

[8]. This 3D beam element, which formulation is developed in [35] , is assumed to be a straight

beam of uniform cross section capable of resisting to axial forces, bending moments about the

two principal axes in the plane of its cross section and twisting moments about its centroid axis.

Also of importance about this 3D beam element is that it accounts for the stress stiffening on

bending which is critical when modelling the contact and messenger wires that are subject to

high tensioning. The one exception for the use of these beam elements are for modelling the

claws and clamps that hold the structure together on the dropper/contact-wire/messenger-wire

and steady-arm/contact-wire junctions; these are modelled as lumped masses.

In order to ensure the correct representation of the wave propagation 4 to 6 elements must

be used in between droppers to appropriately model the contact and messenger wires. There is

no special requirement on the number of elements required to model each dropper, registration

or steady-arm as shown on Table 2.2.

Component Element Type Number of Elements

contact wire Euler Bernoulli Beam 6 between droppers

dropper Euler Bernoulli Beam 1 per each

messenger wire Euler Bernoulli Beam 6 between droppers

stich wire Euler Bernoulli Beam 1 between droppers

steady arm Euler Bernoulli Beam 1 per each

gramps/claws lumped mass -

Table 2.2: Number of elements and element type used to model each component of the catenary

model

20

The typical data required to model a catenary depends on the catenary type, among which

most of the high-speed catenaries used in operation today are of one of the types presented in

Figure 1.2. For a simple catenary, not necessarily reflecting any of the catenaries illustrated in

Figure 1.2, the typical data required to build a finite element model is presented in Table 2.3.

General

Catenary height [m] 1.4 Contact wire height [m] 5.08 Number of spans 25 Number of droppers/span 8 Nº spans at C.W. height 20 Inter-dropper distance [m] 6.75 Span length [m] 50-54 Stagger [m] 0.20

Damping 0.027 Damping 0

Contact Wire Messenger Wire Droppers Steady Arms

Material Cu Bz II (braided) Bz II (braided) Section [mm2] 150 66 12 120 Mass [kg/m] 1.33 0.605 0.11 1.07 Tension [N] 20000 14000 - - Claw with dropper dropper - C.W. Claw mass [kg] 0.195 0.165 - 0.200 Length [m] - - 1.25-1.075 1.24 Angle w/horiz. - - 90º -10º

Table 2.3: Geometric and material properties of a generic simple catenary

Using the data contained in Table 2.3 a finite element model of the generic simple

catenary is obtained, being different views shown in Figure 2.9. Note that the initial and

terminal spans of the catenary have the contact wire in position for the initialization of the

contact with the pantograph. However, the length of these spans is fundamental in the dynamic

calculations due to the wave reflection.

To represent accurately the wire tensioning on the catenary system the axial tension is

added to the catenary by adding constant axial compressive forces, with value equal to the

applied line tensioning force, on each element of the contact, messenger and stich wire, while at

both end nodes of the contact and messenger wire a pinned point constraint is applied as

represented on Figure 2.10. Also a pinned point constraint is added to each end node of the

steady arm element and on the messenger wire node right above a roller constraint, which is free

to move along the catenary length. These boundary conditions not only allow to preserve the

stagger but also to represent the masts action on the catenary.

21

Figure 2.9: Finite element model of a generic catenary with the sag highlighted.

Figure 2.10: Representation of the constraints applied on a general catenary model

0 200 400 600 800 1000 1200-2

-1.5

-1

-0.5

0

0.5

1

1.5y y

X [m]

Y [

m]

0 200 400 600 800 1000 12005

5.5

6

6.5

7

7.5

8y y

X [m]

Z [

m]

0

200

400

600

800

1000

1200

-2

-1.5

-1

-0.5

0

0.5

1

1.5

5

6

7

8

X [m]Y [m]

Z [

m]

22

Note that the data presented here for the catenaries is indicative. Due to confidentiality

reasons it is not possible to reproduce the exact data used for the catenary models developed in

this work [37].

2.3.3 Lumped Mass Pantograph Dynamic Analysis in Finite Elements Application

When using lumped mass pantograph models it is possible to use the same finite element

code to solve the equations of motion of both pantograph and catenary. For this purpose, the

pantograph is considered a linear system and its equations of motion must be assembled in the

same way as the catenary equations, expressed by equation (2.2). The contact forces developed

between the pantograph collector strip and the catenary contact wire are evaluated with the same

contact model used one the multibody pantograph, described on section 4.1, and applied both on

the appropriate beam element of the contact wire and the top mass of the pantograph. Notice

that in this case only a longitudinal velocity of the pantograph is prescribed and no other motion

between the masses occurs besides their expected vertical movement.

In order to produce the pantograph equations of motion, consider the representation of the

lumped mass pantograph model, in Figure 2.11, with three staged lumped masses (m1,m2,m3)

linked by spring/damper suspensions with correspondent model parameters (K1,K2,K3,C1,C2,C3).

Figure 2.11: Representation of the pantograph lumped mass model

The governing equations of motion are derived at each mass as

3 3 3 3 2 3 3 2 03 3( ) ( ) ( ) cm y C y y K y y l m g F t (2.12)

2 2 3 2 3 2 2 1 3 2 3 03 2 2 1 02 2( ) ( ) ( ) ( ) m y C y y C y y K y y l K y y l m g (2.13)

3 3 2 1 2 1 1 0 2 1 2 02 2 1 0 01 1( ) ( ) ( ) ( ) staticm y C y y C y y K y y l K y y l m g F (2.14)

23

being the following restriction used as a boundary condition.

0 0 0constant 0 y y y (2.15)

In addition to the specific pantograph lumped mass model parameters, the equations of

motions involve the gravitation constant g, in the evocation of the weights due to the masses.

The spring free lengths, l01 ,l02, l03 are considered along with the ground height y0, relative to the

catenary model in order to adjust the lumped masses heights. Of importance is the top mass

height, with which contact force Fc(t) is evaluated at each time and applied back on the top

mass, m3. The static uplift force, Fstatic, of the pantograph model, applied on its lower mass,

provides a direct way to regulate the contact force magnitude and its mean value on a dynamic

analysis.

Equations (2.12) to (2.15) can be re-written in vector-matrix form in order to use the same

form as the catenary equations of motion,

3 3 3 3 3 3 3 3 3 3 03

2 2 3 3 2 2 2 3 3 2 2 2 2 3 03 2 02

1 1 2 2 1 1 2 2 1 1 1 2 02 1

0 0 0 0 ( )

0 0

0 0 0 0 (

cm y C C y K K y m g K l F t

m y C C C C y K K K K y m g K l K l

m y C C C y K K K y m g K l K 01 0 )

staticl y F

(2.16)

In this form the equations of motion of the pantograph can be solved in the finite element code

either by adding them to the catenary equations or by solving them separately in a different

finite element code. When the pantograph and catenary equations of motion are solved

independently, i. e. ,

catenary dynamic equilibrium equation: cat cat cat catM a + C v + K d = f (2.17)

pantograph dynamic equilibrium equation: pat pat pat patM y + C y + K y = f (2.18)

being their interaction achieved via the contact force. When interpreted together, the completed

system of equation of motion are written as

cat cat cat cat

patpat pat pat

M 0 C 0 K 0 fa v d+ + =

fy y y0 M 0 C 0 K

(2.19)

In addition to the developed methodology above for the integration of a lumped mass

pantograph model in the finite elements code of the catenary dynamic analysis, it must be noted

that this lumped mass pantograph model can also be easily integrated on a multibody

application. This issue is approached in Chapter 3 of this work.

24

2.3.4 Time integration method

The selection of a 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. The time integration method embraced for this specific

implementation is an implicit Newmark family integration algorithm [36, 38]. This particular

method was chosen due to its unconditional stability nature when used implicitly and its proven

robustness in FEM applications of the type of the ones demonstrated in this work.

To illustrate the developed integration algorithm consider the solution of the linear

dynamic equilibrium equation of the catenary system expressed by equations (2.2). 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

t t t t t t t tM a C v K d f (2.20)

Admitting that the solution of the dynamic equilibrium equation is known at time t, the

Newmark method leads to the displacements and velocities on time t+Δt obtained by

21

2

t t t t t t tt td d v a a (2.21)

1 t t t t t t tv v a a (2.22)

The parameters γ and ζ are determined in order to obtain integration accuracy and stability.

However when γ =1/4, ζ=1/2 and the above stated assumptions are used implicitly to solve the

equilibrium equation this particulate application of the Newmark method is unconditionally

stable and known as “trapezoidal rule”. In order to solve the present system implicitly the

Newmark assumptions expressed in equations (2.21) and (2.22) are rearranged respectively for

t ta and t tv in terms of t td giving,

2

1 1 11

2

t t t t t t ttta d d v a (2.23)

1 22

t t t t t t t t

t

tv d d v a (2.24)

These two relations are then substituted into the equilibrium equation (2.20) which than can be

solved for the displacements t td as,

ˆ ˆˆ t +Δt t+Δt t +Δt t+ΔtKd = f LUd = f (2.25)

where,

25

0 1ˆ a aK K M C (2.26)

0 2 3 1 4 5ˆ t t t t t ta a a a a at +Δt t +Δtf f M d v a C d v a (2.27)

and,

0 1 2 32

4 5 6 7

1 1 1; ; ; 1;

2

1; 2 ; 1 ; 2

a a a at tt

ta a a t a t

(2.28)

The notation LU is used in equation (2.25) to mean a factorization of the stiffness matrix in the

solution of the implied system of equations [39]. Afterwards the accelerations and velocities can

me calculated by using equation (2.23) and (2.24).

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, as the one implemented in this work, because the largest computation

cost that occurs at each integration time step is the solution of the system of linear equations

expressed on equation (2.25). 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 [36, 40]. In this case a LU decomposition is selected.

Taking the advantage on the fact that the effective stiffness matrix K̂ remains constant as also

its factorization products, the factorization is done only once and the same products are used on

the procedure at every time step. This method not only saves computational cost for the

dynamic analysis but also allows to treat the dropper nonlinearity as a compensating force in the

framework of a linear analysis rather than following a nonlinear dynamic analysis approach.

One other aspect of the integration algorithm involves the calculation of the effective

loads vector t̂ +Δtf . As the external loads vector f , expressed in (2.9), is not constant in time the

effective loads vector must be calculated at every integration time step. Moreover the

calculation of the pantograph contact forces and dropper compensation forces, as expressed in

equations (2.10) and (2.11), depend on a close prediction of the node displacements,

t td , and

velocities, t tv ,that would belong to the solution of the dynamic equilibrium equations at time

t t . 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 td and tv 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 and so on. This correction procedure is done iteratively until a

26

good enough convergence is reached where, t t t t dd d and t t t t dv v , being d and

v user defined tolerances. The iterative process is represented in Figure 2.12 together with the

integration algorithm steps. Note that the criteria for convergence of the nodal displacements

and velocities must imply convergence of the force vector also, i.e., the balance of the

equilibrium equation right-hand side contribution of the dropper slacking compensation force

with the left-hand-side product of the dropper stiffness by the nodal displacements in equation

(2.20).

t t t t d

t t t t v

d d

v v

2

1 1 11

2

1 22

t t t t t t t

t t t t t t t t

tt

t

t

a d d v a

v d d v a

ˆt+Δt t+ΔtLUd = f

0 2 3 1 4 5ˆ t t t t t ta a a a a at+Δt t+Δtf f M d v a C d v a

, c

t t t t t tf (d v )

d

t t t tf (d )0

t t t

t t t

i

d d

v v

1 i iallowed iterationsi

t t t t

t t t t

d d

v v

Figure 2.12: Flowchart of the integration algorithm steps as each time step

Experimenting with the computational implementation of the finite element procedure,

outlined here in the context of the catenary dynamic analysis, shows that the maximum number

of iterations allowed, for the correction process to be 4 or higher. If a maximum number of

iterations is set to be below 4 there is the danger that the droppers exhibit residual compression

forces during the dynamic analysis, with all implications that such error has over the evaluation

of the pantograph-catenary contact force.

27

3 General Pantograph Dynamic Analysis and Modelling

The railway roof pantographs are the systems responsible for collecting the energy from the

overhead line. In order to guarantee a smooth operation, without losing contact with the contact

wire or requiring an excessive contact force that, which lead not only to high wear but also large

uplifts of the steady arms, the pantographs must be dynamically responsive to the different

range of frequencies with which they are excited. Furthermore, the pantograph and catenary

must have characteristics that allow for multiple pantograph operation without the degradation

of the contact quality of any of the pantograph collector strips [41]. The use of active control

strategies for the pantograph may lead to an improvement of the pantograph contact, especially

for the trailing pantographs. However many of the prototypes are still experimental [42].

3.1 Characteristics of High-Speed Pantographs

The roof pantographs used in high-speed railway applications are of the type depicted in Figure

1.3. Mechanically they are characterized as mechanisms with three loops ensuring that the

trajectory of head, while lifting the pantograph, is in a straight line, perpendicular to the plane of

the base, while the pantograph head is maintained levelled. The pantographs are always

mounted in the train in a perfect vertical alignment with the centre of the boggies of the vehicle

in order to ensure that during curving the centre of the bow does not deviate from the centre of

the railroad, more than what is expected from the normal railway dynamics. The mechanical

system that guarantees the required characteristics of the trajectory of the pantograph head

during rising is generally made up by a four-bar linkage for the lower stage and another four-bar

linkage for the upper stage. Another linkage between the head and the upper stage of the

pantograph ensures that the bow is always levelled. In order to control the raise of the

pantograph one bar of the lower four-bar linkage is actuated upon by a pneumatic actuator.

The major differences between current pantographs reside not only on the raising

mechanism constituted by the actuator and the lower stage but also in the pantograph head and

its suspension. The numerical methods used to perform the dynamic analysis of the pantograph

must be able to represent the important details of the system, including mechanisms and

compliances and to evaluate their correct dynamics. Two different types of models are generally

used to represent pantographs: lumped mass and multibody. Each of them has advantages and

shortcomings in their use that are discussed hereafter.

3.2 Multibody Dynamic Analysis of Pantographs

A typical multibody model is defined as a collection of rigid or flexible bodies that have their

relative motion constrained by kinematic joints and is acted upon by external forces. The forces

applied over the system components may be the result of springs, dampers, actuators or external

28

applied forces describing gravitational, contact/impact or other forces. A wide variety of

mechanical systems can be modelled as the schematic system represented in Figure 3.1.

Figure 3.1: Generic multibody system.

Let the configuration of the multibody system be described by n Cartesian coordinates q,

and a set of m algebraic kinematic independent holonomic constraints be written in a compact

form as [43].

, tΦ q 0 (2.29)

Differentiating Equation (2.29) with respect to time yields the velocity constraint equation.

After a second differentiation with respect to time the acceleration constraint equation is

obtained

qΦ q υ (2.30)

qΦ q γ (2.31)

where q is the Jacobian matrix of the constraint equations, is the right side of velocity

equations, and is the right side of acceleration equations, which contains the terms that are

exclusively function of velocity, position and time.

The equations of motion for a constrained multibody system (MBS) of rigid bodies are

written as

( ) cMq g g (2.32)

where M is the system mass matrix, q is the vector that contains the state accelerations, g is the

generalized force vector, which contains all external forces and moments, and g(c) is the vector

of constraint reaction equations. The joint reaction forces can be expressed in terms of the

Jacobian matrix of the constraint equations and the vector of Lagrange multipliers

( ) c Tqg Φ λ (2.33)

body 1 body 3

body 2

body ibody n

Damper

Ball jointSpring

Revolute joint

External forces

29

where is the vector that contains m unknown Lagrange multipliers associated with m

holonomic constraints. Substitution of Equation (2.33) in Equation (2.32) yields

TqMq Φ λ g (2.34)

In dynamic analysis, a unique solution is obtained when the constraint equations are

considered simultaneously with the differential equations of motion with proper set of initial

conditions. Therefore, equation (2.31) is appended to equation (2.34), yielding a system of

differential algebraic equations that are solved for q and . This system is given by

Tq r

q

M Φ q g

Φ 0 λ γ (2.35)

In each integration time step, the accelerations vector, q , together with velocities vector,

q , are integrated in order to obtain the system velocities and positions at the next time step.

This procedure is repeated up to final time will be reached. The solution of the multibody

equations of motion and their integration in time is depicted in Figure 3.2. The set of differential

algebraic equations of motion, Equation (2.35) does not use explicitly the position and velocity

equations associated to the kinematic constraints, Equations (2.29) and (2.30), respectively.

Consequently, for moderate or long time simulations, the original constraint equations are rapidly

violated due to the integration process. Thus, in order to stabilize or keep under control the

constraints violation, Equation (2.35) is solved by using the Baumgarte Stabilization Method or

the augmented Lagrangean formulation and the integration process is performed using a

predictor–corrector algorithm with variable step and order. Due to the long simulations time

typically required for pantograph-catenary interaction analysis, it is also necessary to implement

constraint violations correction methods, or even the use of the coordinate partition method for

such purpose [43].

Figure 3.2: Flowchart with the forward dynamic analysis of a multibody system

Build

; ;

;

TqM

g

Solve

2

2

T

tt

q

q q

gM q

0 q

i

i

qy

q

Form1

i

i

i i

qy

q

Integrate

Yes No

ConstraintsCorrect?

Partition?

Build

*

12 ωi

Ti i i

rq

p L

Build

* 12

14

ω

(ω ω )

i

Ti i i

Ti

r

p Lq

p

* *;

u uq q

v v

Partition

uy

u

Form1i i

uy

u

Integrate

Factorize

*

q

LU R

S D

Calculate

( , , )t u v

u v 0

u v

No

Yes

No

* 0

* 0

0i

i

i

q qq q

Initialize

Correct?0 0;q q

Yes

30

Note that the numerical approach described implies the use of Cartesian coordinates. If

other types of coordinates are used, such as joint coordinates, the flowchart shown in Figure 3.2

must suffer the appropriate adaptations.

3.3 Multibody Pantograph Models

A pantograph consists of a collection of bodies and mechanical elements, as depicted in Figure

3.3(a), attached to a railway carbody that is moving along the track, as depicted in Figure 3.3(b).

Two modelling strategies can be used to define the reference motion of the pantograph: guide

the motion of the railway vehicle as a result of the rail-wheel contact and simply fix the

pantograph in the top of the train carbody, as implied in Figure 3.3(b), control the kinematics of

the pantograph base using a kinematic constraint, as specified in Figure 3.3(c) [44]. The first

strategy allows introducing in the pantograph-catenary contact the perturbation associated to the

dynamics of the vehicle and track interaction while the second strategy may include or not such

perturbations depending on how the trajectory associated to the guiding constraint is obtained.

In any case, once the kinematics of the pantograph is compatible with the geometries of the

catenary and railway both strategies are acceptable. In this work the motion of the base of the

pantograph is guided along a tangent track.

The rigid bodies that compose a particular pantograph system are presented in Figure 3.4.

Note that different pantographs may have different topologies, especially in what the head and

the lower stage raising mechanisms is concerned.

(a)

(b)

(c)

Figure 3.3: (a) Pantograph model; (b) Roof mounted pantograph on the vehicle guided on the

track; (c) Pantograph with prescribed base motion

11

RJ-2

RJ-1

RJ-3

SJ-1

SJ-2 SJ-3

SJ-4

22

33

44

55

66

77

mstatic

X

ZY

X

ZY

31

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3.4: The pantograph system: (a) Complete pantograph; (b) Base; (c) Lower arm; (d)

Upper arm; (e) Lower link; (f) Upper link; (g) Head support; (h) Bow

The data required for the multibody model of the pantograph concerns the mass, inertia,

initial positions and initial orientations of all bodies in the system, as shown in Table 3.1. The

type and location of all kinematic joints that connect the different bodies of the system, as

depicted by Table 3.2, and the force elements characteristics, i.e., springs, dampers and

actuators, must also be specified, as in Table 3.3.

ID

Rigid Body

Mass [Kg]

Inertia [Kg.m2] Initial Position [m] Initial Orientation I / I / I x0 / y0 / z0 e1 / e2 / e3

1 Panto. Base 32.65 2.76/ 4.87/ 2.31 0.000/ 0.000/ 0.000 0.00/ 0.00/ 0.002 Lower Arm 32.18 0.31/ 10.43/ 10.65 –0.571/ 0.000/ 0.412 0.00/ 0.17/ 0.003 Upper Arm 15.60 0.15/ 7.76/ 7.86 –0.394/ 0.000/ 1.055 0.00/ –0.18/ 0.004 Lower Link 3.10 0.05/ 0.46/ 0.46 –0.887/ 0.000/ 0.283 0.00/ 0.21/ 0.005 Upper Link 4.51 0.08/ 1.50/ 1.50 –0.357/ 0.000/ 1.003 0.00/ –0.16/ 0.006 Stab. Arm 4.67 0.34/ 0.01/ 0.50 0.553/ 0.000/ 1.418 0.00/ 0.00/ 0.007 Panto. Head 7.80 6.62/ 0.23/ 6.87 0.553/ 0.000/ 1.498 0.00/ 0.00/ 0.00

Table 3.1: Rigid body data of the pantograph multibody model

32

ID Kinematic Joint Connected Bodies Attachment Points Local Coordinates [m]

i j Body i (j/j/) Body j (j/j/j) 1 Revolute joint 1 2 (0.020/0.000/0.132)P (0.820/0.000/0.000)P (0.020/1.000/0.132)Q (0.820/1.000/0.000)Q

2 Revolute joint 2 3 (-0.820/0.000/0.000)P (-1.014/0.000/0.000)P (-0.820/1.000/0.000)Q (-1.014/1.000/0.000)Q

3 Revolute joint 3 6 (1.014/0.000/0.000)P (0.000/0.000/0.000)P (1.014/1.000/0.000)Q (0.000/1.000/0.000)Q

4 Spherical joint 1 4 (-0.259/0.000/0.000)P (0.688/0.000/0.000)P (-/-/-)Q (-/-/-)Q

5 Spherical joint 3 4 (-1.187/0.000/-0.126)P (-0.615/0.000/-0.025)P (-/-/-)Q (-/-/-)Q

6 Spherical joint 2 5 (-0.780/0.000/0.000)P (-1.000/0.000/0.000)P (-/-/-)Q (-/-/-)Q

7 Spherical joint 5 6 (0.962/0.000/0.000)P (0.000/0.000/-0.105)P (-/-/-)Q (-/-/-)Q

8 Revolute-Prismatic joint

6 7 (0.000/0.000/0.010)P (0.000/0.000/0.000)P (1.000/0.000/0.010)Q (0.000/0.000/-1.000)Q

Table 3.2: Kinematic joints used in the pantograph multibody model

ID Force

Element Stiffness Undef.

Length Damp.

Coeffic.Force Bodies Attach Pts Local Coord [m]

[N/m] [m] [N.s/m] [N] i j Body i (I / I / i) Body j (j / j / j) 1 Sp-Damp 2.00 0.459 60.0 0 1 2 0.259/0.000/0.000 0.870/0.000/-0.136

2 Actuator 0 0 0 920 1 2 0.483/0.000/0.125 0.870/0.000/-0.136

3 Sp-Damp 3000.00 0.103 13.0 0 6 7 0.000/0.335/0.000 0.000/ 0.335/0.010

4 Sp-Damp 3000.00 0.103 13.0 0 6 7 0.000/–0.335/0.000 0.000/ -0.335/0.010

Table 3.3: Linear force elements data used in the pantograph multibody model

It should be noted that the data provided for the pantograph model, in this work, which

closely matches the pantograph shown in Figure 3.4, is indicative of the configuration and

properties of a particular type of generic pantograph but that do not, necessarily, reflect any

existing pantograph. However, the data gathered ensures that the dynamic performance of the

model is realistic, but not necessarily validated. Note also that due to confidentiality reasons the

full spectrum of the pantograph data cannot be reported in this work [13].

One of the criticisms to this modelling procedure is that the multibody model of the

pantograph is made of rigid bodies connected by perfect kinematic joints, as the one presented

here. It only has 2 degrees-of-freedom, i.e., the raising of the pantograph and the head

suspension. In fact, laboratory dynamic analysis of real pantographs leads to frequency response

measurements in all pantographs that exhibit the presence of three resonances, implying that a

basic requirement for the pantograph models is to have three degrees of freedom, at least. The

third degree-of-freedom of the multibody pantograph may be associated with the clearance or

with bushings existing in the joints [45, 46] or even with the flexibility of one or more of the

bodies considered in this model [47, 48]. In any case, the minimal requirements for modelling

multibody pantographs that can have a realistic behaviour in the complete frequency range of

their operation, i.e. in the 0-20 Hz range, are still to be identified.

33

3.4 Lumped Mass Pantograph Model

Alternatively to a multi-body pantograph model another modelling method is commonly used

consisting on a lumped mass approach. The lumped mass pantograph model, depicted on Figure

3.5(b), is composed of a simple series of lumped masses linked consequently to a ground by

spring/damper elements. Although in the literature pantograph models are presented with two,

three of more mass stages, for high-speed train applications, there is a minimal requirement of

three stages to well represent the system [49].

a) b) c)

Figure 3.5: Lumped mass pantograph model: (a) Laboratory parameter identification procedure;

(b) Three stage lumped mass model; (c) Parameter values

While the multi-body pantograph models can be built with design data alone, for example

with data obtained from technical drawings complemented with measured physical

characteristics from selected components, the lumped mass pantograph model parameters, as the

ones presented on Figure 3.5(c), must be identified experimentally. In this sense, the lumped

mass pantograph model can be thought as a transfer function in which an experimental

procedure, represented on Figure 3.5 (a) is used. The test rig used for this procedure is also

presented in Figure 3.6. The idea is to excite the contact strips of the desired pantograph to

model with prescribed motions of known frequency and amplitude while measuring the

response of the pantograph namely the contact forces on the collector strip and positions,

velocities and accelerations at prescribed points of the mechanical pantograph. This acquired

data is then used to build the frequency response functions (FRF) of the pantograph. The

lumped mass model is built in such a way that the upper stage mass, m3, and suspension

characteristics, k3 and c3, match those of the prototype, while the intermediary and lower stage

parameters are identified in such way that the FRF of the model is matched to the

experimentally acquired FRF [50].

Z(t)

Excitation bar withprescribed motion

Parameter Valuem1 4.5m2 6.3m3 7.8k1 62k2 8000k3 7000c1 54c2 0c3 30

34

Figure 3.6: Test rig for the experimental identification of the lumped mass pantograph, courtesy

of Politecnico di Milano (PoliMi)

The structures of the lumped mass pantograph model, in the multibody framework,

include 4 rigid bodies connected by 3 translational joints, in the local z direction. In the same

direction, joining each body to another, there is as spring-damper element with the

characteristics depicted in Figure 3.5. A static force is applied to the mass m1 in order to ensure

a proper average contact force during the dynamic analysis.

It is important to note that in spite of the simplicity of their construction and fidelity of

their dynamic response, the lumped mass models are commonly used by operators,

manufacturers and homologation bodies instead of more complex models. The only part of the

lumped mass model that as a physical interpretation is the upper stage, which limits the use of

this type of models for any application that requires modifications on the pantographs structure

or mechanics. From this point of view, multibody pantograph models cannot be replaced

provided that they are able to adequately represent the dynamics of the implied system,

including a match with the FRF experimental data.

35

4 Pantograph-Catenary Interaction

The contact involved in the pantograph-catenary interaction involves the pantograph collector

strip and the catenary contact wire. The efficiency of the electrical current transmission and the

wear of the collector strip and of the contact wire are deeply influenced by the quality of the

contact. This implies that the correct modelling of the contact mechanics involved between

these two systems is crucial for its accurate and efficient evaluation. Furthermore, the catenary

finite element simulation software may also include a lumped mass pantograph model to

interact with the catenary, as described in 2.3.3, or in another approach communicate with a

multibody module that allows the simulation of a pantograph model. In this case, the contact

model is also used as a bridge in the co-simulation environment between the two distinct codes,

as described in section 4.2.

4.1 Contact Modelling

The contact between the collector strip of the pantograph and the contact wire of the catenary,

from the contact mechanics point of view, consists in the contact of a cylinder made of copper

with a flat surface made of carbon having their axis perpendicular as shown in Figure 4.1. The

contact problem can be treated either by a kinematic constraint between the collector strip and

the contact wire 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 [28, 30]. With the

second procedure the contact force defined in function of the relative penetration between the

two cylinders [45, 51]. The use of the kinematic constraint between contact wire and collector

strip forces these elements to be in permanent contact, being this approach valid only if no

contact loss exist. The use of the penalty formulation allows for the loss of contact and it is the

method of choice for what follows.

a) b) c)

Figure 4.1: Pantograph-catenary contact: (a) Pantograph bow and catenary contact wire; (b)

Cross-section of the contact wire; (c) Cross-section of the collector strip

36

4.1.1 Geometric Conditions for Contact

In order to model the contact force using a penalty formulation it is necessary to

geometrically access if there is contact and identify the contact location either on the pantograph

collector strip or on the catenary contact wire. Also it is necessary to calculate the relative

penetration of the contact. For this purpose a three step procedure is implemented at every time

step of the catenary integration algorithm and for each collector strip present on the considered

analysis. In the first step a contact wire 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

element on the contact wire. On the second stage, it is assumed that there is contact and by

geometric interference and use of the shape functions of the catenary finite element, the

potential points of contact on the contact wire and the collector strip are located. At the third

step the relative penetration of the contact is calculated and it is accessed if there is contact

indeed or there is contact loss.

To find the catenary element candidate for contact consider the representation of the top

view of a catenary contact wire element and a pantograph registration strip on Figure 4.2.

IC

AC

ˆ ABu

ˆ IJu

A

B

C

I

Jy

x

Figure 4.2: Representation of the top view of a catenary contact wire element and a pantograph

registration strip

The nodes A and B represent the collector strip extremities and nodes I and J are the

contact wire element node positions at the actual time step in consideration. The node C

represents the intersection, on the xy plane, of two lines defined by de nodes A, B and nodes I, J.

It is possible relate the node C position with the other defined nodes by

ˆ C I IJ ICr r u (2.36)

ˆ C A AB ACr r u (2.37)

37

Where ir defines the vector position of a node i, IC and AC is the norm from node I to C and

A to C respectively, and ˆ MNu is a versor of a generic vector that goes from node M to node N.

Decomposing the vector equations (2.36) and (2.37) on the x and y axes as,

C I IJxx x u IC (2.38)

C I IJyy y u IC (2.39)

C A ABxx x u AC (2.40)

C A AByy y u AC (2.41)

and equalling the expressions, it is possible to form the a equation system presented as

I IJx A ABx

I IJy A ABy

x u IC x u AC

y u IC y u AC (2.42)

where the solution is IC and AC .

Assuming that there is contact and that the collector strip is a rigid body, its potential

point of contact, pantographr , can be calculated as

ˆ pantograph A ABr ACr u (2.43)

where if this position is between de nodes A and B positions then the contact wire element

candidate for contact is found. If not the procedure is restarted for the next element of the

contact wire and so on.

The potential contact position on the contact wire element can be calculated as

( ) catenary i ir r N d (2.44)

Where, as presented on Figure 4.3, ir , corresponds to the contact wire node I position but

without the deformation accounted for, the vector id contains the displacements of the node i

and the matrix ( )N contains the element shape functions [52] in order of , which is the local

element relative position of the contact point in its longitudinal direction defined as

IC

L (2.45)

38

IC

id L

CI

i j

potential point of contact

on the catenary

jd

z

x

J

Figure 4.3: Representation of the potential point of contact on the catenary contact wire element

Finally, to access if there in fact contact in, the relative penetration, , is calculated by

( ) ( ) pantograph pat cateanry catz r z r

(2.46)

where pantographz and cateanryz are the position coordinates of the calculated potential points of

contact on the z axis, and patr and catr correspond to the contact wire and collector strip radius

of their considered circular cross section. If the relative penetration value is positive then there

is contact, if not or the value is null there is a contact loss at the time step taken into

consideration.

4.1.2 Continuous Contact Force Model

The continuous contact force model used here is based on a contact force model with

hysteresis damping for impact in multibody systems. In this work, the Hertzian type contact

force including internal damping can be written as [53]

( )

23(1 )1

4

nN

eF K (2.47)

where K is the generalized stiffness contact, e is the restitution coefficient, is the relative

penetration velocity and ( ) is the relative impact velocity. The proportionality factor K is

obtained from the Hertz contact theory as the external contact between two cylinders with

perpendicular axis. Note that the contact force model depicted by Equation (2.47) is one of the

different models that can be applied. Other continuous contact force models are presented in

references [54-56].

In the application that follows the contact is considered purely elastic, i.e., the restitution

coefficient e=1, the generalized stiffness defined to be K=20000 N/m and power of the

penetration is n=1. The relative penetration, , is evaluated taking into account the nodal

displacements of the beam finite element in which contact occurs and its shape functions. For

39

the efficiency of the computer code it is important that the numbers of the finite elements that

are in contact with each of the collector strips are kept track of so that unnecessary searches for

contact are avoided.

4.2 Co-Simulation of Multibody and Finite Elements

The interaction between the multibody pantograph and the finite elements catenary systems

represents a coupled problem where the pantograph is a nonlinear dynamic system handled with

variable time step and multi-order integrator, while the catenary is modelled as a dynamic linear

system integrated with a Newmark family numerical integrator using a fixed time step.

Generally the dynamic analysis of pantograph-catenary systems is done with both models

using the same formulation, as in the case of the FEM lumped mass pantograph model.

However, if it is expected the pantograph exhibits large displacements and rotations during its

operation it is not advisable to model this system with finite elements method. Nonlinear finite

elements in dynamic analysis of nonlinear systems when compared to multibody methodology

lead to larger computational time costs. Also, in spite of being possible to build a catenary

model in multibody formulation there is no reliable model that can handle all its complex

details. Furthermore the modelling of catenary systems using finite element formulation is

nowadays used by the railways industry, with validated results. For the mentioned reasons, in

the development of this work, both systems are then modelled using distinct formulations, i. e. ,

pantograph multibody models and catenary finite element models.

The analysis of pantograph-catenary interaction is done by two stand-alone and

independent codes, the multibody pantograph and the finite elements catenary, running in a co-

simulation environment. A procedure to run simultaneously a multibody code and a finite

elements code in a co-simulation environment, enabling real-time simulation of the pantograph–

catenary interaction is proposed here.

4.2.1 Co-Simulation Procedure

The structure of the communication between the pantograph and the catenary codes is shown in

Figure 4.4. The multibody pantograph code (PAT) provides the catenary finite element code

(CAT) with the positions and velocities of the pantograph collector strip A and B nodes. These

nodes correspond to the collector strip extremities. With the information provided, the CAT

code calculates the position of the point of contact between the pantograph collector strip and

the catenary contact wire by geometric interference and the contact force using an appropriate

contact model, as described in section 4.1. Following these calculations the CAT code

communicates to the PAT code a point, C, position correspondent to contact point position and

the contact force, Fc. Each code handles independently their equations of motion of their

40

referred sub-system and applies the contact force on the contact point both shared between

them.

Figure 4.4: Co-simulation flowchart between a finite element and a multibody code

The catenary is modelled as a dynamic linear system with its state variable integrated

with a Newmark family numerical integrator [38] using a fixed time step where a prediction of

the positions and velocities of the catenary but also the pantograph are needed. The multibody

pantograph code integration procedure is based on a predictor-corrector algorithm with variable

time step, which in order to proceed with its dynamic analysis needs the contact force and its

application point at different time instants during the integration period. Therefore, one of the

codes has to make a prediction to a forthcoming time, before advancing to a new time step.

Either the PAT predicts a contact point and a force of contact or the CAT predicts the collector

strip position and velocity. In this work it is chosen the PAT code to predict the contact position

and contact force using a simple linear extrapolation based on stored data of the contact force

history during the previous time steps. The reason for being the PAT code to make a prediction

is that PAT code uses a variable time step and an extrapolation/interpolation algorithm that is

already used to estimate the contact position and force on a forthcoming or past time instant that

is not multiple of the fixed CAT code time step.

The compatibility between the two integration algorithms imposes that the state variables

of the two sub-systems are readily available during the integration time and also that a reliable

prediction of the contact forces is available at any given time step. The accuracy of this

prediction relies on having small enough time steps either of the fixed CAT code time step or

the maximum allowed PAT code time step. Also, it is necessary to ensure that the multibody

code variable time step is never larger than the catenary CAT code time step. Accordingly to

these basic rules, a fully integrated communication protocol between CAT and PAT codes is

developed, and its structure is presented in section 4.2.2.

41

4.2.2 Communication Protocol

The interface between the catenary finite elements code and the pantograph multibody code is

built upon the use of two communication files. They not only serve as communication channels

between the two applications but also are the means for their synchronization. The two files

consist of a communication of the collector strip position and velocity (PAT2CAT) from the

multibody pantograph code to the finite elements catenary code and a communication to deliver

the point of contact and contact force (CAT2PAT) from the catenary code to the pantograph

code. Each of this communication files also include a flag that controls the progress of the

integration algorithms so that they stay synchronized.

There are occasions in which one of the algorithms has to wait for other and vice-versa.

The developed communication interface is composed of two stages. In the first stage, the

initialization stage shown in Figure 4.5, the codes exchange input data necessary to their own

initialization procedures. No contact at the catenary is implied or allowed at the start time. In

second stage, depicted in Figure 4.6, data is shared between codes to preform dynamic analysis

using the communication files previously described.

Figure 4.5: Initialization stage flowchart of the communication interface

42

Figure 4.6: Dynamic analysis stage flowchart of the communication interface

4.2.3 Data Exchange Methodology

One of the most critical issues of using co-simulation procedures is the added computational

cost due to data exchange between codes, especially when this data is large or, as is this case of

pantograph-catenary co-simulation, it is accessed frequently. The time spent on data exchange

between applications must be negligible compared to the computation time costs of the two

analyses. The use of physical data files for information exchange, also known as file

input/output, is a robust, well known and very popular methodology. However, for either a

recursive use or for large data sets it leads to slow data exchange when compared to the use of

virtual memory sharing. The data exchange between codes using virtual memory is not

43

straightforward as the codes may be developed in different languages and run in different

computer systems giving rise to memory access issues.

One form of sharing memory between multiple processes is by the use of a memory

mapped file methodology. A memory mapped file is a segment of computer virtual memory

which is mapped in order to have a direct byte-for-byte assignment to a hard disk file or other

resource that the operating system can reference with. Once this correlation is established, or

using a proper term “mapped”, the memory mapped file can be accessed by multiple processes

for reading and writing directly on the virtual memory. The primary benefit of memory mapping

is the increase of input/output performance enabling multiple applications to access

simultaneously a file in the same way they access virtual memory making file reading and

writing much faster.

In this work the memory mapped file mapping, reading and writing is entrusted to an

object oriented class of MATLAB [57] which enables the use of memory mapped files either

directly on a MATLAB application or indirectly on an another language application via a call to

a compatible MATLAB engine [58]. The implementation of the data exchange method on the

work using memory mapping is represented in Figure 4.7. The catenary analysis application is

implemented in MATLAB while the pantograph analysis is developed using FORTRAN. At the

start of a data exchange any of the MATLAB application or engine creates a file and maps it to

virtual memory while the other waits for the file creation. Whenever such file is found, it is

mapped. Having both applications mapped the same file in virtual memory they can

communicate between each other using their common memory mapped file. Note that the

created file only serves has a point of reference for both applications to map the same file in the

virtual memory. Furthermore, during the communication, only the memory mapped files are

changed, the created file is not used further.

Figure 4.7: Representation of the data exchange procedure between applications using memory

mapped files

44

45

5 Applications to Overhead Current Collecting Systems

The computational analysis methodologies developed in this work are implemented in a

software tool that handles the catenary dynamics and its interaction with the pantograph. With

this tool, many scenarios can be built in order produce a wide range of simulations to reach any

particular study of interest. These studies can be applied from operational acceptance to design

optimization and analysis of the dynamic response of a specific system either on the catenary or

on the pantograph or both. With the use of this computer analysis tool, different simulation

scenarios are studied [59-63].

Three case studies, each with its specific interest, were selected to be analysed and

presented on this work. One case addresses a comparison between three pantograph-catenary

pairs that are currently operating in Europe. Another case presents the analysis of multiple

pantograph operation in high-speed trains involving a realistic catenary and a high speed

pantograph model. A final case involves an analysis on a catenary with an overlap section in

order to emphasize the transition between different sections of a catenary.

The case studies evaluated here are analysed in the framework of the application of the

European regulation EN50367 and EN50317 [24, 25] which specify not only the basis for the

data processing but also the rules for pantograph-catenary acceptance.. In particular, it is

specified that the contact force time histories are filtered with a cut-off frequency of 20 Hz

before being post-processed.

As in all other simulation codes, the initial conditions of the analysis must be such that

pantograph and catenary are out of contact. Then, in the initial part of the analysis the

pantographs are raised until their bows touch the contact wire. In order to disregard this

transient part of the dynamic response, only the contact forces developed in a specific interval of

interest are taken into analysis, ensuring in this form that only the steady state of the

pantograph-catenary contact problem is studied

5.1 Pantograph-catenary pairs in current operation

A large majority of the railway lines in current operation have been developed with a

particular pantograph-catenary couple in mind, forming what is called here a national pair. In

high speed networks in Europe, there are among others, the CX-LN2 pair in France, the

ATR95-C270 pair in Italy and the DSA380–Re330 in Germany. All these catenary systems

have different structural characteristics and operate on different settings. On Table 5.2 trough

Table 5.4 the geometric and material characteristics of the LN2, C270 and Re330 catenaries can

be found, being their geometry depicted in Figure 5.1 trough Figure 5.3 with a representation of

their respective finite element models. The pantograph models, in spite of their mechanic

assembly being quite similar at first sight, they have particular differences that influence their

46

dynamic response. The values of these parameters for the lumped mass models are presented on

Table 5.1, showing a considerable difference between pantographs. It’s understandable that to

achieve the best contact quality possible it is important for the catenary and pantograph models

of each pair to fit their counterpart as much as possible.

Table 5.1: Identified lumped mass model parameters for the CX, ATR95 and DSA800

pantographs

LN2

Catenary length [m] 1138 Contact wire height [m] 5.08 Number of spans 24 Number of droppers/span 7-8 Nº spans at C.W. height 20 Inter-dropper distance [m] 4-6.75 Span length [m] 45-49.5 Stagger [m] 0.40

Damping 0.0272 Damping 0

Contact Wire Messenger Wire Droppers Steady Arms

Section [mm2] 150 65.5 12 120 Mass [kg/m] 1.334 0.605 0.11 1.07 Young modulus [GPa] 120 84.7 84.7 84.7 Tension [N] 20000 14000 - - Claw with dropper dropper - C.W. Claw mass [kg] 0.195 0.165 - 0.200 Length [m] - - 1.25-1.075 1.22 Angle w/horiz. - - 90º -10º

Table 5.2: Geometric and material properties of the LN2 catenary

L. M. Pantograph Faiveley CX

Contact ATR95

Stemman DSA380 parameters units

m1 Kg 5.58 10.14 7

m2 Kg 8.78 13.05 11

m3 Kg 7.75 9.45 10.5

k1 N/m 178.45 7247.6 65.7

k2 N/m 15487 30274 6700

k3 N/m 7000 7978.7 20000

c1 N s/m 108.39 225.33 500

c2 N s/m 0.009 0.01 0

c3 N s/m 45.85 87.74 200

47

C270

Catenary length [m] 1425 Contact wire height [m] 5.3 Number of spans 25 Number of droppers/span 9 Nº spans at C.W. height 21 Inter-dropper distance [m] 5.1-6 Span length [m] 57 Stagger [m] 0.40

Damping 0.0272 Damping 0

Contact Wire Messenger Wire Droppers Steady Arms

Section [mm2] 150 117 12.6 134 Mass [kg/m] 1.335 1.0378 0.1173 1.42 Young modulus [GPa] 100 96.8 15.9 84.7 Tension [N] 20000 16000 - - Claw with dropper dropper - C.W. Claw mass [kg] 0.195 0.165 - 0.200 Length [m] - - 1.035-0.555 1.02 Angle w/horiz. - - 90º -11º

Table 5.3: Geometric and material properties of the C270 catenary

Re330

Catenary length [m] 1057 Contact wire height [m] 5.3 Number of spans 17 Number of droppers/span 9 Nº spans at C.W. height 11 Inter-dropper distance [m] 5-6.85 Span length [m] 51-65 Stagger [m] 0.6

Damping Damping

Contact Wire Messenger Wire Stich Wire Dropper

7.5E-7 5E-3 5E-3 1E-5

Contact Wire. Messenger Wire Stich Wire Dropper

0 0 0 0

Steady Arm 1 Steady Arm 0

Contact

Wire

Messenger

Wire

Stich

Wire

Droppers Steady

Arms

Section [mm2] 120 120 35 10 120 Mass [kg/m] 1.075 1.068 0.3115 0.089 1.07 Young modulus [GPa] 120 84.7 84.7 84.7 84.7 Tension [N] 27000 21000 3500 - - Claw with dropper dropper m. wire - c. wire Claw mass [kg] 0.25 0.25 0.25 - 0.25 Length [m] - - - 1.25-1.075 0.65 Angle w/horiz. - - - 90º 6º

Table 5.4: Geometric and material properties of the Re330 catenary

48

Figure 5.1: Representation of the finite element model of the LN2 catenary with the static

deformation accounted for

49

Figure 5.2: Representation of the finite element model of the C270 catenary with the static

deformation accounted for

50

Figure 5.3: Representation of the finite element model of the Re330 catenary with the static

deformation accounted for

51

5.2 Analysis of existing pantograph-catenary national pairs

In order to evaluate the contact quality between the pantograph-catenary pairs, using the

simulation tool developed on this work, a set of four simulations are defined. Two simulations

are assigned to the CX-LN2 pair where the only difference resides in the lumped mass model

formulation used, i. e. , finite element versus multibody models. In the first case, the lumped

mass model is run with the same finite element procedure of the catenary, described in section

2.3.3, and in the second case the same lumped mass model is used but within the multibody

dynamic analysis module setup to run in co-simulation with the finite element application, as

addressed in section 4.2. The other two simulations are assigned to the ATR95-C270 and

DSA380–Re330 pairs. It is important to note that the finite element catenary models presented

here are set with their real specification data. The regular operating train speed on the pairs

taken into consideration is 300 Km/h for all and the set average contact force 157.5 N, as

specified by norm EN50367.

Catenaries are periodic structures and it can be considered that their repetition interval is

the span which contemplates the length between two supports of the catenary. For this reason,

the interval of interest for the analysis results reports chosen for each catenary is not length

based but span based. So, to have a clear picture of each catenary dynamic response and be able

to have a just comparison among them, an interval of interest containing 5 spans of each

catenary is selected. The intervals of interest for the reports on the dynamic analysis are

described in Table 5.5.

Interval of interest

Start [m] End [m] Length [m]LN2 551 798 247 C270 798 1083 285

RE330 496 821 325

Table 5.5: Description of the intervals of interest for the reports of the dynamic analysis used for

each catenary model.

The contact force results along the catenary for the LN2 are presented on Figure 5.4

showing the results obtained for the finite element pantograph model alongside the multibody

lumped mass pantograph model.

52

Figure 5.4: Pantograph-catenary contact force for the CX-LN2 pair with different lumped mass

pantograph formulation, finite element formulation (FEM) and multibody formulation (MB).

These results show a very distinctive similarity among the two simulations. The peaks presented

on the graph actually coincide with the pantograph passage on the droppers for the lowest peaks

and on the steady arms of the support system for the larger peaks. In order to determine the

pantograph acceptance to this catenary and also conclude about the differences between the

finite element and the multibody modelling options a statistical analysis to the contact force

results is presented on Figure 5.5. The statistical values reposted are the maximum and the

minimum, the mean contact force, the standard deviations of the contact force and the statistical

minimum of the contact force.

Figure 5.5: Statistical quantities of the pantograph-catenary contact force for the CX-LN2 pair

with different lumped mass pantograph formulation, finite element formulation (FEM) and

multibody formulation (MB).

The first observation is that both pantograph models lead to the same results. The second

observation is that in both simulations the pantographs pass on all the acceptance norms, as

listed in chapter 1. There is no loss of contact neither the uplift on the steady arms pass the 12

cm threshold. The maximum uplifts on each steady arm registered during the simulation are

shown on Table 5.6 .

53

CX-LN2 Steady Arm Uplifts

SA Number 1 2 3 4 5 6 7 8 9 10 11

Track Length [m] 86.9 132.2 177.7 222.7 267.7 312.7 357.7 402.7 452.2 501.7 551.2

Max Uplift [cm] 0.02 0.04 0.08 0.23 1.49 5.21 4.16 4.34 4.58 4.54 4.54

SA Number 12 13 14 15 16 17 18 19 20 21

Track Length [m] 600.7 650.2 699.7 749.1 798.6 848.1 897.6 947.1 997.1 1046.9

Max Uplift [cm] 4.55 4.54 4.54 4.55 4.54 4.54 1.70 0.03 0.02 0.02

Table 5.6: Maximum steady arm uplifts registered on the CX-LN2 pair

It is important to gather about the difference on modelling the lumped mass pantograph

with a finite element formulation attached to the catenary’s versus the multibody method with

the developed co-simulation procedure. On Figure 5.4 and Figure 5.5 it is observed a very close

resemblance between the results. To have a finer numeric comparison, Table 5.7 shows the

numeric values of the statistical analyses done to each simulation and the absolute deviation

among them. It is within good measure to assert that the contact force statistical results are very

good matches of each other. When looking through the maximum, mean and standard deviation

values, it can be reached that their absolute deviations account for less than 0,08 % of their

corresponded thresholds on the European norm for catenary acceptance at 300 Km/h. This gives

some confidence when accepting one pantograph with either formulation. It should also be

noted that the statistical data, used here to compare both models, result from the statistical

analysis of the contact forces which in turn passed through an independent fast Fourier

transform filtering process

Contact force [N] Maximum  Minimum  Amplitude  Mean StandardDeviation 

Statistical Maximum 

StatisticalMinimum 

CX-LN2 FEM 249.883 102.015 147.868 157.298 38.893 273.976 40.621

CX-LN2 MB 249.965 101.925 148.040 157.314 38.929 274.100 40.527

Absolute deviation 0.082 0.090 0.172 0.016 0.036 0.125 0.093

Table 5.7: Statistical quantities of the pantograph-catenary contact force for the CX-LN2 pair

for different lumped mass pantograph formulations and their absolute deviation

This particular study is of importance because not only shows a mutual agreement using

distinctive dynamic analysis formulations of the lumped mass pantograph but also, with the

most significance, stands for the accuracy of the developed high-speed co-simulation. So it is to

be expected that when a full multibody pantograph model is available the co-simulation

program works as swiftly as with the multibody lumped mass pantograph model. It is also

important to note that, for an additional computational cost, it is possible to have the results

even better correlated by reducing the maximum time step size of the multibody integration

procedure. As the catenary numerical procedure has the most dominant computation work load,

a reasonable tuning of the multibody time step does not interfere much with the overall

computational time as it would on the catenary’s numerical integration procedure. Nevertheless

54

the selection of the integration time step for each of the models is of extreme importance to

obtain an accurate dynamic analysis of each one the systems independently.

The contact force results obtained from the ATR95/C270 pair are presented in Figure 5.6.

Though the contact force periodicity along the catenary spans can be easily observed, the results

show a fair distinction from the CX-LN2 pair. The peaks of the contact force due to the

pantograph passage under the droppers and steady arms are less obvious in the case of this pair.

Not much more can be gathered about the pantograph-catenary contact force presented results,

this is the reason why the European norms rely on thresholds based on statistical quantities

hidden within the contact forces developed along the catenary. The statistical evaluation of these

contact forces is then depicted on Figure 5.7.

Figure 5.6: Pantograph-catenary contact force for the ATR95-C270 pair

Figure 5.7: Statistical quantities of the pantograph-catenary contact force for the ATR95-C270

pair

A close look to the statistical analysis allows concluding of the acceptance of the

pantograph on this catenary. Furthermore, as it is also required by the norms there is no loss of

contact and the steady arm uplifts, presented on Table 5.8, are well inside their 12 cm

acceptance limit.

55

Table 5.8: Maximum steady arm uplifts registered on the ATR95-C270 pair

The contact force results of the DSA800-Re330 pair are show on Figure 5.8 . Once again

it can be reached by observation that the characteristics of the contact force are rather different

than those of the other pairs.

Figure 5.8: Pantograph-catenary contact force for the DSA800-Re330 pair

From the statistical analysis of the contact forces, presented on Figure 5.9, it can be observed

that DSA800 pantograph also passes on the norm requirements. The simulation results also

reveal no contact loss was registered and that the steady arm uplifts are within acceptance, being

their maximum values presented on Table 5.9. It is worth mentioning the very low standard

deviation of the contact force for this pair demonstrating in this form its superior contact

quality.

Figure 5.9: Statistical quantities of the pantograph-catenary contact force for the DSA800-R330

pair

ATR95-C270 Steady Arm Uplifts

SA Number 1 2 3 4 5 6 7 8 9 10 11

Track Length [m] 114.0 171.0 228.0 285.0 342.0 399.0 456.0 513.0 570.0 627.0 684.0

Max Uplift [cm] 0.00 0.00 0.00 0.01 0.03 0.10 0.74 5,35 4.34 4.54 4.52

SA Number 12 13 14 15 16 17 18 19 20 21 22

Track Length [m] 741.0 798.0 855.0 911.9 968.9 1025.9 1082.9 1139.9 1196.9 1253.9 1310.9

Max Uplift [cm] 4.51 4.52 4.53 4.55 4.58 4.61 4.65 4.70 4.77 4.75 2.99

56

DSA800-Re330 Steady Arm Uplifts

SA Number 1 2 3 4 5 6 7

Track length [m] 120.0 171.0 236.0 301.0 366.0 431.0 496.0

Max Uplift [cm] 0.35 0.62 1.55 5.84 6.01 5.92 5.99

SA Number 8 9 10 11 12 13 14

Track length [m] 561.0 626.0 691.0 756.0 821.0 886.0 937.0

Max Uplift [cm] 5.99 5.96 5.95 5.97 5.95 1.97 0.16

Table 5.9: Maximum steady arm uplifts registered on the ATR95-C270 pair

In order to make a clear comparison between the dynamic responses of the pantograph-

catenary pairs, the statistical results are gathered in Figure 5.10. Although all tested pair pass the

European norm for pantograph acceptance it is clear the DSA800-Re330 pair has a better

performance relatively to the other pairs. This can be concluded mainly by the lowest standard

deviation, lower maximum force and higher minimum contact force values of the stated pair.

Figure 5.10: Comparison of the statistical quantities of the pantograph-catenary contact force

between pairs

The standard deviation can be a good indicator of the level of wear enforced on the

catenary’s contact wire and pantograph collector strip. A lower standard deviation means a

lower contact force variation which results not only in less wear but also guarantees the wear is

spread more evenly, avoiding localized damages on the equipment. Another statistical variable

of significance, that was not looked on until now is the statistical minimum. In spite of this

variable not being included on the norms it as practical importance. The statistical minimum is

defined as the mean contact force minus three times the standard deviations. Its physical

significance on this specific problem can be put into terms of possibility of occurring contact

loss. While in a scenario where there is no contact loss a tendency for its occurrence might

remain being the statistical minimum a good indicator for the eventuality. The DSA800-Re330

pair shows the most comforting results in which the statistical minimum is considered. In spite

of the ATR95-C270 pair having a slightly better performance than the CX-LN2 pairs, both of

57

them show a similar contact quality, although their contact force history along the catenary

show a distinct difference between them.

It is important to defer that the Re330 catenary is of the stich wire type and the LN2 and

C270 are of the simple type, being the deference on their performance clear. The Re330 shows

the best results, however the present trend on catenary construction is to build more of the

simple type. One possible reason for choosing such catenary type is economical, as the stich

catenary, is a more complex structure harder to mount and maintain. Its tolerance requirements

upon construction are much tighter and the maintenance cycles much shorter and more

demanding. This makes the stich wire catenaries much more costly than the simple type. Plus,

the simple type catenaries can still improve their contact quality efficiency on new designs by

adding more tension on the wires and developing better materials for them. Also it is important

to note that the contact quality not only depends on the catenary system but also on the

pantograph model and most importantly on their compatibly to one another. For these reasons it

is then worth concluding that there still is room for improvement on the present catenary

systems designs and their pantograph specific compatibility. A numeric simulation tool, like the

one developed on this work, should then be proven valuable to ease the design process of new

systems and help optimizing their interface.

To extent the comparison between the operating pairs a little further, an analysis on the

dropper action remains of interest. The droppers are designed to support the contact wire.

Ideally they would support it as evenly as possible along the catenary avoiding or reducing the

catenary sag as much as possible. In a general catenary span, the first and last droppers tend to

support more the catenary weight than the others, which is clear by exhibiting a higher tension

stress state then the rest. Figure 5.12 shows, for two droppers on the CX-LN2 pair, the resulting

tension effort applied in the dropper and also its lower node displacement along time. The

chosen droppers are located at the start of the span and at middle span. It is noticeable that the

droppers at rest have a distinct stress state among them exhibiting different tension efforts. As

the pantograph passes under each dropper they change their axial effort and the displacement of

the node that connects to the contact wire is very noticeable. Also evident is the disturbance on

the dropper tension stress state as the pantograph move away. It is interesting to note that in this

catenary system the dropper slacking is sporadic and, when it occurs, it is very residual. On the

other hand, by observing Figure 5.12 similar results are obtained for the ATR95-C270 pair. The

dropper slacking of the first and last droppers on the span is evident and no other dropper

slacking is observed. While at rest it is noticeable the difference on their tension state. This

shows that the contact wire is supported differently on each pair, even though both LN2 and

C270 are catenaries of the simple type. In spite of the noticed differences between catenary pairs

it is very interesting to observe the similarity of the dynamic response of the droppers after the

pantograph passage as they both recuperate in a similar fashion and present similar amplitudes

58

of the wire vibrations. This behaviour is related to their damping, as both catenaries are

modelled with the same Rayleigh damping parameter values.

(a) (b)

Figure 5.11: Effort and lower node displacement of the first dropper on the span, (a), and other

at middle span, (b), registered on the CX-LN2 pair.

(a) (b)

Figure 5.12: Effort and lower node displacement of the first dropper on the span, (a), and other

at middle span, (b), registered on the CX-LN2 pair.

59

By observing the dropper results for the DSA800-Re330 pair in Figure 5.13, a very

distinct difference of their behaviour can be observed. In fact for the considered pantograph-

catenary pair, all droppers slack regardless of being on the start, end or middle of the span.

Besides slacking, they also present a more aggressive recovery with a very noticeable peak on

their tension effort.

The entire dropper results presented for all catenary allow concluding that the dynamic

response of each catenary and their pantograph pair is different. This can be related to the

several catenary structural differences and involves not only their geometry and material

properties but also their characteristic damping behaviour. The number of droppers in each span

and their specific locations and spacing effect on how the contact wire weight is supported,

which ultimately as impact in the contact quality.

It is also clear that the catenary systems are very complex structures and their dynamic

response can be greatly modified by changing each of the modelling parameters. Furthermore it

is important to note that the contact quality on a catenary cannot be analysed independently of

the specific pantograph model and its dynamic response must always be accounted when

analysing this structures.

(a) (b)

Figure 5.13: Effort and lower node displacement of the first dropper on the span, (a), and other

at middle span, (b), registered on the DSA800-Re330 pair.

60

5.3 Multiple Pantograph Operation

In high- speed railway vehicles a limiting factor on the operational speed is the current

collection as there is the need to operate with multiple pantographs. However, the contact

quality of the pantograph-catenary interaction is perturbed due to the mutual influence of the

leading and trailing pantographs in each other. In this case, the operation of multiple

pantographs in catenaries with low and moderate damping is considered. Different vehicles of

high-speed trains have different lengths and, consequently, the separation distance between

pantographs may change accordingly. The typical separations between pantographs shown in

Figure 5.14 reflect how multiple train units operate and constitute the scenarios to which the

methodologies proposed in this work are applied.

Figure 5.14: Multiple pantograph operations of high-speed trains with typical distances between

pantographs.

Using the data contained in Table 2.3, but allowing the damping factor α to be 0.0027 or

0.027, finite element models of a generic simple catenary are obtained, being different views of

their static geometry shown in Figure 2.9. Note that, although the catenary model used is a

generic model of a catenary, the data is realistic. Furthermore, several models of the same

catenary are developed with different proportional damping factors to allow studying the

variation of the pantograph-catenary contact quality in face of the structural energy dissipation

of the catenary. The pantograph chosen to interact with this catenary is the Faiveley CX,

presented on Table 5.1.

In all scenarios, considered here, the pantographs move along the catenary in a tangent

track with a velocity of 300 km/h. In the initial part of the analysis the pantographs are raised

until their bows touch the contact wire. In order to disregard this transient part of the dynamic

response, only the contact forces that develop in the pantograph between 400 and 800 m, and

the droppers and steady arms that exist in this range are used in the analysis of results. Figure

61

5.15 shows the characteristics of the contact forces that develop between the pantographs and

catenaries, with different proportional damping.

(a) (b)

Figure 5.15: Pantograph-catenary contact force for several pantograph separations in catenaries

with different proportional damping: (a) =0.00275; (b) =0.0275.

The results show that the amplitude of the contact forces in the trailing pantographs is

always larger than what the leading pantographs exhibit, being that difference higher for lightly

damped catenaries. The contact force results, as shown in Figure 5.15, hide some of the

important results of the analysis that are used in design and in homologation of pantographs. In

Figure 5.16 the statistic values of the contact force are overviewed for the different pantograph

separations, running in the two catenaries considered before.

The first important observation of statistical quantities depicted in Figure 5.16 is that the

standard deviation of the contact force for all pantographs, running on the lightly damped

catenary, is always larger than 30% of the mean contact force. These values imply that the trains

using these pantographs would not be allowed to run at a speed of 300 km/h in the catenary

system. However, the pantographs can be used, with the current operational setup, in the

catenary with normal damping. In both cases, the multiple pantograph operation with a

separation of 200 m shows the worst contact force characteristics for the trailing pantograph,

i.e., the trailing pantograph exhibits larger maximum forces, lower minimum forces and larger

standard deviations. None of the pantographs exhibits any contact loss.

0

50

100

150

200

250

300

454 508 562 616 670

Co

ntac

t Fo

rce

[N]

(filt

ered

0-2

0 H

z)

Track Lenght [m]

Leading (31 m)

Trailing (31 m)

0

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454 508 562 616 670

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ntac

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rce

[N]

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Track Lenght [m]

Leading (31 m)

Trailing (31 m)

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454 508 562 616 670

Con

tact

For

ce [

N]

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Leading (200 m)Trailing (200 m)

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454 508 562 616 670

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ntac

t Fo

rce

[N]

(filt

ered

0-2

0 H

z)

Track Lenght [m]

Leading (200 m)

Trailing (200 m)

62

(a) (b)

Figure 5.16: Statistical quantities of the pantograph-catenary contact force in catenaries with

different proportional damping: (a) =0.00275; (b) =0.0275.

Another characteristic of the contact force that is worth being analysed is its histogram.

Figure 5.17 presents the histograms of all pantographs for all separations considered in this

work. The histograms show that for a lightly damped catenary the contact forces not only have

large variations, as observed also in Figure 5.15, but also that the number of occurrences of

contact forces in each range considered is high, i.e., even away from the mean contact force the

existence of higher or lower contact forces is not sporadic. For a catenary with average damping

the contact force magnitude is closer to the mean contact force. In all cases considered, the

mean contact force is always 150 N, which satisfies the regulations.

-50

0

50

100

150

200

250

300

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (31 m)Trailing (31 m)

0

50

100

150

200

250

300

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (31 m)Trailing (31 m)

-50

0

50

100

150

200

250

300

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (100 m)Trailing (100 m)

0

50

100

150

200

250

300

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (100 m)Trailing (100 m)

-50

0

50

100

150

200

250

300

350

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (200 m)Trailing (200 m)

0

50

100

150

200

250

300

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (200 m)Trailing (200 m)

63

(a) (b)

Figure 5.17: Histograms of the pantograph-catenary contact force in catenaries with different

proportional damping: (a) =0.00275; (b) =0.0275.

One of the reasons why the contact force characteristics has to stay inside a limited range

concerns the potential interference between the pantograph head and the catenary mechanical

components. The steady-arm uplift, shown in Figure 5.18, and the dropper axial force, depicted

in Figure 5.19, are measures of the catenary performance and of its compatibility with the

running pantographs.

0 25 50 75 100 125 150 175 200 225 250 275 3000

2

4

6

8

10

12

14

16

18

Rel

ativ

e F

req

uenc

y [H

z]

Contact Force [N]

Leading (31 m)

Trailing (31 m)

0 25 50 75 100 125 150 175 200 225 250 275 3000

5

10

15

20

25

30

Rel

ativ

e F

req

uenc

y [H

z]

Contact Force [N]

Leading (31 m)

Trailing (31 m)

0 25 50 75 100 125 150 175 200 225 250 275 3000

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Leading (100 m)

Trailing (100 m)

0 25 50 75 100 125 150 175 200 225 250 275 3000

5

10

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ativ

e F

req

uenc

y [H

z]

Contact Force [N]

Leading (100 m)

Trailing (100 m)

0 25 50 75 100 125 150 175 200 225 250 275 3000

2

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ativ

e F

req

uenc

y [H

z]

Contact Force [N]

Leading (200 m)

Trailing (200 m)

0 25 50 75 100 125 150 175 200 225 250 275 3000

5

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30

Rel

ativ

e F

req

uenc

y [H

z]

Contact Force [N]

Leading (200 m)

Trailing (200 m)

64

(a) (b)

Figure 5.18: Typical steady-arm uplift in catenaries with different proportional damping for two

different separations of pantographs: (a) =0.00275; (b) =0.0275.

(a) (b)

Figure 5.19: Typical mid-span dropper forces in catenaries with different proportional damping

for two different separations of pantographs: (a) =0.00275; (b) =0.0275.

The steady-arm uplift is lower than 7 cm in all cases depicted in Figure 5.18. Although

not represented, the maximum uplift of all steady arms of the catenary is also lower than the 12

cm limit allowed for the type of catenary used. The droppers exhibit slacking for the lightly

-0.03

-0.01

0.01

0.03

0.05

0.07

0 1 2 3 4 5 6 7 8

Sup

po

rt U

plif

t [m

]

Time [s]

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Su

ppor

t U

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]

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]

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Dro

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er's

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[N]

Time [s]

65

damped catenary, as seen for the axial forces shown in Figure 5.19. For the catenary with

average damping, only the passage of the trailing pantograph, when the separation is 200m, has

some slacking. It is interesting to notice that the position of the contact wire of the catenary is

disturbed even before the pantograph bow passes. This is because the traveling wave speed is

higher than the train speed. For lightly damped catenaries this disturbance is higher. For longer

catenary sections, it is expected that the trailing pantograph may affect the contact of the leading

pantograph due to the wave traveling speed of the contact wire.

The mutual influence of the pantographs in each other’s contact quality is better

understood when displaying the contact force characteristics as shown in Figure 5.20, for the

lightly damped catenary and Figure 5.21, for the average damped one. In both cases the

statistical values of the contact force of a single pantograph operation are also presented to

better understand the problem.

(a) (b)

Figure 5.20: Statistical quantities associated to the contact force for a catenary with low

damping (=0.00275): (a) Leading pantographs; (b) Trailing pantographs.

(a) (b)

Figure 5.21: Statistical quantities associated to the contact force for a catenary with average

damping (=0.0275): (a) Leading pantographs; (b) Trailing pantographs.

For a lightly damped catenary the perturbation of the trailing pantograph over the leading

pantograph exist but are low. However, the trailing pantograph contact forces are clearly

affected by the leading pantograph, being the influence enhanced by the decrease of the

catenary damping. The tendency exhibited for the lightly damped catenary is the for a

-50

0

50

100

150

200

250

300

350

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (Single)

Leading (31 m)

Leading (100 m)

Leading (200 m)

-50

0

50

100

150

200

250

300

350

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Trailing (Single)

Trailing (31 m)

Trailing (100 m)

Trailing (200 m)

-50

0

50

100

150

200

250

300

350

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Leading (Single)

Leading (31 m)

Leading (100 m)

Leading (200 m)

-50

0

50

100

150

200

250

300

350

Maximum Minimum Amplitude Mean StandardDeviation

StatisticalMinimum

Co

ntac

t Fo

rce

[N]

Trailing (Single)

Trailing (31 m)

Trailing (100 m)

Trailing (200 m)

66

pantograph separation of 31 m the trailing pantograph contact quality suffers, slightly, from the

existence of a trailing pantograph being this influence due to the wave traveling speed in the

contact wire. For a pantograph separation of 200 m the trailing pantograph contact quality is

clearly affected in both lightly and average damped catenaries. The results suggest that the

critical distance between pantographs, at least for the catenary design considered in this work, is

200m. These results are in agreement with the findings of Ikeda who studied the multiple

pantograph operation for the Japanese Shinkansen pantograph-catenary interaction and found

the same critical separation [5]. Thus it is suggested that regardless of the type of construction

of the catenary a critical separation distance between the pantographs exists and that the

distance is close to 200 m.

5.4 Pantograph-Catenary performance with overlap sections

The overlap section of a catenary refers to the spans of a catenary system where two

catenaries sections overlap. The need for overlap sections is due to the fact that the length of the

messenger and contact wires must be finite, in practice not longer than 1.5 Km. This section is

critical since the catenary transition must be as smooth as possible not disturbing the quality of

the current collection and must avoid any loss of contact. The most crucial part of the dynamic

behaviour happens in the transition when the collector strip of the pantograph meets the new

catenary, establishing two contacts on the same strip.

In order to analyse the effect of the overlap section a modelling scenario is built using a

realistic model of the LN2 catenary (TGV Atlantique line). To pair with the catenary the

Faiveley CX pantograph is selected which is modelled as the lumped mass pantograph

represented on Table 5.1.

The method to build a finite element catenary model of two catenary sections with an

overlap section is exactly the same used to build simple section catenaries. The two catenaries

are modelled as independent systems, as they have in practice no physical link between them.

However, they have to be positioned just right so that the overlap positioning stays correctly

modelled. Also it is important to note that the catenary overlapping is not done in the same way

for every type of catenary, i.e., each catenary type has its individual intersecting method. To

have a clear outlook of the LN2 line overlapping arrangement a representation is presented on

Figure 5.22.

For a fair comparison between an overlap and a normal catenary section two intervals of

interest are selected for analysis, each containing ten catenary spans. The intervals of interest are

presented in Table 5.10. In the work it is considered both single pantograph operation and

multiple with the separations of 31, 100, 200 and 400 meters. Note that the simulation with a

400 meter separation can only be accomplished due to the use of a second catenary. Using only

one catenary to analyse this last case would not be accurate since the catenary would not have

67

enough analysable length available after the transient response fading from the initial contact on

the start of the simulation.

Interval of interest

Start [m] End [m] Length [m] Normal section 1242 1778 536 Overlap section 799 1292 493

Table 5.10: Description of the intervals of interest used for each catenary section

Figure 5.22: Schematic of a LN2 overlap section arrangement with projected views

The LN2 catenary main characteristics and modelling data used to build this model are

presented Table 5.11. The resulting catenary mesh with the static deformation already accounted

for is presented on Figure 5.23.

LN2 (TGV Atlantique line)

Catenary height [m] 1.4 Contact wire height [m] 5.08 Number of spans 24-26 Number of droppers/span 7-8 Nº spans at C.W. height 21-23 Inter-dropper distance [m] 6.75 Span length [m] 45-54 Stagger [m] 0.40

Damping 0.0272 Damping 0

Contact Wire Messenger Wire Droppers Steady Arms

Section [mm2] 150 65.5 12 120 Mass [kg/m] 1.334 0.605 0.11 1.07 Young modulus [GPa] 120 84.7 84.7 84.7 Tension [N] 20000 14000 - - Claw with dropper dropper - C.W. Claw mass [kg] 0.195 0.165 - 0.200 Length [m] - - 1.25-1.075 1.22 Angle w/horiz. - - 90º -10º

Table 5.11: Geometric and material properties of the LN2 catenary (TGV Atlantique line)

68

Figure 5.23: Representation of the finite element model of the LN2 catenary (TGV Atlantique

line) with the static deformation already accounted.

69

Starting with the results of the pantographs passage on the regular section, Figure 5.24

depicts the contact forces developed between the pantographs and the catenary. The results

show that the amplitude of the contact forces in the trailing pantographs is always larger than

those in the leading pantographs. However this effect fades away as the pantograph separation

becomes larger.

Figure 5.24: Pantograph-catenary contact force for a single pantograph and multiple

pantographs with several separations in a regular catenary section

As seen before, not much more can be inferred from the contact force. To have a more

detailed analysis of these results, as it is required by the European norms, the statistical values

of the contact force are overviewed for the different pantograph separations, in Figure 5.25. The

most important observation of these statistical results is that all pantographs are approved on the

compatibility acceptance norms. Nevertheless, attention must be paid to the trailing pantographs

at the 200 and 400 meter separations were both exhibit the higher contact force maximums and

standard deviations. The pantograph at 200 meter separation is the most critical, for which the

70

standard deviation threshold has just 12% of leftover margin. None of the pantographs have

contact loss neither their uplift on the steady arms is higher than the norm limits.

Figure 5.25: Statistical quantities of the pantograph-catenary contact force on a single

pantograph and multiple pantographs with several separations in a regular catenary section

In what the contact force results of the pantographs passage on the overlap section are

concerned, Figure 5.26 depicts the contact forces developed between the pantographs and the

catenary. These results show that the amplitude of the contact forces in the trailing pantographs

is always larger than that of the leading pantographs and that this amplitude increase fades away

as the pantograph separation go larger. It is also evident the contact force maximum which

coincides with the pantographs first contact on next catenary section.

71

Figure 5.26: Pantograph-catenary contact force for a single pantograph and multiple

pantographs with several separations in a catenary overlap section

The statistical analysis of the contact forces developed on the overlap section for a single

pantograph and for multiple pantographs, at several separations, is presented on Figure 5.27.

The results show that all pantographs pass on their acceptance requirements, none of the

pantographs have contact losses and the steady arm uplifts are within acceptable limits.

However, as in the normal section results, the trailing pantograph at 200 meter separation is

even in a more critical point for norm acceptance being its standard deviation threshold not

reached by 5%.

72

Figure 5.27: Statistical quantities of the pantograph-catenary contact force on a single

pantograph and multiple pantographs with several separations in a catenary overlap section

In order to make a comparison of the contact quality between the overlap section and the

normal section, the results of the contact force characteristics concerning the two sections types

are better understood when presenting the results as shown in Figure 5.28.As stated before, all

simulation possibilities pass under the European norm for pantograph acceptance, although the

results clearly show that the contact quality on the overlap section worsens, as expected due to

the catenary section shift. This degradation of contact is more accentuated for the multiple

pantographs in close proximity where the most critical case is for the leading pantograph at 31

meter distance from the trailing, which shows that the maximum contact force is closest to its

acceptance limit of 350N and also presents the largest standard deviation. As the distance

between pantographs becomes larger this degradation of the maximum contact force is less

accentuated to the point that at 400 meters distance the leading pantograph exhibits contact

force characteristics very close to its trailing counterpart on the normal section. Even the trailing

pantograph on the overlap section has a slightly better contact quality than the trailing

73

pantograph in the normal section. It is also noticeable, when observing the contact force

maximums on Figure 5.28 and the contact forces along the catenary on Figure 5.26, that for

multiple pantographs on an overlap section the critical pantograph is the leading one which

meets the next catenary section first and eases the trailing pantograph entrance by uplifting the

second sections contact wire.

Figure 5.28: Statistical quantities associated to the contact force between an overlap and a

normal section of the catenary system for different pantograph separations.

To study mutual influence between the leading and the trailing pantographs over a normal

and overlap section of the catenary, the statistical data of the contact forces is graphically

rearranged and presented on Figure 5.29 for the normal section and on Figure 5.30 for the

overlap section. In both cases the statistical values of the contact force of a single pantograph

operation are also presented to better understand the problem. In relation to the contact quality

on the normal section, the perturbations of the trailing pantograph over the leading pantograph

exists but are very low except for the pantograph with a 400 meter separation, where a

74

degradation on the contact is noticeable. It is difficult to reason with certainty about the causes

for this degradation. However several points have to be taken into consideration. First it is

important to take into account that the wave propagation due to contact of either or both the

leading and the trailing pantograph with the contact wire can reflect on the catenary ends. Also

it is understandable that this wave propagation is damped by the catenary damping. However

this effect might not be enough to overcome a bigger perturbation imposed on the line as the

trailing pantograph exhibits one of the worst contact qualities with the highest maximum contact

force. On the other hand, the trailing pantograph contact forces are clearly affected by the

leading pantograph. It is observed that the trailing pantograph for separation of 31 and 100

meters have a slightly better contact quality when compared with its leading counterpart or also

to the single pantograph case. However and much more evident is the negative influence on the

contact quality that the leading pantograph has over the trailing at 200 and 400 meter separation.

These results are consistent with the conclusions taken on section 5.3 where a critical distance

between pantographs is suggested.

Figure 5.29: Statistical quantities associated to the contact force between single, leading and

trailing pantographs on a normal catenary section for different pantograph separations

On the catenary overlap section, where a comparison between the leading and trailing

pantographs can be found on Figure 5.30, it is observed that for the leading pantographs only

the one with a 31 meter separation appears to be influenced by the trailing pantograph. It is

within reason to relate this slight contact quality deterioration of the leading pantograph due to

the mutual influence of the contact wire uplifts caused by both the leading and trailing

pantographs. The uplifts relate the punctual raising of the contact wire on the position of the

pantograph contact. When there are two punctual uplifts close enough, they increase each other

slightly because the contact wire stays more supported. As the pantograph separations becomes

larger, at 100, 200 and 300 meters, the punctual uplifts of both leading and trailing pantographs

do not interfere with each other. At the 31 meter separation their mutual influence results on a

small fading of the contact quality on the leading pantograph if compared to the other partings.

75

Figure 5.30: Statistical quantities associated to the contact force between single, leading and

trailing pantographs on a catenary’s overlap section for different pantograph separations

Until now the contact force developed over the overlap section has been analysed by the sum of

the contact forces that actuate on the pantograph collector strip. When transitioning from one

catenary to another two different contacts occur on the same pantograph, one corresponding to

the contact with the departing catenary section and a second to the contact with the incoming

catenary section. Figure 5.31 presents the contact forces related to each contact made on a

specific pantograph. For simplicity only the results for the leading and trailing pantographs, at

31 meters separation, are shown. However, for all the other pantographs separations the results

are similar. Careful analysis of the presented results show that that while there is never a

complete contact loss for the total of the two contacts developed on the overlap section, the

results register a contact loss and regain of contact on each of the contacts developed on the

departing and incoming catenaries. This event occurs on both leading and trailing pantographs.

To explain this effect it has to be taken into account that the incoming catenary has no imposed

uplift until contact occurs. As the contact on the incoming catenary occurs the contact on the

departing catenary fades until is lost. However, when the incoming catenary gains uplift due to

the new contact its contact wire is raised matching the contact wire of the departing catenary

causing a regain of contact that quickly is loosed again. This occurrence shows the importance

of the uplifts on the catenary transition and it is expected that larger uplifts result on less smooth

line transitions. It is very difficult to validate this numeric result with the presently existent

experimental results. Note that this occurrence takes roughly 0.3 seconds. Limited literature

references are available with overlap section studies. Reference [64] relates that the uplift on the

contact wire has great influence on the dynamic performance of the pantograph in this type of

sections as it has been concluded above with the results of the case study presented here.

76

(a) (b)

Figure 5.31: Discretized contact forces on the catenary overlap section for the leading (a); and

trailing pantograph (b), with 31 meters separation.

From another stand point the influence of the leading contact over the trailing pantograph

is evident on the results. All the trailing pantographs exhibit a better contact quality with,

especially when comparing the contact force maximums between the leading and trailing

pantographs at each separation. These contact force maximums, which can be observed in

Figure 5.26, relate to the moment where the pantograph meets the incoming catenary. So it is

concluded that, at least for this particular catenary and specifically for this overlap section

arrangement the leading pantograph appears to ease the trailing pantograph entrance one the

incoming catenary having a smoother transition on the overlap section, even smother than a

single pantograph transitioning.

77

6 Conclusions and Future Development

The development of catenary and pantograph systems that allow their operation with

higher speeds and better overall contact require that the computer tools used in their analysis

include all modeling features relevant to their analysis. A computational approach based on the

co-simulation of linear finite element and general multibody codes is presented and

demonstrated in the framework of the pantograph-catenary interaction. It was shown that the use

of linear finite elements are enough to allow for the correct representation of the catenary

provided that the wire tension forces are accounted for in the stiffness formulation and that the

droppers slacking is properly represented via the force vector. Minimal requirements for the

catenary finite element modelling include the use of Euler-Bernoulli beam elements with axial

tensioning and geometric stress stiffening for the catenary messenger and contact wire with a

discretization enough to capture the deformation wave traveling of the contact wire. The

dynamic equilibrium of the catenary system after each time step is attained with an iterative

scheme in which displacements, velocities and forces are corrected. A minimum number of

iterations must be set to find the correct dropper compensation forces.

It was also shown that the use of multibody dynamics methods allow capturing all of the

important dynamic features of the pantographs. When multibody pantograph models are used,

the co-simulation between the finite element and multibody codes must be ensured. The contact

model between the pantograph collector strip and the catenary contact wire is used to achieve

the co-simulation. Although other procedures exist, the use of a contact penalty formulation

demonstrates to be enough to obtain all main contact features. The correct use of the co-

simulation procedure, in which the minimal time step with which the problem can be solved is

controlled by the finite element part, it is the challenging part of putting the dynamic simulation

of these systems. The results of several case studies, presented in this work, demonstrate how all

quantities used to characterize the dynamic response of the system are readily available from the

pantograph and catenary simulation tools.

It was shown that catenary systems are very complex structures and can present a very

different dynamic response among them. This important when evaluating the contact quality

that each catenary provide. However to achieve a better contact quality it is important not to

consider the catenary and the pantograph systems independently as a factor of compatibility

between both systems plays a fundamental role.

The application of the procedures to multiple pantograph operations, in high-speed

railway vehicles, allowed the identification of the important quantities of the dynamic response

that are required for the pantograph homologation and for operational decisions. The catenary

damping plays a fundamental role in the pantograph-catenary contact quality. Catenary low

damping leads to higher maximum contact forces, lower minimum contact forces, eventually to

78

contact losses, and to higher standard deviations of the contact forces. All these characteristics

of the contact force lead to the rejection of the operation of multiple pantograph units at the

required speed of 300 km/h in lightly damped catenaries. It is also concluded, from the results

of the analysis, that for operations in average damped catenaries all standard separations

between the pantographs lead to acceptable contact forces. As a general tendency, it was

observed that for smaller pantograph separations the trailing pantograph affects the quality of

the leading pantograph contact due to the wave travelling speed of the contact wire. For larger

pantograph separations it is the leading pantograph that affects adversely the contact quality of

the trailing pantograph. In any case, all results show that the critical separation between leading

and trailing pantographs is 200 m, i.e., it is at this separation that the leading pantograph has a

greater influence in the contact quality of the trailing pantograph.

The numeric software tool developed here is also able to consider catenary overlap

sections which represent a critical section on the catenary systems. These irregularities in the

system can lead to increased contact force variation and thereby contact loss possibility. It was

possible to identity that the uplift on the contact wire imposed by the pantograph-catenary

contact has a great influence on the quality of the contact. The contact degradation is

particularly noticeable for the leading pantograph in multi pantograph operations when close

separations between pantographs are used. Also, within the same reasoning, it was observed that

the first pantograph passage eases the trailing pantograph transition.

Some identified challenges remain for future considerations. One of the major difficult

when developing the catenary finite element model was to develop a catenary mesh that would

match the real contact wire sag when statically deformed. As catenaries are complex structures

this was not easy to achieve as the intricate flexibility and line tensioning of these types of

structures need to be taken into account. Still, there is a need to develop better methodologies on

this concern, especially when a catenary model is to be built from experimental measures that

the statically deformed shape of the catenary finite element model would have to match with

accuracy.

One other aspect which needs to be the aim of further investigation is the identification of

the catenary damping parameters. It was shown that the catenary damping plays a fundamental

role in the pantograph-catenary contact quality, so it’s correct modelling is critical. However, it

is recognized that the estimation of the structural damping of not only the catenary but other

structures is still a technological challenge. Rayleigh damping, also known as proportional

damping, was used to model the developed catenary model. It is even possible to apply this

same methodology locally by addressing different damping parameter to each component. Still,

these damping parameters need to be correctly identified either on current operating catenaries

or catenary in design phase. So it is of importance to find methodologies able to identify the

catenary damping on existent catenaries with experimental testing and validation, plus relate

79

these findings to catenaries that are still in project. Furthermore it should also be considered the

damping on the contact law used for the penalty method to model the contact. The parameters of

the contact law used on this work were set to consider the contact as purely elastic which is a

current practice on present software tools that model catenaries. However, as already shown, a

correct modelling of the catenary model is critical and it is possible that the same applies on the

contact modelling. The contact law used on this work, as with many others that can be easily

included on this tool, is able to take into account damping hysteresis but it remains the problem

to identify their damping parameters correctly.

Spatial pantograph models require the use of multibody dynamics procedures to capture

all their dynamic features. In the particular case of using unidirectional lumped mass pantograph

models the equations of the motion of the system can be solved together with the finite element

equilibrium equations. The lumped mass models result from a laboratory identification of the

system that represents the pantograph prototype and it is, by definition, a validated model.

However, a multibody pantograph model made of rigid bodies and perfect kinematic joints does

not have the minimum features to represent correctly the real system. It is a topic of research to

identify the minimum features required for a multibody pantograph model, being the existence

of bushings and clearances in the joints and eventual flexibility of the bow and arms some of the

modeling features that need to be accounted for.

One point that should be taken into future consideration is the aerodynamic effect on the

catenary due to the direct effects of the wind on the overhead contact line and on the pantograph

components plus the indirect effect due to the additional motion of the carbody imparted to the

base of the pantograph. These phenomenons are estimated to have impact on the contact quality

in particularly wind exposed areas. Although the current numerical simulation tool here

developed is able to handle this effects there are not implemented.

80

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