Railway track dynamic modelling

BLAS BLANCO

Licentiate ThesisStockholm, Sweden 2017

TRITA-AVE 2017:34ISSN 1651–7660ISBN 978–91–7729–440–5

Postal address:Royal Institute of Technology

Farkost och FlygSE–100 44 Stockholm

Visiting address:Teknikringen 8

Stockholm

Contact:blasb@kth.se

Academic thesis with permission by KTH Royal Institute of Technology, Stockholm,to be submitted for public examination for the degree of Licentiate in EngineeringMechanics, on Friday the 9th of June 2017 at 10:15, in Munin, Teknikringen 8,KTH, Stockholm.

© Blas Blanco, 27 April 2017

Tryck: Universitetsservice US AB

Abstract

The railway vehicles are an increasing mean of transportation due to, its reducedimpact on environment and high level of comfort provided. These reasons have con-tributed to settle a positive perception of railway traffic into the European society.In this upward context, the railway industrial sector tackles some important chal-lenges; maintaining low operational costs and controlling the nuisance by-productsof trains operation, the most important being railway noise. Track dynamic plays amain role for both issues, since a significant part of the operational costs are asso-ciated with the track maintenance tasks and, the noise generated by the track canbe dominant in many operational situations. This explains why prediction toolsare highly valued by railway companies.

The work presented in this licentiate thesis proposes methodologies for accurateand efficient modelling of railway track dynamics. Two core axes have led thedevelopment of this task, on one hand, the rail modelling and, on the other hand,the characterisation of the finite length nature of track supports.

Firstly, concerning the rail modelling technique, it has evolved under two majorpremises. On one hand, regarding the frequency domain, it should describe highfrequency behaviour of the rail. In order to accomplish with this first premise, amodel based on Timoshenko beam theory is used, which can accurately account forthe vertical rail behaviour up to 2500Hz. On the other hand, with respect to thetime domain, the response should be smooth and free of discontinuities. This lastcondition is fulfilled by implementation of the Timoshenko local deformation.

Secondly, a model of support that considers its finite length nature is sought. Forthis purpose, a Timoshenko element over elastic foundation is formulated. Thus,the common model of support, which is based on a concentrated connection, issubstituted by a distributed model of support. In this way, several enhancementsare achieved; the temporal contact force response is smoothed and a more realisticshape is obtained, the amplitude of the displacement due to the parametric exci-tation is reduced and the magnitude associated to the ‘pin-pin’ frequency is notoverestimated.

Keywords: Track modelling, Timoshenko element, local deformation, responsediscontinuity, distributed supports.

Sammanfattning

Spårburna fordon är ett ökande transportslag i dagens samhälle. Skälen är främsten begränsad miljöpåverkan men även den höga komforten har bidragit till attge järnvägstrafiken en positiv bild i dagens Europa. Denna uppåtgående trend in-nebär att järnvägsindustrin måste handskas med ett antal viktiga utmaningar. Delsupprätthålla låga driftkostnader men också kunna kontrollera orsakerna till buller-störningar, där spårburet buller är den viktigaste faktorn. Spårdynamiken spelar enavgörande roll i båda dessa frågor, eftersom en betydande del av driftskostnadernaorsakas av spårunderhåll samt att bullret som genereras av rälen många gångerär dominerande. Detta förklarar varför beräkningsverktyg är högt värderade avjärnvägsföretag.

Arbetet som presenteras i denna licentiatuppsats föreslår metoder för exaktoch effektiv modellering av järnvägsspårets dynamik. Två huvudspår har lett tillutvecklingen av denna uppgift; modellering av rälen och karaktärisering av sliprar-nas dynamik. I det första huvudspåret av modellering av räler, har utvecklingenskett under två övergripande villkor; dels att i frekvensdomänen kunna beskriva dethögfrekventa beteendet hos rälen. För att uppnå det första villkoret har en modellbaserad på Timoshenkos balkteori tillämpats. Denna kan beskriva den vertikalaresponsen i rälen upp till ungetär 2500Hz. Det andra villkoret är att impulssvaret,i tidsdomänen, ska vara jämnt och fritt från diskontinuiteter, vilket uppfylls genomimplementering av lokal deformering enligt Timoshenko.

I andra huvudspåret, eftersöks en modell som beaktar sliperns ändliga längd.För detta ändamål har ett Timoshenko-element som placerats på en elastisk grund,formulerats. Den gemensamma stödmodellen, vilken är baserad på en lokal kop-pling, är således ersatt av en utbred stödmodell. På detta sätt uppnås flera förbät-tringar. Det temporala kontaktkraftssvaret blir utjämnat och ett mer realistisktsvar erhålls samt att amplituden för förskjutningen på grund av parametrisk exci-tation minskas och därmed överskattas inte amplituden associerad med ‘pin-pin’-resonansen.

Nyckelord: Räl modellering, Timoshenko-element, lokal deformation, responsdiskontinuitet, utbredda stöd.

Acknowledgements

This work has been performed at the Department of Aeronautical and Vehicle En-gineering of KTH, and, at the Energy and Transport Division of CEIT.

In the first place, I would like to show my gratitude to all the people who havehelped me in some way to reach my goals.

I would like to thank my supervisors, Leif Kari and Asier Alonso, their sugges-tions, guidance and help during this first part of my PhD thesis. As well as, forthe opportunity to perform this work in collaboration with the Marcus WallenbergLaboratory (MWL) in KTH. Also, I would like to thank Nere Gil-Negrete her im-plication with the development of my work and her advice at the tight spots. Also,many thanks to CAF I+D from where my work has been followed with interest andcontributions have been made to improve it.

I am specially grateful to José Germán Giménez without whose knowledge andcommitment, this work would not have been possible.

I thank to all my fellows in CEIT and KTH, for the good moments we havespent together; the cafés, fikas, sagardotegiak and Thursdays of innebandy. Amongall the people who have accompanied me along this time, I would like to thank toLuck who during each visit to Stockholm has thrown me a line, and to my officemates in CEIT; Chun, Esther and Itziar, by creating a friendly and nice place towork. And specially, many thanks to my colleague and friend Miguel, who hasshared with me, since the very beginning of this journey, his passion for life.

Finally, I wish to thank particularly my family whose unconditional supportand trust have given me the strength to carry on.

Dissertation

This thesis consists of two parts. The first part gives an overview of the researcharea and work performed. The second part contains the appended research papers(A and B).

Paper A

Blanco, B., Alonso, A., Gil-Negrete, N., Kari, L., Giménez, J.G. Implementationof Timoshenko element local deflection for vertical track modelling. Submitted toJournal of Sound and Vibration.

J.G. Giménez gave the keys ideas to develop this work. B. Blanco developed andimplemented the model, and wrote the article. A. Alonso followed the work andguided during its progress. J.G. Giménez, A. Alonso, L. Kari and N. Gil-Negretediscussed the results and reviewed the paper.

Paper B

Blanco, B., Alonso, A., Kari, L., Gil-Negrete, N., Giménez, J.G. Distributed sup-port modelling for vertical track dynamic analysis. Submitted to Vehicle SystemDynamics.

Alonso, A. and J.G. Giménez inspired the core idea for this paper and followedthe work progress. B. Blanco developed and implemented the method and wrotethe article. L. Kari and N. Gil-Negrete supervised the work, discussed the ideasand reviewed the paper.

Contents

I OVERVIEW 3

1 Introduction 51.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Medium-high frequency phenomena . . . . . . . . . . . . . . . . . . . 5

1.2.1 Wheel-Rail noise . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Rail corrugation . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Track structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3.1 Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Vertical dynamic behaviour . . . . . . . . . . . . . . . . . . . 9

1.4 Track modelling techniques . . . . . . . . . . . . . . . . . . . . . . . 101.5 Aim of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Rail modelling 132.1 Description of the track model . . . . . . . . . . . . . . . . . . . . . 132.2 Description of the local system . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Static approach of the local system . . . . . . . . . . . . . . . 162.2.2 Dynamic approach of the local system . . . . . . . . . . . . . 16

2.3 Direct numerical integration . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Integration scheme . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4 Temporal response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Distributed support model of rail-pad 253.1 Support modelling methodologies . . . . . . . . . . . . . . . . . . . . 253.2 Distributed support model . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Timoshenko beam over elastic foundation . . . . . . . . . . . 263.2.2 Interaction between sleeper and rail . . . . . . . . . . . . . . 283.2.3 Matrices of TEEF . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Frequency domain simulations . . . . . . . . . . . . . . . . . . . . . . 303.4 Time domain simulations . . . . . . . . . . . . . . . . . . . . . . . . 313.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

CONTENTS 1

4 Resolution based on precalculated IRF 354.1 Numerical convolution product . . . . . . . . . . . . . . . . . . . . . 354.2 Integration scheme of the convolution . . . . . . . . . . . . . . . . . 374.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Conclusion and future work 395.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Bibliography 41

II APPENDED PAPERS 45

Part I

OVERVIEW

3

1Introduction

1.1 Background

Railway dynamic simulations tackle the study and prediction of the movement ofrailway vehicles. For these systems the main source of excitation has its origin inwheel-rail contact forces. The importance of track modelling in the calculation ofcontact forces depends on the frequency range of interest. For low-frequency stud-ies, such as safety, comfort and curve negotiation simulations, the track modellingaccepts coarse simplifications. However, the track requires a description in a highlevel of detail while addressing medium-high phenomena.

The medium-high frequency interaction between the railway vehicle and trackis responsible of noise nuisance and deterioration of wheel and rail contact surfaces.The current context is characterised by the prompt increasing of high-speed rail-way lines, the public awareness of noise nuisance as a cause of stress diseases andinadequate rest, and the ambition of operator and manufacturers to improve thedurability and maintenance of wheels and tracks. This situation has led the effort ofresearchers to achieve a better understanding of the medium-high frequency inter-action phenomena. Accordingly, the track models used in these sorts of simulationsbecome more sophisticated.

The railway track has some remarkable features that make its modelling a com-plex task. The most important features to account for are: a geometry that canbe considered infinite in its longitudinal dimension, dissipative properties period-ically distributed, sources of excitation with a moving nature and the presence ofcomponents with a non-linear behaviour.

1.2 Medium-high frequency phenomena

In order to understand the importance of modelling the medium-high frequencybehaviour of railway systems, the most common undesired by-products of wheel-rail interaction are presented in this section. These are the wheel-rail noise and therail corrugation.

1.2.1 Wheel-Rail noiseThe railway traffic is associated to three main kinds of noise: the engine noise, thewheel-rail noise and the aerodynamic noise, being the second dominant for a velocity

6 CHAPTER 1. INTRODUCTION

range within 30 and 300 km/h [1]. Below and above this velocity range, engine andaerodynamic noise are dominant, respectively. The wheel-rail interaction is themain mechanism of noise generation for most operational conditions in railwaylines, therefore it is the most widely studied.

The wheel-rail noise is caused by contact forces, which excite the track andvehicle, leading to structural vibrations that are propagated to the environmentin the forms of acoustic noise and ground vibration. The most important sourcesof wheel-rail noise are sleepers, rails and wheels (Fig. 1.1 (left)). As it is shownin Fig. 1.1 (right), the dominant source of noise highly depends on the excitationfrequency. Below 500Hz noise mainly comes from sleepers, between 500 to 2000Hzrails play the main role and above 2000Hz the main source are the wheels. It isassumed that track dynamics control wheel-rail noise below 2000Hz.

Depending on the track geometry different types of noise are identified. Ontangent track, structural noise with a broad frequency content occurs, known asrolling noise. It is due to the excitation caused by the irregularities of bothwheel and rail surfaces. In curves, very high noise level can be produced, which isoften dominated by single frequencies. This phenomenon is known as curve squeal.Instead of vertical excitation by a relative displacement associated to irregularities,squeal noise is generated by unsteady lateral creep forces. Another kind of noise isdue to wheel-track singularities, such as wheel-flats or rail joints, known as impactnoise.

Nowadays, the railway noise emission is subjected to regulations established bythe European Commission. It is a response to the public concern about this issue.According to European Environment Agency (EEA) more than 14 million peoplewere exposed to more than 55 dB due to railway noise during 2014 [3]. Some

Figure 1.1: Sources of noise: wheel, rail and track (left) [1]; and their contributiondepending on the excitation frequency (right) [2].

1.3. TRACK STRUCTURE 7

European countries have done their own studies, for instance, in Sweden, the socialcost of railway noise was estimated to be SEK 908 million per year [4].

1.2.2 Rail corrugation

Corrugation is a rail damage mechanism fixed to a specific wavelength from 0.03to 0.3m (Fig. 1.2). According to Ref. [5], 40% of tracks are prone to developcorrugation, due to they are subjected to traffic conditions that normally do notundergo important variations. It results in the predominance of some excitingfrequencies, leading to a fixed pattern deterioration. The corrugation affects to thewheel and rail lifetime, it highly increases the level of generated noise and in somecircumstances, it leads to rail replacement to avoid safety issues.

Figure 1.2: Example of ‘Heavy haul’ corrugation [6].

Rail corrugation tends to be more frequent in curve sections, although it is alsopresent in tangent tracks. Classification of this phenomenon is not an easy taskbecause the mechanisms causing corrugation are diverse and they are not completelyunderstood. The classification proposed by Ref. [6] is presented in Tab. 1.1. TheP2 resonance is associated to the resonance of the vehicle as an unsprung mass onthe track stiffness (typically close to the Hertz contact resonance).

1.3 Track structure

In this section, the track structure and its main dynamic features are described. Atfirst, a description is presented of the different track components that are subjectof modelling. This description is followed by an explanation of the vibration modesassociated to the vertical dynamic behaviour of the track.

8 CHAPTER 1. INTRODUCTION

Table 1.1: Corrugation classification according to Ref. [6].

Type Wavelength-fixing mechanism Frequency (Hz)Heavy haul P2 resonance 50 - 100Light rail P2 resonance 50 - 100Other P2 resonance P2 resonance + torsional modes 50 - 100Trackform specific Sleeper resonance 50 - 100Rutting Second torsion mode of axle 250 - 400Roaring rails Pinned-Pinned rail resonance 400 - 1200

1.3.1 Components

The function of the track is to guide trains in a safe and economical manner. Thetrack is built by assembling of the following components:

• Rails: They have a double function; providing a running surface and, simul-taneously, bearing capacity. As running surface, it must be smooth for thetrain wheels, allowing a quiet running and a low level of vibrations. Verticalload is the main force acting over a rail. Therefore, rails are derived from anI-profile, to provide bearing capacity in that direction. The most commonrail on main lines in Europe is the UIC60 rail.

• Rail-pads and fastening: Rail-pads protect sleepers from wear and impactdamage. They are placed between the rail and sleepers, providing of an elasticconnection between both. The fastening is introduced by clips. Rail-padsincrease the track flexibility and they introduce part of the track dissipativeproperties. Soft rail-pads permit larger deflection of the rail and a betterdistribution of the load at the sleepers. Moreover, the softer the rail-pad is thebetter is the isolation of sleepers and ballast from high-frequency vibrations.The rail-pad is composed by rubber material with non-linear behaviour thatis known to be influenced by the preload.

• Sleepers: They are the supports of the rail, transmitting forces from the raildown to the ballast bed. Their other task is to maintain the characteristicdimensions between both rails; being the gauge, level, and the alignment ofthe track. Most typical sleepers are those formed by monobloc of concrete,although, bi-bloc configuration is also used. Regarding the material, theycan be made not only with concrete, but also wood or mixed; using concretefor the outer and steel for the central parts. Normally, a constant distancebetween sleepers of 0.5− 0.65m is used.

• Ballast: It is typically made of course stone, in such a way that sleeperscan be embedded into it. The ballast layer supports the vertical and lateral

1.3. TRACK STRUCTURE 9

forces. It provides a considerable flexibility and dissipation to the system. Itsmodelling is complex, due to its granular nature and the characterisation ofits properties.

It also worth noting the existence of ballastless tracks. In this sort of tracksballast is substituted by concrete slabs. It has a higher installation cost, but incontrast, it provides a great accuracy and longevity, which has a direct impact overthe maintenance cost. This kind of track is a particularly appropriate choice forplacements where the maintenance tasks are obstructed, e.g. tunnels, bridges andstations. Several configurations exist for ballastless tracks, some of them neither usesleepers while connecting directly the rail to the slab with one or two intermediatepads. If the sleepers are not removed they can be embedded on the slab, withoutany intermediate element or with resilient pads between the slab and sleepers.

1.3.2 Vertical dynamic behaviourFollowing, a general description of the vertical dynamic behaviour of the track isintroduced in order to understand the role played by each component. The presenceof these components gives rise to several resonant frequencies, which are presentedin Fig. 1.3. It shows the frequency response of a periodically supported track dueto a static load.

1

2

3

1

2

3

Figure 1.3: Track receptance at the exciting point. Mid-span excitation (–). Exci-tation above a sleeper (- -). The mode shapes are qualitatively represented.

10 CHAPTER 1. INTRODUCTION

The three resonances presented in Figure 1.3 are:

1. Sleepers and rail vibrating in phase: In this case the rail and sleepers act likethe mass and ballast acts both as the spring and as a mass. Therefore, railand sleepers are moving like a mass elastically connected to the ballast. Thisresonance normally takes place between 50 to 300Hz.

2. Sleepers and rail vibrating out of phase: In this resonance the rail and sleepersact like independent masses connected by the railpads, which act as springs.Thus, the vibrating masses move in opposite directions. This resonance isfound between approximately 200 to 600Hz, mainly depending on the railpadstiffness.

3. Rail resonance: The frequency where this resonance occurs is called ‘pin-pin’frequency. In this vibrating motion the movement is chiefly located in the rail,and the supports resemble rigid joints. It first happens when the wavelengthof the rail bending waves is twice the sleeper spacing, which generally isaround 1000Hz. This vibration is poorly damped; since, the displacement islocated in the rail, and its associated damping is mainly due to its own internaldamping. The response to an excitation at this frequency strongly dependson exciting position. It is maximum at the mid of a span, and minimumabove the sleeper, leading to a resonance and an anti-resonance, respectively.

Excitation of the presented dynamic behaviour is mainly caused by the irregu-larities at the contact surfaces. The wavelength irregularities encompasses a broadrange, from 0.03 to even 2m and, in turn, train operational speeds can cover from10 to 300 km/h. Therefore, the frequency range in which a track is excited is verywide, mainly from 1 to 2800Hz. Other source of excitation is caused by changes ofthe foundation stiffness, which is known as parametric excitation. The most com-mon example of this kind of excitation is due to the different flexibility between apoint at the mid-span and another above the sleeper. It arises the so-called sleeperpassing frequency, f = V/l, where V is the vehicle speed and l is the distancebetween sleepers. Other examples of varying flexibility are, transitions betweenballast to slab tracks and changes on the rail sections and on the type of supports.These cases are very common in switches, crossing and stations.

1.4 Track modelling techniques

Track modelling has been a field of profound research during the last decades.Excellent reviews of track modelling can be found in Refs. [7, 8], in which a firstdistinction is made between frequency and time domain modelling. The frequencydomain modelling is normally addressed by analytical methodologies [9–12], becausethey entail a low computational cost. On the other hand, time domain modelling

1.5. AIM OF THE THESIS 11

usually makes use of modal superposition [13–15] and finite element methodologies[16–18], which can tackle more diverse cases including non-linearities.

Other major classification can be done depending on the rail modelling tech-nique. In this respect, beam models based on Euler-Bernoulli and Timoshenko the-ories can be found. The Euler-Bernoulli theory neglects shear strain and rotatoryinertia of the beam, therefore it is accurate for studies not exceeding a frequencyaround 500Hz. On the contrary, Timoshenko beam theory takes into account bothshear strain and rotatory inertia, and it is able to describe accurately vertical andlateral dynamics up to approximately 2500 and 1500Hz, respectively. Above thesefrequencies, Timoshenko theory is not accurate since the rail cross-section can notbe longer taken as non-deformable. Models describing the rail cross-section defor-mation can be found in Refs. [19–22].

With respect to the models that make use of the Timoshenko beam theory,analytical methodologies have been developed that account for the moving natureof the load and periodic foundation [23, 24] for the frequency domain. In spite oftheir low computational cost, analytical methods require assumptions which is notmet in a wide range of cases. Numerical techniques increase the computational cost,but they are able to perform simulations considering non-linear components [25],jointed tracks [26], sleeper voids, rail and sleeper lift-off [27], foundation transitions[28] and finite length of supports [29].

1.5 Aim of the thesis

The aim of this PhD project is to develop a track model that enables for the studyof the medium-high frequency range. Therefore, the model can be used to performnoise emission prediction and corrugation assessment in future works. The designof this model is performed according to following requirements:

• The model should be valid for the range of frequencies where noise emissionfrom the rail is dominant. According to Ref. [21], beam models can predictsuccessfully noise emission since, rail cross-sectional deformation takes placesin a range of frequencies where the rail is not the dominant source of noise.Hence, modelling the rail according to Timoshenko beam theory is an enoughaccurate approach.

• The model should be able to perform time domain simulations. In this way,the influence of singularities and non-linear behaviour can be studied.

• The response should be smooth and continuous. The presence of discontinu-ities at the temporal response difficulties the convergence of the resolution, inspite of using implicit integration and small time-steps. It is due to contactsystems are very stiff mathematical problems [30].

• Finally, the computational cost of the resolution scheme should be kept aslow as possible.

12 CHAPTER 1. INTRODUCTION

1.6 Organization of the thesis

This thesis is organised as follows: Chapter 2 deals with the rail modelling, inwhich the accuracy and continuity of temporal response are enhanced by imple-mentation of the local deformation for the Timoshenko beam element. Chapter 3is focused on the development of a new track support model based on a distributedfoundation, in order to substitute the traditional support modelling that has someinherent drawbacks. Chapter 4 studies a resolution strategy based on the numeri-cal convolution product to reduce the computational cost of temporal simulations.Finally, in Chapter 5 the conclusion of this work and the future lines of researchare presented.

2Rail modelling

This chapter addresses the rail modelling by using Timoshenko elements. Thetemporal response is enhanced with the implementation of the local system defor-mation. At first, the decomposition of the whole system into the global and localsystems is presented. Once the problematic is exposed, the equations defining thelocal system are developed for both static and dynamic approaches. The resolutionmethodology based on a Newmark integrator scheme is subsequently shown. Fi-nally, the results are analysed. This chapter refers to the work presented in PaperA.

2.1 Description of the track model

The conventional Timoshenko element (TIM4) [31] has four degrees of freedom(DOF), representing vertical displacement, w, and bending rotation, θ, for eachnode. These nodal displacements are normally referred as, u(e) = [w1 θ1 w2 θ2].The static Timoshenko beam governing equations are

Q = κAG

(−θ + ∂w

∂x

), (2.1)

M = −EI ∂θ∂x, (2.2)

where Q is the shear strength, κ is the Timoshenko coefficient, A is the section ofthe area, G is the shear modulus, x is the longitudinal coordinate,M is the bendingmoment, E is the Young’s modulus and I is the section second moment of inertia.From Eq. (2.1) and (2.2) the interpolation function for TIM4 can be derived as

Nw1(ξ) = 11 + Φ

[1− 3ξ2 + 2ξ3 + Φ(1− ξ)

], (2.3)

Nθ1(ξ) = ξ

1 + Φ

[L(1− 2ξ + ξ2) + Φ

2 (1− ξ)L], (2.4)

Nw2(ξ) = 11 + Φ

[3ξ2 − 2ξ3 + Φξ

], (2.5)

Nθ2(ξ) = ξ

1 + Φ

[L(ξ2 − ξ)− Φ

2 (1− ξ)L], (2.6)

14 CHAPTER 2. RAIL MODELLING

where L is the element length, ξ = x/L and Φ = 12EI/GAκL2. The presented in-terpolation functions are analogous to those of the Euler–Bernoulli element, whichare based on the Hermite cubic polynomials. Additionally, the Timoshenko interpo-lation functions consider the influence of shear by means of Φ, which characterisesthe ratio of bending and shear stiffness. If the beam is considered infinitely stiff toshear strength, Φ becomes zero, and the interpolation functions resemble those ofthe Euler-Bernoulli element. The mass and stiffness matrices are derived from theinterpolation functions, and they can be found in Ref. [31]. The model of trackintroduces periodic supports as concentrated connections periodically separated bya distance, l. Each support is modelled by a mass and two Maxwell-Voigt models,representing the sleeper mass and the elasticity of both rail-pad and ballast, respec-tively. For the cases here studied, vehicle is simplified as a moving mass attachedto the vehicle by a non-linear Hertzian spring, Kc. The notation for the systemparameters is presented in Fig. 2.1.

The equations of motion are stated by assembling the Timoshenko elementarymatrices together with the supports. Time domain simulations are performed byiteratively solving the equation system with a numerical integrator. However, if thesurfaces are considered perfectly smooth or with small irregularities, the responseat the contact position shows an irregular behaviour. It is presented as sharpdiscontinuities, each time the load overtakes a node. This non-physical responsehas its origin in two reasons. Firstly, underestimation of shear strain at the loadpositions. The formulation of TIM4 interpolation functions, assumes vanishingshear strain at the medium point between the element nodes. Therefore, when aload is applied at the middle of an element, after extrapolation as nodal forces,the resulting displacement field will lead to a very low shear strain at the loadposition, albeit it is at its maximum. Secondly, the TIM4 element does not imposecontinuity of the vertical deflection first derivative at their nodes, creating a shearincompatibility between adjacent elements.

Both problems have been tackled in Refs. [17, 26]. In Ref. [26] the shear under-estimation is corrected with hat-functions, and shear compatibility is guaranteed byusing TIM7 elements. This model forces to redefinition of the system equation ateach iteration and, on the other hand, it increases the number of DOF. In Ref. [17],the shear strain underestimation is corrected by introducing an equivalent contactstiffness. In spite of obtaining a significant enhancement, the shear incompatibilityis not corrected and a large number of TIM4 elements is required to reduce it.

Implementation of the local element Timoshenko deformation may diminish thediscontinuity to a negligible level. Furthermore, it also reduces the computationcosts by reducing the required number of elements. The core idea for this de-velopment is widely known in the structure analysis field. The displacement ofa structure subjected to loads acting on positions between nodes can be studiedby superposition of two systems. The first system is formed by the loads actingon a structure derived from imposing fixed ends at the element nodes, hereaftercalled local system. Once it is solved, reactions at the fixed ends are introducedas nodal forces at the initial structure, being a second system, denoted as global

2.2. DESCRIPTION OF THE LOCAL SYSTEM 15

system. The superposition of both solutions makes up the real deformation. Thissolution is exact only for static problems in which the analysis of the local systemis restricted to the element in which the load acts. For dynamic problems, the localsystem is excited by external loads and the inertia of the global system, thus itrequires to be described in elements adjacent to the loaded one. It gives rises totwo approaches of the local system, one static and another dynamic. In Figure 2.1the split of the track problem into global and local systems is pointed out.

Global system

Local system

Whole system

Ms

K C

K b

p

b C

p

Ms

K C

K b

p

b C

p

Ms

K C

K b

p

b C

p

fcfc

Kc

M w

V

fext

Dynamic approachStatic approach

lQ

Q2

1

M1 M2 L

L

Figure 2.1: Split into the global and the local system. [Paper A Fig. 1]

2.2 Description of the local system

The approaches of the local system are defined analytically through the Timoshenkogoverning beam equations. At first, the static approach is stated, and secondly,the dynamic approach is developed, which uses the static approach and adds newequations to the system.

16 CHAPTER 2. RAIL MODELLING

2.2.1 Static approach of the local systemThe local system described statically is only represented by the displacement field ofa Timoshenko beam with clamped ends loaded by a point force, fa. The expressionsthat describe the vertical deflection for this problem are

wA(x, xa) = faL3

6EI

{x

L

[(L− xaL

)(x2

L2 −Φ2

)+ 3x

2a

L2 −x3a

L3 − 2xaL

]

+ xa(L− xa)2L3(1 + Φ)

[(2(L− xa) + LΦ)

(x3

L3 − 3 x2

L2 + 2 xL

)

+ (2xa + LΦ)(x

L− x3

L3

)]}, (2.7)

wB(x, xa) = faL3

6EI

{xaL

[(Φ2x− LL

)+ 3 x

2

L2 −x3 + xx2

a

L3 − 2 xL

+ x2a

L2

]

+ xa(L− xa)2L3(1 + Φ)

[(2(L− xa) + LΦ)

(x3

L3 − 3 x2

L2 + 2 xL

)

+ (2xa + LΦ)(x

L− x3

L3

)]}, (2.8)

where xa is the loading position. Equations (2.7) and (2.8) are the vertical dis-placement field for both sides of the beam with respect to the load position, wherewA is on the left side and wB is on the right side. Assessment of this approach isdone in Fig. 2.2. It shows the vertical displacement field of a periodically supportedrail obtained by three different models. Two of them do not introduce the localsystem, and they represent the rail with two and four elements per bay. The load isintroduced at xa = 0.25l. Comparing both solutions, it is observed how the modelwith half elements underestimates the displacement at the load position. The thirdcase uses the global solution of the two elements per bay model, and superposesthe static local system. For this case, the displacement field agrees with that fromthe four element per bay model. In conclusion, the static approach of the localsystem represents accurately the real shear strain of the Timoshenko beam at theload position.

2.2.2 Dynamic approach of the local systemThis approach addresses the problem from a different perspective, in which thedynamic behaviour of the local system is described. It is stated that the smallerthe finite beam element is, the higher are its natural frequencies; thus, leadingto an almost negligible effect over the solution when the size is largely reduced.However, this involves a very fine discretisation in order to get continuous resultswhile increasing the computational costs.

2.2. DESCRIPTION OF THE LOCAL SYSTEM 17

7 8 9 10 11 12 13Position (m)

-4

-3

-2

-1

0

Ver

tical

dis

plac

emen

t (m

)

×10-4

9 9.2 9.4 9.6-5

-4

-3

-2×10-4

2 elements per bay

2 elements per bay + local sys.

4 elements per bay

af f

a

Figure 2.2: Contribution of the local system to the static deformation of the globalsystem. Supports are denoted with �. [Paper A Fig. 2]

2.2.2.1 Dynamic study of a Timoshenko beam with clamped ends

In this development the local system is described by modal superposition of thevibration modes corresponding to a Timoshenko beam with clamped ends. Themathematical process starts from the coupled differential equations for transversefree vibration according to Timoshenko beam theory

− ∂Q(x, t)∂x

+ ρA∂2w(x, t)∂t2

= 0, (2.9)

− ∂M(x, t)∂x

+Q(x, t)− Iρ∂2θ(x, t)∂t2

= 0. (2.10)

where ρ is the material density. The functions describing the vertical displacementrespond to the next expressions

W (x) = P1 cos(λ1x) + P2 sin(λ1x) + P3 cosh(λ2x) + P4 sinh(λ2x), (2.11)

and for ω > ω0

W (x) = P1 cos(λ1x) + P2 sin(λ1x) + P3 cos(λ3x) + P4 sin(λ3x), (2.12)

18 CHAPTER 2. RAIL MODELLING

0 0.05 0.1 0.15

Position (m)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Nor

mal

ised

dis

plac

emen

t (-)

(a) Vertical vibration mode shapes.

0 0.05 0.1 0.15

Position (m)

-2

0

2

4

6

8

10

Nor

mal

ised

ben

ding

rot

atio

n (-

)

(b) Bending vibration mode shapes.

Figure 2.3: First four modes of vibration of a double clamped Timoshenko beam.Length of 15 cm. Frequencies: (–, black) 6,196Hz (·, green) 12,284Hz (-·-, red)18,160Hz (- -, blue) 18,645Hz. [Paper A Fig. 3]

where, ω0 =√κAG/Iρ is the cut-off frequency. The definition of Pi and λi is

given in Paper A. The natural frequencies of the vibration modes are those whichannul det(Λ), where Λ represents the system of equations derived from imposingthe clamped ends boundary conditions. Once the natural frequencies has beenobtained, the vibration mode can be fully defined by solving ΛP = 0, where Pgathers the coefficients of (2.11) and (2.12). The vibration modes are normalizedby imposing √∫ L

0(ρAW 2 (x) + ρIΘ2 (x)) dx = 1. (2.13)

Figure 2.3 presents the four vibration mode shapes of a 0.15m Timoshenko beamwith the geometric characteristics of the UIC60 rail profile. In Paper A, the dynamicapproach formulation is explained in detail.

2.2.2.2 Residual flexibility

To consider the total static flexibility of the local system by modal superposition,a high number of vibrations modes is required. This necessity can be lightened byusing a residual flexibility, defined as

FRes(x) = FSta(x)− FDyn(x) = FSta(x)−Nm∑i=1

W 2i (x)ω2i

, (2.14)

where FSta and FDin, are the flexibilities of the local static and dynamic approaches,respectively. The term Nm is the number of vibration modes taken into account.

2.2. DESCRIPTION OF THE LOCAL SYSTEM 19

The definition of FSta can be obtained from Eq. (2.7) imposing wAa(x, xa = x)/P ,

FSta(ξ) = L3

6EI

{ξ

[(1− ξ)

(ξ2 − Φ

2

)+ 3ξ2 − ξ3 − 2ξ

]

+ ξ(1− ξ)2L(1 + Φ)

[L (2(1− ξ) + Φ)

(ξ3 − 3ξ2 + 2ξ

)+ L(2ξ + Φ)

(ξ − ξ3)]}, (2.15)

The static and dynamic local flexibilities for several number of vibration modes aredisplayed in Fig. 2.4.

0 0.05 0.1 0.15Position (m)

0

5

10

15

Flex

ibili

ty (

m/N

)

×10-11

1 vibration mode3 vibration modes6 vibration modes20 vibration modesStatic

Figure 2.4: Static and dynamic flexibility. L = 0.15 cm. [Paper A Fig. 4]

2.2.2.3 Mass-cross terms

On the contrary to the static approach, the dynamic one requires the descriptionof the local system in those elements adjacent to the loaded one. This is due toan interaction between the local and global systems throughout mass-cross terms.These terms arise from the non-orthogonality between the modal shapes of localmodes and the TIM4 interpolation functions. The inclusion of this interaction leadsto the shear strain compatibility between elements, hence to a large reduction ofthe non-physical slope discontinuity. The mass-cross term between the i-th naturalfrequency and a nodal displacement j is defined as

mcrossij =

∫L

ρAWi(x)Nvj(x)dx+∫L

ρIΘi(x)Nbj(x)dx, (2.16)

where Nvj and Nbj are the vertical and rotational interpolation functions of theTimoshenko element, respectively. The expressions for Nvj are collected in Eqs.

20 CHAPTER 2. RAIL MODELLING

(2.3) to (2.6), and from them, Nbj is defined as

Nbj(x) = N Ivj(x) + ΦL2

12 N IIIvj (x). (2.17)

2.3 Direct numerical integration

The time domain simulations are performed by direct numerical integration of theequations of motion. At first, the equations of motion are stated and secondly theintegration process is described.

2.3.1 Equations of motionThe equations of motion are represented in a general form as

Mu + Cu + Ku = fc + fext, (2.18)

where M, C and K are the mass, damping and stiffness matrices, respectively, fcand fext are the contact and external force vectors, respectively, and u is the systemDOF displacements. The interaction between rail and wheel is described in termsof the interpenetration, δ = wr + wirr − ww, where wr and ww, are the verticaldisplacements of rail and wheel at the contact point, respectively, and wirr is thesurfaces irregularities. The contact force reads

fc = Kcδn+1 + Ccδ δ > 0, (2.19)

and fc = 0 for δ ≤ 0, where, Kc is the Hertzian contact coefficient and Cc is thecontact damping. The parameter n becomes 0 or 1/2 depending on whether thecontact is taken as linear or non-linear, respectively. For the case studied here theinfluence of the contact damping is considered negligible. The forces due to thecontact interaction can be expressed in matrix form as

fc = −

−1NΨ

fc = −

−1NΨ

Kcδn+1 = −

−1NΨ

Kcδn (wG.r + wLoc.r + wirr − ww)

= −

−1NΨ

Kcδn

−1NΨ1

T

wwuGuLwirr

= Kcδn

−1 NT Ψ 1N −NNT −NΨT −NΨ −ΨNT −ΨΨT −Ψ

wwuGuLwirr

= Kcδ

n

−1 NT ΨN −NNT −NΨT

Ψ −ΨNT −ΨΨT

wwuGuL

+Kcδn

1−N−Ψ

wirr = −KcδnEu + firr.

(2.20)

2.3. DIRECT NUMERICAL INTEGRATION 21

The subscripts G and L in wr and u refer to the global and local systems, respec-tively. The vector N is used to extrapolate the nodal forces of the loaded elementfrom the contact force, by using the interpolation function of TIM4, Eqs. (2.3)to (2.6). The vector Ψ gathers the local system contribution to the vertical dis-placement, i.e. the shape functions of the local modes and a unitary value for theresidual or static flexibility. Introducing (2.20) into (2.18) the equation of motionbecomes

Mu + Cu + (K +KcδnE) u = fext + firr. (2.21)

The global system equations are assembled straightforwardly with the mass andstiffness TIM4 matrices representing the rail. Together with rail global matrices,the periodic concentrated connections are represented by the stiffness and dampingterms of ballast and rail-pad, and by the sleeper mass. With respect to the localsystem, for the case of the static approach it is described as

Klocwloc = fc, (2.22)

where Kloc = F−1Sta. For the case of the dynamic approach of the local system, each

mode of vibration, i, at each element, j provides an equation with the general form

ηij + 2ζωiηij + ω2i ηij + Mcross

i u(e)j = fcWi(x), (2.23)

together with a equation analogous to (2.22) and associated to the residual flexibil-ity. The variable ζ is the modal damping and, the vector Mcross

i gathers the masscross terms for the nodal displacements at a certain vibration mode.

2.3.2 Integration schemeThe equations of motion are solved via an iterative scheme based on the New-mark integrator, which is commonly used in railway simulations [17, 26, 29]. TheNewmark equations are defined as

uk+1 = uk + ∆tuk + ∆t2

2 [(1− 2β) uk + 2βuk+1] , (2.24)

uk+1 = uk + ∆t [(1− γ) uk + γuk+1] , (2.25)

which establish a linear relation between the dynamic state of the system at theknown instant tk and at the unknown instant tk+1, weighted by the Newmarkparameters, γ and β. After substituting (2.24) and (2.25) into (2.21), the equationsof motion become

z (uk+1) = Auk+1 + B1uk + B2uk + B3uk − fext − firr = 0, (2.26)

equalling the number of equations and unknowns. This development is not ableto describe the DOF associated to the static local system or the residual flexi-bility, where the introduction should be done separately by performing the next

22 CHAPTER 2. RAIL MODELLING

transformation A =[A′(1,m)×(1,m−1) A′(1,m)×m/∆t2β

], where

A′ = M + ∆tγC + ∆t2β (K +KcδnE) , (2.27)

andm refers to the DOF describing the static local system or the residual flexibility.At instant tk+1 the m DOF is only characterised by its displacement, and at theinstant tk its state is null. The Bi matrices, which weight the influence of theacceleration, velocity and displacement at the instant tk, are

B1 = ∆t(1− γ)C + ∆t2

2 (1− 2β) (K +KcE) , (2.28)

B2 = C + ∆t (K +KcδnE) , (2.29)

B3 = (K +KcδnE) . (2.30)

The convergence of the system is reached by using a Newton–Raphson iterativeprocess

zk+1.r = zk+1.r − J−1r zk+1.r, (2.31)

where J is the Jacobian matrix. At a linear contact, J = A, otherwise secondorder terms of the interpenetration arise. These terms are exposed in Paper A,which permit analytic calculation of the Jacobian matrix. Furthermore, the use ofA enables for fast convergence of the equations, with more iterations, but insteadavoiding the calculation and inversion of the Jacobian matrix at each r-th iteration.

2.4 Temporal response

In order to evaluate the capacity of the methodology to obtain continuous results,temporal simulations are performed and the contact force response is studied. Thetrack parameters are those used by Ref. [17] (Tab. 2.1), which previously hastackled the same problem. The results are presented in Fig. 2.5.

Figure 2.5a shows the temporal response of the contact force when only theglobal system is introduced and obtained by implementation of the static approachof the local system. It is observed how a significant enhancement is reached withthe static approach, which represents accurately the shear strain at the contactposition. However, the discontinuity is still noticeable each time the load overtakesa node. As it has been previously explained, the cause of this slope discontinuityis due to at the lack of a continuity for the shear deformation. Fig. 2.5b shows theresults obtained by the dynamic approach, together with those results provided bythe static approach of the local system. Once the dynamic approach is introducedthe discontinuities become insignificant.

The convergence of the solution for a low number of elements per bay is tested inFig. 2.6. The performance of the dynamic approach of the local system is assessed,by comparing the results for two and eight elements per bay. For two elements,

2.4. TEMPORAL RESPONSE 23

10.2 10.4 10.6 10.8 11 11.20.8

0.9

1

1.1

1.2

Con

tact

for

ce f

acto

r

Distance (m)

Static approachConventional Timoshenko element

Sleeper positions

(a) Local static approach and conventionalmodel with 8 elements per bay.

10.2 10.4 10.6 10.8 11 11.2

Distance (m)

0.8

0.9

1

1.1

1.2

Static approachDynamic approach

Sleeper positions

(b) Local dynamic and static approaches.Dynamic: 4 elements per bay, 5 local sys-tems and 6 vibrations modes per local sys-tem. Static: 8 elements per bay.

Figure 2.5: Wheel/rail dynamic force in steady-state interaction at 148 km/h forthe track parameters of Tab. 2.1. [Paper A Fig. 5]

Table 2.1: Parameters for the track in Ref. [17]

.

Young’s modulus (GPa) E 207 Rail area (cm2) A 71.7Moment of inertia (cm4) I 2350 Mass density (kg/m) ρ 56Pad damping (kNs/m) Cp 21.8 Sleeper spacing (m) l 0.79Pad stiffness (MN/m) Kp 200 Shear coefficient κ 0.34Ballast damping (kNs/m) Cb 21.8 Wheel mass (kg) Mw 500Ballast stiffness (MN/m) Kb 31.6 Sleeper mass (kg) Ms 50Contact coeff. (GN/m3/2) Kc 100 External force (kN) fext 82

the result agrees with that from eight elements per bay, while the discontinuityremains in a low level. Therefore, the requirements of the track model regardingthe discretisation are significantly reduced when the dynamic approach is used, ascompared to those from the static approach. This fact brings another importantadvantage, since accurate and continuous results can be obtained with a lowercomputational cost.

The results presented in Fig. 2.6 for two elements per bay are obtained describ-ing the local system at 5 elements and using 6 modes of vibration for each localsystem. The influence of the dynamic local system configuration is studied in PaperA into deeper level of detail.

24 CHAPTER 2. RAIL MODELLING

10.2 10.4 10.6 10.8 11 11.2

Distance (m)

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Con

tact

for

ce f

acto

r

2 Elements per bay8 Elements per bay

Figure 2.6: Convergence of the results obtained with the dynamic approach for adifferent number of elements per bay. [Paper A Fig. 7a]

2.5 Conclusion

The methodology based on the implementation of the local system has successfullyaddressed the continuity problems of the temporal simulations with Timoshenkoelements. The formulation has been derived in an analytic way for both staticand dynamic approaches of the local system. For the static approach it is foundthat, in spite of correcting the shear strain underestimation, this approach forces touse a high number of elements to reduce shear incompatibilities between adjacentelements. On the other hand, the dynamic approach corrects shear underestima-tion and reduces the shear incompatibility by using a low number of element, andwithout resorting to finite elements with a higher number of DOF (TIM7). This, to-gether with its accuracy and computational efficiency, make the dynamic approachof the local system an effective solution for time-domain simulations.

3Distributed support model of rail-pad

As it is exposed in Section 1.3.2, the presence of vibration modes introduced byperiodic foundation makes necessary its modelling when medium-high frequencyrange is of interest. In spite of periodic foundation track modelling is a generalisedtrend currently, some models do not take into account the support length in anyextent, and others point out its influence with rotational interactions between sleep-ers and rail. The goal of this chapter is to develop an accurate model of rail-padsupport, which accounts in a realistic way for the support length. It is presentedand validated via frequency and time domain simulations. This chapter refers tothe work presented in Paper B.

3.1 Support modelling methodologies

Most railway track models account for the interaction between sleepers and railby means of Maxwell-Voigt connections. In this way, the support is modelled asa concentrated force acting on the rail. It neglects the finite length nature ofthe support, even when it encompasses around a quarter part of the rail length.Moreover, concentrated support modelling is known to cause exaggeration of the‘pin-pin’ vibration mode [32]. Accordingly, in Refs. [29, 33] its strong influence onthe frequency response is proved by distributing the rail-pad stiffness along thoserail sections over the supports.

An attempt to describe the finite length nature in concentrated supports is basedon the inclusion of a rotatory interaction between rail and sleepers [24, 34–36]. Amore realistic approach of the finite support length is proposed in Ref. [29]. Inthat model, multiple concentrated connections are introduced per each support bydividing the rail length over supports in several elements.

The model of support presented here accounts for its finite length in a com-pact, analytical and efficient way. It is called distributed support model since itis characterised throughout the distributed stiffness of the pad, kd = Kp/Ls. Thesupport is modelled with a Timoshenko element over elastic foundation, hereafterdenoted as TEEF. Figure 3.1 presents the schemes of the supports models that arethe subjects of study in this chapter.

26 CHAPTER 3. DISTRIBUTED SUPPORT MODEL OF RAIL-PAD

Ferrara support Distributed supportConcentrated support

Figure 3.1: Different support modellings. [Paper B Fig. 1]

3.2 Distributed support model

TEEF introduces two new features with respect to the previous models. Firstly, theinfluence of the elastic foundation over the Timoshenko beam displacement field isimplemented analytically into the stiffness and mass matrix. Secondly, the verticalmovement of the sleeper is related to the displacement and rotation of the rail.Thus, a fifth DOF arises resulting in another interpolation function, Ns.

The starting point of the distributed support model is the formulation pro-posed in Ref. [37]. That work introduces the foundation stiffness by the supportinteraction stiffness matrix S(e)

k , calculated by using the virtual work principle as

IVW + δu(e)Tkd

∫L

N(e)EF

TN(e)

EFdxu(e) = IVW + δu(e)TS(e)

k u(e) = EVW, (3.1)

where IVW and EVW are the internal and external virtual work, respectively. InRef. [37] the effect of the foundation over the interpolation functions is not consid-ered. However, it was found that definition of S(e)

k using the TIM4 interpolationfunctions (from (2.3) to (2.6)), leads to an inconsistent results in the track re-sponse. Thus, the vector N(e)

EF collects the exact interpolation functions for theTimoshenko beam over an elastic function. Moreover, the deductions of N(e)

EF andNs are exposed, and from them the matrices of TEEF.

3.2.1 Timoshenko beam over elastic foundation

The equilibrium equations of a Timoshenko beam over elastic foundation, are thoseof TIM4, Eqs. (2.1) and (2.2), but in this case, the shear strength derivative is no

3.2. DISTRIBUTED SUPPORT MODEL 27

null, due to the distributed shear introduced by the foundation,

q = −QI = −kdw, (3.2)

where the upper index denotes the spatial derivative. After some mathematicaltransformations, the differential governing equation can be expressed only in termsof the vertical deflection and its derivatives

wIV − kd

κAGwII + kd

EIw = 0. (3.3)

In parallel, bending rotation can be expressed as function of vertical deflection as

θ = wI

(1− kdEI

(κAG)2

)+ EI

κAGwIII. (3.4)

The detailed process followed to obtain these expressions can be found in PaperB. The solution for differential equation (3.3) has vanishing particular solution andthe homogeneous solution reads

w(x) = A1er1x +A2er2x +A3er3x +A4er4x, (3.5)

where ri are the characteristic roots of (3.3).

r1 = β + iα ; r2 = β − iα ; r3 = −β + iα ; r4 = −β − iα. (3.6)

After some trigonometric transformations, (3.5) can be expressed as

w(x) = ch (βx) [C1c (αx) + C2s (αx)] + sh (βx) [C3c (αx) + C4s (αx)] , (3.7)

where the abbreviations c, s, ch and sh, are used to denote trigonometric func-tions. Equation (3.7), represents the general expression of the TEEF interpolationfunctions. Defining the rail vertical displacement and rail bending rotation as

w(x) = C1g1(x) + C2g2(x) + C3g3(x)3 + C4g4(x), (3.8)θ(x) = C1h1(x) + C2h2(x) + C3h3(x)3 + C4h4(x), (3.9)

the coefficients of (3.7), Ci, can be obtained for each interpolation function bysolving the equations system

g1(0) g2(0) g3(0) g4(0)h1(0) h2(0) h3(0) h4(0)g1(L) g2(L) g3(L) g4(L)h1(L) h2(L) h3(L) h4(L)

C1C2C3C4

=

w(0)θ(0)w(L)θ(L)

, (3.10)

for the set of boundary conditions

w(0) = 1 ; θ(0) = 0 ; w(L) = 0 ; θ(L) = 0⇒ Ciw1 ⇒ NEFw1, (3.11)

w(0) = 0 ; θ(0) = 1 ; w(L) = 0 ; θ(L) = 0⇒ Ciθ1 ⇒ NEFθ1, (3.12)

w(0) = 0 ; θ(0) = 0 ; w(L) = 1 ; θ(L) = 0⇒ Ciw2 ⇒ NEFw2, (3.13)

w(0) = 0 ; θ(0) = 0 ; w(L) = 0 ; θ(L) = 1⇒ Ciθ2 ⇒ NEFθ2. (3.14)

28 CHAPTER 3. DISTRIBUTED SUPPORT MODEL OF RAIL-PAD

Figure 3.2 shows the level of disagreement between TIM4 interpolation functionsand NEF as the distributed stiffness increases.

0 0.2 0.4 0.6 0.8 1x/L

0

1

2

3

4

5

(N1−N

EF

1)×100

(a) Difference between NEF1 and N1.

0 0.2 0.4 0.6 0.8 1x/L

0

2

4

6

8

(N2−N

EF

2)N

max

2×100

(b) Difference between NEF2 and N2.

Figure 3.2: Difference between TEEF and TIM4 interpolation functions for dif-ferent distributed stiffness. The beam length is 0.5m. Distributed stiffness:(− · −) 100 MN/m2, (− −) 200 MN/m2, (-×-) 300 MN/m2, (-∗-) 400 MN/m2, (-◦-) 600 MN/m2, (-+-) 1000 MN/m2. Beam parameters corresponding to the UIC60rail. [Paper B Fig. 3]

3.2.2 Interaction between sleeper and railThe rail displacement field is expressed in terms of the four nodal displacementsand the sleeper vertical movement, ws, since the distributed elastic foundation leadsto a direct interaction between all the rail sections over the pad, and the sleeper.Therefore, the interpolation of vertical beam deflection becomes

wr =[w1 θ1 w2 θ2 ws

] [NEFw1

NEFθ1

NEFw2

NEFθ2

Ns]T

= u(e)EFsN

TEFs, (3.15)

where Ns is the interpolation function associated to the sleeper. The function Nshas the form of the homogeneous solution, presented in Eq. (3.7), plus a non-vanishing particular solution wp = ws. By setting a unitary sleeper movement ws,the coefficients of the homogeneous solution can be obtained by imposing vanishingboundaries conditions

w(0) = 0 ; θ(0) = 0 ; w(L) = 0 ; θ(L) = 0⇒ Ciws ⇒ NEFws. (3.16)

The influence of the rail-pad stiffness in the interpolation function Ns is presentedin Fig. 3.3.

3.2. DISTRIBUTED SUPPORT MODEL 29N

s

Stiffness (MN/m)ξ (-)

01

0.02

0.04

500

0.06

400

0.08

0.5 300

0.1

200100

0 0

Figure 3.3: Interpolation function of the sleeper displacement, Ns, for a 0.5m TEEFwith the parameters of the UIC60 rail. [Paper B Fig. 4]

3.2.3 Matrices of TEEF

From the general definition of the stiffness matrix according to the Timoshenkobeam theory [37], the TEEF stiffness matrix is defined as

K(e)EF =

∫L

BTEFbEIBEFbdx+

∫L

BTEFshGAκBEFshdx+ S(e)

ks , (3.17)

where the first and second terms of the equation right side are the bending andshear stiffness matrices, respectively. The matrix S(e)

ks introduces over S(e)k the

sleeper DOF. The vectors B relate displacement and strain as ε = Bu, which isdefined for bending and shear strains as

BEFb = dNEFb

dx =[NII

EF

(1− kdEI

(κAG)2

)+ EI

κAGNIV

EF

], (3.18)

and

BEFsh =(

NIEF

kd

κAG−NIII

EF

)EI

κAG, (3.19)

respectively. The matrix S(e)ks can be obtained analogously to the expression given in

Eq. (3.1). The virtual work principle is adapted accounting for a moving foundation

30 CHAPTER 3. DISTRIBUTED SUPPORT MODEL OF RAIL-PAD

base, resulting in the expression

Sks = kd

∫L

NEFw1

NEFθ1

NEFw2

NEFθ2

NEFw1

NEFθ1

NEFw2

NEFθ2

T

NEFw1

(Ns − 1)NEFθ1

(Ns − 1)NEFw2

(Ns − 1)NEFθ2

(Ns − 1)Sim. N2

s − 2Ns + 1

dx =

S(e)k

Sw1sSθ1sSw2sSθ2s

Sim. Sss

.(3.20)

Finally, the stiffness matrix of the TEEF is stated as

K(e)EFs =

K(e)EF

Sw1sSθ1sSw2sSθ2s

Sim. Sss

. (3.21)

Accordingly, the mass matrices of the TEEF is defined as

M(e)EF = ρA

∫L

N(e)EF

TN(e)

EFdx+ ρI

∫L

N(e)EFb

TN(e)

EFbdx, (3.22)

where N(e)EFbs can be deduced from Eq. (3.18).

3.3 Frequency domain simulations

The receptance matrix of the track,

H (ω) =(−ω2M + iωC + K

)−1, (3.23)

enables for the study of the frequency domain. Figure 3.4 shows the results infrequency domain for three types of support modelling presented in Fig. 3.1, wheretrack parameters gathered in Table 3.1 are taken from Ref. [34]. This model hasbeen taken as reference because it makes use of vertical and rotatory interactionfor the support modelling.

It is observed how the support modelling affects to the frequency behaviour ofthe track from 200Hz to the ‘pin-pin’ frequency, at which the amplitude differenceis the most significant. Concentrated models cause deviation of the ‘pin-pin’ reso-nance amplitude, producing an overestimation for excitation at the mid-span (Fig.3.4a) and underestimation for excitation above a sleeper (Fig. 3.4b). With respectto the amplitude obtained by the distributed model of support, the concentratedmodels amplify by a factor of 1.92 and 6.25, for the cases of vertical and rotatoryinteractions and only vertical interaction, respectively. The impact over the anti-resonance observed for excitation above the sleeper is less significant. However,neither of the concentrated support models are able to correct the underestimationof this amplitude.

3.4. TIME DOMAIN SIMULATIONS 31

102 103

Frequency (Hz)

10-9

10-8

Am

plitu

de (

m/N

)

Rail CVS (×)Rail CVRS (o)Rail DS (∗)Sleeper CVSSleeper DS

(a) Excitation and response at the mid-span.

101 102 103

Frequency (Hz)

10-10

10-9

10-8

Rail CVS (×)Rail CVRS (o)Rail DS (∗)Sleeper CVSSleeper DS

(b) Excitation and response above thesleeper.

Figure 3.4: Vertical response in frequency domain of both rail and sleeper fordifferent types of support modellings. CVS: Concentrated vertical support. CVRS:Concentrated vertical-rotatory support. DS: Distributed support. [Paper B Fig. 6]

Table 3.1: Parameters for the track in Ref. [34]

Young’s modulus (GPa) E 210 Inertia moment (cm4) I 3055Cross-sectional area (cm2) A 76.9 Mass density (kg/m3) ρ 7850Shear coefficient κ 0.4 Sleeper spacing (m) l 0.6Pad stiffness (MN)/m Kp 350 Pad damping (Ns/m) Cp 48Ballast stiffness (MN/m) Kb 70 Ballast damping (kNs/m) Cb 47Wheel mass (kg) Mw 550 Sleeper mass (kg) Ms 160External force (kN) fext 100 Contact stiffness (GN/m) Kc 1.028Support length (m) Ls 0.16

3.4 Time domain simulations

Time domain simulations are performed by means of the procedure exposed inSection 2.3. Regarding the local deformation treated in Chapter 2, here it is imple-mented using only the static approach. It is derived by solving the displacementfield of a point loaded Timoshenko beam over elastic foundation with clamped ends.This development is given in Paper B. The extension to a dynamic approach of thelocal system for the TEEF is not on the focus of this work.

The steady-state response for smooth surfaces without irregularities is presentedin Fig. 3.5. In this case, the only source of excitation is the varying stiffnessbetween adjacent supports, i.e. parametric excitation. The displacement at thecontact point (Fig. 3.5a) shows a certain level of stiffening due to the inclusion ofthe finite length nature of the support, either way, as a distributed support, or, as

32 CHAPTER 3. DISTRIBUTED SUPPORT MODEL OF RAIL-PAD

6 6.5 7 7.5 8 8.5 9Distance (m)

-9

-8.8

-8.6

-8.4

-8.2

-8

-7.8

-7.6

-7.4

-7.2V

ertic

al d

ispl

acem

ent (

m)

×10-4

DSCVRSCVS

(a) Vertical rail displacement steady-state re-sponse at the contact position for a wheelmoving at 24m/s.

6 6.5 7 7.5 8 8.5 9Distance (m)

0.95

1

1.05

1.1

1.15

Con

tact

for

ce (

N)

×105

CVSCVRSDS

(b) Contact force steady-state response for awheel moving at 24m/s.

Figure 3.5: Temporal response of the wheel-rail system due to purely parametricexcitation at 24m/s for tracks with different support modellings. DS: Distributedsupport. CVS: Concentrated vertical support. CVRS: Concentrated vertical rota-tory support. [Paper B Fig. 9]

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8Distance (m)

0.95

1

1.05

1.1

1.15

Con

tact

for

ce (

N)

×105

Smooth surfaces2 µ m roughness amplitude4 µ m roughness amplitude

Figure 3.6: Steady-state response for a corrugated rail and the distributed sup-port modelling. Wheel speed: 44.4m/s. Irregularity wavelength: 4.1 cm. Thedistributed support position and its expanse are indicated with �. [Paper B Fig.12]

a concentrated rotatory interaction. However, the amplitude of the displacementis only reduced by the distributed support. Regarding the contact force presentedin Fig. 3.5b, the contribution of the rotatory interaction is insignificant, while thedistributed support has an important impact on the continuity of the results. Itis observed how the concentrated connections give rise to an abrupt disruption ofthe contact forces each time the load overtakes a support. However, the temporalresponse for the distributed support experiences a much softer transition, onlyperturbed by the shear incompatibilities between elements, which could be solvedby using the dynamic approach of the local system for the TEEF.

3.5. CONCLUSION 33

6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8Distance (m)

1

1.05

1.1

1.15

Con

tact

for

ce (

N) ×105

Smooth surfaces2 µ m roughness amplitude4 µ m roughness amplitude

Figure 3.7: Steady-state response for a corrugated rail and the concentrated connec-tion support modelling. Wheel speed: 44.4m/s. Irregularity wavelength: 4.1 cm.The point support position is indicated with N. [Paper B Fig. 13]

The ‘roaring-rails’ corrugation type is supposed to be caused by the vibrationof the rail in its ‘pin-pin’ resonance. Therefore, it is of interest to study the tempo-ral response of the contact forces under the excitation of a sinusoidal irregularity,which excites the system at that frequency. Figs. 3.6 and 3.7 display the temporalresponse for the distributed model and the concentrated model with vertical androtatory interactions, respectively. Again, the distributed modelling provides a re-sponse absent of the discontinuity caused by the concentrated support, leading toa wavy shape, which is suppose to be in a greater agreement with the nature of thesystem.

3.5 Conclusion

In this chapter a model of support that accounts for the finite length nature of rail-pads is implemented. This problem has been successfully tackled from an analyticalpoint of view leading to the formulation of the Timoshenko element over elasticfoundation (TEEF). The methodology has been proved to avoid the deviation of thetrack frequency response, mainly at the ‘pin-pin’ frequency. It is worth highlightingthat ‘pin-pin’ resonance amplitude obtained by using a rotatory interaction, is stillmisdirected as compared to that provided by the distributed approach. The mostsignificant advantages of the distributed model of support with respect to othersare its computational efficiency, since it only uses one element per support, and itscapacity to obtain continuous results for purely parametric excitation.

4Resolution based on precalculated IRF

This chapter tackles the formulation of an efficient resolution strategy. Previously,in Section 2.3, a resolution scheme is proposed based on the direct numerical inte-gration of the equations of motion through Newmark equations. That methodologybecomes slower as the system size increases. Here, a strategy based on the usage ofprecalculated impulse response functions (IRF), also known as Green’s functions,is proposed to speed up time domain simulations. The convolution between theIRF and contact forces has been used for models based on, periodically supportedEuler-Bernoulli beam [36], continuously supported Timoshenko beam [38] and pe-riodically supported Timoshenko beam [34, 39]. Those works propose analyticexpressions of the IRF leading to a fast calculation. On the contrary, the method-ology proposed in this work is numerical, involving a larger computational cost ofthe IRF precalculus. However, it has advantages, since its formulation is simpleand enables an easy introduction of the model of support proposed in Chapter 3.This methodology is exposed in part in Paper A.

4.1 Numerical convolution product

The convolution product between the IRF and the contact force is expressed as

w(xj , t) =∫ t

0h(xi, xj , t− τ)fc(xi, τ)dτ. (4.1)

It provides the vertical response at the point xj and at time t, for a excitation atxi. The impulse response, h(xi, xj , t), is obtained by an inverse Fourier transformof the cross receptance, H(xi, xj , ω), here approached as

h(e(i), e(j), tij

)= 1N

N∑n=0

H(e(i), e(j), ωn

)eiωnt∆ω, (4.2)

where e(i) is the excitation element and e(j) is the response element. The receptancematrix, H(ω), is obtained according to Eq. (3.23). The matrix h

(e(i), e(j), tij

)relates the DOF of the exciting element with those of the element where the response

36 CHAPTER 4. RESOLUTION BASED ON PRECALCULATED IRF

is studied

h(e(i), e(j), tij) =

h11(e(i), e(j), tij) h12(e(i), e(j), tij) · · ·

h21(e(i), e(j), tij) h22(e(i), e(j), tij) · · ·...

.... . .

4×4

. (4.3)

Equation (4.1) is expressed in terms of h and the interpolation functions as

w(xj , t) = N(ξj)Nk∑k=0

h(e

(i)k , e(j), tk

)NT(ξk)fk∆t, (4.4)

where Nk is the number of previous impulses and ∆t is the time discretisation. Fora constant speed, other level of precalculus can be introduced by using the movingIRF, hm. It is formulated as

hm(xi, t, d0) = N(ξj , d0)h(e(i)(xi), e(j)(xi, t, d0), t)N(ξi), (4.5)

where d0 is the distance at t = 0 between the exciting and response points. In thiscase, the rail displacement reads

w(xj , t, d0) =Nk∑k=1

hm (xk, tk, d0) fk∆t. (4.6)

Figure 4.1 shows IRF for a static case, and for two different speeds moving cases,where d0 = 0.

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Time (s)

-5

0

5

10

15

20

Res

pons

e (m

/N)

×10-7

Static30 m/s60 m/s

(a) Excitation above a sleeper.

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Time (s)

-1

-0.5

0

0.5

1

1.5

2

×10-6

Static30 m/s60 m/s

(b) Excitation at the mid-span.

Figure 4.1: Comparison between the static and moving IRF, for 30 and 60m/s.Track parameters from Ref. [34].

4.2. INTEGRATION SCHEME OF THE CONVOLUTION 37

4.2 Integration scheme of the convolution

The contact force is calculated in each iteration through the interpenetration be-tween wheel and rail, δ. A static force, fext, acting over the wheel is introduced.The wheel equilibrium equation reads

−Mwww(t) + fc(t) = −fext. (4.7)

By means of Newmark equations (2.24) and (2.25) the kinetic state of the mass isdescribed. The rail vertical displacement at a certain instant t is obtained by usingthe precalculated IRF as

wr(t) = −Nk−1∑k=1

hm (xk, tk, d0) fk∆t− (hm0 + hloc)f(t)∆t

= wpr − (hm0 + hloc)f(t)∆t, (4.8)

where Nk is the total number of impulses, wpr gathers the contribution of all the

previous impulses resulted from the convolution product, hm0 = hm (x(t), 0, 0) isthe instantaneous moving impulse response and hloc = FSta/∆t is the contributionof the local system, FSta being its flexibility (2.15).

An iterative process is used over the Newmark equations and Eqs. (4.7) and(4.8), since the connection between the wheel and rail is based on a non-linearspring. The number of equations is drastically reduced, from a number equals tothe system DOF to only four equations.

4.3 Results

The validity of the numerical convolution is tested by comparison with the resultsfrom Ref. [34] (Fig. 4.2). The simulations are based on a model composed by 61bays with 6 elements per bay. The impulse response is obtained for a maximum

15 16 17 18

105

100

95

90

Distance (m)

Co

nta

ct f

orc

e (k

N)

Figure 4.2: Results provided by Mazilu [34].

38 CHAPTER 4. RESOLUTION BASED ON PRECALCULATED IRF

6 6.5 7 7.5 8 8.5 9Distance (m)

0.9

0.95

1

1.05C

onta

ct f

orce

fac

tor

(-)

Static approachConvolutionDynamic approach

Figure 4.3: Wheel-rail contact force for a vehicle moving at 24m/s for parameterspresented at Tab. 3.1. Local static approach.

frequency of 2500Hz. The temporal response of the contact force is shown inFig. 4.3, obtained by means of the numerical convolution and the direct numericalintegration for both static and dynamic approaches. Not including the precalculus,the numerical convolution is found to be much faster than the direct numericalintegration. It is one and two orders of magnitude faster than resolutions based ondynamic and static approaches of the local system, respectively.

5Conclusion and future work

5.1 Summary

In order to perform predictions of wheel-rail noise generation and corrugationgrowth, an accurate modelling of the track is necessary. In this connection, thisthesis has developed two new modelling techniques to enhance the prediction oftrack dynamics. Firstly, a finite element methodology for the modelling of the rail,and secondly, a distributed model of support to substitute the common modelsbased on concentrated connections.

The rail modelling was tackled by means of Timoshenko finite elements. Thischoice was motivated by two reasons, the versatility of finite element methods,and the capacity of Timoshenko theory to predict the rail behaviour within thefrequency range of interest. However, the conventional modelling with Timoshenkoelements involves problems of continuity and accuracy. In this regard, a descriptionof the local deformation of the Timoshenko finite element can improve significantlythe temporal response. Two approaches have been formulated to describe the localdeformation, one static and another dynamic. The static approach provides enoughaccuracy to the model, but the discontinuities remain. On the other hand, thedynamic approach has achieved accurate results and an almost total eliminationof the continuity problems. Moreover, the computational cost is reduced since itmakes use of a reduced number of elements per bay.

Proceeding with the development of techniques for the medium-high frequencytrack modelling, the description of supports has been studied. Most track modelsintroduce periodic supports as concentrated connections between the rail and sleep-ers. However, it is known to result in misdirection of the frequency response, and asudden disrupt of the contact force temporal response. To address these problems, adistributed support model has been formulated and implemented as a finite elementthat accounts for the rail-pad stiffness. It substitutes the traditional Timoshenkoelement in those rail sections over supports. The support model has been assessedby means of frequency and time domain simulations. It has been concluded thatdistributed support model corrects the frequency response, specially around the‘pin-pin’ frequency, and moreover, it eliminates the discontinuity introduced by theconcentrated models in time domain simulations. Furthermore, the methodology iscompact and computationally efficient.

Finally, a very efficient solving strategy has been formulated for time domainsimulations. It is based on the convolution product between the contact force and

40 CHAPTER 5. CONCLUSION AND FUTURE WORK

impulse response functions (IRF). Together with it, the static approach of the localsystem is implemented. The IRF are obtained via the receptance matrix of themodel based on Timoshenko elements. In spite of the initial cost associated tothe precalculus of IRF, this methodology has proved to be very computationallyefficient.

5.2 Future work

The future lines of research of this PhD project will be targeted to:

• Formulating the dynamic approach of the local system for the TEEF. Inthis way, the track model based on distributed supports will be enhanced byeliminating the shear incompatibilities between elements.

• Extending the numerical convolution solving strategy to both the dynamiclocal system approach and the distributed support model. Therefore, contin-uous and accurate results can be obtained through fast temporal simulations.

• Assessing the impact of non-linear rail-pad behaviour on the system response.

• Introduction of the developed track model into a global noise prediction tool,which involves medium-high frequency models of both wheel-set dynamicsand wheel-rail contact interaction.

• Evaluation of the acoustic power generated by railway systems under differenttrack modelling conditions.

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Part II

APPENDED PAPERS

45