Railway track dynamic modelling
BLAS BLANCO
Licentiate Thesis
Stockholm, Sweden 2017
TRITA-AVE 2017:34 ISSN 1651–7660
ISBN 978–91–7729–440–5
Postal address:
Royal Institute of Technology Farkost och Flyg SE–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 Engineering Mechanics, 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 reduced impact 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-products of trains operation, the most important being railway noise. Track dynamic plays a main 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 can be dominant in many operational situations. This explains why prediction tools are highly valued by railway companies.
The work presented in this licentiate thesis proposes methodologies for accurate and efficient modelling of railway track dynamics. Two core axes have led the development 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 major premises. On one hand, regarding the frequency domain, it should describe high frequency behaviour of the rail. In order to accomplish with this first premise, a model based on Timoshenko beam theory is used, which can accurately account for the vertical rail behaviour up to 2500 Hz. On the other hand, with respect to the time domain, the response should be smooth and free of discontinuities. This last condition is fulfilled by implementation of the Timoshenko local deformation.
Secondly, a model of support that considers its finite length nature is sought. For this purpose, a Timoshenko element over elastic foundation is formulated. Thus, the common model of support, which is based on a concentrated connection, is substituted by a distributed model of support. In this way, several enhancements are achieved; the temporal contact force response is smoothed and a more realistic shape 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 not overestimated.
Keywords: Track modelling, Timoshenko element, local deformation, response
Sammanfattning
Spårburna fordon är ett ökande transportslag i dagens samhälle. Skälen är främst en begränsad miljöpåverkan men även den höga komforten har bidragit till att ge järnvägstrafiken en positiv bild i dagens Europa. Denna uppåtgående trend in-nebär att järnvägsindustrin måste handskas med ett antal viktiga utmaningar. Dels upprätthålla låga driftkostnader men också kunna kontrollera orsakerna till buller-störningar, där spårburet buller är den viktigaste faktorn. Spårdynamiken spelar en avgörande roll i båda dessa frågor, eftersom en betydande del av driftskostnaderna orsakas av spårunderhåll samt att bullret som genereras av rälen många gånger är dominerande. Detta förklarar varför beräkningsverktyg är högt värderade av järnvägsföretag.
Arbetet som presenteras i denna licentiatuppsats föreslår metoder för exakt och effektiv modellering av järnvägsspårets dynamik. Två huvudspår har lett till utvecklingen av denna uppgift; modellering av rälen och karaktärisering av sliprar-nas dynamik. I det första huvudspåret av modellering av räler, har utvecklingen skett under två övergripande villkor; dels att i frekvensdomänen kunna beskriva det högfrekventa beteendet hos rälen. För att uppnå det första villkoret har en modell baserad på Timoshenkos balkteori tillämpats. Denna kan beskriva den vertikala responsen i rälen upp till ungetär 2500 Hz. Det andra villkoret är att impulssvaret, i tidsdomänen, ska vara jämnt och fritt från diskontinuiteter, vilket uppfylls genom implementering 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 realistiskt svar 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, respons
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 have helped 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, for the opportunity to perform this work in collaboration with the Marcus Wallenberg Laboratory (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 and contributions have been made to improve it.
I am specially grateful to José Germán Giménez without whose knowledge and commitment, this work would not have been possible.
I thank to all my fellows in CEIT and KTH, for the good moments we have spent together; the cafés, fikas, sagardotegiak and Thursdays of innebandy. Among all the people who have accompanied me along this time, I would like to thank to Luck who during each visit to Stockholm has thrown me a line, and to my office mates in CEIT; Chun, Esther and Itziar, by creating a friendly and nice place to work. And specially, many thanks to my colleague and friend Miguel, who has shared with me, since the very beginning of this journey, his passion for life.
Finally, I wish to thank particularly my family whose unconditional support and trust have given me the strength to carry on.
Dissertation
This thesis consists of two parts. The first part gives an overview of the research area 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. Implementation
of Timoshenko element local deflection for vertical track modelling. Submitted to
Journal of Sound and Vibration.
J.G. Giménez gave the keys ideas to develop this work. B. Blanco developed and implemented the model, and wrote the article. A. Alonso followed the work and guided during its progress. J.G. Giménez, A. Alonso, L. Kari and N. Gil-Negrete discussed 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 System
Dynamics.
Alonso, A. and J.G. Giménez inspired the core idea for this paper and followed the work progress. B. Blanco developed and implemented the method and wrote the article. L. Kari and N. Gil-Negrete supervised the work, discussed the ideas and reviewed the paper.
Contents
I
OVERVIEW
3
1 Introduction 5
1.1 Background . . . 5
1.2 Medium-high frequency phenomena . . . 5
1.2.1 Wheel-Rail noise . . . 5
1.2.2 Rail corrugation . . . 7
1.3 Track structure . . . 7
1.3.1 Components . . . 8
1.3.2 Vertical dynamic behaviour . . . 9
1.4 Track modelling techniques . . . 10
1.5 Aim of the thesis . . . 11
1.6 Organization of the thesis . . . 12
2 Rail modelling 13 2.1 Description of the track model . . . 13
2.2 Description of the local system . . . 15
2.2.1 Static approach of the local system . . . 16
2.2.2 Dynamic approach of the local system . . . 16
2.3 Direct numerical integration . . . 20
2.3.1 Equations of motion . . . 20
2.3.2 Integration scheme . . . 21
2.4 Temporal response . . . 22
2.5 Conclusion . . . 24
3 Distributed support model of rail-pad 25 3.1 Support modelling methodologies . . . 25
3.2 Distributed support model . . . 26
3.2.1 Timoshenko beam over elastic foundation . . . 26
3.2.2 Interaction between sleeper and rail . . . 28
3.2.3 Matrices of TEEF . . . 29
3.3 Frequency domain simulations . . . 30
3.4 Time domain simulations . . . 31
CONTENTS 1
4 Resolution based on precalculated IRF 35
4.1 Numerical convolution product . . . 35 4.2 Integration scheme of the convolution . . . 37 4.3 Results . . . 37
5 Conclusion and future work 39
5.1 Summary . . . 39 5.2 Future work . . . 40
Bibliography 41
Part I
OVERVIEW
1
Introduction
1.1
Background
Railway dynamic simulations tackle the study and prediction of the movement of railway vehicles. For these systems the main source of excitation has its origin in wheel-rail contact forces. The importance of track modelling in the calculation of contact forces depends on the frequency range of interest. For low-frequency stud-ies, such as safety, comfort and curve negotiation simulations, the track modelling accepts coarse simplifications. However, the track requires a description in a high level of detail while addressing medium-high phenomena.
The medium-high frequency interaction between the railway vehicle and track is 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 and inadequate rest, and the ambition of operator and manufacturers to improve the durability and maintenance of wheels and tracks. This situation has led the effort of researchers to achieve a better understanding of the medium-high frequency inter-action phenomena. Accordingly, the track models used in these sorts of simulations become 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 can be considered infinite in its longitudinal dimension, dissipative properties period-ically distributed, sources of excitation with a moving nature and the presence of components with a non-linear behaviour.
1.2
Medium-high frequency phenomena
In order to understand the importance of modelling the medium-high frequency behaviour of railway systems, the most common undesired by-products of wheel-rail interaction are presented in this section. These are the wheel-wheel-rail noise and the rail corrugation.
1.2.1
Wheel-Rail noise
The railway traffic is associated to three main kinds of noise: the engine noise, the wheel-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 and aerodynamic noise are dominant, respectively. The wheel-rail interaction is the main mechanism of noise generation for most operational conditions in railway lines, therefore it is the most widely studied.
The wheel-rail noise is caused by contact forces, which excite the track and vehicle, leading to structural vibrations that are propagated to the environment in the forms of acoustic noise and ground vibration. The most important sources of wheel-rail noise are sleepers, rails and wheels (Fig. 1.1 (left)). As it is shown in Fig. 1.1 (right), the dominant source of noise highly depends on the excitation frequency. Below 500 Hz noise mainly comes from sleepers, between 500 to 2000 Hz rails play the main role and above 2000 Hz the main source are the wheels. It is assumed that track dynamics control wheel-rail noise below 2000 Hz.
Depending on the track geometry different types of noise are identified. On tangent track, structural noise with a broad frequency content occurs, known as
rolling noise. It is due to the excitation caused by the irregularities of both wheel and rail surfaces. In curves, very high noise level can be produced, which is often 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 is
due to wheel-track singularities, such as wheel-flats or rail joints, known as impact
noise.
Nowadays, the railway noise emission is subjected to regulations established by the European Commission. It is a response to the public concern about this issue. According to European Environment Agency (EEA) more than 14 million people were 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 contribution depending on the excitation frequency (right) [2].
1.3. TRACK STRUCTURE 7
European countries have done their own studies, for instance, in Sweden, the social cost 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 0.03 to 0.3 m (Fig. 1.2). According to Ref. [5], 40% of tracks are prone to develop corrugation, due to they are subjected to traffic conditions that normally do not undergo important variations. It results in the predominance of some exciting frequencies, leading to a fixed pattern deterioration. The corrugation affects to the wheel and rail lifetime, it highly increases the level of generated noise and in some circumstances, 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 also present in tangent tracks. Classification of this phenomenon is not an easy task because the mechanisms causing corrugation are diverse and they are not completely understood. The classification proposed by Ref. [6] is presented in Tab. 1.1. The P2 resonance is associated to the resonance of the vehicle as an unsprung mass on the 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. At first, a description is presented of the different track components that are subject of modelling. This description is followed by an explanation of the vibration modes associated 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 - 100
Light rail P2 resonance 50 - 100
Other P2 resonance P2 resonance + torsional modes 50 - 100
Trackform specific Sleeper resonance 50 - 100
Rutting Second torsion mode of axle 250 - 400
Roaring 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. The track 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 the train wheels, allowing a quiet running and a low level of vibrations. Vertical load is the main force acting over a rail. Therefore, rails are derived from an I-profile, to provide bearing capacity in that direction. The most common rail on main lines in Europe is the UIC60 rail.
• Rail-pads and fastening: Rail-pads protect sleepers from wear and impact damage. They are placed between the rail and sleepers, providing of an elastic connection between both. The fastening is introduced by clips. Rail-pads increase the track flexibility and they introduce part of the track dissipative properties. Soft rail-pads permit larger deflection of the rail and a better distribution of the load at the sleepers. Moreover, the softer the rail-pad is the better is the isolation of sleepers and ballast from high-frequency vibrations. The rail-pad is composed by rubber material with non-linear behaviour that is known to be influenced by the preload.
• Sleepers: They are the supports of the rail, transmitting forces from the rail down to the ballast bed. Their other task is to maintain the characteristic dimensions between both rails; being the gauge, level, and the alignment of the track. Most typical sleepers are those formed by monobloc of concrete, although, bi-bloc configuration is also used. Regarding the material, they can be made not only with concrete, but also wood or mixed; using concrete for the outer and steel for the central parts. Normally, a constant distance between sleepers of 0.5 − 0.65 m is used.
• Ballast: It is typically made of course stone, in such a way that sleepers can 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. Its modelling is complex, due to its granular nature and the characterisation of its properties.
It also worth noting the existence of ballastless tracks. In this sort of tracks ballast is substituted by concrete slabs. It has a higher installation cost, but in contrast, it provides a great accuracy and longevity, which has a direct impact over the maintenance cost. This kind of track is a particularly appropriate choice for placements where the maintenance tasks are obstructed, e.g. tunnels, bridges and stations. Several configurations exist for ballastless tracks, some of them neither use sleepers while connecting directly the rail to the slab with one or two intermediate pads. If the sleepers are not removed they can be embedded on the slab, without any intermediate element or with resilient pads between the slab and sleepers.
1.3.2
Vertical dynamic behaviour
Following, a general description of the vertical dynamic behaviour of the track is introduced in order to understand the role played by each component. The presence of these components gives rise to several resonant frequencies, which are presented in Fig. 1.3. It shows the frequency response of a periodically supported track due to 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 like the mass and ballast acts both as the spring and as a mass. Therefore, rail and sleepers are moving like a mass elastically connected to the ballast. This resonance normally takes place between 50 to 300 Hz.
2. Sleepers and rail vibrating out of phase: In this resonance the rail and sleepers act like independent masses connected by the railpads, which act as springs. Thus, the vibrating masses move in opposite directions. This resonance is found between approximately 200 to 600 Hz, mainly depending on the railpad stiffness.
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 wavelength of the rail bending waves is twice the sleeper spacing, which generally is around 1000 Hz. This vibration is poorly damped; since, the displacement is located in the rail, and its associated damping is mainly due to its own internal damping. The response to an excitation at this frequency strongly depends on exciting position. It is maximum at the mid of a span, and minimum above 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 irreguirregu-larities encompasses a broad range, from 0.03 to even 2 m and, in turn, train operational speeds can cover from 10 to 300 km/h. Therefore, the frequency range in which a track is excited is very wide, mainly from 1 to 2800 Hz. Other source of excitation is caused by changes of the 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 a point at the mid-span and another above the sleeper. It arises the so-called sleeper passing frequency, f = V /l, where V is the vehicle speed and l is the distance between sleepers. Other examples of varying flexibility are, transitions between ballast 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 first distinction is made between frequency and time domain modelling. The frequency domain modelling is normally addressed by analytical methodologies [9–12], because they 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 rotatory inertia of the beam, therefore it is accurate for studies not exceeding a frequency around 500 Hz. On the contrary, Timoshenko beam theory takes into account both shear strain and rotatory inertia, and it is able to describe accurately vertical and lateral dynamics up to approximately 2500 and 1500 Hz, respectively. Above these frequencies, Timoshenko theory is not accurate since the rail cross-section can not be 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 nature of the load and periodic foundation [23, 24] for the frequency domain. In spite of their low computational cost, analytical methods require assumptions which is not met 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 study of the medium-high frequency range. Therefore, the model can be used to perform noise emission prediction and corrugation assessment in future works. The design of this model is performed according to following requirements:
• The model should be valid for the range of frequencies where noise emission from the rail is dominant. According to Ref. [21], beam models can predict successfully noise emission since, rail cross-sectional deformation takes places in 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 enough accurate 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, in spite of using implicit integration and small time-steps. It is due to contact systems are very stiff mathematical problems [30].
• Finally, the computational cost of the resolution scheme should be kept as low 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, in which the accuracy and continuity of temporal response are enhanced by imple-mentation of the local deformation for the Timoshenko beam element. Chapter 3 is focused on the development of a new track support model based on a distributed foundation, in order to substitute the traditional support modelling that has some inherent 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 research are presented.
2
Rail modelling
This chapter addresses the rail modelling by using Timoshenko elements. The temporal response is enhanced with the implementation of the local system defor-mation. At first, the decomposition of the whole system into the global and local systems is presented. Once the problematic is exposed, the equations defining the local system are developed for both static and dynamic approaches. The resolution methodology based on a Newmark integrator scheme is subsequently shown. Fi-nally, the results are analysed. This chapter refers to the work presented in Paper A.
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 each node. These nodal displacements are normally referred as, u(e) = [w
1θ1 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 of the area, G is the shear modulus, x is the longitudinal coordinate, M is the bending moment, 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) = 1 1 + Φ1 − 3ξ 2+ 2ξ3+ Φ(1 − ξ) , (2.3) Nθ1(ξ) = ξ 1 + Φ L(1 − 2ξ + ξ2) +Φ 2(1 − ξ)L , (2.4) Nw2(ξ) = 1 1 + Φ3ξ 2− 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, which are based on the Hermite cubic polynomials. Additionally, the Timoshenko interpo-lation functions consider the influence of shear by means of Φ, which characterises the ratio of bending and shear stiffness. If the beam is considered infinitely stiff to shear strength, Φ becomes zero, and the interpolation functions resemble those of the Euler-Bernoulli element. The mass and stiffness matrices are derived from the interpolation functions, and they can be found in Ref. [31]. The model of track introduces periodic supports as concentrated connections periodically separated by a 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 attached to the vehicle by a non-linear Hertzian spring, Kc. The notation for the system
parameters is presented in Fig. 2.1.
The equations of motion are stated by assembling the Timoshenko elementary matrices together with the supports. Time domain simulations are performed by iteratively solving the equation system with a numerical integrator. However, if the surfaces are considered perfectly smooth or with small irregularities, the response at the contact position shows an irregular behaviour. It is presented as sharp discontinuities, each time the load overtakes a node. This non-physical response has its origin in two reasons. Firstly, underestimation of shear strain at the load positions. The formulation of TIM4 interpolation functions, assumes vanishing shear strain at the medium point between the element nodes. Therefore, when a load 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 load position, albeit it is at its maximum. Secondly, the TIM4 element does not impose continuity of the vertical deflection first derivative at their nodes, creating a shear incompatibility 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 by using TIM7 elements. This model forces to redefinition of the system equation at each 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 contact stiffness. In spite of obtaining a significant enhancement, the shear incompatibility is not corrected and a large number of TIM4 elements is required to reduce it.
Implementation of the local element Timoshenko deformation may diminish the discontinuity to a negligible level. Furthermore, it also reduces the computation costs by reducing the required number of elements. The core idea for this de-velopment is widely known in the structure analysis field. The displacement of a structure subjected to loads acting on positions between nodes can be studied by superposition of two systems. The first system is formed by the loads acting on a structure derived from imposing fixed ends at the element nodes, hereafter called local system. Once it is solved, reactions at the fixed ends are introduced as 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. This solution is exact only for static problems in which the analysis of the local system is restricted to the element in which the load acts. For dynamic problems, the local system is excited by external loads and the inertia of the global system, thus it requires to be described in elements adjacent to the loaded one. It gives rises to two approaches of the local system, one static and another dynamic. In Figure 2.1 the 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 fc fc Kc Mw V fext Dynamic approach Static approach l Q Q2 1 M1 M 2 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 Timoshenko governing beam equations. At first, the static approach is stated, and secondly, the dynamic approach is developed, which uses the static approach and adds new equations to the system.
16 CHAPTER 2. RAIL MODELLING
2.2.1
Static approach of the local system
The local system described statically is only represented by the displacement field of a Timoshenko beam with clamped ends loaded by a point force, fa. The expressions that describe the vertical deflection for this problem are
wA(x, xa) = faL3 6EI ( x L L − xa L x2 L2 − Φ 2 + 3x 2 a L2− x3 a L3 − 2 xa L +xa(L − xa) 2L3(1 + Φ) " (2(L − xa) + LΦ) x 3 L3 − 3 x2 L2 + 2 x L + (2xa+ LΦ) x L− x3 L3 #) , (2.7) wB(x, xa) = faL3 6EI ( xa L Φ 2 x − L L + 3x 2 L2 − x3+ xx2 a L3 − 2 x L + x2 a L2 +xa(L − xa) 2L3(1 + Φ) " (2(L − xa) + LΦ) x3 L3 − 3 x2 L2 + 2 x L + (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, where
wA is on the left side and wB is on the right side. Assessment of this approach is
done in Fig. 2.2. It shows the vertical displacement field of a periodically supported rail obtained by three different models. Two of them do not introduce the local system, and they represent the rail with two and four elements per bay. The load is introduced at xa= 0.25l. Comparing both solutions, it is observed how the model with half elements underestimates the displacement at the load position. The third case uses the global solution of the two elements per bay model, and superposes the static local system. For this case, the displacement field agrees with that from the four element per bay model. In conclusion, the static approach of the local system represents accurately the real shear strain of the Timoshenko beam at the load position.
2.2.2
Dynamic approach of the local system
This approach addresses the problem from a different perspective, in which the dynamic behaviour of the local system is described. It is stated that the smaller the finite beam element is, the higher are its natural frequencies; thus, leading to 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 results while increasing the computational costs.
2.2. DESCRIPTION OF THE LOCAL SYSTEM 17 7 8 9 10 11 12 13 Position (m) -4 -3 -2 -1 0 Vertical displacement (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
a
f
f
a
Figure 2.2: Contribution of the local system to the static deformation of the global system. 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 the vibration modes corresponding to a Timoshenko beam with clamped ends. The mathematical process starts from the coupled differential equations for transverse free 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 displacement respond to the next expressions
W (x) = P1cos(λ1x) + P2sin(λ1x) + P3cosh(λ2x) + P4sinh(λ2x), (2.11)
and for ω > ω0
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 Normalised displacement (-)
(a) Vertical vibration mode shapes.
0 0.05 0.1 0.15 Position (m) -2 0 2 4 6 8 10
Normalised bending rotation (-)
(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) 6,196 Hz (·, green) 12,284 Hz (-·-, red) 18,160 Hz (- -, blue) 18,645 Hz. [Paper A Fig. 3]
where, ω0 = pκAG/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 which annul det(Λ), where Λ represents the system of equations derived from imposing the clamped ends boundary conditions. Once the natural frequencies has been obtained, the vibration mode can be fully defined by solving ΛP = 0, where P gathers the coefficients of (2.11) and (2.12). The vibration modes are normalized by imposing
s Z L
0
(ρAW2(x) + ρIΘ2(x)) dx = 1. (2.13)
Figure 2.3 presents the four vibration mode shapes of a 0.15 m Timoshenko beam with the geometric characteristics of the UIC60 rail profile. In Paper A, the dynamic approach 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 by using a residual flexibility, defined as
FRes(x) = FSta(x) − FDyn(x) = FSta(x) −
Nm X i=1 W2 i(x) ω2 i , (2.14)
where FStaand FDin, are the flexibilities of the local static and dynamic approaches,
2.2. DESCRIPTION OF THE LOCAL SYSTEM 19
The definition of FStacan 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 are displayed in Fig. 2.4. 0 0.05 0.1 0.15 Position (m) 0 5 10 15 Flexibility (m/N) ×10-11 1 vibration mode 3 vibration modes 6 vibration modes 20 vibration modes Static
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 description of the local system in those elements adjacent to the loaded one. This is due to an interaction between the local and global systems throughout mass-cross terms. These terms arise from the non-orthogonality between the modal shapes of local modes and the TIM4 interpolation functions. The inclusion of this interaction leads to the shear strain compatibility between elements, hence to a large reduction of the non-physical slope discontinuity. The mass-cross term between the i-th natural frequency and a nodal displacement j is defined as
mcrossij = Z L ρAWi(x)Nvj(x)dx + Z L ρIΘi(x)Nbj(x)dx, (2.16)
where Nvj and Nbj are the vertical and rotational interpolation functions of the
20 CHAPTER 2. RAIL MODELLING
(2.3) to (2.6), and from them, Nbj is defined as
Nbj(x) = NvjI (x) +
ΦL2
12 N
III
vj(x). (2.17)
2.3
Direct numerical integration
The time domain simulations are performed by direct numerical integration of the equations of motion. At first, the equations of motion are stated and secondly the integration process is described.
2.3.1
Equations of motion
The equations of motion are represented in a general form as
M¨u + C ˙u + Ku = fc+ fext, (2.18)
where M, C and K are the mass, damping and stiffness matrices, respectively, fc
and fextare the contact and external force vectors, respectively, and u is the system
DOF displacements. The interaction between rail and wheel is described in terms of the interpenetration, δ = wr+ wirr− ww, where wr and ww, are the vertical
displacements of rail and wheel at the contact point, respectively, and wirr is the
surfaces irregularities. The contact force reads
fc= Kcδn+1+ Cc˙δ δ > 0, (2.19)
and fc = 0 for δ ≤ 0, where, Kc is the Hertzian contact coefficient and Cc is the
contact damping. The parameter n becomes 0 or 1/2 depending on whether the contact is taken as linear or non-linear, respectively. For the case studied here the influence of the contact damping is considered negligible. The forces due to the contact interaction can be expressed in matrix form as
fc= − −1 N Ψ fc= − −1 N Ψ Kcδn+1= − −1 N Ψ Kcδn(wG.r+ wLoc.r+ wirr− ww) = − −1 N Ψ Kcδn −1 N Ψ 1 T ww uG uL wirr = Kcδn −1 NT Ψ 1 N −NNT −NΨT −N Ψ −ΨNT −ΨΨT −Ψ ww uG uL wirr = Kcδn −1 NT Ψ N −NNT −NΨT Ψ −ΨNT −ΨΨT ww uG uL + Kcδn 1 −N −Ψ wirr= −KcδnEu + firr. (2.20)
2.3. DIRECT NUMERICAL INTEGRATION 21
The subscripts G and L in wrand u refer to the global and local systems,
respec-tively. The vector N is used to extrapolate the nodal forces of the loaded element from 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 the residual or static flexibility. Introducing (2.20) into (2.18) the equation of motion becomes
M¨u + C ˙u + (K + KcδnE) u = fext+ firr. (2.21)
The global system equations are assembled straightforwardly with the mass and stiffness TIM4 matrices representing the rail. Together with rail global matrices, the periodic concentrated connections are represented by the stiffness and damping terms of ballast and rail-pad, and by the sleeper mass. With respect to the local system, for the case of the static approach it is described as
Klocwloc= fc, (2.22)
where Kloc= FSta−1. 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+ 2ζωiη˙ij+ ωi2ηij+ Mcrossi 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 Mcrossi gathers the mass cross terms for the nodal displacements at a certain vibration mode.
2.3.2
Integration scheme
The 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]. The Newmark equations are defined as
uk+1= uk+ ∆t ˙uk+ ∆t2
2 [(1 − 2β) ¨uk+ 2β ¨uk+1] , (2.24) ˙
uk+1= ˙uk+ ∆t [(1 − γ) ¨uk+ γ ¨uk+1] , (2.25) which establish a linear relation between the dynamic state of the system at the known instant tk and at the unknown instant tk+1, weighted by the Newmark parameters, γ and β. After substituting (2.24) and (2.25) into (2.21), the equations of motion become
z (¨uk+1) = A¨uk+1+ B1u¨k+ B2u˙k+ B3uk− fext− firr= 0, (2.26)
equalling the number of equations and unknowns. This development is not able to 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 =hA0(1,m)×(1,m−1) A0(1,m)×m/∆t2βi, where
A0 = M + ∆tγC + ∆t2β (K + KcδnE) , (2.27)
and m 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 the instant tk its state is null. The Bi matrices, which weight the influence of the acceleration, 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 iterative process
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 second order terms of the interpenetration arise. These terms are exposed in Paper A, which permit analytic calculation of the Jacobian matrix. Furthermore, the use of
A enables for fast convergence of the equations, with more iterations, but instead
avoiding 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. The track parameters are those used by Ref. [17] (Tab. 2.1), which previously has tackled the same problem. The results are presented in Fig. 2.5.
Figure 2.5a shows the temporal response of the contact force when only the global system is introduced and obtained by implementation of the static approach of the local system. It is observed how a significant enhancement is reached with the static approach, which represents accurately the shear strain at the contact position. However, the discontinuity is still noticeable each time the load overtakes a node. As it has been previously explained, the cause of this slope discontinuity is due to at the lack of a continuity for the shear deformation. Fig. 2.5b shows the results obtained by the dynamic approach, together with those results provided by the static approach of the local system. Once the dynamic approach is introduced the discontinuities become insignificant.
The convergence of the solution for a low number of elements per bay is tested in Fig. 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 11.2 0.8 0.9 1 1.1 1.2
Contact force factor
Distance (m)
Static approach
Conventional Timoshenko element
Sleeper positions
(a) Local static approach and conventional model 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 approach Dynamic approach Sleeper positions
(b) Local dynamic and static approaches. Dynamic: 4 elements per bay, 5 local 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 for the 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 71.7
Moment of inertia (cm4) I 2350 Mass density (kg/m) ρ 56
Pad damping (kNs/m) Cp 21.8 Sleeper spacing (m) l 0.79
Pad stiffness (MN/m) Kp 200 Shear coefficient κ 0.34
Ballast damping (kNs/m) Cb 21.8 Wheel mass (kg) Mw 500
Ballast stiffness (MN/m) Kb 31.6 Sleeper mass (kg) Ms 50
Contact coeff. (GN/m3/2) K
c 100 External force (kN) fext 82
the result agrees with that from eight elements per bay, while the discontinuity remains in a low level. Therefore, the requirements of the track model regarding the discretisation are significantly reduced when the dynamic approach is used, as compared to those from the static approach. This fact brings another important advantage, since accurate and continuous results can be obtained with a lower computational cost.
The results presented in Fig. 2.6 for two elements per bay are obtained describ-ing the local system at 5 elements and usdescrib-ing 6 modes of vibration for each local system. The influence of the dynamic local system configuration is studied in Paper A 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
Contact force factor
2 Elements per bay 8 Elements per bay
Figure 2.6: Convergence of the results obtained with the dynamic approach for a different number of elements per bay. [Paper A Fig. 7a]
2.5
Conclusion
The methodology based on the implementation of the local system has successfully addressed the continuity problems of the temporal simulations with Timoshenko elements. The formulation has been derived in an analytic way for both static and dynamic approaches of the local system. For the static approach it is found that, in spite of correcting the shear strain underestimation, this approach forces to use a high number of elements to reduce shear incompatibilities between adjacent elements. On the other hand, the dynamic approach corrects shear underestima-tion and reduces the shear incompatibility by using a low number of element, and without resorting to finite elements with a higher number of DOF (TIM7). This, to-gether with its accuracy and computational efficiency, make the dynamic approach of the local system an effective solution for time-domain simulations.
3
Distributed support model of rail-pad
As it is exposed in Section 1.3.2, the presence of vibration modes introduced by periodic foundation makes necessary its modelling when medium-high frequency range is of interest. In spite of periodic foundation track modelling is a generalised trend currently, some models do not take into account the support length in any extent, 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-pad support, which accounts in a realistic way for the support length. It is presented and validated via frequency and time domain simulations. This chapter refers to the work presented in Paper B.
3.1
Support modelling methodologies
Most railway track models account for the interaction between sleepers and rail by means of Maxwell-Voigt connections. In this way, the support is modelled as a concentrated force acting on the rail. It neglects the finite length nature of the 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 on the frequency response is proved by distributing the rail-pad stiffness along those rail sections over the supports.
An attempt to describe the finite length nature in concentrated supports is based on the inclusion of a rotatory interaction between rail and sleepers [24, 34–36]. A more realistic approach of the finite support length is proposed in Ref. [29]. In that model, multiple concentrated connections are introduced per each support by dividing 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 it is characterised throughout the distributed stiffness of the pad, kd= Kp/Ls. The
support is modelled with a Timoshenko element over elastic foundation, hereafter denoted as TEEF. Figure 3.1 presents the schemes of the supports models that are the subjects of study in this chapter.
26 CHAPTER 3. DISTRIBUTED SUPPORT MODEL OF RAIL-PAD
Ferrara support Distributed support
Concentrated 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, the influence of the elastic foundation over the Timoshenko beam displacement field is implemented analytically into the stiffness and mass matrix. Secondly, the vertical movement 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 support interaction stiffness matrix S(e)k , calculated by using the virtual work principle as
IVW + δu(e)Tkd
Z
L
N(e)EFTN(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. In Ref. [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 interpolation functions (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 the Timoshenko beam over an elastic function. Moreover, the deductions of N(e)EFand
Ns 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 those of 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 mathematical transformations, the differential governing equation can be expressed only in terms of the vertical deflection and its derivatives
wIV− kd
κAGw
II+ 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 κAGw III. (3.4)
The detailed process followed to obtain these expressions can be found in Paper B. The solution for differential equation (3.3) has vanishing particular solution and the homogeneous solution reads
w(x) = A1er1x+ A2er2x+ A3er3x+ A4er4x, (3.5)
where riare 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 interpolation functions. 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 by solving 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) C1 C2 C3 C4 = 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 ⇒ N EF w1, (3.11) w(0) = 0 ; θ(0) = 1 ; w(L) = 0 ; θ(L) = 0 ⇒ Ciθ1 ⇒ N EF θ1 , (3.12) w(0) = 0 ; θ(0) = 0 ; w(L) = 1 ; θ(L) = 0 ⇒ Ciw2 ⇒ N EF w2, (3.13) w(0) = 0 ; θ(0) = 0 ; w(L) = 0 ; θ(L) = 1 ⇒ Ciθ2 ⇒ N EF θ2 . (3.14)
28 CHAPTER 3. DISTRIBUTED SUPPORT MODEL OF RAIL-PAD
Figure 3.2 shows the level of disagreement between TIM4 interpolation functions and NEFas the distributed stiffness increases.
0 0.2 0.4 0.6 0.8 1 x/L 0 1 2 3 4 5 (N 1 − N E F 1 )× 10 0
(a) Difference between NEF
1 and N1. 0 0.2 0.4 0.6 0.8 1 x/L 0 2 4 6 8 (N 2 − N E F 2 )N m a x 2 × 10 0 (b) Difference between NEF 2 and N2.
Figure 3.2: Difference between TEEF and TIM4 interpolation functions for dif-ferent distributed stiffness. The beam length is 0.5 m. 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 UIC60
rail. [Paper B Fig. 3]
3.2.2
Interaction between sleeper and rail
The rail displacement field is expressed in terms of the four nodal displacements and the sleeper vertical movement, ws, since the distributed elastic foundation leads
to 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 NwEF1 N EF θ1 N EF w2 N EF θ2 Ns T
= u(e)EFsNTEFs, (3.15)
where Ns is the interpolation function associated to the sleeper. The function Ns
has 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 vanishing boundaries conditions
w(0) = 0 ; θ(0) = 0 ; w(L) = 0 ; θ(L) = 0 ⇒ Ciws ⇒ N EF
ws. (3.16) The influence of the rail-pad stiffness in the interpolation function Nsis presented
3.2. DISTRIBUTED SUPPORT MODEL 29 N s Stiffness (MN/m) ξ (-) 0 1 0.02 0.04 500 0.06 400 0.08 0.5 300 0.1 200 100 0 0
Figure 3.3: Interpolation function of the sleeper displacement, Ns, for a 0.5 m TEEF
with 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 Timoshenko beam theory [37], the TEEF stiffness matrix is defined as
K(e)EF= Z
L
BTEFbEIBEFbdx +
Z
L
BTEFshGAκBEFshdx + S (e)
ks, (3.17)
where the first and second terms of the equation right side are the bending and shear stiffness matrices, respectively. The matrix S(e)ks introduces over S(e)k the sleeper DOF. The vectors B relate displacement and strain as = Bu, which is defined for bending and shear strains as
BEFb= dNEFb dx = " NIIEF 1 − kdEI (κAG)2 ! + EI κAGN IV EF # , (3.18) and BEFsh= NIEF kd κAG− N III 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 Z L NEF w1 NEF θ1 NEF w2 NEF θ2 NEF w1 NEF θ1 NEF w2 NEF θ2 T NEF w1 (Ns− 1) NEF θ1 (Ns− 1) NEF w2 (Ns− 1) NEF θ2 (Ns− 1) Sim. N2 s − 2Ns+ 1 dx = S(e)k Sw1s Sθ1s Sw2s Sθ2s Sim. Sss . (3.20) Finally, the stiffness matrix of the TEEF is stated as
K(e)EFs= K(e)EF Sw1s Sθ1s Sw2s Sθ2s Sim. Sss . (3.21)
Accordingly, the mass matrices of the TEEF is defined as
M(e)EF= ρA Z
L
N(e)EFTN(e)EFdx + ρI Z
L
N(e)EFbTN(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 in frequency domain for three types of support modelling presented in Fig. 3.1, where track parameters gathered in Table 3.1 are taken from Ref. [34]. This model has been taken as reference because it makes use of vertical and rotatory interaction for the support modelling.
It is observed how the support modelling affects to the frequency behaviour of the track from 200 Hz to the ‘pin-pin’ frequency, at which the amplitude difference is 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 respect to the amplitude obtained by the distributed model of support, the concentrated models amplify by a factor of 1.92 and 6.25, for the cases of vertical and rotatory interactions 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 underestimation of this amplitude.
3.4. TIME DOMAIN SIMULATIONS 31 102 103 Frequency (Hz) 10-9 10-8 Amplitude (m/N) Rail CVS ( ×) Rail CVRS (o) Rail DS (∗) Sleeper CVS Sleeper 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 CVS Sleeper DS
(b) Excitation and response above the sleeper.
Figure 3.4: Vertical response in frequency domain of both rail and sleeper for different 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 3055
Cross-sectional area (cm2) A 76.9 Mass density (kg/m3) ρ 7850
Shear coefficient κ 0.4 Sleeper spacing (m) l 0.6
Pad stiffness (MN)/m Kp 350 Pad damping (Ns/m) Cp 48
Ballast stiffness (MN/m) Kb 70 Ballast damping (kNs/m) Cb 47
Wheel mass (kg) Mw 550 Sleeper mass (kg) Ms 160
External force (kN) fext 100 Contact stiffness (GN/m) Kc 1.028
Support length (m) Ls 0.16
3.4
Time domain simulations
Time domain simulations are performed by means of the procedure exposed in Section 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 displacement field 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 the local system for the TEEF is not on the focus of this work.
The steady-state response for smooth surfaces without irregularities is presented in Fig. 3.5. In this case, the only source of excitation is the varying stiffness between adjacent supports, i.e. parametric excitation. The displacement at the contact point (Fig. 3.5a) shows a certain level of stiffening due to the inclusion of the 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 9 Distance (m) -9 -8.8 -8.6 -8.4 -8.2 -8 -7.8 -7.6 -7.4 -7.2 Vertical displacement (m) ×10-4 DS CVRS CVS
(a) Vertical rail displacement steady-state re-sponse at the contact position for a wheel moving at 24 m/s. 6 6.5 7 7.5 8 8.5 9 Distance (m) 0.95 1 1.05 1.1 1.15 Contact force (N) ×105 CVS CVRS DS
(b) Contact force steady-state response for a wheel moving at 24 m/s.
Figure 3.5: Temporal response of the wheel-rail system due to purely parametric excitation at 24 m/s for tracks with different support modellings. DS: Distributed support. 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 8 Distance (m) 0.95 1 1.05 1.1 1.15 Contact force (N) ×105 Smooth surfaces 2 µ m roughness amplitude 4 µ m roughness amplitude
Figure 3.6: Steady-state response for a corrugated rail and the distributed sup-port modelling. Wheel speed: 44.4 m/s. Irregularity wavelength: 4.1 cm. The distributed support position and its expanse are indicated with . [Paper B Fig. 12]
a concentrated rotatory interaction. However, the amplitude of the displacement is only reduced by the distributed support. Regarding the contact force presented in Fig. 3.5b, the contribution of the rotatory interaction is insignificant, while the distributed support has an important impact on the continuity of the results. It is observed how the concentrated connections give rise to an abrupt disruption of the contact forces each time the load overtakes a support. However, the temporal response for the distributed support experiences a much softer transition, only perturbed by the shear incompatibilities between elements, which could be solved by 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 8 Distance (m) 1 1.05 1.1 1.15 Contact force (N) ×105 Smooth surfaces 2 µ m roughness amplitude 4 µ m roughness amplitude
Figure 3.7: Steady-state response for a corrugated rail and the concentrated connec-tion support modelling. Wheel speed: 44.4 m/s. Irregularity wavelength: 4.1 cm. The point support position is indicated withN. [Paper B Fig. 13]
The ‘roaring-rails’ corrugation type is supposed to be caused by the vibration of 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 temporal response for the distributed model and the concentrated model with vertical and rotatory interactions, respectively. Again, the distributed modelling provides a re-sponse absent of the discontinuity caused by the concentrated support, leading to a wavy shape, which is suppose to be in a greater agreement with the nature of the system.
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 analytical point of view leading to the formulation of the Timoshenko element over elastic foundation (TEEF). The methodology has been proved to avoid the deviation of the track frequency response, mainly at the ‘pin-pin’ frequency. It is worth highlighting that ‘pin-pin’ resonance amplitude obtained by using a rotatory interaction, is still misdirected as compared to that provided by the distributed approach. The most significant advantages of the distributed model of support with respect to others are its computational efficiency, since it only uses one element per support, and its capacity to obtain continuous results for purely parametric excitation.
