Contribution of vehicle/track dynamics to the ground vibrations induced by the Brussels tramway

Transcription

1 Contribution of vehicle/track dynamics to the ground vibrations induced by the Brussels tramway G. Kouroussis 1, O. Verlinden 1, C. Conti 1 1 Université de Mons UMONS, Faculty of Engineering, Department of Theoretical Mechanics, Dynamics and Vibrations, Place du Parc, 2 7 Mons (BELGIUM) Georges.Kouroussis@umons.ac.be Abstract In the recent years, the railway network has been considerably developed, especially in Belgium, where it becomes one of the heaviest of Europe. This development is certainly a reliable and long-term solution in terms of pollution and traffic jam, but it unfortunately increases vibratory nuisances in the neighbourhood. The paper focuses on the T2 LRV tram operating in Brussels for which comprehensive measurements have been made in the past in order to analyse and reduce the level of the generated vibrations. A prediction model is derived: the compound vehicle/track/soil system is simulated according to a decoupled approach, considering in a first step the vehicle/track subsystem and, secondly, the response of the soil to the ground forces determined in the first step. Various simulations have been performed in order to assess the effect of the roughness or local unevenness like a rail defect on the soil vibration level. All simulated results show a good agreement with site measurements, when available. 1 Introduction Like the major European capitals, Brussels constitutes a heavily urban territory with a few more than one million of residents. The railway network in the Brussels Capital Region (16 km 2 ) consists not only of the urban tramway network but also of the intercity and international train lines, conveying everyday 13 people working in Brussels and in the surroundings. Although railway transport appears as the most promising solution to traffic congestion, the development of new lines is confronted to the availability of convenient areas and the mistrust of the dwellers likely to be submitted to new nuisances. These difficulties have been largely encountered during the implementation of the new RER network. Presently, tramway represents 2% of urban transport in Brussels. A few years ago, the Belgian urban public transport company (STIB for Société des Transports Intercommunaux de Bruxelles ) progressively replaced the old PCC7 trams by the T2 LRV in order to serve the nineteen municipalities of the Brussels Capital Region. Developed by Bombardier Transport, this multicar tramway is characterized by a low floor design, imposing namely bogies involving independent rotating wheels and motors mounted directly inside the wheels. Some studies have been started from 1994, after having observed important vibratory nuisances in the neighbourhood of this new tram. Two great national projects have been developed, in order to propose solutions so as to alleviate the vibratory level in the surrounding buildings: 3489

2 349 PROCEEDINGS OF ISMA21 INCLUDING USD21 The VLIM project [1] (by the Department of Civil Engineering of KUL) has focused on the soil, studying the design and efficiency of isolating screens. The TRANSDYN project [2] (by the Department of Theoretical Mechanics, Dynamics and Vibrations of UMONS) concentrated on the vehicle by assessing the benefit brought by resilient wheels. Both studies have allowed to characterise the investigated site, in terms of soil and track parameters. Whatever the investigated solution, it is interesting to dispose of a model able to predict, from the design stage of a vehicle or of a track, the efforts transmitted by the vehicle to the track/soil system and the vibration propagation in the surroundings. Such a model could be used either by train constructors or railway operators to test the efficiency of anti-vibration solutions on the vehicle and/or the track respectively. Such a model first requires a good representation of the generation of the vibratory nuisances, which is actually initiated in the wheel/rail contact. A good representation of the wheel-rail contact is then necessary to get a proper representation of the forces transmitted to the track and, subsequently to the ground. Yet, various authors [3, 4] make the assumption of a stationary contact force, neglecting the dynamics of the vehicle. In a more appropriate way, other authors consider the dynamics of the vehicle/track system. Lu et al. [], and Lombaert at al. [6], include the effect of the vertical track irregularity through a non-stationary random excitation. Auersch [7] has studied the vehicle/track dynamics by using a spectral representation of the contact force, determined by two different methods, with the emphasis on influence of the train speed. Garden and Stuit [8] have adopted a modular approach, based on a convolution of the force signal from the vehicle/track model (sleeper contribution) with the impulse response of the soil, performed in the time domain. Concerning the wave propagation in the soil, the most popular method is the boundary element method due namely to its numerical efficiency and its ability to naturally treat infinite domains. On the other hand, it is practically limited to linear formulations and simple geometries. The finite element method (FEM) is by contrast able to model a soil with complex geometries, as far as proper boundary conditions are applied on the domain border. The simulation can also be performed in time domain, which appears well adapted to represent the transient nature of wave propagation and opens the way to nonlinear contributions in any part of the model. Of course, it must be mentioned that the FEM method demands more computational resources but Kouroussis et al. [9, 1] have recently shown that time domain allows to reduce the domain size without loss of accuracy, which makes the FEM method applicable even on usual computers. Another advantage is that FEM software s are nowadays commonly used in industry. The purpose of this paper is to study the vibration generation induced by the T2 tram from the points of view of either the train constructor and the train operator. The vehicle, the track and the soil are all considered, according to the approach recently proposed by Kouroussis et al. A particular attention is brought on the excitation mechanisms which in tramways, come principally from the irregularities of the rail and wheel surfaces. The latter can be local (rail defects, crossings) or distributed (overall unevenness). The dynamic characteristics of the tram and the site are presented. The paper compares the effect of local and distributed irregularities at various speeds and evaluates the efficiency of resilient wheels. The obtained results show a good agreement with available site measurements. 2 Modelling approach In order to be able to reproduce the generation and the propagation of the vibrations induced by railway vehicles, the vehicle, the track and the soil must all be considered. The simulation approach referred to in this paper is based on the assumption that the whole system can be studied in two successive steps: the vehicle/track subsystem is first simulated, yielding the ground forces applied to the soil subsystem in a second step. Both subsystems are simulated in the time domain. This assumes that the vehicle/track subsystem and

3 RAILWAY DYNAMICS AND GROUND VIBRATIONS 3491 the soil are decoupled, which has been demonstrated to be valid [11, 12] as soon as the ballast is significantly more flexible than the foundation. The flow chart in Figure 1 shows the various calculation steps and the connection of the two sub problems. The main concern of the approach is to allow a relatively detailed modelling of the vehicle. d p d b d s m k p k b k s step 1 Dynamic study of the vehicle/track subsystem z x v step 2 Dynamic study of the soil subsystem y z x infinite elements region of interest (finite element modelling) Figure 1: Description of the proposed prediction model, according to a decoupling between the ballast and the soil The vehicle is modelled according to a classical multibody approach: i.e. a combination of rigid bodies and interconnection elements like multidirectional springs and dampers. The track is modelled by means of a planar two layered finite element model consisting of a flexible rail, defined by its Young s modulus E r, its geometrical moment of inertia I r, its section A r and its density ρ r, and materialized by classical beam elements. The track is attached to the ground, at each sleeper, by two successive spring damper systems representing the railpad and the foundation, with an intermediary lumped mass m representing the sleeper. The track interacts with the vehicle through the vertical component of the wheel-rail contact force, determined according to the classical Hertzian theory. Only the vertical motion of the track and the vehicle

4 3492 PROCEEDINGS OF ISMA21 INCLUDING USD21 are considered. The soil is modelled by means of the finite element software ABAQUS where infinite elements are added at the border of the region of interest (FIEM approach). The excitation of the soil is given by the forces applied by the sleepers to the ground, and issued from the simulation of the vehicle track system. For both subsystems, the simulation is performed in time domain. The soil model and the track/soil interaction are finely described in [9, 11]. 3 Track irregularity 3.1 Overall unevenness The distributed irregularity of the track is described by means of its PSD (power spectrum density) S zz, expressed in terms of the spatial frequency φ (number of cycles per unit of length). The corresponding PSD S zz in the frequency domain is easily deduced as where v is the speed of the vehicle. S zz (f) = S zz(φ) v (1) From various representations found in the literature, the functions proposed by Garg et Dukkipati [13] were retained because they present a wide classification (6 classes from good to very bad quality) and are based on the large data collected in the U.S. by the Federal Railway Administration (FRA). As the vertical motion is considered in the simulation and a 2D model is derived for the track, only the vertical profile is taken into account, and the cross level is neglected. The gauge and the lateral alignment do not intervene. The vertical profile P SD is then expressed in the following way ( S zz (φ) = Aφ2 2 φ 2 + φ 2 1) φ ( ) 4 φ 2 + φ 2 (2) 2 with A the roughness constant and φ 1 and φ 2 two cutoff spatial frequencies. In the original reference [13], parameter A is referred to be expressed in cpf/inch instead of cpf inch. As this could induce errors during unit conversion, Table 1 presents proper values of A, φ 1 and φ 2 in SI units, with their own variance σ 2. The evolution of the PSD of the vertical profile vs spatial fresquency is presented in Figure 2(a) for quality classes 1 to 6. According to Garg and Dukkipati, this representation should be limited to a wavelength range from 1. m to 3 m. parameters Table 1: Parameters intervening in Eq. (2) track classes symbols units A [1 6 m] φ 1 [1 3 m 1 ] φ 2 [1 2 m 1 ] σ 2 [1 4 m 2 ] As the simulation is performed in time domain, the P SD must be converted to an actual profile expressed in terms of position x. If φ is the resolution retained for the spatial frequency, the profile can be written

5 RAILWAY DYNAMICS AND GROUND VIBRATIONS 3493 according to the following Fourier series h(x) = k 2 φszz (k φ) cos(k φx + ϕ k ) (3) where ϕ k is determined randomly according to a uniform distribution between π and π. Figure 2(b) illustrates the actual vertical profile related to the six classes. PSD [m 2 /m 1 ] domain of validity class 1 class 2 class 3 class 4 class class 6 Random unevenness [m] 1 x class 1 class 2 class 3 class 4 class class Spatial frequency φ [m 1 ] (a) Power spectral density Distance [m] (b) Spatial evolution Figure 2: Generated unevennesses, based on the model of Garg and Dukkipati 3.2 Local defect Along the network, tramways also encounter local defects, such as rail joints, crossings or switch gears, which induce a deterministic rail profile that will be added to the distributed irregularity. As the exact geometry of existing local defects is not easy to measure, an artificial stepwise discontinuity has been considered in the TRANSDYN project so as to make easier the comparison between numerical and experimental results. The discontinuity corresponds to a steel plate, of thickness h and length l (Figure 3), welded to the rail. The vertical motion induced at the centre of the wheel is also displayed on Figure 3 and is represented, during wheel climbing by h defect (x) = Rwheel 2 (x x l ) 2 + h R wheel (4) with l = h (2 R wheel h). with R wheel the wheel radius, x the position of the centre of the wheel, x the position of the wheel when it hits the defect and l the distance between x and the start of the defect. Of course, after wheel climbing (x > x + l ), the height is constant h defect (x) = h while the wheel fall is obtained symmetrically from the climbing. During the test related to the tram of Brussels [2], the height h and the length l of the plate were equal to 1 and mm respectively. The length was chosen sufficiently short to excite all rigid body modes of the vehicle. Indeed, Figure 4 representing the defect spectrum for different defect lengths at a speed of 3 km/h shows that, at the given speed, all frequencies up to 7 Hz are equally excited. It is of interest to note that a shorter defect length increases the bandwidth but, as the power is distributed over a larger frequency band, decreases the low frequency level.

6 3494 PROCEEDINGS OF ISMA21 INCLUDING USD21 PSfrag O O v R wheel P A Q h x P l l step discontinuity on the rail head Figure 3: Modelling of the passage under the stepwise Amplitude [m 2 /Hz] x l = 1 mm l = mm l = 1 mm l = mm Frequency [Hz] Figure 4: Influence of the wedge length l on the excitation frequency 3.3 Contact force The wheel/rail normal contact force N is calculated from the penetration d according to the well-know Hertzian theory N = K Hz d 3/2 ; () where coefficient K Hz is determined from the radii of curvature of the wheel and rail surfaces and the elastic properties of their materials. The penetration d derives from the relative position of the wheel with respect to the rail and then depends on their configuration parameters.

7 RAILWAY DYNAMICS AND GROUND VIBRATIONS Study case 4.1 The tramway T2 in Brussels The T2 LRV tram running in Brussels is a medium sized system, with several interesting peculiarities. Figure presents the studied configuration, composed of a small center car surrounded by two large ones. The end cars are supported by one BA2 bogie while the center car lies on a BR4 4 one. Both bogies are equipped with independent wheels, the BA2 one having the particularity to be articulated so as to allow each wheel to remain tangent to the rail. All wheels are motorized but not the small wheels of the BA2 bogie. For the latter, the motors are placed directly inside the wheels, which makes them very heavy. Each bogie comprises rubber primary and air-spring secondary suspensions. The axle loads and the principal dimensions are given in Figure, revealing a total vehicle mass of 32 tonnes for a length of 22.8 meters. The speed of the vehicle is limited to 7 km/h t 3.4 t 4.2 t 4.2 t 3.4 t 8.3 t Figure : Tram T2 As all bodies of the vehicle model move with a constant driving velocity along the track, the longitudinal motion is assumed to be known a priori and does not need to be involved in the model. Moreover, small pitch angles can be assumed so that the governing equations of the vehicle model are reduced to their linearised form [M v ] { q v } + [C v ] { q v } + [K v ] {q v } = {f v }, (6) where the subscript v denotes the vehicle. Vector f v includes the gravitational forces acting on each body and the wheel/rail contact forces. We will see that the vehicle modes play an important role in the generation of ground vibrations. Figure 6 presents the main dynamic modes of the front car derived from the model: the car bounce mode at lowfrequency (1.7 Hz), the bogie bounce (2.4 Hz) and pitch (26.6 Hz) modes, and the rear axle hop mode at higher frequency. 4.2 Site configuration The Brussels tramway network consists of 39% of ballasted tracks, 18% of clinker pavement and 28% for natural soil pavement essentially. The selected site (Haren) is logically ballasted but, more importantly, has the advantage to have been studied in great detail in the past. The EBT Vignole rail is regularly supported by wood sleepers. As our model is planar, only the vertical profile irregularity is considered. The latter corresponds, according to the railway operator STIB, to a medium quality (class 3 according to [13]). The railpad and ballast dynamic parameters (mass, stiffness and

8 3496 PROCEEDINGS OF ISMA21 INCLUDING USD21 (a) car bounce mode (f = 1.7 Hz / ξ = 32%) (b) bogie bounce mode (f = 2.4 Hz / ξ = 14%) (c) bogie pitch mode (f = 26.6 Hz / ξ = 8%) (d) rear axle hop mode (f = 66.1 Hz / ξ = 27%) Figure 6: Modal analysis of the T2 leading bogie damping) were obtained by updating the numerical vertical numerical receptances from the experimental ones. Various experimental set up were used: impact tests (Figure 7(a)) realized with hammers of various weights, harmonic tests through an unbalanced motor (Figure 7(b)), for characterising the low-frequency behaviour, static loading with the help of a driving machine (Figure 7(c)) applying a controlled force on the track (track settlement value). Supplementary measurements, performed by Van Den Broeck [14] in the context of the VLIM project, have also been exploited. Table 2 gathers all the identified track parameters. (a) Impulse hammer (b) Unbalanced motor (c) Static loading Figure 7: Means of characterisation of the track (Haren site) Let us note that the stiffness kb and damping d b correspond to the ballast and the ground in series coefficients determined from the identification, actually 1 k b 1 d b = 1 k b + 1 k f (7) = 1 d b + 1 d f. (8) In order to derive the ballast stiffness k b and damping d b coefficients, the soil characteristics (k f, d f ) are estimated by means of the Wolf s formulas [1] [ Gb ( a ) ].7 k f = = 123 MN/m (9) (1 ν) b abρ d f = 1.48 πg k f = 177 kns/m (1)

9 RAILWAY DYNAMICS AND GROUND VIBRATIONS 3497 giving the apparent stiffness of a rectangular rigid plate with 2a and 2b dimensions (b < a), lying on a homogeneous half-space (defined by its shear modulus G, its density ρ and its Poisson s ratio ν). The ballast parameters are de facto equal to k b = 32.1 MN/m and d b = 2 kns/m. The hypothesis of decoupling is therefore verified. Table 2: Track properties, derived from TRANSDYN project [2] A r 63.8 cm 2 d b 4 kns/m L.72 m E r 21 GN/m 2 d p 3 kns/m m 4.42 kg I r 1988 cm 4 kb 2. MN/m K Hz 92.9 GN/m 3/2 ρ r 78 kg/m 3 k p 9 MN/m In the TRANSDYN project, only a homogeneous soil could be considered due to the use of approximated Green s functions for modelling the ground response [2]. With the recently adopted FIEM approach, this limitation no longer exists. The soil model then consists of five horizontal layers, lying over a halfspace (Table 3), whose properties are extracted from the data collected during VLIM project. Table 3: Soil characteristics and layering, according to the VLIM project [1, 14] E ρ ν c P c S c R β h i Layer 1 61 MPa 1876 kg/m m/s 12 m/s 114 m/s.4 s 1.2 Layer 2 84 MPa 1876 kg/m m/s 17 m/s 133 m/s.4 s 1.8 Layer MPa 1876 kg/m m/s 26 m/s 247 m/s.4 s 1. Layer MPa 1876 kg/m 3.27 m/s 28 m/s 269 m/s.4 s 1. Layer 4 MPa 1876 kg/m m/s 3 m/s 29 m/s.4 s 1. Halfspace 46 MPa 1992 kg/m m/s 286 m/s 274 m/s.4 s 4.3 Validation of the proposed model For the purpose of validation, the results of the model have been compared to their experimental counterparts, when the vehicle comes up against a local rail defect. Figures 8 and 9 present typical results (in terms of velocity at soil surface) at 2 m from the track, for constant speeds of 2 km/h and 3 km/h, respectively. Globally, a good agreement exists between experimental and simulated results. The impact of each wheel on the local defect is clearly emphasized on the figures, both in experimental and in computed curves. For example, at 3 km/h, each axle crosses the defect at t = 1.4 s, t = 1.6 s, t = 2.3 s, t = 2. s, t = 3.2 s and t = 3.4 s. The maximum level is also well predicted, despite a background noise appearing in the measurement, lightly hiding the shape of each impact. A recent paper [11] has demonstrated that the vibrations are essentially dominated by the bogie pitch mode. This result can be reproduced only if the simulation involves a sufficiently detailed model of the vehicle.

10 3498 PROCEEDINGS OF ISMA21 INCLUDING USD Velocity [mm/s] Velocity [mm/s] Time [s] (a) Experimental Time [s] (b) Simulated Figure 8: Vertical ground velocity at 2 m from the track, during the passing of the tram at speed v = 2 km/h 1 1 Velocity [mm/s] Velocity [mm/s] Time [s] (a) Experimental Time [s] (b) Simulated Figure 9: Vertical ground velocity at 2 m from the track, during the passing of the tram at speed v = 3 km/h 4.4 Vibration level vs vehicle speed As far as the local defect is concerned, both simulations and experiments have brought the surprising result that the vibratory level diminishes when the vehicle speed varies from 2 to 3 km/h. The model allows to verify this trend on a larger velocity range, as illustrated on Figure 1(a), which plots the vertical peak particle velocity P P V (maximum absolute amplitude of the vertical velocity signal), in function of the distance from the track and the tram speed. It turns out that the P P V regularly decreases not only with the distance but also with the speed. This unexpected result can be explained from the frequency spectrum of the defect: when the speed increases, the spectrum is spread over a larger frequency band so that its average value diminishes (Figure 11). The typical vehicle eigenfrequencies, are then excited to a lesser extent and the overall vibration level diminishes. This is combined with a change of the spatial frequency with speed. For instance, at velocities v of 2 km/h and 3 km/h, the dominant bogie pitch mode at 26.6 Hz corresponds to spatial frequencies of 3.7 rad/m and 2. rad/m, respectively. The overall unevenness intervenes in a lesser degree. Considering that the level of its P SD diminishes gradually with the spatial frequency, a speed increase is translated by a diminution of the spatial frequency and, consequently an increase of the PSD level in the frequency domain. On the other hand, the quasi-static track deflection has its importance that grows up. These two great points naturally suggest that the vibratory

11 RAILWAY DYNAMICS AND GROUND VIBRATIONS PPV [mm/s] 1 PPV [mm/s] Distance from the track [m] Vehicle speed [km/h] (a) During the passing on the local defect 1 Distance from the track [m] Vehicle speed [km/h] (b) During the passing on a rough rail (without local defect) Figure 1: Vertical peak particle velocity, in function of the distance from the track and of the tram speed Amplitude of S(f) [m 2 /Hz] x Hz 26.6 Hz 66 Hz 1 km/h 2 km/h 3 km/h 4 km/h km/h 7 km/h Frequency [Hz] (a) In the frequency domain Amplitude of S(φ)/v [m 2 /Hz] x rad/m 3 rad/m 2 km/h 3 km/h Spatial frequency φ [m 1 ] (b) In the spatial domain Figure 11: Influence of the local defect P SD with the tram speed level increases with the tram speed, as presented in Figure 1(b). The level is indeed smaller than the one imposed by the local defect but, at high speed, the order of magnitude becomes the same. Note that the levels obtained at low speeds and for a local defect are important and widely exceed the recommended values imposed by the usual standards expected to assess the degradation risk of surrounding structures [16, 17]. 4. Benefit of resilient wheels The large vibration level produced by the T2 tramway is mainly due to the weight of the motors which are not suspended as they are directly mounted on the wheel. A solution to reduce the ground vibrations then consists in equipping the tramway with resilient wheels, where the rolling tread and the hub are separated by a layer of soft material. Figure 12 presents the vertical ground velocities induced by the original vehicle and by a vehicle whose motor wheels are resilient. The curves show the ground response when the leading bogie (motor and trailing wheels) comes up against the local discontinuity. Experimental values corroborate the numerical prediction and show that the vibration level induced by the resilient wheels (8.3 tonnes per axle) becomes comparable to the one generated by the trailing wheels (3.4 tonnes per axle).

12 3 PROCEEDINGS OF ISMA21 INCLUDING USD21 1 Numerical Experimental 1 Numerical Experimental Velocity [mm/s] Velocity [mm/s] Time [s] (a) Without modification Time [s] (b) With modification Figure 12: Vertical ground velocity at 2 m from the track, when the T2 tram crosses the discontinuity at v = 3 km/h (leading bogie contribution) Conclusion The prediction of rail traffic induced vibrations often relies on the hypothesis that the vehicle can be reduced to a sequence of constant axle loads. This condition brings the possibility to model the track and the soil as a whole, but neglects the vehicle/track dynamics and its impact on the soil response. However, in some cases, the vehicle dynamics brings a dominant contribution in the soil response, which cannot be reproduced if the vehicle is not included in the model with enough details. The T2 tram operating in Brussels constitutes a perfect example of such a situation. Recently, the authors have proposed a methodology for the modelling of the overall vehicle/track/soil system. The analysis is performed in two successive steps: firstly the track/soil subsystem which yields the ground forces and secondly the soil response to these forces. The approach assumes that the vehicle/track and soil subsystems are decoupled and is valid as far as the foundation is sufficiently rigid with respect to the ballast. This methodology has been applied on the T2 tram of Brussels, and more especially on the site of Haren which has been the subject of several complementary investigations. Namely, several in situ measurements are available and were exploited to derive the mechanical properties of the track and the soil. A comparison with the measurements, when the vehicle comes up against an artificial local defect, has allowed to validate the model and the proposed methodology. In turn, the model has been used to study the influence of the tram velocity and has demonstrated the benefit brought by resilient wheels, itself corroborated by experimental measurements. In the case of the local defect, it is surprisingly observed, both numerically and experimentally, that the vibration level induced in the soil decreases with speed. However this phenomenon could be explained by an analysis in the frequency domain. References [1] G. Degrande, G. De Roeck, W. Dewulf, P. Van den Broeck, and M. Verlinden. Design of a vibration isolating screen. In P. Sas, editor, Proceedings ISMA 21, Noise and Vibration Engineering, Vol. II, pages , Leuven (Belgium), [2] B. de Saedeleer, S. Bilon, S. Datoussaïd, and C. Conti. Vibrations induced by urban railway vehicles modeling of the vehicle/track system. In Proceeding of the transport and environment study days of the BSMEE, Mons, [3] V. V. Krylov. Effect of track properties on ground vibrations generated by high-speed trains. Acustica acta Acustica, 84(1):78 9, 1998.

13 RAILWAY DYNAMICS AND GROUND VIBRATIONS 31 [4] X. Sheng, C. J. C. Jones, and D. J. Thompson. Prediction of ground vibration from trains using the wavenumber finite and boundary element methods. Journal of Sound and Vibration, 293(3 ):7 86, 26. [] F. Lu, Q. Gao, J. H. Lin, and F. W. Williams. Non stationary random ground vibration due to loads moving along a railway track. Journal of Sound and Vibration, 298:3 42, 26. [6] G. Lombaert, G. Degrande, J. Kogut, and S. François. The experimental validation of a numerical model for the prediction of railway induced vibrations. Journal of Sound and Vibrations, 297(3 ):12 3, 26. [7] L. Auersch. Theoretical and experimental excitation force spectra for railway-induced ground vibration: vehicle track soil interaction, irregularities and soil measurements. Vehicle System Dynamics, 48(2):23 261, 21. [8] W. Gardien and H. G. Stuit. Modelling of soil vibrations from railway tunnels. Journal of Sound and Vibration, 267:6 619, 23. [9] G. Kouroussis, O. Verlinden, and C. Conti. Ground propagation of vibrations from railway vehicles using a finite/infinite-element model of the soil. Proc. IMechE, Part F: J. Rail and Rapid Transit, 223(F4):4 413, 29. [1] G. Kouroussis, O. Verlinden, and C. Conti. Efficiency of the viscous boundary for time domain simulation of railway ground vibration. In Proceeding of the 17th International Congress on Sound and Vibration, Cairo (Egypt), 21. [11] G. Kouroussis, O. Verlinden, and C. Conti. On the interest of integrating vehicle dynamics for the ground propagation of vibrations: the case of urban railway traffic. Vehicle System Dynamics, (in press), 21. [12] G. Kouroussis. Modélisation des effets vibratoires du trafic ferroviaire sur l environnement. PhD thesis, Faculté Polytechnique de Mons, 29. [13] V. K. Garg and R. V. Dukkipati. Dynamics of Railway Vehicle Systems. Academic Press, Toronto (Canada), [14] P. Van Den Broeck. A prediction model for ground-borne vibrations due to railway traffic. PhD thesis, Katholieke Universiteit te Leuven, 21. [1] J. P. Wolf. Foundation Vibration Analysis Using Simple Physical Models. Prentice Hall, New Jersey (USA), [16] Deutsches Institut für Normung. DIN 41-3: Structural vibrations Part 3: Effects of vibration on structures, [17] Association Suisse de Normalisation. SN-64312a: Les ébranlements Effet des ébranlements sur les constructions, 1992.

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