A hybrid modelling approach for predicting ground

Journal of Sound and Vibration 335 (2015) 147–173

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A hybrid modelling approach for predicting ground vibration from trains N. Triepaischajonsak, D.J. Thompson n Institute of Sound and Vibration Research, University of Southampton, Southampton, SO17 1BJ, UK

a r t i c l e i n f o

abstract

Article history: Received 27 January 2014 Received in revised form 18 September 2014 Accepted 23 September 2014 Handling Editor: G. Degrande Available online 18 October 2014

The prediction of ground vibration from trains presents a number of difficulties. The ground is effectively an infinite medium, often with a layered structure and with properties that may vary greatly from one location to another. The vibration from a passing train forms a transient event, which limits the usefulness of steady-state frequency domain models. Moreover, there is often a need to consider vehicle/track interaction in more detail than is commonly used in frequency domain models, such as the 2.5D approach, while maintaining the computational efficiency of the latter. However, full time-domain approaches involve large computation times, particularly where threedimensional ground models are required. Here, a hybrid modelling approach is introduced. The vehicle/track interaction is calculated in the time domain in order to be able t account directly for effects such as the discrete sleeper spacing. Forces acting on the ground are extracted from this first model and used in a second model to predict the ground response at arbitrary locations. In the present case the second model is a layered ground model operating in the frequency domain. Validation of the approach is provided by comparison with an existing frequency domain model. The hybrid model is then used to study the sleeper-passing effect, which is shown to be less significant than excitation due to track unevenness in all the cases considered. & 2014 Elsevier Ltd. All rights reserved.

1. Introduction As the demand for transport increases, the railways operate faster, longer, heavier and more frequent trains. As a result of this, greater noise and vibration is generated, which is of increasing concern for local residents, particularly as people become more environmentally aware. Moreover, where new lines are proposed, noise and vibration are important aspects that require careful consideration in the planning stage and often form the basis of objections to such developments. While noise is covered by national and European legislation, there is much less regulation covering vibration. Vibration transmitted through the ground can be experienced either as feelable vibration in the range 4–80 Hz or as low frequency ground-borne noise in the range 30–250 Hz [1]. Unlike airborne sound, vibration and ground-borne noise vary strongly from one location to another due to large variations in soil properties and ground geometry. Although often associated with underground lines in cities, both ground vibration and ground-borne noise can also be experienced for tracks on the ground surface, particularly where the soil is relatively soft. Where noise barriers are installed to shield airborne sound, this can increase the sensitivity of residents to ground vibration and ground-borne noise. n

Corresponding author. E-mail address: [email protected] (D.J. Thompson).

http://dx.doi.org/10.1016/j.jsv.2014.09.029 0022-460X/& 2014 Elsevier Ltd. All rights reserved.

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N. Triepaischajonsak, D.J. Thompson / Journal of Sound and Vibration 335 (2015) 147–173

The main excitation mechanisms of ground vibration and ground-borne noise are usually identified as the moving quasistatic load and dynamic excitation due to the wheel and track unevenness (roughness) [2,3]. The moving quasi-static load causes local deflection of the track under each wheel. However, provided that the train speed remains lower than the wave speed in the ground, the vibration produced by this mechanism does not propagate away from the track and its effect is limited to low frequencies and locations very close to the track. Sheng et al. [2] showed that, for the vibration generated by a train at speeds well below the wave speed of the ground, the dynamic component is much more important. Similar conclusions were reached by Lombaert and Degrande [3]. In addition to these two mechanisms, vibration can be caused by parametric excitation due to variations in track support stiffness. One example of this is the sleeper-passing effect where variations in track stiffness within a sleeper span can lead to excitation at the sleeper-passing frequency. Various modelling techniques can be used to investigate the generation and propagation of ground vibration from source to receiver. These may be based on empirical, analytical or numerical methods. These methods almost all operate in the frequency domain. Empirical methods are often used in practical situations due to the difficulties in obtaining reliable estimates of ground properties and other parameters. Various empirical approaches have been proposed [4,5] and in the US, the Federal Transit Administration (FTA) guidance manual [6] recommends the use of such an empirical method to predict ground-borne vibration associated with a transportation project. However, these empirical models cannot readily be used to study parameter variations and so, for this, analytical or numerical approaches are preferred. Simple analytical models represent the ground as a homogeneous half-space or an infinite whole space [7,8]. In practice, however, the ground usually has a layered structure, with a layer of softer weathered material at the surface and one or more stiffer layers underneath. At sufficient depth the lowest layer can often be represented as a homogeneous half-space. In the frequency range of interest for railway ground vibration the layered structure of the ground has important effects on the propagation of surface vibration [1]. Kausel and Roësset [9] present analytical expressions for stiffness matrices which are expressed in terms of wavenumbers in the ground, which can be used to represent these layered grounds. This analytical ground model has been extended by Sheng et al. [10,11] to include a model of the track coupled with a layered ground. Dynamic train/track interaction due to various excitation mechanisms can be included using such models. In the semianalytical model of Sheng et al. [2], the track is represented as an infinite, layered beam resting on one or more elastic layers overlying a three-dimensional half-space. The track and ground are modelled in the wavenumber domain and movement of the train is included through a transformation from a stationary to a moving frame of reference. The train itself is described in terms of multiple rigid-body systems. The model includes the effects of moving quasi-static loads and the dynamic excitation due to the vertical irregular profile of the rails as the sources of excitation. The output is given in the form of displacement power spectra of the track and the ground. This model is known as TGV (Train-induced Ground Vibration) and has been validated against measurements in [2,12]. For trains running in tunnels it is important to include the tunnel structure in the model. An analytical model was developed by Forrest and Hunt [13] for a railway tunnel of circular cross-section. Hussein and Hunt [8] developed this approach further as the Pipe-in-Pipe (PiP) model. However, such analytical models for the vibration from surface or underground railways are limited in scope to simple geometry. To represent arbitrary geometry of ground or structures, numerical models are required, such as those based on finite or boundary elements. Finite element (FE) models are used for example in [14,15] in time-domain simulations of the effects of trains running at high speeds. To simulate the infinite nature of the soil, the FE model has to be extended to relatively large distances and appropriate absorbing boundaries have to be implemented which leads to long computation times. The use of boundary elements (BE) for the soil directly includes the radiation condition and avoids the need for including a large FE mesh for the soil. A fully three-dimensional finite element-boundary element model has been developed by Galvin et al. [16]. The quasi-static and the dynamic excitation mechanisms due to a train passage can be considered using this model based on a time-domain approach. However, the computation time required for the calculation of threedimensional dynamic soil–structure interaction remains large. In [17] a combined FE–BE model is presented in the frequency domain in which the Green's functions for the layered soil are directly included in the BE formulation. The ground and built structures can often be considered to be homogeneous in the direction along the track. Twodimensional models can be used, at least for relative studies [18], but these lack the full three-dimensional propagation. To minimise the time consumed for the prediction using three-dimensional models, while avoiding the limitation of a two-dimensional model, many authors have used a so-called 2.5D approach [19,20]. In such a situation a sequence of twodimensional models can be used, each of which corresponds to a particular wavenumber in the track direction. Sheng et al. [20,21] developed such a model based on coupled wavenumber finite and boundary element methods to predict ground vibration from trains both in tunnels and on the ground surface. Similar models have also been presented in [22,23], while in [24] a 2.5D approach was used with FE and infinite elements. Although many of the models discussed operate in the frequency domain, excitation models in the time domain are required to take account of transient excitation due to specific features on the track or parametric excitation due to spatiallyvarying track or ground properties. Moreover, there is often a need to consider vehicle/track interaction in more detail than is commonly used in frequency domain models, such as the 2.5D approach, while maintaining the computational efficiency of the latter. Particularly where nonlinearities are present in the track or vehicle components or in the wheel/rail interaction, a time-domain approach is essential. Although this is possible using a finite or boundary element approach, as in [14–16], in this paper a more efficient hybrid approach is introduced. It combines a simpler wheel/track interaction model working


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in the time-spatial domain, introduced in Section 2, and a separate ground model; in the present case the latter is an analytical layered ground model working in the wavenumber–frequency domain [9], described in Section 3. The two models are coupled via the forces acting on the ground beneath the track as described in Section 4. This hybrid approach is validated by comparing the results with those of Sheng's semi-analytical model [2,25] for the case of excitation by track unevenness and the quasi-static moving loads, as well as with simpler beam models, as described in Section 5. Results are presented and discussed in Sections 6 and 7. The hybrid model can be used to study a number of other excitation mechanisms; in the present paper parametric excitation due to sleeper-passing effects is considered in Section 8. 2. Time-domain vehicle–track interaction model A number of established models are available to calculate the wheel/track interaction in the time domain, including several commercial vehicle dynamics packages. For the present purpose it is important to include the track dynamics, including the distribution of dynamic loads via the rails and sleepers to the ground, but the ground itself will not be explicitly included at this stage. To represent the vehicle, several wheelsets are required together with the appropriate static loads. Suspension springs and sprung masses (bogies and vehicle body) could also be included but will only have an influence at frequencies below about 10 Hz [26]. Use is made of a model based on the work of Nielsen and Igeland [27], as implemented by Croft [28]. The track is represented using a finite element model with beams for the rails, masses for the discrete sleepers and spring–damper elements for the rail pads and ballast. A series of wheels, represented by their unsprung mass and static axle load, run along the track and vibration is induced by the unevenness of the rail and wheel as well as by the moving axle loads. Although the track and vehicle models are linear, the fact that the vehicle is moving along the track requires that the model operates in the time domain. In the present application the rail is represented by Euler–Bernoulli beam elements, as the intended frequency range is limited to below 250 Hz. The Euler–Bernoulli beam model can be used reliably to predict the rail vertical dynamic response for frequencies up to 500 Hz, above which a Timoshenko beam model would be required to give satisfactory results [1]. The finite element model of the track is truncated to a certain number of sleeper bays; here 60 bays are used. The rail is connected by rail pads to discrete masses representing the sleepers at an equal spacing of 0.6 m, although this distance can also be varied along the track if required. Rail pads and ballast are modelled as spring–damper elements, with a simple viscous damping model. The rotational stiffness of the rail pads about the transverse axis is also included. The ballast stiffness and damping are modified to include the effects of the ground; this will be described in Section 4. Only vertical displacement and rotation in the vertical plane are considered; lateral effects are not included. To reduce calculation times, a modal summation approach is used [27] and the track response is calculated initially in modal coordinates; the track model itself is linear allowing use of modal summation. Here, modes with a natural frequency up to 3 kHz are included as otherwise the shape of the quasi-static response is not captured adequately. For the rail, four beam finite elements are used in each sleeper span. Although Nielsen and Igeland [27] and Croft [28] used this model to represent a single rail, in the present paper the track parameters are chosen to represent both rails as a single composite beam and the vehicle unsprung masses represent the whole wheelset, i.e. both wheels and the axle. To study low frequency vibration, and particularly the response to quasi-static moving loads, signals with a relatively long duration are required. Due to the presence of transient effects at the start and end of the finite track model only the

kH2

Fe2

Fe1

Mw2

Mw1

v

kH1 r A

A kp, cp Ms Base

keq, ceq

Fig. 1. Schematic view of the vehicle/track interaction model with ‘circular’ track. Points A are joined together. Only two of the four wheels are shown. Fe1 and Fe2 are the static axle loads, Mw1 and Mw2 are the wheelset masses, kH1 and kH2 are the Hertz contact stiffnesses, r is the unevenness amplitude, kp and cp are the rail pad stiffness and damping, Ms is the sleeper mass, keq and ceq are the equivalent ballast stiffness and damping and v is the train speed.

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central portion of the track response is usable. It is therefore desirable to make the track much longer but the increase in the number of degrees of freedom would be prohibitive. Instead, therefore, the track has been modified to make it ‘circular’ to reduce the effect of the finite length while retaining the same number of degrees of freedom. The modified model is shown schematically in Fig. 1. The two ends of the track are connected together, i.e. the clamped boundaries are omitted and instead the last node and the first node are constrained to have the same displacement and rotation. Such a circular model was previously used by Ripke [29] and Baeza and Ouyang [30]. This avoids the effects of the boundaries and also allows the vehicle to run several times around the track. The equations of motion are given in [27] and are not reproduced here. The system of equations includes modal amplitudes and velocities for the track and physical displacements and velocities for the vehicle. They are solved using a variable order Adams–Bashforth–Moulton solver [31,32]. The time step is set to 1 ms. The profile of track unevenness in the spatial domain is generated from a one-third octave spectrum by forming a series of sine waves with random phase. Their amplitude is chosen so that the resulting spectrum matches the measured unevenness spectrum in each one-third octave band. This profile is used as the excitation, with each wheel passing over it with the appropriate delay. Where the wheels pass over the ‘circular’ track more than once, the unevenness signal is not Table 1 The parameters used for different types of ground. Parameters of ground

Ground 1 (soft)

P-wave speed (m/s) S-wave speed (m/s) Density (kg m 3) Young's modulus (MPa) Shear modulus (MPa) Poisson's ratio Damping loss factor Layer depth

Ground 2 (stiff)

Upper layer

Half-space

Upper layer

Half-space

240 120 1800 69.12 25.92 0.333 0.1 3.0 m

700 350 2000 653.3 245.0 0.333 0.1 Infinite

500 250 1800 300.0 112.5 0.333 0.1 3.0 m

700 350 2000 653.3 245.0 0.333 0.1 Infinite

−7

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10

−8

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Receptance, m/N

10 10

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Frequency, Hz

Fig. 2. Receptances from layered ground model (a) at 0 m for soft soil; (b) at 0 m for stiff soil; (c) at 8 m for soft soil; (d) at 8 m for stiff soil, with different loading area diameters: —, 0.3 m; , 0.6 m; , 1.2 m; , 2.4 m.


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repeated but a longer signal is formed covering the whole duration. The response can also be calculated using a perfectly smooth track. In each case the unevenness profile has a 5 mm spatial sampling. The track response output from the model can be used to determine the forces acting on the ground beneath each sleeper. These will be used together with a ground model, described in the next section, to give a prediction of the ground vibration due to the passing train, as described in Section 4. 3. Layered ground model in frequency–wavenumber domain To model the propagation of vibration through the ground, an axisymmetric layered ground model is used to obtain transfer receptances (displacement per unit force as a function of frequency) from each sleeper position to a chosen receiver position. The layered ground model used here is based on equations presented by Kausel and Roësset [9]. This is an efficient method for predicting ground vibration due to point or line loads. It is based on a transfer matrix approach relating displacements and internal stresses at a given interface (between layers) to those at neighbouring interfaces. The model is used to represent an elastic layered half-space and is used to obtain transfer receptances from each sleeper position to a receiver position according to the radial distance. Both the source and the receiver may be located at any depth but in the present work only positions on the ground surface have been considered. The force is applied as a constant pressure over an area which, for the axi-symmetric case is circular. This is necessary as the response to a point load is otherwise infinite at the load application point. Two different sets of parameters are used here to represent the ground, as listed in Table 1. These two layered half-space models of the ground, although not based on actual sites, are chosen as typical examples of what is likely to occur in practice. Both have a softer layer of depth 3 m overlying a stiffer half-space, the half-space being the same in the two cases for convenience. The most important parameter for the ground response on the surface is the shear wave velocity (S-wave speed); the compressional wave velocity (P-wave speed) is much less important. The damping is represented by a constant loss factor of 0.1. The effect of the loading area used in the axi-symmetric ground model on the receptance at the driving point is shown in Fig. 2 for the two types of ground. The vibration response at the excitation point reduces as the diameter of the loading area is increased, see Fig. 2(a,b), since the applied force is distributed over a larger area. However, the loading area has little influence on the response at a distance of 8 m away from the excitation point, at least below 20 Hz, as shown in Fig. 2(c,d). −7

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0.6 0.4 0.2

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1 Distance, m

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1 Distance, m

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Fig. 3. Transfer receptances from layered ground model for 0.6 m diameter loading region to different distances (a) for soft soil; (b) for stiff soil: —, 0 m; , 0.6 m; , 1.2 m; , 1.8 m; (c) for soft soil at 1 Hz, (d) for stiff soil at 1 Hz: —, response to load; , average over each sleeper bay; , edges of each sleeper bay.

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For the softer soil a broad peak occurs in the transfer receptance at about 20 Hz. This corresponds to the cut-on frequency of the layered ground, above which waves propagate in the upper layer with little influence of the underlying half-space [1]. The results at 8 m for different loading area diameters diverge above a certain frequency that depends on the diameter. They drop initially when the diameter is equal to about half the wavelength of shear waves in the upper soil layer. For the soft soil this occurs at about 25 Hz for a 2.4 m diameter, 50 Hz for 1.2 m etc. For the stiffer soil the corresponding frequencies are twice as high. Therefore, the transfer receptance is independent of loading area size only if the wavelength is greater than about twice the diameter. In the present application, the sleeper applies a dynamic load to the ground through the ballast. The sleeper is relatively rigid in this frequency range and has a length of 2.5 m and a typical width of 0.25 m. The load acting on the ground is distributed over an area that is larger than this due to spreading of the load through the ballast layer. To approximate this, four circular load regions of diameter 0.6 m will be used beneath each sleeper, as will be described in Section 4. Comparing Fig. 2(a) and (b) it can be seen that the receptance at 0 m of the stiffer ground is around a factor of 4 (12 dB) lower than that of the softer ground, in accordance with the square of the ratio of shear wave speeds in the upper layer (see Table 1). However, from Fig. 2(c) and (d) the transfer receptances at 8 m of the two grounds are equal at low frequencies, where transmission is dominated by the underlying half-space, this being identical for the two grounds. They differ by a factor of 4 above around 20–30 Hz, where waves have cut on in the upper layer of soil. Above 100 Hz the transfer receptance of the softer soil is generally lower than that of the stiffer soil due to a higher influence of material damping as the wavelength is shorter. The possible effect of interactions between adjacent sleepers has also been considered approximately by calculating the transfer receptance to distances of 0.6, 1.2, 1.8 m etc from the centre of the loading region. The diameter of each load region is set to 0.6 m, corresponding to the sleeper spacing. The results are shown in Fig. 3(a,b). The transfer receptance at a point 0.6 m away from the force point is already about 14 dB (factor of 5) lower than the receptance at 0 m for both soil types considered, apart from the peak at 20 Hz for the soft soil where the difference is about 10 dB (factor of 3). Fig. 3(c,d) shows the transfer receptance at 1 Hz plotted against distance from the centre of the loading region of diameter 0.6 m. Clearly the deflection close to the boundary between the loading region and the zone beneath the adjacent sleeper is greater than at the centre of this sleeper but the average receptance over this second sleeper span is similar to the receptance at its centre. There is a 10 dB difference between the average receptances of the excited region and the adjacent sleeper, which is slightly smaller than from centre to centre. Based on this evidence, it is considered sufficient to include only the receptances at the driving point to model the ground beneath each sleeper in the hybrid model; coupling terms between adjacent sleepers though the ground are neglected. For use with the wheel/track interaction model, the results from the layered ground model are pre-calculated at a series of distances equally spaced from 0 to 100 m at a spacing of 0.2 m. These are expressed as transfer mobilities (velocity per unit force) by multiplying the receptance by iω, where ω is the circular frequency. To calculate the response at various distances perpendicular to the track due to the forces at all sleepers, an interpolation is applied to the pre-calculated data to determine the transfer mobility for other distances as required. The interpolation must take separate account of magnitude and phase. The loading area diameter used here is again 0.6 m. 4. Hybrid model The transfer mobilities, obtained from the layered ground model in Section 3, are combined with the forces beneath the sleepers, obtained from the wheel/track interaction model in Section 2, to give the response at the receiver location. This hybrid model thus combines the time-domain wheel/track interaction model with a layered ground model in the

F F u1 kb

u1

cb keq

ceq

u2 Yg

Fig. 4. (a) The coupled system of ballast and layered ground, (b) equivalent spring–damper system.


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wavenumber–frequency domain. A similar hybrid approach could utilise a different ground model or indeed a different vehicle/track interaction model. For the case of a track in a tunnel the high impedance of the tunnel invert forms a suitable impedance discontinuity between the track and the underlying structure and ground. The track model could therefore be used with a rigid foundation to obtain the forces acting on the tunnel structure. For the case of a track on the ground surface, however, as considered here, such an impedance discontinuity is not necessarily present. The approach taken is to replace the ballast by an equivalent spring and damper representing the ballast and the ground. 4.1. Equivalent stiffness and damping loss factor In the finite element model, see Fig. 2, as the ground is not explicitly included, the “ballast” spring and damper should be chosen to represent the influence of the ground as well as the ballast. Consider the system shown in Fig. 4(a). The ballast support consists of a spring, kb , and damper, cb . Beneath this is the layered ground, represented by its mobility Y g which can be obtained from the model of Section 3. The objective is to find an equivalent spring and damper, as shown in Fig. 4(b), that represent this system as closely as possible for use in the time-domain wheel/track interaction model. For harmonic motion, the equations of motion can be written as F ¼ ðkb þ cb iωÞðu1 u2 Þ F¼

iωu2 Yg

(1) (2)

where F is the applied force and u1 and u2 are the amplitudes of the displacement above and below the spring/damper. Eliminating u2 this gives F kb iω cb ω2 ¼ ¼ keq þ iωceq u1 iω þ kb Y g þcb iωY g

(3)

The real part of this equation gives the equivalent stiffness keq and the imaginary part gives the equivalent damping coefficient ceq . The ground mobility Yg is obtained using the model of Section 3 [9] in which the load is applied over a circular area. To represent the whole area under the sleeper, however, the force under each sleeper is divided into four equal components and applied to the ground at positions 70.9 m and 70.3 m relative to the track centreline, as shown in Fig. 5. These positions are chosen to represent the whole track width, as the sleepers are around 2.5 m wide and will load the ground over at least this width. To allow for the four separate components, Eq. (3) is applied for each of the four springs, i.e. with kb and cb representing one quarter of the ballast stiffness and damping, together with Yg for the ground. Then the overall equivalent stiffness and damping keq and ceq in Fig. 4 are formed by multiplying by 4. Results for the equivalent stiffness and damping are shown in Fig. 6 for the two grounds considered here. These are based on an overall ballast stiffness kb of 1.89 108 N/m and damping cb of 1.32 105 Ns/m in each case, as listed in Table 2. Fig. 6 (e,f) shows the magnitude of the dynamic stiffness of the ballast, ground and the equivalent stiffness (in each case representing the total over four springs under the sleeper). For the soft ground, the ground and ballast stiffnesses are similar in magnitude, whereas the stiff ground is approximately a factor of 4 stiffer than the ballast. There is some frequency dependence in the results of Fig. 6(a–d), especially for the equivalent damper. This is due in part to the frequency

Fig. 5. Representation of the ballast by four springs beneath each sleeper.

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Fig. 6. Equivalent ballast/ground stiffness (a) for soft soil; (b) for stiff soil, and equivalent damping (c) for soft soil; (d) for stiff soil. —, overall value (in frequency range considered); , overall value (outside chosen frequency range); , average value. (e) Modulus of dynamic stiffness for soft soil: —, ground; , ballast; , overall value. (f) Same for stiff ground.

dependence of the layered ground, see Fig. 3. Although the two models cannot match over the whole frequency range, the average values (between 10 and 100 Hz) of equivalent stiffness and damping are used to replace kb and cb in the finite element model to represent the combined ballast and ground support. These and other track parameters are listed in Table 2. In principle it would be possible to include a frequency-dependent stiffness in the time domain model, although this cannot be done directly; instead this would require a higher order model with additional degrees of freedom to be fitted to the frequency-dependent behaviour which would introduce other complications. The model used here is therefore preferred. 4.2. Forces acting on ground The finite element track model produces time histories of interaction forces, wheel displacements and velocities, and track modal displacements and velocities. Using the modal summation method, the latter can be used to find the displacements and velocities of each node point in the track model as a function of time. To extract the forces acting at the ground interface beneath each sleeper, F i , the displacements and velocities of each sleeper, ui and u_ i , are multiplied by


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Table 2 Parameters used to represent the track in the hybrid model (for two rails). Parameters

Soft ground

Stiff ground

Rail Bending stiffness, Nm2 Loss factor Mass per unit length, kg/m

1.26 107 0 120

1.26 107 0 120

Rail pad Stiffness per sleeper, N/m Damping factor, Ns/m

2.1 108 1.84 104

2.1 108 1.84 104

Sleeper Mass per sleeper, kg Spacing, m

294 0.6

294 0.6

Ballast Ballast stiffness, N/m Ballast damping, Ns/m Equivalent stiffness, N/m Equivalent damping, Ns/m

1.89 108 1.32 105 7.88 107 1.16 105

1.89 108 1.32 105 1.44 108 1.15 105

the equivalent ballast stiffness, keq , and damping, ceq , used in the model F i ¼ keq ui þ ceq u_ i

(4)

F i for each sleeper is equally distributed over four circular regions as shown in Fig. 5. This yields a time history of the force acting on the ground below each sleeper. Note that this force is the same in the ballast and acting on the ground provided that the springs and dampers can be treated as massless. Although the ballast is not massless it is a reasonable approximation below 100 Hz (calculations have been carried out using the TGV model with and without ballast mass and the results are indistinguishable). Examples of these force time histories are given in Section 7 below. 4.3. Coupling approach Having obtained the force time histories these are converted to the frequency domain to allow them to be combined with the ground mobilities. The auto and cross power spectral densities of the forces acting at the ground surface, F i , are formed and written as the matrix SFF at each excitation frequency. It is important to include all the cross spectral densities as these include information about the relative phase of each force and hence account for movement of the wheels along the track. The auto and cross power spectral densities of the forces can be found from [33] i 1h n SF i F j ðf Þ ¼ F~ i ðf ÞF~ j ðf Þ (5) T where F~ i ðf Þ is the discrete Fourier transform of the force F i ðt Þ and T is the analysis length. As the force F i in Eq. (4) is distributed over four circular regions, F i ðt Þ here is taken as one quarter of the force coming from Eq. (4). A single segment is used in determining the spectral densities, with a window function applied to the force time-histories. The effect of the window function is discussed in Section 7. Combining these force spectra from the wheel/track interaction model with the mobilities of the ground, the auto spectral density of the ground response velocity at the receiver position Svv is given by [33] Svv ¼ Y H SFF Y

(6)

where Y is a vector of ground mobilities from the four force positions beneath each sleeper to the response position, obtained from the model described in Section 3 and H is the Hermitian transpose (complex conjugate transpose). For a model with 300 sleepers (see Section 7.1), Y is a vector of dimensions 1200 1 and SFF is a matrix of dimensions 1200 1200. The velocity response Svv is finally converted to one-third octave bands and expressed as an average over the passage time of the vehicle. 4.4. Input data for the vehicle and unevenness The parameters used to represent the vehicle are shown in Table 3. Four wheelsets are used to represent two adjacent bogies (at the ends of adjacent coaches), as shown in Fig. 7. Each wheelset is linked to the rail by a Hertzian contact spring. In the present case this spring is chosen to be linear but the method allows a nonlinear spring to be used; this would be important for predicting the response to impacts, e.g. at rail joints. Parameters are also listed for a freight vehicle considered in Section 8.2 which has a different axle spacing. The contact spring is included between the wheel and rail using a linear

156

N. Triepaischajonsak, D.J. Thompson / Journal of Sound and Vibration 335 (2015) 147–173 Table 3 Properties used to represent the vehicles. Parameters

Passenger vehicle

Freight vehicle

Wheelset mass (kg) Axle load (kN) Speed of trains (m/s) Vehicle length (m) Distance between outer wheelsets of adjacent vehicles (m) Bogie wheelbase (m) Linearised Hertzian contact stiffness, two wheels (N/m)

1200 108.3 25 23 4.4 1.6 2.42 109

1200 100 25 n/a 4.2 1.8 2.42 109

2.6 m

4.4 m

2.6 m

Fig. 7. Spacing between wheelsets used in the model of a passenger vehicle. The vehicle length is assumed to be 23 m.

stiffness. In practice, it has a higher impedance than either the wheel or the track impedances at ground vibration frequencies and so it does not influence the results. An unevenness profile has been generated based on a measured one-third octave spectrum [12], as shown in Fig. 8. This was measured using two systems: at wavelengths greater than 1 m it is based on the output of a track recording coach that measures the loaded profile of the track; at shorter wavelengths it is obtained from a rail roughness trolley which measures the unloaded profile of the railhead. Compared with other measurements of rail roughness in the literature, these rails are relatively smooth. This can be seen from a comparison with the reference curve from ISO 3095:2013 [34], also shown in Fig. 8, which is a limit for a test site to be used for acceptance testing of the noise of new vehicles (in ISO 3095 this is only defined for wavelengths shorter than 0.4 m). At wavelengths longer than about 1 m the unevenness spectrum reflects the track geometry rather than rail surface irregularities; this was measured in loaded condition. This does not distinguish between different sources of unevenness such as rail straightness, corrugation or track bed unevenness. 5. Other models used for comparison To demonstrate the validity of the hybrid model, several alternative models are used for comparison. The semi-analytical vehicle–track–ground model of Sheng et al. [2,25] is used for verification of the ground response while separate beam models are also used to verify the quasi-static and dynamic components of rail vibration. These models are introduced in this section. The results are given in Sections 6 and 7. 5.1. TGV model The results of the model developed here will be compared with the frequency–wavenumber semi-analytical model of Sheng et al. [2,25] known as Train-induced Ground Vibration (TGV). In this model the track is represented as an infinite, layered beam resting on a ground consisting of one or more elastic layers overlying a three-dimensional half-space of ground material. This model includes both the moving quasi-static loads and the vertical irregular profile of the rails as the sources of excitation. The vertical dynamic behaviour of the train is modelled using a multi-body system with both primary and secondary suspensions. In the present calculations, however, the vehicle is represented only by its unsprung masses and axle loads, as listed in Table 3. The train is set to one vehicle (four wheelsets) representing the bogies at the ends of adjacent vehicles (as in the hybrid model, see Fig. 7). The track unevenness spectrum used is given in Fig. 8. The track parameters used in the TGV model are chosen to correspond as closely as possible to those used in the hybrid model (see Table 2). However, there are some small differences: the damping of the rail pad is represented by a constant loss factor of 0.15 and that of the ballast by a loss factor of 0.5. The damping coefficients listed in Table 2 are chosen to be equivalent to these loss factors at 270 and 110 Hz respectively; these are the natural frequencies of the rail mass on the pad stiffness and the rail and sleeper mass on the ballast stiffness. The ground is modelled using the parameters listed in Table 1.


157

70

Roughness level, dB re 1µm

60 50 40 30 20 10 0 −10 102

101

100

10−1

Wavelength, m

Fig. 8. Track unevenness spectrum in one-third octave bands obtained from Steventon site [12]. Rail roughness limit curve from ISO 3095:2013 [33].

In the TGV model [25] the track is assumed to contact the ground over a particular width, over which the tractions are assumed to be uniformly distributed. The contact width is chosen as 3 m in the present analysis. Although slightly wider than the corresponding width for the hybrid model (2.4 m) this difference has no significant effect on the results. The ballast mass can be included in TGV but is set to 0 here to correspond more closely to the parameters used in the hybrid model. Calculations are performed using 180 excitation frequencies, logarithmically spaced between 0.25 and 120 Hz in addition to the quasi-static load at 0 Hz. The response is calculated at 400 ground frequencies with linear spacing 0.25 Hz and maximum frequency 100 Hz. Due to the movement of the train each excitation frequency causes response at a range of ground frequencies, see [25]. The results are finally converted into one-third octave form. The ground response is calculated in the wavenumber domain with 2048 wavenumbers in both the axial and transverse directions. The maximum wavenumbers are taken as 12.9 rad/m in the axial direction and 32.2 rad/m in the transverse direction. 5.2. Beam model for quasi-static component of rail vibration To study the quasi-static component of rail vibration, consider the static deflection of a beam on an elastic foundation to a vertical load applied at x¼0. The response of a beam on elastic foundation to a static load of magnitude F0 is given by [1] uðxÞ ¼

F 0 β βjxj e cos βjxjþ sin βjxj 2s

(7)

1=4 , s is the support stiffness per unit length and EI is the bending stiffness. The response to the train can where β ¼ s=4EI then be calculated by superposing the response for each wheel position within the train. To calculate the combined support stiffness, the rail pad stiffness and the equivalent ballast stiffness are combined in series and then divided by the sleeper spacing. For the soft ground and the parameters in Table 2, this gives s¼95.5 MN/m2. Fig. 9 shows the track deflection as a function of distance for four axles, each with a load of 108 kN. The location of the axles is as shown in Fig. 7. For a train moving at speed v, the distance x can be converted to the corresponding time t¼x/v. From this time signal, the spectrum of the rail response can be calculated. This is expressed in one-third octave bands as an average over the length of the train (23 m). Fig. 10(a) compares the results for a vehicle of four axles and a single axle. From this it is clear that the peaks and dips are associated with the axle spacing. The distance between bogie centres is 7.0 m. At a train speed of 25 m/s this corresponds to a frequency of 3.6 Hz, which can be seen as a peak in the spectrum in the 3.15 Hz band; there is a corresponding dip at the 1.6 Hz band where the two bogies are out of phase. The bogie wheelbase is 2.6 m which corresponds to a frequency of 9.6 Hz. This can be seen as a peak in the 8 and 10 Hz bands. There is a corresponding dip at 5 Hz where the two wheels in a bogie are out of phase. A further peak occurs at 20 Hz. Fig. 10(b) shows the results for four different values of support stiffness. It is clear that the results depend strongly on the support stiffness below about 16 Hz (for this train speed) but are independent of the stiffness at higher frequencies. 5.3. Dynamic component of rail vibration due to track unevenness To provide an independent check of the component of rail vibration due to track unevenness, a simple frequency domain wheel/rail interaction model is used [1]. The interaction force amplitude F at circular frequency ω is given by F¼

r

αr þ αw þ αc

(8)

158


−0.1 0

Deflection, mm

0.1 0.2 0.3 0.4 0.5 0.6 0.7 −20

−15

−10

−5

0

5

10

15

20

Distance, m

Fig. 9. Quasi-static component of rail vibration based on beam on elastic foundation for four axles (soft ground, s ¼ 95.5 MN/m2).

Rail velocity, dB re 10−9 m/s

140

130

120

110

100

90

80 100

101

102

Frequency, Hz

Rail velocity, dB re 10−9 m/s

140

130

120

110

100

90

80 100

101

102

Frequency, Hz

Fig. 10. Quasi-static component of rail vibration based on beam on elastic foundation shown in one-third octave bands. (a) For various numbers of axles (soft ground, s ¼ 95.5 MN/m2). —, four axles; , single axle. (b) For various values of support stiffness: —, 30 MN/m2; 100 MN/m2; , 300 MN/m2; , 1000 MN/m2.

where r is the unevenness amplitude, αr is the track receptance, αw is the wheel receptance and αc is the receptance of the contact spring. In the present case this model is applied to the whole track and both wheels of the wheelset. The wheelset is modelled as an unsprung mass Mw with receptance αw ¼ 1=ðω2 M w Þ and the contact spring has receptance αc ¼ 1=kH where kH is the linearised contact stiffness (combining both wheels). The unevenness is represented by its one-third octave band spectrum, as shown in Fig. 8. For the rail receptance, the track is modelled using a beam on a continuous foundation of rail pads, sleeper and ballast [1,35]. The receptance from this model is given in Section 5.4 below. The contact force is compared with the other models in Section 6.


159

This track model is then extended to calculate the average rail response during a pass-by. The average vibration during the passage of a wheel is calculated allowing for the rate of decay of vibration with distance [1], u2r ¼

1 L

Z

1 1

ur ðxÞ2 2 dx

(9)

where ur(x) is the rail vibration amplitude at a distance x from the excitation point and L is the vehicle length. Although the vibration is normalised by the time taken for the passage of the ‘train’, i.e. the vehicle of length L ¼23 m, the integration in Eq. (9) is extended to 71 to capture the whole response of the pass-by. The results of this coupled receptance model are shown in Section 6 below.

5.4. Track receptance Fig. 11 compares the track receptance calculated using two of the above models. The first is TGV, described in Section 5.1, in which the track is attached to a layered ground. The second model is the beam-on-elastic-foundation model [1,35], which uses the parameters listed in Table 2, i.e. those used in the hybrid model. The ‘ballast’ stiffness in this model represents the combination of ballast and ground, as described in Section 4.1. Although calculated using the model introduced in Section 5.3, this is equivalent to the receptance that will be used in the hybrid model (apart from the effect of discrete sleepers which is not significant for the response due to track unevenness in this frequency range). Results are shown for the two grounds, the parameters of which are listed in Table 1. The wheelset receptance is also shown for comparison. The beam-on-elastic-foundation model is stiffness-controlled at low frequency before rising to a highly damped peak at 70–100 Hz; this is the resonance of the track mass on the ballast stiffness. The results for the layered ground models, however, have more complex behaviour. It can be seen that the results for the stiffer ground agree better between the two models than those for the softer ground. For the soft ground the effect of the layering leads to a peak in ground receptance just below 20 Hz (see Fig. 2) which is reflected in a peak in the track receptance at about 16 Hz. The beam-on-elasticfoundation model cannot include this feature unless the frequency-dependence of the equivalent stiffness can also be included, see Fig. 6. The track model could be extended to include additional layers of mass and stiffness within the ballast layer to mimic this behaviour [36], but this has not been attempted here.

−6

−6

10

Receptance, m/N

Receptance, m/N

10

−7

10

−8

10

−9

10

−7

10

−8

10

−9

0

10

1

10

10

2

10

0

10

2

10

Frequency, Hz

pi

pi

pi/2

pi/2

Phase, rad

Phase, rad

Frequency, Hz

1

10

0 −pi/2 −pi

0 −pi/2 −pi

0

10

1

10

Frequency, Hz

2

10

0

10

1

10

2

10

Frequency, Hz

Fig. 11. Track receptance (a) for soft soil; (b) for stiff soil. —, beam on elastic foundation model; , TGV model of track on layered ground; , wheelset receptance.

160


The difference between the two models at low frequency corresponds to about 1.8 dB for the soft ground (difference of 22 per cent in amplitude), whereas for the stiff ground it is only 0.4 dB (5 per cent difference). Despite the omission of the effect of ground layering, the use of the equivalent stiffness gives a receptance that is a reasonable compromise over the frequency range. Although the two layered grounds have a difference in stiffness of a factor of 4 (see Fig. 3), due to the influence of the ballast the equivalent stiffness of the ballast and ground has a smaller difference, approximately a factor of 2 (see Table 2). Furthermore the resulting track receptance at low frequencies only differs by a factor of about 1.5 due to the influence of the rail pads and the rail bending stiffness. −0.1

Displacement, mm

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

Time, s

Fig. 12. Deflection of the rail at a point in the centre of the track model during the passage of four wheelsets. Results from time-domain model with smooth track for soft ground.

140

Velocity level, dB re 10−9 ms/

130 120 110 100 90 80 70 60 100

101

102

Frequency, Hz

Velocity level, dB re 10−9 m/sc

140 130 120 110 100 90 80 70 60 100

101

102

Frequency, Hz

Fig. 13. Comparison of rail responses for smooth track in one-third octave bands. (a) Soft ground, (b) stiff ground. —, Hybrid model for smooth track; , TGV due to quasi-static loads; , static beam model.


161

6. Track vibration results In this section a comparison is made of the rail vibration responses predicted using the various models. The rail responses are caused by a combination of quasi-static and dynamic loads. To study these in the hybrid model, three calculations are performed: with a smooth track including the axle loads to represent the quasi-static excitation; with a rough track but without the axle loads to represent excitation due to track unevenness; and with a rough track and including the axle loads to represent the total. Note that the smooth track case, as it includes the wheelset mass, also includes the sleeper-passing effect and is not purely the result of quasi-static excitation. 6.1. Rail vibration for smooth track To check the responses due to the moving loads for a smooth track, the rail deflection obtained from the wheel/track interaction model of Section 2 is first compared with the static beam-on-elastic foundation model from Section 5.2. The response of the rail during the passage of the four axles over the smooth track in the time-domain model is shown in Fig. 12 for one location in the centre of the track section. Comparing this with the static response shown in Fig. 9, it can be seen that they agree very well, although Fig. 12 is plotted against time while Fig. 9 is plotted against distance. These two scales correspond to one another (apart from the origin) for the current train speed of 25 m/s. Taking the one-third octave spectrum of this response and comparing it with the result obtained from the static response in Fig. 9, the results shown in Fig. 13 are obtained. Results are shown here for both layered ground models, expressed as the average over the length of the vehicle (23 m). Also shown is the quasi-static response spectrum obtained from the TGV model. At low frequencies the results are almost identical, with the result from TGV being slightly higher below 4 Hz for the soft ground. This is due to the fact that the track is slightly stiffer in the simpler beam models than in TGV (see Fig. 11). Note that the modal basis used for the track extends to modes with natural frequencies up to 3 kHz as explained in Section 2; if this is limited to a smaller range it has been found (results not shown here) that the shape of the quasi-static response is not reproduced correctly. For example, including modes up to 800 Hz causes the spectrum of the response to be too low above 16 Hz for this speed.

80

RMS load, dB re 1 N

70

60

50

40

30

20 100

101

102

Frequency, Hz

80

RMS load, dB re 1 N

70

60

50

40

30

20 100

101

102

Frequency, Hz

Fig. 14. One-third octave spectra of contact force. (a) Soft ground, (b) stiff ground. —, Hybrid model; , TGV; , coupled receptance model.

162


At higher frequencies, the amplitude of the quasi-static component falls and the result from TGV agrees with the simple model of Section 5.2. However, the results from the time-domain model from Section 2, which includes the dynamic effects of the wheelset mass, exhibit a peak at 40–50 Hz. This is caused by the sleeper-passing effect. For a discretely supported rail, the track stiffness varies within a sleeper span; when a train passes over the sleepers at a spacing λ, with a train speed v, this causes the wheelset to oscillate vertically, inducing a response at a frequency f¼ v/λ and its harmonics in the wheel response and hence in the interaction force. If the wheelset mass were not included in this model, the wheelset oscillation would not cause any additional interaction force and the rail response spectrum would follow that of the static beam model. The sleeper pitch of 0.6 m corresponds to a frequency of 41.7 Hz for the train speed of 25 m/s. This is obtained in the present smooth track case if the wheelset masses are included in the interaction model. The results from the continuously supported track models do not contain this sleeper-passing effect. The responses of the two layered ground models are similar; the stiffer ground has a slightly lower response below 10 Hz, consistent with the effect of support stiffness seen in Fig. 10(b). 6.2. Component of rail vibration due to track unevenness The calculated contact force spectrum due to excitation by track unevenness is shown in one-third octave form in Fig. 14 for the two ground models. Results are shown from the time-domain model of Section 3 (labelled ‘hybrid model’) with the axle loads set to 0, the simple coupled receptance model of Section 5.3 and the TGV model. Very good agreement can be seen. At low frequency, where the vehicle receptance is much greater than that of the track, the contact force in Eq. (8) is approximately given by F

r

αw

¼

r ω2 M w

(10)

where Mw is the wheelset unsprung mass. This relation is independent of the track receptance. The features in the spectrum are largely determined by the shape of the unevenness spectrum, see Fig. 8. At higher frequency Eq. (10) no longer holds. The vehicle receptance is mass-controlled while the track receptance is stiffness-controlled. These have equal magnitude at around 40–60 Hz, which is identified as the coupled vehicle–track

Velocity level, dB re 10−9 m/s

140 130 120 110 100 90 80 70 60 100

101

102

Frequency, Hz


140 130 120 110 100 90 80 70 60 100

101

102

Frequency, Hz

Fig. 15. Comparison of rail responses in one-third octave bands due to track unevenness. (a) Soft ground, (b) stiff ground. —, Hybrid model; , TGV; , coupled receptance model.


163

resonance and leads to a peak in the contact force. From Fig. 11, the track receptance in the TGV model is lower than that from the simple beam models in the region 30–100 Hz. Consequently the coupled vehicle–track resonance occurs at a higher frequency in the TGV model, leading to the differences seen in the contact force above 30 Hz. The component of vibration due to track unevenness predicted using the time-domain model is compared with the dynamic component of vibration predicted using TGV and the coupled receptance model in Fig. 15. Generally good agreement is seen. Differences with the TGV model are seen between 30 and 80 Hz due to the differences in contact force noted above. At low frequencies, the spectra from TGV and the time domain model show the same ‘modulation’ of the

140


130 120 110 100 90 80 70 60 100

101

102

Frequency, Hz

140


130 120 110 100 90 80 70 60

100

101

102

Frequency, Hz

Fig. 16. Comparison of rail responses in one-third octave bands. (a) Soft ground, (b) stiff ground. —, Hybrid model total; , hybrid model due to track unevenness; , hybrid model smooth track; , TGV total.

Force (offset for each sleeper)

6

x 10

5

5

4

3

2

1

0 0

1

2

3

4

5

6

Time, s

Fig. 17. Time histories of forces beneath sleepers for smooth track, soft ground case. Every fifth sleeper is shown, results offset vertically for clarity.

164


spectrum associated with the axle passing frequencies as seen in the quasi-static response [3]. The coupled receptance model does not contain the load motion and therefore does not show this ‘modulation’ of the spectrum. Fig. 16 compares the dynamic components of rail vibration obtained from the time-domain model with track unevenness and for the smooth track. The total response is also shown, which includes both the moving axle loads and the excitation from track unevenness. For this speed of 25 m/s, the rail response is dominated by the response to moving axle loads below 20 Hz, whereas above 25 Hz the component due to unevenness excitation dominates. Also shown is the overall vibration predicted by TGV. This shows good agreement with the hybrid model apart from the differences already noted around the vehicle–track coupled resonance. x 105

Force (offset for each sleeper)

15

10

5

0

0

1

2

3

4

6

5

Time, s

Fig. 18. Time histories of forces beneath sleepers for smooth track, soft ground case after unwrapping. Every tenth sleeper is shown, for clarity results are offset vertically and regions set to zero are omitted.

5

x 10

4

4

Force

3

2

1

0

−1 0

1

2

3

4

5

6

1

2

Time, s

x 104

4

4

3

3

2

2

1

1

0

0

−1 0

x 104

5

Force

Force

5

1

2

3 Time, s

4

5

6

−1

0

3

4

5

6

Time, s

Fig. 19. Effect of windowing on force signals for smooth track; results shown for every fortieth sleeper. (a) Rectangular, (b) Tukey (60% flat), (c) Hanning.


165

7. Ground vibration results The vibration at the ground surface during the passage of the train is calculated using the hybrid method described in Section 4, combining the time-domain calculation of track response with the frequency-domain ground response model. 7.1. Forces acting on the ground From the track vibration, the forces acting on the ground beneath each sleeper are calculated from the sleeper displacement and velocity and the corresponding equivalent stiffness and damping of the ballast, as described in Section 4.2. The time histories of these forces are shown in Fig. 17. Results are shown for every fifth sleeper and they are offset vertically for clarity. As the four wheelsets pass, the force at each sleeper has four peaks. However, these are repeated up to four times as the train runs around the ‘circular’ track. It is found that the track length of 60 sleeper bays (36 m) is sufficient for the force to have reduced to zero at 730 sleeper bays either side of the middle of the train. However, if a longer train is considered, the length of the track section included would have to be increased. To form a set of time histories representing a train passing once along a longer section of track, the force data from four laps is ‘unwrapped’. At any given time the forces from the circular track model correspond only to one lap, i.e. 60 sleepers. These forces are mapped onto a longer section of track consisting of 5 60 sleepers, including 30 before and 30 after the section traversed. At any time, the 60 sleepers centred on the current position of the train are assigned non-zero forces and the remainder are set to zero. Fig. 18 shows the forces after this ‘unwrapping’ process. Here only every tenth sleeper is shown and regions that are set to zero are omitted for clarity. Through the unwrapping process the original 60 sleepers are extended to 300 and the train only passes each of these sleepers once. For a smooth rail as shown here, the time histories of forces on the ground could be obtained simply by simulating one lap and copying the result to the different sleepers along a track model with extended length; however for a track with unevenness the response during different laps is not the same and a full calculation is required.

130


120 110 100 90 80 70 60 50 100

10

1

102

Frequency, Hz

90


80 70 60 50 40 30 20

10

0

10

1

102

Frequency, Hz

Fig. 20. Ground responses in one-third octave bands of softer soil site for smooth track at (a) 0 m and (b) 8 m away from the track with various window functions. —, Hanning; , Tukey (60% flat); , Tukey (90% flat); , Rectangular.

166


7.2. Windowing A window function has been used to avoid errors due to the truncation of the force time-histories. Examples of the corresponding force time histories using three different window functions areshown in Fig. 19.These are rectangular, Tukey and Hanning windows. The Hanning window [33] in Fig. 19(c) has the form 12 1 cos 2 ð2π t=TÞ where T is the length of the window and t is time. This can give a good smoothing of the transients at the ends of the signal but loses energy away from the centre. For a random signal this can be compensated by an increase in amplitude but in the present case of a transient, where the centre of the window is opposite the response point, such compensation would over-emphasise the central part of the response and is not valid. The rectangular window in Fig. 19(a) consists of a weighting of unity over the whole duration, i.e. no window, and suffers from truncation of the force signals at the ends. The Tukey window in Fig. 19(b) is a compromise consisting of a unity weighting over most of the duration, with the (1–cos2)/2 weighting applied to the ends, in this case 20 per cent of the length at each end. This avoids the loss of energy found for the Hanning window while also avoiding the effects of the transients at the ends. 7.3. Effect of length of signal and windowing Fig. 20 compares spectral results obtained when different window functions are used in the analysis of the forces acting on the ground for a smooth track. The vehicle is allowed to travel four laps in each case. As shown in Fig. 20(a), the choice of window function has no influence on the response at 0 m (beneath the track), which is dominated by the forces on sleepers close to the response point. However, as seen in Fig. 20(b), the response at 8 m away is much higher for the Tukey and rectangular windows than for the Hanning window. Comparing these results with the TGV results in the next section it becomes clear that only the Hanning window gives the correct spectrum below 10 Hz. The rectangular window causes large errors due to transients at the ends of the signals and the Tukey windows only partly compensate for this effect. For the response to unevenness excitation, however (not shown here), it is found that the Hanning window leads to lower responses due to a loss of energy, especially for larger distances from the track. The Tukey window avoids this loss of

130


120 110 100 90 80 70 60 50 100

101 Frequency, Hz

102

101 Frequency, Hz

102

90


80 70 60 50 40 30 20 10 0 10

Fig. 21. Ground responses in one-third octave bands of softer soil site for smooth rail track at (a) 0 m and (b) 8 m away from the track based on various track lengths. —, 4 laps; , 2 laps; , 1 lap. A Hanning window is used in each case.


167

energy while also avoiding the problems introduced by the rectangular window. The results presented in the next sections therefore use the Hanning window for the response for the smooth track case and the Tukey window with 60 per cent flat region for the response to unevenness excitation. For the total response the Tukey window offers the best compromise as the unevenness excitation dominates the response in the region where the window most affects the smooth track response. To investigate the effect of the track length on the ground response, Fig. 21 shows results when the vehicle is allowed to travel one, two and four laps of the circular track. The corresponding lengths of the track are 36, 72 and 144 m respectively. The length of the track considered has little effect on the ground deflection at the position underneath the track. On the other hand at 8 m away from the track the length considered has a significant effect on the results even though the Hanning window is used and clearly the length should be as large as possible to avoid errors. However, since the quasi-static response is of secondary importance at this distance, and the sleeper passing effects are not affected, the length of four laps appears sufficient.

7.4. Comparison with TGV model Fig. 22 shows the responses due to track unevenness, without the static axle loads, at three different distances from the track. Comparisons are made with the results of TGV. It can be seen that the agreement is generally very good. The differences are greater around 30–80 Hz due to the differences in contact force spectra, see Fig. 14. The responses for a smooth track at three different distances are plotted in Fig. 23 and compared with the corresponding results from TGV. The quasi-static response drops rapidly with distance from the track. For example at 8 m it has dropped by 30–40 dB at 1 Hz and by over 70 dB at 5 Hz. The result from the hybrid model agrees with TGV at 8 m up to around 10 Hz. At higher distances the level at low frequencies is too low to be calculated correctly, probably due to truncation of the signal, but it is so small compared with the component due to unevenness that this is not significant. At higher frequencies the response from the hybrid model increases due to the sleeper-passing effect, as described in Section 6.1. In particular, a peak can be seen at the 40 Hz band for both ground models and all distances. The track in the TGV model is continuously supported and so the results from this model do not contain the sleeper-passing effect.

130 Velocity level, dB re 10−9 m/s

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Fig. 22. Ground responses in one-third octave bands due to unevenness excitation for (a) soft soil and (b) stiff soil. Results plotted for different distances from the track: , 0 m; —, 8 m; , 16 m. Thick lines: hybrid model; thin lines: TGV model. A Tukey window with 60% flat region is used.

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Fig. 23. Ground responses in one-third octave bands due to moving axle loads for (a) soft soil and (b) stiff soil. Results plotted for different distances from the track: , 0 m; —, 8 m; , 16 m. Thick lines: hybrid model, smooth track; thin lines: TGV model, quasi-static response. A Hanning window is used.

8. Sleeper-passing effect Having established that the hybrid model gives results in good agreement with conventional frequency domain models, it is used in this section to investigate the extent of the sleeper-passing effect in the ground vibration at various distances from the track. The effects on this of changes in various parameters are also considered.

8.1. Vehicle speed Comparing the responses in Fig. 23 for a smooth track with those due to unevenness excitation in Fig. 22 it is clear that the response due to the sleeper-passing effect at 40 Hz is considerably lower than that due to unevenness excitation. The level difference between them in the 40 Hz band is mostly around 20 dB, varying somewhat with location. The differences are slightly smaller for the stiffer soil but still mostly greater than 17 dB. These, and subsequent results discussed in this section, are summarised in Table 4. Recall, from Section 4.4, that the track unevenness used is considered to be from a relatively smooth rail. As the speed of the train increases, the effect of the quasi-static component dominates the vibration response close to the track up to higher frequencies. Nevertheless, at distances further from the track the effect of the quasi-static component remains much smaller than the response due to track unevenness (as long as the train speed remains lower than the wave speed in the ground). Fig. 24(a) shows the total ground response at 8 m away from the track obtained from the hybrid model for various speeds of the train, while Fig. 24(b) shows the corresponding results for a smooth track. Results are shown for the softer soil at eight different train speeds, logarithmically spaced between 12.5 and 63 m/s. These speeds are all well below the wave speeds in the ground. The quasi-static component of vibration dominates up to 4 Hz at 63 m/s but at higher frequencies the component due to track unevenness remains much larger. The peaks at low frequencies correspond to the axle spacing, see Section 5.2. These peaks clearly shift to higher frequencies as the speed increases.


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The sleeper-passing frequency increases with train speed, and is indicated by the markers in Fig. 24(b). As seen above, when the speed of the train is 25 m/s the sleeper-passing frequency occurs at 40 Hz; this corresponds to the resonance of the unsprung mass bouncing on the track stiffness, as seen by the peak in the dynamic load spectrum in Fig. 14. The ground response for the smooth track has its maximum amplitude at this frequency but so does the component of vibration due to track unevenness. Fig. 25 shows the amplitude of the peaks in the response at 8 m in the one-third octave band containing sleeper-passing frequency and the corresponding total vibration in the same frequency band. From this it is clear that the response due to track unevenness remains around 20 dB higher than the sleeper-passing effect at all speeds. Table 4 Level difference (in dB) between response including roughness and without roughness at the sleeper-passing frequency at different distances from the track.

Soft ground, 12.5 m/s Soft ground, 16 m/s Soft ground, 20 m/s Soft ground, 25 m/s Soft ground, 32 m/s Soft ground, 40 m/s Soft ground, 50 m/s Soft ground, 63 m/s Stiff ground, 25 m/s Soft ground, doubled axle load 25 m/s Freight axle spacing, soft ground 25 m/s Soft ground, 25 m/s, reduced pad stiffness Soft ground, 25 m/s, increased pad stiffness Soft ground, 25 m/s, reduced ballast stiffness Soft ground, 25 m/s, increased ballast stiffness

0m

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17.7 17.3 19.4 20.0 23.0 25.0 21.3 24.6 14.2 13.8 17.4 19.3 21.8 24.4 19.3

18.9 17.4 18.6 24.3 18.0 21.4 20.3 19.2 17.7 18.3 15.3 23.9 25.9 29.3 23.5

22.9 20.9 23.9 21.8 23.1 18.3 24.7 21.1 19.2 16.0 20.8 21.7 23.2 25.9 21.1

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Fig. 24. Ground responses in one-third octave bands at 8 m away from track for various speeds of the train: (a) total, (b) smooth track. Thinner lines: , 12.5 m/s; , 16 m/s; , 20 m/s; —, 25 m/s; thicker lines: , 32 m/s; , 40 m/s; , 50 m/s; —, 63 m/s. Markers indicate sleeper-passing frequency.

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Table 4 summarises the level differences between the response to unevenness excitation and that due to the sleeper-passing effect at various distances from these results. In all cases the response due to track unevenness exceeds the sleeper-passing effect by more than 14 dB at all distances from the track, with some differences caused by the random nature of the unevenness profile. 8.2. Axle load and axle spacing The amplitude of the vibration for the smooth track is proportional to the axle load. Doubling the value of axle load gives the results shown in Fig. 26. As expected, when the axle load is doubled the vibration due to the moving axle load increases


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Fig. 25. Ground responses at 8 m away from track in one-third octave band containing sleeper-passing frequency for various speeds of the train: —, total; , smooth track.

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Fig. 26. Ground responses in one-third octave bands at (a) 0 m and (b) 8 m: —, total excitation; , smooth track. Thin lines: reference axle load; thick lines: doubled axle load.


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Fig. 27. Ground responses in one-third octave bands for freight vehicle on soft ground (a) at 0 m, (b) at 8 m. —, total excitation; , smooth track.

by 6 dB, including the response at the sleeper-passing frequency. The component of the response due to track unevenness, however, is not affected by the change in axle load apart from very small changes in the contact stiffness, which are neglected here. The total response is therefore only affected at low frequency where the quasi-static excitation is dominant. The level difference between the total and responses and the smooth track response at the sleeper-passing frequency reduces by 6 dB but remains more than 14 dB. Fig. 27 shows results for a different axle spacing. This is a freight vehicle with a bogie wheelbase of 1.8 m and a separation between axles 2 and 3 of 4.2 m. Unlike the reference case, these distances correspond to a whole number of sleeper spans. The axle load is assumed to be 100 kN. Comparing these results with those for the reference vehicle, it can be seen that the characteristic modulation of the spectrum at low frequencies changes due to the axle spacing, with a peak at 4 Hz instead of 3 Hz and a dip at 6.3 Hz instead of 5 Hz. For this case also, however, the dynamic component of response due to track unevenness is 15–20 dB higher than the sleeper-passing effect at 40 Hz. 8.3. Rail pad and ballast stiffness Finally the effect of changing the rail pad and ballast stiffnesses is considered. The rail pad stiffnesses (for two rails) are increased or reduced by a factor of 3 to give 70 and 630 MN/m which represent very soft baseplates and typical medium stiffness rail pads respectively. Results are calculated for the soft soil and listed in Table 4. In each case the ground response due to unevenness excitation remains around 20 dB greater than the sleeper-passing effect. Results are also listed in Table 4 for a three-fold increase or decrease in ballast stiffness. Again the component of vibration due to track unevenness remains dominant. 9. Conclusions A wheel/track interaction model operating in the time and spatial domain has been combined with a frequency-domain layered ground model to form a hybrid model for predicting ground vibration from trains. To increase the effective length, the track has been made ‘circular’. The advantage of this compared with making the track longer is that the number of

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degrees of freedom is not increased but a longer track response is obtained by allowing the vehicle to travel more than one lap. Comparisons are made with an existing frequency–wavenumber domain model to validate the hybrid model. Simple beam-on-elastic-foundation models are also used for comparison with the rail response. The comparisons made among these models show a good agreement for both dynamic and quasi-static components of the rail and ground response. Some differences occur due to differences in the rail receptance, which affects the contact force in the vicinity of the coupled vehicle–track resonance frequency. This highlights the need to match the track receptance correctly in the relevant frequency range. The matching achieved here could be improved for a layered ground by including further layers of mass and stiffness in the track support. Meanwhile the quasi-static component of vibration relies on the correct estimation of the static stiffness of the track. The model has been used to study the effect of the static axle load in exciting the ground at the sleeper-passing frequency. It has been found that, for a range of speeds and other parameters, the sleeper-passing effect at various distances from the track is at least 14 dB less than the response due to track unevenness in the corresponding one-third octave band, despite using an unevenness corresponding to a relatively smooth track. The fact that this effect is sometimes seen in experimental data, e.g. [37], suggests that there may be other factors that are not taken into account in the present model. The main advantage of the hybrid approach introduced here is a reduction in calculation time compared with a direct fully-coupled time-domain model. The same approach could be used in principle with other vehicle/track interaction models, such as vehicle dynamics packages, provided that the responses of individual sleepers can be determined. Other, more complex ground models could also be included in the approach.

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