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Modelling vibrations caused by tram movement on slab track line
Mathematical and Computer Modelling 54 (2011) 280–291
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Modelling vibrations caused by tram movement on slab track line Julia Real a , Pablo Martínez a,∗ , Laura Montalbán a , Antonio Villanueva b a
Departamento de Ingeniería e Infraestructura de los Transportes, Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Valencia, 14 Camino de Vera, 46022 Valencia, Spain b
INECO, 1-8 Capitán Haya, 28020 Madrid, Spain
article
info
Article history: Received 5 November 2010 Received in revised form 7 February 2011 Accepted 7 February 2011 Keywords: Ground vibrations Slab track Tram Timoshenko beam Fourier Transform
abstract The recent growth in the use of the railway and particularly that of the tram in urban areas highlights the need to investigate and mitigate the harmful effects associated with these means of transport. The vibrations caused by passing vehicles can be a source of vibration and may damage buildings close to the line. This article aims to develop an analytical mathematical model for predicting the ground vibrations caused by the passing of trams along a slab track. The model is based on the wave equation and is solved in the frequency domain through the Fourier Transform. The loads caused by the vehicle are calculated using a quarter car model. The model is calibrated and validated with real data collected along the tram network of Alicante (Spain), providing a useful tool for future research on the transmission of ground vibrations. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The growing need for mobility, along with worries about the environment and questions of sustainability mean public bodies are increasingly deciding in favour of developing rail borne transport systems. This drive is clearly shown not only in interurban transport links but also in the growth of trams and subways as the main means of urban transport. Many European cities have decided to develop and introduce their own tram networks as a way of improving urban mobility and reducing traffic congestion [1]. The expansion of tram networks is generally beneficial to the population; however any associated environmental impacts such as the vibrations generated by the passing of a vehicle should be evaluated before developing or extending an urban network. When a vehicle moves along the line it produces vibrations which are transmitted to the surrounding area through the air and the ground. The airborne vibration is felt in the form of sound which is not only irritating but could become harmful to public health. On the other hand, ground vibrations may reach the foundations of nearby buildings and introduce structurally damaging stresses and loads. Both effects become more important in the areas around tramways because their lines are generally closer to buildings and people. Therefore, a better understanding of the vibrations generated by these vehicles would allow a more efficient assessment and mitigation of any possible negative effects. In this context, the main objective of this paper is to develop, calibrate and validate an analytical model capable of predicting the ground vibrations under the line. The model presented here is an improvement of previous works in certain aspects such as the load modelling and the adaptation to slab tracks (see literature review below). It represents the first step
∗
Corresponding author. E-mail addresses: [email protected] (J. Real), [email protected] (P. Martínez), [email protected] (L. Montalbán), [email protected] (A. Villanueva). 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.02.010
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
281
Nomenclature a A Ak c2 cLj cTj C E Ej G hj I k1 k2 m1 m2 Pi Q V
λj λ∗j µj µ∗j ρ ρj υ υj ω¯ i
Track gauge Rail cross-sectional area Rail shear cross-sectional area Train primary damping coefficient Ground longitudinal wave velocity (j = 1, 2, 3) Ground shear wave velocity (j = 1, 2, 3) Equivalent ballast coefficient Rail Young modulus Ground Young modulus (j = 1, 2, 3) Rail shear modulus Layer thickness (j = 1, 2, 3) Rail inertia in y-axis direction Equivalent track spring constant Train primary spring constant Unsprung mass per axle Sprung mass per axle Harmonic load magnitude Static load per axle Train velocity First Lamé parameter (j = 1, 2, 3) First damping coefficient (j = 1, 2, 3) Second Lamé parameter (j = 1, 2, 3) Second damping coefficient (j = 1, 2, 3) Rail mass density Ground mass density (j = 1, 2, 3) Rail Poisson coefficient Ground Poisson coefficient (j = 1, 2, 3) Harmonic load frequency
within a wider study on the generation, transmission and effects of vibrations on structures near to a tramway in the city of Alicante (Spain): Project TRAVIESA. The article is structured in the following way: an initial literature review on previous work in this field is followed by the mathematical development of the model and its solution using the Fourier Transform. The model is then calibrated and validated using real vibration data collected on Line 1 of the Alicante tram network run by FGV (Ferrocarrils de la Generalitat Valenciana). The article closes with an analysis of the results and a presentation of the more important conclusions drawn. 2. Literature review A great deal of research has taken place over recent years on the generation and transmission of vibrations associated with railways. Much of the work concentrates on either the numerical or analytical mathematical modelling of this phenomenon. Numerical methods have the advantage of being adaptable to different configurations of the area being studied and allow the introduction of track irregularities or other localized factors. Examples can be found in [2,3], who used a finite elements method (FEM) to predict ground vibrations on the surface and in tunnels, respectively. Another noteworthy case was proposed by Katou et al. [4], who ran a three dimensional finite difference model (FDM) in the context of high speed trains. Celebi [5] made a comparison between a boundary elements method (BEM) and a flexible volume method (FVM), and Galvín and Domínguez [6] carried out a comprehensive study of the BEM method for railways. Analytical methods provide continuity to the solution in all the domains along with a greater physical consistency because their formulation directly depends on the physical parameters of the materials being considered. Those advantages have been taken into account when choosing an analytical method for this paper. A typical hypothesis in the railway field is to consider the rail as a beam supported on viscoelastic terrain subjected to variable loads in time. The general proposition of this hypothesis can be found in [7] and its application to the railway in the works of Muscolino and Palmeri [8], Schevenels et al. [9] and Metrikine and Vrouwenvelder [10]. This hypothesis is also assumed for the model developed in this paper. The way in which the equations are solved using analytical methods is worthy of further discussion. Some authors, like Mazilu [11] directly solve the problem in the time domain using Green’s function, but in general some type of integral transformation is used such as the Laplace Transform [7] or the Wavelet transform [12,13].
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Fig. 1. Cross section of the railway (provided by GTP).
However, many authors prefer to use the Fourier Transform for solving equations in the frequency domain, because it allows the vibration spectrum to be studied and provides information on the main frequencies and amplitudes of the phenomenon. Examples of this procedure can be found in [10,12] as well as in [14]. These studies (particularly [10,12]) are the base of the method presented in this paper, but some aspects have been analysed to improve the modelling of the phenomenon. First of all, the way loads are formulated should be considered. In the case of numerical models, some authors have implemented a full account of the loads induced by a train [15,16]. However, in the case of analytical models, most of the reviewed authors, [8,10,12] choose to simulate a single harmonic point load which contemplates certain variability of the load but not the load distribution produced by a railway vehicle with multiple axles. This issue is addressed in this paper by means of introducing a set of point loads which are a closer approximation to the train actual effect over the track. Moreover, the load formulation used in this paper allows the study of both harmonic and static loads [17]. In this way dynamic loads caused by track irregularities are included in the model. This represents another improvement from previous works. Secondly, previous studies [12] use the Euler–Bernoulli beam equation to model the rail as a beam, while this paper applies the extended Timoshenko beam equation, hence including the rail strain due to shear stress within the global modelling of the vibration. Finally, it is worth noting that most of the reviewed research concentrates on the vibrations generated by conventional railway lines. Comparatively, there is not as much bibliography available on the vibrations caused by tramway networks typically built using a slab track typology. In this way, the application of the method presented here to a tramway line represents an interesting line of research because the negative consequences of the vibrations (noise, structural damage) have greater importance in an urban environment. 3. Methodology A detailed description of the methodology used in the development, calibration and validation of the model presented in this article follows below. 3.1. Mathematical model In the context of this article, vibration is a phenomenon which propagates energy through a continuous physical medium causing said medium to deform and suffer stress. To reproduce this phenomenon, an analytical formulation is developed based on the previous works of Metrikine and Vrouwenvelder [10], Koziol et al. [12] and Salvador et al. [17]. A cross section of the track being studied is shown in Fig. 1. The model is applied to a two dimensional cut (length and depth) in the ground under the track, and it evaluates the vertical and longitudinal displacements generated by the passing of a mobile load (the tram). The cross section is simplified by taking the two rails as a single beam and ignoring any discontinuity or irregularity in the Y axis. Fig. 2 shows the longitudinal section of the ground and the X and Z axis considered. The ground is assumed to be made up of three horizontal layers of viscoelastic, homogeneous and isotropic materials. The model is therefore linear and the superposition principle can be applied. Layer 1 corresponds to the surface layer of concrete HA-25, layer 2 is the layer which the rails sit on (also made of HA-25) and layer 3 is an average of the layer of HM-10 and the layer of levelling concrete. This latter layer is assumed to extend indefinitely in depth (Boussinesq half space) to simulate that the wave is not affected by any interaction in this direction. Both the rail and elastomeric covering are modelled as a beam of negligible thickness located at the interface between the first and second layer to simulate the fact that the rail is embedded in the concrete. The mechanical behaviour of the rail
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283
Fig. 2. Geometry of the problem.
is defined by the Timoshenko beam theory [18], producing the following system of equations:
∂ 2w ∂ ∂w ρA 2 = Ak G −θ + q(x, t ) ∂t ∂x ∂x ∂ ∂θ ∂w ∂ 2θ EI + Ak G −θ ρI 2 = ∂t ∂x ∂x ∂x
(1)
where G is the shear modulus, w is the vertical displacement and θ is the angular displacement. The mechanical characteristics of the modelled rail are averaged between the steel and the elastomer. The ground dynamics are governed by the following wave equation, expressed in vectorial terms:
(λˆ + µ)∇ ˆ x,z (∇x,z d) + µ∇ ˆ x2,z d = ρ
∂ 2d ∂t2
(2)
ˆ and µ where d is the displacement vector, ρ is the density of the material and λ ˆ are operators describing the viscoelasticity of the ground, defined in the following way: ∂ ∂t (3) ∗ ∂ µ ˆ =µ+µ ∂t where λ and µ are Lamé parameters and λ* and µ∗ are damping coefficients for the ground which must be calibrated using λˆ = λ + λ∗
experimental data. The loads acting on the model are due to the interaction between wheel and rail. Two groups can generally be distinguished: - Load due to the weight of the vehicle. This is a quasi-static load which moves along the rail at velocity V . - Harmonic loads with different amplitudes and frequencies coming from the vehicle (considered as a system of masses and springs) and caused by certain imperfections and/or discontinuities in both the rail and the wheel. These could be defects such as wear and tear on the rail, rail corrugation, wear on the wheels, rail joints, etc. The resulting wave transmitted along the track and through the ground is formed from the superposition of the waves generated by all the loads described above. In this way, the effect of loads caused by track and wheel irregularities is taken into account in this otherwise completely linear model, providing a solution closer to reality [17]. The harmonic loads were modelled here using an auxiliary vehicle model called quarter car model, as defined by Melis [19]. Fig. 3 shows the layout of the model, representing an axle of the tram. Where m1 corresponds to the unsprung mass per axle of the vehicle, m2 to the sprung mass per axle, k2 and c2 represent the tram primary spring constant and primary damping coefficient respectively and k1 represents the equivalent track spring constant, the latter being another parameter which needs to be calibrated. The auxiliary model and known vehicle data provide the different harmonic loads acting on the ground, each one expressed in accordance with the following equation: Fi (t ) = Pi cos(ω ¯ it )
(4)
where Pi represents the amplitude and ω ¯ i the harmonic load frequency of the i-th load Fi . The resulting force is equal to the sum of all the harmonics, which agrees with the ground linearity hypothesis assumed beforehand. Owing to the absence of data on line defects, the frequencies ω ¯ i of the main harmonics have been obtained approximately from the average line vibration spectra, as explained in Section 4. It should be noted that the previous formulation is not valid for ω ¯ i = 0, which is the case corresponding to the static load. Therefore, the static load has to be modelled independently by calculating the rail deformation due to the passing of
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Fig. 3. Quarter car model.
the load in accordance with the Zimmermann model [20] and assuming that the terrain under the rail deforms to the same extent. The resulting vibration is added to that obtained for the rest of the loads, again following the phenomenon’s linearity hypothesis. Once the leading equations for the phenomenon and the loads acting on the ground have been defined, boundary conditions must be formulated to delimit the problem and permit its solution (5). Note that from here on, the sub indexes 1, 2 and 3 refer to the first, second and third layer of the line section, respectively: u1 ( x , h 1 , t ) = 0
(5a)
u2 ( x , h 1 , t ) = 0 v1 (x, h1 , t ) = w(x, t ) v2 (x, h1 , t ) = w(x, t ) σxz1 (x, 0, t ) = 0 σzz1 (x, 0, t ) = 0 u2 (x, h1 + h2 , t ) = u3 (x, h1 + h2 , t ) v2 (x, h1 + h2 , t ) = v3 (x, h1 + h2 , t ) σxz2 (x, h1 + h2 , t ) = σxz3 (x, h1 + h2 , t ) σzz2 (x, h1 + h2 , t ) = σzz3 (x, h1 + h2 , t ) u3 (x, ∞, t ) = v3 (x, ∞, t ) = σxzg (x, ∞, t ) = σzzg (x, ∞, t ) = 0.
(5b) (5c) (5d) (5e) (5f) (5g) (5h) (5i) (5j) (5k)
In accordance with Metrikine and Vrouwenvelder [10], this group of boundary conditions is enough to provide the stationary solution for Eq. (2) and any initial conditions for the problem need not be defined. 3.2. Solution The model, once formulated as shown in Section 3.1, is solved as follows. According to Koziol et al. [12], Eq. (2) can be expressed as a function of the Lamé potentials in such a way that the horizontal and vertical displacements of the ground and the corresponding stresses are as follows:
∂ϕ ∂ψ + ∂x ∂z ∂ϕ ∂ψ v= − ∂z ∂x 2 2 ∂ ϕ ∂ 2ψ ∂ ϕ ∂ 2ψ σzz = λˆ + + 2 µ ˆ − ∂ x2 ∂ z2 ∂ z2 ∂ x∂ z 2 2 2 ∂ ψ ∂ ψ ∂ ϕ σxz = µ ˆ 2 − 2 + . ∂ x∂ z ∂x ∂ z2
u=
(6)
(7)
By way of this transformation, the vectorial equation (2) becomes two scalar equations. The following step consists of applying the Fourier Transform on both the X axis and the time domain:
˜
f˜ (k, z , ω) =
∫
∞ −∞
∫
∞ −∞
f (x, z , t )ei(ωt −kx) dxdt
(8)
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
f ( x, z , t ) =
1 4π
∞
∫ 2
∫
∞
285
˜
f˜ (k, z , ω)ei(kx−ωt ) dkdω
(9)
−∞
−∞
where (9) represents the inverse transform. Applying (8) to Eq. (2), previously expressed as a function of (6) and (7), produces the following system of ordinary differential equations which are dependent only on Z: d2 ϕ˜˜ dz 2
− R2L ϕ˜˜ = 0
˜
˜ d2 ψ dz 2
−
R2T
(10)
ψ˜˜ = 0
where the coefficients RL and RT are: R2L = k2 − R2T
ω2 cL2 −
iω(λ∗ +2µ∗ )
=k −
cT2 −
iωµ∗
ρ
ω2
2
(11)
.
ρ
Representing the wave transmission velocity in the domain of the wave number and the frequency. This is why they depend on cL and cT , which, in turn, are the velocities of the longitudinal and transverse wave in the time domain, as defined below:
(λ + 2µ) ρ µ cT = . ρ cL =
(12)
It is now possible to solve system (10), obtaining for each layer:
ϕ˜˜ j = Aj1 (k, ω)eRLj z + Aj2 (k, ω)e−RLj z ψ˜˜ j = Aj3 (k, ω)eRTj z + Aj4 (k, ω)e−RTj z
(13)
where the sub index j refers to the j-th layer. Therefore, considering (13) and applying the Fourier Transform defined in (8) to Eqs. (6) and (7) the following expression of displacements and stresses is obtained: u˜˜ j (k, z , ω) = ik(Aj1 (k, ω)eRLj z + Aj2 (k, ω)e−RLj z ) + RTj (Aj3 (k, ω)eRjj z − Aj4 (k, ω)e−Rjj z )
(14a)
v˜˜ j (k, z , ω) = RLj (Aj1 (k, ω)eRLj z − Aj2 (k, ω)e−RLj z ) − ik(Aj3 (k, ω)eRTj z + Aj4 (k, ω)e−RTj z )
(14b)
σ˜˜ zz j (k, z , ω) = Cj1 (Aj1 (k, ω)eRLj z + Aj2 (k, ω)e−RLj z ) + Cj2 (Aj3 (k, ω)eRTj z − Aj4 (k, ω)e−RTj z )
(14c)
σ˜˜ xz j (k, z , ω) = Dj1 (Aj1 (k, ω)e
(14d)
RLj z
− Aj2 (k, ω)e
−RLj z
) + Dj2 (Aj3 (k, ω)e
RTj z
+ Aj4 (k, ω)e
−RTj z
)
where:
˜
˜
˜ˆ )R2 − λˆ j k ˆ j + 2µ Cj1 = (λ j Lj ˜ˆ RTj Cj2 = −2ikµ j ˜ˆ RLj Dj1 = 2ikµ j
(15)
˜ˆ (k2 + R2 ). Dj2 = µ j Tj Leaving only the different coefficients Aj,p to be found and the problem will be solved. The boundary conditions defined in (5) now need to be transformed to the frequency domain as well as the wave number using (8), making: u˜˜ 1 (k, h1 , ω) = 0
(16a)
u˜˜ 2 (k, h1 , ω) = 0
(16b)
˜˜ x, t ) v˜˜ 1 (k, h1 , ω) = w(
(16c)
˜˜ x, t ) v˜˜ 2 (k, h1 , ω) = w(
(16d)
σ˜˜ xz1 (k, 0, ω) = 0
(16e)
σ˜˜ zz1 (k, 0, ω) = 0
(16f)
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u˜˜ 2 (k, h1 + h2 , ω) = u˜˜ 3 (k, h1 + h2 , ω)
(16g)
v˜˜ 2 (k, h1 + h2 , ω) = v˜˜ 3 (k, h1 + h2 , ω)
(16h)
σ˜˜ xz2 (k, h1 + h2 , ω) = σ˜˜ xz3 (k, h1 + h2 , ω)
(16i)
σ˜˜ zz2 (k, h1 + h2 , ω) = σ˜˜ zz3 (k, h1 + h2 , ω)
(16j)
u˜˜ 3 (k, ∞, ω) = v˜˜ 3 (k, ∞, ω) = σ˜˜ xzg (k, ∞, ω) = σ˜˜ zzg (k, ∞, ω) = 0.
(16k)
The Timoshenko beam equation (1) is also transformed using Fourier:
Ak Gk2
1+
Ak G I ρω2 − Ak G − EIk2
˜˜ k, ω) = q˜˜ (k, ω) − aσzz 2(k, h1 , ω). − ρ Aω2 w(
(17)
Therefore, by taking into account the boundary conditions (16), the beam equation (17) and the Eqs. (14), the following algebraic system can be proposed: MA = q
(18)
where: A = {Aj,p }T j = 1, 2, 3 p = 1, 2, 3, 4
q= 0
0
1
0
0
0
0
0
0
0
T
q˜˜ (k, ω)
(19)
M = {Mm,n } m = 1, 2, . . . , 10 n = 1, 2, . . . , 10. Note that the system (18) has a dimension of 10 × 10 instead of 12 × 12. This is because condition (16k) necessarily implies that A3,1 = A3,3 = 0. The terms of the matrix M are detailed in the Appendix. The system thus expressed is solved using Kramer’s law to obtain the various coefficients Aj,p so that Eqs. (14) are completely defined. The last step is to apply the inverse transform (9) to Eqs. (14) to get the ground displacements and stresses in the time domain. However, the mathematical complexity of the expressions solved in the frequency domain considerably complicates the analytical solution of this process. Therefore, instead of applying the transform (9), the calculation is simplified using the steps described below. Firstly, given that the objective is to obtain only the stationary solution, a unique point can be fixed (e.g. x = 0, z = 0) to reduce the number of variables. Next, the expression of the transform for sinusoid functions is considered, which in the case of vertical displacement yields:
vj (0, 0, t ) =
Fi 4π
∫
∞
[˜vj (k, kV − Ω )e−i(kV −Ω ) + v˜ j (k, kV + Ω )e−i(kV +Ω ) ]dk
(20)
−∞
where vj is the vertical displacement, V is the speed of the load displacement and Ω is the frequency for each of the harmonics considered. Finally, the integral (20) is transformed into a summation by discretizing variable k (wave number) so that:
vj (0, 0, t ) =
N Pi −
2VT n=1
[˜vj (k(n), k(n)V − Ω )e−i(k(n)V −Ω ) + v˜ j (k(n), k(n)V + Ω )e−i(k(n)V +Ω ) ].
(21)
The acceleration comes from twice deriving expression (21) with respect to time. Note that this is the acceleration due to one axle of the tram. The final step to obtain the complete solution of the problem is to consider the static load. As stated previously, the effect of this load is studied separately using Zimmermann’s formulation, according to which the deformation of a rail on a continuous support and under a point load is: Q
4EI
.
aC − x x x e L cos + sin (22) 2bC 4EI L L where Q is the applied load, a the track width, E the Young modulus of the track, I the inertia of the rail, C an equivalent ballast coefficient for the material under the line, and L is an elastic length defined as: y0 =
L=
4
aC
4
(23)
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Table 1 Parameters of the model.
λ∗j µ∗j
Damping coefficients for each layer
k1
Track spring constant
Fig. 4. Tram’s structure.
Eq. (22) can be expressed as a function of time, considering speed V for the tram so that x = Vt. This yields an equation which provides the deformation of point x = 0 for a load displacing at speed V . Twice deriving this expression leaves acceleration at x = 0 due to static load, which added to the acceleration obtained using (21) for the harmonic loads provides the complete ground accelerogram for the passing of a single axle. Finally, this accelerogram is replicated with the gap corresponding to the distance between the different axles, hence giving the ground acceleration at a specific point due to the entire tram. 3.3. Calibration and validation Once the model has been built, a series of parameters (Table 1), with as yet unknown values must be calibrated. The rest of the model’s track and tram parameters are already known data or are easily obtained from simple calculus. A particular case taken from the bibliography [21] is the ballast coefficient C required by Zimmermann’s model. The parameters presented in Table 1 are calibrated by comparing the accelerations provided by the model with an accelerogram measured directly on Line 1 of the Alicante tram system. The values of the parameters are modified until a suitable fit is obtained. This fit is made qualitatively by checking the following criteria: -
Value of the maximum and minimum peaks. Average time between peaks. Time the signal takes to grow to the first peak. Attenuation time from the last peak.
The first two of these receive the greatest weight because they refer to the area with greatest amplitude (both positive and negative) in the vibration signal, which could have most influence in the area surrounding the line. Once the model is calibrated, it is validated using another accelerogram measured under the same conditions. Once again the model’s fit to the data is evaluated using the same criteria as explained before. 4. Data collection All the data used for the calibration and validation of the model come from measurements taken on Line 1 of the Alicante tram network. The measurements were taken when the tram passed by and when the line was empty using 2 FiberSensingTM triaxial accelerographs based on optical fibre technology. These sensors measure the variation of light wavelength caused by ground acceleration in terms of distance (nanometres). These values are then transformed into acceleration through the sensors sensitivity provided by the manufacturer. The sensors used have a sensitivity of 0.851 nm/g and 0.060 nm/g respectively and were placed at about 30 cm from the rail. The data obtained, once turned into acceleration, is plotted in the form of accelerograms used for model calibration and validation. In addition to this, frequency spectra were also obtained by applying the Discrete Fourier Transform (DFT) to the data. These spectra were used to obtain the main frequencies (i.e. those giving greater and distinctive peaks of acceleration) of the signal which were assumed to be related to the different rail defects present on that specific line. These frequencies (ω ¯ i ) were used as input data for the auxiliary quarter car model to obtain the different harmonic loads to be implemented in the main model. As explained before, this aims to introduce nonlinear effects due to track irregularities in a linear model. The tram in service on Line 1 is a BOMBARDIER FLEXITY OUTLOOK with a total length of 32.37 m and a maximum service speed of 70 km/h. Each tram consists of 5 modules and three bogies as shown in Fig. 4. The lead and rear bogies are motorized whilst the central bogie is not. Data referring to the size, shape and mass of the vehicle have been provided by the Alicante tram network operating company FGV.
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J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291 acceleration (m/s2)
5
10
20
30
40
time (s)
50
-5
Fig. 5. Modelled accelerations. acceleration (m/s2)
8 6 4 2 0
22
24
26
28
30
32
34
time (s)
-2 -4 -6
Fig. 6. Measured accelerations.
5. Analysis of results The results obtained and their evaluation within the framework of the data collected is presented below. Fig. 5 shows the accelerogram (acceleration in m/s2 vs. time in seconds) provided by the calibrated model. The graph clearly shows how the signal grows up to a series of maxima and later drops off, simulating the passing of a tram over the study’s reference point. The maxima can be broken down into 6 consecutive peaks grouped in pairs, corresponding to the six axles (three bogies). These peaks reach a magnitude of about 8 m/s2 when positive and around 7 m/s2 when negative which agree with the observed data presented below. Fig. 6 represents the accelerogram recorded on the line, showing maximum and minimum observed accelerations more or less corresponding to those produced by the model. The track measurements show more acceleration peaks than the model. These secondary peaks do not correspond to any axles but are probably due to other effects which the model cannot simulate because of its predetermined hypothesis (homogeneity, isotropy, etc.). In fact, during the measuring phase, the welded joints between the rails were noticed to be quite irregular and could cause knocks as the vehicle passes over them, explaining the additional peaks in the signal. This kind of defect is not harmonic but rather singular and is thus not properly modelled by the set of harmonic loads defined. Moreover, only those track defects with greater amplitudes were included in the model and other harmonics may have been missed. Fig. 7 compares the two accelerograms (model in red, measurement in green) and shows a good fit between the peaks from the axles as well as in the attenuation of the wave. Certain discrepancies appear in the growth of the wave as the model wave presents a small oscillation which is not present in the measured wave. This could be due, once again, to the model’s basic hypothesis, because the formulation used corresponds only to the stationary solution of the wave equation and any transitory phenomena are lost in the calculations. In any case, the comparison criteria explained in Section 3.3 mean that the fit between model and data is sufficiently satisfactory to consider the model as calibrated. More details on these criteria can be found in Table 2, which also shows the discrepancy in the wave growth period mentioned above. The values of the calibrated parameters in accordance with the stated criteria are presented in Table 3. It is worth mentioning here that during calibration parameter λ* was found not to influence the result of the model and therefore its value does not need to be adjusted.
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289
acceleration (m/s2)
5
10
20
30
40
time (s)
-5
Fig. 7. Comparison between modelled and measured accelerations (model in red, data in green). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 2 Calibration comparison criteria. Criteria
Model
Data
Difference (%)
Maximum peak (m/s2 ) Minimum peak (m/s2 ) T between peaks (s) Growth (s) Attenuation (s)
8.56 6.68 1.43 3.8 3.8
7.76 7.12 1.5 3 4
10 6 4.6 26 5
Table 3 Calibrated model parameters. Layer 1
λ∗ (Pa) µ∗ (Pa)
15E6
k1 (N/m)
Layer 2 No influence 15E6 2E9
Layer 3 10E6
acceleration (m/s2)
5
10
20
30
40
time (s)
-5
Fig. 8. Validation of the model (model in red, data in green). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
However, parameter µ∗ does notably affect the form of the signal, but because only data on surface vibrations is available only the parameter corresponding to the first layer has been truly calibrated. The value of µ∗ in the other layers is assumed to be similar to the surface layer because all the layers are made up of concrete with similar mechanical characteristics. The value of parameter k1 has a noteworthy influence on the amplitude of the signal’s peaks. The model was validated by comparing the modelled signal (with the calibrated parameters) with another measured signal obtained under the same conditions. Fig. 8 shows this comparison (model in red, measurement in green). Once again a good fit can be seen for the signal both in the amplitude of the peaks and in the time between the peaks. Nevertheless, there appear to be some discrepancies in the attenuation period similar to those already seen during the signal’s growth. Table 4 shows the comparison criteria for the validation.
290
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291 Table 4 Comparison criteria for validation. Criteria
Model
Data
Difference (%)
Maximum peak (m/s2 ) Minimum peak (m/s2 ) T between peaks (s) Growth (s) Attenuation (s)
8.56 6.68 1.43 3.8 3.8
6.63 6.31 1.3 3.1 3
29 6 10 23 27
According to those results, the validation fit appears to be slightly worse than the calibration fit. Along with the already mentioned discrepancies in the growth and attenuation times (which, as explained in Section 3.3, are less important criteria) there appears to be an overestimation of the maximum peak which has to be investigated. The fact that the disagreement appears only in the estimation of the maximum peaks and not in the minima could indicate that this discrepancy is likely due to a certain irregularity in the measurement rather than in the model. Although all the data were collected during a unique session and under similar conditions, two accelerometers with differing sensitivities were used, which, without doubt, could influence the magnitude of the acceleration registered in each case. Furthermore, given that an overestimation of the maximum vibration peak stays on the safe side, the model was considered to be acceptably validated. 6. Conclusions This article has presented the research done to develop an analytical model to evaluate the ground vibrations under a track caused by the passing of a tram, along with the model’s calibration and validation using real vibration data registered on Line 1 of the Alicante tram system. The model shows some advances on previous work, such as taking into account both harmonic loads and static loads in reproducing the displacement caused by the passing of a track borne vehicle. The load distribution corresponding to different axles is also considered by the model. Another advance is the application of the model to slab track, instead of to a more conventional ballast based track, leading the way to future research on vibration in urban areas where embedded rails are more commonly used. The model is based on the hypothesis of linearity and homogeneity in the materials and therefore has certain limitations, such as the impossibility of including discontinuities or localized defects in the track, and not reproducing secondary order phenomena. Anisotropy of the ground, which is likely found at some points even in a rather homogeneous concrete slab, is also not modelled. All these limitations would require some changes in the model to somehow include them or even the implementation of a fully non-linear model in order to be completely addressed. The calculation of the harmonic loads could be further improved as the input data for the auxiliary model (quarter car) were approximately obtained from the measurement of vibrations. A direct measurement of the track’s geometry and its defects would allow a much more precise definition of these loads and thus a better result for the analytical model. Despite the limitations described, the proposed model quite accurately reproduces the phenomenon of track vibration, both in the magnitude of the peaks and in the approximate growth and attenuation of the signal. Therefore, it can be seen as a useful tool for further, wider research on the generation and transmission of ground vibrations and their effects on surrounding buildings. Acknowledgements The authors would like to thank GTP (Ente Gestor de la Red de Transporte y de Puertos de la Generalitat) for providing the track cross sections; and FGV (Ferrocarrils de la Generalitat Valenciana) for providing the vehicle technical data. They would also like to thank the company EDILON-SEDRA for providing the data on the elastomer installed on the line. Appendix. Matrix elements (10×10)
M
M1(5×5) M3(5×5)
=
ikeRL1 h1 0 M1 = K¯ RL1 eRL1 h1 RL1 eRL1 h1 D1,1
M2(5×5) M4(5×5)
ike−RL1 h1 0 −K¯ RL1 e−RL1 h1 −RL1 e−RL1 h1 −D1,1
RT 1 eRT 1 h1 0 −K¯ ikeRT 1 h1 −ikeRT 1 h1 D1,2
−RT 1 e−RT 1 h1 0
−K¯ ike−RT 1 h1 −ike−RT 1 h1 D1,2
0 RL2 h1
ike aC2,1 eRL2 h1 RL2 h1 −RL2 e 0
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
0
ike−RL2 h1 M2 = aC2,1 e−RL2 h1 −RL2 h1 RL2 e
0
C
C1,1 0 0 0 0
1,1
0 M3 = 0 0 0
0 RT 2 eRT 2 h1 aC2,2 eRT 2 h1 ikeRT 2 h1 0 C1,2 0 0 0 0
−RL2 (h1 +h2 )
ike −R e−RL2 (h1 +h2 ) M4 = L2 −R (h +h ) −D2,1 e L2 1 2 C2,1 e
0 0 0 0
K¯ = Ak Gk2
−RL2 (h1 +h2 )
1+
0 0 0 0 0
ike−RT 2 h1 0
−C1,2
0
0
−RT 2 e−RT 2 h1 −aC2,2 e−RT 2 h1
0
RL2 (h1 +h2 )
0 0 0 0 0
ike RL2 eRL2 (h1 +h2 ) RL2 (h1 +h2 ) D2,1 e C2,1 eRL2 (h1 +h2 ) 0
RT 2 e
291
RT 2 (h1 +h2 )
−ikeRT 2 (h1 +h2 ) D2,2 eRT 2 (h1 +h2 ) C2,2 e
RT 2 (h1 +h2 )
Ak G I ρω2 − Ak G − EIk2
0
−RT 2 (h1 +h2 )
−R T 2 e −ike−RT 2 (h1 +h2 ) −R (h1 +h2 )
D2,2T 2
−R (h +h ) −C2,2 T 2 1 2
0
−ike
−RL3 (h1 +h2 )
RL3 e−RL3 (h1 +h2 )
D3,1 e−RL3 (h1 +h2 )
−C3,1 e−RL3 (h1 +h2 )
0
−RT 3 (h1 +h2 )
RT 3 e ike−RT 2 (h1 +h2 )
−RT 3 (h1 +h2 ) −D3,2 e C3,2 e−RT 3 (h1 +h2 )
− ρ Aω2 .
References [1] CityRailways, CityRailways Magazine, (Online) Issue 2, Available at: http://www.cityrailways.net/issue2/ (accessed 12/03/10), 2009. [2] X. Sheng, C.J.C. Jones, D.J. Thompson, Prediction of ground vibration from trains using the wavenumber finite and boundary element methods, Journal of Sound and Vibration 293 (2006) 575–586. doi:10.1016/j.jsv.2005.08.040. [3] Y.B. Yang, H.H. Hung, Soil vibrations caused by underground moving trains, Journal of Geotechnical and Geoenvironmental Engineering 11 (134) (2008) 1633–1644. [4] M. Katou, T. Matsuoka, O. Yoshioka, Y. Sanada, T. Miyoshi, Numerical simulation study of ground vibrations using forces from wheels of a running high-speed train, Journal of Sound and Vibration 318 (2008) 830–849. doi:10.1016/j.jsv.2008.04.053. [5] E. Celebi, Three-dimensional modelling of train-track and sub-soil analysis for surface vibrations due to moving loads, Applied Mathematics and Computation 179 (2006) 209–230. [6] P. Galvín, J. Domínguez, Analysis of ground motion due to moving surface loads induced by high-speed trains, Engineering Analysis with Boundary Elements 31 (2007) 931–941. [7] F.F. Çalim, Dynamic analysis of beams on viscoelastic foundation, European Journal of Mechanics A/Solids 28 (2009) 469–476. [8] G. Muscolino, A. Palmeri, Response of beams resting on viscoelastically damped foundation to moving oscillators, International Journal of Solids and Structures 44 (2007) 1317–1336. [9] M. Schevenels, G. Lombaert, G. Degrande, D. Clouteau, The wave propagation in a beam on a random elastic foundation, Probabilistic Engineering Mechanics 22 (2007) 150–158. [10] A.V. Metrikine, A.C.W.M. Vrouwenvelder, Surface ground vibration due to a moving train in a tunnel: two-dimensional model, Journal of Sound and Vibration 234 (1) (2000) 43–66. [11] T. Mazilu, Green’s functions for analysis of dynamic response of wheel/rail to vertical excitation, Journal of Sound and Vibration 306 (2007) 31–58. doi:10.1016/j.jsv.2007.05.037. [12] P. Koziol, C. Mares, I. Esat, Wavelet approach to vibratory analysis of surface due to a load moving in the layer, International Journal of Solids and Structures 45 (2008) 2140–2159. [13] E.G. Florez, S. Cardona i Foix, L. Jordi Nebot, Time-frequency analysis of vibration signals taken on the foot from rail during pass of train, Sciencia et Technica 35 (2007) 243–247. [14] W.I. Liao, T.J. Teng, C.S. Yeh, A method for the response of an elastic half-space to moving sub-Rayleigh point loads, Journal of Sound and Vibration 284 (1–2) (2005) 173–188. doi:10.1016/j.jsv.2004.06.005. [15] L. Auersch, The excitation of ground vibration by rail traffic: theory of vehicle–track–soil interaction and measurements on high-speed lines, Journal of Sound and Vibration 284 (2005) 103–132. doi:10.1016/j.jsv.2009.10.007. [16] G. Lombaert, G. Degrande, Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains, Journal of Sound and Vibration 319 (3–5) (2009) 1036–1066. doi:10.1016/j.jsv.2005.08.059. [17] P. Salvador, J. Real, C. Zamorano, A. Villanueva, A procedure for the evaluation of vibrations induced by the passing of a train and its application to real railway traffic, Mathematical and Computer Modelling 53 (1–2) (2011) 42–54. [18] J.M. Gere, in: Paraninfo Thomson (Ed.), Timoshenko: Resistencia de Materiales (Timoshenko: Mechanics of Materials), Madrid, 2004. [19] M. Melis, Introducción a la Dinámica Vertical de la vía y Señales Digitales en Ferrocarriles (Introduction to Vertical Track Dynamics and Digital Signals in Railways). Ed. UPM, Madrid, 2008. [20] A. López, Infraestructuras Ferroviarias (Railway Infrastructures). Ed. UPC, Barcelona, 2006. [21] S. Anacleto Lupiáñez, 2009, Modelización de la Interacción Vía-Tranvía (Modellization of the Track–Tramway Interaction), M.Sc. Thesis, Universitat Politècnica de Catalunya (UPC).
Contents lists available at ScienceDirect
Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm
Modelling vibrations caused by tram movement on slab track line Julia Real a , Pablo Martínez a,∗ , Laura Montalbán a , Antonio Villanueva b a
Departamento de Ingeniería e Infraestructura de los Transportes, Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos, Universidad Politécnica de Valencia, 14 Camino de Vera, 46022 Valencia, Spain b
INECO, 1-8 Capitán Haya, 28020 Madrid, Spain
article
info
Article history: Received 5 November 2010 Received in revised form 7 February 2011 Accepted 7 February 2011 Keywords: Ground vibrations Slab track Tram Timoshenko beam Fourier Transform
abstract The recent growth in the use of the railway and particularly that of the tram in urban areas highlights the need to investigate and mitigate the harmful effects associated with these means of transport. The vibrations caused by passing vehicles can be a source of vibration and may damage buildings close to the line. This article aims to develop an analytical mathematical model for predicting the ground vibrations caused by the passing of trams along a slab track. The model is based on the wave equation and is solved in the frequency domain through the Fourier Transform. The loads caused by the vehicle are calculated using a quarter car model. The model is calibrated and validated with real data collected along the tram network of Alicante (Spain), providing a useful tool for future research on the transmission of ground vibrations. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction The growing need for mobility, along with worries about the environment and questions of sustainability mean public bodies are increasingly deciding in favour of developing rail borne transport systems. This drive is clearly shown not only in interurban transport links but also in the growth of trams and subways as the main means of urban transport. Many European cities have decided to develop and introduce their own tram networks as a way of improving urban mobility and reducing traffic congestion [1]. The expansion of tram networks is generally beneficial to the population; however any associated environmental impacts such as the vibrations generated by the passing of a vehicle should be evaluated before developing or extending an urban network. When a vehicle moves along the line it produces vibrations which are transmitted to the surrounding area through the air and the ground. The airborne vibration is felt in the form of sound which is not only irritating but could become harmful to public health. On the other hand, ground vibrations may reach the foundations of nearby buildings and introduce structurally damaging stresses and loads. Both effects become more important in the areas around tramways because their lines are generally closer to buildings and people. Therefore, a better understanding of the vibrations generated by these vehicles would allow a more efficient assessment and mitigation of any possible negative effects. In this context, the main objective of this paper is to develop, calibrate and validate an analytical model capable of predicting the ground vibrations under the line. The model presented here is an improvement of previous works in certain aspects such as the load modelling and the adaptation to slab tracks (see literature review below). It represents the first step
∗
Corresponding author. E-mail addresses: [email protected] (J. Real), [email protected] (P. Martínez), [email protected] (L. Montalbán), [email protected] (A. Villanueva). 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.02.010
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
281
Nomenclature a A Ak c2 cLj cTj C E Ej G hj I k1 k2 m1 m2 Pi Q V
λj λ∗j µj µ∗j ρ ρj υ υj ω¯ i
Track gauge Rail cross-sectional area Rail shear cross-sectional area Train primary damping coefficient Ground longitudinal wave velocity (j = 1, 2, 3) Ground shear wave velocity (j = 1, 2, 3) Equivalent ballast coefficient Rail Young modulus Ground Young modulus (j = 1, 2, 3) Rail shear modulus Layer thickness (j = 1, 2, 3) Rail inertia in y-axis direction Equivalent track spring constant Train primary spring constant Unsprung mass per axle Sprung mass per axle Harmonic load magnitude Static load per axle Train velocity First Lamé parameter (j = 1, 2, 3) First damping coefficient (j = 1, 2, 3) Second Lamé parameter (j = 1, 2, 3) Second damping coefficient (j = 1, 2, 3) Rail mass density Ground mass density (j = 1, 2, 3) Rail Poisson coefficient Ground Poisson coefficient (j = 1, 2, 3) Harmonic load frequency
within a wider study on the generation, transmission and effects of vibrations on structures near to a tramway in the city of Alicante (Spain): Project TRAVIESA. The article is structured in the following way: an initial literature review on previous work in this field is followed by the mathematical development of the model and its solution using the Fourier Transform. The model is then calibrated and validated using real vibration data collected on Line 1 of the Alicante tram network run by FGV (Ferrocarrils de la Generalitat Valenciana). The article closes with an analysis of the results and a presentation of the more important conclusions drawn. 2. Literature review A great deal of research has taken place over recent years on the generation and transmission of vibrations associated with railways. Much of the work concentrates on either the numerical or analytical mathematical modelling of this phenomenon. Numerical methods have the advantage of being adaptable to different configurations of the area being studied and allow the introduction of track irregularities or other localized factors. Examples can be found in [2,3], who used a finite elements method (FEM) to predict ground vibrations on the surface and in tunnels, respectively. Another noteworthy case was proposed by Katou et al. [4], who ran a three dimensional finite difference model (FDM) in the context of high speed trains. Celebi [5] made a comparison between a boundary elements method (BEM) and a flexible volume method (FVM), and Galvín and Domínguez [6] carried out a comprehensive study of the BEM method for railways. Analytical methods provide continuity to the solution in all the domains along with a greater physical consistency because their formulation directly depends on the physical parameters of the materials being considered. Those advantages have been taken into account when choosing an analytical method for this paper. A typical hypothesis in the railway field is to consider the rail as a beam supported on viscoelastic terrain subjected to variable loads in time. The general proposition of this hypothesis can be found in [7] and its application to the railway in the works of Muscolino and Palmeri [8], Schevenels et al. [9] and Metrikine and Vrouwenvelder [10]. This hypothesis is also assumed for the model developed in this paper. The way in which the equations are solved using analytical methods is worthy of further discussion. Some authors, like Mazilu [11] directly solve the problem in the time domain using Green’s function, but in general some type of integral transformation is used such as the Laplace Transform [7] or the Wavelet transform [12,13].
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J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
Fig. 1. Cross section of the railway (provided by GTP).
However, many authors prefer to use the Fourier Transform for solving equations in the frequency domain, because it allows the vibration spectrum to be studied and provides information on the main frequencies and amplitudes of the phenomenon. Examples of this procedure can be found in [10,12] as well as in [14]. These studies (particularly [10,12]) are the base of the method presented in this paper, but some aspects have been analysed to improve the modelling of the phenomenon. First of all, the way loads are formulated should be considered. In the case of numerical models, some authors have implemented a full account of the loads induced by a train [15,16]. However, in the case of analytical models, most of the reviewed authors, [8,10,12] choose to simulate a single harmonic point load which contemplates certain variability of the load but not the load distribution produced by a railway vehicle with multiple axles. This issue is addressed in this paper by means of introducing a set of point loads which are a closer approximation to the train actual effect over the track. Moreover, the load formulation used in this paper allows the study of both harmonic and static loads [17]. In this way dynamic loads caused by track irregularities are included in the model. This represents another improvement from previous works. Secondly, previous studies [12] use the Euler–Bernoulli beam equation to model the rail as a beam, while this paper applies the extended Timoshenko beam equation, hence including the rail strain due to shear stress within the global modelling of the vibration. Finally, it is worth noting that most of the reviewed research concentrates on the vibrations generated by conventional railway lines. Comparatively, there is not as much bibliography available on the vibrations caused by tramway networks typically built using a slab track typology. In this way, the application of the method presented here to a tramway line represents an interesting line of research because the negative consequences of the vibrations (noise, structural damage) have greater importance in an urban environment. 3. Methodology A detailed description of the methodology used in the development, calibration and validation of the model presented in this article follows below. 3.1. Mathematical model In the context of this article, vibration is a phenomenon which propagates energy through a continuous physical medium causing said medium to deform and suffer stress. To reproduce this phenomenon, an analytical formulation is developed based on the previous works of Metrikine and Vrouwenvelder [10], Koziol et al. [12] and Salvador et al. [17]. A cross section of the track being studied is shown in Fig. 1. The model is applied to a two dimensional cut (length and depth) in the ground under the track, and it evaluates the vertical and longitudinal displacements generated by the passing of a mobile load (the tram). The cross section is simplified by taking the two rails as a single beam and ignoring any discontinuity or irregularity in the Y axis. Fig. 2 shows the longitudinal section of the ground and the X and Z axis considered. The ground is assumed to be made up of three horizontal layers of viscoelastic, homogeneous and isotropic materials. The model is therefore linear and the superposition principle can be applied. Layer 1 corresponds to the surface layer of concrete HA-25, layer 2 is the layer which the rails sit on (also made of HA-25) and layer 3 is an average of the layer of HM-10 and the layer of levelling concrete. This latter layer is assumed to extend indefinitely in depth (Boussinesq half space) to simulate that the wave is not affected by any interaction in this direction. Both the rail and elastomeric covering are modelled as a beam of negligible thickness located at the interface between the first and second layer to simulate the fact that the rail is embedded in the concrete. The mechanical behaviour of the rail
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
283
Fig. 2. Geometry of the problem.
is defined by the Timoshenko beam theory [18], producing the following system of equations:
∂ 2w ∂ ∂w ρA 2 = Ak G −θ + q(x, t ) ∂t ∂x ∂x ∂ ∂θ ∂w ∂ 2θ EI + Ak G −θ ρI 2 = ∂t ∂x ∂x ∂x
(1)
where G is the shear modulus, w is the vertical displacement and θ is the angular displacement. The mechanical characteristics of the modelled rail are averaged between the steel and the elastomer. The ground dynamics are governed by the following wave equation, expressed in vectorial terms:
(λˆ + µ)∇ ˆ x,z (∇x,z d) + µ∇ ˆ x2,z d = ρ
∂ 2d ∂t2
(2)
ˆ and µ where d is the displacement vector, ρ is the density of the material and λ ˆ are operators describing the viscoelasticity of the ground, defined in the following way: ∂ ∂t (3) ∗ ∂ µ ˆ =µ+µ ∂t where λ and µ are Lamé parameters and λ* and µ∗ are damping coefficients for the ground which must be calibrated using λˆ = λ + λ∗
experimental data. The loads acting on the model are due to the interaction between wheel and rail. Two groups can generally be distinguished: - Load due to the weight of the vehicle. This is a quasi-static load which moves along the rail at velocity V . - Harmonic loads with different amplitudes and frequencies coming from the vehicle (considered as a system of masses and springs) and caused by certain imperfections and/or discontinuities in both the rail and the wheel. These could be defects such as wear and tear on the rail, rail corrugation, wear on the wheels, rail joints, etc. The resulting wave transmitted along the track and through the ground is formed from the superposition of the waves generated by all the loads described above. In this way, the effect of loads caused by track and wheel irregularities is taken into account in this otherwise completely linear model, providing a solution closer to reality [17]. The harmonic loads were modelled here using an auxiliary vehicle model called quarter car model, as defined by Melis [19]. Fig. 3 shows the layout of the model, representing an axle of the tram. Where m1 corresponds to the unsprung mass per axle of the vehicle, m2 to the sprung mass per axle, k2 and c2 represent the tram primary spring constant and primary damping coefficient respectively and k1 represents the equivalent track spring constant, the latter being another parameter which needs to be calibrated. The auxiliary model and known vehicle data provide the different harmonic loads acting on the ground, each one expressed in accordance with the following equation: Fi (t ) = Pi cos(ω ¯ it )
(4)
where Pi represents the amplitude and ω ¯ i the harmonic load frequency of the i-th load Fi . The resulting force is equal to the sum of all the harmonics, which agrees with the ground linearity hypothesis assumed beforehand. Owing to the absence of data on line defects, the frequencies ω ¯ i of the main harmonics have been obtained approximately from the average line vibration spectra, as explained in Section 4. It should be noted that the previous formulation is not valid for ω ¯ i = 0, which is the case corresponding to the static load. Therefore, the static load has to be modelled independently by calculating the rail deformation due to the passing of
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J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
Fig. 3. Quarter car model.
the load in accordance with the Zimmermann model [20] and assuming that the terrain under the rail deforms to the same extent. The resulting vibration is added to that obtained for the rest of the loads, again following the phenomenon’s linearity hypothesis. Once the leading equations for the phenomenon and the loads acting on the ground have been defined, boundary conditions must be formulated to delimit the problem and permit its solution (5). Note that from here on, the sub indexes 1, 2 and 3 refer to the first, second and third layer of the line section, respectively: u1 ( x , h 1 , t ) = 0
(5a)
u2 ( x , h 1 , t ) = 0 v1 (x, h1 , t ) = w(x, t ) v2 (x, h1 , t ) = w(x, t ) σxz1 (x, 0, t ) = 0 σzz1 (x, 0, t ) = 0 u2 (x, h1 + h2 , t ) = u3 (x, h1 + h2 , t ) v2 (x, h1 + h2 , t ) = v3 (x, h1 + h2 , t ) σxz2 (x, h1 + h2 , t ) = σxz3 (x, h1 + h2 , t ) σzz2 (x, h1 + h2 , t ) = σzz3 (x, h1 + h2 , t ) u3 (x, ∞, t ) = v3 (x, ∞, t ) = σxzg (x, ∞, t ) = σzzg (x, ∞, t ) = 0.
(5b) (5c) (5d) (5e) (5f) (5g) (5h) (5i) (5j) (5k)
In accordance with Metrikine and Vrouwenvelder [10], this group of boundary conditions is enough to provide the stationary solution for Eq. (2) and any initial conditions for the problem need not be defined. 3.2. Solution The model, once formulated as shown in Section 3.1, is solved as follows. According to Koziol et al. [12], Eq. (2) can be expressed as a function of the Lamé potentials in such a way that the horizontal and vertical displacements of the ground and the corresponding stresses are as follows:
∂ϕ ∂ψ + ∂x ∂z ∂ϕ ∂ψ v= − ∂z ∂x 2 2 ∂ ϕ ∂ 2ψ ∂ ϕ ∂ 2ψ σzz = λˆ + + 2 µ ˆ − ∂ x2 ∂ z2 ∂ z2 ∂ x∂ z 2 2 2 ∂ ψ ∂ ψ ∂ ϕ σxz = µ ˆ 2 − 2 + . ∂ x∂ z ∂x ∂ z2
u=
(6)
(7)
By way of this transformation, the vectorial equation (2) becomes two scalar equations. The following step consists of applying the Fourier Transform on both the X axis and the time domain:
˜
f˜ (k, z , ω) =
∫
∞ −∞
∫
∞ −∞
f (x, z , t )ei(ωt −kx) dxdt
(8)
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
f ( x, z , t ) =
1 4π
∞
∫ 2
∫
∞
285
˜
f˜ (k, z , ω)ei(kx−ωt ) dkdω
(9)
−∞
−∞
where (9) represents the inverse transform. Applying (8) to Eq. (2), previously expressed as a function of (6) and (7), produces the following system of ordinary differential equations which are dependent only on Z: d2 ϕ˜˜ dz 2
− R2L ϕ˜˜ = 0
˜
˜ d2 ψ dz 2
−
R2T
(10)
ψ˜˜ = 0
where the coefficients RL and RT are: R2L = k2 − R2T
ω2 cL2 −
iω(λ∗ +2µ∗ )
=k −
cT2 −
iωµ∗
ρ
ω2
2
(11)
.
ρ
Representing the wave transmission velocity in the domain of the wave number and the frequency. This is why they depend on cL and cT , which, in turn, are the velocities of the longitudinal and transverse wave in the time domain, as defined below:
(λ + 2µ) ρ µ cT = . ρ cL =
(12)
It is now possible to solve system (10), obtaining for each layer:
ϕ˜˜ j = Aj1 (k, ω)eRLj z + Aj2 (k, ω)e−RLj z ψ˜˜ j = Aj3 (k, ω)eRTj z + Aj4 (k, ω)e−RTj z
(13)
where the sub index j refers to the j-th layer. Therefore, considering (13) and applying the Fourier Transform defined in (8) to Eqs. (6) and (7) the following expression of displacements and stresses is obtained: u˜˜ j (k, z , ω) = ik(Aj1 (k, ω)eRLj z + Aj2 (k, ω)e−RLj z ) + RTj (Aj3 (k, ω)eRjj z − Aj4 (k, ω)e−Rjj z )
(14a)
v˜˜ j (k, z , ω) = RLj (Aj1 (k, ω)eRLj z − Aj2 (k, ω)e−RLj z ) − ik(Aj3 (k, ω)eRTj z + Aj4 (k, ω)e−RTj z )
(14b)
σ˜˜ zz j (k, z , ω) = Cj1 (Aj1 (k, ω)eRLj z + Aj2 (k, ω)e−RLj z ) + Cj2 (Aj3 (k, ω)eRTj z − Aj4 (k, ω)e−RTj z )
(14c)
σ˜˜ xz j (k, z , ω) = Dj1 (Aj1 (k, ω)e
(14d)
RLj z
− Aj2 (k, ω)e
−RLj z
) + Dj2 (Aj3 (k, ω)e
RTj z
+ Aj4 (k, ω)e
−RTj z
)
where:
˜
˜
˜ˆ )R2 − λˆ j k ˆ j + 2µ Cj1 = (λ j Lj ˜ˆ RTj Cj2 = −2ikµ j ˜ˆ RLj Dj1 = 2ikµ j
(15)
˜ˆ (k2 + R2 ). Dj2 = µ j Tj Leaving only the different coefficients Aj,p to be found and the problem will be solved. The boundary conditions defined in (5) now need to be transformed to the frequency domain as well as the wave number using (8), making: u˜˜ 1 (k, h1 , ω) = 0
(16a)
u˜˜ 2 (k, h1 , ω) = 0
(16b)
˜˜ x, t ) v˜˜ 1 (k, h1 , ω) = w(
(16c)
˜˜ x, t ) v˜˜ 2 (k, h1 , ω) = w(
(16d)
σ˜˜ xz1 (k, 0, ω) = 0
(16e)
σ˜˜ zz1 (k, 0, ω) = 0
(16f)
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u˜˜ 2 (k, h1 + h2 , ω) = u˜˜ 3 (k, h1 + h2 , ω)
(16g)
v˜˜ 2 (k, h1 + h2 , ω) = v˜˜ 3 (k, h1 + h2 , ω)
(16h)
σ˜˜ xz2 (k, h1 + h2 , ω) = σ˜˜ xz3 (k, h1 + h2 , ω)
(16i)
σ˜˜ zz2 (k, h1 + h2 , ω) = σ˜˜ zz3 (k, h1 + h2 , ω)
(16j)
u˜˜ 3 (k, ∞, ω) = v˜˜ 3 (k, ∞, ω) = σ˜˜ xzg (k, ∞, ω) = σ˜˜ zzg (k, ∞, ω) = 0.
(16k)
The Timoshenko beam equation (1) is also transformed using Fourier:
Ak Gk2
1+
Ak G I ρω2 − Ak G − EIk2
˜˜ k, ω) = q˜˜ (k, ω) − aσzz 2(k, h1 , ω). − ρ Aω2 w(
(17)
Therefore, by taking into account the boundary conditions (16), the beam equation (17) and the Eqs. (14), the following algebraic system can be proposed: MA = q
(18)
where: A = {Aj,p }T j = 1, 2, 3 p = 1, 2, 3, 4
q= 0
0
1
0
0
0
0
0
0
0
T
q˜˜ (k, ω)
(19)
M = {Mm,n } m = 1, 2, . . . , 10 n = 1, 2, . . . , 10. Note that the system (18) has a dimension of 10 × 10 instead of 12 × 12. This is because condition (16k) necessarily implies that A3,1 = A3,3 = 0. The terms of the matrix M are detailed in the Appendix. The system thus expressed is solved using Kramer’s law to obtain the various coefficients Aj,p so that Eqs. (14) are completely defined. The last step is to apply the inverse transform (9) to Eqs. (14) to get the ground displacements and stresses in the time domain. However, the mathematical complexity of the expressions solved in the frequency domain considerably complicates the analytical solution of this process. Therefore, instead of applying the transform (9), the calculation is simplified using the steps described below. Firstly, given that the objective is to obtain only the stationary solution, a unique point can be fixed (e.g. x = 0, z = 0) to reduce the number of variables. Next, the expression of the transform for sinusoid functions is considered, which in the case of vertical displacement yields:
vj (0, 0, t ) =
Fi 4π
∫
∞
[˜vj (k, kV − Ω )e−i(kV −Ω ) + v˜ j (k, kV + Ω )e−i(kV +Ω ) ]dk
(20)
−∞
where vj is the vertical displacement, V is the speed of the load displacement and Ω is the frequency for each of the harmonics considered. Finally, the integral (20) is transformed into a summation by discretizing variable k (wave number) so that:
vj (0, 0, t ) =
N Pi −
2VT n=1
[˜vj (k(n), k(n)V − Ω )e−i(k(n)V −Ω ) + v˜ j (k(n), k(n)V + Ω )e−i(k(n)V +Ω ) ].
(21)
The acceleration comes from twice deriving expression (21) with respect to time. Note that this is the acceleration due to one axle of the tram. The final step to obtain the complete solution of the problem is to consider the static load. As stated previously, the effect of this load is studied separately using Zimmermann’s formulation, according to which the deformation of a rail on a continuous support and under a point load is: Q
4EI
.
aC − x x x e L cos + sin (22) 2bC 4EI L L where Q is the applied load, a the track width, E the Young modulus of the track, I the inertia of the rail, C an equivalent ballast coefficient for the material under the line, and L is an elastic length defined as: y0 =
L=
4
aC
4
(23)
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287
Table 1 Parameters of the model.
λ∗j µ∗j
Damping coefficients for each layer
k1
Track spring constant
Fig. 4. Tram’s structure.
Eq. (22) can be expressed as a function of time, considering speed V for the tram so that x = Vt. This yields an equation which provides the deformation of point x = 0 for a load displacing at speed V . Twice deriving this expression leaves acceleration at x = 0 due to static load, which added to the acceleration obtained using (21) for the harmonic loads provides the complete ground accelerogram for the passing of a single axle. Finally, this accelerogram is replicated with the gap corresponding to the distance between the different axles, hence giving the ground acceleration at a specific point due to the entire tram. 3.3. Calibration and validation Once the model has been built, a series of parameters (Table 1), with as yet unknown values must be calibrated. The rest of the model’s track and tram parameters are already known data or are easily obtained from simple calculus. A particular case taken from the bibliography [21] is the ballast coefficient C required by Zimmermann’s model. The parameters presented in Table 1 are calibrated by comparing the accelerations provided by the model with an accelerogram measured directly on Line 1 of the Alicante tram system. The values of the parameters are modified until a suitable fit is obtained. This fit is made qualitatively by checking the following criteria: -
Value of the maximum and minimum peaks. Average time between peaks. Time the signal takes to grow to the first peak. Attenuation time from the last peak.
The first two of these receive the greatest weight because they refer to the area with greatest amplitude (both positive and negative) in the vibration signal, which could have most influence in the area surrounding the line. Once the model is calibrated, it is validated using another accelerogram measured under the same conditions. Once again the model’s fit to the data is evaluated using the same criteria as explained before. 4. Data collection All the data used for the calibration and validation of the model come from measurements taken on Line 1 of the Alicante tram network. The measurements were taken when the tram passed by and when the line was empty using 2 FiberSensingTM triaxial accelerographs based on optical fibre technology. These sensors measure the variation of light wavelength caused by ground acceleration in terms of distance (nanometres). These values are then transformed into acceleration through the sensors sensitivity provided by the manufacturer. The sensors used have a sensitivity of 0.851 nm/g and 0.060 nm/g respectively and were placed at about 30 cm from the rail. The data obtained, once turned into acceleration, is plotted in the form of accelerograms used for model calibration and validation. In addition to this, frequency spectra were also obtained by applying the Discrete Fourier Transform (DFT) to the data. These spectra were used to obtain the main frequencies (i.e. those giving greater and distinctive peaks of acceleration) of the signal which were assumed to be related to the different rail defects present on that specific line. These frequencies (ω ¯ i ) were used as input data for the auxiliary quarter car model to obtain the different harmonic loads to be implemented in the main model. As explained before, this aims to introduce nonlinear effects due to track irregularities in a linear model. The tram in service on Line 1 is a BOMBARDIER FLEXITY OUTLOOK with a total length of 32.37 m and a maximum service speed of 70 km/h. Each tram consists of 5 modules and three bogies as shown in Fig. 4. The lead and rear bogies are motorized whilst the central bogie is not. Data referring to the size, shape and mass of the vehicle have been provided by the Alicante tram network operating company FGV.
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J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291 acceleration (m/s2)
5
10
20
30
40
time (s)
50
-5
Fig. 5. Modelled accelerations. acceleration (m/s2)
8 6 4 2 0
22
24
26
28
30
32
34
time (s)
-2 -4 -6
Fig. 6. Measured accelerations.
5. Analysis of results The results obtained and their evaluation within the framework of the data collected is presented below. Fig. 5 shows the accelerogram (acceleration in m/s2 vs. time in seconds) provided by the calibrated model. The graph clearly shows how the signal grows up to a series of maxima and later drops off, simulating the passing of a tram over the study’s reference point. The maxima can be broken down into 6 consecutive peaks grouped in pairs, corresponding to the six axles (three bogies). These peaks reach a magnitude of about 8 m/s2 when positive and around 7 m/s2 when negative which agree with the observed data presented below. Fig. 6 represents the accelerogram recorded on the line, showing maximum and minimum observed accelerations more or less corresponding to those produced by the model. The track measurements show more acceleration peaks than the model. These secondary peaks do not correspond to any axles but are probably due to other effects which the model cannot simulate because of its predetermined hypothesis (homogeneity, isotropy, etc.). In fact, during the measuring phase, the welded joints between the rails were noticed to be quite irregular and could cause knocks as the vehicle passes over them, explaining the additional peaks in the signal. This kind of defect is not harmonic but rather singular and is thus not properly modelled by the set of harmonic loads defined. Moreover, only those track defects with greater amplitudes were included in the model and other harmonics may have been missed. Fig. 7 compares the two accelerograms (model in red, measurement in green) and shows a good fit between the peaks from the axles as well as in the attenuation of the wave. Certain discrepancies appear in the growth of the wave as the model wave presents a small oscillation which is not present in the measured wave. This could be due, once again, to the model’s basic hypothesis, because the formulation used corresponds only to the stationary solution of the wave equation and any transitory phenomena are lost in the calculations. In any case, the comparison criteria explained in Section 3.3 mean that the fit between model and data is sufficiently satisfactory to consider the model as calibrated. More details on these criteria can be found in Table 2, which also shows the discrepancy in the wave growth period mentioned above. The values of the calibrated parameters in accordance with the stated criteria are presented in Table 3. It is worth mentioning here that during calibration parameter λ* was found not to influence the result of the model and therefore its value does not need to be adjusted.
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
289
acceleration (m/s2)
5
10
20
30
40
time (s)
-5
Fig. 7. Comparison between modelled and measured accelerations (model in red, data in green). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Table 2 Calibration comparison criteria. Criteria
Model
Data
Difference (%)
Maximum peak (m/s2 ) Minimum peak (m/s2 ) T between peaks (s) Growth (s) Attenuation (s)
8.56 6.68 1.43 3.8 3.8
7.76 7.12 1.5 3 4
10 6 4.6 26 5
Table 3 Calibrated model parameters. Layer 1
λ∗ (Pa) µ∗ (Pa)
15E6
k1 (N/m)
Layer 2 No influence 15E6 2E9
Layer 3 10E6
acceleration (m/s2)
5
10
20
30
40
time (s)
-5
Fig. 8. Validation of the model (model in red, data in green). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
However, parameter µ∗ does notably affect the form of the signal, but because only data on surface vibrations is available only the parameter corresponding to the first layer has been truly calibrated. The value of µ∗ in the other layers is assumed to be similar to the surface layer because all the layers are made up of concrete with similar mechanical characteristics. The value of parameter k1 has a noteworthy influence on the amplitude of the signal’s peaks. The model was validated by comparing the modelled signal (with the calibrated parameters) with another measured signal obtained under the same conditions. Fig. 8 shows this comparison (model in red, measurement in green). Once again a good fit can be seen for the signal both in the amplitude of the peaks and in the time between the peaks. Nevertheless, there appear to be some discrepancies in the attenuation period similar to those already seen during the signal’s growth. Table 4 shows the comparison criteria for the validation.
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J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291 Table 4 Comparison criteria for validation. Criteria
Model
Data
Difference (%)
Maximum peak (m/s2 ) Minimum peak (m/s2 ) T between peaks (s) Growth (s) Attenuation (s)
8.56 6.68 1.43 3.8 3.8
6.63 6.31 1.3 3.1 3
29 6 10 23 27
According to those results, the validation fit appears to be slightly worse than the calibration fit. Along with the already mentioned discrepancies in the growth and attenuation times (which, as explained in Section 3.3, are less important criteria) there appears to be an overestimation of the maximum peak which has to be investigated. The fact that the disagreement appears only in the estimation of the maximum peaks and not in the minima could indicate that this discrepancy is likely due to a certain irregularity in the measurement rather than in the model. Although all the data were collected during a unique session and under similar conditions, two accelerometers with differing sensitivities were used, which, without doubt, could influence the magnitude of the acceleration registered in each case. Furthermore, given that an overestimation of the maximum vibration peak stays on the safe side, the model was considered to be acceptably validated. 6. Conclusions This article has presented the research done to develop an analytical model to evaluate the ground vibrations under a track caused by the passing of a tram, along with the model’s calibration and validation using real vibration data registered on Line 1 of the Alicante tram system. The model shows some advances on previous work, such as taking into account both harmonic loads and static loads in reproducing the displacement caused by the passing of a track borne vehicle. The load distribution corresponding to different axles is also considered by the model. Another advance is the application of the model to slab track, instead of to a more conventional ballast based track, leading the way to future research on vibration in urban areas where embedded rails are more commonly used. The model is based on the hypothesis of linearity and homogeneity in the materials and therefore has certain limitations, such as the impossibility of including discontinuities or localized defects in the track, and not reproducing secondary order phenomena. Anisotropy of the ground, which is likely found at some points even in a rather homogeneous concrete slab, is also not modelled. All these limitations would require some changes in the model to somehow include them or even the implementation of a fully non-linear model in order to be completely addressed. The calculation of the harmonic loads could be further improved as the input data for the auxiliary model (quarter car) were approximately obtained from the measurement of vibrations. A direct measurement of the track’s geometry and its defects would allow a much more precise definition of these loads and thus a better result for the analytical model. Despite the limitations described, the proposed model quite accurately reproduces the phenomenon of track vibration, both in the magnitude of the peaks and in the approximate growth and attenuation of the signal. Therefore, it can be seen as a useful tool for further, wider research on the generation and transmission of ground vibrations and their effects on surrounding buildings. Acknowledgements The authors would like to thank GTP (Ente Gestor de la Red de Transporte y de Puertos de la Generalitat) for providing the track cross sections; and FGV (Ferrocarrils de la Generalitat Valenciana) for providing the vehicle technical data. They would also like to thank the company EDILON-SEDRA for providing the data on the elastomer installed on the line. Appendix. Matrix elements (10×10)
M
M1(5×5) M3(5×5)
=
ikeRL1 h1 0 M1 = K¯ RL1 eRL1 h1 RL1 eRL1 h1 D1,1
M2(5×5) M4(5×5)
ike−RL1 h1 0 −K¯ RL1 e−RL1 h1 −RL1 e−RL1 h1 −D1,1
RT 1 eRT 1 h1 0 −K¯ ikeRT 1 h1 −ikeRT 1 h1 D1,2
−RT 1 e−RT 1 h1 0
−K¯ ike−RT 1 h1 −ike−RT 1 h1 D1,2
0 RL2 h1
ike aC2,1 eRL2 h1 RL2 h1 −RL2 e 0
J. Real et al. / Mathematical and Computer Modelling 54 (2011) 280–291
0
ike−RL2 h1 M2 = aC2,1 e−RL2 h1 −RL2 h1 RL2 e
0
C
C1,1 0 0 0 0
1,1
0 M3 = 0 0 0
0 RT 2 eRT 2 h1 aC2,2 eRT 2 h1 ikeRT 2 h1 0 C1,2 0 0 0 0
−RL2 (h1 +h2 )
ike −R e−RL2 (h1 +h2 ) M4 = L2 −R (h +h ) −D2,1 e L2 1 2 C2,1 e
0 0 0 0
K¯ = Ak Gk2
−RL2 (h1 +h2 )
1+
0 0 0 0 0
ike−RT 2 h1 0
−C1,2
0
0
−RT 2 e−RT 2 h1 −aC2,2 e−RT 2 h1
0
RL2 (h1 +h2 )
0 0 0 0 0
ike RL2 eRL2 (h1 +h2 ) RL2 (h1 +h2 ) D2,1 e C2,1 eRL2 (h1 +h2 ) 0
RT 2 e
291
RT 2 (h1 +h2 )
−ikeRT 2 (h1 +h2 ) D2,2 eRT 2 (h1 +h2 ) C2,2 e
RT 2 (h1 +h2 )
Ak G I ρω2 − Ak G − EIk2
0
−RT 2 (h1 +h2 )
−R T 2 e −ike−RT 2 (h1 +h2 ) −R (h1 +h2 )
D2,2T 2
−R (h +h ) −C2,2 T 2 1 2
0
−ike
−RL3 (h1 +h2 )
RL3 e−RL3 (h1 +h2 )
D3,1 e−RL3 (h1 +h2 )
−C3,1 e−RL3 (h1 +h2 )
0
−RT 3 (h1 +h2 )
RT 3 e ike−RT 2 (h1 +h2 )
−RT 3 (h1 +h2 ) −D3,2 e C3,2 e−RT 3 (h1 +h2 )
− ρ Aω2 .
References [1] CityRailways, CityRailways Magazine, (Online) Issue 2, Available at: http://www.cityrailways.net/issue2/ (accessed 12/03/10), 2009. [2] X. Sheng, C.J.C. Jones, D.J. Thompson, Prediction of ground vibration from trains using the wavenumber finite and boundary element methods, Journal of Sound and Vibration 293 (2006) 575–586. doi:10.1016/j.jsv.2005.08.040. [3] Y.B. Yang, H.H. Hung, Soil vibrations caused by underground moving trains, Journal of Geotechnical and Geoenvironmental Engineering 11 (134) (2008) 1633–1644. [4] M. Katou, T. Matsuoka, O. Yoshioka, Y. Sanada, T. Miyoshi, Numerical simulation study of ground vibrations using forces from wheels of a running high-speed train, Journal of Sound and Vibration 318 (2008) 830–849. doi:10.1016/j.jsv.2008.04.053. [5] E. Celebi, Three-dimensional modelling of train-track and sub-soil analysis for surface vibrations due to moving loads, Applied Mathematics and Computation 179 (2006) 209–230. [6] P. Galvín, J. Domínguez, Analysis of ground motion due to moving surface loads induced by high-speed trains, Engineering Analysis with Boundary Elements 31 (2007) 931–941. [7] F.F. Çalim, Dynamic analysis of beams on viscoelastic foundation, European Journal of Mechanics A/Solids 28 (2009) 469–476. [8] G. Muscolino, A. Palmeri, Response of beams resting on viscoelastically damped foundation to moving oscillators, International Journal of Solids and Structures 44 (2007) 1317–1336. [9] M. Schevenels, G. Lombaert, G. Degrande, D. Clouteau, The wave propagation in a beam on a random elastic foundation, Probabilistic Engineering Mechanics 22 (2007) 150–158. [10] A.V. Metrikine, A.C.W.M. Vrouwenvelder, Surface ground vibration due to a moving train in a tunnel: two-dimensional model, Journal of Sound and Vibration 234 (1) (2000) 43–66. [11] T. Mazilu, Green’s functions for analysis of dynamic response of wheel/rail to vertical excitation, Journal of Sound and Vibration 306 (2007) 31–58. doi:10.1016/j.jsv.2007.05.037. [12] P. Koziol, C. Mares, I. Esat, Wavelet approach to vibratory analysis of surface due to a load moving in the layer, International Journal of Solids and Structures 45 (2008) 2140–2159. [13] E.G. Florez, S. Cardona i Foix, L. Jordi Nebot, Time-frequency analysis of vibration signals taken on the foot from rail during pass of train, Sciencia et Technica 35 (2007) 243–247. [14] W.I. Liao, T.J. Teng, C.S. Yeh, A method for the response of an elastic half-space to moving sub-Rayleigh point loads, Journal of Sound and Vibration 284 (1–2) (2005) 173–188. doi:10.1016/j.jsv.2004.06.005. [15] L. Auersch, The excitation of ground vibration by rail traffic: theory of vehicle–track–soil interaction and measurements on high-speed lines, Journal of Sound and Vibration 284 (2005) 103–132. doi:10.1016/j.jsv.2009.10.007. [16] G. Lombaert, G. Degrande, Ground-borne vibration due to static and dynamic axle loads of InterCity and high-speed trains, Journal of Sound and Vibration 319 (3–5) (2009) 1036–1066. doi:10.1016/j.jsv.2005.08.059. [17] P. Salvador, J. Real, C. Zamorano, A. Villanueva, A procedure for the evaluation of vibrations induced by the passing of a train and its application to real railway traffic, Mathematical and Computer Modelling 53 (1–2) (2011) 42–54. [18] J.M. Gere, in: Paraninfo Thomson (Ed.), Timoshenko: Resistencia de Materiales (Timoshenko: Mechanics of Materials), Madrid, 2004. [19] M. Melis, Introducción a la Dinámica Vertical de la vía y Señales Digitales en Ferrocarriles (Introduction to Vertical Track Dynamics and Digital Signals in Railways). Ed. UPM, Madrid, 2008. [20] A. López, Infraestructuras Ferroviarias (Railway Infrastructures). Ed. UPC, Barcelona, 2006. [21] S. Anacleto Lupiáñez, 2009, Modelización de la Interacción Vía-Tranvía (Modellization of the Track–Tramway Interaction), M.Sc. Thesis, Universitat Politècnica de Catalunya (UPC).