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Wavelet analysis of beam-soil structure response for fast moving train

This paper presents a wavelet based approach for the vibratory analysis of beam-soil structure related to a point load moving along a beam resting on the surface. The model is represented by the Euler-Bernoulli equation for the beam, elastodynamic equation of motion for the soil and appropriate boundary conditions. Two cases are analysed: the model with a half space under the beam and the model where the supporting medium has a finite thickness. Analytical solutions for the displacements are obtained and discussed in relation to the used boundary conditions and the type of considered loads: harmonic and constant. The analysis in time-frequency and velocity-frequency domains is carried out for realistic systems of parameters describing physical properties of the model. The approximate displacement values are determined by applying a wavelet method for a derivation of the inverse Fourier transform. A special form of the coiflet filter used in numerical calculations allows to carry out analysis without loss of accuracy related to singularities appearing in wavelet approximation formulas, when dealing with standard filters and complex dynamic systems.


Introduction
The problem analysed in this paper is related to the abatement of vibrations generated by high-speed trains. The investigation of the traffic interaction with the environment plays a significant role in the construction of new tracks and trains [1,2,3] and is one of most important subjects of EU research in the area of surface transportation.
Modern high-speed trains move with velocities which increase very dangerously the level of vibrations [4,5]. The Rayleigh velocity is usually treated as a critical velocity and the theoretical investigations and experimental tests show that the ground vibrations grow significantly when the trains' speed reaches this critical value [6,7]. Although many results concerning railway tracks built on the surface were described in the literature [3,6], new analytical and numerical approaches to the problem of vibrations abatement are still needed, with special interest in ground vibration analysis [7,8,9].
The type of the load and properties of the supporting layer strongly influence the level of accuracy of numerical approximations of displacement. Two different cases of the same model are presented in Model 2 this paper: the first one with a half space under the beam and the second one when the supporting layer has finite thickness. The complexity of solutions is discussed in relation to the velocity and frequency of the load. The level of vibrations generated by the load is computed in the cases of harmonic and constant point loads moving along the beam resting on the surface of viscoelastic medium. The case of the harmonic load is very important for the analysis of surface vibrations generated by moving load and usually shows a strong response of the system for even its small velocities. It does not show any singularities in mathematical formulas and can be effectively solved by using carefully adopted calculation methods. In order to avoid numerical instabilities appearing for some systems of parameters, a wavelet approximation method [10] and a specially chosen coiflet filter [11] are used for derivation of the inverse Fourier transform. This method is very effective for high velocities and high load frequencies and allows to reduce numerical problems for calculations that prevent effective analysis of the system due to the big amount of time needed for calculations [12].

A load moving along a beam
An infinitely long beam is resting on viscoelastic surface and the point load is moving along the beam.
ρ , a and ) (⋅ δ are the vertical displacement of the beam, the vertical stress, the vertical point load, the bending stiffness, the mass per unit length of the beam, the thickness of the beam in y direction and the Dirac delta function, respectively.
The motion of the viscoelastic layer can be described by the following elastodynamic Navier's equation: , ρ and λ , µ are the displacement vector, the mass density of the soil and Lame' constants, respectively. The boundary and continuity conditions are: The interpretation of these conditions can be described in the following way: the beam does not move horizontally, the vibrations of the beam and the vertical displacements of the soil at the surface are the same and the radiated waves vanish at h z = (model 1) and at the infinity (model 2). The Fourier transform can be used for obtaining the steady state response of the surface for these systems [4]. The described model presents the system based on the assumptions formulated in [4,12] where the problem of ground vibrations due to a load moving in a tunnel is investigated for harmonic and constant loads. Accordingly the reformulated equations and conditions lead to the investigation of the soil response for the load moving along the beam resting on the surface similar to the models published in [8,13,9] where the main direction of investigation is focused on derivation of critical velocities. One should note that the present paper analyses the two-dimensional model, whereas the analysis in three dimensions would allow to solve the problem in a more general case with y direction and possibly simplify some calculations by introduction of other systems of coordinates.

Analytical solution
The formulas for displacements and stress components can be rewritten by using Lame' potentials: Instead of equation (2) one can write two scalar equations: Applying the Fourier transforms gives the following representation of the system in the transform domain: velocities of the longitudinal and the shear waves in the layer, respectively. The boundary and continuity conditions (3)-(5) can be rewritten accordingly: The solutions for equations (9b) are: where coefficients 1 A and 3 A become zeros when the supporting layer has infinite thickness (model 2). Then one can derive the solutions for displacements and stresses in the transform domain: and hence, by using boundary conditions, one obtains the system of algebraic equations with respect to i A that can be solved accordingly to the Cramer's rule: with D and j D the associated determinants. The solution for vertical displacement is represented by the equation: When the moving load is harmonically varying in time with the load frequency Ω , ) cos( ) ( 0 t P t P Ω = , formula (14) takes the following form: The case of 0 = Ω can be treated as a load constant in time. For this type of load the integrand in (15) has a strong singularity that prevents direct numerical calculation of the vertical displacement for both models. Therefore, in this paper, the displacements for relatively low load frequency are derived in order to approximate the constant point load. Instead of classical numerical integration, the alternative method [10] based on wavelet approximation of the inverse Fourier transform is used for calculation of integral (15). This method was previously successfully applied for the solution of number problems related to ground born vibrations analysis and investigation of interactions of fast railway transportation with the environment [12,15,16]. This method is very efficient for the analysis of ground and beam vibrations generated by loads moving with velocities near critical values and relatively high or low load frequencies [12,15]. The wavelet approach allows to analyse detailed features of dynamic systems which might be lost when using numerical integration.
The method adopted in this paper for calculation of integral (15)  differs from any integer and therefore allows to take into account all terms of approximating sequence (16), as opposed to coiflet filters used in previous publications [12,15,16]. In those cases, another singularity appeared in wavelet formulas and due to that fact, an additional approximating procedure was needed in order to eliminate troublesome terms. This modified system of coiflets improved the approximation accuracy and eliminated doubts related to analytical nature of the used sequences.
For the coiflets, the approximating formula (16) takes the following form: f is used in this paper, due to the fact that considered, relatively low, load frequencies shows strong sensitivity of the system for its dynamic changes. Numerical simulations show that for 16 > n the sequence (17) stabilises and further terms do not differ significantly. One should note that a similar procedure can be applied for a wide spectrum of daublets and coiflets filters which were specially constructed for numerical applications. These carefully considered and modified, if needed, systems of coefficients give approximation with appropriate accuracy depending on the nature of the analysed systems.
Numerical simulations show (Fig. 3) that vertical displacement for the constant load can be approximated by the harmonic one with relatively low load frequency. One can show that the response characteristics stabilises when the frequency load increases below is a half space (Figs. 1 and 2), especially for high velocities near critical values (Fig. 5). Therefore one should analyse very carefully models with finite supporting layer [4] because complex dynamic variations of the system can give wrong results when applying this assumption. However the waves' reflections might influence obtained results for this model, the propagation of reflected waves can be negligible due to viscoelastic properties of layer and its appropriately chosen thickness for the approximating procedure.     Figure 6 shows that the maximum amplitude of vertical vibrations decreases for higher velocities when the frequency load is equal to 10 10 − = Ω which means that the assumption of the thickness m 10 = h for the supporting layer is not realistic and leads to wrong results. This tendency is reversed when the value h increases (Fig. 7) and these correct observations of the influence of velocity on train induced vibrations are confirmed by the analysis of more realistic model 2 (Fig. 8) and by the theory [3,4,6,7,12]. The wavelet method adopted in this paper for the derivation of vertical displacements allows further detailed analysis of the system, especially in the area of high velocities and the point load constant in time.  The maximum amplitude derived by using the coiflet based wavelet approximation for a wide range of velocities shows that critical value of velocity for the constant load can be estimated around m/s 80 for model 2 (Fig. 9). One can observe that the vibrations can reach the relatively high level of mm 40 when the constant load moves with velocity near critical value.

Conclusions
The problem of viscoelastic soil vibrations generated by a load moving along a beam resting on the surface was analysed in two cases: with finite thickness of supporting layer and with half space under the beam. A special wavelet method was adopted for the derivation of vertical displacements at the surface. This method allowed to carry out the analysis in the area of high velocities and low frequencies leading to the approximation of the point load constant in time. The applied estimation allowed to alleviate the problem of singularities appearing in the integrated formulas when the constant load was considered. The modified coiflets filter improved the accuracy of wavelet approximation compared to previously published results. The parametric analysis of the system was carried out depending on a number of factors, e.g. the thickness of supporting layer and the velocity of moving load and the critical velocity was numerically estimated for the constant load and a half space under the beam. The analysis of this system can be extended by using the wavelet approach for more complex situations.