Keywords: double-beam system, nonlinear foundation, Adomian polynomials, wavelet expansion.

The problem of the dynamic response of an infinite double-beam system resting on a nonlinear viscoelastic foundation subjected to an oscillating moving load is discussed. The problem considered is of great theoretical and practical significance in railway engineering. There are many models presented in the literature for the rail tracks on a rigid foundation. A number of researchers have derived a closed form solution for special cases of linear models using various mathematical approaches such as Green's function technique or Fourier and Laplace transforms [1,2]. Nonlinear foundation models need numerical or semi-analytical solution procedures. A commonly applied technique is a perturbation method [3,4]. The solutions obtained by using this method, which depends on a small parameter, are computationally difficult and usually have low accuracy.

In this paper, the linear double-beam model is extended with an assumption of the nonlinear stiffness of the bottom layer. The upper beam is forced by a moving load representing a train. The theoretical model consists of the coupled nonlinear system of fourth order partial differential equations with homogeneous boundary conditions. The new closed form solution for displacements is derived using modern mathematical tools. The novelty of the method developed is based on Adomian's decomposition [5] and a special type of wavelet expansion, using filters of the coiflet type [6,7]. With such an idea the small parameter approach can be overcome as well as the difficulties related to direct or numerical calculation of Fourier integrals. In order to secure the convergence of the Adomian series, a special convergence condition is taken into account and an error index is defined. The numerical examples in the time domain show the usefulness of the method presented that allows the parametric analysis of the model investigated.

1

M.F.M. Hussein, H.E.M. Hunt, "Modelling of floating-slab tracks with continuous slabs under oscillating moving loads", J. of Sound and Vibration, 297, 37-54, 2006. doi:10.1016/j.jsv.2006.03.026

2

L. Fryba, "Vibrations of Solids and Structures under Moving Loads", Thomas Telford Ltd., London, 1999.

3

Y.H. Kuo, S.Y. Lee, "Deflection of non-uniform beams resting on a nonlinear elastic foundation", Computers and Structures, 51, 513-519, 1994.

4

M.H. Kargarnovin, D. Younesian, D.J. Thompson, C.J.C. Jones, "Response of beams on nonlinear viscoelastic foundations to harmonic moving loads", Computers and Structures, 83, 1865-1877, 2005. doi:10.1016/j.compstruc.2005.03.003

5

G. Adomian, "Solving Frontier Problems of Physics: The Decomposition Method", Kluwer, Boston, MA, 1994.

6

L. Monzon, G. Beylkin, W. Hereman, "Compactly supported wavelets based on almost interpolating and nearly linear phase filters (coiflets)", Appl. Comput. Harm. Anal., 7, 184-210, 1999. doi:10.1006/acha.1999.0266

7

P. Koziol, "Wavelet approach for the vibratory analysis of beam-soil structures", VDM Verlag Dr. Müller, Saarbrucken, 2010.

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