Keywords: Euler-Bernoulli beam, moving load, nonlinear foundation, modified Adomian polynomials, wavelet expansion.

The investigation of physical phenomena associated with modern transport shows the lack of approaches enabling effective analysis of the behaviour of dynamic systems. Therefore the development of new methods for vibration analysis is of importance for the prediction of dynamics. The assumption of nonlinear properties of the foundation is essential for modelling of load-beam-foundation systems. One can find some solutions of nonlinear problems in the literature but the published results are usually based on numerical approaches or perturbation methods and cannot be used effectively for parametrical analyses. This paper presents an application of the wavelet based approach using a modified Adomian's decomposition combined with a coiflet approximation. The method is adopted for the Euler-Bernoulli beam resting on a nonlinear viscoelastic foundation and subjected to a series of moving loads representing the train. The simplified model of the Euler-Bernoulli beam does not reflect some important physical features appearing in such systems but it can be successfully used for showing the essential properties of the semi-analytical method developed. The vehicle is modelled as a finite sum of moving loads harmonic in time and distributed at separated intervals. The classical cubic representation of nonlinearity is introduced in the foundation stiffness. The steady-state response for transverse displacement of the beam is obtained by applying the Galilean co-ordinate system. The assumptions introduced regarding the load and nonlinear properties lead to complexity of the computations. In this case, numerical simulations might give solutions with errors and the proposed semi-analytical method is an alternative approach allowing effective parametrical analysis. The Adomian solution is compared with results obtained by using a more classical perturbation method based on the small parameter approach. Comparative studies performed for various sets of physical parameters supplement the investigation presented. The modified wavelet based solution for the nonlinear problem of the Euler-Bernoulli beam subjected to a finite sum of moving loads and its comparison with the solution obtained using the perturbation method are the main novelty of this paper.

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