Keywords: moving load, stochastic systems, wavelet approximation.

This paper is concerned with the analysis of bending waves in an infinite beam resting on an elastic randomly inhomogeneous foundation and subjected to a distributed harmonic moving load. The fact that velocity of trains grows together with new tracks and vehicles design, leads to the need for the estimation of critical speeds and more detailed analysis of their influence on the surrounding environment. The improvement of existing systems and the formulation of new models better reflecting real situations become necessary for the understanding of phenomena arising from high speed train construction. The methods of solution developed should allow for the analysis of parametrically complex systems leading to possibility of response prediction for tracks built on ground which cannot be characterised precisely enough.

This paper presents a new solution for the beam response to a moving load for the case when supporting elastic foundation is described by a homogeneous random function representing the subgrade reaction. The beam is modelled by the Euler-Bernoulli equation and the load is described as harmonically varying in time, distributed on some interval and moving in a horizontal direction. The stochastic properties of foundation are characterised by a second order stationary random function with correlation function belonging to the class of smooth functions. This type of correlation function implies a high regularity of the stochastic solution of the system [1]. The modelling approach presented leads to difficulties in both, analytical and numerical calculations. Classical integration methods and commonly used stochastic methods of solution become ineffective due to the complex representation of formulas appearing in the calculations. A special wavelet based approximation algorithm [2] combined with the first order smoothing approximation [3] is adopted in this paper allowing a new closed form solution to be found. The modified coiflet filter [4] used for the numerical calculations improves the accuracy of the approximation. The approach developed allows the derivation of the stochastic Green's function for the fourth order stochastic differential equation giving a new analytical solution for the moving load model considered. The results obtained significantly extend the class of solutions for stochastic dynamic systems and are an important contribution to the field.

1

K. Sobczyk, "Stochastic Differential Equations with Applications to Physics and Engineering", Kluwer Academic Publ., Dordrecht 1991.

2

P. Koziol, Z. Hryniewicz, "Analysis of bending waves in beam on viscoelastic random foundation using wavelet technique", International Journal of Solids and Structures, 43, 6965-6977, 2006. doi:10.1016/j.ijsolstr.2006.02.018

3

R.C. Bourret, "Propagation of random perturbed fields", Canad. J. Phys., 40, 782-790, 1962.

4

G. Beylkin, R. Coifman, V. Rokhlin, "Fast wavelet transforms and numerical algorithms", Communications on Pure and Applied Mathematics, 44, 141-183, 1991. doi:10.1002/cpa.3160440202

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