Keywords: moving loads, elastic waves, damping mechanisms, finite elements, boundary elements.

Moving sources occur in a variety of problems, in particular in the fields of road and railway traffic. The generation of ground vibration due to moving loads or vehicles is twofold: Firstly, wheel flats, surface roughness or changing stiffness of the soil/structure along the road or track cause random or deterministic periodic vibrations. Secondly, a moving mass travelling at critical velocity may cause a sonic boom effect in the surrounding environment. This may be observed, for example, when a train hits the Rayleigh wave speed of a soil deposit. The response of the road or track structure and the underlying or surrounding soil is strongly dependent on the properties of the materials. Therefore, it may be impossible to predict the response at a given site from field measurements at a similar site. In order to estimate, for example, the level of ground borne noise, a mathematical model of the vehicle-structure-soil interaction problem is necessary.

A variety of models may be used in order to model infinite structures and viscoelastic media. The aim of this paper is to discuss the formulation of such models in a local frame of reference following a moving source, such as a vehicle. The emphasis is put on the formulation of finite element method and boundary element method models for unbounded domains. Further, the problems involved in the description of material dissipation in the moving frame of reference are addressed.

The finite element method is adaptable for the analysis of most problems in the dynamics of structures and materials. However, finite elements are not well suited for the analysis of wave radiation problems. This is due to the fact that only a finite part of an infinite structure or medium can be described. Hence, artificial boundaries arise in a model of an unbounded structure or medium. Absorption of the wave energy at these artificial boundaries may be achieved by the implementation of transmitting boundary conditions. This was done for an elastic half-space in the work by Krenk et al. [3], while Andersen et al. [1] treated a beam on a Kelvin foundation. Both models were formulated in the moving frame of reference following a vehicle.

Boundary elements have an inherent ability to radiate waves, since the Green's functions are adopted as weight functions. However, the method is very expensive in terms of computer time. Currently, the analysis of realistic soil-structure interaction problems is restricted to the frequency domain, since the solution in the time domain involves a numerical evaluation of the convolution integral over the entire time history. Andersen and Nielsen [2] compared the boundary element results for a moving load on the surface of an elastic half-space with the results obtained by Sheng et al. [4], who suggested a formulation in the horizontal wavenumber domain. The two methods have been found to provide very similar results. The wavenumber-frequency-domain method provides a much faster solution than the boundary element method, but it may only be used for the analysis of horizontally stratified soil.

Material damping models pose a problem in a moving-frame-of-reference description. In the frequency domain, the damping is related to the frequency of vibration at a material point, whereas in the time domain, the damping is related to the motion of the material point. These quantities are not immediately available in the moving coordinate frame, in which the apparent frequency of excitation and response includes a wavenumber term due to convection. A means of dealing with damping in the moving frame of reference is described in the paper.

Numerical examples are given, which illustrate the performance of transmitting boundary conditions in a finite element model of an unbounded soil. Further, it has been found that the combined finite element and boundary element methods are adaptable for the analysis of such problems as forces moving along a railway track on an embankment over a subsoil.

1

L. Andersen, S.R.K. Nielsen and P.H. Kirkegaard, "Finite element modelling of infinite Euler beams on Kelvin foundations exposed to moving loads in convected co-ordinates", Journal of Sound and Vibration, 241(4), 587-604, 2001. doi:10.1006/jsvi.2000.3314

2

L. Andersen and S.R.K. Nielsen, "Boundary element analysis of the steady-state response of an elastic half-space to a moving force on its surface", Engineering Analysis with Boundary Elements, 27(1), 23-38, 2003. doi:10.1016/S0955-7997(02)00096-6

3

S. Krenk, L. Kellezi, S.R.K. Nielsen and P.H. Kirkegaard, "Finite elements and transmitting boundary conditions for moving loads", In "Proceedings of the 4th European Conference on Structural Dynamics, Eurodyn'99", Praha, 447-452, 1999.

4

X. Sheng, C.J.C. Jones and M. Petyt, "Ground vibration generated by a harmonic load acting on a railway track", Journal of Sound and Vibration, 225(1), 3-28, 1999. doi:10.1006/jsvi.1999.2232

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