Dynamic behaviour of a layer of discrete particles, Part 1: Analysis of body waves and eigenmodes

This paper is one in a series of two, and discusses the body wave propagation and the eigenmodes for a layer of discrete particles. This configuration is a representation for a ballast layer used in ordinary railway tracks. The discrete nature of the ballast is simulated via an elastic nine-cell square lattice. After deriving the equations of motion for the lattice, the long-wave approximation for the equations of motion is compared with the equations of motion for a classic elastic continuum. This comparison yields relations between the macroscopic continuum parameters and the microscopic lattice parameters. Then, the dispersion curves that characterize the eigenmodes of the discrete layer are derived. The differences between these dispersion curves and those of a continuous layer are elucidated. By means of a parametric study, the influence is shown of the particle diameter and thickness of the layer on the dynamic response. In an accompanying paper, the layer response to a moving, harmonically vibrating load is analyzed, which is a representation for a ballast track that is subjected to an instantaneous train axle passage.