Dynamic behaviour of a layer of discrete particles, Part 2: Response to a uniformly moving, harmonically vibrating load

The steady state response of a discrete, granular layer to a uniformly moving, harmonically vibrating load is elaborated. This model simulates the response of a ballast layer to an instantaneous train axle passage. After deriving the boundary value problem and illuminating the corresponding solution procedure, a parametric study is carried out that considers cases that are relevant for railway practice. In the parametric study, the examination of the displacement pattern is combined with the analysis of the kinematic characteristics of the waves generated by the load. The latter analysis requires the derivation of an infinite number of so-called kinematic invariants. Accordingly, the emergence of typical "discrete waves" is demonstrated, which cannot be captured by a continuum model. The parametric study elucidates how the response is influenced by the particle size, the layer viscosity and the load frequency. In conclusion, the influence of the load velocity on the steady state response is studied, revealing the magnitudes of the critical velocities that characterise resonance of the layer vibrations.