Semi-Analytical Approach and Green’s Function Method: A Comparison in the Analysis of the Interaction of a Moving Mass on an Infinite Beam on a Three-Layer Viscoelastic Foundation at the Stability Limit—The Effect of Damping of Foundation Materials

In this paper, the interaction of a mass moving uniformly on an infinite beam on a three-layer viscoelastic foundation is analyzed with the objective of determining the lowest velocity at the stability limit, called, in this context, the critical velocity. This issue is important for rail transport and, in particular, for the high-speed train, because the moving mass is the basic model of a vehicle, and the infinite beam on a three-layer viscoelastic foundation is the usual mechanical representation of the railway track. In addition to this, the advantages and disadvantages of the two implemented methods, namely, the semi-analytical approach and the Green’s function method, are summarized in terms of computational time, the precision of the obtained results, limitations, and the feasibility of implementation. All results are presented in a dimensionless form to cover a wide range of possible scenarios. Some results may be considered academic, however, results related to a particular railway track are also included. Particular attention is paid to the influence of the damping of materials in the foundation upon the critical velocity of the moving mass. Regarding the semi-analytical approach, it is demonstrated that the critical velocities can be obtained in an exact manner by tracing the branches of the so-called instability lines in the velocity–moving-mass plane. This analysis can be maintained within the real domain. As for the time series, they can be determined by a numerical inverse Laplace transform. Moreover, thanks to the analytical form of the final result in the Fourier domain, each value corresponding to a specific time instant can be obtained directly, that is, without the previous time history. Regarding the Green’s function method, this is used to verify a few points delimiting the stable and unstable regions of the moving mass with the help of the D-decomposition approach. Additionally, a numerical algorithm based on the Green’s function and convolution integral written for dimensionless quantities is used to calculate the time series of the moving mass. In addition to identifying the critical velocity of the moving mass, its connection with the critical velocity of the moving force is emphasized, and the possibility of validating the results on long finite beams using modal expansion is presented and described.