Keywords: trains, ground vibrations, embedded embankment, experiment, theory, integral equations, effect ditch.

The present paper considers the calculation of vibrations caused by a moving train and the investigation of the effect of train speed, oscillatory behaviour and, particularly, the track properties on the vibrational level. The surface motions generated by the moving train are computed with the help of a Green’s function formulation and integral-equation methods. The results of the model that we have developed to compute these vibrations, are compared with experimental data, from which a qualitative insight is obtained into the vibration problem.

We model the train as a sequence of oscillating, vertical point forces, moving over a track, embedded in an elastic, horizontally layered, three-dimensional half space. Each of the these forces representing the train can be considered as a superposition of stationary point sources, all having appropriate phases. Hence, the wavefield generated by a moving source can be represented in terms of the slowness transforms of the wavefield generated by a stationary force. We can model the rail embankment by introducing a region into the geometry consisting of material with different density and/or Lamé parameters than the embedding medium. It is known that both mass density and Lamé parameter contrasts can have an important impact on the vibration level. In our study, we have first introduced an embankment geometry into the problem by considering a mass-density contrast only. For this type of contrast, the resulting scattering problem can be formulated as a domain-integral equation. This integral equation is solved numerically in an efficient manner.

In order to validate our theoretical results, we performed an experiment to measure seismograms due to moving trains. Several passages of trains have been recorded. The experimental data show some interesting features. First, it can be observed that a small ditch located between the embankment and the meadow causes a strong damping of the waves propagating away from the track. Furthermore, in the embankment, it is observed that waves are reflected by the ditch. These waves contribute significantly to the vibration level in the embankment.

By taking a high-density contrast in the model (i.e., the embankment consists of material with a high mass density with respect to its surroundings), we create a low-velocity embankment. This low-velocity embankment behaves as a waveguide, especially for the high-frequency waves. Outside mainly low-frequency waves are observed, while inside the embankment, both low-frequency and high-frequency waves (trapping) are present. Thus, the simulated results show phenomena comparable to the experimental ones. The strong decay in wave amplitudes, which is visible in the experimental data, is also obtained here. Moreover, reflections of high-frequency waves inside the embankment are present too. Concluding, the model describes in a qualitative manner the waveguide behaviour of the embankment although the origin of this waveguide behaviour is different for both cases. In the modelling approach, the embankment acting as a waveguide is caused by the choice of the soil parameters, contrary to what we have seen in the experiment where the waveguide behaviour is caused by the geometry (viz the presence of the ditch).

With this model we can study the effect of different materials used in the embankment and the influence of the resonance frequencies of the train on the propagation of the waves and the intensity level of ground motions.

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