Keywords: moving load, critical velocity, non-symmetric operator, wave dispersion, contact waves, visco-elastic continuum.

Effects of dynamic interaction of a railroad and a moving high speed load are investigated for several decades, among many others [1,2]. The contemporary needs are focused on the detection of possible critical operating speeds of high-speed trains. Regarding the dynamic processes of the system track-subsoil various mathematical models of analytical or numerical character have been discussed. While the numerical approach provides very detailed quantitative results, it does not enable any insight into the structure of the mechanism investigated. On the other hand the analytical models usually suffer from a problematic simplification of real physical conditions.

The paper attempts to cover this gap abandoning Winkler or Pasternak type of the mass-less subsoil and introducing subsoil in the form of a visco-elastic layer of a constant thickness and uniformly distributed mass. Mathematical model of the subsoil consists of Lamé equations in a two-dimensional continuum. Elastic constants are introduced as differential operators modeling visco-elasticity of the subsoil. The track is considered as the Timoshenko infinite beam. Contact of both parts is regarded as shear free.

This model provides the critical velocities approaching the subsoil compress or shear wave velocities or the subsoil-beam contact wave velocity. They are often far below the operating speeds of high-speed trains and special measures should be adopted for track projects. The load speed is considered as critical, if the inverse transformation integral either loses the existence (zero viscosity) or proves to be an extreme absolute value (positive viscosity).

Qualitative results have been obtained by this analytical investigation: (i) When the internal viscosity is respected, wave amplitudes remain within certain limits when passing through any critical speed. Nevertheless, the wave type and structure change are significant. While the sub-critical response has rather a wave character, in super-critical regime a dominating shock wave propagates from the point of the load with very low damping in space. This wave is dangerous for the neighbouring infrastructure and dynamic stability of the car; (ii) The system behaves as a low pass filter in time and space for certain combinations of input parameters. On the other hand some wave-bands vanish rapidly in space and therefore they can be neglected when analyzing the system as a whole; (iii) The influence of the subsoil layer thickness is limited by asymptotes related with individual types of critical speeds and with the properties of the relevant half-plane.

1

L. Frýba, "Vibration of solids and structures under moving loads", Academia, Praha, 1972.

2

Y.B. Yang, H.H. Hung, "Wave Propagation for Train-induced Vibrations", World Scientific, London, 2009.

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