Transition radiation in an inhomogeneous and non-linear one-dimensional system

Abstract

When a source moves along a straight line with constant velocity and acts on an inhomogeneous medium, energy is radiated away from the source. This phenomenon is referred to as transition radiation. The energy radiation can also be generated by the change of velocity or path of the source (radiation in a non-uniform motion), as well as when it moves faster than waves can propagate through the medium (Vavilov-Cherenkov or Mach radiation).
Transition radiation is of high importance for both use railways and high-speed railways because the velocity of the trains may approach the wave velocity of the soil in the substructure. This leads to an amplified radiation of waves which cause plastic deformation in the railway track in the vicinity of the transition zone.
In the first part of this thesis, the solution for the system composed of an infinite beam resting on inhomogeneous Winkler foundation with a smooth transition in stiffness and subjected to a constant moving load is obtained. The proposed solution is computed for a finite system incorporating a set of non-reflective boundaries such that it mimics the behaviour of the infinite system. The response is obtained using a combination of a transform method (i.e. the Laplace transform) and a numerical method (i.e. the Finite Difference Method). With the obtained solution, the transition radiation and the influence of the length of the transition zone on it are analysed.
The response obtained for the system with smooth transition in stiffness exhibits a change in magnitude when compared to the response obtained for an abrupt transition. Increasing the length of the transition zone causes a decrease in the magnitude of the excited free waves at the transition. For a very long transition zone, the overall behaviour of the response changes, leading to a significantly shorter radiated pulse, implying that the excited free waves are more localized in space. In addition to the change in magnitude, also a phase shift is observed between the two responses.
In the second part of this thesis, the solution for the system composed of an infinite beam resting on inhomogeneous and non-linear Winkler foundation, and subjected to a constant moving load is obtained. The Winkler-stiffness behaviour is assumed to be piecewise linear, and consequently, the system behaves linearly between non-linear events. This enables the solution to be obtained using a mixed time-frequency approach combined with the Finite Difference Method for the spatial discretization. The computed response incorporates the non-reflective boundaries and the smooth transition in stiffness, obtained in the first part of the thesis. The solution is used to determine the possibility of plastic deformation introduced by the transition radiation, and the parameters influencing it.
The plastic deformation exhibited in the transition zone is observed to be a consequence of a constructive interference of the excited free waves and the approaching eigenfield. The influence of two parameters on the resulting plastic deformation is analysed, namely the length of the transition zone and the velocity of the load relative to the wave velocity in the system. Increasing the length of the transition zone reduces the magnitude of the excited free waves, therefore leads to a less pronounced constructive interference between the excited free waves and the eigenfield, and thus to a decrease in magnitude of the plastic deformations. Surprisingly, in the case of the load velocity close to the wave velocity in the system, enlarging the length of the transition zone does not lead to significant reduction in the plastic deformation exhibited.
The solution method presented in this thesis includes inhomogeneous foundation with a non-linear behaviour and the infinite extent of the model is correctly accounted for. Therefore, it could be used for the initial design of transition zones. Given the stiffness jump that has to be bridged between the two domains, the optimum length of the transition zone can be obtained such that minimum damage results in the railway track.