Keywords: moving load, moving mass, transversal vibration, transition radiation, normal-mode analysis, natural frequencies, orthonormal mode shapes.

The rapid growth of high-speed railway networks and the evolution of train vehicles capable to operate at more than 500km/h, has given rise to a number of related problems that have motivated a significant amount of scientific work. The attenuation of train-induced vibrations, aggravated by inhomogeneities in the track foundation, certainly belongs to the issues still demanding further attention.

In this contribution, the transversal vibrations originated by a load moving uniformly along a beam resting on a visco-elastic foundation are studied. The beam can be composed from various parts, called structural elements, with constant properties. The methodology can handle discontinuities in the foundation stiffness parameter, in the flexural rigidity of the beam, in the mass per unit length or in the damping coefficient. It is assumed that there is only a finite number of discontinuities. Special attention is paid to the additional vibrations, conventionally referred to as transition radiation, which arises as the point load traverses the region of such a discontinuity. Reflected waves at the extremities of the full beam are avoided by the introduction of infinite structural elements.

The load is introduced as a set of time varying forces, moving at constant velocity, thus keeping the same distances between themselves. The effect of the load mass is analysed using two approaches, in a discrete way as a mass inclusion according to its actual position, and, in a continuous way, as a distributed mass in the region of the current position of the set of moving forces.

The governing equation of the problem is solved by normal-mode analysis. Solution is expressed analytically in the form of an infinite sum of orthonormal natural modes multiplied by a generalized coordinate of the displacement. Normal modes and natural frequencies are obtained numerically exploiting the concept of global dynamic stiffness matrix. This ensures that the frequencies obtained are exact in the sense that they do not contain the error inherent to the standard finite element method. The methodology has restrictions neither on velocity magnitude nor on material or mass damping coefficients. The effect of axial force, shear distortion and rotary inertia are also evaluated. Procedures are programmed in Maple.

The approach looks simple, though; numerical expression of the results is not straightforward. A general procedure for numerical implementation is given and verified. To illustrate the utility of the methodology, several analyses are performed. First, the probability that an admissible upward displacement will be exceeded, when the load passes by with a certain velocity obeying the normal distribution, is determined. In this problem the given structure has an intermediate part of adaptable foundation stiffness, which is optimized in a parametric way. Important conclusions about the intermediate stiffness are made. In the second analysis, the effect of the mass of the load on the transition radiation is studied.

The results obtained can have direct application in the design of high-speed railway lines.

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