ABSTRACT: In this contribution, a new method for a deflection shape determination of an infinite beam on a visco-elastic foundation traversed by uniformly moving mass is presented. The method invokes the dynamic stiffness matrix concept and for the sake of simplicity the results are shown on Euler-Bernoulli beams. The solution is presented in the context of a review of some methods for solution of uniformly moving mass and load problems on finite and infinite beams. Advantages and disadvantages of these methods are summarized. KEY WORDS: Moving load; Moving mass; Eigenvalue expansion; Dynamic stiffness matrix. 1 INTRODUCTION Dynamic analyses of beam structures under moving loads have attracted the engineering and scientific community from the middle of the 19th century, when railway construction began. Increasing demands on the railway network capacity leads to a necessity of better understanding of dynamic phenomena related to train-track-soil interactions and therefore questions regarding the moving load and moving mass problems are the still important subjects in nowadays investigations. New modelling approaches, as well as their solving methods, are needed to perform simulations that could reflect important features of dynamic systems. In this context analytical and semi-analytical solutions have the undoubted advantage of possibility of direct sensibility analysis on parameters involved in the problem. Moving force problem is far simpler. It has a semi- analytical or analytical solution available for finite as well as infinite beams. Generalizations affecting the beam theory and foundation models, like extension from the Euler-Bernoulli theory to the Timoshenko-Rayleigh theory, or generalizations of Winkler foundation to Pasternak or other foundation models, introduction of foundations of finite depth or alterations from viscous to hysteretic damping models do not present substantial difficulty [1], except in cases when numerical solution of complex frequencies is necessary. In finite beams eigenvalue expansion techniques can be used and in infinite beams either Fourier transform or the concept of the dynamic stiffness matrix can be exploited [2, 3]. In the latter case two semi-infinite beams are connected by the continuity conditions at the load application point. Such a solution can easily be extended to the moving force with harmonic component [4] or non-uniform foundation [5, 6]. The inertial effects of both the beam and the moving vehicle were studied as early as in 1929 by Jeffcott [7] by the method of successive approximations. The moving mass problem does not have fully analytical solution. Analysing finite beams, it is seen that the governing equations in modal space remain coupled [3]. There is however a classical work [8], which is often taken as a bench-mark solution, but this solution does not consider all effects at the contact point as already depicted by others [9]. There are other papers repeating the same error [10], some of them corrected by Letters to the Editor [11]. If a steady-state solution exists for an infinite beam, then it exactly matches the solution for the moving force and the mass has no contribution as indicated in [2, 12]. If the solution is not steady, there is an oscillation around the steady-state deflection and the amplitude and frequency of this oscillation has to be determined. In this paper a new method for their determination is presented. 2 PROBLEM STATEMENT Let a uniform motion of a constant vertical force and a mass along a horizontal beam on a linear visco-elastic foundation be assumed (Figure 1). The foundation is modelled as homogeneous distributed spring-and-dashpot sets. Simplifications for the analysis of vertical vibrations are outlined as follows: (i) the beam obeys linear elastic Euler-Bernoulli theory; (ii) the beam damping is proportional to the velocity of vibration; (iii) the beam and mass are in continuous contact; (iv) no other loading is added; (v) the vertical displacement is measured from the equilibrium deflection position caused by the beam mass; (vi) the velocity is maintained constant and no restriction is imposed on its magnitude. Figure 1. Structure under consideration. On the Moving Mass versus Moving Load Problem Z. Dimitrovová 1 1 Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, and LAETA, IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal email: zdim@fct.unl.pt P v x M w , kc , EI m