Calculation of force distribution for a periodically supported beam subjected to moving loads

Abstract

In this study, a novel model for a periodically supported beam subjected to moving loads was developed using a periodicity condition on reaction forces. This condition, together with Fourier transforms and Dirac combs properties, forms a relation between the beam displacement and support reaction forces. This relation explains the force distribution to the supports, and holds for any type of support and foundation behaviors. Based on this relation, a system equivalence for a periodically supported beam is presented in this paper. An application to nonballasted viscoelastic supports is presented as an example and the results clearly match the existing model. Next, an approximation of realtime responses was developed for the moving loads as periodical series. The comparison shows that this approximation can be used for a limited number of loads if the distances between loads are sufficiently large. The system equivalence for a periodically supported beam is efficient for supports with linear behavior, and could be developed for other behaviors.

Effects of railway track design on the expected degradation: Parametric study on energy dissipation

“…It is therefore very apt for parametric analyses aiming at fundamental understanding. Frequencydomain models have been applied in the past with different degrees of complexity and with different aims, ranging from very simple beam-on-elastic-foundation (BOEF) models [18][19][20][21][22] to more sophisticated twoor three-layer linear beam models accounting for discrete sleeper spacing and transient loading [23][24][25][26][27][28] and finally fully three-dimensional beam or plate on halfspace models (eventually with stratification), the latter ones with the aim of predicting environmental vibration due to moving trains [29][30][31][32][33][34]. Given the scope of the present study and its preliminary character, it is chosen to adopt a model that is on one hand no more complicated than strictly necessary, on the other hand able to describe all individual track components and take into account energy dissipation in the substructure or subsoil.…”

Analytical Model of the Dynamics of Railway Sleeper

“…In the steady-state, we suppose that all sleepers have the same response but with a delay which is equal to the time for the moving forces to cover the distance between two sleepers. By using this condition, Hoang et al [1] have shown a relation between the beam displacement (w r ) and reaction force (R) at the sleeper position as follows : Figure 1: Periodically supported beam subjected to moving loads where K(!) and Q(!)…”

“…In addition, the sleeper dynamic response is important because it affects the stability of the railway track and the maximum speed of the train. Substantial researches using analytic methods for rail track have been carried out, for example : the effect of wheel-rail contact forces [17,18] or the vibration of the railway track (by considering the model of a ballasted track with discrete supports [8,9,13,14,1] or the model of the infinite beam placed on a continuous foundation [10,7,6,11]). The studies of each component of the rail track were performed on the rail [19,16,15], or on the ballast [20,22].…”

“…The dynamic equation of the beam in the frequency domain by using the Fourier transform. Then by using the relation between the reaction forces and displacement of the beam [1] in the frequency domain and the Dirac delta distribution, the dynamic equation for the beam (sleeper) together with the rails and foundation is written and solved by using Green functions. The method of Green function to obtain the response of a beam structure to a moving mass has been successfully applied previously [25,26].…”

Analysis of transverse vibration and stability issues of discrete-continuous elastic systems with nonlinearly variable parameters

“…This proposal was developed in joint work [2][3][4][5][6][7]. It must be clarified that Bernstein's previously proposed "spectral function" method was used only to analyze systems with fixed parameters without considering friction [10,13]. At work [6] Zoryj L.M.…”