Dynamic response of periodic infinite structure to arbitrary moving load based on the Finite Element Method

Dynamics of repetitive structures subjected to moving loads is a common problem in railway engineering. Bridges, rails or catenaries are the most representative periodic structures, on which the train acts as a moving excitation. Usually, these structures are long enough to consider that their dynamic response is in permanent regime. In this work we present a method to obtain the steady-state solution of an infinite periodic structure subjected to a periodic moving load running at constant speed 𝑉𝑉.This problem has been dealt with in the literature by different approaches. Analytical models [1], two-and-a-half dimensional (2.5D) Finite Element models [2] and the Wave Finite Element Method (WFEM) [3] are found to be used. The method proposed in this work is valid for any generic periodic structure because it is modelled by the classical Finite Element Method. It is mathematically simpler and more efficient compared to WFEM, and it avoids the numerical problems that arise when WFEM is applied to catenaries.The proposed method consists of solving the dynamic interaction problem on a single repetitive block of the structure in which the periodicity condition is applied. Each block of length 𝐿𝐿 is excited by the same load. Thus, the periodicity condition states that the solution at the left boundary of the block is the same as at the right boundary but advanced a period 𝑇𝑇=𝐿𝐿/𝑉𝑉. This condition is imposed in the frequency domain and a procedure to shift into the time domain is presented.