Closed-form solution for mode superposition analysis of continuous beams on flexible supports under moving harmonic loads

Abstract In this paper, a closed-form solution of the moving harmonic load problem for continuous Euler–Bernoulli beam systems is presented. The generality of the boundary conditions is taken into account by solving the characteristic equation of the system, obtaining its natural frequencies and mode shapes. The undetermined coefficient method is applied to solve the governing differential equation of motion, determining the base functions of the solution space of the problem. For vertical vibrations, three numerical examples of footbridges are presented. The main contribution of this paper is to provide the closed-form solution of the moving harmonic load problem applied to continuous footbridges including the phase angle in the load definition. In this way, it is possible to find the solution in time domain of the harmonic component of any load spectra.