Dynamical response of a Timoshenko beams on periodical nonlinear supports subjected to moving forces

The periodically supported Timoshenko beam subjected to moving forces has been investigated by numerous researches. The existed models have been developed for linear supports, and this article presents a new one for nonlinear supports. By using a periodic condition and the Fourier series development, the dynamic equation of the Timoshenko beam leads to a relation between the beam displacements and the reaction forces of the supports. This relation does not depend on the support behaviour and it exists also for the Euler-Bernoulli beam. Then, the responses can be obtained by combining this relation and the constitutive law of the supports. A numerical method based on discretization of the time and frequency responses has been developed for nonlinear supports. Moreover, the influence of the beam model has been studied with numerical examples of linear and nonlinear supports. The results show that the Timoshenko beam should be used for the moving forces with high speed and/or the supports with large stiffness.