Nonlinear vibrations of beams with fractional derivative elements crossed by moving loads

This paper addresses the evaluation of the time-domain response of nonlinear beams endowed with a fractional derivative element crossed by moving loads. Nonlinearities originate from the assumption of moderately large displacements of the beam. Following a Galerkin-type solution procedure, beam transversal displacement is represented in terms of the linear modes of vibration and time-dependent generalized coordinates. A novel step-by-step integration scheme, labeled improved pseudo-force method (IPFM), is developed for the numerical solution of the set of coupled nonlinear fractional differential equations governing the time-dependent generalized coordinates. The proposed procedure stems from the extension of a recently developed step-by-step scheme for the dynamic analysis of fractional single-degree-of-freedom systems. The IPFM involves the following main steps: i) to apply the Grünwald–Letnikov approximation of the fractional derivative; ii) to treat terms depending on the unknown values of the response as pseudo-forces; iii) to handle nonlinearities by performing iterations at each time step. Numerical results are presented to assess the accuracy of the IPFM as well as to investigate the influence of the fractional derivative order and coefficient on nonlinear beam vibrations under moving loads.