Nonlinear random vibrations of beams with fractional derivative elements

This paper deals with nonlinear random vibrations of a beam comprising a fractional derivative element; the nonlinear term arises from the assumption of moderately large beam displacements. It is shown that the beam response can be determined reliably via an optimal statistical linearization procedure. Specifically, the solution is obtained by utilizing an appropriate iterative representation of the stochastic response spectrum, which involves the linear modes of vibration of the beam. Such a representation allows retaining the nonlinearity in the time-dependent part of the response, which, in turn, is linearized in a stochastic mean square sense. The reliability of the proposed approximate solution is assessed in relation to the results of relevant Monte Carlo simulations. In this regard, a boundary integral method (BIM)-based algorithm is employed, in conjunction with a Newmark integration scheme, for estimating the beam response from spectrum-compatible realizations of the excitation, while accounting for the memory effect related to the fractional derivative element. The presented analytical and Monte Carlo results pertain to simply supported beams excited by a random uniform load of a given power spectral density function (colored white noise). Nevertheless, problems involving other boundary conditions can be solved readily by the method presented herein.