Nonlinear random vibrations of plates endowed with fractional derivative elements

This paper deals with the problem of determining the nonlinear response of a plate endowed with fractional derivative elements and exposed to random loads. It shows that an approximate solution of the nonlinear fractional partial differential equation governing the plate vibrations can be obtained via a statistical linearization based approach. The approach is implemented by considering a time-dependent representation of the response involving the eigen-functions of the linear problem. This representation allows deriving a nonlinear fractional ordinary differential equation governing the variation of the time-dependent part of the response, which is linearized in a mean square sense. Then, an iterative procedure provides the response statistics and power spectral density functions. Next, a Boundary Element Method is proposed for conducting relevant Monte Carlo data. The method is developed in conjunction with a Newmark integration scheme for estimating the response in the time domain given spectrum compatible realizations of the excitation. Monte Carlo data and statistical linearization solutions are calculated for square plates with simply supported stress-free edges, but problems involving other boundary conditions can be solved by the proposed approach.