Random vibration of strongly nonlinear system with fractional derivative

Most systems in practice have fractional characteristics, and it is necessary to use Fractional Derivatives (FDs) to model the system. Unfortunately, due of the memory effect of FD, solving fractional-order systems is challenging, particular for the systems under random excitations. In this paper, a dimension-reduced approach that combines physical prior knowledge with a data-driven strategy to determine the random response of the strongly nonlinear systems containing with FD element is proposed. By employing memoryless transformation, the extended order augmented equation with approximate integer coefficients is first obtained. Importantly, the corresponding Fokker-Planck-Kolmogorov (FPK) equation—a (2n+mn)-dimensional partial differential equation is derived. Subsequently, the stationary FPK equation with high dimension of the extended memoryless system is separated into the low-dimensional FPK equations for the interested physical variables, in which the Equivalent Drift Coefficients (EDCs) and Equivalent Diffusion Coefficients (EDFs) of the decoupled FPK equations are obtained with Radial Basis Function Neural Network (RBFNN) regression algorithm by using a small amount of prior data from the direct Monte Carlo Simulation (MCS). By doing so, a high-dimensional FPK equation is reduced to a one- or two-dimensional partial differential equation, from which the numerical solution can be computed quite easily. To examine the examination the effectiveness of the proposed procedure, three numerical examples with FD term are analyzed, and compared with the pertinent MCS of the original system. The results show that the proposed method has the advantage of good accuracy and relatively small amount of calculation time.