Approximate response determination of nonlinear oscillators with fractional derivative elements subjected to combined periodic and evolutionary stochastic excitations

In this paper, an approximate analytical method is proposed to determine the response of nonlinear oscillators with fractional derivative elements subjected to combined periodic and evolutionary stochastic excitations. This is done by combining a memory-free formulation with a linearization framework to treat both the nonlinearity and the fractional derivative elements of the system. Specifically, assuming that the system response is written as the sum of a periodic and a stochastic components, the system governing equation of motion is equivalently cast into a corresponding set of a nonlinear fractional deterministic differential sub-equation and a nonlinear fractional stochastic differential sub-equation. The fractional deterministic sub-equation is subsequently transformed into a set of coupled linear equations with integer-order derivatives solely, by relying on the memory-free formulation. On the other hand, a combination of the statistical linearization and the stochastic averaging methods is employed to treat the nonlinear fractional stochastic sub-equation subjected to the evolutionary excitation. Finally, the oscillator response displacement consisting of the mean and the variance of the periodic and the stochastic response components, respectively, is obtained by solving simultaneously the set of equations derived by applying the memory-free formulation and linearization treatments. The proposed framework can treat nonlinear oscillators with fractional derivative elements subjected to combined periodic and non-stationary stochastic excitations characterized by arbitrary evolutionary power spectrum forms, even of the non-separable kind. Its accuracy and effectiveness are demonstrated by numerical examples pertaining to nonlinear oscillators with fractional derivative elements subjected to periodic and stochastic excitation described by both separable and non-separable power spectrum forms, while Monte Carlo simulation data are also used for comparison.