Response of nonlinear oscillators with fractional derivative elements under evolutionary stochastic excitations: A Path Integral approach based on Laplace's method of integration

In this paper, an approximate analytical technique is developed for determining the non-stationary response amplitude probability density function (PDF) of nonlinear/hysteretic oscillators endowed with fractional element and subjected to evolutionary excitations. This is achieved by a novel formulation of the Path Integral (PI) approach. Specifically, a stochastic averaging/linearization treatment of the original fractional order governing equation of motion yields a first-order stochastic differential equation (SDE) for the oscillator response amplitude. Associated with this first-order SDE is the Chapman-Kolmogorov (CK) equation governing the evolution in time of the non-stationary response amplitude PDF. Next, the PI technique is employed, which is based on a discretized version of the CK equation solved in short time steps. This is done relying on the Laplace's method of integration which yields an approximate analytical solution of the integral involved in the CK equation. In this manner, the repetitive integrations generally required in the classical numerical implementation of the procedure are avoided. Thus, the non-stationary response amplitude PDF is approximately determined in closed-form in a computationally efficient manner. Notably, the technique can also account for arbitrary excitation evolutionary power spectrum forms, even of the non-separable kind. Applications to oscillators with Van der Pol and Duffing type nonlinear restoring force models, and Preisach hysteretic models, are presented. Appropriate comparisons with Monte Carlo simulation data are shown, demonstrating the efficiency and accuracy of the proposed approach.