THE JOINT RESPONSE-EXCITATION PDF EVOLUTION EQUATION. NUMERICAL SOLUTIONS FOR THE LONG-TIME, STEAD-STATE RESPONSE OF A HALF OSCILLATOR

The response excitation theory, introduced in Athanassoulis & Sapsis (2006) and Sapsis & Athanassoulis (2008), is a new powerful method that can be used for treating nonlinear systems with arbitrary polynomial non-linearities excited by colored, Gaussian or non Gaussian, stochastic processes. Since the assumption of delta-correlated excitation is overcome, this formulation is appropriate for macroscopic stochastic dynamical systems, where the excitation correlation time is comparable with the system’s relaxation time. In these cases, the response is not Markovian and, thus, a reduction to Fokker-PlanckKolmogorov equations is not generally possible. Besides, the filtering approach, is not generally applicable for non Gaussian excitation. The joint Response-Excitation Probability Density Function (REPDF) evolution equation is derived by finite dimensional projection of a Functional Differential Equation for the joint, response-excitation characteristic functional. However, it is a peculiar equation, involving two times and two probability arguments (one for the excitation and one for the response), and partial derivatives only with respect to one (excitation) time and one (excitation) probability argument. This equation alone is a necessary differential constraint, that cannot provide a unique joint REPDF. However, if it is further supplemented by the marginal compatibility constraints, and analytically solved, local, linearized problems, accounting for the local, response-excitation, correlation structure, becomes uniquely solvable. In this paper we present, for the first time, a numerical solution to this equation, in the long time, based on an appropriate Kernel Density (KD) representation of the REPDF, and a Galerkin-type numerical scheme to calculate the KD coefficients. This scheme is used to derive the probabilistic characteristics of the responses of a half oscillator, subject to asymptotically stationary, strongly colored, Gaussian excitation. The obtained PDF is favorably compared with results from a Monte Carlo simulation.