On the dynamic response of non-linear systems with parameter uncertainties

This paper presents a procedure for obtaining the dynamic response of non-linear systems with parameter uncertainties. Consideration is given to systems with polynomial non-linearity subjected to deterministic excitation. The uncertain parameters are modeled as time-independent random variables. The set of orthogonal polynomials associated with the probability density function is used as the solution basis, and the response variables are expanded in terms of a finite sum of these polynomials. A set of deterministic non-linear differential equations is derived using the weighted residual method. The discrete-time solutions to the equation set are evaluated numerically using a step-by-step time-integration scheme and the response statistics are determined. Application of the proposed method is illustrated through the analysis of non-linear single-degree-of-freedom structural systems exhibiting uncertain stiffnesses. Both hardening and softening stiffness characteristics are examined. The accuracy of the results is validated by direct numerical integration.