On the Dynamic Response of Nonlinear Uncertain Systems

This thesis presents an approach for performing second moment analyses of nonlinear dynamic systems with parameter uncertainty. 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. When a deterministic excitation source is considered, the response variables are expanded in terms of a finite sum of these polynomials with time-dependent coefficients. The weighted residual method is employed to derive a set of deterministic nonlinear differential equations that can be solved numerically for evaluations of response statistics. This solution approach is further extended to nonlinear continuous systems involving inhomogeneous random media. A discrete representation is obtained via a spatial discretization procedure for the continuous response variables as well as the random continuum. Thus, the continuous random system can then be treated as in the case of the discrete random systems. The solution approach is applied to a study of a nonlinear random shear-beam model subjected to a near-field earthquake ground motion. The response uncertainty for nonlinear uncertain systems subjected to external stochastic excitation is also investigated. A general solution procedure based on equivalent linearization is presented. In this solution methodology, the instantaneous equivalent stiffness and damping matrices are approximated as quadratic random functions. The resulting Liapunov system with explicit random coefficients can then be solved using the newly developed solution approach. Applications to single-degree-of-freedom uncertain systems are given and the accuracy of the results is validated.