Nonlinear Response to Polynomial of Filtered Gaussian Process

Abstract

A method for the evaluation of the response statistical moments of nonlinear second-order dynamic systems with linear damping and polynomial restoring force subjected to non-Gaussian input in the form of a polynomial of a filtered Gaussian process is presented. Itô's stochastic differential equations for the augmented system of the structure and the shaping filter are derived. The solution is looked for by using a moment equation approach. The equations for the response statistical moments are written by applying Itô's differential rule. Because of the nonlinearities in polynomial form, the moment equations form an infinite hierarchy so that a closure scheme is needed. An iterative closure scheme is proposed. First, the system is partially linearized, substituting a linear restoring force for the nonlinear one, while retaining the nonlinear excitation. In this way, the response of the linearized system is non-Gaussian. In the second step, the moment equations of the actual nonlinear system are closed by approximately expressing the hierarchical moments by means of the estimates obtained for the linearized system. The first numerical analysis regards a Duffing oscillator subject to wind pressure: the results compare well with Monte Carlo simulation, and it is shown that the response moments allow the evaluation of other important response statistics. The second example concerns an offshore tower excited by Morison forces approximated by a cubic polynomial. Again, the proposed approach performs well.