Functional-perturbational analysis of non-linear systems under random external and parametric excitations

Abstract

A functional-perturbational type approach based on the Volterra-Wiener functional expansion is developed to analyze the non-linear response of dynamic systems to random excitations. The approach is applicable for non-linear systems under parametric excitation provided that the system is also subjected to external excitation. An example of a cantilever beam subjected to axial and lateral excitations is considered. For the linear case the response is determined up to second order approximation. The mean square response is compared with the results of the method of moments. It is found that the response predicted by the functional approach represents an asymptotic approximation to the moment method solution. The condition of mean square stability does not emerge directly from the functional solution. However, one can establish the stability boundary of the system if the expansion is expressed by a closed form analytical function. The non-linear mean square response is determined and also compared with the Gaussian closure scheme solution. The functional solution gives real positive mean square response for all external excitation levels while the Gaussian closure scheme predicts positive mean square response only for a relatively lower spectral density level of external excitation.