Response of Linear and Non-Linear Structural Systems under Gaussian or Non-Gaussian Filtered Input
Many types of loadings acting on engineering structures possess random and dynamic characteristics. Even though the study of random vibration, using the concepts of stochastic process theory, is a relatively new engineering discipline, interest in this fi
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RESPONSE OF LINEAR AND NON-LINEAR STRUCTURAL SYSTEMS UNDER GAUSSIAN OR NON-GAUSSIAN FILTERED INPUT
G. Muscolino University of Messina, Messina, Italy
1 INTRODUCTION. Many types of loadings acting on engineering structures possess random and dynamic characteristics. Even though the study of random vibration, using the concepts of stochastic process theory, is a relatively new engineering discipline, interest in this field has grown rapidly in the last few decades. The result is a very extensive literature. However, while in the random vibration of linear structures there are now several papers which cover both theoretical and practical aspects and a number of text-books which give a good overview of the subject [ 1-4] is available, the study of structures which present non-linearities is more recent and exact solutions are available for few special cases only. Furthermore, while a comprehensive linear theory exists, no correspondent general theoretical framework for non-linear problems has been formulated due to the complexity of these problems.
F. Casciati (ed.), Dynamic Motion: Chaotic and Stochastic Behaviour © Springer-Verlag Wien 1993
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G. Muscolino
The most commonly method used to calculate the response of weakly non-linear systems is the statistical or equivalent linearization method [5]. The main idea of this method, which actually requires relatively little numerical efforts, is to evaluate the Gaussian response properties of an equivalent linear system. However, it becomes immediately apparent that this methods gives accurate results especially for weakly non-linear systems. An other method used for sufficiently small non-linearities is the perturbation method [6], the basic idea of this method is to expand the solution of the non-linear set of equations in terms of a small scaling parameter which characterizes the magnitude of the non-linear terms. This method is not very extensively used for the poor convergence properties and excessive computational requirements in several cases. More general methods which give more accurate results are the methods based on the non-Gaussian closures. The basic idea of these methods is to assume the non-Gaussian probability density function of the response of non-linear systems as a series expansion with adjustable parameters of a Gaussian probability density function. The adjustable parameters depend on the statistical moments of the response which are generally governed by an infinite hierarchy of coupled differential equations. The closure approximation truncates the infmite terms series expansion in order to obtain a soluble set of equations. The simplest level of closure is the Gaussian one in which higher order statistical moments are expanded in terms of the first and second moments, as if the random processes involved were Gaussian distributed. The Gaussian closure for purely external excitations gives results identical to statistical linearization. Improvements in accuracy can be obtained by using higher order level of closure. By no means it can be sta
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