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By Umberto Alibrandi, Giuseppe Ricciardi (auth.), Isaac Elishakoff, Christian Soize (eds.)

Table of contents: Stochastic tools in nonlinear structural dynamics.- Stochastic types of uncertainties in computational structural dynamics and structural acoustics.- the story of stochastic linearization strategies: over part a century of progress.- entire modeling of doubtful platforms utilizing fuzzy set theory.- Bounding uncertainty in civil engineering: theoretical heritage and applications.- mixed tools in nondeterministic mechanics.

In this publication the present state-of-the-art of nondeterministic mechanics in its quite a few types is gifted. the themes variety from stochastic difficulties to fuzzy units; from linear to nonlinear difficulties; from particular methodologies to combos of varied thoughts; from theoretical issues to sensible functions. it truly is particularly designed to light up some of the elements of the 3 methodologies (probabilistic or stochastic modelling, fuzzy units established research, antioptimization of constructions) to accommodate numerous uncertainties and deepen the dialogue in their professionals and cons.

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5 and Fig. 6 the displacement variance and the relative error are plotted, in the case of softening behavior of the oscillator (0 < I < 1). From these figures the lack of convergence of the series expansion used by the NGSL is evident. vI = 4 , 6 , 8) . vI = 4 , the error is lesser than 5% in the entire range 0 < I < l. The NGSL applied to the double-well Duffing oscillator. consider the following Duffing oscillator: Let us U. Alibrandi and G. 2 0 Figure 5. Displacement variance versus I for the softening power-law oscillator.

12 Measures of non Gaussianity: kurtosis and negentropy The determination of the directions of maximum independence requires a quantitative measure of nongaussianity of the random variables Yi. The simplest one is given from the kurtosis "'( . ), which for zero mean and unit variance random variables is defined as (134) and it is strictly related to the fourth moment of Yi. The kurtosis is zero for Gaussian Yi, while for non Gaussian random variables can assume positive or negative values. Typically, the random variables with positive kurtosis (called supergaussian distributions) have "spiky" pdf around the origin and heavy tails, such as the Laplace distribution; conversely, the PDF of the random variables with negative kurtosis (known as subgaussian distributions) are "flat" around the origin and have very low values at the tails, and then "'(Yi) { > = 0 0 <0 supergaussian pdf gaussian pdf subgaussian pdf (135) Consequently usually as a measure of non Gaussianity it is used not the 50 U.

Muscolino, G. Ricciardi, and P. Cacciola. Monte carlo simulation in the stochastic analysis of non-linear systems under external stationary poisson white noise input. Int. 1. , 38:141-48, 2003. A. Naess and J. M. Johnsen. Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Probabilistic Engineering Mechanics, 8(2):91-106, 1993. H. J. Pradlwarter. Non-gaussian linearization - an efficient tool to analyze nonlinear mdof-systems. Journal of Nuclear Engineering and Design, 128: 175-92, 1991.

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