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This article is an update of an earlier paper by Ibrahim (1987) and is aimed at reviewing the work published during the last decade in the area of vibration of structures with parameter uncertainties. Different types of uncertainty modeling are described in terms of material and geometric properties. These models are considered in terms of Gaussian or non-Gaussian distributions. Computational stochastic algorithms including stochastic finite element methods and Monte Carlo simulation are dominating a major part of current activities. Recent analytical developments of the random eigenvalue problem are reviewed with reference to typical structural elements. These developments include the implementation of statistical energy analysis, stochastic boundary element methods, and interval algebra. Other topics include forced vibration of single-and multi-degree-offreedom systems including nonlinear systems, localization in disordered periodic structures, and experimental results. Computational stochastic mechanics has found several industrial applications including aerospace, automotive and composite structural elements. The review also covers developments in the areas of statistical modeling of high frequency vibrations. There are 183 references.

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SUMMARYThe forced harmonic vibration analysis of portal frames consisting of viscously damped beams with spatial stochastic variation of mass and sti ness properties is considered. The analysis is based on the assembly of element stochastic dynamic sti ness matrices. The solution involves inversion of the global dynamic sti ness matrix, which, in this case, turns out to be a complex-valued symmetric random matrix. Three alternative approximate procedures, namely, random eigenfunction expansion method, complex Neumann expansion method and combined analytical and simulation method are used to invert the matrix. The performance of these approximate procedures is evaluated using Monte Carlo simulation results.

The problem of identifying parameters of nonlinear vibrating systems using spatially incomplete, noisy, time-domain measurements is considered. The problem is formulated within the framework of dynamic state estimation formalisms that employ particle filters. The parameters of the system, which are to be identified, are treated as a set of random variables with finite number of discrete states. The study develops a procedure that combines a bank of self-learning particle filters with a global iteration strategy to estimate the probability distribution of the system parameters to be identified. Individual particle filters are based on the sequential importance sampling filter algorithm that is readily available in the existing literature. The paper develops the requisite recursive formulary for evaluating the evolution of weights associated with system parameter states. The correctness of the formulations developed is demonstrated first by applying the proposed procedure to a few linear vibrating systems for which an alternative solution using adaptive Kalman filter method is possible. Subsequently, illustrative examples on three nonlinear vibrating systems, using synthetic vibration data, are presented to reveal the correct functioning of the method. r