In recent years, extensive research has been reported about a method which is called the generalized polynomial chaos expansion. In contrast to the sampling methods, e.g., Monte Carlo simulations, polynomial chaos expansion is a nonsampling method which represents the uncertain quantities as an expansion including the decomposition of deterministic coefficients and random orthogonal bases. The generalized polynomial chaos expansion uses more orthogonal polynomials as the expansion bases in various random spaces which are not necessarily Gaussian. A general review of uncertainty quantification methods, the theory, the construction method, and various convergence criteria of the polynomial chaos expansion are presented. We apply it to identify the uncertain parameters with predefined probability density functions. The new concepts of optimal and nonoptimal expansions are defined and it demonstrated how we can develop these expansions for random variables belonging to the various random spaces. The calculation of the polynomial coefficients for uncertain parameters by using various procedures, e.g., Galerkin projection, collocation method, and moment method is presented. A comprehensive error and accuracy analysis of the polynomial chaos method is discussed for various random variables and random processes and results are compared with the exact solution or/and Monte Carlo simulations. The method is employed for the basic stochastic differential equation and, as practical application, to solve the stochastic modal analysis of the microsensor quartz fork. We emphasize the accuracy in results and time efficiency of this nonsampling procedure for uncertainty quantification of stochastic systems in comparison with sampling techniques, e.g., Monte Carlo simulation.

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In order to get a better insight into the function of the human middle ear it is necessary to simulate its dynamic behaviour by means of the finite-element method. Three-dimensional measurements of the surfaces of the tympanic membrane and of the auditory ossicles malleus, incus and stapes are carried out and geometrical models are created. On the basis of these data, finite-element models are constructed and the dynamic behaviour of the combinations tympanic membrane with malleus in its elastic suspensions and stapes with annular ligament is simulated. Natural frequencies and mode shapes are computed by modal analysis. These investigations showed that the ossicles can be treated as rigid bodies only in a restricted frequency range from 0 to 3.5 kHz.

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In this paper, the application of generalized polynomial chaos expansion in stochastic structural modal analysis including uncertain parameters is investigated. We review the theory of polynomial chaos and relating error analysis. A general formulation for the representation of modal problems by the polynomial chaos expansion is derived. It shows how the modal frequencies and modal shapes are influenced by the parameter uncertainties. The key issues that arise in the polynomial chaos simulation of modal analysis are discussed for two examples: a discrete 2-DOF system and continuous model of a microsensor. In both cases, the polynomial chaos expansion is used for the approximation of uncertain parameters, eigenfrequencies and eigenvectors. We emphasize the accuracy and time efficiency of the method in estimation of the stochastic modal responses in comparison with the sampling techniques, such as the Monte Carlo simulation.