Stochastic Eigenvalue Problem with Polynomial Chaos

A non statistical eigenvalues extraction algorithm that uses generalized polynomial chaos is presented for solving real and distinct stochastic eigenvalue problems. An iterative procedure, similar yet ecient from to the vector iteration method used for solving deterministic problems is developed. The proposed method uses eigenvalues obtained using the mean values of system properties and the eigenvalue shift theorem. This algorithm yields not only the fundamental but any higher eigenvalue. The uncertainties in the system inputs as well as the system response (eigenvalues and eigenvectors)are represented by the WienerAskey orthogonal polynomial functions. Galerkin projection is applied in the probabilty space to minimize the weighted residual of the error. Free vibration response of a discrete three degrees-of-freedom system and that of a simply supported beam in the presence of uncertainties in material properties is analyzed. First three eigenvalues of both types of systems are obtained. The convergence of eigenvalue polynomial coecients is studied. The accuracy of the proposed method is compared with the results obtained from Monte Carlo simulation using Latin Hypercube Sampling (LHS). Excellent agreement is observed.

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