Random eigenvalue problems for bending vibrations of beams

Silke Mehlhose, Jürgen vom Scheidt and Ralf Wunderlich

Zeitschrift für Angewandte Mathematik und Mechanik 79 (1999), 693-702

Abstract :
The paper deals with the determination of statistical characteristics of eigenvalues for a class of ordinary differential operators with random coefficients. This problem arises from the computation of eigenfrequencies for the bending vibrations of beams possessing random geometry and material properties. Representations of eigenvalues are found by applying the Ritz method and perturbation results for matrix eigenvalue problems. Approximations of the probability density function and the moments of the random eigenvalues are given by means of expansions in powers of the correlation length of weakly correlated random functions which are used for modelling the random terms. The eigenvalue statistics determined analytically are compared favourably with Monte-Carlo simulations.