Probability Distributions of Natural Frequencies of Uncertain Dynamic Systems

This article presents a polynomial dimensional decomposition method for calculating the probability distributions of random eigenvalues commonly encountered in dynamic systems. The method involves a hierarchical decomposition of a multivariate function in terms of variables with increasing dimensions, a broad range of orthonormal polynomial bases consistent with the probability measure for a Fourier-polynomial expansion of component functions, and an innovative dimension-reduction integration for calculating the expansion coefficients. The new decomposition does not require sample points, yet it generates a convergent sequence of lower-variate estimates of the probability distributions of eigensolutions. Numerical results, including frequency distributions of a piezoelectric transducer, indicate that the decomposition method developed provides accurate, convergent, and computationally efficient estimates of the tail probabilistic characteristics of eigenvalues.