Probability Distributions of Natural Frequencies of Uncertain Dynamic Systems

Abstract

This article presents a polynomial dimensional decompositionmethod for calculating the probability distributions of random eigenvalues commonly encountered in dynamic systems. The method involves a hierarchical decom-position 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. Nomenclature Aj = jth random coefficient matrix BN, L2 = Borel field on RN and Hilbert space, respectively Ci1 iSj1 jS = coefficient for S-variate component function c, X = mean and covariance of X, respectively E = expectation operator f = general nonlinear function fixi = marginal probability density of Xi fXx = joint probability density of X KX = random stiffness matrix KiX = ith random stiffness of spring k = truncation parameter L = size of random coefficient matrices m = mass matrix m = degree of orthonormal polynomial basis N = number of random variables n = number of Gaussian quadrature points R = positive integer ranging from S to N RN, CN = N-dimensional real and complex vector spaces, respectively S = positive integer ranging from 1 to N X = input random vector Xi = ith random variable ij = coefficient for univariate component function of i1i2j1j2 = coefficient for bivariate component function of X = random eigenvalue function i1i2i3j1j2j3 = coefficient for trivariate component function of I;i = imaginary part of ith component of i1 iS = S-variate component function of R;i = real part of ith component of ̂RX = R-variate approximation of ~SX = S-variate polynomial approximation of 0 = first coefficient of polynomial decomposition

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