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A wavelets-based method is developed to estimate the evolutionary power spectral density (EPSD) of nonstationary stochastic processes. The method relies on the property that the continuous wavelet transform of a nonstationary process can be treated as a stochastic process with EPSD given in terms of the EPSD of the process in a closed form. This yields an equation in the frequency domain relating the instantaneous mean-square value of the wavelet transform to the EPSD of the process. A number of these equations are considered, each related to a certain scale of the wavelet transform, in conjunction with representing the target EPSD as a sum of time-independent shape functions modulated by time-dependent coefficients; the squared moduli of the Fourier transforms of the wavelets associated with the selected scales are taken as shape functions. This leads to a linear system of equations which is solved to determine the unknown time-dependent coefficients; the same system matrix applies for all time instances. Numerical examples demonstrate the accuracy and computational efficiency of the proposed method. DOI: 10.1061/(ASCE)0733-9399(2004)130:8(952) CE Database subject headings: Stationary processes; Earthquakes; Spectra; Spectral density function; Stochastic processes.

Random vibrations of nonlinear systems subjected to Gaussian input are investigated by a technique based on statistical quadratization, and cubicization. In this context, and depending on the nature of the given nonlinearity, statistics of the stationary response are obtained via an equivalent system with a polynomial nonlinearity of either quadratic or cubic order, which can be solved by the Volterra series method. The Volterra series response is expanded in a trigonometric Fourier series over an adequately long interval T, and exact expressions are derived for the Fourier coefficients of the second- and third-order response in terms of the Fourier coefficients of the first-order, Gaussian response. By using these expressions, statistics of the response are determined using the statistics of the Fourier coefficients of the first-order response, which can be readily computed since these coefficients are independent zero-mean Gaussian variables. In this manner, a significant reduction of the computational cost is achieved, as compared to alternative formulations of quadratization and cubicization methods where rather prohibitive multifold integrals in the frequency domain must be determined. Illustrative examples demonstrate the reliability of the proposed technique by comparison with data from pertinent Monte Carlo simulations. CE Database keywords: Oscillators; Nonlinear systems; Random vibration; Gaussian process.