Spectral approach to equivalent statistical quadratization and cubicization methods for nonlinear oscillators

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