Equivalent Statistical Quadratization for Nonlinear Systems

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Abstract
The statistical linearization method is often inadequate for estimating spectral properties of random responses of nonlinear systems. This is due to the fact that the power spectra of responses of linear systems span only the frequency range of the excitation spectrum, whereas significant responses outside this range are possible for nonlinear systems. As an extension of the linearization method, a statistical “quadratization” method is presented. Specifically, the statistics of the nonlinear system are obtained from an “equivalent” quadratic system, which is constructed by replacing the nonlinearity by polynomials up to quadratic order. The nonlinear equivalent system has a form for which the solution can be approximated by using the Volterra series method. The method is formulated for a nonlinear single‐degree‐of‐freedom oscillator under stationary, Gaussian excitation. Stationary, non‐Gaussian response statistics are obtained. The non‐Gaussian response probability distribution is approximated by a third‐order Gram‐Charlier expansion. To demonstrate the applicability of the method, solutions are obtained for two sample systems. The corresponding results compare well with relevant Monte Carlo simulation data. Further, it is shown that the quadratization method is notably superior to the linearization method for the considered systems.

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