Equivalent Statistical Quadratization and Cubicization for Nonlinear Systems

A summary is given of the development of two techniques for handling system and excitation nonlinearities: equivalent statistical quadratization and equivalent statistical cubicization. Depending upon the nature of a given nonlinearity, one of these procedures may be employed to approximate it by a quadratic or cubic polynomial. In this manner, the nonlinearity is preserved and the response transfer functions are attainable using a Volterra functional series approach. When the parent input processes are characterized by appropriate spectra, integration in the frequency domain yields the desired spectra, bispectra or higher-order cumulants of the response. Using the system moment information corresponding to the response cumulants, a moment-based Hermite transformation yields probability density functions for the non-Gaussian processes. Also, the force and response spectra exhibit the appropriate secondary peaks corresponding to the particular system and excitation characteristics. The accurate prediction of both extremes and response power spectral densities is a notable improvement over equivalent statistical linearization. All results compare well with those obtained using a time-domain simulation.