Two-stage optimal stochastic linearization in analyzing of non-linear stochastic dynamic systems

A novel stochastic linearization approach is developed to predict the second-moment response of non-linear systems under stochastic parametric and external excitations. The present approach is realized by a two-stage optimization: the first stage of optimal linearization modeling and the second stage of parameters optimization. Five examples, including two polynomial oscillators, one hysteretic Bouc-Wen oscillator under stochastic external excitation, and two polynomial oscillators under stochastic parametric and external excitations are selected to illustrate the present approach. The validity of the present approach is validated by some approximate solutions, exact solutions, and Monte Carlo simulations. For system non-linearity, which can be approximated as a full-states linear combination in the Gaussian linearization model, the present approach offers a more accurate prediction of the second moment than that by the Gaussian linearization method. The two-stage optimal Gaussian linearization method incorporates the merits of Gaussian linearization method in the first stage and the SPEC-alternative in the second stage.