Stochastic linearization: what is available and what is not

Abstract

Stochastic equivalent linearization methods are the most popular among all approximation methods for the dynamics of a nonlinear system under random excitation. A complete presentation of these methods can be found in Roberts, J. B., Spanos, P. D., Random Vibration and Statistical Linearization. J. Wiley & Sons, 1990 [5]. Despite the fact they were introduced 40 years ago, the first justification, concerning the so-called “true linearization”, was proposed by Kozin (Kozin, F., The Method of Statistical Linearization for Non-Linear Stochastic Vibrations. In Nonlinear Stochastic Dynamic Engineering Systems, ed. F. Ziegler, G. I. Schue ̈ller. Springer Verlag, 1987.) [4] in 1987. The so called “Gaussian linearization” is the most used of all. The goal of this contribution is to present a mathematical approach recently introduced in Bernard, P., Wu, L., Stochastic Linearization: The Theory, to appear [2], to the problem of stochastic linearization based on the use of a large deviation principle. This approach can be considered as an extension of Kozin's work (Kozin, F., The Method of Statistical Linearization for Non-Linear Stochastic Vibrations. In Nonlinear Stochastic Dynamic Engineering Systems, ed. F. Ziegler, G. I. Schue ̈ller. Springer Verlag, 1987.) [4]. Several linearization methods are justified, among which the “true linearization”. The “Gaussian linearization” unfortunately cannot be justified. Moreover, an example from Alaoui, M., Bernard, P., Asymptotic Analysis and Linearization of the Randomly Perturbated Two-wells Duffing Oscillator, [1] shows that it can give rise to wrong results. This fact was also noticed by Grundmann in this international conference on uncertain structures.