ABSTRACT
In general, very limited classes of nonlinear dynamic systems possess exact solutions and therefore various approximate methods have been proposed for their solutions over the last three decades. Among these methods the statistical or equivalent linearization (SL) and their derivatives presented in chapter 4 seem to be the most popular ones due mainly to their simplicity and their adaptability to multi-degree-offreedom (mdof) systems. However, the solution of the response statistics other than the second moment, such as the auto-correlation by the SL technique may not be reliable because it is based on the minimization of the mean square error. For example, the difference of the tail ranges between the exact probability density of the response of a Duffing oscillator under Gaussian white noise excitation and the approximate result obtained by the SL for a response greater than three times the standard deviation may be by a factor as large as 250 [5.1 ]. This, in tum, means that for high threshold levels the prediction of crossing statistics by the SL may be seriously incorrect. It has also been shown that for nonlinear damped systems SL can give the first excursion probability in error by several orders of magnitude [5.2]. Moreover, for self-sustained or parametrically excited systems the error may even be larger due to the non-Gaussian property of the response [5.3, 5.4).
