tailieunhanh - Diffstochastic equivalent linearization based on the fokker plank equation approach
In the paper a technique for determining the coefficients of the linearized equivalent equation based on the Fokker-Piank equation approach is presented. The investigation is then applied to Duffing and Vanderpol oscillations under a zero mean Gaussian white noise. | T~p chl CO' hqc Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 1 (6 - 11) DIFFSTOCHASTIC EQUIVALENT LINEARIZATION BASED ON THE FOKKER-PLANK EQUATION APPROACH NGUYEN DONG ANH - NGUYEN Due TINH Institute of Mechanics, Hanoi Vietnam 1. Introduction Stochastic equivalent linearization is the most popular approach to the approximate analysis of non-linear syste~s under random excitations. Over many years the original version of Gaussian equivalent linearization (GEL) has been developed by many authors, see . [Atalik & Utku, 1976], [Casciati & Faraveilli, 1986], [Roberts & Spanos, 1990], [Zhang et a!., 1991], [Anh & Schiehlen, 1995]. In order to improve the accuracy of GEL different techniques have been proposed, see . [6, 7, 8]. In the paper a technique for determining the coefficients of the linearized equivalent equation based on the Fokker-Piank equation approach is presented. The investigation is then applied to Duffing and Vanderpol oscillations under a zero mean Gaussian white noise. 2. Equivalent linearization criterion Consider a single-degree-of-freedom mechanical system, whose motion is described by the equation: () wherein the symbols have their customary meanings, f is a non-linear function of x and X, w, h, u are positive constants, and e is a positive parameter. The random excitation e(t) is a Gaussian white noise process of unit intensity E(e(t), e(t + r)) = 5(r), () where E(.) denotes the ei.:pectation operation, and 5(r) is Dirac-Delta function. Following the linearization method, we introduce new linear terms in the expression of the equation () () The linearized equation takes the form () There are some criteria for determining the coefficients p., A, see . [6, 7, 8j. In the paper an alternative approach to GEL is proposed as follows. 6 According to the classical approach of the averaging method the state coordinates (x, :i;) are transformed to the pair of amplitude and phase (a, cp) by the change :i:(t)
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