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Since the effect of some nonlinear terms is lost during the first order averaging procedure, the higher order stochastic averaging method is developed to predict approximately the response of linear and lightly nonlinear systems subject to weakly external excitation of second order coloured noise random processes. Application to Dufling oscillator is considered. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 21, 1999, No 2 (65 - 74) THE INFLUENCE OF SECOND ORDER NARROW-BAND COLORED NOISES ON NON-LINEAR RANDOM VIBRATIONS NGUYEN DONG ANH, NGUYEN DUC TINH Institute of Mechanics, NCST of Vietnam ABSTRACT. Since the effect of some nonlinear terms is lost during the first order averaging procedure, the higher order stochastic averaging method is developed to predict approximately the response of linear and lightly nonlinear systems subject to weakly external excitation of second order coloured noise random processes. Application to Dufling oscillator is considered. 1. Introduction For many years the well-known averaging method, originally given by Krylov and Bogoliubov and then developed by Mitropolskii (Bogoliubov and Mitropolskii, 1961) has proved to be a very powerful approximate tool for investigating deterministic weakly nonlinear vibration problems. In the field of random vibration the averaging method was extended by Stratonovich (1963) ·and has a mathematically rigorous proof by Khasminskii (1963). At.present, the stochastic averaging method (SAM) is widely used in different problems of stochastic mechanics such as vibration, ~tability and reliability problems (see e.g. Ariaratnam & Tam,1979; Bolotin·, 1.984; Ibrahim, 1985; Lin & Cai, 1995; Roberts & Spanos, 1986; Zhu, 1988). It should be noted that principally only first order SAM ha.S been applied in practice and usually to systems subject to white noise or wideband random processes. It is well-known, however, the effect of some non-linear terms is lost during the first order averaging procedure. In order to over come this insufficiency, different averaging procedures for obtaining approximate solutions have been developed (see e.g. Mitropolskii et al, 1992; Red-Horse & Spanos, 1992; Sri Namachchivaya & Lin, 1988; Zhu & Lin, 1994; Zhu et al, 1997). Recently, a higher order averaging procedure using Fokker-Planck (FP) equation was developed in (Anh, 1993, .
