TAILIEUCHUNG - Higher order stochastic averaging method for mechanical systems having two degrees of freedom

Higher order stochastic averaging method is widely used for investigating single-degree-of-freedom nonlinear systems subjected to white and coloured random noises. In this paper the method is further developed for two-degree-of-freedom systems. An application to a system with cubic damping is considered and the second approximation solution to the Fokker-Planck (FP) equation is obtained. | Vietnam Journal of Mechanics, VAST, Vol. 26 , 2004, No 2 (103 - 110) HIGHER ORDER STOCHASTIC AVERAGING METHOD FOR MECHANICAL SYSTEMS HAVING TWO DEGREES OF FREEDOM NGUYEN D ue TINH Mining Techni cal College, Quang Ninh ABSTRACT . Higher order stochastic averaging method is widely used for investigating single-degree-of-freedom nonlinear systems subjected to white and coloured random noises. In this paper the method is further developed for two-degree-of-freedom systems. An application to a system with cubic damping is considered and the second approximation solution to the Fokker-Planck (FP) equation is obtained. 1 Introduction The stochastic averaging method was extended by Stratonovich (1963) and has a mathematically rigorous proof by Khasminskii (1963) . At present, t he stochastic averaging met hod (SAM) is widely used in different problems of stochastic mechanics , such as vibration, stability and reliability problems (see . Ariaratnam and Tam , 1979; Bolot in, 1984; Roberts and Spanos, 1986; Zhu, 1988) . It should be noted that principally only first order SAM has been applied in practice and usually to systems subj ect to white noise or wideband random processes. However , the effect of some terms is lost during the first order averaging procedure. In order to overcome this insufficiency, different averaging procedures for obtaining approximate solutions have been developed ~see . Mitropolskii et al, 1992; Red-Horse and Spanos, 1992; Zhu and Lin, Hl94 ; Zhu et al, 1997). Recently. a higher order averaging procedure using FP equation is developed in (Anh, 1993) and t hen applied to the systems having one degree of freedom under white noise and coloured noise excitations (Anh and Tinh, 1995; Tinh, 1999) . In t he present paper this procedure is further developed to lightly nonlinear systems subject to white noise excitations . An application to the system with cubic damping is considered. 2 Higher approximate solutions to FP .

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