ABSTRACT
The response of general nonlinear oscillator under parametric random excitations and external random excitations has been extensively studied in the last three decades. The foundation of the development has been installed earlier by Rayleigh [3.1], Fokker [3.2], and Smoluchowski [3.3], for example. In general, no exact solution can be found. When the excitations can be idealized as Gaussian white noises, in which case, the response of the system can be represented by a Markovian vector and the probability density function of the response is described by the FPK equation, exact stationary solution can be obtained. The solution of the FPK equation has been reported in the literature [3.4-3.15]. The following approach is that presented by To and Li [3.15]. It seems that the latter approach gives the broadest class of solvable reduced FPK equations. It is based on the systematic procedure of Lin and associates [3.11-3.14], and the application of the theory of elementary or integrating factor for first order ordinary differential equations. In Ref. [3.14] the solution of the reduced FPK equation is obtained by applying the theory of generalized stationary potential which is less restrictive than that employing the concept of detailed balance [3.12]. The latter is similar to that of Graham and Haken [3.16]. The basic idea of the concept of Graham and Haken is to separate each drift coefficient into reversible and irreversible parts.
