By Cho W.S. To

It is a systematic presentation of a number of sessions of analytical concepts in non-linear random vibration. The publication additionally encompasses a concise therapy of Markovian and non-Markovian options of non-linear differential equations.

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**Sample text**

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 theresponse is described by the FPK equation, exact stationary solution can be obtained. 15]. 15]. tI seems that the latter approach gives the broadest class of solvable reduced FPK equations. 14], and the application of the theory of elementary or integrating factor for first order ordinary differential equations.

The elements of [B] are Bij. 2) in which p(X,t) is the joint transition probability density function or simply referred to as transition probability density, and D is the matrix of excitation intensities whose ij'th element is Dij =BSij where Sij are the cross-spectral densities of the white noise processes. 4a,b) It is understood that the first and second derivate moments are evaluated at X = x. 5) The FPK equation is invariant under translations in time t. 3), respectively, as where L* is the adjoint operator to L.

As they are used for illustration only the probability density function of every case is considered. Example I. Consider the model in Ref. (8) is an arbitrary function, and 8 is the total energy (I-2) Applying the same symbols as in the method presented above, the two Itô stochastic differential equations for Eq. (I-1) are 28 Nonlinear Random Vibration (I-3) (I-4) where B(t) or B is a unit Wiener process. The corresponding reduced FPK equation becomes (I-5) The first and second derivate moments are divided into two parts as those in the procedure described above except that A2 (2) = - g(x1) is chosen in accordance with Eq.