The Determination of Stochastic Moments of Phase Coordinates in a Nonlinear Structural Model

The problem of stochastic dynamics of a nonlinear finite element model is considered in this article using the expansion of the solution in the truncated orthogonal basis of eigenvectors of the statistically linearized system. It is assumed that the nonlinearity present in the system does not lead to a considerable change in the system’s dynamic behavior. It only makes a significant quantitative amendment to the probabilistic characteristics in relation to the linear model. In order for the latter to maintain its physical meaning, all nonlinear characteristics are represented as the sum of the linear and nonlinear components. The additive external action is taken as stationary or quasi-stationary. The system of equations relative to the main coordinates written in the Cauchy form is brought to the canonic form using the generating filters. The well-known differential equations of the method of moments are used to solve the problem within the correlation theory framework, providing that the relationship between the eigenvalues and eigenvectors, and the unknown stochastic moments, through which the coefficients of statistical linearization are expressed, is known. To reveal this relationship, the expansions of eigenvalues and eigenvectors to power series with respect to coefficients of statistical linearization are used. These expansions are considered as variations of the elements of the stiffness matrix of the linear model. The linear or quadratic approximation is taken into account. The stochastic moments that constitute the coefficients of statistical linearization are calculated through an iterative procedure. Practice shows that this procedure converges in two or three approximations.  The results are illustrated by an example.

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