An extension of the Kazakov relationship for non-Gaussian random variables and its use in the non-linear stochastic dynamics

A generalization for non-Gaussian random variables of the well-known Kazakov relationship is reported in this work. If applied to the stochastic linearization of non-linear systems under non-Gaussian excitations, this relationship allows us to define the significance of the linearized stiffness coefficient. It is the sum of that one known in the literature (the mean of the tangent stiffness) and of terms taking into account the non-Gaussianity of the response. Moreover, the relationship here given is used for finding alternative formulae between the moments and the quasi-moments. Lastly, it is used in the framework of the moment equation approach, coupled with a quasi-moment neglect closure, for solving non-linear systems under Gaussian or non-Gaussian forces. In this way an iterative procedure based on the solution of a linear differential equation system, in which the values of the response mean and variance are those of the precedent iteration, is originated. It reveals a good level of accuracy and a fast convergence.