Stochastic stability of parametric oscillations of elastic shells

There is a powerful mathematical apparatus which appeared up on the basis of the theory of Brownian motion of Markov processes and processes of diffusional type nowadays. It allows deciding intricate dynamic problems taking into account fluctuation processes. A classic result in this area is the article of O.O. Andronov, L.S. Pontryagin and O.A. Vitt, in which firstly the methods of the theory of Markov processes were applied to research of problems of statistical dynamics of the nonlinear systems. Later, the strict mathematical theory of stochastic differential equations of Ito was presented in an article of I.I. Gikhman and A.V. Skorokhodov. An important step in application of this theory to research of dynamic problems of the elastic systems was become researches of R.L. Stratonovich. These researches were based on combination of Krilov-Bogolyubov method of averaging with the method of theory of Markov processes. The strict ground of this approach was done by R.Z. Khasminski. A significant contribution to the development of the theory of stochastic systems and the introduction of probabilistic methods for the calculation of structures was made by V.V. Bolotin and his followers. V.V. Bolotin performed significant work on the application of probabilistic methods to the calculation of structures. Also important are studies of stochastic parametric oscillations of various systems by Dimentberg, V.I. Klyatskin and others. From the beginning of 80-th of the last century the scientists of Structural and Theoretical mechanics department of the Kyiv National University of Construction and Architecture were engaged in development of the numeral research of stochastic stability of elastic systems. Results of numerical researches of stability of parametric oscillations of the cylindrical and shallow shells under different stochastic influences were presented at this article. Parametric oscillations models of the shells were formed on the basis of the asymptotic or functional approaches and Monte-Carlo method using the calculation procedures of finite element analysis software. Stochastic stability of elastic shells was formulated as stability in probability, on average and with respect to the moment functions of different order. The critical values of stochastic load intensity and the regions of stochastic stability of shells were obtained by Runge-Kutta method of the fourth order and the continuation by parameter method.