On the Theory of Fluctuations of Strongly Nonlinear (vibroimpact) Systems

During the operation of a large group of technical devices and mechanisms with impact interactions, such as clock mechanisms, systems associated with vibro-driven piles and wells drilling for oil and gas production, pneumatic impact mechanisms, vibrating mills, cyclic automation mechanisms, impact interaction mechanisms due to the presence of gaps and others, there exist parameters, such as the restitution coefficient, adjusting gap, that do not remain constant but change randomly from impact to impact. However, most of the research results of strongly nonlinear systems contain a deterministic formulation. In this paper, we propose a numerical-analytical method for studying the dynamics of strongly nonlinear systems, taking into account fluctuations in system parameters. In this case, the mathematical apparatus of the point mappings method of Poincare surfaces is widely used. Random deviations of parameters can be both delta- and non-delta-correlated. The concepts of stochastic stability are introduced, which made it possible to determine not only the optimal values of system parameters, but also to find, to some extent, dangerous/safe areas of the stability regions boundaries of the systems motion modes.