Dynamics of a vibration system, taking into account the hereditary type friction and the mobility of a vibration limiter

Abstract

The paper investigates the dynamics of some vibration systems taking into account the hereditary-type dry friction and mobility of a vibration limiter. The interaction of the vibration limiter and the vibration system occurs either according to Newton’s hypothesis (an absolutely rigid limiter) or softly (the limiter is mobile). A general mathematical model (MM) of systems has been developed, which is highly non-linear nonautonomous system with variable structure. A numerical-analytical approach using the mathematical apparatus of the point mapping method is implemented to study the dynamics of the MM. The peculiarity in the research approach is that the point mapping is not formed by the classical method (mappings of the Poincaré surface into itself), but by the duration of relative rest of the vibration system, which greatly facilitated the process of point mapping and its detailed study. The presence of floating boundaries of slip-motion plates required the creation of an original approach in constructing point mapping and interpreting the results obtained. The structure of the phase portrait of the MM as a function of the characteristics of the sliding and state friction forces, as well as on the type and position of the limiter was studied using the developed research methodology and the created software product. Based on the character of changes in bifurcation diagrams, it was possible to determine the main laws of changes in the motion regimes (occurrence of random complexity via the period possible transfer to chaos doubling process) when changing the parameters of the vibration system (amplitude and frequency of periodic action, the form of the functional dependence describing the change in the coefficient of friction of relative rest. Analytical results with and without a vibration limiter are compared.