Dynamics of frictional systems with memory

Dynamics of a number of shock-vibrating systems, accounting for dry friction forces with memory, is analyzed. The vibration limiter and the vibrational system interact according to Newton’s hypothesis. The mathematical model (MM) of the systems is a strongly nonlinear non-autonomous system with a variable structure. A special feature of the approach of the present study is that a point mapping is formed on the basis of the time of relative rest of the system. The moments of the transition of an image point from the surfaces of sliding motions that correspond to the relative rest of the system are determined each time from the solution of nonlinear equations taking into account functional dependence of the coefficient of friction of relative rest and the previous history of the process. This necessitated the development of an original approach to point mapping and to the interpretation of the obtained results. A new investigation methodology and an original software product have been developed for analyzing the phase portrait structure of the MM, depending on the characteristics of sliding and static friction forces, as well as on the type and location of the limiter. The character of the change of the bifurcation diagrams enables to determine the main laws of the process of change of the motion regimes (from periodic motion regimes of arbitrary complexity to the transition to chaos via the period-doubling process) when varying the parameters of the vibrational system (the amplitude and frequency of the periodic effect, the form of the functional relation describing the change of the coefficient of friction of relative rest). The dynamic characteristics of the systems with and without accounting for the vibration limiter are compared in the paper.