On the theory of a non-autonomous vibro-impact system with memory in the frictional force
The dynamics of a non-autonomous vibro-impact system is investigated in this work. The system is composed of two vibrating bodies, which experience frictional resistance that is described with the help of a modified Coulomb friction model with memory. The mathematical model that describes the dynamics of the vibro-impact system can be classified as a dynamical system with variable structure. It consists of a system of ordinary differential equations and some functional relations. The model is analyzed by means of application of the point mapping technique. This technique allows to study the structure of the phase space of the system and its dependence on (i) the varying in time static coefficient of friction, (ii) the parameters of a harmonic force that acts on the system, and (iii) the position of a stopper that limits the system displacement. Numerical study of the system dynamics allowed for identification of the main regimes of the system motion and their intermittency. For example, periodical regimes of high complexity were found as well as the transition to chaos through the period doubling bifurcation. Additionally, the use of symbolic computations helped to uniquely interpret the obtained bifurcation diagrams.