Dynamics of a frictional system consisting of a rough body situated on a rough belt moving at a constant velocity is studied. The vibrations are limited by an elastic obstacle. The Coulomb-Hammonton dry friction characteristic is chosen, according to the hypothesis of Ishlinskiy and Kragelskiy, in the form of hereditary-type friction, where the coefficient of friction of relative rest (CFRR) is a monotone non-decreasing continuous functionof the time of relative rest at the previous analogous time interval. A mathematical model and the structure of its phase space are presented, as well as the equations of point maps of the Poincare surface and the results of studying the dynamic characteristics of the parameters of the system (velocity of the belt, type of the functional relation of CFRR, rigidity of the elastic obstacle, etc.). A software product has been developed which makes it possible to find complex periodic motion regimes, as well as to calculate bifurcation diagrams used for determining the main variations of the motion regime from periodic to chaotic (the period doubling scenario).