Periodic Solution And Stick-Slip Bifurcation For A Class Of Systems With Dry Friction

The stick-slip behavior in friction oscillators is very complicated due to the non-smooth of the dry friction, which is the basic form of motion of dynamical systems with friction. In this thesis, the stick-slip periodic solution in a single-degree-of-freedom oscillator with dry friction is investigated in detail, and for a two-degree-of-freedom system with dry friction, the necessary condition for the stick-slip boundary is obtained. The main results of the research involve:1. In a single-degree-of-freedom dynamical system with dry friction, under the assumption of kinetic friction being the Coulomb friction, the existence condition of the stick-slip periodic solution is analytically derived which gives out a new result in a class of friction systems. Moreover, the time and states of motion on the boundary of the stick and slip motions are semi-analytically obtained in a single stick-slip period. Finally, the theoretical results are validated by numerical simulation. At the same time, the relationship between the ratio T/2πwith period T as numerator and the parameterξnearby the bifurcation curve is obtained. Since the system is symmetrical with respect to the velocity, the case for v0>0 can be discussed in the same way as that of the case v0<0 which has been investigated in this paper, and it is clear that the similar conclusions hold.2. In a two-degree-of-freedom dynamical system with dry friction, the non-smooth bifurcation and chaos phenomena are investigated. By the theoretical analysis, the necessary condition which must be satisfied for the stick-slip orbits in the boundary of stick and slip motions is obtained, and in the switching manifold, we find out the possible area where the slip motion will happen. Using the numerical simulation, it is found that the system possesses steady periodic behaviors when the parameterμm takes rather large values. However, a large number of chaos phenomena appear when the parameterμm is descended to a certain extent. In the parameter region of chaos, there are also lots of periodic phenomena which include complicated periodic motions besides the simple ones.