Periodic Solution And Dynamics Research Of The Dry Friction Systems

In this paper, the periodic solutions and the dynamics behavior of a kind of dry friction Filippov systems are analyzed based on differential inclusion theory.The chapter one focuses on the domestic and foreign academic research status and development trend of the dry friction dynamical systems, the set-valued map and the differential inclusion theory,and summarizes the research approach and the stability theory of the periodic solutions of nonlinear dynamical systems.In chapter two, a class of dry friction model is derived after describeing several different types of dry friction force, by the nondimensionalization of system differential equations, the dynamical behavior of the dry friction system are analyzed and the conditions of the stick-slip motion are investigated. Using set-valued mappings the nonsmooth differential equations are transformed into the differential inclusions, and the stability and the existence of periodic solutions of the Fillppov differential inclusions are discussed. Finally the general algorithm of Lyapunov exponent in dry friction system is given.In chapter three, the differential equations of the single degree of freedom system is presented. The existence conditions of the stick-slip motion are studied, and the existence of periodic solutions of the single degree of freedom dry friction were proved. By using Floquet theorem, the stability of the periodic solution is studied with the calculation of jump matrix, the global base solution matrix and the floquet multipliers. Fourth-order runge-kutta method is applied in numerical calculation of dimensionless equations of the system. In case of varying the values of one parameter, simulate and obtaine bifurcation diagram, Lyapunov exponent spectrum, the system phase diagram, poincare map and these simulation results confirm the kinetic analysis above.A class of two degrees of freedom dry friction system model is analysed in chapter four. The existence conditions of the stick-slip motion of the mass block are derived by researching characteristics of system, as well as analysing of the block stick-slip boundary surface. The stability of periodic solution of the system is investigated by means of the calculation the fundamental matrix, jumping matrix and floquet multipliers based on the characteristic roots of global fundamental solution matrix. Different values of parameters are used in numerical simulation of the system to determine the type of motion of the system, then apply the identification method above to calculate the global base solution eigenvalues, the stability of the periodic solution of the model is studied based on the mode of eigenvalues.In chapter five, a class of three degrees of freedom of the brake system is discussed. Numerical simulation is conducted to describe the dynamic properties as condition of stick-slip motion and the system stick-slip interface. The numerical simulation results are utilized to analsys the dynamic motion characteristics under different parameter changes, combined with the phase diagram and poincare map of the dynamic characteristics of the system to make further research. The inclusions in this chapter are: 1) Dynamic characteristics of the system can be impact under the changes of excitation frequency and showed a single-cycle, multi-cycle phenomenon of entanglement, bifurcation and chaos. 2) With the enlargement of amplitude of harmonic excitation force, the system sequentially showed a process that start with monocycle to multicycle end up with chaos.