Research About The Solutions Of Two Piecewise Smooth System Models

In this paper, we mainly focus on one particular form of nonlinear system, piecewise smooth system, explore the switch ability of the trajectory on the boundary, especially when the boundary is time-varying, its Floquet coefficient has great difference with the fixed boundary. Consider the various combination forms of trajectory, one new equilibrium type is obtained. One important research aspect of dynamic system is its closed orbit and bifurcation, for the abrupt change in the crossing process, the bifurcation of piecewise smooth system has its own characteristics, this paper investigates its closed orbit and bifurcation in-depth, including the equilibrium bifurcation, limit cycle, and the formation of homoclinic and heteroclinic orbits, some conclusions are achieved.After exploring some characteristics of piecewise smooth system, we investigate the transmission and brake systems in depth and propose two important models, 6-DOF friction spring-oscillator model and 2-DOF meshing oscillator model. Researchers mainly adopt numericalcomputation or FE method to simulate or just ignore the non-smooth components, so these results are different from the actual operation in some ways. To the auto transmission, for the change of contact ratio and the existing of gear backlash, 10 sub-domains are obtained, and the boundaries are consisted of one time-varying boundary and two fixed boundaries; in the brake system, for the fluctuation of hydraulic pressure, it has multiple time-varying boundaries,consider the transition of static and kinetic frictions, the system includes 4 sub-domains, the paper explores its bifurcation and crossing situation, employs a new research method and has some reference for other high-dimensional coupled systems.