Research On Grazing Bifurcation And Singularity In Vibro-impact Systems

Vibration System as a common dynamic system is closely related to people’s daily life, which plays a very important role in the field of engineering and industry. Especially in recent years, it was focused for researching and attracted extensive attention of many scholars, engineerings and technical personnels.As a common nonlinear dynamical system, it has nonlinear and singularity problems which caused by factors such as the collision and impact, making the dynamic response of the system becomes quite complex. Because of its non-smooth vector field, which making some traditional research methods in smooth system are no longer applicable to such systems. Thus it needs to explore some new theoretical approaches to study its periodic solutions and bifurcation behavior. Therefore, it has some difficulties in theory study.Firstly, the paper gives a study for a class of two degrees of freedom rigid collision vibration systems with clearance, and analysis the dynamic behavior which near the grazing orbital of the system, which combining the characteristics of the movement obtained the analytical conditions of the vibration system that has grazing orbital by analytical method, and deduced the zero time discontinuous mapping(ZDM) and the Poincare section discontinuous mapping with mapping and series expansion method, and gave further compound to the smooth map corresponding to no collision periodic trajectories with discontinuous mapping pushed, then get segment mapping expressions of the grazing bifurcation, and gives a more detailed analysis for its singularity of Jacobian matrix, which found singularity items only exist in the trace of system Jacobian matrix, and singularity items appear when satisfy the corresponding conditions only. Finally, numerical simulation was gived for the system by the fourth-order Runge-Kutta algorithm. It was combined bifurcation diagram and phase diagram to find that parameter bifurcation will appear discontinuous from a constant cycle track to the transition point in next, and system corresponding grazing motion collision will occur in these discontinuous points.Then on the basis of the first system model, adding the elastic to the constraint, namely two degrees of freedom elastic collision vibration system, which is studied the second basic model in this paper, and used the same methods to analyze its dynamic behavior around the grazing orbital, deduced mathematical expressions of zero-time discontinuous mapping in system in detail, which given its segment mapping expressions for grazing bifurcation combined discontinuous mapping deduced. Through the system Jacobian matrix singularity analysis, it’s found that have exactly the same conclusion with rigidity vibration system, that is to say, the singularity of the system will not have too much impact when the change of the constraints for the system. Finally, though the theoretical analysis and numerical simulation, the Poincare mapping singularity and non-differentiable were found that the grazing collision point made global stable manifold theorem failure which applied to smooth dynamical systems, and caused strange attractor geometry changed and generated chaotic phenomenon, further illustrated the complex dynamic behavior of grazing collision which is closely related to collision vibration.