Co-Dimension-Two Grazing Bifurcations In Impact Oscillators

The grazing behavior is a special motion state of the systems. Impact oscillators give rise to complex dynamics by grazing incidence. In this thesis, firstly, a class of n-degree-of-freedom impact oscillators are investigated in detail. The stability conditions and co-dimension-two grazing bifurcations of grazing periodic orbits of systems are discussed, then deduced theoretical results are used to research the single-degree-of-freedom vibro-impact systems and the two-degree-of-freedom vibro-impact systems, respectively. Finally, the co-dimension-two grazing bifurcations of systems are analyzed. The main results of the thesis are as follows.1. A class of n-degrees of freedom vibro-impact systems with unilateral rigid constraint are studied. At first, by virtue of implicit function theorem and Taylor expansion, Jocabian matrix of local Poincare map Psmooth is obtained. Then the discontinuity map DM on Poincare section is derived by detailed analysis. And Poincare map P∞is obtained by combination of discontinuity map DM and local Poincare map Pmooth. The sufficient conditions for determining stability of grazing periodic orbits and co-dimension-two grazing bifurcations are obtained by applying Poincare map P∞to analyze the stability of grazing periodic orbits. Finally, the analytic expression of co-dimension-two grazing bifurcation is simplified.2. A class of single-degree-of-freedom vibro-impact systems with unilateral rigid constraint are investigated. After analyzing, the grazing condition for a periodic motion of system is obtained, and then the grazing curve is described by numerical simulation. Applying the theoretical results of Chapter two to a single-degree-of-freedom concrete system, co-dimension-two grazing bifurca-tions are discussed analytically and numerically which show in good agreement. Finally, the dynamics of the co-dimension-two grazing points and points in the vicinity of them are analyzed. The existence of the co-dimension-two grazing points is confirmed by numerical simulation. Although the ways for studying the system are different, the theoretical results obtained in this thesis are the same as in the literature.3. For a class of two-degree-of-freedom vibro-impact systems with unilateral rigid constraint, the equations of motion are uncoupled by using modal matrix and the general solution of system is obtained through the modal superposition method. Then, through calculation, the condition for existence of grazing periodic motion of system is obtained and so is the grazing curve. Consequently, the condition for existence of grazing periodic motion is verified by numerical simulation. The system with two-degree-of-freedom is different from that with single-degree-of-freedom. With one more degree of freedom, the former becomes more complex. Therefore it is more difficult to get the concise analytical criterion for co-dimension-two grazing bifurcations occurring. Finally, applying theoretical results of Chapter two to the two-degree-of-freedom model, the co-dimension-two grazing bifurcations are studied by numerical simulation, and the co-dimension-two grazing bifurcation points are found numerically. The unfolding of the co-dimension-two grazing bifurcation of system is also given.