Studies On Grazing Bifurcations In A Cantilever Bean System

In this dissertation, periodic grazing motions in a cantilever beam system are studied. By method of numerical simulations, two cases of grazing bifurcations are given. And on the basis of Nordmark's discontinuity mapping method, a local-discontinuity-mapping near the grazing trajectory is established. Then the mapping is used to analyze the grazing bifurcations. Also the numerical simulations are used to verify the theoretical results.Firstly, the grazing periodic motions in this cantilever beam system with one constrain are discussed. At the beginning, two cases of grazing bifurcations are obtained: one is from period one motion to period two motion, the other is from period one motion to chaos. And, in our analysis, a group of co-existence attractors is found, i.e. a period one attractor and two chaos attractors.Secondly, a normal form of discontinuity mapping of a piecewise smooth dynamic system which is the generalization of the above cantilever beam system is given. And the derived normal form is used to give out the local discontinuity mapping of the cantilever beam system discussed previously, by this means, the grazing bifurcation of the cantilever beam system is studied. It is shown that the results of numerical simulation about the system agree with that of local discontinuity mapping.