Determination Of The Critical Parameters In A Vibro-impact System And Amplitude Analysis Of Invariant Circles

The vibro-impact systems are extensively applied in mechanical engineering field. Impacts leads to strong nonlinearity such that the dynamical behaviors of the vibro-impact systems become very complex. And vibro-impact systems are usually high dimensional and multi-parameter systems. In this thesis, the analyses of critical parameters and the invariant circles are adressed.Based on the Schur-Cohn criterion, a method of determining the stable area and its boundary of criticality in the parameter space is proposed.With numerically estimating the varying range of the parameters in advance, this method is applied to an inertial shaker model. The stable area and its boundary of the vibro-impact system are determined. The numerical simulation verified the method. Further, the disturbances of the critical bifurcation point give rise to different kinds of bifurcations in the simulations. And the Grazing-sticking motion is found. Based on the Poincare map, a three-order Taloy series which can be viewed as an iterated map is calculated at the fixed point. The dynamical behavior of the iterated map is studied and compared with that of the simulation systems. By applying the Theory of Centre Manifold and Normal Forms, the existence of invariant circle in vibro-impact systems is proved and its amplitude is estimated theoretically.