Parameter-State Space Global Dynamical Studies Of Collision Vibration System

Collision,impact,clearance and other non-smooth factors exist widely in the field of nature and engineering.The study and control of non-smooth dynamics are becoming more and more important and challenging.The thesis devotes to the syntheses on the vibro-impact systems in the parameter-state space.The syntheses composes of the sensitivity analysis of the bifurcation parameter,the mechanism that dedicates to qualitative change of the basins of attraction for coexisting attractors,and the schemes for the bifurcation control and chaos control.The methods for discontinuous mapping are employed to study zero-time discontinuous mapping.It also gives the normal section discontinuous mapping of the collision surface of both continuous,and the discontinuous vector field at the grazing points of piecewise-smooth system.The conditions of codimension-2 grazing bifurcation in piecewise smooth system are surveyed.The thesis steps into,with the help of the improved Poincare-type cell mapping method,multi-branch coexistence and annihilation mechanism as well as qualitative change mechanism of the attraction domain in the non-smooth elastic collision system.Whereby,the global dynamics of the parameter space and state space in both rigid collision vibration and the elastic collision systems are presented.One most contribution of the thesis is the proposition of a parameter sensitivity analysis method for non-smooth dynamic system in virtue of multi-parameter stability theory.Hereby the bifurcation parameter sensitivity of rigid collision vibration system and elastic collision system can be performed.Moreover,the prediction and control of periodic multiplicative bifurcation are testing with a class of cubic symmetric discrete chaotic systems.The main contents of the thesis are given as follows:The first part of the thesis introduces the purpose,the significance and the main contents of the research.It also summarizes the development history and the current research status of non-smooth systems,especially that of the non-smooth bifurcations.The primary theoretical foundations for non-smooth dynamical systems are tiled with a circuitous manner.The classification and the analytical methods,including numerical and theoretical methods,of non-smooth differential systems are described in the same section.The discontinuous maps at the grazing points of rigid collision vibration system and piecewise smooth systems,together with the establishment of complex maps of periodic orbits with grazing bifurcations are described.It gives the normal section discontinuous mapping of the collision surface of both continuous,and the discontinuous vector field at the grazing points of piecewise-smooth systems.The conditions of codimension-2 grazing bifurcation in piecewise smooth system are surveyed.Based on the discontinuous mapping theory,the composite zero-time discontinuous mapping(ZTDM)and the normal section discontinuous mapping of the collision surface (NSDM)in the cases of continuous and discontinuous vector fields at the grazing points of piecewise smooth dynamical system are studied respectively.It manifests the validity of using low-order and high-order composite NSDM in studying the grazing bifurcation.The end of this chapter presents the studies on the co-dimension-two grazing bifurcation of a piecewise smooth dynamical system with a discontinuous vector field at grazing points.Meanwhile the conditions for co-dimension-two grazing bifurcation of a system with a discontinuous vector field at a grazing point are shown.The second part devotes to the analytical expression of the solution for the impact vibration system with clearance and pre-loaded spring.By using the improved Poincare type cell-to-cell mapping method,the attractors and their attraction domains are obtained on the time Poincare cross sections and the position Poincare cross sections.The coexisting attractors generated from saddle node bifurcations,period-doubling bifurcations,boundary collision bifurcations are illustrated.Moreover,it illustrates the attractor annihilation caused by global bifurcation such as boundary crisis,basin boundary metamorphoses and interior crisis.Different parameter spaces of non-smooth hybrid collision systems and elastic collision systems are distinguished.The distributions of various dynamics in these systems are demonstrated.By employing the multi-parameter stability theory,the third part presents the method of partial derivatives of system parameters as the eigenvalues of Jacobian matrices being of simple,semi-simple and non-deficient respectively.A method for calculating the sensitivity of the bifurcation and state parameters of non-smooth dynamical systems is presented.The Poincare mapping of the system is derived to establish the Floquet matrix of rigid collision vibration system and elastic collision systems.The partial derivatives of the Floquet matrix of each system are obtained for each of parameter vectors,and the parameters which are most sensitive to different bifurcations are identified by disturbing the eigenvalues of Floquet matrix.The parameters which have obvious influence on the dynamic characteristics of the system are identified effectively from the whole bifurcation parameters and the state parameter groups.The distribution of various kinds of dynamic motion in rigid collision vibration system and piecewise smooth collision vibration system is illustrated on the parameter spaces of these two systems.In the low-frequency region ?<1,the q=i/1(i=2,3,…)subharmonic periodic motion induced by grazing bifurcation exists in the parameter domains.The limit value of the quotient of the natural derivative of grazing bifurcation point difference between two adjacent periods in the subharmonic periodic motion is calculated as 1.In the parametric plane(?,?),there are abundant dynamic phenomena such as "periodic peak","ring" island,"shrimp-shaped" island and "chaotic eye" in rigid collision vibration and piecework smooth collision vibration systems.The distribution regions of coexisting attractors induced by different reasons in the parameter-state space are portrayed in virtue of bifurcation parameter sensitive singularity.The attractor coexistence regions induced by saddle bifurcation usually appears inside the periodic motion.The attractor coexistence regions(CA-GB)formed by saddle bifurcation,which is induced by period-doubling bifurcation,usually appear around the period-doubling bifurcation line.The final part of the thesis focuses on the bifurcation prediction and control method based on Lyapunov exponent and radial basis function neural network.The bifurcation control schemes are proposed for a class of piecewise smooth collision vibration systems with clearance and pre-loaded spring.An efficient method on predicting the appearance of bifurcations in terms of Lyapunov exponents is proposed.The controller for suppressing period-doubling bifurcations is designed on the basis of RBF neural network.The adaptive hybrid gravitational search algorithm(AHGSA)combined with RBF neural network is to optimize the parameters in bifurcation controller.To control the period-doubling bifurcations,the fitness function is formulated on the maximum Lyapunov exponent of the corresponding bifurcation points.