Analysis Of Discontinuous Dynamical Behaviors In A Vibro-impact System

Discontinuous dynamical systems extensively exist in mechanics and engineer-ing.Because of the clearances and relative motions between the moving parts,there are numerous impact phenomena in mechanical systems,which results in complicated discontinuous dynamics.As one kind of discontinuous dynamical sys-tem,vibro-impact system has received great attentions of researchers because of its practical applications in mechanical engineering.Therefore,the study of vibro-impact system has important theoretical and practical value.Recently,one of the new developments in discontinuous dynamical systems,i.e.,the theory of flow switchability for discontinuous dynamical systems,has been found.Such theory regards impact phenomena are the discontinuous dynamical behaviors that occur in time-varying domains and time-varying boundaries,and investigates the com-plex motion switching mechanism in mechanical systems by using the G-functions,which illustrates the dynamical behaviors of discontinuous dynamical systems from a new point of view.In this paper,the discontinuous dynamical behaviors in a two-degree-of-freedom vibro-impact system with multiple constraints are investigated by using the flow switchability theory for discontinuous dynamical systems.The analytical conditions for the onset and vanishing of stick motions and the grazing flows appearing are presented mathematically,and the the periodic motions with different mapping structures are analytically predicted as well.The main contents of this paper are as follows:In Chapter 1,the background and significance of research on the vibro-impact system in mechanical engineering are stated.The concepts of G-functions and decision theorems of flow switchability to a discontinuous boundary in the theory of flow switchability for discontinuous dynamical systems are presented.In Chapter 2,the physical model,i.e.,a two-degree-of-freedom vibro-impact system with multiple constraints,is introduced firstly.Due to the interaction be-tween the two masses in this discontinuous system,the following four cases are taken into account:both the masses are free-flight;one of the two masses is sticking;and both the masses are sticking.According to the discontinuity caused by impact,different domains and boundaries are defined in absolute and relative coordinates.Then based on the theory of flow switchability for discontinuous dynamical systems,the analytical conditions of motion switching,such as the onset and vanishing of stick motions,grazing motions appearing,are obtained through the G-functions.The physical explanations are also given.By using the mapping dynamics theory,the periodic motions with different mapping structures in the vibro-impact sys-tem are analytically predicted on the basis of the defined switching sets and four-dimensional mappings.Lastly,the numerical simulations,such as displacement-time history,velocity-time history,phase trajectory and G-function-time history,are carried out through MATLAB to illustrate the analytical conditions of motion switchability and the periodic motions for a better understanding of the complex dynamical behaviors in the vibro-impact system.Chapter 3 concludes this paper and provides an outlook of the theories and simulations of the two-degree-of-freedom vibro-impact system with multiple con-straints that can be studied in future.