Analysis Of Discontinuous Dynamical Behaviors For A Class Of 2-DOF Friction-influenced Oscillators With Multiple Elastic Constraints

Discontinuous dynamical systems widely exist in mechanical engineering and industry.Because there is often a gap between two or more objects in a discontin-uous dynamic system,impact phenomena often occur in such system,in addition,friction phenomena also also occur in mechanical systems when two masses in contac-t and moving with each other.Such discontinuous dynamical systems may have complicated dynamic behaviors and also have practical applications in enginceering,so many scholars are interested in it and have done a lot of research on such sys-tems through geometry,numerical analysis and experimental methods,etc.,and such investigations have great significance in theoretical guidance and practical application.In recent,years,the new development on the study of discontinuous dynamical systems is obtained,for instance,the G-function is used to study the motion switching on the discontinuous boundary as an important tool in the flow switchability theory of discontinuous dynamical systems.Based on such new the-ory,the discontinuous dynamical behaviors in a 2-DOF(two-degree-of-freedom)friction-influenced oscillator with multiple elastic constraints are investigated in this paper.In addition,the analytical conditions of motion switching in such system and the research results of periodic motion are given.The main contents of the paper are as follows:In Chapter 1,the background and present situation of research on the friction-induced system with impact are introduced.The basic concept of the flow transition theory of discontinuous dynamical system and the decision theorem of flow trans-formation at discontinuous boundary are bricfly introduced.In Chapter 2,firstly,the physical model studied in this paper,i.e.,a,2-DOF friction-influenced oscillator with multiple elastic constraints,is introduced.The mechanical model is composed of two masses driven by periodic forces,which are on the two conveyor belts with constant speeds separately and are connected by two linear springs and two dampers.Because of the mutual effect between the two masses and the elastic constraints on both sides of the first mass,the following five cases should be considered:both the masses have nonsliding motion;one of the two masses has sliding motion;both the masses have sliding motion simultaneously;and the first mass with elastic constraints has stick motion.Secondly,different domains and boundaries for such system are defined based on the friction discontinuity and multiple elastic constraints,and the state vectors and field vectors of such system are introduced.Based on the above domains and boundaries,the analytical condi-tions for passable motion,sliding motion,grazing motion and stick motion in the friction-induced system are obtained by means of G-functions on the corresponding boundaries.In addition,the mapping structures are presented and the periodic mo-tions of such system are described by the basic mappings.Finally,the time histories of displacement,velocity,G-functions and the corresponding trajectories in phase plane for the two masses are illustrated numerically for a better understanding of the complex dynamical behaviors of such system.In Chapter 3,the research contents of this paper are summarized,and the problems and theories that can be further studied in the future are pointed out.