Research On The Complex Dynamics Of Discontinuous Dynamic Systems With Collisions

Discontinuity is a common phenomenon in the real world.Many models in technological production and engineering control have shown discontinuous charac-teristics,such as the transmission of gears,the heat exchangers,the joints of robots and the connections of machine chassis.The appearance of discontinuity makes the dynamic behaviors of systems more complex,so the study for such discontinuous systems have more theoretical significance and practical application value.There are many reasons for the appearance of discontinuity,and impact is one of the most important ones.The theory of discontinuous dynamical systems is a new method in recent years,which can make the study clearer and more concise.This article mainly has the following contents:1.There are two kinds of impacts:direct impact and oblique impact.Direct impact is the motion that the normal direction of the impact surface is parallel to the direction of motion.And the other impacts are oblique impacts.The previous research are mostly about direct impact,but the oblique impact is more common.The appearance of oblique impact will make the system strongly nonlinear and nonsmooth.So in Chapter 3,the periodic N—n motion of an oblique impact system with single degree of freedom is investigated by the theory of discontinuous dynamical systems.The analytical conditions for the existence and local stability of the periodic N—n motion are obtained.All of these are under the assumption that the mass moved with very small angular amplitude.Once such an assumption was not true,the method above would be invalid.The discrete mapping method is used to solve this problem.The analytical conditions for the existence,the local stability of the periodic N—n motion are given through the discrete mapping method.The numerical simulation shows that the oblique impact will make the dynamic behavior of a single degree of freedom system more complex.2.In engineering,the multi-degree of freedom impact system is more common.Taking a two degrees of freedom impact system as an example,we studied the com-plex dynamical behavior of an oblique impact system by the theory of discontinuous dynamical systems in Chapter 4.The occurrence or disappearance conditions of sticking motion and grazing motion on the separation boundaries are given in de-tails.The conditions here are necessary and sufficient,which generate better results than those obtained with only sufficient conditions.The results appropriately in-terpret the physical phenomenon of this oblique impact system,hence validate our conclusions.As a supplement,we also give the analytic conditions of the existence of periodic motions.Numerical simulations for sticking motion and grazing motion are presented at last.3.In Chapter 5,taking population differential system with state-dependent im-pulse as an example,we study the dynamical behaviors of the impulsive differential system by using the theory of discontinuous dynamical systems.The necessary and sufficient conditions are obtained for the trajectory direction,which are better than the previous work.Furthermore,the existence of period-1 solution is investigated for a special impulsive population differential system in details.Compared with the traditional geometric theory,the trajectory direction of differential system can be studied here only through simple algebraic calculation.And the existence of the pe-riodic motion can be solved by algebraic equations,which provides great convenience for the computer programming and simulation.4.As a supplement,we studied the dynamical behavior of a practical engineer-ing machinery-an impacting shaker system.Based on reasonable assumptions,the machine is simplified to a two degrees of freedom direct impacting system.By us-ing the theory of discontinuous dynamical systems,the analytical conditions for the occurrence and disappearance of the sticking motion and grazing motion are given.The existence of periodic motion of the system is studied,and the corresponding stability analysis is given.The main problem solved here is how to adopt reasonable assumptions to simplify a practical problems to a mathematical model.