Dynamical Analysis Of Oscillators In Several Mechanical Systems

In the field of mechanical engineering and daily life, discontinuous dynamical systems are widespread, in which the discontinuities caused by the friction of the contact surface and the switchable control law in the system are the most common.Due to the discontinuity of the studied system, the theory and model of the dis-continuous dynamical system need to be analyzed in order to better explain the dynamics of discontinuous dynamical systems in mechanical engineering and life behavior.The theoretical study of this system is beneficial to the normal operation of mechanical production. In this paper, two physical models will be studied by using the theory of discontinuous dynamical system. The first system is the oscillator model with the two degrees of freedom on the variable speed conveyor, the discon-tinuity of the system is caused by the presence of friction between the vibrator and the conveyor belt. The second system consists of a conversion model controlled by a transition on a constant speed conveyor. The discontinuity of the system is determined by the friction between the conveyor belt and the oscillator, and the control law of the system. We divide the domain and the boundary according to the discontinuity of the dynamical system and analyze the analytical conditions,construct the mapping structure to carry on the analytic prediction and the numer-ical simulation better. The numerical simulation is given on the basis of selecting the appropriate initial value and system parameters. The whole article is divided into three chapters.In Chapter 1, we introduce the research background and research significance of discontinuous dynamical systems, and give the corresponding flow switchability theories, concepts and several related lemmas.In Chapter 2, the 2 -DOF friction-induced oscillator with two harmonic ex-citations on the variable speed belt. is introduced, and the flow conversion theory of the discontinuous dynamical system is used to analyze the dynamic behavior of the system. Based on the discontinuity caused by frictional forces, the domain-s and boundaries of this system are defined. The analytic conditions of passable motion, onset and disappearance of stick motion, and the analytical conditions of grazing motion are proposed based on the division of the domains and the bound-aries. The basic mapping is constructed to describe the motion of the vibrator,and the analytic prediction of periodic motion is given by the mapping structures,and the appropriate initial values and system parameters are selected for numerical simulation to illustrate the stick and nonstick motion.In Chapter 3, based on the theory of flow switchiblity of discontinuous dy-namical systems, the dynamical behavior of the vibrator on constant velocity belt with conversion control law is studied. The different domains and boundaries of this system are defined according to the discontinuity of friction and the definition of transition control law. Based on the division of domain and boundary, the ana-lytical conditions of traversing motion, starting or disappearing motion of stick or sliding motion and analytical conditions of grazing motion are analyzed theoreti-cally. Basic mappings are introduced to describe the motion of the oscillator. The mapping structure gives the analytical prediction of the periodic motion. Finally,different initial conditions are given under the set system parameters to simulate the passable motion and the grazing motion on different boundaries. Through the velocity and force response of this motion, the analytical conditions of the motion switching can be verified in such a discontinuous system.