Flow Conversion And Periodicity Of Two Types Of Discontinuous Dynamic Systems With Asymmetric Damping Properties And Flow Barriers

Discontinuous dynamical systems exist everywhere in the real world.Discontinuous features often exist in a large number of technology production and engineering control models.Compared with the continuous systems,discontinuous dynamical systems have more complex dynamical behaviors.There are many factors leading to the discontinuity.such as friction,impact and the changes of the vector fields.Moreover,one of the most important factors that leads to the discontinuity in system is the friction.Friction is a complex nonlinear physical phenomenon.It exists widely in t,he mechanical engineering.and is the basic form of mutual transformation of all kinds of energy.In the process of scientific and technological production,sometimes people utilize the friction to work,and sometimes in order to improve the accuracy of the mechanical systems,it is necessary to eliminate the adverse effects of friction to the greatest extent.By abstracting and modelling the friction appearing in practical problems,and studying the dynamical behaviors of the model,it is helpful to the better understanding of the dynamical behaviors of this kind of discontinuous dynamical system.Meanwhile,it is of great significance in saving resources,optimizing design and improving the performance of machines.There have been many studies on the discontinuities caused by friction,but the dynamical behaviors of the systems are mainly studied by special analytical,numerical and experimental methods.The discussions on the local singularity of the flow and the prediction of periodic motions are still not adequate.In recent years,with the gradual formation and perfection of the theory of discontinuous dynamical systems,the dynamical behaviors of many models have been well analyzed and predicted,which provides a theoretical reference for solving practical problems.In this paper,two kinds of discontinuous dynamical systems-the system with asymmetric damping properties and the system with flow barriers on the boundary are investigated using the flow switchability theory and the flow barrier theory of discontinuous dynamical systems,and the results of flow switchability and periodic motions are given.The main contents of this paper are as follows1.Represented by the car suspension system,the complex dynamical behaviors of a kind of discontinuous system with piecewise linear damping characteristics are investigated using the flow switchability theory and the mapping dynamics of the discontinuous dynamical systems.Different from the most previous studies,the examined system in this paper has the asymmetric damping properties.The damping coefficients of the system depend on the velocity direction of the oscillator.Based on the movements of the oscillator,the phase space is divided into several sub-domains and their boundaries.By introducing the state vectors and vector fields,the vector forms of the motion equations for the oscillator in each sub-domain are given.The dot product functions of the position vector difference and the normal vector-G functions on the discontinuous boundaries are defined.By using the sign of G functions and its change regulations,and then by the sign of the forces acting on the oscillator,the sufficient and necessary conditions for the flow switchability,the occurring and vanishing of the stick motion and the grazing flows are obtained and proved analytically.According to the theoretical analysis,the sign of the force production can be used to determine the passibility and stick motions on the separation boundary.From the ranges of the phase angle in which the grazing motions occur,it can be seen that the ranges are different for the grazing motions in each subdomain,which provides an effective criterion to determine the grazing motions Switching planes and three basic mappings are defined to describe the different motions in each domain,including two non-stick mappings and a stick mapping,and the governing equations of these basic mappings are also given.Using appropriate mapping structures,the periodic motions of the oscillator are analytically predicted.Further the theoretical analysis on stability and bifurcations of periodic motions are conducted by Jacobian matrices of the mapping structures and their eigenvalues.Finally,by selecting appropriate parameters,numerical simulations of periodic and grazing motions for the quarter car suspension system with dual-rate dampers are given to verify the analytical conditions,which provides the information for the selecting of parameters in the car suspension system2.The complex dynamical behaviors of a nonlinear,periodically forced friction oscillator with flow barriers on a belt moving with a constant velocity is studied using the flow switchability theory and the flow barrier theory of the discontinuous dynamical systems.The singularities of the flow near the separation boundary will be changed when the flow barriers exist on the separation boundary.In previous studies on discontinuous dynamical systems with flow barriers,the motion equation for the oscillator is linear However,in this paper,the spring and viscous damper included in the examined friction oscillator have linear and nonlinear coefficients,the motion equation is nonlinear.To avoid computational errors,the nonlinear friction between the oscillator and the belt is approximated by a piecewise linear,kinetic friction model with the static friction force And then the motion equation for the oscillator is given by Newton's second law.The phase space is divided into several sub-domains and their boundaries according to the velocity of the oscillator.The vector forms of the motion equations for the oscillator in each sub-domain are given by introducing the state vectors and vector fields.At the initial moment when the oscillator moves relative to the belt,the maximum static friction force is different from the kinetic friction force,so the boundary flow barriers exist in this dynamical systems.The lower and upper limits of the boundary flow barriers are given according to the range of the static friction.G functions for the flow barrier are defined through the dot product of the vector fields and the normal vector.By the sign of G functions,and then by the sign of the forces acting on the oscillator,the analytical conditions for the flow switchability and the grazing motions to the boundary are obtained using the flow switchability theory and the flow barrier theory of the discontinuous dynamical systems.The necessary and sufficient conditions for the onset and vanishing of the stick motion on the discontinuous boundary with flow barriers are also developed.The theoretical results show that the flow barriers have strong influence on the local singularity of the flow near the separation boundary:from the obtained force criteria,the critical value of the boundary flow barriers is zero when the stick motion disappears,and then becomes non-zero.That is to say,the non-friction forces acting on the oscillator need to overcome the maximum static friction to produce a new relative motion between the oscillator and the belt.The above results are fundamentally different from the corresponding ones of the friction oscillator without flow barriers.On this basis,according to the switchability of the flow to the boundary,switching sets on the boundaries and the basic mappings are defined,including the global mappings and the local mappings.The periodic motions of the oscillator are predicted analytically through the general mapping structures obtained by basic maps,and the theoretical analysis for stability and bifurcations of periodic flows are done by Jacobian matrices of the mapping structures and their eigenvalues.In order to further explain the dynamical behaviors of discontinuous dynamical systems with flow barriers,on the basis of previous theoretical analysis results about flow switchability and periodic motions of the discontinuous dynamical systems with flow barriers on the boundary,the complex dynamical behaviors of a friction oscillator with flow barriers on a belt moving with a constant velocity is studied,and the oscillator is subjected to two periodic excitations with different frequencies.By selecting appropriate parameters,the numerical simulations of the periodic motions with and without stick motions and grazing motions in such a piecewise friction model are given to verify the analytical conditions.From the theoretical analysis and the numerical simulation,it can be seen that the grazing phenomenon is identical to the friction oscillator without flow barriers,which shows that the grazing phenomenon is independent of the flow barriers although the flow barriers exist on the separation boundary because of the static friction between the oscillator and the belt.