Study On Modeling And Periodic Solution Of Non-smooth Double Pendulum

Many non-smooth characteristics,such as collision,impact,clearance,dry friction and so on,exist widely in engineering practice.The strong nonlinearity caused by the non-smooth phenomena makes the dynamic response of nonlinear dynamical system very complex.At present,the research of the low-dimensional non-smooth dynamical system has made some progress.However,it is difficult to generalize the theory and describe the geometry of the high-dimensional non-smooth system.In fact,many problems in engineering practice need to be described by high-dimensional non-smooth dynamical system.Therefore,it is of great theoretical significance and practical application value to construct the model of non-smooth system,develop the theory of analytical method of high-dimensional non-smooth system,and study the composition characteristics of highdimensional non-smooth track.This paper focuses on the construction of collision model,the analytical method of the non-smooth periodic solution,numerical simulation analysis and collision experimental design.The content of this paper is organized as follows.In introduction,the research background,research status,research content and innovation of the subject are discussed.In chapter 1,the preparatory knowledge are introduced,which gives the basic concept of dynamic system,non-smooth dynamical system,the concept of phase diagram,Chaos,Poincaré section,bifurcation diagram,the classification of non-smooth system,and the introduction of research methods.In chapter 2,the theoretical modeling of non-smooth double pendulum is developed.Based on the smooth double pendulum model of hinge connection,the non-smooth double pendulum model of coupling excitation symmetry constraint is constructed by installing simple harmonic excitation device and symmetrical rigid oblique plate on the base.Lagrangian Function Method is used to obtain the control equation as the differential equation of parametric excitation pulse.Because the equation of the system contains time-varying parameters,the system motion is affected by external excitation and internal structural parameters,and the dynamic behavior is more complex,and the dynamic behavior is more complex.However,the coupling of the non-smoothness caused by collision and the high-dimensional nonlinear makes the theoretical analysis of the system very difficult.Therefore,based on the improved numerical simulation program,the dynamic behavior of the three special cases of the coupled excitation rigid constrained non-smooth double pendulum model is analyzed.The numerical simulation results show that the system presents gradually complex motion due to the change of constraint conditions and external excitation forms.In particular,each system has different forms of collision periodic solutions,which provides ideas for the subsequent theoretical study of periodic solutions.In chapter 3,the existence conditions and analytical expressions of the periodic solutions for the two-sided non smooth pendulum model with horizontal excitation are discussed.The existence conditions and analytical expressions of periodic solutions for this kind of non-smooth high-dimensional system are obtained by using the modal analysis method and matrix theory.The coefficient in the expression is related to the collision recovery coefficient,which shows that the collision will affect the form of the periodic solution.In chapter 4,for the non-smooth double pendulum model of unilateral collision under horizontal excitation,the specific boundary conditions are given,and the periodic solutions of the system are classified.By using the modal analysis method and matrix theory,the existence conditions and the analytical expressions of the periodic solutions of the system with the small angle motion are detected.The coefficient in the expression is related to the collision recovery coefficient,which reflects the impact on the periodic solution.Using MATLAB software,the calculation program of smooth system is improved.Furthermore,the phase diagram,time history diagram and Poincaré section of the system are obtained,and the periodic solution of the system under given parameters is found.In chapter 5,the concept and scope of application of high dimensional subharmonic Melnikov functions are generalized.An appropriate reversible transformation is constructed to decouple the coupled system for the unilateral impact non-smooth double pendulum model under horizontal excitation,and the general coupled system is decoupled into Hamiltonian system.The decoupling transformation changes the collision law determined by non-smooth factors.The energy coordinate transformation is introduced to transformer the decoupled system into an integrable and smooth equation.On this basis,the subharmonic Melnikov function of the periodic solution of the non-smooth system is given.The non-smooth subharmonic Melnikov function is applied to study the existence of periodic solutions of non-smooth systems.The parameter region for the existence of subharmonic periodic solutions is given,and the influence of parameters on the existence of subharmonic periodic solutions is discussed.In chapter 6,the experimental research is carried out.Aiming at the non-smooth double pendulum model of unilateral impact under vertical excitation established,the experiment is designed and the unilateral single point impact experiment is performed.The experimental results and the numerical simulation results are verified by drawing the phase diagram,time history diagram and Poincaré section diagram of the system.The conclusion shows that the numerical results are in good agreement with the experimental results.On the one hand,the correctness of the previous numerical simulation program is confirmed,on the other hand,the rationality of the experimental design is verified.