Mechanism Analysis And Simulation Study Of Chaotic Rotation Of The Cylinder Waterwheel

Chaos,as a new theory widely used in modern times,has been widely concerned by scholars for decades,among which the most widespread influence is the research of chaos based on the Lorenz system.Generally speaking,people study chaos from two perspectives.One is to study the behavior of the solution of the nonlinear system and carry out numerical simulation with the computer,the other is to carry out physical experiment and to get the appropriate mathematical model,such as chaotic waterwheel experiment.There are few Chinese literatures on physical experiments,and mathematicians rarely connect physical phenomena with mathematical mechanisms.In this paper,the mathematical model of the Cylinder waterwheel is established,the chaotic synchronization analysis and the high frequency term analysis are carried out,the internal dynamic mechanism and energy conversion of the model are discussed.Through theoretical analysis and the numerical simulation,a reasonable explanation and analysis are given for the actual rotation of the Cylinder waterwheel.Firstly,Lorenz waterwheel,Malkus waterwheel and the Cylinder waterwheel are introduced.The research history and development status of chaos are summarized,and the innovation points and structure of this paper are expounded.The basic knowledge of bifurcation and chaos theory,mechanics,energy science,ordinary differential equations and mathematical analysis applied in this paper are summarized.Secondly,from the mechanical point of view,according to the torque balance theorem and the conservation of mass law,the mathematical model of the cylinder wheel is deduced,and the theoretical analysis is carried out.The symmetry of system,invariance,dissipation and the existence of attractor are studied.The equilibrium points and its local stability are discussed.When and which rotation phenomenon occurs in the model system is fully explained.global stability of the system is analyzed,and a large number of numerical simulations are performed.The internally abundant chaotic behavior of the system is showed,and the correctness of the theoretical analysis is verified.Based on theoretical analysis and numerical simulation results,the chaotic rotation phenomenon of water wheel is explained.Then,the method of chaos synchronization is used to verify that the mathematical model derived in this paper can be synchronized with the experimental model of the Cylinder waterwheel when the coupling parameters are appropriate,which shows that the mathematical model is correct to describe the rotation phenomenon of the Cylinder waterwheel.The important approximate process in the derivation of mathematical model is explained rationally by the analysis of high frequency terms,and the omission in the approximate process has no significant effect on the generation and development of the chaotic behavior of the system,which shows that the mathematical model obtained by this method is reasonable.Finally,the mechanism of chaos and its energy evolution of mathematical model system are discussed by means of dynamic mechanism analysis and energy conversion method.the system is changed as the Kolmogorov system,the subsistent torque is respectively combined,to discuss a variety of torque mode dynamic state of the system,and to explore the main reason for the system to produce chaos.The existing torque models and torque sizes of the actual rotating Cylinder waterwheel are discussed,to analyze the role of various torque to the Cylinder waterwheel actual rotation phenomenon,and to discuss the intrinsic mechanical mechanics of the chaotic rotating of the Cylinder waterwheel.The energy variation diagram,attractor diagram,state variable trajectory and other simulation diagrams are drawn for various torque modes,and the correctness of theoretical analysis is intuitively verified by the images.