Nonlinear Dynamics Of A Class Of Bistable Composite Plate Structures

Under the background of physical and mechanical experiments,this paper mainly explores the nonlinear dynamics of a class of bistable composite plate structures,and studies the bistable Duffing system with two slow variables,which can transform between two steady States with the change of excitation parameters.We combine the bifurcation and chaos theory,fourth-order Runge-Kutta method,Melnikov method and other theoretical knowledge to study the dynamic response behavior of bistable dynamic system with damping effect,and do numerical simulation research by making time series diagram,plane phase diagram,Poincare cross-section diagram,bifurcation diagram and maximum Lyapunov exponent diagram,and analyze the dynamic mechanism under different parameters theoretically.In the first two chapters of this paper,the fixed points of undisturbed and disturbed systems are studied,and the existence conditions and generation mechanism of chaos in the system are discussed.In chapter three,a class of fast and slow Duffing system with periodic parameter excitation is studied,mainly considering the fixed point chaotic attractor and singular non-chaotic attractor of the system.In the last two chapters,Euler discretized bistable Duffing equation,studied the phenomenon of period-doubling bifurcation and attractor steady-state coexistence,and analyzed the nonlinear behavior of the system under dual-frequency excitation based on Melnikov method.The innovation of this paper is to discuss the bistable characteristics and dynamic behavior of a class of fast and slow Duffing system with parametric excitation in theory by combining the existing experimental data and physical models,and to give the functional equation corresponding to chaotic threshold based on Melnikov method,so as to combine theory with experiment.The results are as follows:(1)Hamiltonian function,homoclinic trajectory equation and bistable potential well are calculated.Chaos occurs when the homoclinic orbit branch breaks.Based on Melnikov method,the lemma of chaos in the sense of Smale horseshoe is given.(2)Fixed point chaos will show the bistable phenomenon of single branch existence or double branch combination.With the change of excitation frequency,chaotic attractor will become another kind of singular non-chaotic attractor.(3)By analyzing the perioddoubling bifurcation with Euler discrete Duffing system,several kinds of transition modes are found by numerical simulation,including the bistable phenomenon of coexistence of periodic attractor and chaotic attractor.(4)When two different excitation frequencies are in integer ratio,the system changes from non-chaotic to chaotic and appears bistable.Once the excitation parameters are changed,the system will lose stability.Therefore,as a new composite material with two steady-state structures,it can provide important theoretical guidance for engineering design and nonlinear control.