Research On Dynamics Of Duffing Oscillator With Bilateral Rigid Constrains

As an important part of nonlinear dynamic systems,vibro-impact systems exist widely in mechanical systems with the clearance.The research on nonlinear dynamics behavior of vibroimpact systems has important scientific significance to promote the development of nonlinear dynamics and practical value to solve practical engineering problems.This paper considers a single-degree-of-freedom Duffing oscillator with bilateral rigid constraints.The dynamics of the system is studied,including symmetry,local bifurcation,grazing bifurcation and global dynamics.Firstly,the symmetry of the Duffing oscillator with bilateral rigid constraints is considered.When the moving portrait transversely collides with the left and right constraint surfaces,the Poincaré mapping of the system is constructed.The symmetry of the Poincarémapping is analyzed,and the Poincaré mapping can be expressed as the second iteration of another virtual mapping.A symmetric fixed point and antisymmetric fixed points are defined by the virtual mapping.The shooting method in the general sense and the shooting method for the symmetric orbits are introduced,and by means of the discontinuous mapping and the shooting method,the periodic solution is obtained,and its stability is analyzed.By numerical simulation,the saddle-node bifurcation,the pitchfork bifurcation and the period doubling bifurcation of the system are analyzed,and the virtual mapping is used to capture a pair of antisymmetric periodic orbits.The pitchfork bifurcation of the system is analyzed by the central manifold theorem.Considering the influence of parameter perturbation on the symmetry of the system,the bifurcation of the asymmetric periodic motion can be described by a codimension two cusp bifurcation.The typical symmetry breaking phenomenon occurs in the pitchfork bifurcation process,and the hysteresis and jump phenomena occur near the cusp bifurcation.Subsequently,the expression of Poincaré-section discontinuous mapping is derived in detail,and the compound Poincaré mapping of unilateral grazing orbit and the compound Poincaré mapping of bilateral grazing orbit are constructed,which are applied to the dynamical analysis of the bilateral rigid constrained Duffing oscillator's unilateral grazing bifurcation and bilateral grazing bifurcation.The grazing periodic orbit and Jacobi matrix of the Poincarémapping are calculated by the shooting method.The bifurcation graphs obtained by compound Poincaré mapping are compared with the bifurcation graphs obtained by direct simulation,and the bifurcation types of the two bifurcations are the same.Dynamic phenomena such as chaos introduced by grazing,periodic addition grazing bifurcation and discontinuous grazing bifurcations in unilateral grazing bifurcations,and the bilateral discontinuous grazing bifurcation are found.The existence of symmetric periodic n-1-1 orbits in bilateral grazing neighborhood is given,and the grazing stability criterion with unilateral constraint is extended to the grazing bifurcation with bilateral constraint,which is verified by numerical results.Finally,the global dynamics of the vibro-impact system is studied by considering the transversal intersection of the periodic orbit and the collision surfaces.The method of calculating the unstable manifolds of general two dimensional mapping is given,and the stable manifolds of the map are calculated along the direction of stable eigenvector of inverse mapping.The Poincaré section is chosen after the impact at the right stop,the vibro-impact system is represented by the Poincaré mapping.The method of calculating invariant manifolds is applied to the single-degree-of-freedom bilateral rigid constrained Duffing oscillator,and the stable and unstable manifolds of the system are obtained by numerical simulation.When the stable manifold and the unstable manifold intersect transversely,there exists a Smale horseshoe invariant set by using the Smale-Birkhoff theorem in the system.Combined with Chapter 2,when the external excitation amplitude increases,the path from the double impact periodic orbit to chaos can be summarized as: a symmetric fixed point? a pair of antisymmetric fixed points? a pair of antisymmetric chaotic attractors? a symmetric chaotic attractor.The symmetry restoring bifurcation occurs in the system.The Lyapunov exponent of the system is calculated numerically,and the Lyapunov exponent graph with parameters corresponds to the global bifurcation graph.