Nonlinear Dynamics Of A Class Of Geometrical Nonlinear System And Its Application

As the modern new type mechanical equipment toward large-scale, rapid development direction of miniaturization, high speed, large displacement, large deformation caused by increasingly prominent in the geometrical nonlinear vibration in engineering problems. Base on the SD oscillator of bistable state, We propose a geometrical nonlinear model with tristable state and geometric nonlinear vibration isolation system with quadristable state and high-order quazi-zero-stiffness(HOQZS) characteristics which accurately depict various kinds of typical mechanical model with geometric nonlinear characteristics in engineering. The analysis of equilibrium bifurcation, chaos prediction, two parameters co-dimension three bifurcation, Hopf bifurcation, primary resonance response, force transmissibility and power flow provide a theoretical basis for design of HOQZS isolator. The main contants are as follows:First of all, we propose a novel nonlinear geometrical system with tristable states and strong irrational nonlinearities having smooth and discontinuous(SD) characteristics depending on the values of a smoothness parameter. The equilibrium stability and the complex bifurcations of the unperturbed system are investigated. The bifurcation sets of the equilibria in parameter space are constructed to demonstrate transitions in the multiple well dynamics. The Melnikov method is employed to obtain the analytical criteria of chaotic thresholds for the singular closed orbits of homoclinic, homo-heteroclinic, cuspidal heteroclinic and tangent homoclinic orbits of the perturbed system for both smooth and discontinuous regimes. The numerical simulations of the periodic solution and chaos have proved the effectiveness of the theoretical results.Secondly, we investigate the global bifurcations and multiple bucklings of a geometrical nonlinear oscillator with tristable state and a pair of strong irrational nonlinear restoring forces. The equilibrium stabilities of multiple snap-through buckling system under static loading are analysed. It is found that the exhibits complex bifurcations of two parameters codimension-three at the catastrophe point. The universal unfolding for the codimension-three bifurcation is also found to be equivalent to a nonlinear viscous damped system. The bifurcation diagrams and the corresponding codimension-three behaviours are obtained by employing subharmonic Melnikov functions for the existing singular closed orbits of homoclinic, tangent homoclinic, homo-heteroclinic and cuspidal heteroclinic, respectively. The Hopf bifurcation of damping system was investigated near the catastrophe point in parameter space, and the Hopf bifurcation surfaces diagram are obtained and presented for the existed single well, double well and triple well potential, respectively. Finally, the coexistence of limit cycle in single, double and triple limit cycle phase diagram are given using the fourth order Runge-Kutta method. The numerical simulation verified the efficiency of the theoretical results.Once more, we propose three geometical nonlinear isolators with bistable of LQZS,, tristable of cubic(CQZS) and quadristable of quintic QZS(QQZS) state respectively which having the smooth and discontinuous characteristics depending on the value of parameters. It is shown that the quadristable nonlinear oscillator of exhibits the complex equilibrium bifurcations of single, double, triple and quadruple well properties, and the singular closed orbits of homoclinic, heteroclinic and homo-heteroclinic types as well for both smooth and discontinuous cases.Finally, harmonic balance method is employed to investigate the QZS mechanism of the isolator with LQZS, CQZS and QQZS characteristic when the mass is subjected to an external harmonic excitation. The analytical results of force transmissivity of QQZS system are obtained by varying the system parameters. It is found that the QQZS characteristic can greatly enlarge the isolation frequency band and improve the low frequency isolation performance. Furthermore, the power flow characteristics and the maximum kinetic energy of this isolation system are examined for a better assessment of the isolation performance. The results provide a new insight into the understanding of nonlinear isolation mechanism and also demonstrate significant benefits of the HOQZS property to vibration attenuation in engineering. The chaotic behaviours are also investigated numerically for the perturbed system under the perturbation of both viscous-damping and external excitation.