Grazing bifurcation analysis of a relative rotation system with backlash non-smooth characteristic

Grazing bifurcation of a relative rotation system with backlash non-smooth characteristic is studied along with the change of the external excitation in this paper. Considering the oil film, backlash, time-varying stiffness and time-varying error, the dynamical equation of a relative rotation system with a backlash non-smooth characteristic is deduced by applying the elastic hydrodynamic lubrication（EHL） and the Grubin theories. In the process of relative rotation, the occurrence of backlash will lead to the change of dynamic behaviors of the system, and the system will transform from the meshing state to the impact state. Thus, the zero-time discontinuous mapping（ZDM） and the Poincare mapping are deduced to analyze the local dynamic characteristics of the system before as well as after the moment that the backlash appears（i.e.,the grazing state）. Meanwhile, the grazing bifurcation mechanism is analyzed theoretically by applying the impact and Floquet theories. Numerical simulations are also given, which confirm the analytical results.

Bifurcation Analysis of a Nonlinear Vibro-Impact System with an Uncertain Parameter via OPA Method

In this paper,the bifurcation properties of the vibro-impact systems with an uncertain parameter under the impulse and harmonic excitations are investigated.Firstly,by means of the orthogonal polynomial approximation(OPA)method,the nonlinear damping and stiffness are expanded into the linear combination of the state variable.The condition for the appearance of the vibro-impact phenomenon is to be transformed based on the calculation of themean value.Afterwards,the stochastic vibro-impact systemcan be turned into an equivalent high-dimensional deterministic non-smooth system.Two different Poincarésections are chosen to analyze the bifurcation properties and the impact numbers are identified for the periodic response.Consequently,the numerical results verify the effectiveness of the approximation method for analyzing the considered nonlinear system.Furthermore,the bifurcation properties of the system with an uncertain parameter are explored through the high-dimensional deterministic system.It can be found that the excitation frequency can induce period-doubling bifurcation and grazing bifurcation.Increasing the randomintensitymay result in a diffusion-based trajectory and the impact with the constraint plane,which induces the topological behavior of the non-smooth system to change drastically.It is also found that grazing bifurcation appears in advance with increasing of the random intensity.The stronger impulse force can result in the appearance of the diffusion phenomenon.

Bursting oscillations as well as the bifurcation mechanism in a non-smooth chaotic geomagnetic field model

Based on the chaotic geomagnetic field model, a non-smooth factor is introduced to explore complex dynamical behaviors of a system with multiple time scales. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamic behaviors. Due to the existence of non-smooth factors, different types of bifurcations are presented in spiking states, such as grazing-sliding bifurcation and across-sliding bifurcation. In addition, the non-smooth fold bifurcation may lead to the appearance of a special quiescent state in the interface as well as a non-smooth homoclinic bifurcation phenomenon. Due to these bifurcation behaviors, a special transition between spiking and quiescent state can also occur.

Bifurcation and Chaos in a Dynamical System of Rub-impact Rotor

Nonlinear vibration characteristics of a rub impact Jeffcott rotor are investigated. The system is two dimensional, nonlinear, and periodic. Fourier series analysis and the Floquet theory are used to perform qualitative global analysis of the dynamical system. The governing ordinary differential equations are also integrated using a numerical method to give the quantitative result. This preliminary study revealed the chaotic feature of the system. After the rub impact, as the rotating speed is increased three kinds of routes to chaos are found, that is, from a stable periodic motion through period doubling bifurcation, grazing bifurcation, and quasi periodic bifurcation to chaos.