Unfolding And Control Of Degenerate Grazing Bifurcations In Impacting Oscillators

As common scientific problems,impacting oscillations are widely encountered in the field of mechanical engineering with clearance.Affected by the non-smooth factor induced by impact,the dynamic responses of impacting oscillators are quite sensitive to the parameter change and more complex phenomena than generic smooth oscillators can be exhibited in impacting oscillators.Therefore,it shall be of great significance to promote the further development of the non-smooth dynamics and enhance its potential application in the mechanisms with clearance,by investigating the nonlinear dynamic mechanisms and research methods,revealing the widely existing instability phenomena and implementing effective control in the impacting oscillators.At present,research on the grazing singularity of impacting oscillator is one of the challenges in the field of non-smooth dynamics.Here,grazing indicates the tangency contact between the trajectory of impacting oscillator and its rigid constraints,leading to uncertainties in the system dynamic response.Grazing can induce the singularity of Poincaré maps of impacting oscillator and have essential effects on both the generation and evolution of the system response.In the literature,grazing phenomena have been avoided in many papers which analyzed the motion stability,bifurcation and chaos of impacting oscillators.Actually,in-depth understandings on the grazing phenomena and relevant non-smooth dynamics are still required.Therefore,it would be quite necessary to do deep research on such challenging problem,i.e.,analysis of grazing singularity.Focused on a special type of co-dimension-two grazing bifurcations in the impacting oscillators,i.e.,degenerate grazing bifurcation,this dissertation performs systematic and in-depth studies on the bifurcation mechanisms and control of degenerate grazing.The main work of this dissertation are as follows:(1)The existing theory of local zero time discontinuity mapping is extended to order two,in order to overcome the drawback that the lowest order truncation cannot restore the full information of near-grazing dynamics in the specified parameter region.The newly derived higher-order discontinuity mapping lays the theoretical foundation for this dissertation.Based on the derived higher-order discontinuity mapping,theoretical analyses on the existence of the near-grazing impact period-one motion are performed.And the calculation index corresponding to the degenerate grazing bifurcation point is proposed accordingly.And the proposed method can be extended to other impacting oscillators with multi rigid constraints.(2)The physical meanings of real roots of the scalar existence equation of the impact period-one motion are discussed and the corresponding coexistence phenomena near the degenerate grazing points are revealed as well.Based on the derived higher-order discontinuity mapping,perturbation analysis of the characteristic equation of impact period-one motion is performed and one truncated characteristic equation is obtained accordingly.Using the derived truncation,the stability and bifurcations of the impact period-one motion can be easily discussed.The novel dynamic phenomena of NeimarkSacker bifurcations and related co-dimension-two bifurcations are revealed for the first time in a two-degree-of-freedom impact oscillator.And the corresponding conjugate eigenvalues required for Neimark-Sacker bifurcations are found to be generated in two main ways,i.e.,the interaction of eigenvalues or induced by grazing bifurcations.(3)To reveal the evolution laws of system responses in the parameter region where the basic impact period-one motions are unstable,two-parameter bifurcation diagrams in the vicinity of certain degenerate grazing points are computed numerically by using the parallel computing technology.On this basis,one-parameter bifurcation diagrams,shooting method and cell mapping method are used to discuss some typical evolution processes and promote the in-depth understanding into the obtained two-parameter bifurcation diagrams.It can be summarized that,the standard bifurcation scenarios reported in the literature can be induced by the subcritical period-doubling bifurcation of the impact period-one motion.While for the unreported bifurcation scenarios and novel dynamic behaviors,they shall be induced by the supercritical period-doubling or the subcritical Neimark-Sacker bifurcations of the impact period-one motion.(4)To suppress the discontinuous jump phenomena arising in the neighborhood of degenerate grazing bifurcation points of the impacting oscillators,the control strategy of degeneration of eigenvalue is proposed to stabilize the impact period-one motion.Under the proposed strategy,the originally grazing-induced singular eigenvalue can be controlled into the unit circle and continuous transition phenomenon from the nonimpact period motion to the impact period-one motion via grazing bifurcation can be achieved accordingly.The proposed control strategy has overcome the drawback of the existing strategy based on the grazing stability criterion,under which the continuous transition phenomena between the simple and predictable impact periodic motions cannot be observed.