Study On Near-grazing Dynamics Of Impact Oscillators

The grazing bifurcations are the special dynamical phenomena that can lead to sudden change of dynamic responses of vibro-impact oscillators when the grazing contacts between the trajectory and constraint surface occur.Under grazing conditions,singularity will arise in the Poincaré mapping of near-grazing motions.This kind of singularity can have a great influence on the formation and characteristics of complex dynamic responses such as bifurcations and chaos.In this thesis,the grazing bifurcations which can be widely found in the single-degree-of-freedom linear impact oscillator and nonlinear impact oscillators with rigid clearance are studied from the theoretical and numerical point of view.The main contents of this thesis are as follows:In order to study the grazing bifurcations of a single-degree-of freedom linear impact oscillator,we first discuss the general solution form of the non-impact differential motion equation of the oscillator under different damping states.Then we derived the compound Poincaré mapping with the aid of the zero-time-discontinuity-mapping in the neighborhood of the grazing point.Numerical simulation results verify the validity of using low order compound Poincaré mapping to study the grazing bifurcation.Moreover,in view of controlling the grazing induced chaos in the single-degree-of freedom linear impact oscillator,the existence and stability corresponding to the impact periodic motions are studied by using the low order compound Poincaré mapping.The conditions where the co-dimension-two grazing bifurcations arise are determined.Based on the special dynamical properties of such co-dimension-two grazing bifurcation points,two different control schemes are designed to suppress the grazing induced chaos respectively.The original complex motion of the system is effectively controlled to a simple impact periodic motions.The feasibility of the proposed control strategy is verified by numerical simulations.To study the grazing bifurcation of a single-degree-of-freedom nonlinear impact oscillator,the first order approximate periodic solution and stability of the steady state motion of the Van de pol oscillator are analyzed by using the multi-scale method.Furthermore,using the numerical analysis method such as the shooting method,the difference method and the interpolation method to locate the fixed point of the periodic motion on the Poincaré section and determine the periodic motion trajectories of the system.And the critical grazing value of the periodic motion trajectories is obtained,and then the grazing bifurcation behavior of the periodic motion is analyzed.In addition,the grazing bifurcation of quasi-periodic motion ofthis system is investigated.Finally,we show the detailed derivation of the local first-order normal form discontinuity mappings.The compound first-order Poincaré mapping that can be used to characterize the near-grazing dynamics is constructed correspondingly.However,given the first-order compound Poincaré mapping can not fully show the complex periodic motion or chaos of the oscillator in the neighborhood of the co-dimension-two grazing bifurcation point,the first-order normal form discontinuous mapping is extended to the two order,and the corresponding two-order compound Poincaré mapping is constructed.The validity of theoretical derivation is verified by numerical simulations.