Grazing Bifurcations Of Impact Motion Of Cantilever Beam With Bilateral Constraints And Its Impulsive Control

The cantilever beam system with impact is a kind of typical non-smooth dynamic system.It has a wide range of practical engineering applications in vibro-impact systems.Collision can lead to a special grazing phenomenon between system trajectory and constraint surface.This phenomenon will cause complex dynamic behaviors of the impact system.In this thesis,the grazing dynamics and control of a cantilever beam impact system with bilateral constraints are studied.The main contents are as follows:The non-smooth dynamics of two kinds of cantilever beam systems with bilateral elastic and bilateral rigid constraints are studied.Firstly,based on the dynamic equation of the impact cantilever beam and the definition of the grazing point,the existence condition of the double grazing periodic motion is analyzed.Secondly,the high-order discontinuous mapping with parameters near the double grazing trajectory is derived by choosing the zero velocity as Poincar(?) section.Then,a new composite piecewise normal mapping is established by combining smooth flow mapping with higher-order discontinuous mapping.Finally,the validity of the higher-order normal mapping is verified by comparing the bifurcation diagrams of the lower-order normal mapping and the higher-order normal mapping,and the numerical simulations further reveal the grazing dynamics of the impact cantilever beam with the bilateral constraints.The impulsive control of the grazing dynamics of the impact cantilever beam with the bilateral constraints is studied.According to the characteristics of rigid collision,a statedependent impulse control method is developed.The Poinca(?) mapping of the periodic impact motion of controlled systems is established,and the stability control conditions of the periodic impact motion are presented.Based on the control conditions,the chaotic motion near the grazing motion is controlled into a stable periodic orbit by selecting an appropriate control gain.Based on the theory of discontinuous mapping,the stability criteria of grazing bifurcation and the two possible discontinuous grazing bifurcation situations for the impact cantilever beam with the bilateral constraints are presented.Based on the stability criteria of grazing orbit,the control region of stability of grazing bifurcation is obtained by developing a impulse control method with fixed moments,and the control of jumping discontinuous grazing bifurcation is realized.