碰撞振动系统不连续擦边分岔的线性反馈控制

Linear feedback control of the discontinuous grazing bifurcation in a vibro-impact system

不连续擦边分岔通常导致系统响应直接跳跃到碰撞周期运动或大幅值的混沌带。为了抑制擦边点处的跳跃现象并保证控制后系统的响应仍能保持为简单的周期运动,本文基于单自由度振碰系统中一系列孤立的退化擦边点能使未碰周期运动可以连续转迁进入碰撞周期运动这一特殊的动力学性质,设计了一类线性反馈控制器并利用零时间不连续映射的方法,将单自由度系统中大幅值的混沌带控制到稳定的碰撞周期运动,抑制了擦边点处的跳跃现象。数值仿真结果表明,本文提出的控制策略简单而有效。

Discontinuous grazing bifurcation can cause the instant jump of the system responses to the 1/n impact periodic motions or even large-amplitude narrow-band chaos. The purpose of the research is to suppress the jump phenomena at the discontinuous grazing points and ensure that the system responses after control can be simple periodic motions of 1/n order. Based on the special dynamic properties of continuous transition of the system responses from the non-impact periodic motions to the 1/n impact periodic motions caused by a set of isolated degenerate grazing points in the single degree-of-freedom vibro-impact system, a proper linear feedback controller is designed to suppress the narrow-band chaos to 1/n impact periodic motions to avoid the jump phenomena at the grazing points by using zero-time discontinuity mapping method. Numerical simulation shows that the proposed control strategy is simple and effective.