中文摘要：

对一类含间隙碰撞振动系统Poincaré映射的Hopf-Hopf交互分岔进行了反控制研究。首先,基于碰撞振动系统建立了六维Poincaré映射,由于六维映射相应雅克比矩阵的特征值没有解析的表达式,这使得由特征值特性描述的传统临界分岔准则在确定控制增益中具有很大的局限性。针对这个局限性,建立了包含特征值分布条件、横截条件和非共振条件的显式临界准则。所建立的准则与传统的分岔准则等价,但并不依赖雅克比矩阵特征值的直接计算。然后,针对碰撞的不连续特性导致的隐式Poincaré映射在闭环系统控制设计中的困难,发展了一种基于原碰撞系统的线性反馈控制方法。最后,数值分析给出了在指定的参数点所设计的映射Hopf-Hopf交互分岔的环面解,进一步验证了理论分析的正确性。

英文摘要：

Anti-controlling Hopf-Hopf interaction bifurcation on Poincarémap in a three-degree-of-freedom vibratory system with clearance is addressed in this paper.Firstly,a six-dimensional Poincarémap of the vibro-impact system is established.As the analytical expressions of all eigenvalues of Jacobi matrix for six-dimensional map are unavailable,the classical critical criteria of Hopf-Hopf interaction bifurcation described by the properties of eigenvalues have a great limitation in obtaining control gains.Aiming at the limitation,a new bifurcation criterion including eigenvalue assignment,transversality condition and nonresonance condition is established.The established criteria are equivalent to the classical critical criteria,but they do not depend on eigenvalue computations of Jacobi matrix.Then,for the difficulty of implicit Poincarémap created by discontinuity of vibroimpact motions in anti-control design of bifurcation of the close-loop system,a linear feedback control method is developed.Finally,numerical simulation shows that a torus solution of Hopf-Hopf interaction bifurcation on Poincarémap is created in a desired parameter location and verifies theoretical analysis.