三自由度含间隙碰撞振动系统Neimark-Sacker分岔的反控制

分岔反控制作为传统分岔控制的逆问题, 其目的是在预先指定的系统参数点通过控制主动设计出具有所期望特性的分岔解. 以一类三自由度含间隙双面碰撞振动系统为研究对象, 在不改变原系统平衡解结构的前提下, 考虑到在碰撞振动系统反控制过程中由Poincaré映射的隐式特点和传统的映射Neimark-Sacker分岔临界准则带来的困难, 通过对原系统施加线性反馈控制器并利用不直接依赖于特征值计算的Neimark-Sacker分岔显式临界准则研究了此系统的分岔反控制问题. 首先对原系统施加线性反馈控制, 建立闭环控制系统的六维Poincaré映射. 由于六维映射的雅克比矩阵的特征值没有解析的表达式, 利用高维映射Neimark-Sacker分岔的显式临界准则, 获得了系统出现拟周期碰撞振动运动的控制参数区域. 然后采用中心流形-正则形方法分析了拟周期分岔解的稳定性. 数值仿真结果表明本文方法可以在指定的系统参数点通过控制设计出稳定的拟周期碰撞运动.

Anti-controlling Neimark-Sacker bifurcation of a three-degree-of-freedom vibration system with clearance

Anti-control of bifurcation, as an inverse problem of conventional bifurcation analysis, is aimed at creating a certain bifurcation with desired dynamic properties at a pre-specified system parameter location via control. The main purpose of this paper is to address the problem of anti-control of Neimark-Sacker bifurcation of a three-degree-of-freedom vibro-impact system with clearance (i.e., the second Hopf bifurcation of the original system), which may be viewed as a design approach to creating a quasi-periodic impact motion (or torus solution) at a specified system parameter location via control. Firstly, in the premise of no change of periodic solutions of the original system, when the difficulties that are brought about by the implicit Poincaré map of the vibro-impact system are considered, a linear feedback controller is added to the original system and a six-dimensional Poincaré map of the close-loop control system is established. In order to design a desired bifurcation solution by control, the multiple control gains are used to tune the existence of this bifurcation based on the corresponding critical criterion. However, for six-dimensional map of the vibro-impact system in the paper, the analytical expressions of all eigenvalues of Jacobi matrix with respect to parameters are unavailable. This implies that when the classical critical criterion described by the properties of eigenvalues is used, we have to numerically compute eigenvalues point by point and check their properties to search for the control gains. So, the numerical calculation is a laborious job in the process of determining the control gains. To overcome the difficulty originating from the classical bifurcation criterion, the explicit critical criterion without using eigenvalue calculation of high-dimensional map is used to obtain the controlling parameters area when quasi-periodic impact motion occurs. Then, the stability of quasi-periodic bifurcation solution is analyzed by utilizing the center manifold and normal formal theory. Finally the numerical experiments verify that the stable quasi-periodic impact motion can be generated at a designated system parameter point by the proposed control.