【作者】 张海燕；

【导师】 张伟； 王洪兰；

【作者基本信息】 北京工业大学 ， 机械设计与理论， 2002， 硕士

【摘要】 主动式电磁轴承(AMBs)被广泛地应用于工业和航空航天工程中，但由于电磁控制力是被控对象的位移和控制电流的非线性函数，因而构成了一个非线性机电系统。非线性力的作用使转子在某些参数域中产生相当大的振动，因此分析该类系统的非线性振动特性和稳定性一直是电磁轴承-转子动力学研究的重要课题。在这类系统中含有极其丰富和复杂的动力学行为，如分叉、分形和混沌动力学等。本文研究了电磁轴承-转子系统的非线性动力学，表明电磁轴承-转子系统在某些参数区域内可以出现全局分叉和混沌运动。 本文的研究内容和所获得的主要结果有以下几个方面： (1)综述了磁悬浮研究的历史，介绍了主动式电磁轴承-转子系统在工程实际中的应用，主动式电磁轴承的研究现状，总结了近十年来国内外对电磁轴承-转子系统非线性动力学的研究进展和取得的成果，指出了电磁轴承发展的趋势及进行非线性动力学研究的必要性。介绍了高维扰动Hamilton系统的全局摄动方法。 (2)当考虑转子质量时，电磁轴承-转子系统在水平方向和垂直方向的动力学方程是不同的，并导致系统的非线性特性不同。因此首先建立了考虑转子质量的二自由度刚体电磁轴承-转子系统模型，电磁轴承具有八个极对，从而使承载能力增加。得到了电磁轴承-转子系统二自由度的非线性动力学方程。 (3)利用多尺度方法，研究了电磁轴承-转子系统的1／3亚谐共振和1／2亚谐共振情况下的非线性动力学，得出了两种共振情况下的频幅响应方程。利用数值模拟方法得出了1／3亚谐共振和1／2亚谐共振情况下的局部分叉和振幅-频率响应曲线。 (4)研究了电磁轴承-转子系统在变刚度情况下的全局分叉和混沌动力学。利用多尺度方法得出了主参数共振情况下的平均方程，利用Normal Form理论对具有双零特征值和一对纯虚特征值的平均方程进行了简化，得到了平均系统的Normal Form。利用全局摄动法研究了电磁轴承-转子系统的全局分叉和混沌动力学，利用数值模拟方法分别对平均方程和原方程进行了分析，得到了的描述系统混沌运动的相图和波形图，从而验证了理论结果的正确性。上述研究说明电磁轴承-转子系统可以出现混沌运动，并且混沌运动具有初值敏感性。更多还原

【Abstract】 Active Magnetic Bearings (AMDs) have been widely used in the engineering, for example, the mechanical engineering, the aeronautic and astronautics engineering. Because most of the components in AMBs are of the nonlinear characteristics, the dynamics in AMBs is very complicated. The electromagnetic force is a nonlinear function with respect to the displacement of the rotor and the controlling electric current. The nonlinear electromagnetic force may cause the large oscillations of the rotor in some parameter regions. Thus, the studies on the properties of the nonlinear dynamics and the stability for the rotor-AMBs system play an important role in the engineering. There are abundant and complicated dynamical behaviors in the rotor-AMBs system, such as the local and global bifurcations and the chaotic dynamics. In this dissertation, we investigate the nonlinear dynamics in the rotor-AMBs system. The two-degree-of-freedom nonlinear system with cubic nonlinearities will be used to explore the bifurcations and chaotic dynamics in the rotor-AMBs system with eight pole pairs. The results obtained by the dissertation show that there exist the chaotic motions in some parameter regions.The research contents and the major results obtained in this dissertation are as follows.(1) We give a review on the researches for the rotor-AMBs system. The applications and developments of the rotor-AMBs system in the engineering are showed in recent years. In particularly, we present the results for the studies of nonlinear dynamics in the rotor-AMBs system. We also point out the trend of the rotor-AMBs system in future and the necessity of the study on the nonlinear dynamics of the rotor-AMBs system.(2) Due to the effect of the rotor weight, the equations of motion in the horizontal and vertical directions and the nonlinear properties are different. Therefore, a rigid model with two-degree-of-freedom is established after the rotor weight is considered. Because there are the eight pole pairs, the capacity supporting load in AMBs is greatly increased. The dimensionless equations of the rotor-AMBs in horizontal and vertical directions are also obtained.(3) We investigate the nonlinear dynamics of the rotor-AMBs system in 1/3 and 1/2 subharmonic resonances. Using the multiple method of scale, the averaged equations of the rotor-AMBs system are obtained. The amplitude-frequency response equations and the local bifurcations are respectively analyzed in the two resonant cases. The numerical simulations are given to obtain the amplitude-frequency response curve.(4) The global bifurcations and chaotic dynamics are investigated when the rotor-AMBs system has the time-varying stiffness. The multiple method of scale is used to obtain the averaged equations in primary parameter resonance. From the averaged equations, the theory of normal form is applied to find the explicit formulas of normalform associated with a double zero and a pair of pure imaginary eigenvalues with the aid of the Maple program. Based on the normal form obtained above, a global perturbation method is utilized to give the analysis for the global bifurcations and chaotic dynamics of the rotor-AMBs system. The global bifurcations analysis indicates that there exist the heteroclinic bifurcations and the Silnikov-type homoclinic orbit in the averaged equations. These mean that the chaotic motions can occur in the rotor-AMBs system with the time-varying stiffness. The numerical simulations verify the analytical prediction.更多还原