电磁轴承—转子系统的非线性动力学
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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。
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 normal
form 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.
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。
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 normal
form 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.
引文
1.U.J. Na and A. Palazzolo, Optimized realization of fault-tolerant heteropolar magnetic bearings,Journal of Vibration and Acoustics, 122: p209-221,2000.
2.J.W. Beams, High Rotation Speeds, J. Appl. Phys, 8:p795-806, 1937.
3.H.S. Kemper, chewebende Aufhangung durch eletromagneticshe Kraft,eine Moglichkeit fur eine grundsatzlich neue Fortbewegungsart, Eleketrotechn.Z.59:p391-395,1938.
4.J.W. Beams, The production of high centrifugal fields, J. Appl. Phys, p886-890, 1946.
5．杨艳丽和汪希平，电磁轴承技术及其应用开发，中国机械工程学报，28(3)：p24-29，1996．
6．陈立群，张艳鸣和谢友柏，电磁轴承的发展与应用，西北轻工业学院学报，14(3)：p103-110，1996．
7．沈钺和虞烈，电磁轴承及控制系统的设计原理及参数选择，机械工程学报，369(11)：p62-64，2000．
8．汪希平和陈学军，陀螺效应对电磁轴承系统设计的影响，机械工程学报，37(4)：p48-52，2001．
9．赵雷，张德魁，杨作兴等，电磁轴承最优刚度与系统结构参数关系的研究，机械工程学报，36(12)：p62-64，2000．
10．袁崇军等，电磁轴承-转子稳定性分析与控制器参数设计，西安交通大学学报 30：p7-14，1996．
11.M. Chinta and A. B. Palazzolo, Stability and bifurcation of rotor motion in a magnetic bearing, Journal of Sound and Vibration 214:p793-803,1998.
12.S. Akishita, T. Morimura and S. Hamaguchi, Vibration control of magnetically suspended flexible rotor by the use of optimal regulator, In: Proceedings of 2nd International Symposium on Magnetic Bearing, p147-154, 1990.
13.C. W. Lee and J. S. Kim, Modal testing and suboptimal vibration control of flexible rotor bearing system by using a magnetic beating. ASME Journal of Dynamic Systems,Measurement, and Control,114(6):p244-252,1992.
14.K. Nonami, Active vibration control of a flexible rotor using H_∞ control theory. JSME International Journal,Series C, 35(3):p393-399,1992.
15.J.Y.Huang, Magnetic bearing control using fuzzy logic, IEEE Transactions on Industry Applications, 31(5): p1492-1497, 1995.
16.H.M. Chen and P. Lewis, Pule-based damping control for magnetic beatings, In: Proceedings of 3nd International Symposium on Magnetic Bearig, p25-34, 1992.
17.W.M. Cui and K. Nonami, H_∞ control of flexible rotor-magnetic bearing systems, In:Proceedings of 3nd International Symposium on Magnetic Bearig, p505-514, 1992.
18．景敏卿，郝金平，谢友柏等，基于DSP的电磁轴承控制器硬件设计，中国机械工程学报，32(7)：p45-48，1999．
19．沈钺和虞烈，四自由度电磁轴承-转子系统的数学模型与仿真，机械工程学报，301(2)：p34-39，2000．
20．汪希平，张直明，于良等，轴向磁悬浮轴承的力学特性分析，应用力学学报，17(3)：p29-33，2000．
21.L.N. Virgin, T. F. Walsh and J. D. Knight, Nonlinear beahavior of a magnetic bearing system,Transaction of the ASME 117: p582-588, 1999.
22．汪希平，袁崇军和谢友柏，电磁轴承系统的控制策略和功放电路的设计，西安交通大学学报 29：p68-74，1995．
23.G. Petela and K. Botros, Magnetic bearing control of flexible shaft vibrations based on fulty-access velocity-displacement feedback. ASME Journal of Engineering for Gas Turbines and Power,117:p188-197,1995.
24.P.K.Sinha, Electromagnetic suspension: new results using neural networks. IEEE Transaction on Magnetics, 29(6):2791-2793,1993.
25.K.Nonami and H.Yamaguchi, Robust control of magnetic bearing systems by means of sliding mode control, In: Proceedings of 3nd International Symposium on Magnetic Bearing,p537-546,1992.
26.K. Matsuda, Self-sensing magnetic bearings using the differential transformer principle,Journal of JSME, Series, C, 63(609):p1441-1447,1997.
27.M.D.Noh,E.H.Malsen, Self-sensing magnetic bearings using parameter estimation. IEEE Transactions on Instrumentation and measurement, 46(1):p45-50,1997.
28.K.K.Sivadasan, Analysis of self-sensing active magnetic bearings working on inductance measurement principle. IEEE Transactions on Magnetics, 32(2):p329-334,1996.
29.K.K.Sivadasan, Analysis of self-sensing magnetic bearings using parameter estimation. IEEE Transactions on Instrumentaion and Measurement,46(1):p45-50,1997.
30.Vischer D, Bleuler H. Self-sensing active magnetic levitation. IEEE Transaction on Magnetics 29(2):p1276-1281,1993.
31.T.Mizuno, K. Araki and H. Bleuler, Stability analysis of self-sensing magnetic bearing controllers. IEEE Transactions on Control Systems Technology,4(5):p572-579,1996.
32．陈安华，刘德顺和朱萍玉，非线性转子碰摩振动周期响应的分叉与振幅突变，中国机械工程学报，12(2)：p126-131，2001．
33．徐健康和张华容，电磁悬浮转子的非线性控制，振动工程学报 6(4)：p386-393，1993．
34．王洪礼和吴志强，磁力轴承转子系统的Hopf分叉，天津大学学报27(6)，1994．
35.L.N. Virgin, T. F. Walsh and J. D. Knight, Nonlinear behavior of a magnetic bearing system, Journal of Engineering for Gas Turbines and Power, Transactions of the ASME,117:p582-588,1995.
36．王洪礼和吴志强，磁力轴承参数对转子运动稳定性的影响，非线性振动、分叉及混沌，天津大学出版社，p654-659，1992．
37.J.C. Ji, L.Yu and A.Y.T. Leung, Bifurcation behavior of a rotor in active magnetic bearings, Journal of Sound and Vibration 235, p133-151,2001.
38.J.C. Ji and L.Yu, Drop Dynamics of a high speed unbalanced rotor in active magnetic bearing machinery,Mechanics of Structures and Machines 28, p185-200, 2000.
39.T.Ishii and R.G. Kirk, Transient response technique applied to active magnetic bearing machinery duing rotor drop, Journal of Vibration and Acoustics, Transactions of the ASME 118, p154-163, 1996.
40.J.C.Ji and C. H. Hansen, Nonlinear oscillations of a rotor in active magnetic bearings,Journal of Sound and Vibration 240, p599-612, 2001.
41．武新华和张新江，弹性转子-轴承系统的非线性动力学研究，中国机械工程 12(11)：p1221-1224，2001．
42．刘恒，虞烈和谢友柏，非线性不平衡轴承转子系统全局特性及其稳定性准则的研究，机械工程学报 35(2)：p61-65，1999．
43．汪希平，电磁轴承系统的参数设计与应用研究，西安交通大学博士论文，1994．3．
44．毛军红，主动型电磁轴承系统研究及控制器设计，西安交通大学硕士论文，1990．5．
45．王海英，电磁轴承—转子系统的稳定性分析，西安交通大学硕士学位论文，1995，4．
46.J.Guckenheimer, Noise in chaotic systems, Nature, 298(358), 1982.
47.V.K.Melnilov, On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc.,12:p1-57,1963.
48.P.J.Holmes,A nonlinear oscillatior with a strange attractor,Philos.Trans.Roy.Soc.London Ser.A.292:p419-448,1979.
49.S.N.Chow,J.K.Hale and J.Mallet-Paret,An example of bifurcation to homoclinic orbits,Journal of Differential Equations 37:p351-373,1980.
50.B.D.Greeenspan and P.J.Holmes,Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators, SIAM J. Math. Anal 15:p69-97
51.P.J.Holmes and J.E.Marsden,Horseshoes in perturbation of Hamiltonian systems with two degrees of freedom,Comm.Math.Phys.82:p523-544,1982.
52.P.J.Holmes and J.E.Marsden,Melnilov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems,J.Math.Phys 23:p523-544,1982.
53.G.Kovacic and S.Wiggins,Orbits homoclinic to resonance with an application to chaos in a model of the forced and demped sine-Gordon equation,Physica D 57:p185-225,1992.
54.Z.C.Feng and P.R.Sethna,Global bifurcations in the motion of parametrically excited thin plate,Nonlinear Dynamics 4:p389-408,1993.
55.Z.C.Feng and S.Wiggins,On the existence of chaos in a parametrically forced mechanical systems with broken O(2) symmetry,Zeitschriftfuer Angewandte Mathematik und Physikalische 44,201-248,1993
56.Z.C. Feng and K. M. Liew, Global bifurcations in parametrically excited systems with zero-to-one-internal resonance, Nonlinear Dynamics 21:p249-263,2000.
57.W.Zhang, Z. M. Liu and P. Yu, Global dynamics of a parametrically and extemally excited thin plate, Nonlinear Dynamics 24: p245-268, 2001.
58.W.Zhang and Y. Tang, Global dynamics of the cable under combined parametrical and external excitation, International Journal of Nonlinear Mechanics 37:p505-526,2000.
59.G.Kovacic, Singular perturbation theory for homoclinic orbits in a class of near intergrable dissipative systems, SIME J. Math. Anal. 26:p1611-1643,1995.
60．闻邦椿，顾家柳，夏松波和王正，高等转子动力学，机械工业出版社，2000．
61．张伟，含有参数激励的非线性动力系统的高余维和复杂动力学，天津大学博士论文，1997，5．
62．周纪卿和朱因远，非线性振动，西安交通大学出版社，1997．
63.D.Laier and R. Markert, Simulation of nonlinear effects on magnetically suspended rotors, Proceedings of the 1st International Conference on Engineering Computation and Computer Simulation ECCS-1, Changsha, China,Vol.1,p473-482, 1995.
64．郑兆昌，机械振动(中册)，北京：机械工业出版社，1986．
65．钱伟长，奇异摄动理论及其在力学中的应用，北京科学出版社，1981．
66.A.H.Nayfeh,Applied Nonlinear Dynamics, New York, 1995.
67．王沫然，MATLAB 5.X与科学计算，清华大学出版社，2000．
68．W.H.普雷斯，B.P.弗拉内里等，数值方法大全，兰州大学出版社，1992．
69.A.Heck,Introduction to Maple,Springer-Verlag New York Berlin Heidelberg, 1993.
70．陈予恕，唐云等，非线性动力学中的现代分析方法，科学出版社，1992．
71.P.YU,W.ZHANG and Q.BI,Vibration analysis on a thin plate with the aid of computation of normal forms, International Journal of Non-linear Mechanics, 2000.
72.S.Wiggins,Global Bifurcations and Chaos-Analytical Methods.New York:Springer-Verlag,1988.
73．张旺和王世鎏，自动控制原理，北京理工大学出版社，1994．
74.A.H.Nayfeh,Perturbation Methods,New York:Wiley-Interscience,1973.
75．胡海岩，应用非线性动力学。航空工业出版社。
76.G. Kaovacic and S.Wiggins, Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D 57: p185-225, 1992.
2.J.W. Beams, High Rotation Speeds, J. Appl. Phys, 8:p795-806, 1937.
3.H.S. Kemper, chewebende Aufhangung durch eletromagneticshe Kraft,eine Moglichkeit fur eine grundsatzlich neue Fortbewegungsart, Eleketrotechn.Z.59:p391-395,1938.
4.J.W. Beams, The production of high centrifugal fields, J. Appl. Phys, p886-890, 1946.
5．杨艳丽和汪希平，电磁轴承技术及其应用开发，中国机械工程学报，28(3)：p24-29，1996．
6．陈立群，张艳鸣和谢友柏，电磁轴承的发展与应用，西北轻工业学院学报，14(3)：p103-110，1996．
7．沈钺和虞烈，电磁轴承及控制系统的设计原理及参数选择，机械工程学报，369(11)：p62-64，2000．
8．汪希平和陈学军，陀螺效应对电磁轴承系统设计的影响，机械工程学报，37(4)：p48-52，2001．
9．赵雷，张德魁，杨作兴等，电磁轴承最优刚度与系统结构参数关系的研究，机械工程学报，36(12)：p62-64，2000．
10．袁崇军等，电磁轴承-转子稳定性分析与控制器参数设计，西安交通大学学报 30：p7-14，1996．
11.M. Chinta and A. B. Palazzolo, Stability and bifurcation of rotor motion in a magnetic bearing, Journal of Sound and Vibration 214:p793-803,1998.
12.S. Akishita, T. Morimura and S. Hamaguchi, Vibration control of magnetically suspended flexible rotor by the use of optimal regulator, In: Proceedings of 2nd International Symposium on Magnetic Bearing, p147-154, 1990.
13.C. W. Lee and J. S. Kim, Modal testing and suboptimal vibration control of flexible rotor bearing system by using a magnetic beating. ASME Journal of Dynamic Systems,Measurement, and Control,114(6):p244-252,1992.
14.K. Nonami, Active vibration control of a flexible rotor using H_∞ control theory. JSME International Journal,Series C, 35(3):p393-399,1992.
15.J.Y.Huang, Magnetic bearing control using fuzzy logic, IEEE Transactions on Industry Applications, 31(5): p1492-1497, 1995.
16.H.M. Chen and P. Lewis, Pule-based damping control for magnetic beatings, In: Proceedings of 3nd International Symposium on Magnetic Bearig, p25-34, 1992.
17.W.M. Cui and K. Nonami, H_∞ control of flexible rotor-magnetic bearing systems, In:Proceedings of 3nd International Symposium on Magnetic Bearig, p505-514, 1992.
18．景敏卿，郝金平，谢友柏等，基于DSP的电磁轴承控制器硬件设计，中国机械工程学报，32(7)：p45-48，1999．
19．沈钺和虞烈，四自由度电磁轴承-转子系统的数学模型与仿真，机械工程学报，301(2)：p34-39，2000．
20．汪希平，张直明，于良等，轴向磁悬浮轴承的力学特性分析，应用力学学报，17(3)：p29-33，2000．
21.L.N. Virgin, T. F. Walsh and J. D. Knight, Nonlinear beahavior of a magnetic bearing system,Transaction of the ASME 117: p582-588, 1999.
22．汪希平，袁崇军和谢友柏，电磁轴承系统的控制策略和功放电路的设计，西安交通大学学报 29：p68-74，1995．
23.G. Petela and K. Botros, Magnetic bearing control of flexible shaft vibrations based on fulty-access velocity-displacement feedback. ASME Journal of Engineering for Gas Turbines and Power,117:p188-197,1995.
24.P.K.Sinha, Electromagnetic suspension: new results using neural networks. IEEE Transaction on Magnetics, 29(6):2791-2793,1993.
25.K.Nonami and H.Yamaguchi, Robust control of magnetic bearing systems by means of sliding mode control, In: Proceedings of 3nd International Symposium on Magnetic Bearing,p537-546,1992.
26.K. Matsuda, Self-sensing magnetic bearings using the differential transformer principle,Journal of JSME, Series, C, 63(609):p1441-1447,1997.
27.M.D.Noh,E.H.Malsen, Self-sensing magnetic bearings using parameter estimation. IEEE Transactions on Instrumentation and measurement, 46(1):p45-50,1997.
28.K.K.Sivadasan, Analysis of self-sensing active magnetic bearings working on inductance measurement principle. IEEE Transactions on Magnetics, 32(2):p329-334,1996.
29.K.K.Sivadasan, Analysis of self-sensing magnetic bearings using parameter estimation. IEEE Transactions on Instrumentaion and Measurement,46(1):p45-50,1997.
30.Vischer D, Bleuler H. Self-sensing active magnetic levitation. IEEE Transaction on Magnetics 29(2):p1276-1281,1993.
31.T.Mizuno, K. Araki and H. Bleuler, Stability analysis of self-sensing magnetic bearing controllers. IEEE Transactions on Control Systems Technology,4(5):p572-579,1996.
32．陈安华，刘德顺和朱萍玉，非线性转子碰摩振动周期响应的分叉与振幅突变，中国机械工程学报，12(2)：p126-131，2001．
33．徐健康和张华容，电磁悬浮转子的非线性控制，振动工程学报 6(4)：p386-393，1993．
34．王洪礼和吴志强，磁力轴承转子系统的Hopf分叉，天津大学学报27(6)，1994．
35.L.N. Virgin, T. F. Walsh and J. D. Knight, Nonlinear behavior of a magnetic bearing system, Journal of Engineering for Gas Turbines and Power, Transactions of the ASME,117:p582-588,1995.
36．王洪礼和吴志强，磁力轴承参数对转子运动稳定性的影响，非线性振动、分叉及混沌，天津大学出版社，p654-659，1992．
37.J.C. Ji, L.Yu and A.Y.T. Leung, Bifurcation behavior of a rotor in active magnetic bearings, Journal of Sound and Vibration 235, p133-151,2001.
38.J.C. Ji and L.Yu, Drop Dynamics of a high speed unbalanced rotor in active magnetic bearing machinery,Mechanics of Structures and Machines 28, p185-200, 2000.
39.T.Ishii and R.G. Kirk, Transient response technique applied to active magnetic bearing machinery duing rotor drop, Journal of Vibration and Acoustics, Transactions of the ASME 118, p154-163, 1996.
40.J.C.Ji and C. H. Hansen, Nonlinear oscillations of a rotor in active magnetic bearings,Journal of Sound and Vibration 240, p599-612, 2001.
41．武新华和张新江，弹性转子-轴承系统的非线性动力学研究，中国机械工程 12(11)：p1221-1224，2001．
42．刘恒，虞烈和谢友柏，非线性不平衡轴承转子系统全局特性及其稳定性准则的研究，机械工程学报 35(2)：p61-65，1999．
43．汪希平，电磁轴承系统的参数设计与应用研究，西安交通大学博士论文，1994．3．
44．毛军红，主动型电磁轴承系统研究及控制器设计，西安交通大学硕士论文，1990．5．
45．王海英，电磁轴承—转子系统的稳定性分析，西安交通大学硕士学位论文，1995，4．
46.J.Guckenheimer, Noise in chaotic systems, Nature, 298(358), 1982.
47.V.K.Melnilov, On the stability of the center for time periodic perturbations. Trans. Moscow Math. Soc.,12:p1-57,1963.
48.P.J.Holmes,A nonlinear oscillatior with a strange attractor,Philos.Trans.Roy.Soc.London Ser.A.292:p419-448,1979.
49.S.N.Chow,J.K.Hale and J.Mallet-Paret,An example of bifurcation to homoclinic orbits,Journal of Differential Equations 37:p351-373,1980.
50.B.D.Greeenspan and P.J.Holmes,Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators, SIAM J. Math. Anal 15:p69-97
51.P.J.Holmes and J.E.Marsden,Horseshoes in perturbation of Hamiltonian systems with two degrees of freedom,Comm.Math.Phys.82:p523-544,1982.
52.P.J.Holmes and J.E.Marsden,Melnilov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems,J.Math.Phys 23:p523-544,1982.
53.G.Kovacic and S.Wiggins,Orbits homoclinic to resonance with an application to chaos in a model of the forced and demped sine-Gordon equation,Physica D 57:p185-225,1992.
54.Z.C.Feng and P.R.Sethna,Global bifurcations in the motion of parametrically excited thin plate,Nonlinear Dynamics 4:p389-408,1993.
55.Z.C.Feng and S.Wiggins,On the existence of chaos in a parametrically forced mechanical systems with broken O(2) symmetry,Zeitschriftfuer Angewandte Mathematik und Physikalische 44,201-248,1993
56.Z.C. Feng and K. M. Liew, Global bifurcations in parametrically excited systems with zero-to-one-internal resonance, Nonlinear Dynamics 21:p249-263,2000.
57.W.Zhang, Z. M. Liu and P. Yu, Global dynamics of a parametrically and extemally excited thin plate, Nonlinear Dynamics 24: p245-268, 2001.
58.W.Zhang and Y. Tang, Global dynamics of the cable under combined parametrical and external excitation, International Journal of Nonlinear Mechanics 37:p505-526,2000.
59.G.Kovacic, Singular perturbation theory for homoclinic orbits in a class of near intergrable dissipative systems, SIME J. Math. Anal. 26:p1611-1643,1995.
60．闻邦椿，顾家柳，夏松波和王正，高等转子动力学，机械工业出版社，2000．
61．张伟，含有参数激励的非线性动力系统的高余维和复杂动力学，天津大学博士论文，1997，5．
62．周纪卿和朱因远，非线性振动，西安交通大学出版社，1997．
63.D.Laier and R. Markert, Simulation of nonlinear effects on magnetically suspended rotors, Proceedings of the 1st International Conference on Engineering Computation and Computer Simulation ECCS-1, Changsha, China,Vol.1,p473-482, 1995.
64．郑兆昌，机械振动(中册)，北京：机械工业出版社，1986．
65．钱伟长，奇异摄动理论及其在力学中的应用，北京科学出版社，1981．
66.A.H.Nayfeh,Applied Nonlinear Dynamics, New York, 1995.
67．王沫然，MATLAB 5.X与科学计算，清华大学出版社，2000．
68．W.H.普雷斯，B.P.弗拉内里等，数值方法大全，兰州大学出版社，1992．
69.A.Heck,Introduction to Maple,Springer-Verlag New York Berlin Heidelberg, 1993.
70．陈予恕，唐云等，非线性动力学中的现代分析方法，科学出版社，1992．
71.P.YU,W.ZHANG and Q.BI,Vibration analysis on a thin plate with the aid of computation of normal forms, International Journal of Non-linear Mechanics, 2000.
72.S.Wiggins,Global Bifurcations and Chaos-Analytical Methods.New York:Springer-Verlag,1988.
73．张旺和王世鎏，自动控制原理，北京理工大学出版社，1994．
74.A.H.Nayfeh,Perturbation Methods,New York:Wiley-Interscience,1973.
75．胡海岩，应用非线性动力学。航空工业出版社。
76.G. Kaovacic and S.Wiggins, Orbits homoclinic to resonance with an application to chaos in a model of the forced and damped sine-Gordon equation, Physica D 57: p185-225, 1992.