弹性转子—轴承系统非线性动力学行为及控制
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摘要
旋转机械在机械、动力、交通、航空航天及空间技术领域中占有极其重要的地位，也是国民经济的关键装备之一。随着生产与科学技术的迅速发展，对于转子系统非线性动力学行为的研究，已经发展成为当前相关领域的一个热点。基于这一背景，本文以弹性转子一轴承系统为研究对象，就其故障消除问题展开研究，取得了一些有益的结果。
旋转机械在运行过程中，事故时有发生。研究表明，非线性振动是导致系统故障的主要原因。只要通过微扰，将混沌等复杂行为转化为简单的同步周期运动，即可确保机器正常运转。基于这一思路，本文并没有像传统的反馈控制法那样，求出不稳定的目标周期轨道，而是直接将位移反馈到系统中去，再用数值模拟研究其非线性动力学行为，借助相图、分岔图、Poincare映射图分析系统的运动形态，检验控制的效果。结果显示，所用的方法实现了预先设想，由于该法只需较小的反馈增益(小于0.5)，且是双输入的，因此在实验中便于操作。
本文首先介绍了一种精确的短轴承非稳态非线性油膜力的解析公式，然后介绍了用于分析转子—轴承系统在非线性因素作用下运动形态的方法。
其次在上面提供的理论基础上，建立弹性转子—轴承系统动力学模型，采用龙格—库塔算法来求解系统的运动微分方程，画出了数值模拟图，结果显现系统中存在丰富的周期、概周期甚至混沌运动。
第三，在上述模型中引入位移反馈，仍然利用龙格—库塔算法求解并作出了受控系统的数值模拟图。结果表明这种控制方法可以将混沌、概周期以及高周期运动转化为同步周期运动。
最后，对本文中所取得的结果进行了总结，同时指出了本文的创新之处及存在的不足。
Rotating machinery plays a very important role in machinery, power, communication, aerospace, space technology and so on. It is also one of main equipments in national economy. Along with the rapid development of production, science and technology, research on some problems of nonlinear dynamical behaviors of a rotor system has evolved to be a top point in related fields nowadays. Based on this background, the method removing abnormal behaviors consisting in the flexible rotor-short journal bearing system is studied in this paper, and some useful results are got.
During the operation of rotating machinery, serious accidents often happen. In fact, the irregular behaviors of machinery are mainly caused by its nonlinear vibrations So, the normal operation of system can be obtained if its complicated motions such as chaos are converted into simple synchronous periodic motions by small perturbations. Based on this idea, the displace feedback terms are introduced into the system directly in this thesis, instead of solving the unsteady periodic target orbit in traditional feedback control method. Then, the nonlinear dynamical behaviors of the controlled flexible rotor-short journal bearing system are studied by using numerical simulations.
The motion characteristics and the control effects are analyzed by phase diagram, bifurcation diagram and Poincare maps. Researches show the results agree with the imagination. Furthermore, the approach adopted is easy to be used in experiment, for it only needs a small feedback-gain (less than 0.5), and only two variables are required to be controlled.
Firstly, the analytical formula of the unsteady oil film force of the short journal bearing, is cited in this thesis. In addition, the methods for the research of the motion characteristics of system are introduced.
Secondly, based on the theories above, the dynamical model of the short journal bearing-flexible rotor system is established, and the motion differential equation of the system is solved by using the Runge-kutta method, then, the diagrams about uncontrolled system are plotted by numerical simulations. Results show the various forms of periodic, quasi-periodic and chaotic motions.
Thirdly, two displace-feedback terms are added to the model mentioned, the motion characteristics of the short journal bearing-flexible rotor system are investigated under the consideration of control, and its motion differential equation is solved by the Runge-Kutta method, then, the diagrams about controlled system are plotted by numerical simulations. Results show this control method can transform chaotic, quasi-periodic and multiple periodic motions into synchronous periodic motions.
Finally, the results we've obtained are summarized in the end of this thesis. Also some creativities as well as existing problems are pointed out.
旋转机械在运行过程中，事故时有发生。研究表明，非线性振动是导致系统故障的主要原因。只要通过微扰，将混沌等复杂行为转化为简单的同步周期运动，即可确保机器正常运转。基于这一思路，本文并没有像传统的反馈控制法那样，求出不稳定的目标周期轨道，而是直接将位移反馈到系统中去，再用数值模拟研究其非线性动力学行为，借助相图、分岔图、Poincare映射图分析系统的运动形态，检验控制的效果。结果显示，所用的方法实现了预先设想，由于该法只需较小的反馈增益(小于0.5)，且是双输入的，因此在实验中便于操作。
本文首先介绍了一种精确的短轴承非稳态非线性油膜力的解析公式，然后介绍了用于分析转子—轴承系统在非线性因素作用下运动形态的方法。
其次在上面提供的理论基础上，建立弹性转子—轴承系统动力学模型，采用龙格—库塔算法来求解系统的运动微分方程，画出了数值模拟图，结果显现系统中存在丰富的周期、概周期甚至混沌运动。
第三，在上述模型中引入位移反馈，仍然利用龙格—库塔算法求解并作出了受控系统的数值模拟图。结果表明这种控制方法可以将混沌、概周期以及高周期运动转化为同步周期运动。
最后，对本文中所取得的结果进行了总结，同时指出了本文的创新之处及存在的不足。
Rotating machinery plays a very important role in machinery, power, communication, aerospace, space technology and so on. It is also one of main equipments in national economy. Along with the rapid development of production, science and technology, research on some problems of nonlinear dynamical behaviors of a rotor system has evolved to be a top point in related fields nowadays. Based on this background, the method removing abnormal behaviors consisting in the flexible rotor-short journal bearing system is studied in this paper, and some useful results are got.
During the operation of rotating machinery, serious accidents often happen. In fact, the irregular behaviors of machinery are mainly caused by its nonlinear vibrations So, the normal operation of system can be obtained if its complicated motions such as chaos are converted into simple synchronous periodic motions by small perturbations. Based on this idea, the displace feedback terms are introduced into the system directly in this thesis, instead of solving the unsteady periodic target orbit in traditional feedback control method. Then, the nonlinear dynamical behaviors of the controlled flexible rotor-short journal bearing system are studied by using numerical simulations.
The motion characteristics and the control effects are analyzed by phase diagram, bifurcation diagram and Poincare maps. Researches show the results agree with the imagination. Furthermore, the approach adopted is easy to be used in experiment, for it only needs a small feedback-gain (less than 0.5), and only two variables are required to be controlled.
Firstly, the analytical formula of the unsteady oil film force of the short journal bearing, is cited in this thesis. In addition, the methods for the research of the motion characteristics of system are introduced.
Secondly, based on the theories above, the dynamical model of the short journal bearing-flexible rotor system is established, and the motion differential equation of the system is solved by using the Runge-kutta method, then, the diagrams about uncontrolled system are plotted by numerical simulations. Results show the various forms of periodic, quasi-periodic and chaotic motions.
Thirdly, two displace-feedback terms are added to the model mentioned, the motion characteristics of the short journal bearing-flexible rotor system are investigated under the consideration of control, and its motion differential equation is solved by the Runge-Kutta method, then, the diagrams about controlled system are plotted by numerical simulations. Results show this control method can transform chaotic, quasi-periodic and multiple periodic motions into synchronous periodic motions.
Finally, the results we've obtained are summarized in the end of this thesis. Also some creativities as well as existing problems are pointed out.
引文
[1] Ott E, Grebogi C, Yorke J A. Controlling Chaos. Phys. Rev. Lett, 1990,64(11):1196-1199
[2] Ditto W L, Rauseo S N, Spano M L. Experimental control of chaos. Phys. Rev. Lett, 1990,65(26):3211-3214
[3] Hunt E R. Stabilizing high-period orbits in a chaotic system: The Diode Resonator. Phys. Rev. Lett, 1991,67(15):1953-1966
[4] Pyragas K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A, 1992,170(6):421-428,1993,181:201
[5] Pyragas K. Experimental control of chaos by delayed self-controlling feedback. Phys. Lett. A, 1993,180:99-102
[6] Matias M A, Guemez J. Stabilizing of chaos by proportional pulses in the system variables. Phys. Rev. Lett, 1994,72:1455
[7] Chen G, Dong X.. On feedback control of chaotic nonlinear dynamic systems, Int. J. Bifurcation & chaos, 1992,2:407-411
[8] Liu Y, Barbosa L C, Rios, J R et al, Supression of chaos in a CO_2 laser by feedback. Phys. Lett. A, 1994,193:259
[9] 方天华．实现混沌同步的非线性变量反馈控制法．原子能科学技术，1998，32，(1)：59-64
[10] 罗晓曙，刘慕仁，方锦清等．一种基于系统变量的线性和非线性变换实现混沌控制的方法．物理学报 2000，49(5)：849-853
[11] Luo X S, Fang J Q, Wang L H, et al. A Strategy of Chaos Control and a Unified Mechanism for Several Kinds of Chaos Control Methods. ACTA Physics Sinica, 1999,8(12):895-901
[12] J. Guemez M. A. Matias. Phys. Lett. A, 1993,181:29
[13] Hubler A., Luscher, Naturwissenschaften, 1989,76:67
[14] Jackson E A, Hubler A. Physica, 1990,D40:407
[15] Sinha S, Biswa D. Phys. Rev. Lett, 1993,71:2010
[16] 张文，崔升，徐小峰等．动载轴承非稳态非线性油膜力的一般数学模型．工程力学进展．1998：158-167
[17] 孙保苍，周传荣．轴承-转子系统的分岔与混沌特性研究．振动、测试与诊断．2004，24(3)：192-196
[18] 刘延柱，陈立群．非线性振动．第一版．北京：高等教育出版社，2001
[19] 陈予恕．非线性振动系统的分叉和混沌理论．第一版．北京：高等教育出版社，1993
[20] 胡海岩．应用非线性动力学．第一版．北京：航空工业出版社，2000
[21] 胡岗，萧井华，郑之刚．混沌控制．第一版．上海：上海科技教育出版社，2000
[22] 方锦清．驾驭混沌与发展高新技术．第一版．北京：原子能出版社，2002
[23] 王光瑞，于熙玲，陈式刚．混沌的控制、同步与利用．第一版．北京：国防工业出版社，2001
[24] 陈式刚．映象与混沌．第一版．北京：国防工业出版社，1992
[25] 王光瑞，陈光旨．非线性常微分方程的混沌运动．第一版．南宁：广西科学技术出版社，1995
[26] 陈式刚．圆映射．第一版．上海：上海科技教育出版社，1998
[27] Shinbrot T, Grebogi C, Ott E, Yorke J A. Using small perturbations to control chaos. Nature, 1993,363(3):411-417
[28] Ott E, Spano M L. Controlling chaos. Physics Today. 1995, 34
[29] 詹姆斯．格莱克．混沌：开创新科学．第一版．上海：上海译文出版社，1990
[30] 方锦清．非线性系统中混沌控制与同步及其应用前景(一)．物理学进展，1996，16(1)：1-74
[31] 方锦清．非线性系统中混沌控制方法、同步原理及其应用前景(二)．物理学进展，1996，16(2)：137-202
[32] 方锦清，姚伟光．非线性系统理论基础、算法及其应用．第一版．北京：核工业研究生部，1993
[33] 方锦清．混沌控制及其应用前景．自然杂志，1993，15(9，10)：6-9
[34] 方锦清．混沌控制及其应用前景．科技导报，1994，71(5)：23-28
[35] 方锦清．迅速发展中的混沌控制与同步．自然杂志，18(1996)291
[36] 方锦清．超混沌、混沌的控制与同步．科技导报，1996，94(4)：6-8
[37] 方锦清．非线性控制与混沌控制论——略谈与现代控制理论的结合．自然杂志．1998，20(3)：147-152
[38] Dressier U, Gregor Nitsche. Controlling chaos using time delay coordinates. Phys. Rev. Lett, 1992,68(1): 1261-1264
[39] Bielawski S, Derozier D, Glorieux P. Controlling unstable periodic orbits by a delayed continuous feedback. Phys. Rev. E, 1994,49(2):971-973
[40] 方锦清．激光及其高阶级联系统中的超混沌同步与控制．强激光与粒子束，1996，8(1)：133-145
[41] 罗晓曙，方锦清等．一种新的基于系统变量延迟反馈的控制混沌方法．物理学报2000，49(8)：1423-1427
[42] Chen G R, Dong X N. From Chaos to Order, Methodologies, Perspectives and Applications. World Scientific Series on Nonlinear Science, Singapore, World Scientific, 1998
[43] Wang Hu-Zheng, et al. The Handbook of modem engineering mathematics. Press of Hua Zhong University of Science and Technology, 1988,3:948-961
[44] 韩曾普．自适应控制．第一版．北京：清华大学出版社，1995
[45] Bernardo M D. An adaptive approach to the control and synchronization and continuous-time chaotic systems. Inter. J. Bifurcation & Chaos, 1996,6(3):557-568
[46] Bernardo M D. A purely adaptive controller and synchronize and control chaotic systems. Phys. Lett, 1996,214:139-144
[47] 虞烈，刘恒．轴承．转子系统动力学．第一版．西安：西安交通大学出版社，2001
[48] 张文．转子动力学理论基础．第一版．北京：科学出版社，1990
[49] 闻邦椿，武新华，丁千，韩清凯等．故障旋转机械非线性动力学的理论与试验．第一版．北京：科学出版社，2004
[50] 闻邦椿，刘树英，张纯宇．机械振动学．第一版．北京：冶金工业出版社，2000
[51] 席慧玲．有限宽轴承-转子系统非线性动力学行为的若干问题研究(硕士学位论文)．镇江：江苏大学，2005
[52] 邱飞宇．耦合故障转子系统非线性动力学行为数值模拟(硕士学位论文)．镇江：江苏大学，2006
[2] Ditto W L, Rauseo S N, Spano M L. Experimental control of chaos. Phys. Rev. Lett, 1990,65(26):3211-3214
[3] Hunt E R. Stabilizing high-period orbits in a chaotic system: The Diode Resonator. Phys. Rev. Lett, 1991,67(15):1953-1966
[4] Pyragas K. Continuous control of chaos by self-controlling feedback. Phys. Lett. A, 1992,170(6):421-428,1993,181:201
[5] Pyragas K. Experimental control of chaos by delayed self-controlling feedback. Phys. Lett. A, 1993,180:99-102
[6] Matias M A, Guemez J. Stabilizing of chaos by proportional pulses in the system variables. Phys. Rev. Lett, 1994,72:1455
[7] Chen G, Dong X.. On feedback control of chaotic nonlinear dynamic systems, Int. J. Bifurcation & chaos, 1992,2:407-411
[8] Liu Y, Barbosa L C, Rios, J R et al, Supression of chaos in a CO_2 laser by feedback. Phys. Lett. A, 1994,193:259
[9] 方天华．实现混沌同步的非线性变量反馈控制法．原子能科学技术，1998，32，(1)：59-64
[10] 罗晓曙，刘慕仁，方锦清等．一种基于系统变量的线性和非线性变换实现混沌控制的方法．物理学报 2000，49(5)：849-853
[11] Luo X S, Fang J Q, Wang L H, et al. A Strategy of Chaos Control and a Unified Mechanism for Several Kinds of Chaos Control Methods. ACTA Physics Sinica, 1999,8(12):895-901
[12] J. Guemez M. A. Matias. Phys. Lett. A, 1993,181:29
[13] Hubler A., Luscher, Naturwissenschaften, 1989,76:67
[14] Jackson E A, Hubler A. Physica, 1990,D40:407
[15] Sinha S, Biswa D. Phys. Rev. Lett, 1993,71:2010
[16] 张文，崔升，徐小峰等．动载轴承非稳态非线性油膜力的一般数学模型．工程力学进展．1998：158-167
[17] 孙保苍，周传荣．轴承-转子系统的分岔与混沌特性研究．振动、测试与诊断．2004，24(3)：192-196
[18] 刘延柱，陈立群．非线性振动．第一版．北京：高等教育出版社，2001
[19] 陈予恕．非线性振动系统的分叉和混沌理论．第一版．北京：高等教育出版社，1993
[20] 胡海岩．应用非线性动力学．第一版．北京：航空工业出版社，2000
[21] 胡岗，萧井华，郑之刚．混沌控制．第一版．上海：上海科技教育出版社，2000
[22] 方锦清．驾驭混沌与发展高新技术．第一版．北京：原子能出版社，2002
[23] 王光瑞，于熙玲，陈式刚．混沌的控制、同步与利用．第一版．北京：国防工业出版社，2001
[24] 陈式刚．映象与混沌．第一版．北京：国防工业出版社，1992
[25] 王光瑞，陈光旨．非线性常微分方程的混沌运动．第一版．南宁：广西科学技术出版社，1995
[26] 陈式刚．圆映射．第一版．上海：上海科技教育出版社，1998
[27] Shinbrot T, Grebogi C, Ott E, Yorke J A. Using small perturbations to control chaos. Nature, 1993,363(3):411-417
[28] Ott E, Spano M L. Controlling chaos. Physics Today. 1995, 34
[29] 詹姆斯．格莱克．混沌：开创新科学．第一版．上海：上海译文出版社，1990
[30] 方锦清．非线性系统中混沌控制与同步及其应用前景(一)．物理学进展，1996，16(1)：1-74
[31] 方锦清．非线性系统中混沌控制方法、同步原理及其应用前景(二)．物理学进展，1996，16(2)：137-202
[32] 方锦清，姚伟光．非线性系统理论基础、算法及其应用．第一版．北京：核工业研究生部，1993
[33] 方锦清．混沌控制及其应用前景．自然杂志，1993，15(9，10)：6-9
[34] 方锦清．混沌控制及其应用前景．科技导报，1994，71(5)：23-28
[35] 方锦清．迅速发展中的混沌控制与同步．自然杂志，18(1996)291
[36] 方锦清．超混沌、混沌的控制与同步．科技导报，1996，94(4)：6-8
[37] 方锦清．非线性控制与混沌控制论——略谈与现代控制理论的结合．自然杂志．1998，20(3)：147-152
[38] Dressier U, Gregor Nitsche. Controlling chaos using time delay coordinates. Phys. Rev. Lett, 1992,68(1): 1261-1264
[39] Bielawski S, Derozier D, Glorieux P. Controlling unstable periodic orbits by a delayed continuous feedback. Phys. Rev. E, 1994,49(2):971-973
[40] 方锦清．激光及其高阶级联系统中的超混沌同步与控制．强激光与粒子束，1996，8(1)：133-145
[41] 罗晓曙，方锦清等．一种新的基于系统变量延迟反馈的控制混沌方法．物理学报2000，49(8)：1423-1427
[42] Chen G R, Dong X N. From Chaos to Order, Methodologies, Perspectives and Applications. World Scientific Series on Nonlinear Science, Singapore, World Scientific, 1998
[43] Wang Hu-Zheng, et al. The Handbook of modem engineering mathematics. Press of Hua Zhong University of Science and Technology, 1988,3:948-961
[44] 韩曾普．自适应控制．第一版．北京：清华大学出版社，1995
[45] Bernardo M D. An adaptive approach to the control and synchronization and continuous-time chaotic systems. Inter. J. Bifurcation & Chaos, 1996,6(3):557-568
[46] Bernardo M D. A purely adaptive controller and synchronize and control chaotic systems. Phys. Lett, 1996,214:139-144
[47] 虞烈，刘恒．轴承．转子系统动力学．第一版．西安：西安交通大学出版社，2001
[48] 张文．转子动力学理论基础．第一版．北京：科学出版社，1990
[49] 闻邦椿，武新华，丁千，韩清凯等．故障旋转机械非线性动力学的理论与试验．第一版．北京：科学出版社，2004
[50] 闻邦椿，刘树英，张纯宇．机械振动学．第一版．北京：冶金工业出版社，2000
[51] 席慧玲．有限宽轴承-转子系统非线性动力学行为的若干问题研究(硕士学位论文)．镇江：江苏大学，2005
[52] 邱飞宇．耦合故障转子系统非线性动力学行为数值模拟(硕士学位论文)．镇江：江苏大学，2006