高维非线性动力系统的降维方法与分岔研究
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摘要
本文研究了基于切比雪夫正交多项式理论和非线性Galerkin法的高维非线性动力学系统降维技术及非线性油膜力、密封力作用下转子-轴承系统的非线性动力学问题。
复杂动力系统往往呈现出高维、非线性、强耦合的特点,这类系统的动力学特性即使采用数值方法研究也存在着很多困难。非线性非自治系统的动力学行为模式往往由低维流形上的动力学所控制,因而降维简化就成为从理论上揭示其动力学现象的必要步骤和前提。本文利用平移后的切比雪夫正交多项式理论求得一个线性项随时间周期变化的非线性非自治系统的Liapunov-Floquet变换矩阵,将其变换为线性项时不变的系统,再利用非线性Galerkin法对此系统进行降维。通过对降维前后系统的时间历程图和相轨迹图的对比,说明降维成功。
针对单圆盘Jeffcott转子-轴承系统,建立其在非线性油膜力和密封力作用下的动力学方程,其中油膜力采用了Capone在1991年提出的短轴承假设下的非线性油膜力模型,该模型具有较好的精度和收敛性,密封力采用了Muszynska非线性模型。本文利用龙格-库塔法求解上述非线性转子-轴承动力学系统,得到了系统的周期响应和非线性动力学特性,利用最大值法绘制了系统随转子转速和圆盘质量偏心变化的分岔图。在所研究的转速变化范围内,通过绘制Poincare截面图、轨迹图、相图等,发现随着转子转速和圆盘质量偏心的变化,系统发生周期运动、倍周期运动以及准周期运动,进而进入混沌状态,而后又变为二周期运动,最后以周期运动结束。
Based on the Chebyshev orthogonal polynomials theory and the nonlinear Galerkin method, the order reduction of high order nonlinear dynamics and the dynamical characteristics of rotor-bearing systems with nonlinear oil film force and sealing force are studid in this paper.
Complex dynamical systems always present high dimension, nonlinear and strong coupling characteristics. For these systems, there are many difficulties even if we use the numerical methods to study their dynamical characteristics. The nonlinear non-autonomous systems’dynamical behavior patterns are often controlled by the low-dimensional manifolds on the dynamics. Thus the order reduction becomes the necessary steps and precondition to reveal their dynamical phenomenon theoretically. In this paper the shifted Chebyshev orthogonal polynomials are used to obtain the Liapunov-Floquet transition matrix of a nonlinear non-autonomous system whose linear part is periodic, and to transfer the system to a new one with time invariant linear part. Then, we use the nonlinear Galerkin method to make this system’s dimension lower. Through comparing the time trace and the phase diagram of the reduced model with the original model, we can find the order reduction is successful.
The dynamical equations of a single-disc Jeffcott rotor-bearing system with nonlinear oil film force and sealing force are established, where the nonlinear oil film force model put forward by Capone in 1991 under the short-bearing assumption is used to make the model has better accuracy and convergence. About the nonlinear sealing force, the Muszynska model is used. The Runge-Kutta method is used to study the nonlinear dynamical equations and the periodic response and nonlinear dynamical characteristics of the system are obtained. The bifurcation map with respect to the rotating speed and disc mass eccentricity is drawn by using the maximum method. Among the rotating speed range, by drawing the diagrams of Poincare section, trajectory map and phase trait, we can learn that with the changes of rotating speed and the mass eccentricity, the system has periodic, periodic doubling and quasi-period motion, then becomes chaos, next be changed to double period motion, and periodic motion at last.
复杂动力系统往往呈现出高维、非线性、强耦合的特点,这类系统的动力学特性即使采用数值方法研究也存在着很多困难。非线性非自治系统的动力学行为模式往往由低维流形上的动力学所控制,因而降维简化就成为从理论上揭示其动力学现象的必要步骤和前提。本文利用平移后的切比雪夫正交多项式理论求得一个线性项随时间周期变化的非线性非自治系统的Liapunov-Floquet变换矩阵,将其变换为线性项时不变的系统,再利用非线性Galerkin法对此系统进行降维。通过对降维前后系统的时间历程图和相轨迹图的对比,说明降维成功。
针对单圆盘Jeffcott转子-轴承系统,建立其在非线性油膜力和密封力作用下的动力学方程,其中油膜力采用了Capone在1991年提出的短轴承假设下的非线性油膜力模型,该模型具有较好的精度和收敛性,密封力采用了Muszynska非线性模型。本文利用龙格-库塔法求解上述非线性转子-轴承动力学系统,得到了系统的周期响应和非线性动力学特性,利用最大值法绘制了系统随转子转速和圆盘质量偏心变化的分岔图。在所研究的转速变化范围内,通过绘制Poincare截面图、轨迹图、相图等,发现随着转子转速和圆盘质量偏心的变化,系统发生周期运动、倍周期运动以及准周期运动,进而进入混沌状态,而后又变为二周期运动,最后以周期运动结束。
Based on the Chebyshev orthogonal polynomials theory and the nonlinear Galerkin method, the order reduction of high order nonlinear dynamics and the dynamical characteristics of rotor-bearing systems with nonlinear oil film force and sealing force are studid in this paper.
Complex dynamical systems always present high dimension, nonlinear and strong coupling characteristics. For these systems, there are many difficulties even if we use the numerical methods to study their dynamical characteristics. The nonlinear non-autonomous systems’dynamical behavior patterns are often controlled by the low-dimensional manifolds on the dynamics. Thus the order reduction becomes the necessary steps and precondition to reveal their dynamical phenomenon theoretically. In this paper the shifted Chebyshev orthogonal polynomials are used to obtain the Liapunov-Floquet transition matrix of a nonlinear non-autonomous system whose linear part is periodic, and to transfer the system to a new one with time invariant linear part. Then, we use the nonlinear Galerkin method to make this system’s dimension lower. Through comparing the time trace and the phase diagram of the reduced model with the original model, we can find the order reduction is successful.
The dynamical equations of a single-disc Jeffcott rotor-bearing system with nonlinear oil film force and sealing force are established, where the nonlinear oil film force model put forward by Capone in 1991 under the short-bearing assumption is used to make the model has better accuracy and convergence. About the nonlinear sealing force, the Muszynska model is used. The Runge-Kutta method is used to study the nonlinear dynamical equations and the periodic response and nonlinear dynamical characteristics of the system are obtained. The bifurcation map with respect to the rotating speed and disc mass eccentricity is drawn by using the maximum method. Among the rotating speed range, by drawing the diagrams of Poincare section, trajectory map and phase trait, we can learn that with the changes of rotating speed and the mass eccentricity, the system has periodic, periodic doubling and quasi-period motion, then becomes chaos, next be changed to double period motion, and periodic motion at last.
引文
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2陈予恕.非线性振动.北京:高等教育出版社, 2002:1~57
3陈予恕,曹登庆,吴志强.非线性动力学理论及其在机械系统应用中的若干进展.宇航学报. 2007, 28(4):794~804
4王文达,史宝珍.大型超超临界火电机组现状和发展趋势.上海电气. 2004
5陆启韶.非线性复杂系统的动力学问题.新疆大学学报. 2001, 18(2):141~146
6张文.转子动力学理论基础[M].北京:科学出版社, 1990:
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12 S. Wiggings. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, 1990
13 Wu Zhiqiang, Chen Yushu, Bi Qinsheng. Normal Form of the Nosemi-Simple Bifurcation Problem. Applied Mathematics and Mechanics. 1997, 18(4):349~354
14 Chen Yushu, Sun Hongjun. Complex Inner Product Averaging Method for Calculating Normal Form of ODE. Applied Mathematics and Mechanics.2001, 22(12):1368~1374
15 G. Rega, H. Troger. Dimension Reduction of Dynamical Systems: Methods, Models, Applications. Nonlinear Dynamics. 2005, 41:1~15
16 S. C. Sinha, S. Redkar, V. Deshmukh. Order Reduction of Paramerically Exicited Nonlinear Systems: Techniques and Applications. Nonlinear Dynamics. 2005, 41:237~273
17 E. Pesheck, C. Pierre. A New Galerkin-Based Approach for Accurate Non-Linear Normal Modes through Invariant Manifolds. Journal of Sound andVibration. 2002, 249(5):971~973
18 H. G. Matthies, M. Meyer. Nonlinear Galerkin Methods for the Model Reduction of Nonlinear Dynamical Systems. Computers and Structures. 2003, 81:1277~1286
19 I. Georgiou. Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods. Nonlinear Dynamics. 2005, 41:69~110
20 P. Sundararajan, S. T. Noah. An Algorithm for Response and Stability of Large Order Non-Linear Systems—Application to Rotor Systems. Journal of Sound and Vibration. 1998, 214(4):695~723
21 F. Kwasniok. The Reduction of Complex Dynamical Systems Using Principal Interaction Patterns. Physica D. 1996, 96:28~60
22 K. Hasselmann. PIPs and POPs: The Reduction of Complex Dynamical Systems Using Principal Interaction Patterns. Journal of Geophysical Research. 1988, 93:11015~11021
23 G. Kerschen, J. C. Golinval, A. F. Vakakis. The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview. Nonlinear Dynamics. 2005, 41:147~169
24 S. W. Shaw, C. Pierre. Normal Modes for Nonlinear Vibratory Systems. Journal of Sound and Vibration. 1993, 164(1):85~124
25吴志强,陈予恕.非线性模态的分类和新的求解方法.力学学报. 1996, 28(3):298~307
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27 A. H. Nayfeh., C. Chin, S. A. Nayseh. Nonlinear Normal Modes of a Cantilever Beam. Journal of Vibration and Acoustics. 1995, 117:477~481
28 A. H. Nayfeh. On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems . Journal of Vibration and Control. 1995, 1:389~430
29 K. Yagasaki, M. Sakata, K. Kimura. Dynamics of a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation. ASME Journal of Applied Mechanics. 1990, 112:209~217
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31 Wang Zaihua, Hu Haiyan. An Energy Analysis of Local Dynamics of a Delayed Oscillator near a Hopf Bifurcation. Nonlinear Dynamics. 2006, 46:149~159
32 Wang Huailei, Hu Haiyan. Bifurcation Analysis of a Delayed Dynamic System via Method of Multiple and Shooting Technique. International Journal of Bifurcation and Chaos. 2005, 15(2):425~450
33 Wang Huailei, Wang Zaihua, Hu Haiyan. Hopf Bifurcation of an Oscillator with Quadratic and Cubic Nonlinearities and with Delayed Velocity Feedback. Acta Mechanica Sinica. 2004, 20(4):426~434
34徐鉴,陆启韶,王乘. Van der Pol-Duffing时滞系统的稳定性和Hopf分岔.力学学报. 2000, 32(1):112~116
35郑吉兵,孟光,谢建华.时变小扰动Hamilton系统的Hopf分岔.力学学报. 2001, 33(2):215~223
36贺群,徐伟,方同等. Duffing系统随机分岔的全局分析.力学学报. 2003, 35(4):452~460
37张广军,徐建学,姚宏.含噪双稳杜芬镇子方程的分岔与随机分析.力学学报. 2006, 38(2):283~288
38 M. Botman. Experiments on Oil Film Dampers for Turbo-Machinery. ASME Journal of Engineering for Power. 1976, 393~400
39 A. G. Holmes. A Periodic Behavior of a Rigid Shaft in Short Bearings. International Journal for Numerical Methods in Engineering. 1978, 12:695~702
40 R. Brancati, M. Russo. On the Stability of Periodic Motions of an Unbalanced Rigid Rotor on Lubricated Journal Bearings. Nonlinear Dynamics. 1996, 10:175~186
41 R. D. Brown, P. Addison, A.H.C. Chan. Chaos in the Unbalance Response of Journal Bearings. Nonlinear Dynamics. 1994, 5:421~432
42刘恒,陈绍汀.非线性系统全局动态特性分析的PCM法及其在转子轴承系统中的应用.应用力学学报. 1995, 12(3):7~15
43郭丹,陈绍汀.非线性动力系统全局分析的变胞胞映射法与转子-轴承系统的全局稳定性.应用力学学报. 1996,13(4):8~15
44文成秀,姚玉玺,闻帮椿.动力系统的点映射-胞映射综合法.振动工程学报. 1997, 10(4):413~419
45 G. Adiletta, A. R. Guido, C. Rossi. Chaotic Motions of a Rigid Rotor in ShortJournal Bearings. Journal of Nonlinear Dynamics. 1996, 10:251~269
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63 D. E. Bently, A. Muszynska. Perturbation Tests of Bearing/Seal for Evaluation of Dynamic Coefficients, Symposium on Rotor Dynamical Instability. Summer Annual Conference of the ASME Applied Mechanics Division, Houston, Texas, 1983
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65 D. W. Childs. Dynamic Analysis of Turbulent Annular Seals Based on Hirs’Lubrication Equation. ASME Journal of Lubrication Technology. 1983, 105:429~436
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67 A. Steindl, H. Troger. Methods for Dimension Reduction and Their Application in Nonlinear Dynamics. International Journal of Solids and Structures. 2001, 38:2131~2147
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69席慧玲.有限宽轴承-转子非线性动力学行为的若干问题研究.江苏大学硕士学位论文. 2005
2陈予恕.非线性振动.北京:高等教育出版社, 2002:1~57
3陈予恕,曹登庆,吴志强.非线性动力学理论及其在机械系统应用中的若干进展.宇航学报. 2007, 28(4):794~804
4王文达,史宝珍.大型超超临界火电机组现状和发展趋势.上海电气. 2004
5陆启韶.非线性复杂系统的动力学问题.新疆大学学报. 2001, 18(2):141~146
6张文.转子动力学理论基础[M].北京:科学出版社, 1990:
7焦映厚,陈照波,夏松波等.非线性转子动力学的研究现状与展望.哈尔滨工业大学学报. 1999, 31(3):1~4
8黄文虎,武新华,焦映厚等.非线性转子动力学研究综述.振动工程学报. 2000, 13(4):497~509
9黄文虎,夏松波,焦映厚等.旋转机械非线性动力学设计基础理论与方法.北京:科学出版社, 2006
10 V. I. Arnold. Bifurcation Theory and Catastrophe Theory. Springer, 1999
11 J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York, 1983
12 S. Wiggings. Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, 1990
13 Wu Zhiqiang, Chen Yushu, Bi Qinsheng. Normal Form of the Nosemi-Simple Bifurcation Problem. Applied Mathematics and Mechanics. 1997, 18(4):349~354
14 Chen Yushu, Sun Hongjun. Complex Inner Product Averaging Method for Calculating Normal Form of ODE. Applied Mathematics and Mechanics.2001, 22(12):1368~1374
15 G. Rega, H. Troger. Dimension Reduction of Dynamical Systems: Methods, Models, Applications. Nonlinear Dynamics. 2005, 41:1~15
16 S. C. Sinha, S. Redkar, V. Deshmukh. Order Reduction of Paramerically Exicited Nonlinear Systems: Techniques and Applications. Nonlinear Dynamics. 2005, 41:237~273
17 E. Pesheck, C. Pierre. A New Galerkin-Based Approach for Accurate Non-Linear Normal Modes through Invariant Manifolds. Journal of Sound andVibration. 2002, 249(5):971~973
18 H. G. Matthies, M. Meyer. Nonlinear Galerkin Methods for the Model Reduction of Nonlinear Dynamical Systems. Computers and Structures. 2003, 81:1277~1286
19 I. Georgiou. Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods. Nonlinear Dynamics. 2005, 41:69~110
20 P. Sundararajan, S. T. Noah. An Algorithm for Response and Stability of Large Order Non-Linear Systems—Application to Rotor Systems. Journal of Sound and Vibration. 1998, 214(4):695~723
21 F. Kwasniok. The Reduction of Complex Dynamical Systems Using Principal Interaction Patterns. Physica D. 1996, 96:28~60
22 K. Hasselmann. PIPs and POPs: The Reduction of Complex Dynamical Systems Using Principal Interaction Patterns. Journal of Geophysical Research. 1988, 93:11015~11021
23 G. Kerschen, J. C. Golinval, A. F. Vakakis. The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview. Nonlinear Dynamics. 2005, 41:147~169
24 S. W. Shaw, C. Pierre. Normal Modes for Nonlinear Vibratory Systems. Journal of Sound and Vibration. 1993, 164(1):85~124
25吴志强,陈予恕.非线性模态的分类和新的求解方法.力学学报. 1996, 28(3):298~307
26陈予恕,吴志强.非线性模态理论的研究进展.力学进展. 1997, 27(3):289~300
27 A. H. Nayfeh., C. Chin, S. A. Nayseh. Nonlinear Normal Modes of a Cantilever Beam. Journal of Vibration and Acoustics. 1995, 117:477~481
28 A. H. Nayfeh. On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems . Journal of Vibration and Control. 1995, 1:389~430
29 K. Yagasaki, M. Sakata, K. Kimura. Dynamics of a Weakly Nonlinear System Subjected to Combined Parametric and External Excitation. ASME Journal of Applied Mechanics. 1990, 112:209~217
30 Hu Haiyan, Wang Zaihua. Dynamics of Controlled Mechanical Systems withDelayed Feedback. Heidelberg: Spriger-Verlag, 2002:1~294
31 Wang Zaihua, Hu Haiyan. An Energy Analysis of Local Dynamics of a Delayed Oscillator near a Hopf Bifurcation. Nonlinear Dynamics. 2006, 46:149~159
32 Wang Huailei, Hu Haiyan. Bifurcation Analysis of a Delayed Dynamic System via Method of Multiple and Shooting Technique. International Journal of Bifurcation and Chaos. 2005, 15(2):425~450
33 Wang Huailei, Wang Zaihua, Hu Haiyan. Hopf Bifurcation of an Oscillator with Quadratic and Cubic Nonlinearities and with Delayed Velocity Feedback. Acta Mechanica Sinica. 2004, 20(4):426~434
34徐鉴,陆启韶,王乘. Van der Pol-Duffing时滞系统的稳定性和Hopf分岔.力学学报. 2000, 32(1):112~116
35郑吉兵,孟光,谢建华.时变小扰动Hamilton系统的Hopf分岔.力学学报. 2001, 33(2):215~223
36贺群,徐伟,方同等. Duffing系统随机分岔的全局分析.力学学报. 2003, 35(4):452~460
37张广军,徐建学,姚宏.含噪双稳杜芬镇子方程的分岔与随机分析.力学学报. 2006, 38(2):283~288
38 M. Botman. Experiments on Oil Film Dampers for Turbo-Machinery. ASME Journal of Engineering for Power. 1976, 393~400
39 A. G. Holmes. A Periodic Behavior of a Rigid Shaft in Short Bearings. International Journal for Numerical Methods in Engineering. 1978, 12:695~702
40 R. Brancati, M. Russo. On the Stability of Periodic Motions of an Unbalanced Rigid Rotor on Lubricated Journal Bearings. Nonlinear Dynamics. 1996, 10:175~186
41 R. D. Brown, P. Addison, A.H.C. Chan. Chaos in the Unbalance Response of Journal Bearings. Nonlinear Dynamics. 1994, 5:421~432
42刘恒,陈绍汀.非线性系统全局动态特性分析的PCM法及其在转子轴承系统中的应用.应用力学学报. 1995, 12(3):7~15
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