两自由度干摩擦自激振动系统的Hopf-Hopf分岔分析
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摘要
工程领域方面，随处可见到干摩擦引起的振动现象，其引起的自激振动对机械系统有着重要的影响。为了研究干摩擦的振动行为，各种不同的理论模型被建立起来，但是多数理论模型仅仅是进行一些定性的分析或者通过数值模拟方法，用来解释所研究的问题，对干摩擦模型的系统研究尚属少见。研究表明，干摩擦引起的振动现象是丰富多样的，因而对干摩擦模型的系统研究是很有必要的。另一方面，干摩擦系统可归属于非光滑动力学系统的一种，其研究反映了当代人们对非光滑动力学系统研究的广泛关注和重视。
规范型的方法是利用坐标变换在平衡点附近把常微分方程化简，一般来说，得到的范式方程较为简洁，并且还能提供原方程定性性态方面的重要信息。正是由于求规范型的方法具有规范、统一等特点，更适用于电子计算机计算。随着电子技术的发展，范式理论得到较大的发展并大量应用在工程实际中，成为研究常微分方程动态分岔的基本工具。
本文主要研究了一类两自由度干摩擦自激振动系统的Hopf-Hopf分岔，并基于PB范式理论推导出了系统非共振情况下的三阶范式。首先，考虑到实际应用的广泛性，将干摩擦函数取为向量场非连续的滑动摩擦力函数；由于非连续干摩擦函数无法按照一般的PB范式理论进行推导，从而提出了摩擦力函数的在平衡点附近的级数展开方法，将摩擦力函数化为平衡点附近的连续函数；然后通过PB范式理论推导了这类两自由度干摩擦自激振动系统在非共振情况下的范式，获得了三阶范式系数的显式表达式。根据三阶范式系数的显式表达式，通过理论分析和逻辑推理，无需数值计算即可判断出不变圈和环面分岔的稳定性，从而简化了相图分析。在分析系统三阶范式的基础上发现，系统在小阻尼和临界速度的情况下，只出现不稳定的Hopf圈或Hopf-Hopf不变环面，也不会出现1：1强共振条件；最后，通过范式方程，分析了系统Hopf分岔及Hopf-Hopf环面分岔现象，计算了各种分岔的稳定域及参数范围，并给出分岔方程的各个系数，得到系统三阶范式的各种分岔区间。这些结果揭示了干摩擦系统复杂的动力学现象。级数展开方法及PB范式理论具有普遍性，可用于分析系统各种情况下的Hopf分岔及Hopf-Hopf环面分岔现象。本文所有理论结果均通过数值模拟证实了其正确性。
The vibration induced by the dry friction universally exists in engineering. The self-excited vibration caused by the dry frictions has significant impacts to the mechanical structures. In order to reveal these phenomena, many models have been established. In most of these models, there were just few of qualitative analysis or some calculations, but systematic investigation on the dry friction models exists scantly in literature. Whereas the vibration phenomena caused by the dry frictions is abundant and multiform, it is necessary to do thorough studies. In addition, the dry friction vibration is classified as a sort of non-smooth dynamics systems, so the research on this aspect reflects the people's attention and widespread interesting in the non-smooth dynamics systems.
In the theory of normal form, the ordinary differential equations are simplified through the method of coordinate transformation. In general, the normal form equations derived are simple, and they can keep the important qualities information of original equation moreover. Because the computation of normal form which is canonical and uniform, is convenient for computer calculation, the normal form theory has been made a great progress and widely applied in physic with the development of electronic technology, and has became an elementary tool for the dynamic research of ordinary differential equations.
In this paper, the Hopf-Hopf bifurcations of a two-degreed-freedom self-excited system with dry friction were studied, and the three order truncated normal form in nonresonance was obtained using the PB normal form theory. First, considering the extensive use in practice, the dry friction function was chosen as a discontinuous vector field slip-friction function. Because of it's discontinuity, the friction function can't be transformed to the normal form equation as usual; hence, the series expanding method was proposed so that the discontinuous friction function is transformed to continuous function near the balance point. In this way, the normal form of the two-degreed-freedom self-excited system in nonresonance case is educed, and the explicit expression of coefficient of the system's three order truncated normal form is obtained. Based on the explicit expression of coefficient, the stability of the invariant circle or torus bifurcation can be judged through theory analysis and logical deduction without any numerical calculation, also the complexity of the phase diagram analysis is reduced. From the system's three order truncated normal form analysis, it was shown that, the systems with small damp and small critical velocity only possess of instability Hopf invariant circle and instability Hopf-Hopf torus, and the 1:1 strong resonant case is impossible to appear. Finally, by virtue of normal form equations, the Hopf bifurcation and Hopf-Hopf torus bifurcation phenomena were analyzed in detail, the stability region and corresponding parameter range of each bifurcation are calculated, the coefficient and related bifurcation regions of the system's three order truncated normal form are obtained. The results reveal the complicate dynamic actions of the dry friction system. The method of series expanding and PB normal form theory, which may be generalized, can be applied to analysis the Hopf and Hopf-Hopf bifurcation of the system in all cases. The numerical simulations verify the correctness of all theory results of this paper.
规范型的方法是利用坐标变换在平衡点附近把常微分方程化简，一般来说，得到的范式方程较为简洁，并且还能提供原方程定性性态方面的重要信息。正是由于求规范型的方法具有规范、统一等特点，更适用于电子计算机计算。随着电子技术的发展，范式理论得到较大的发展并大量应用在工程实际中，成为研究常微分方程动态分岔的基本工具。
本文主要研究了一类两自由度干摩擦自激振动系统的Hopf-Hopf分岔，并基于PB范式理论推导出了系统非共振情况下的三阶范式。首先，考虑到实际应用的广泛性，将干摩擦函数取为向量场非连续的滑动摩擦力函数；由于非连续干摩擦函数无法按照一般的PB范式理论进行推导，从而提出了摩擦力函数的在平衡点附近的级数展开方法，将摩擦力函数化为平衡点附近的连续函数；然后通过PB范式理论推导了这类两自由度干摩擦自激振动系统在非共振情况下的范式，获得了三阶范式系数的显式表达式。根据三阶范式系数的显式表达式，通过理论分析和逻辑推理，无需数值计算即可判断出不变圈和环面分岔的稳定性，从而简化了相图分析。在分析系统三阶范式的基础上发现，系统在小阻尼和临界速度的情况下，只出现不稳定的Hopf圈或Hopf-Hopf不变环面，也不会出现1：1强共振条件；最后，通过范式方程，分析了系统Hopf分岔及Hopf-Hopf环面分岔现象，计算了各种分岔的稳定域及参数范围，并给出分岔方程的各个系数，得到系统三阶范式的各种分岔区间。这些结果揭示了干摩擦系统复杂的动力学现象。级数展开方法及PB范式理论具有普遍性，可用于分析系统各种情况下的Hopf分岔及Hopf-Hopf环面分岔现象。本文所有理论结果均通过数值模拟证实了其正确性。
The vibration induced by the dry friction universally exists in engineering. The self-excited vibration caused by the dry frictions has significant impacts to the mechanical structures. In order to reveal these phenomena, many models have been established. In most of these models, there were just few of qualitative analysis or some calculations, but systematic investigation on the dry friction models exists scantly in literature. Whereas the vibration phenomena caused by the dry frictions is abundant and multiform, it is necessary to do thorough studies. In addition, the dry friction vibration is classified as a sort of non-smooth dynamics systems, so the research on this aspect reflects the people's attention and widespread interesting in the non-smooth dynamics systems.
In the theory of normal form, the ordinary differential equations are simplified through the method of coordinate transformation. In general, the normal form equations derived are simple, and they can keep the important qualities information of original equation moreover. Because the computation of normal form which is canonical and uniform, is convenient for computer calculation, the normal form theory has been made a great progress and widely applied in physic with the development of electronic technology, and has became an elementary tool for the dynamic research of ordinary differential equations.
In this paper, the Hopf-Hopf bifurcations of a two-degreed-freedom self-excited system with dry friction were studied, and the three order truncated normal form in nonresonance was obtained using the PB normal form theory. First, considering the extensive use in practice, the dry friction function was chosen as a discontinuous vector field slip-friction function. Because of it's discontinuity, the friction function can't be transformed to the normal form equation as usual; hence, the series expanding method was proposed so that the discontinuous friction function is transformed to continuous function near the balance point. In this way, the normal form of the two-degreed-freedom self-excited system in nonresonance case is educed, and the explicit expression of coefficient of the system's three order truncated normal form is obtained. Based on the explicit expression of coefficient, the stability of the invariant circle or torus bifurcation can be judged through theory analysis and logical deduction without any numerical calculation, also the complexity of the phase diagram analysis is reduced. From the system's three order truncated normal form analysis, it was shown that, the systems with small damp and small critical velocity only possess of instability Hopf invariant circle and instability Hopf-Hopf torus, and the 1:1 strong resonant case is impossible to appear. Finally, by virtue of normal form equations, the Hopf bifurcation and Hopf-Hopf torus bifurcation phenomena were analyzed in detail, the stability region and corresponding parameter range of each bifurcation are calculated, the coefficient and related bifurcation regions of the system's three order truncated normal form are obtained. The results reveal the complicate dynamic actions of the dry friction system. The method of series expanding and PB normal form theory, which may be generalized, can be applied to analysis the Hopf and Hopf-Hopf bifurcation of the system in all cases. The numerical simulations verify the correctness of all theory results of this paper.
引文
1 J.J. Sinou, F. Thouverez, L. Jezequel. Analysis of friction and instability by thecenter manifold theory for a non-linear sprag-slip model. Journal of Sound and Vibration, 2003, 265:527-559
2 Yong Li, Z.C. Feng. Bifurcation and chaos in friction-induced vibration. Communications in Nonlinear Science and Numerical Simulation, 2004, 9:633-647
3 U. Galvanetto, S.R. Bishop. Characterization of the dynamics of a fourdimensional stick-slip system by a scalar variable. Chaos, Solitons and Fractals, 1995, 5:2171-2179
4 U. Galvanetto. Seme discontinuous bifurcations in a two-block stick-slip system. Journal of Sound and Vibration, 2001, 248(4): 653-669
5 谢建华．一类碰撞振动系统的余维二分叉和Hopf分叉．应用数学和力学，1996，17(1)：63-72
6 Blazejczyk-Okolewska and all. Chaotic mechanics in systems with impacts and friction. World Scientific, Singapore, 1999
7 Chunguang Li, Guanrong Chen, Xiaofeng Liao, Juebang Yu. Hopf bifurcation in an Internet congestion control model. Chaos, Solitons and Fractals 19 (2004) 853-862
8 韩福景，杨绍普，郭京波．参数激励下受电弓系统的分岔与混沌，石家庄铁道学院学报，2004，17(1)：25-29
9 陈少华，孙明光．油轮“鱼尾”运动及Hopf分岔算法，海洋工程，1994，12(1)：26-41
10 刘菲，杨翊仁．不可压缩流中二元翼运动的分支问题，西南交通大学学报，2002，37(增刊)：95-97
11 陆启韶．分岔与奇异性．上海科技教育出版社，1995
12 Golubitsky M. Stewart I. and Schaeffer D G. Singularities and Groups in Bifurcation Theory. New York: Springer-Yerlag, 1988
13 Iooss G, Los JE. Quasi-genericity of bifurcations to high dimensional invariant tori for maps. Commun Math Phys, 1988. 119:453-500
14 J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983
15 Yu, P. and Bi.Q.S. Analysis of non-linear dynamics and bifurcation of a double pendulum. Journal of Sound and Vibration, 217, 1998:691-736.
16 Q. Bi and P. Yu. Symbolic computation of normal forms for semi-simple cases Journal of Computational and Applied Mathematics, 1999, 102(2): 195-220.
17 J. H. Xie. Codimension two bifurcations and Hopf bifurcations of an impacting vibrating system. Applied Mathematics and Mechanics, 1996, 17: 65-75.
18 G. L. Wen. Codimension-2 Hopf bifurcation of a two-degree-of-freedom vibroimpact system, Journal of Sound and Vibration, 2001, 242: 475-485.
19 Furter J E. Hopf bifurcation at non-semisimple eigenvalues: a singularity theory approach. In international Series of Numerical Math. 104, E Allgower,K. Bohme&M. Golubitsky(eds.), Birkhauseer, 1992:135-145
20 Golubitsky M, Marsden J E, Stewart l, and Dellnitz M. The constrained Liapunov-Schmidt procedure and periodic orbits, Normal Form and Homoclinic Chaos. W F. Langford & W K. Nagata (eds.), Fields institute Communications 4, Amer. Math. Soc., Providence, 1995
21 Knobloch E. and Proctor M R E. The double Hopf bifurcation with 2:1 resonance. Proc R Soc. Lond. A, 1988, 415:61-90
22 Chossat P. and Dias F. The 1:2 resonance with 0(2) symmetry and its application to hydrodynamics. J. Nonlin. 1995, 5:105-129
23 Leblanc V G. and Langford W F. Classification and Unfolding of 1:2 Resonant Hopf Bifurcation. Arch. Ratimal Mech. Anal, 1996, 136:305-357
24 G.W. Luo, J.H. Xie. Bifurcations and chaos in a system with impacts. Physica D, 2001, 148: 183-200.
25 罗冠炜，谢建华．一类冲击振动系统在强共振条件下的亚谐分叉与Hopf分叉．爆炸与冲击，2003，23(1)：67-73
26 罗冠炜，谢建华，孙训方．高维映射的Hopf分叉分析及其在冲击振动系统中的应用．振动工程，1999，12(3)：360-366
27 高金锋，张晓．倍周期分岔和环面分岔对电力系统电压稳定性的影响．郑州大学学报(工学版)，2005，26(2)：27-31
28 H. Poincare. Les Methodes Nouvelles de la Mecanique Celeste, 1889, Paris: Gauthier-Villars.
29 A.D. Brjuno. Analytical forms of differential equations Ⅰ. Transactions of the Moscow Mathematical Society, 1971, 25:132-198.
30 A.D. Brjuno. Analytical forms of differential equations Ⅱ. Transactions of the Moscow Mathematical Society, 1972, 25:199-299.
31 V.I. Arnold. Mathematical Methods of Classical Mechanics. Berlin: Springer-Verlag, 1978
32 J. Moser. Stable and Random Motion in Dynamical Systems. Princeton, New Jersey: Hermann Weyl Lectures, 1973
33 T.L. Johnson and R. H. Rand. On the existence and bifurcation of minimal normal modes. Journal of Nonlinear Mechanics, 1979, 14:1-14.
34 M.N. Hamdon and T.D. Burton. Analysis of Forced Nonlinear Undamped Oscillators by a Time Transformation Method. Journal of Sound and Vibration, 1986, 110: 223-232.
35 L. Hsu. Analysis of Critical and Post-critical Behavior of Nonlinear Dynamical Systems by the Normal Form Method, Part Ⅰ: Normalization Formulate. Journal of Sound and Vibration, 1983, 89:169-181.
36 L. Hsu. Analysis of Critical and Post-critical Behavior of Nonlinear Dynamical Systems by the Normal Form Method, Part Ⅱ: Divergence and Flutter. Journal of Sound and Vibration, 1983, 89:181-194.
37 J.E. Marsden and M. Mccracken. The Hopf Bifurcation and Its Applications. Applied Mathematical Sciences. Berlin: Springer-Verlag, 1976, 19
38 Yu P. Computation of normal forms via a perturbation technique. Journal of Sound and Vibration, 1998, 211:19-38.
39 张伟，陈予恕，共轭算子法和非线性动力系统的高阶规范形，应用数学和力学， 1997,18(5):421-430.
40 杨绍普，郭文武．平均法与Hopf分叉的规范型系数．石家庄铁道学院学报，2000，13(3)：52-55
41 Takens F. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier, 1973, 23:163-165
42 Ushiki S. Normal forms for singularities of vector fields. Japan. J. Appl. Math., 1984, 1:1-37
43 Yu P. Computation of the simplest normal forms with perturbation parameters based on Lie transform and rescaling. J. Comput. Appl. Math., 2002, 144:359-373
44 Yu P.A simple and efficient method for computing center manifold and normal forms associated with semi simple cases. Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis at press, 2002
45 Yu P and Yuan Y. The simplest normal form for the singularity of a pure imaginary pair and a zero eigenva]ue Continuous, Discrete and Impulsive Systems (DCDIS) Series B: Applications and Algorithms, 2001, 8:219-249
46 Yu P and Yuan Y An efficient method for computing the simplest normal forms of vector fields Int. J. Bifurcation Chaos 2003,13
47 文明，邓子辰．干摩擦LQ控制系统在简谐激励下响应的递推算法．振动与冲击，2004，23(1)：41
48 白鸿柏，黄协清．干摩擦振动系统响应计算方法研究．振动工程学报，1998，11(4)：472
49 李宇燕，黄协清，毛文雄．金属橡胶三次非线性干摩擦系统振动响应计算方法研究．航空材料学报，2004，24(6)：56
50 韩茂安．动力系统的周期解与分支理论．科学出版社，2002：53-68
51 Arnold VI. Geometrical Methods in the Theory of Ordinary Differential Equations. New York: Springer-Verlag, 1983
52 Mardesic P, Rousseau C and Toni B. Linearization of isochronous centers. J. Diff. Eqns., 1995, 121: 67-108
53 F. Takens. Singularities of vector fields. Publ. Math. IHES, 1974, 43:47-100.
54 Y.A. Kugnetsov. Elements of Applied Bifurcation Theory. Springer-Verlag, 1998
55 高普云．非线性动力学-分叉、混沌与孤立子．国防科技大学出版社，2005：165-168
56 刘延柱，陈立群．非线性振动．高等教育出版社，2001：304-305
2 Yong Li, Z.C. Feng. Bifurcation and chaos in friction-induced vibration. Communications in Nonlinear Science and Numerical Simulation, 2004, 9:633-647
3 U. Galvanetto, S.R. Bishop. Characterization of the dynamics of a fourdimensional stick-slip system by a scalar variable. Chaos, Solitons and Fractals, 1995, 5:2171-2179
4 U. Galvanetto. Seme discontinuous bifurcations in a two-block stick-slip system. Journal of Sound and Vibration, 2001, 248(4): 653-669
5 谢建华．一类碰撞振动系统的余维二分叉和Hopf分叉．应用数学和力学，1996，17(1)：63-72
6 Blazejczyk-Okolewska and all. Chaotic mechanics in systems with impacts and friction. World Scientific, Singapore, 1999
7 Chunguang Li, Guanrong Chen, Xiaofeng Liao, Juebang Yu. Hopf bifurcation in an Internet congestion control model. Chaos, Solitons and Fractals 19 (2004) 853-862
8 韩福景，杨绍普，郭京波．参数激励下受电弓系统的分岔与混沌，石家庄铁道学院学报，2004，17(1)：25-29
9 陈少华，孙明光．油轮“鱼尾”运动及Hopf分岔算法，海洋工程，1994，12(1)：26-41
10 刘菲，杨翊仁．不可压缩流中二元翼运动的分支问题，西南交通大学学报，2002，37(增刊)：95-97
11 陆启韶．分岔与奇异性．上海科技教育出版社，1995
12 Golubitsky M. Stewart I. and Schaeffer D G. Singularities and Groups in Bifurcation Theory. New York: Springer-Yerlag, 1988
13 Iooss G, Los JE. Quasi-genericity of bifurcations to high dimensional invariant tori for maps. Commun Math Phys, 1988. 119:453-500
14 J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. New York: Springer-Verlag, 1983
15 Yu, P. and Bi.Q.S. Analysis of non-linear dynamics and bifurcation of a double pendulum. Journal of Sound and Vibration, 217, 1998:691-736.
16 Q. Bi and P. Yu. Symbolic computation of normal forms for semi-simple cases Journal of Computational and Applied Mathematics, 1999, 102(2): 195-220.
17 J. H. Xie. Codimension two bifurcations and Hopf bifurcations of an impacting vibrating system. Applied Mathematics and Mechanics, 1996, 17: 65-75.
18 G. L. Wen. Codimension-2 Hopf bifurcation of a two-degree-of-freedom vibroimpact system, Journal of Sound and Vibration, 2001, 242: 475-485.
19 Furter J E. Hopf bifurcation at non-semisimple eigenvalues: a singularity theory approach. In international Series of Numerical Math. 104, E Allgower,K. Bohme&M. Golubitsky(eds.), Birkhauseer, 1992:135-145
20 Golubitsky M, Marsden J E, Stewart l, and Dellnitz M. The constrained Liapunov-Schmidt procedure and periodic orbits, Normal Form and Homoclinic Chaos. W F. Langford & W K. Nagata (eds.), Fields institute Communications 4, Amer. Math. Soc., Providence, 1995
21 Knobloch E. and Proctor M R E. The double Hopf bifurcation with 2:1 resonance. Proc R Soc. Lond. A, 1988, 415:61-90
22 Chossat P. and Dias F. The 1:2 resonance with 0(2) symmetry and its application to hydrodynamics. J. Nonlin. 1995, 5:105-129
23 Leblanc V G. and Langford W F. Classification and Unfolding of 1:2 Resonant Hopf Bifurcation. Arch. Ratimal Mech. Anal, 1996, 136:305-357
24 G.W. Luo, J.H. Xie. Bifurcations and chaos in a system with impacts. Physica D, 2001, 148: 183-200.
25 罗冠炜，谢建华．一类冲击振动系统在强共振条件下的亚谐分叉与Hopf分叉．爆炸与冲击，2003，23(1)：67-73
26 罗冠炜，谢建华，孙训方．高维映射的Hopf分叉分析及其在冲击振动系统中的应用．振动工程，1999，12(3)：360-366
27 高金锋，张晓．倍周期分岔和环面分岔对电力系统电压稳定性的影响．郑州大学学报(工学版)，2005，26(2)：27-31
28 H. Poincare. Les Methodes Nouvelles de la Mecanique Celeste, 1889, Paris: Gauthier-Villars.
29 A.D. Brjuno. Analytical forms of differential equations Ⅰ. Transactions of the Moscow Mathematical Society, 1971, 25:132-198.
30 A.D. Brjuno. Analytical forms of differential equations Ⅱ. Transactions of the Moscow Mathematical Society, 1972, 25:199-299.
31 V.I. Arnold. Mathematical Methods of Classical Mechanics. Berlin: Springer-Verlag, 1978
32 J. Moser. Stable and Random Motion in Dynamical Systems. Princeton, New Jersey: Hermann Weyl Lectures, 1973
33 T.L. Johnson and R. H. Rand. On the existence and bifurcation of minimal normal modes. Journal of Nonlinear Mechanics, 1979, 14:1-14.
34 M.N. Hamdon and T.D. Burton. Analysis of Forced Nonlinear Undamped Oscillators by a Time Transformation Method. Journal of Sound and Vibration, 1986, 110: 223-232.
35 L. Hsu. Analysis of Critical and Post-critical Behavior of Nonlinear Dynamical Systems by the Normal Form Method, Part Ⅰ: Normalization Formulate. Journal of Sound and Vibration, 1983, 89:169-181.
36 L. Hsu. Analysis of Critical and Post-critical Behavior of Nonlinear Dynamical Systems by the Normal Form Method, Part Ⅱ: Divergence and Flutter. Journal of Sound and Vibration, 1983, 89:181-194.
37 J.E. Marsden and M. Mccracken. The Hopf Bifurcation and Its Applications. Applied Mathematical Sciences. Berlin: Springer-Verlag, 1976, 19
38 Yu P. Computation of normal forms via a perturbation technique. Journal of Sound and Vibration, 1998, 211:19-38.
39 张伟，陈予恕，共轭算子法和非线性动力系统的高阶规范形，应用数学和力学， 1997,18(5):421-430.
40 杨绍普，郭文武．平均法与Hopf分叉的规范型系数．石家庄铁道学院学报，2000，13(3)：52-55
41 Takens F. Normal forms for certain singularities of vector fields. Ann. Inst. Fourier, 1973, 23:163-165
42 Ushiki S. Normal forms for singularities of vector fields. Japan. J. Appl. Math., 1984, 1:1-37
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