非线性动力学双参量奇异性方法及其工程应用
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摘要
动力系统的分岔理论与方法作为非线性动力学的重要组成部分,在工程技术领域中得到了广泛的应用。对于含参数系统,当参数变动并经过某些临界值时,系统的定性性态会发生突然变化——分岔。近年来,人们提出了多种研究动力系统分岔问题的方法。在定性分析方面,奇异性理论得到了广泛的重视和应用。奇异性理论是研究约化方程分岔特性的一种有效而全面的方法,它使我们能够用统一的、明确的方法处理各种复杂的分岔问题。但是迄今为止奇异性理论在分岔问题中的应用主要集中在单状态变量、单分岔参数的动力系统中。随着科学技术的发展与进步,实际工程系统的动力学问题涉及到多参数、多状态变量的复杂系统,这使得非线性动力学的分岔研究面临挑战。
本文针对两状态变量、两分岔参数动力系统进行了分岔研究,并依之分析了输电线路的舞动机理及其优化控制等工程问题,以及一类生化反应系统的分岔问题:
1.对于实际的动力学系统,往往有很多的参数,究竟选哪个参数作为分岔参数,哪个参数会对解的结构引起定性的变化,是十分重要的问题。本文给出了一种选择主要分岔参数的方法。当系统参数受到小扰动时,系统的解结构可能会发生变化,而这一变化可通过系统的Frechet导数矩阵的特征值反映出来。因此将Frechet导数矩阵的特征值在临界值附近展开后可讨论参数变化对特征根的影响。对于特征根为单根和半简的情况,该方法尤为简单。对于特征根为亏损的情况,该方法虽略复杂,但同样适用。另外该方法还可推广到具有周期系数的动力系统中。
2.对于多自由度系统,常存在内共振。人们通常通过消元法、比值法、消元法与比值法相结合等方法将分岔方程约化为单状态变量分岔方程。研究发现,将多状态变量方程约化为一个状态变量系统以后会丢失了一些分岔特性。因此本文将奇异性理论的基本思想推广到了两状态变量系统的分岔分析之中。而对于多参数系统,例如化工系统、电力系统等所含有的实际物理参数很多,而且一些参数具有相同的地位,也就是说这些参数的变化都可能引起系统动力学行为的定性变化——分岔,因此都可作为分岔参数。本文将奇异性理论的基本思想推广到了两分岔参数系统的分岔分析之中,并给出了含有两分岔参数系统的转迁集的计算方法。
3.将两状态变量系统的奇异性理论应用到了输电线路舞动系统之中。应用Hamilton原理建立了输电线路舞动二维模型,其中考虑了变形引起的几何非线性以及空气流引起的空气动力非线性。通过多尺度法得到其分岔方程。应用奇异性理论的到其转迁集,可以看到在不同的参数区域,系统会出现分岔、滞后等不同的分岔模式。分岔、跳跃等现象都可能会引起输电线路的张力的突然变化,这对输电线路的强度来说是不利的,极可能造成输电线路的破坏。另外对输电线路舞动的一维模型进行了研究,得到其起舞的临界风速及振动幅值的解析解。还考虑了扭转对输电线路舞动的影响。为了评估防舞器的防舞效果,对加压重防舞器、动力减震器以及失谐摆的舞动优化控制技术进行了分析。因此本论文为输电线路设计和舞动控制奠定了理论基础。
4.分别将单分岔参数、两分岔参数系统的奇异性理论应用到了Duffing-van der Pol系统之中。比较发现,两分岔参数系统的分岔特性比单参数分岔系统的分岔特性要丰富很多,因此对于多参数系统,尤其几个分岔参数同样重要时,单将一个参数作为分岔参数是不够的。另外将两分岔参数系统的奇异性理论应用到了一类生化反应系统当中,分析了其分岔特性。
Bifurcation theory and methods of the dynamical systems are the important parts of nonlinear dynamics and widely applied in the eigneering fields. For the systems with parameters, when the parameters change, the dynamical behavior may be aroused change—bifurcation. In theses years, many methods for bifurcation analysis of the dynamical systems have been proposed. Among theses methods, singularity theory is of much imporatance and has been widely applied as a quanlative analysis method. Singularity theory is an effective method to study the reduced eqations of the dynamical systems, which can solve the bifurcation problems uniformly and definitely. But up to now, singularity theory is mainly applied in the dynamical systems with one bifurcation parameter and one state variable. As the development of the science technology, there are more and more dynamical systems with multiple bifurcation parameters and multiple state variables. Therefore, the bifurcation analysis of such systems is challenged.
In this dissertation, we pay our attention to the bifurcations of the dynamical systems with two bifurcation parameters and the ones with two state variables. The mechanics of the galloping and optimal control of the transmission line are analyzed, and the bifurcation of a class of biochemical reaction model is studied.
1. For the actual systems, there are many structural parameters. Which parameter can be considered as bifurcation parameter and which parameter will arouse the change of the solution structure of the system are two important issues. In this dissertation, a method to find the main bifurcation parameter of the dynamical systems is given. As known that when the parameter is subject to some small perturbations the solution structure maybe changes and this change can be reflected by the eigenvalues of the Frechet derivatives matrix of the system. Therefore, expanding the eigenvalues of the Frechet derivatives matrix near the critical value, the effects of the parameters can be discussed. For the cases of simple eigenvalue and semi-simple eigenvalue, this method is easy to operate. For the case of defective eigenvalue, although this method has some complexity, it is applicable as well. Furhermore, this method can be extended to the dynamical systems with periodic coefficients.
2. For the system with multiple DOFs, there maybe exists internal resonance. Usually the bifurcation equations can be reduced to the one with one state variable by the elimination method, the proportion method or combine of these two methods. After study, it can be found that some bifurcation properties are lost if the system was reduced. Therefore the singularity theory is developed to the bifurcation analysis of the dynamical systems with two state variables. For the systems with multiple parameters, such as chemical systems and power systems, there are many physical parameters and some parameters are of the same importance, i.e. the change of each important parameter maybe arouse the change of the dynamical behavior—bifurcation. Therefore both two parameters may be considered as bifurcation parameters. In this dissertation the singularity theory is developed to the bifurcation analysis of the dynamical systems with two parameters and the transition sets are given.
3. Singularity theory with two state variables is applied in the galloping of the transmission line. The model of the transmission line with two DOFs is constructed by using Hamilton principle after considering the initial location, the geometric nonlinearity caused by the deformation and the aerodynamic nonlinearity caused by the flow. The bifurcation equations are obtained by multiscale method. After singularity analysis the transition sets of the system are obtained. It is found that in different persistent regions there exist different bifurcation and hysteresis modals. As known that bifurcation and hysteresis modals maybe arouse the abrupt changes of the tension of the transimission line which are disadvantage. The model of the transmission line with one DOF is studied. The critical wind speed and the analytical solution of the amplitude of the transmission line are obtained. Except that, the effects of the torsional motion are considered. For anti-galloping, the optimal control of the masses, dynamic vibration absorber and detuning pendulum to the transmission line are studied, which can provide a theoretical basis for the design and control of the transmission line.
4. The singularity theory with one bifurcation parameter and two parameters are both applied in Duffing-van der Pol system under multi-frequency excitations. After comparison, it can be found that the bifurcation properties of the system with two bifurcation parameters are much more than the system with one parameter. So for the system with multiple structural parameters, especially some parameters are of the same importance, only one parameter is considered as bifurcation parameter is not enough to bifurcation analysis. Additionally, the singularity theory with two bifurcation parameters is applied in a class of biochemical reaction model and the bifurcation properties are analyzed.
本文针对两状态变量、两分岔参数动力系统进行了分岔研究,并依之分析了输电线路的舞动机理及其优化控制等工程问题,以及一类生化反应系统的分岔问题:
1.对于实际的动力学系统,往往有很多的参数,究竟选哪个参数作为分岔参数,哪个参数会对解的结构引起定性的变化,是十分重要的问题。本文给出了一种选择主要分岔参数的方法。当系统参数受到小扰动时,系统的解结构可能会发生变化,而这一变化可通过系统的Frechet导数矩阵的特征值反映出来。因此将Frechet导数矩阵的特征值在临界值附近展开后可讨论参数变化对特征根的影响。对于特征根为单根和半简的情况,该方法尤为简单。对于特征根为亏损的情况,该方法虽略复杂,但同样适用。另外该方法还可推广到具有周期系数的动力系统中。
2.对于多自由度系统,常存在内共振。人们通常通过消元法、比值法、消元法与比值法相结合等方法将分岔方程约化为单状态变量分岔方程。研究发现,将多状态变量方程约化为一个状态变量系统以后会丢失了一些分岔特性。因此本文将奇异性理论的基本思想推广到了两状态变量系统的分岔分析之中。而对于多参数系统,例如化工系统、电力系统等所含有的实际物理参数很多,而且一些参数具有相同的地位,也就是说这些参数的变化都可能引起系统动力学行为的定性变化——分岔,因此都可作为分岔参数。本文将奇异性理论的基本思想推广到了两分岔参数系统的分岔分析之中,并给出了含有两分岔参数系统的转迁集的计算方法。
3.将两状态变量系统的奇异性理论应用到了输电线路舞动系统之中。应用Hamilton原理建立了输电线路舞动二维模型,其中考虑了变形引起的几何非线性以及空气流引起的空气动力非线性。通过多尺度法得到其分岔方程。应用奇异性理论的到其转迁集,可以看到在不同的参数区域,系统会出现分岔、滞后等不同的分岔模式。分岔、跳跃等现象都可能会引起输电线路的张力的突然变化,这对输电线路的强度来说是不利的,极可能造成输电线路的破坏。另外对输电线路舞动的一维模型进行了研究,得到其起舞的临界风速及振动幅值的解析解。还考虑了扭转对输电线路舞动的影响。为了评估防舞器的防舞效果,对加压重防舞器、动力减震器以及失谐摆的舞动优化控制技术进行了分析。因此本论文为输电线路设计和舞动控制奠定了理论基础。
4.分别将单分岔参数、两分岔参数系统的奇异性理论应用到了Duffing-van der Pol系统之中。比较发现,两分岔参数系统的分岔特性比单参数分岔系统的分岔特性要丰富很多,因此对于多参数系统,尤其几个分岔参数同样重要时,单将一个参数作为分岔参数是不够的。另外将两分岔参数系统的奇异性理论应用到了一类生化反应系统当中,分析了其分岔特性。
Bifurcation theory and methods of the dynamical systems are the important parts of nonlinear dynamics and widely applied in the eigneering fields. For the systems with parameters, when the parameters change, the dynamical behavior may be aroused change—bifurcation. In theses years, many methods for bifurcation analysis of the dynamical systems have been proposed. Among theses methods, singularity theory is of much imporatance and has been widely applied as a quanlative analysis method. Singularity theory is an effective method to study the reduced eqations of the dynamical systems, which can solve the bifurcation problems uniformly and definitely. But up to now, singularity theory is mainly applied in the dynamical systems with one bifurcation parameter and one state variable. As the development of the science technology, there are more and more dynamical systems with multiple bifurcation parameters and multiple state variables. Therefore, the bifurcation analysis of such systems is challenged.
In this dissertation, we pay our attention to the bifurcations of the dynamical systems with two bifurcation parameters and the ones with two state variables. The mechanics of the galloping and optimal control of the transmission line are analyzed, and the bifurcation of a class of biochemical reaction model is studied.
1. For the actual systems, there are many structural parameters. Which parameter can be considered as bifurcation parameter and which parameter will arouse the change of the solution structure of the system are two important issues. In this dissertation, a method to find the main bifurcation parameter of the dynamical systems is given. As known that when the parameter is subject to some small perturbations the solution structure maybe changes and this change can be reflected by the eigenvalues of the Frechet derivatives matrix of the system. Therefore, expanding the eigenvalues of the Frechet derivatives matrix near the critical value, the effects of the parameters can be discussed. For the cases of simple eigenvalue and semi-simple eigenvalue, this method is easy to operate. For the case of defective eigenvalue, although this method has some complexity, it is applicable as well. Furhermore, this method can be extended to the dynamical systems with periodic coefficients.
2. For the system with multiple DOFs, there maybe exists internal resonance. Usually the bifurcation equations can be reduced to the one with one state variable by the elimination method, the proportion method or combine of these two methods. After study, it can be found that some bifurcation properties are lost if the system was reduced. Therefore the singularity theory is developed to the bifurcation analysis of the dynamical systems with two state variables. For the systems with multiple parameters, such as chemical systems and power systems, there are many physical parameters and some parameters are of the same importance, i.e. the change of each important parameter maybe arouse the change of the dynamical behavior—bifurcation. Therefore both two parameters may be considered as bifurcation parameters. In this dissertation the singularity theory is developed to the bifurcation analysis of the dynamical systems with two parameters and the transition sets are given.
3. Singularity theory with two state variables is applied in the galloping of the transmission line. The model of the transmission line with two DOFs is constructed by using Hamilton principle after considering the initial location, the geometric nonlinearity caused by the deformation and the aerodynamic nonlinearity caused by the flow. The bifurcation equations are obtained by multiscale method. After singularity analysis the transition sets of the system are obtained. It is found that in different persistent regions there exist different bifurcation and hysteresis modals. As known that bifurcation and hysteresis modals maybe arouse the abrupt changes of the tension of the transimission line which are disadvantage. The model of the transmission line with one DOF is studied. The critical wind speed and the analytical solution of the amplitude of the transmission line are obtained. Except that, the effects of the torsional motion are considered. For anti-galloping, the optimal control of the masses, dynamic vibration absorber and detuning pendulum to the transmission line are studied, which can provide a theoretical basis for the design and control of the transmission line.
4. The singularity theory with one bifurcation parameter and two parameters are both applied in Duffing-van der Pol system under multi-frequency excitations. After comparison, it can be found that the bifurcation properties of the system with two bifurcation parameters are much more than the system with one parameter. So for the system with multiple structural parameters, especially some parameters are of the same importance, only one parameter is considered as bifurcation parameter is not enough to bifurcation analysis. Additionally, the singularity theory with two bifurcation parameters is applied in a class of biochemical reaction model and the bifurcation properties are analyzed.
引文
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63 Liu Z S, Chen S H. An accurate method for computing eigenvector derivatives for free-free structures. Int. Journal of Computers and Structures. 1994, 52(6): 1135-1143
64韩万芝,宋大同,陈塑寰.计算特征向量摄动量的混合基展开法.固体力学学报. 1995, 12(2): 52-56
65陈塑寰.结构动态设计的矩阵摄动理论.科学出版社.北京. 1999
66 Seyranian A P, Mailybaev A A. Multiparameter Stability Theory with Mechanical Application. World Scientific. Singapore. 2003
67李欣业.多自由度内共振系统的非线性模态及其分岔.博士论文.天津大学. 2000
68杨前彪,徐鉴.水平悬臂输液管的内共振现象.第八届全国振动理论及应用学术会议论文集.上海. 2003
69唐友刚,郑俊武,董艳秋等.船舶内共振动力学行为的研究.中国造船. 1998, 4(143): 19-26
70李养成.光滑映射的奇点理论.科学出版社.北京. 2002
71 M.I.Kazakevich, A.G.Vasilenko. Closed analytical solution for galloping aeroelastic self-oscillations. Journal of Wind Engineering and Industrial Aerodynamics. 1996, 65(1-3): 353-360
72 O.Nigol, P.G.Buchan. Conductor galloping partⅠ: Den Hartog mechanism. IEEE Transmission on Power Apparatus and Systems. 1981, 100(2): 699-707
73 O.Nigol, P.G.Buchan. Conductor galloping partⅡ: torsional Mechanism. IEEE Transmission on Power Apparatus and Systems. 1981, 100(2): 708-720
74 P.Yu, N.Popplewell, A.H.Shah. Instability trends of inertially coupled galloping partⅡ: periodic vibration. Journal of Sound and Vibration. 1995, 183(4): 679-691
75 Y.M.Desai, P.Yu, A.H.Shah, N.Popplewell. Perturbation-based finite element analysis of transmission line galloping. Journal of Sound and Vibration. 1996, 191(4): 469-489
76 P.Yu, A.H.Shah, N.Popplewell. Inertially coupled of iced conductors. Transactions of the ASME. 1992,59: 140-145
77 John H.G.Macdonald, Guy L.Larose. Two-degree-of-freedom inclined cable galloping part 2: analysis and prevention for arbitrary frequency ratio. Journal of Wind Engineering and Industrial Aerodynamics. 2008, 96(3): 308-326
78 John H.G.Macdonald, Guy L.Larose. Two-degree-of-freedom inclined cable galloping part 1:general formulation and solution for perfectly tuned system. Journal of Wind Engineering and Industrial Aerodynamics. 2008, 96(3): 291-307
79 H.Jicai, Z.Song, M.Jianguo, W.Shijing. Model for comprehensive simulation of overhead high voltage power transmission line galloping and protection. IEEE Conference on Electrical Insulation and Dielectric Phenomena. 2006: 190-193
80 J.-L., W.Jianwei, L.Lien. Overhead electrical transmission line galloping. A full multi-span 3-DOF model, some applications and design recommendations. IEEE Transactions on Power Delivery. 1998, 13(3): 909-916
81郭应龙,李国兴,尤传永.输电线路舞动.中国电力出版社.北京. 2003
82赵高煜.大跨越高压输电线路分裂导线覆冰舞动的研究.博士论文.华中科技大学. 2005
83何锃,赵高煜.分裂导线扭转舞动分析的动力学建模.工程力学, 2001, 18(2): 126-134
84雷川丽,段炜佳,侯镭等.架空输电线舞动的计算机仿真.高电压技术. 2007, 23(10): 178-183
85楼文娟,孙珍茂,吕翼.扰流防舞器与气动阻尼片的防舞效果.电网技术, 2010, 34(2): 200-204
86肖晓晖,郭应龙,吴晶.压重防舞器配置方案有效性的仿真计算.电力建设, 1998, (6): 25-28
87郭应龙,李国兴等.输电线路舞动.北京:中国电力出版社, 2003
88胡德山,苑舜,陶文秋.阻尼失谐摆防舞器的研究.东北电力技术, 2009, (3): 13-16
89陈兰荪,王东达.一个生物化学反应的振动现象.数学物理学报. 1985, 3: 261-266
90沃松林.一个生物化学反应模型的定性分析.生物数学学报. 1995, 10(2): 92-96
91王洪礼,郭树起.双CSTR连续发酵及其混沌行为.振动工程学报. 1999, 12(4): 468-474
92柴俊,张正娣.三变量CSTR化学反应的复杂动力学行为分析.动力学与控制学报. 2007, 5(1): 34-38
93成志清,烫德满.速度型切削颤振的非线性分析.南京航空学院学报, 1991, 23(1): 46-55
94刘习军,陈予恕.机床速度型切削颤振的非线性研究.振动与冲击. 1999, 18(2): 5-10
95刘习军,王立刚,贾启芬.一种由于摩擦引起的车床切削颤振.工程力学. 2005, 22(1): 107-112
96肖炏,郭永基,唐云,廖浩辉.典型电力系统模型的双参数分岔分析.电力系统自动化. 2000, 24(6): 1-6
97蒋平,顾伟,严伟佳,唐国庆.基于多参数分岔分析方法的多机系统动态负荷裕度研究.电工技术学报. 2007, 22(3): 107-114
98 Nafyeh A H, Mook D L. Nonlinear oscillations. New York: Wiley Interscience. 1979. 325-328
99陈予恕.非线性振动.北京:高等教育出版社. 2002. 201-208
100 Holmes P, Rand D. Phase portraits and bifurcation of nonlinear oscillator: ( )x +α+γx 2 x +βx +δx3 = 0. International Journal of Nonlinear Mechanics. 1980. 15(6): 449-458
101 Maccari Attilio. Approximate solution of a class of nonlinear oscillatiors in resonance with a periodic excitation. Nonlinear Dynamics. 1998. 15(4): 329-343
102董建宁,申永军,杨绍普.多频激励作用下Duffing-van der Pol系统的分岔分析.石家庄铁道学院学报. 2006. 19(1): 62-66
103王连球,王永亮.一类可逆生化反应模型的定性分析.湖南工程学院学报. 2001, 11(1): 92-94
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50叶敏,陈予恕.非线性参数激励系统的动力分岔研究.力学学报. 1995, 25(2): 169-175
51杨绍普,陈予恕.一类单自由度滞后-自激振动系统的分叉.振动工程学报. 1991, 4(2): 95-101
52金栋平,陈予恕.滞后非线性系统的分岔和奇异性.天津大学学报. 1997, 30(3): 299-304
53张雪峰,李韶华,杨绍普. Van der Pol滞后系统在多频激励下的动力学分析.石家庄铁道学院学报. 2002, 15(3): 36-40
54陈芳启,吴志强,陈予恕.黏弹性圆柱形壳动力学高余维分岔、普适开折问题.力学学报. 2001, 33(5): 661-668
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57 Chen Y S, Langford W F. The subharmonic bifurcation solution of nonlinear Mathieu’s equation and Euler dynamic bulking problems. Acta Mechanica Sinica. 1988, 4(4): 350-362
58朱照宣.非线性动力学中的浑沌.力学进展. 1984, 14(2):129-146
59胡海昌.多自由度结构固有振动理论.科学出版社.北京1987
60 William B B. An improved computational technique for perturbations of the generalized symmetric linear algebraic eigenvalue problem. InternationalJournal for Numerical Methods in Engineering. 1987, 24:529-541
61陈塑寰,徐涛.线性振动亏损系统的矩阵摄动理论.力学学报. 1992, 24(6): 747-752
62陈塑寰,刘中生.振型一阶导数的高精度截尾模态展开法.力学学报. 1993, 25(4): 427-434
63 Liu Z S, Chen S H. An accurate method for computing eigenvector derivatives for free-free structures. Int. Journal of Computers and Structures. 1994, 52(6): 1135-1143
64韩万芝,宋大同,陈塑寰.计算特征向量摄动量的混合基展开法.固体力学学报. 1995, 12(2): 52-56
65陈塑寰.结构动态设计的矩阵摄动理论.科学出版社.北京. 1999
66 Seyranian A P, Mailybaev A A. Multiparameter Stability Theory with Mechanical Application. World Scientific. Singapore. 2003
67李欣业.多自由度内共振系统的非线性模态及其分岔.博士论文.天津大学. 2000
68杨前彪,徐鉴.水平悬臂输液管的内共振现象.第八届全国振动理论及应用学术会议论文集.上海. 2003
69唐友刚,郑俊武,董艳秋等.船舶内共振动力学行为的研究.中国造船. 1998, 4(143): 19-26
70李养成.光滑映射的奇点理论.科学出版社.北京. 2002
71 M.I.Kazakevich, A.G.Vasilenko. Closed analytical solution for galloping aeroelastic self-oscillations. Journal of Wind Engineering and Industrial Aerodynamics. 1996, 65(1-3): 353-360
72 O.Nigol, P.G.Buchan. Conductor galloping partⅠ: Den Hartog mechanism. IEEE Transmission on Power Apparatus and Systems. 1981, 100(2): 699-707
73 O.Nigol, P.G.Buchan. Conductor galloping partⅡ: torsional Mechanism. IEEE Transmission on Power Apparatus and Systems. 1981, 100(2): 708-720
74 P.Yu, N.Popplewell, A.H.Shah. Instability trends of inertially coupled galloping partⅡ: periodic vibration. Journal of Sound and Vibration. 1995, 183(4): 679-691
75 Y.M.Desai, P.Yu, A.H.Shah, N.Popplewell. Perturbation-based finite element analysis of transmission line galloping. Journal of Sound and Vibration. 1996, 191(4): 469-489
76 P.Yu, A.H.Shah, N.Popplewell. Inertially coupled of iced conductors. Transactions of the ASME. 1992,59: 140-145
77 John H.G.Macdonald, Guy L.Larose. Two-degree-of-freedom inclined cable galloping part 2: analysis and prevention for arbitrary frequency ratio. Journal of Wind Engineering and Industrial Aerodynamics. 2008, 96(3): 308-326
78 John H.G.Macdonald, Guy L.Larose. Two-degree-of-freedom inclined cable galloping part 1:general formulation and solution for perfectly tuned system. Journal of Wind Engineering and Industrial Aerodynamics. 2008, 96(3): 291-307
79 H.Jicai, Z.Song, M.Jianguo, W.Shijing. Model for comprehensive simulation of overhead high voltage power transmission line galloping and protection. IEEE Conference on Electrical Insulation and Dielectric Phenomena. 2006: 190-193
80 J.-L., W.Jianwei, L.Lien. Overhead electrical transmission line galloping. A full multi-span 3-DOF model, some applications and design recommendations. IEEE Transactions on Power Delivery. 1998, 13(3): 909-916
81郭应龙,李国兴,尤传永.输电线路舞动.中国电力出版社.北京. 2003
82赵高煜.大跨越高压输电线路分裂导线覆冰舞动的研究.博士论文.华中科技大学. 2005
83何锃,赵高煜.分裂导线扭转舞动分析的动力学建模.工程力学, 2001, 18(2): 126-134
84雷川丽,段炜佳,侯镭等.架空输电线舞动的计算机仿真.高电压技术. 2007, 23(10): 178-183
85楼文娟,孙珍茂,吕翼.扰流防舞器与气动阻尼片的防舞效果.电网技术, 2010, 34(2): 200-204
86肖晓晖,郭应龙,吴晶.压重防舞器配置方案有效性的仿真计算.电力建设, 1998, (6): 25-28
87郭应龙,李国兴等.输电线路舞动.北京:中国电力出版社, 2003
88胡德山,苑舜,陶文秋.阻尼失谐摆防舞器的研究.东北电力技术, 2009, (3): 13-16
89陈兰荪,王东达.一个生物化学反应的振动现象.数学物理学报. 1985, 3: 261-266
90沃松林.一个生物化学反应模型的定性分析.生物数学学报. 1995, 10(2): 92-96
91王洪礼,郭树起.双CSTR连续发酵及其混沌行为.振动工程学报. 1999, 12(4): 468-474
92柴俊,张正娣.三变量CSTR化学反应的复杂动力学行为分析.动力学与控制学报. 2007, 5(1): 34-38
93成志清,烫德满.速度型切削颤振的非线性分析.南京航空学院学报, 1991, 23(1): 46-55
94刘习军,陈予恕.机床速度型切削颤振的非线性研究.振动与冲击. 1999, 18(2): 5-10
95刘习军,王立刚,贾启芬.一种由于摩擦引起的车床切削颤振.工程力学. 2005, 22(1): 107-112
96肖炏,郭永基,唐云,廖浩辉.典型电力系统模型的双参数分岔分析.电力系统自动化. 2000, 24(6): 1-6
97蒋平,顾伟,严伟佳,唐国庆.基于多参数分岔分析方法的多机系统动态负荷裕度研究.电工技术学报. 2007, 22(3): 107-114
98 Nafyeh A H, Mook D L. Nonlinear oscillations. New York: Wiley Interscience. 1979. 325-328
99陈予恕.非线性振动.北京:高等教育出版社. 2002. 201-208
100 Holmes P, Rand D. Phase portraits and bifurcation of nonlinear oscillator: ( )x +α+γx 2 x +βx +δx3 = 0. International Journal of Nonlinear Mechanics. 1980. 15(6): 449-458
101 Maccari Attilio. Approximate solution of a class of nonlinear oscillatiors in resonance with a periodic excitation. Nonlinear Dynamics. 1998. 15(4): 329-343
102董建宁,申永军,杨绍普.多频激励作用下Duffing-van der Pol系统的分岔分析.石家庄铁道学院学报. 2006. 19(1): 62-66
103王连球,王永亮.一类可逆生化反应模型的定性分析.湖南工程学院学报. 2001, 11(1): 92-94