电磁场中杆和板的分岔和混沌运动
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摘要
分岔和混沌是非线性系统最重要而又最基本的特性,几乎在所有涉及非线性科学的领域中,都存在着分岔现象和混沌运动。论文在分析和总结非线性微分动力系统分岔和混沌研究现状的基础上对电磁场中的细长压杆和导电梁式板的非线性行为进行了研究。论文主要研究了以下几方面的内容
首先,总结了当前国内外学者对分岔现象及混沌运动问题的研究现状,阐述了产生叉形分岔、Hopf分岔和鞍结分岔三种平衡点分岔和闭轨分岔、同宿轨(异宿轨)分岔的条件。给出了Hamilton系统的同宿轨及异宿轨的计算方法。阐述了Melnikov方法,并给出了几种常用的Melnikov函数。
其次,对电磁场中细长压杆进行受力分析,得到了电磁场作用下细长压杆的运动方程,并针对静力学、线性动力学和非线性动力学三种模型,进行了分岔特性研究。
最后,用Melnikov方法研究了在横向磁场和机械载荷共同作用下的梁式板的非线性行为,首先对无扰动系统进行了分析,由无扰系统是Hamilton系统,根据其Hamilton能量函数求得了它的闭轨道参数方程,并给出了振幅和其系统能量之间的关系,并进一步计算了扰动系统出现次谐分岔的Melnikov函数,由Melnikov函数存在简单零点得知该系统存在Smale马蹄变换意义下的混沌,并进一步得到存在混沌运动时横向磁场强度与外界机械力幅之间的关系。
Bifurcation and Chaos is the most important and most basic features in nonlinear systems,almost all involved in the field of nonlinear science, there is a bifurcation phenomena and chaotic motion. Based on the analysis and summary of nonlinear differential bifurcation and chaotic dynamical systems on the basis of the status quo in the electromagnetic conductivity slender bar and beam-plate nonlinear behavior were studied. This paper mainly on the following aspects of content.
Firstly, summed up the current domestic and foreign scholars on the bifurcation phenomenon of chaotic motion and the issue of the status quo, have fork bifurcation expounded, Hopf bifurcation and saddle-node bifurcation balance bifurcation and three closed orbit bifurcation, homoclinic orbit (heteroclinic track) bifurcation conditions. Hamilton systems and gives the homoclinic tracks and the heteroclinic orbit calculation. Melnikov function briefly introduced the solution.
Secondly, the electromagnetic field in a slender bar Analysis, have been under slender bar electromagnetic field equations of motion and against static mechanics, linear dynamics and nonlinear dynamics three models, a study of bifurcation.
Finally, Melnikov method used in the horizontal magnetic field and mechanical loads under common beam-plate nonlinear behavior, first of all, on non-disturbance system were analyzed by the system is unperturbed Hamiltonian system, in accordance with its energy function Hamilton obtained its close orbital parameters equation, and gives the amplitude and its system the relationship between volume and further disturbances system in the calculation of the bifurcation of the market Melnikov function, the simple existence of Melnikov function of the system that existed Smale horseshoe transform the sense of chaos, and further chaos when transverse magnetic field presence with the outside world mechanical strength of the relationship between the rate.
首先,总结了当前国内外学者对分岔现象及混沌运动问题的研究现状,阐述了产生叉形分岔、Hopf分岔和鞍结分岔三种平衡点分岔和闭轨分岔、同宿轨(异宿轨)分岔的条件。给出了Hamilton系统的同宿轨及异宿轨的计算方法。阐述了Melnikov方法,并给出了几种常用的Melnikov函数。
其次,对电磁场中细长压杆进行受力分析,得到了电磁场作用下细长压杆的运动方程,并针对静力学、线性动力学和非线性动力学三种模型,进行了分岔特性研究。
最后,用Melnikov方法研究了在横向磁场和机械载荷共同作用下的梁式板的非线性行为,首先对无扰动系统进行了分析,由无扰系统是Hamilton系统,根据其Hamilton能量函数求得了它的闭轨道参数方程,并给出了振幅和其系统能量之间的关系,并进一步计算了扰动系统出现次谐分岔的Melnikov函数,由Melnikov函数存在简单零点得知该系统存在Smale马蹄变换意义下的混沌,并进一步得到存在混沌运动时横向磁场强度与外界机械力幅之间的关系。
Bifurcation and Chaos is the most important and most basic features in nonlinear systems,almost all involved in the field of nonlinear science, there is a bifurcation phenomena and chaotic motion. Based on the analysis and summary of nonlinear differential bifurcation and chaotic dynamical systems on the basis of the status quo in the electromagnetic conductivity slender bar and beam-plate nonlinear behavior were studied. This paper mainly on the following aspects of content.
Firstly, summed up the current domestic and foreign scholars on the bifurcation phenomenon of chaotic motion and the issue of the status quo, have fork bifurcation expounded, Hopf bifurcation and saddle-node bifurcation balance bifurcation and three closed orbit bifurcation, homoclinic orbit (heteroclinic track) bifurcation conditions. Hamilton systems and gives the homoclinic tracks and the heteroclinic orbit calculation. Melnikov function briefly introduced the solution.
Secondly, the electromagnetic field in a slender bar Analysis, have been under slender bar electromagnetic field equations of motion and against static mechanics, linear dynamics and nonlinear dynamics three models, a study of bifurcation.
Finally, Melnikov method used in the horizontal magnetic field and mechanical loads under common beam-plate nonlinear behavior, first of all, on non-disturbance system were analyzed by the system is unperturbed Hamiltonian system, in accordance with its energy function Hamilton obtained its close orbital parameters equation, and gives the amplitude and its system the relationship between volume and further disturbances system in the calculation of the bifurcation of the market Melnikov function, the simple existence of Melnikov function of the system that existed Smale horseshoe transform the sense of chaos, and further chaos when transverse magnetic field presence with the outside world mechanical strength of the relationship between the rate.
引文
1高普云编著.非线性动力学.国防科技大学出版社,2005:2-76
2刘式达,刘式适.非线性动力学和复杂现象.气象出版社,1989:228-237
3黄文虎主编.一般力学(动力学,振动与控制)最近进展.科学出版社,1994:87-106
4张琪昌,王洪礼.分岔与混沌理论及应用.天津大学出版社,2005:169-179
5 B. P. Bezruchko, R. N. Ivanov, V. I. Ponomarenko, E. P. Seleznev. Experimental Study of Bifurcations in Systems with Rapidly Varying Parameters. Technical Physics Letters, 2004,28(6):471-474
6 M. Golubitsky and D. Schaeffer. A Theory for Imperfect Bifurcation Theory Via Singularity Throty. Comm Pure Appl, 1979,31(2):21-98
7陈予恕.我国非线性振动研究的发展—十年回顾与今后展望(1979-1989).全国第五届非线性会议论文集,1989:756-778
8 M. Marek and M. Kubicek. Computational Methods in Bifurcation Theory and Dissipative Structures. Spinger-Verlag, 1983:226-236
9韩强编著.弹塑性系统的动力屈曲和分叉.科学出版社,2000:109-119
10黄润生,黄浩编著.混沌及其应用.武汉大学出版社,2005:33-51
11季颖,毕勤胜.非零平衡强非线性Vander-pol系统的奇异性分析及分岔.广西师范大学学报,2005,23(2):13-16
12林文惠.分岔理论的若干进展.北京大学博士论文,2002:4-17
13符文斌.非线性动力系统的分岔控制研究.湖南大学博士学位论文,2004:16-22
14 M. Golubitsky and G. Schaeffer. Singularities and Groups in Bifurcation Theory. Spinger Verlag, 1985,32(7):561-565
15刘秉正.非线性动力学与混沌基础.东北师范大学出版社,1995:48-65
16 P. Yu, W. Zhang, B. Qin.Vibration Analysis on a Thin Plate with the Aid of Computation of Normal Forms. International Journal Nonlinear Mechanics, 2001,36(4):597-627
17王树禾.微分方程模型与混沌.中国科学技术大学出版社,1999:78-94
18陈予恕,唐云编.非线性动力学中的现代分析方法.科学出版社,2000:162-209
19 P. Yu, Y. Yuan. The Simplest Normal Form for the Ingularity of a Pure Imaginary Pair and a Zero Eigenvalue Dynamics of Continuous Discrete and Impulsive Systems Series B. Applications & Algorithms, 2001,8:219-249
20朱因远,周纪卿.非线性振动和运动稳定性.西安交通大学出版社,2002:104-118
21李鸿光.若干工程非线性振动系统的动力学特性及分岔与混沌运动研究.东北大学博士论文,2003:17-25
22 W. X. Yuan. Research on the Relation of Chaos Activity Characteristics of Chaos Activity Characteristics of Cardiac System with the Evolution of Species. Chinese Science Bulletin, 2003,47(24):178-183
23陆启韶.分岔与奇异性.上海科技教育出版社,1995:1-46
24韩强,张善元,杨桂通.一类非线性动力系统混沌运动的研究.应用数学和力学,1999, 20(8):776-782
25迟东璇,汪锐.一类非线性动力系统的混沌研究.东北大学学报,2004,25(4):345-348
26 C. D. Andereck, S. S. Liu and H. L. Swinney. Flow Regimes in a Circular Ouette System with Independently Rotating Cylinders. Fluid Mech, 1986,(164):155-183
27王海滨,周叉和,郑晓静.超导磁体感应电流及其对电磁弹性动力稳定性的影响.核聚变与等离子体物理,2003,23(1):1-6
28 W. F. Langford, R. Tagg. Kostelich, H. Swinney and M. Golubitsky. Primaty Instability and Bicriticality in Flow Between Counter Rotating Cylinders. Phys Fluids, 1998,(31): 776-785
29叶建军.一类非线性结构混沌运动的研究.西南交通大学博士论文,2001:30-35
30杜建成.非线性板壳的分叉和混沌运动.华南理工大学硕士论文,2003:16-28
31 M. Golubitsky, I. Stewart and D. G. Schaeffer. Singularities and Groups in Bifurcation Theory. Spinger-Verlag, 1988:102-106
32王知人,王平,白象忠.载流薄板磁弹性动力失稳临界状态的判别.工程数学学报, 2006,23(1):133-138
33 D. H. Sattinger. Topics on Stability and Bifurcation Theory, Springer Lectures Notes in Mathematics. New York:Spinger-Verlag, 1973,309(6):69-73
34 G. Nicolis and I. Prigogine. Self-organization in Non-equilibrium Systems. John Wiley& Sons, 1977,67(2):556-565
35杨绍普,申永军编著.滞后非线性系统的分岔与奇异性.科学出版社,2003:11-15
36 R. Roche, and B. Austrusson. Experimental Tests on Buckling of Torispherical Heads and Methods of Plastic Bifurcation Analyses. Trans J Press, 1986:342-345
37 P. Shahrzad, and M. Mahzoon. Limit Cycle Flutter of Airfoils in Steady and Unsteady Flows. Journal of Sound and Vibration, 2002,256(2):117-121
38陈予恕编著.非线性振动.天津科学技术出版社,2003.7
39 P. Y. Yuan. Computation of the Simplest Normal Forms with Perturbation Parameters Based on Lie Transform and Rescaling. Journal of Compute and Applied Mathematics, 2002,144(2):359-373
40韩强,黄小清,宁建国编著.高等板壳理论.科学出版社,2002:120-150
41陈爱利,王广瓦.一类三阶非线性常微分方程零解的稳定性.徐州师范大学学报, 2006,24(1):23-26
42 S. N. Chow, C. Z Lin, D Wang. Normal Forms and Bifurcations of Planar Vector Fields. Cambridge University Press, 1994:378-388
43 D. H. Sattinger. Branching in the Presence of Symmetry, CBMSNSF Conference Notes. Slam Press, 1983:779-782
44 S. Y. Lee and Y. H. Kuo. Non-conservative Stability of Intermediated Spring Supported Colimns with an Elastically Restrained and Support Solids Structures. Phys Fluids, 2004,28(9):1129-1137
45李继彬,赵晓华,刘正荣.广义哈密顿系统理论及其应用.科学出版社,1997:106-119
46 J. Williamson. On an Algebraic Problem, Concerning the Normal Forms of Linear Dynamical Systems. Amer of Math, 1963,58:141-163
47龙运佳.混沌振动研究方法与实践.清华大学出版社,1997:23-26
48陈占清,孙明贵等.从非线性动力学的视角认识细长压杆的稳定性.力学与实践,2005,27(2):40-43
49厉彦菊,陈占清.秦越.细长受压杆的分岔特性研究.山东科技,2006,19(1):6-10
50 R. I. Bogdanov. Versal Deformations of a Singular Point on the Plane in the Case ofZero Eigenvalues. Functional Anal Appl, 1975,9:144-145
51胡宇达,姜鑫,刘跃伟.横向磁场中机械载荷作用梁式薄板的非线性主共振.振动与冲击,2006,25(4):88-90
52胡宇达,杜国君,王国伟.电磁与机械载荷作用下导电梁式板的超谐波共振.动力学与控制学报,2004,4(2):35-37
53高本庆.椭圆函数及其应用.国防工业出版社,1991:83-98
54王竹溪,郭敦仁.特殊函数论.科学出版社,1979:367-380
55刘式适,刘式达.特殊函数.气象出版社,2002:634-673
56张年梅,杨桂通.非线性弹性梁的动态次谐分叉与混沌运动.非线性动力学报,1996, 3(2):265-274
57邱平,王新志,叶开沅.非线性弹性地基上的圆薄板的分岔与混沌问题.应用数学和力学,2003,24(8):779-784
58 P. F. Byrd and M. Duo. Frechman.Handbook of Elliptic Intergrals for Engineers and Physicsts. New York:Springer-Verlag, 1954
59刘式适,刘式达.物理学中的非线性方程.北京大学出版社,2001:388-390
2刘式达,刘式适.非线性动力学和复杂现象.气象出版社,1989:228-237
3黄文虎主编.一般力学(动力学,振动与控制)最近进展.科学出版社,1994:87-106
4张琪昌,王洪礼.分岔与混沌理论及应用.天津大学出版社,2005:169-179
5 B. P. Bezruchko, R. N. Ivanov, V. I. Ponomarenko, E. P. Seleznev. Experimental Study of Bifurcations in Systems with Rapidly Varying Parameters. Technical Physics Letters, 2004,28(6):471-474
6 M. Golubitsky and D. Schaeffer. A Theory for Imperfect Bifurcation Theory Via Singularity Throty. Comm Pure Appl, 1979,31(2):21-98
7陈予恕.我国非线性振动研究的发展—十年回顾与今后展望(1979-1989).全国第五届非线性会议论文集,1989:756-778
8 M. Marek and M. Kubicek. Computational Methods in Bifurcation Theory and Dissipative Structures. Spinger-Verlag, 1983:226-236
9韩强编著.弹塑性系统的动力屈曲和分叉.科学出版社,2000:109-119
10黄润生,黄浩编著.混沌及其应用.武汉大学出版社,2005:33-51
11季颖,毕勤胜.非零平衡强非线性Vander-pol系统的奇异性分析及分岔.广西师范大学学报,2005,23(2):13-16
12林文惠.分岔理论的若干进展.北京大学博士论文,2002:4-17
13符文斌.非线性动力系统的分岔控制研究.湖南大学博士学位论文,2004:16-22
14 M. Golubitsky and G. Schaeffer. Singularities and Groups in Bifurcation Theory. Spinger Verlag, 1985,32(7):561-565
15刘秉正.非线性动力学与混沌基础.东北师范大学出版社,1995:48-65
16 P. Yu, W. Zhang, B. Qin.Vibration Analysis on a Thin Plate with the Aid of Computation of Normal Forms. International Journal Nonlinear Mechanics, 2001,36(4):597-627
17王树禾.微分方程模型与混沌.中国科学技术大学出版社,1999:78-94
18陈予恕,唐云编.非线性动力学中的现代分析方法.科学出版社,2000:162-209
19 P. Yu, Y. Yuan. The Simplest Normal Form for the Ingularity of a Pure Imaginary Pair and a Zero Eigenvalue Dynamics of Continuous Discrete and Impulsive Systems Series B. Applications & Algorithms, 2001,8:219-249
20朱因远,周纪卿.非线性振动和运动稳定性.西安交通大学出版社,2002:104-118
21李鸿光.若干工程非线性振动系统的动力学特性及分岔与混沌运动研究.东北大学博士论文,2003:17-25
22 W. X. Yuan. Research on the Relation of Chaos Activity Characteristics of Chaos Activity Characteristics of Cardiac System with the Evolution of Species. Chinese Science Bulletin, 2003,47(24):178-183
23陆启韶.分岔与奇异性.上海科技教育出版社,1995:1-46
24韩强,张善元,杨桂通.一类非线性动力系统混沌运动的研究.应用数学和力学,1999, 20(8):776-782
25迟东璇,汪锐.一类非线性动力系统的混沌研究.东北大学学报,2004,25(4):345-348
26 C. D. Andereck, S. S. Liu and H. L. Swinney. Flow Regimes in a Circular Ouette System with Independently Rotating Cylinders. Fluid Mech, 1986,(164):155-183
27王海滨,周叉和,郑晓静.超导磁体感应电流及其对电磁弹性动力稳定性的影响.核聚变与等离子体物理,2003,23(1):1-6
28 W. F. Langford, R. Tagg. Kostelich, H. Swinney and M. Golubitsky. Primaty Instability and Bicriticality in Flow Between Counter Rotating Cylinders. Phys Fluids, 1998,(31): 776-785
29叶建军.一类非线性结构混沌运动的研究.西南交通大学博士论文,2001:30-35
30杜建成.非线性板壳的分叉和混沌运动.华南理工大学硕士论文,2003:16-28
31 M. Golubitsky, I. Stewart and D. G. Schaeffer. Singularities and Groups in Bifurcation Theory. Spinger-Verlag, 1988:102-106
32王知人,王平,白象忠.载流薄板磁弹性动力失稳临界状态的判别.工程数学学报, 2006,23(1):133-138
33 D. H. Sattinger. Topics on Stability and Bifurcation Theory, Springer Lectures Notes in Mathematics. New York:Spinger-Verlag, 1973,309(6):69-73
34 G. Nicolis and I. Prigogine. Self-organization in Non-equilibrium Systems. John Wiley& Sons, 1977,67(2):556-565
35杨绍普,申永军编著.滞后非线性系统的分岔与奇异性.科学出版社,2003:11-15
36 R. Roche, and B. Austrusson. Experimental Tests on Buckling of Torispherical Heads and Methods of Plastic Bifurcation Analyses. Trans J Press, 1986:342-345
37 P. Shahrzad, and M. Mahzoon. Limit Cycle Flutter of Airfoils in Steady and Unsteady Flows. Journal of Sound and Vibration, 2002,256(2):117-121
38陈予恕编著.非线性振动.天津科学技术出版社,2003.7
39 P. Y. Yuan. Computation of the Simplest Normal Forms with Perturbation Parameters Based on Lie Transform and Rescaling. Journal of Compute and Applied Mathematics, 2002,144(2):359-373
40韩强,黄小清,宁建国编著.高等板壳理论.科学出版社,2002:120-150
41陈爱利,王广瓦.一类三阶非线性常微分方程零解的稳定性.徐州师范大学学报, 2006,24(1):23-26
42 S. N. Chow, C. Z Lin, D Wang. Normal Forms and Bifurcations of Planar Vector Fields. Cambridge University Press, 1994:378-388
43 D. H. Sattinger. Branching in the Presence of Symmetry, CBMSNSF Conference Notes. Slam Press, 1983:779-782
44 S. Y. Lee and Y. H. Kuo. Non-conservative Stability of Intermediated Spring Supported Colimns with an Elastically Restrained and Support Solids Structures. Phys Fluids, 2004,28(9):1129-1137
45李继彬,赵晓华,刘正荣.广义哈密顿系统理论及其应用.科学出版社,1997:106-119
46 J. Williamson. On an Algebraic Problem, Concerning the Normal Forms of Linear Dynamical Systems. Amer of Math, 1963,58:141-163
47龙运佳.混沌振动研究方法与实践.清华大学出版社,1997:23-26
48陈占清,孙明贵等.从非线性动力学的视角认识细长压杆的稳定性.力学与实践,2005,27(2):40-43
49厉彦菊,陈占清.秦越.细长受压杆的分岔特性研究.山东科技,2006,19(1):6-10
50 R. I. Bogdanov. Versal Deformations of a Singular Point on the Plane in the Case ofZero Eigenvalues. Functional Anal Appl, 1975,9:144-145
51胡宇达,姜鑫,刘跃伟.横向磁场中机械载荷作用梁式薄板的非线性主共振.振动与冲击,2006,25(4):88-90
52胡宇达,杜国君,王国伟.电磁与机械载荷作用下导电梁式板的超谐波共振.动力学与控制学报,2004,4(2):35-37
53高本庆.椭圆函数及其应用.国防工业出版社,1991:83-98
54王竹溪,郭敦仁.特殊函数论.科学出版社,1979:367-380
55刘式适,刘式达.特殊函数.气象出版社,2002:634-673
56张年梅,杨桂通.非线性弹性梁的动态次谐分叉与混沌运动.非线性动力学报,1996, 3(2):265-274
57邱平,王新志,叶开沅.非线性弹性地基上的圆薄板的分岔与混沌问题.应用数学和力学,2003,24(8):779-784
58 P. F. Byrd and M. Duo. Frechman.Handbook of Elliptic Intergrals for Engineers and Physicsts. New York:Springer-Verlag, 1954
59刘式适,刘式达.物理学中的非线性方程.北京大学出版社,2001:388-390