电磁场中板、壳的振动分岔
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摘要
电磁固体力学是研究在弹性固态物质中,电磁场与变形场相互作用的理论。板壳磁弹性非线性振动问题在工程实际中比较常见,其控制方程的建立和求解具有很大难度,因此研究板壳磁弹性非线性振动具有重要的理论和实际意义。
论文首先在建立薄板磁弹性非线性振动方程的基础上,得到梁式板的非线性振动方程,运用中心流形定理和方法对其分岔行为进行了具体分析,画出了分岔图,并运用Melnikov函数方法得到发生全局分岔的条件。
其次,对梁式板的振动方程引入参数进行变换,运用多尺度方法求出梁式板主共振下的二阶定常解,用奇异性理论得到分岔方程并求出转迁集,导电梁式板在不同的转迁集的条件下发生分岔的条件是不相同的,求解可得到歧集和双极限点集内分岔条件是不存在,分岔只发生在滞后集内,在滞后集条件下求出了分岔条件。
最后,将两边简支的边界条件带入圆柱薄壳振动方程,得到了圆柱薄壳的带立方和平方项的振动微分方程,用多尺度方法求出主共振解的分岔方程,分析了系统运动中解的稳定性,运用奇异性理论得到分岔方程的转迁集,导电圆柱薄壳在不同的转迁集区域内有不同的分岔条件,圆柱薄壳在歧集和滞后集内分岔条件不存在。分岔只发生在双极限点集内,在双极限点集的条件下求出了分岔条件。
Electromagnetism solid mechanics study electromagnetic field and deformation field interoperable theory of flexibility solid matter, Plate and Shell magnetoelasticity nonlinear vibrational subject is relatively common of engineering reality, Establish and Solve of governing equation have very great difficulty, So to study Plate and Shell magnetoelasticity nonlinear vibrational subject have important theory and reality meaning.
This paper firstly according to equation of magnetoelasticity nonlinear vibration, Gain nonlinear vibrational equation of Beam-plate, Specific analyse bifurcation factor of Beam-plate by Center-manifold-theorem, And gain exist factor of global bifurcation by Melnikov function.
Secondly adhibit parameter to vibrational equation of the conductive Beam-plate and transform, Gain master syntonic single and second order stable common solution by Multi-scale means, Gain equation of bifurcation and Anslyze behaviour of bifurcation by Strangeness-theory, Different change sum of sequences have different bifurcation of the conductive Beam-plate.volume. Solving and Gain factor of bifurcation is not exist of fork volume and double limit point volume, Bifurcation only exist of lag volume, Gain factor of bifurcation at lag volume field.
Finally adhibit boundary condition to vibrational equation of simply supported on both bides of Column-thin-shell, Gain vibrational differential equation of Column-thin-shell and This equation have cubic and squared term, Find master syntonic solutional equation of bifurcation. Gain equation of bifurcation and Anslyze behaviour of bifurcation by Strangeness-theory, Different change sum of sequences have different bifurcation volume of the conductive Column-thin-shell. Factor of bifurcation is not exist of fork volume and lag volume, Bifurcation only exist of double limit point volume, Gain factor of bifurcation at double limit point volume field.
论文首先在建立薄板磁弹性非线性振动方程的基础上,得到梁式板的非线性振动方程,运用中心流形定理和方法对其分岔行为进行了具体分析,画出了分岔图,并运用Melnikov函数方法得到发生全局分岔的条件。
其次,对梁式板的振动方程引入参数进行变换,运用多尺度方法求出梁式板主共振下的二阶定常解,用奇异性理论得到分岔方程并求出转迁集,导电梁式板在不同的转迁集的条件下发生分岔的条件是不相同的,求解可得到歧集和双极限点集内分岔条件是不存在,分岔只发生在滞后集内,在滞后集条件下求出了分岔条件。
最后,将两边简支的边界条件带入圆柱薄壳振动方程,得到了圆柱薄壳的带立方和平方项的振动微分方程,用多尺度方法求出主共振解的分岔方程,分析了系统运动中解的稳定性,运用奇异性理论得到分岔方程的转迁集,导电圆柱薄壳在不同的转迁集区域内有不同的分岔条件,圆柱薄壳在歧集和滞后集内分岔条件不存在。分岔只发生在双极限点集内,在双极限点集的条件下求出了分岔条件。
Electromagnetism solid mechanics study electromagnetic field and deformation field interoperable theory of flexibility solid matter, Plate and Shell magnetoelasticity nonlinear vibrational subject is relatively common of engineering reality, Establish and Solve of governing equation have very great difficulty, So to study Plate and Shell magnetoelasticity nonlinear vibrational subject have important theory and reality meaning.
This paper firstly according to equation of magnetoelasticity nonlinear vibration, Gain nonlinear vibrational equation of Beam-plate, Specific analyse bifurcation factor of Beam-plate by Center-manifold-theorem, And gain exist factor of global bifurcation by Melnikov function.
Secondly adhibit parameter to vibrational equation of the conductive Beam-plate and transform, Gain master syntonic single and second order stable common solution by Multi-scale means, Gain equation of bifurcation and Anslyze behaviour of bifurcation by Strangeness-theory, Different change sum of sequences have different bifurcation of the conductive Beam-plate.volume. Solving and Gain factor of bifurcation is not exist of fork volume and double limit point volume, Bifurcation only exist of lag volume, Gain factor of bifurcation at lag volume field.
Finally adhibit boundary condition to vibrational equation of simply supported on both bides of Column-thin-shell, Gain vibrational differential equation of Column-thin-shell and This equation have cubic and squared term, Find master syntonic solutional equation of bifurcation. Gain equation of bifurcation and Anslyze behaviour of bifurcation by Strangeness-theory, Different change sum of sequences have different bifurcation volume of the conductive Column-thin-shell. Factor of bifurcation is not exist of fork volume and lag volume, Bifurcation only exist of double limit point volume, Gain factor of bifurcation at double limit point volume field.
引文
1黄润生,黄浩(第2版).混沌及其应用.武汉大学出版社, 2005:157-169
2郝柏林.分岔、混沌、奇怪吸引子、湍流及其它.物理学进展, 2000:37-213
3高普云.非线性动力学.国防科技大学出版社, 2005:1-5
4 I. G. Vardolakis. Bifurcation analysis in geo-mechanics. Chapman and Hall, 1993:2-9
5 Y. Chen and A. T. Leung. Bifurcation and chaos engineering. New York:Springer Verlag, 1998:3-7
6 F. Taken. Detecting strange sttractors in turbulence. Lect Notes in Math, 1981, (898): 366-381
7于晋臣.非线性动力系统的分岔研究. [北京交通大学学位论文]. 2006:1-2
8 P. Yu, W. Zhang, Q. S. Bi. Vibration analysis on a thin plate with the aid of computation of normal forms. International Journal Nonlinear Mechanics, 2001,36(4):597-627
9 M. Golubitsky and D. Schaeffer. A Theory for Imperfect Bifurcation Theory Via Singularity Throty. Comm Pure Appl Math, 1979, (32):21-98
10 M. Marek and M. Kubicek. Computational Methods in Bifurcation Theory and Dissipative Structures. Spinger-Verlag, 1983:3-6
11韩强.弹塑性系统的动力屈曲和分叉.北京:科学出版社, 2000:109-119
12郝柏林.从抛物线谈起-混沌动力学引论.上海:上海科技教育出版社, 1993:3-9
13盛昭瀚,马海军.非线性动力系统分析引论.北京:科学出版社, 2001:1-9
14闻邦椿,李以农,韩清凯.非线性振动理论中的解析方法及工程应用.沈阳:东北大学出版社, 2001:211-212
15 Y. H. Pao, C. S. Yeh. A linear ztheory for soft ferromagnetic elastic bodies. International Journal of Engineering Science, 1973,11(4):415-436
16 F. C. Moon. Magneto-Solid Mechanics. New York: John Wiley and Sons, 1984:24-30
17 A. C. Erigen. Theory of electric-magnetic plates. International Journal of Engineering Science, 1989,27(4):363-375
18周又和,郑晓静.软铁磁薄板磁弹性屈曲的理论模型.力学学报, 1996,28(6):651-660
19王璋奇,施传立,杨以涵.导体板电磁弹性振动的一般方程.华北电力学院学报, 1995,22(1):57-63
20徐健学.麦克斯韦尔电磁应力张量和电磁固体耦合动力学.上海交通大学学报, 1990,24(6):16-22
21 J. S. Lee. Dynamic stability of conducting beam-plates in transverse magnetic fields. Journal of Engineering Mechanics, 1996,122(2):89-94
22郑晓静,刘信恩.铁磁导电梁式板在横向均匀磁场中的动力特性分析.固体力学学报, 2000,21(3):243-250
23胡宇达,白象忠.磁场中圆柱壳体的轴对称振动.工程力学, 1999,16(5):47-52
24 Yuda Hu, Yaoling Xu, Xiangzhong Bai. The nonlinear axisymmetric vibration of a cylindrical shell in a longitudinal magnetic field. ICVE’98, Dalian, 1998:320-323
25 F. C. Moon, Y. H. Pao. Vibration and dynamic instability of a beam-plate in a transverse magnetic field. J. Appl. Mech, 1969, (36):141-149
26 Q. S. Lu, C. W. S. To. Dynamic stability and bifurcation of an alternating load and magnetic field excited magnetoelastic beam. Journal of Sound and Vibration, 1995, 181(5):873-891
27 Xiaojing Zheng, Jianping Zhang, Youhe Zhou. Dynamic stability of a cantilever conductive plate in transverse impulsive magnetic field. International Journal of solids and structures, 2005, (42):2417-2430
28 Xiaojing Zheng, Jianping Zhang, Youhe Zhou. Dynamic stability of a cantilever conductive plate in transverse impulsive magnetic field. International Journal of Solids and Structures, 2005, (42):2416-2428
29高原文,缑新科,周又和.脉冲磁场激励下铁磁梁式板动力响应特征研究.振动工程学报, 2005,18(3):314-317
30 G. Rajesh, Kavasseri. Analysis of subharmonic oscillations in ferroresonance circuit. Electrical Power and Energy Systems, 2005, (28):207-214
31胡波.载流板壳磁弹性稳定问题分析. [燕山大学硕士论文]. 2005:8-17
32胡宇达,邱家俊.电磁力作用下发电机定子端部绕组的两自由度主共振.固体力学学报, 2004,25(4):439-445
33胡宇达.磁场中理想传导薄板的非线性磁弹性一般方程.非线性动力学学报, 1999,6(4):338-342
34胡宇达,杜国君.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, 2004,2(4):35-38
35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
36赵将,胡宇达.恒定磁场中简支圆柱壳的磁弹性振动分析.动力学与控制学报, 2006,4(3):267-271
37 R. A. Ditaranto, M. Lessen. Coriolis acceleration effect on the vibration of a rotating thin-walled circular cylinder. Journal of Applied Mechanics, 1964:700-701
38 A. V. Srinivasan, G. F. Lauterbach. Traveling waves in rotating cylindrical shells. Journal of Engineering for Industry, 1971:1229-1232
39 J. R. Leher, S. C. Batterman. Nonlinear static and dynamic deformations of shells of revolution. Int. J. Non-linear Mechanics, 1974, (9):501-519
40 A. H. Nayfeh, A. Raouf. Non-linear oscillation of circular cylindrical shells. Int J. solids Structures, 1987,23(12):1625-1638.
41江晓禹,张相周.圆柱壳的非线性自由振动分析.应用力学学报, 1993,10(3):94-97
42 A. H. Nayfeh, S. A. Nayfeh. On nonlinear modes of continuous systems. Journal of Vibration and Acoustics, 1994, (116):129-136
43 A. H. Nayfeh, S. A. Nayfeh. Nonlinear normal modes of a continuous system with quadratic nonlinearities. Journal of Vibration and Acoustics, 1995, (117):199-205
44 A. H. Nayfeh, S. A. Nayfeh, C. Chin. Nonlinear normal modes of a cantilever. Journal of Vibration and Acoustics, 1995, (117):477-481
45 Charming Chin, A. H. Nayfeh. Bifurcation and chaos in externally excited circular cylinical shells. Journal of Applied Mechanics, 1996, (63):565-574
46王炳雷,田晓耕,沈亚鹏.磁场对铁磁结构动态特性的影响.实验力学, 2005,20(3): 335-343
47孙慧静.非线性系统分岔混沌理论及其应用. [北京交通大学硕士论文]. 2006:3-7
48陈士华,陆君安.混沌动力学初步.武汉:武汉水利出版社, 2000:37-213
49张琪昌,王洪礼,竺致文等.分岔与混沌理论及应用.天津:天津大学出版社, 2005:58-223
50刘式适,刘式达.物理学中的非线性方程.北京:北京大学出版社, 2000:129-135
51С.А.Амбарцумян,Г.Е.Багдасарян,М.В.Белубекян.Магнитоупругостьтонкихоболочекипластин.Москва:Наука, 1977:15-30
52徐芝纶.弹性力学.北京:高等教育出版社, 2003:10-21
53陈予恕,唐云等.非线性动力学中的现代分析方法.北京:科学出版社, 2000:28-173
54 C. D. Andereck, S. S. Liu and H. L. Swinney. Flow Regimes in a Circular Couette System with Independently Rotating Cylinders. J. Fluid Mech, 1986, (164):155-183
55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
56 Gaetano Napoil, Stefano Turzi. On the determination of nontrivial equilibrium configurations close to a bifurcation point. Computers and Mathematics with Applications, 2007, (4):8-10
57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
58白象忠.板壳磁弹性理论基础.北京:机械工业出版社, 1996:156-158
59 J. S. Lee. Dynamic stability of conducting beam-plates in transverse magnetic fields. Journal of engineering Mechanics, 1996,122(2):89-94
60 R. C. A. Hompeson, T. Mullin. Routes to chaos in a magnetoelastic beam. Chaos Solitons and Fractals, 1987,8(4):681-687
61赵将.圆柱壳体的非线性磁弹性共振问题研究. [燕山大学硕士论文]. 2006:14-15
62刘秉正,彭建华.非线性动力学.北京:高等教育出版社, 2004:25-46
2郝柏林.分岔、混沌、奇怪吸引子、湍流及其它.物理学进展, 2000:37-213
3高普云.非线性动力学.国防科技大学出版社, 2005:1-5
4 I. G. Vardolakis. Bifurcation analysis in geo-mechanics. Chapman and Hall, 1993:2-9
5 Y. Chen and A. T. Leung. Bifurcation and chaos engineering. New York:Springer Verlag, 1998:3-7
6 F. Taken. Detecting strange sttractors in turbulence. Lect Notes in Math, 1981, (898): 366-381
7于晋臣.非线性动力系统的分岔研究. [北京交通大学学位论文]. 2006:1-2
8 P. Yu, W. Zhang, Q. S. Bi. Vibration analysis on a thin plate with the aid of computation of normal forms. International Journal Nonlinear Mechanics, 2001,36(4):597-627
9 M. Golubitsky and D. Schaeffer. A Theory for Imperfect Bifurcation Theory Via Singularity Throty. Comm Pure Appl Math, 1979, (32):21-98
10 M. Marek and M. Kubicek. Computational Methods in Bifurcation Theory and Dissipative Structures. Spinger-Verlag, 1983:3-6
11韩强.弹塑性系统的动力屈曲和分叉.北京:科学出版社, 2000:109-119
12郝柏林.从抛物线谈起-混沌动力学引论.上海:上海科技教育出版社, 1993:3-9
13盛昭瀚,马海军.非线性动力系统分析引论.北京:科学出版社, 2001:1-9
14闻邦椿,李以农,韩清凯.非线性振动理论中的解析方法及工程应用.沈阳:东北大学出版社, 2001:211-212
15 Y. H. Pao, C. S. Yeh. A linear ztheory for soft ferromagnetic elastic bodies. International Journal of Engineering Science, 1973,11(4):415-436
16 F. C. Moon. Magneto-Solid Mechanics. New York: John Wiley and Sons, 1984:24-30
17 A. C. Erigen. Theory of electric-magnetic plates. International Journal of Engineering Science, 1989,27(4):363-375
18周又和,郑晓静.软铁磁薄板磁弹性屈曲的理论模型.力学学报, 1996,28(6):651-660
19王璋奇,施传立,杨以涵.导体板电磁弹性振动的一般方程.华北电力学院学报, 1995,22(1):57-63
20徐健学.麦克斯韦尔电磁应力张量和电磁固体耦合动力学.上海交通大学学报, 1990,24(6):16-22
21 J. S. Lee. Dynamic stability of conducting beam-plates in transverse magnetic fields. Journal of Engineering Mechanics, 1996,122(2):89-94
22郑晓静,刘信恩.铁磁导电梁式板在横向均匀磁场中的动力特性分析.固体力学学报, 2000,21(3):243-250
23胡宇达,白象忠.磁场中圆柱壳体的轴对称振动.工程力学, 1999,16(5):47-52
24 Yuda Hu, Yaoling Xu, Xiangzhong Bai. The nonlinear axisymmetric vibration of a cylindrical shell in a longitudinal magnetic field. ICVE’98, Dalian, 1998:320-323
25 F. C. Moon, Y. H. Pao. Vibration and dynamic instability of a beam-plate in a transverse magnetic field. J. Appl. Mech, 1969, (36):141-149
26 Q. S. Lu, C. W. S. To. Dynamic stability and bifurcation of an alternating load and magnetic field excited magnetoelastic beam. Journal of Sound and Vibration, 1995, 181(5):873-891
27 Xiaojing Zheng, Jianping Zhang, Youhe Zhou. Dynamic stability of a cantilever conductive plate in transverse impulsive magnetic field. International Journal of solids and structures, 2005, (42):2417-2430
28 Xiaojing Zheng, Jianping Zhang, Youhe Zhou. Dynamic stability of a cantilever conductive plate in transverse impulsive magnetic field. International Journal of Solids and Structures, 2005, (42):2416-2428
29高原文,缑新科,周又和.脉冲磁场激励下铁磁梁式板动力响应特征研究.振动工程学报, 2005,18(3):314-317
30 G. Rajesh, Kavasseri. Analysis of subharmonic oscillations in ferroresonance circuit. Electrical Power and Energy Systems, 2005, (28):207-214
31胡波.载流板壳磁弹性稳定问题分析. [燕山大学硕士论文]. 2005:8-17
32胡宇达,邱家俊.电磁力作用下发电机定子端部绕组的两自由度主共振.固体力学学报, 2004,25(4):439-445
33胡宇达.磁场中理想传导薄板的非线性磁弹性一般方程.非线性动力学学报, 1999,6(4):338-342
34胡宇达,杜国君.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, 2004,2(4):35-38
35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
36赵将,胡宇达.恒定磁场中简支圆柱壳的磁弹性振动分析.动力学与控制学报, 2006,4(3):267-271
37 R. A. Ditaranto, M. Lessen. Coriolis acceleration effect on the vibration of a rotating thin-walled circular cylinder. Journal of Applied Mechanics, 1964:700-701
38 A. V. Srinivasan, G. F. Lauterbach. Traveling waves in rotating cylindrical shells. Journal of Engineering for Industry, 1971:1229-1232
39 J. R. Leher, S. C. Batterman. Nonlinear static and dynamic deformations of shells of revolution. Int. J. Non-linear Mechanics, 1974, (9):501-519
40 A. H. Nayfeh, A. Raouf. Non-linear oscillation of circular cylindrical shells. Int J. solids Structures, 1987,23(12):1625-1638.
41江晓禹,张相周.圆柱壳的非线性自由振动分析.应用力学学报, 1993,10(3):94-97
42 A. H. Nayfeh, S. A. Nayfeh. On nonlinear modes of continuous systems. Journal of Vibration and Acoustics, 1994, (116):129-136
43 A. H. Nayfeh, S. A. Nayfeh. Nonlinear normal modes of a continuous system with quadratic nonlinearities. Journal of Vibration and Acoustics, 1995, (117):199-205
44 A. H. Nayfeh, S. A. Nayfeh, C. Chin. Nonlinear normal modes of a cantilever. Journal of Vibration and Acoustics, 1995, (117):477-481
45 Charming Chin, A. H. Nayfeh. Bifurcation and chaos in externally excited circular cylinical shells. Journal of Applied Mechanics, 1996, (63):565-574
46王炳雷,田晓耕,沈亚鹏.磁场对铁磁结构动态特性的影响.实验力学, 2005,20(3): 335-343
47孙慧静.非线性系统分岔混沌理论及其应用. [北京交通大学硕士论文]. 2006:3-7
48陈士华,陆君安.混沌动力学初步.武汉:武汉水利出版社, 2000:37-213
49张琪昌,王洪礼,竺致文等.分岔与混沌理论及应用.天津:天津大学出版社, 2005:58-223
50刘式适,刘式达.物理学中的非线性方程.北京:北京大学出版社, 2000:129-135
51С.А.Амбарцумян,Г.Е.Багдасарян,М.В.Белубекян.Магнитоупругостьтонкихоболочекипластин.Москва:Наука, 1977:15-30
52徐芝纶.弹性力学.北京:高等教育出版社, 2003:10-21
53陈予恕,唐云等.非线性动力学中的现代分析方法.北京:科学出版社, 2000:28-173
54 C. D. Andereck, S. S. Liu and H. L. Swinney. Flow Regimes in a Circular Couette System with Independently Rotating Cylinders. J. Fluid Mech, 1986, (164):155-183
55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
56 Gaetano Napoil, Stefano Turzi. On the determination of nontrivial equilibrium configurations close to a bifurcation point. Computers and Mathematics with Applications, 2007, (4):8-10
57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
58白象忠.板壳磁弹性理论基础.北京:机械工业出版社, 1996:156-158
59 J. S. Lee. Dynamic stability of conducting beam-plates in transverse magnetic fields. Journal of engineering Mechanics, 1996,122(2):89-94
60 R. C. A. Hompeson, T. Mullin. Routes to chaos in a magnetoelastic beam. Chaos Solitons and Fractals, 1987,8(4):681-687
61赵将.圆柱壳体的非线性磁弹性共振问题研究. [燕山大学硕士论文]. 2006:14-15
62刘秉正,彭建华.非线性动力学.北京:高等教育出版社, 2004:25-46