杆与板的磁弹性屈曲分岔和混沌分析
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摘要
随着现代高新科技的发展,杆、板和壳等结构元件处于电磁场环境中的情况已是屡见不鲜,这种电磁场与力学场相互耦合的一个基本特征就是非线性特性。这种非线性特性表现出来的力学行为比较复杂,直接影响系统运行的安全性以及可靠性。由此引起了人们广泛的研究兴趣。
以往,对磁弹性屈曲问题的研究主要集中在软铁磁性薄板屈曲理论模型的建立及修正;以Tokamak核聚变反应堆环向磁场载流线圈为代表的对载流线圈及载流杆件的稳定性研究。对于工程中经常遇到的在强磁场作用下的载电流薄板等结构元件的屈曲研究还比较少见,对于电磁场环境下的载电流非铁磁性薄板等结构元件的屈曲分岔和混沌运动的研究尚未见诸文献。
本文正是以此为出发点,采用理论分析、数值计算,开展了对电磁场作用下的杆、板的屈曲、分岔及混沌运动的研究工作。研究内容概括为以下几个部分。
(1)将载流构件磁弹性屈曲问题由线圈、杆件拓展到载流薄板。首先根据载电流薄板的磁弹性非线性运动方程,物理几何方程,洛仑兹力的表达式,电动力学方程,得出了载电流薄板在电磁场与机械荷载共同作用下的磁弹性动力屈曲方程。然后,应用Galerkin原理将方程整理为标准的马丢方程,求解屈曲问题就变为求解马丢方程,即按照系数λ和η的本征值关系得出了磁弹性失稳临界状态的判据。最后,通过具体算例,得出了载流薄板在四边简支、三边简支一边自由、对边简支对边固定、四边固定四种边界条件下的磁弹性动力屈曲方程以及失稳临界状态时相关参量之间的关系曲线,并对其变化规律进行了分析讨论。
(2)应用Lagrange描述法,从非线性动力学的视角讨论了电磁场中受压细长杆的分岔特性。分别就电磁场中受压细长杆的静力学模型、线性动力学模型和非线性动力学模型讨论了其分岔特性。得到了电磁场与机械载荷共同作用下,产生屈曲分岔的条件、分岔点和分岔类型;并以两端铰支的铝质梁和非铁磁钢梁为例,采用Matlab计算程序,得到了受压细长杆在静力学模型、线性动力学模型及非线性动力学模型下,产生分岔的临界载荷值、临界磁场强度值及杆长之间的关系曲线和变化规律。
(3)在横向磁场作用下,考虑机械场与磁场的相互耦合作用,并同时考虑洛仑兹力及洛仑兹力矩,得到载流薄板的几何非线性磁弹性动力平衡方程。通过讨论平衡态的稳定性,得到了电磁场与机械载荷共同作用下的四边简支、四边固定载流矩形薄板的静力屈曲分岔条件、分岔点和分岔类型。应用L-S约化方法,得到了相同边界条件下薄板的动力屈曲分岔条件。并采用具体算例,计算得到了发生分岔的临界载荷值,以及临界载荷与电流密度、磁场强度及板的几何尺寸之间的关系曲线和变化规律。
(4)根据薄板大挠度弯曲的物理及几何方程,考虑洛伦兹力及洛伦兹力矩的影响,建立了在横向磁场和机械载荷共同作用下的几何非线性四边简支和四边固定载流矩形非铁磁性薄板的运动方程,利用Melnikov函数法,从理论上给出了这一磁弹性动力系统在不同情况时,可能发生Smale马蹄意义下混沌的临界条件。
(5)针对横向电磁场中非铁磁简支条形板,应用Galerkin原理得到条形板的几何非线性及物理非线性动力方程,利用Melnikov函数法对其单模态位移模式下的混沌运动进行了理论分析,得到了可能发生混沌的临界条件。利用平均法求得条形板在双模态下的分岔点,并讨论了分岔点的稳定性情况。从理论上讨论了利用单、双模态位移模式模拟非线性行为的差异。
综上所述,电磁场、机械场等多个物理场共同耦合作用下的结构元件中蕴涵了相当丰富、复杂的动力学行为。因此,无论是通过理论研究还是实验研究,都具有理论和实际应用价值。同时,所得结果也可供相关电磁结构的可靠性设计时参考。
It is very common that the rods, plates and shells work in an electromagnetic environment as structural components with the development of modern advanced technology. One of the basic characters when an electromagnetic field and a mechanical field coupled is nonlinear. The mechanical behavior due to the nonlinear feature is very complex. It will affect the safety and reliability of systems significantly. So it attracted lots of researching interesting.
The study of the magneto-elastic buckling problem was focus on the following two aspects previously: Finding the theoretical model of the buckling problem of soft ferromagnetic thin plate and its amendment; The stability study about the current-carrying coil and rods in the magnetic field of Tokamak fusion reactor as examples. However, it is on less of researching on the buckling of current-carrying components, like thin current-carrying plates, working in a strong magnetic field. It is almost no references for buckling Bifurcation and Chaos about current-carrying components working in electromagnetic filed.
Considering the situation talking above, this paper showed some studying work using theoretical analysis and numerical computation about buckling, bifurcation and chaotic motion of current-carrying rods and thin plates. Research summarized in the following parts.
(1)The analysis of the magnetic-elasticity buckling problem has been extended from coils and rods to non-ferromagnetic thin current-carrying plate here. Based on the non-linear magnetic-elasticity equations of motion, physical equations, geometric equations, expressions of Lorenz forces and electro-dynamic equations, the magnetic-elasticity dynamic buckling equation of a current plate under the action of a mechanical load in a magnetic field is derived. Then the buckling equation is changed into a standard form of the Mathieu equation using the Galerkin method. Thus, to solve the buckling problem is to solve the Mathieu equation. According to the eigenvalue relation of the coefficientsλandηin the Mathieu equation, the criterion equation for the buckling problem is also obtained here. As examples, the magnetic-elasticity buckling equation of a thin current-carrying plate applied four kinds of boundaries, simply supported, simply supported at three edges, simply supported and fixed opposite and fixed are obtained. The relation curves of the instability state and with variations in some parameters are also shown in this paper. The calculation results and the effects of the relative parameters are also discussed.
(2)The bifurcation characters of a long slender bar in an electromagnetic field were discussed applying Lagrange description from the perspective of nonlinear dynamics. The bifurcation characters were discussed at the situation state mechanical load, linear and nonlinear dynamic load compression model individual. The buckling bifurcation conditions, bifurcation point and the type of bifurcation were gotten when the long slender bar was under the action of mechanical load in a magnetic field. A hinged aluminum beam and a non-ferromagnetic steel beam were taken as computational example. Their critical load and magnetic intensity value of causing bifurcation were carried out. The relationship curves between these critical values and the length of the beam were shown in the paper.
(3)The bifurcation conditions, bifurcation point and bifurcation type of a thin current-carrying plate simply supported or fixed at all edges were obtained through discussed the stability of equilibrium when the plate applied the mechanical load in a electromagnetic field. The dynamic bifurcation conditions, bifurcation point and bifurcation type of the same plate were also obtained using LS method. At the same time, the critical load was carried out and the relation curves and the variation rules between critical load and the current density, the intensity of magnetic field and the geometry length of the plate were shown in the paper.
(4)The nonlinear equations of motion of a plate simply supported or fixed at all edges were established when the plate applied mechanical load in a transverse magnetic field. The balance points of the nonlinear dynamic system were gotten in the case of non-disturbance. The critical condition of chaos in the sense of Smale horseshoe of the system was also obtained using Melnikov function method when a disturbance happened.
(5)For a strip plate in a transverse electromagnetic field, the chaotic motion under the single-mode displacement was analyzed and obtained its critical chaotic conditions using Melnikov function method. More, the bifurcation point of the strip plate under the dual-mode displacement was obtained and the stability of bifurcation point was discussed using the averaging method. The differences of the simulation to the nonlinear behavior of a system using single-, dual-mode displacement mode were analyzed theoretically.
To sum up, it contains very rich and complex dynamic behavior when structural components are in electromagnetic, mechanical and many other physical fields at the same time. Therefore, whether it is through theoretical research or experimental study, it has the theoretical and practical application value. At the same time, the results obtained here can also be used as references to the reliability design for related electromagnetic structure.
以往,对磁弹性屈曲问题的研究主要集中在软铁磁性薄板屈曲理论模型的建立及修正;以Tokamak核聚变反应堆环向磁场载流线圈为代表的对载流线圈及载流杆件的稳定性研究。对于工程中经常遇到的在强磁场作用下的载电流薄板等结构元件的屈曲研究还比较少见,对于电磁场环境下的载电流非铁磁性薄板等结构元件的屈曲分岔和混沌运动的研究尚未见诸文献。
本文正是以此为出发点,采用理论分析、数值计算,开展了对电磁场作用下的杆、板的屈曲、分岔及混沌运动的研究工作。研究内容概括为以下几个部分。
(1)将载流构件磁弹性屈曲问题由线圈、杆件拓展到载流薄板。首先根据载电流薄板的磁弹性非线性运动方程,物理几何方程,洛仑兹力的表达式,电动力学方程,得出了载电流薄板在电磁场与机械荷载共同作用下的磁弹性动力屈曲方程。然后,应用Galerkin原理将方程整理为标准的马丢方程,求解屈曲问题就变为求解马丢方程,即按照系数λ和η的本征值关系得出了磁弹性失稳临界状态的判据。最后,通过具体算例,得出了载流薄板在四边简支、三边简支一边自由、对边简支对边固定、四边固定四种边界条件下的磁弹性动力屈曲方程以及失稳临界状态时相关参量之间的关系曲线,并对其变化规律进行了分析讨论。
(2)应用Lagrange描述法,从非线性动力学的视角讨论了电磁场中受压细长杆的分岔特性。分别就电磁场中受压细长杆的静力学模型、线性动力学模型和非线性动力学模型讨论了其分岔特性。得到了电磁场与机械载荷共同作用下,产生屈曲分岔的条件、分岔点和分岔类型;并以两端铰支的铝质梁和非铁磁钢梁为例,采用Matlab计算程序,得到了受压细长杆在静力学模型、线性动力学模型及非线性动力学模型下,产生分岔的临界载荷值、临界磁场强度值及杆长之间的关系曲线和变化规律。
(3)在横向磁场作用下,考虑机械场与磁场的相互耦合作用,并同时考虑洛仑兹力及洛仑兹力矩,得到载流薄板的几何非线性磁弹性动力平衡方程。通过讨论平衡态的稳定性,得到了电磁场与机械载荷共同作用下的四边简支、四边固定载流矩形薄板的静力屈曲分岔条件、分岔点和分岔类型。应用L-S约化方法,得到了相同边界条件下薄板的动力屈曲分岔条件。并采用具体算例,计算得到了发生分岔的临界载荷值,以及临界载荷与电流密度、磁场强度及板的几何尺寸之间的关系曲线和变化规律。
(4)根据薄板大挠度弯曲的物理及几何方程,考虑洛伦兹力及洛伦兹力矩的影响,建立了在横向磁场和机械载荷共同作用下的几何非线性四边简支和四边固定载流矩形非铁磁性薄板的运动方程,利用Melnikov函数法,从理论上给出了这一磁弹性动力系统在不同情况时,可能发生Smale马蹄意义下混沌的临界条件。
(5)针对横向电磁场中非铁磁简支条形板,应用Galerkin原理得到条形板的几何非线性及物理非线性动力方程,利用Melnikov函数法对其单模态位移模式下的混沌运动进行了理论分析,得到了可能发生混沌的临界条件。利用平均法求得条形板在双模态下的分岔点,并讨论了分岔点的稳定性情况。从理论上讨论了利用单、双模态位移模式模拟非线性行为的差异。
综上所述,电磁场、机械场等多个物理场共同耦合作用下的结构元件中蕴涵了相当丰富、复杂的动力学行为。因此,无论是通过理论研究还是实验研究,都具有理论和实际应用价值。同时,所得结果也可供相关电磁结构的可靠性设计时参考。
It is very common that the rods, plates and shells work in an electromagnetic environment as structural components with the development of modern advanced technology. One of the basic characters when an electromagnetic field and a mechanical field coupled is nonlinear. The mechanical behavior due to the nonlinear feature is very complex. It will affect the safety and reliability of systems significantly. So it attracted lots of researching interesting.
The study of the magneto-elastic buckling problem was focus on the following two aspects previously: Finding the theoretical model of the buckling problem of soft ferromagnetic thin plate and its amendment; The stability study about the current-carrying coil and rods in the magnetic field of Tokamak fusion reactor as examples. However, it is on less of researching on the buckling of current-carrying components, like thin current-carrying plates, working in a strong magnetic field. It is almost no references for buckling Bifurcation and Chaos about current-carrying components working in electromagnetic filed.
Considering the situation talking above, this paper showed some studying work using theoretical analysis and numerical computation about buckling, bifurcation and chaotic motion of current-carrying rods and thin plates. Research summarized in the following parts.
(1)The analysis of the magnetic-elasticity buckling problem has been extended from coils and rods to non-ferromagnetic thin current-carrying plate here. Based on the non-linear magnetic-elasticity equations of motion, physical equations, geometric equations, expressions of Lorenz forces and electro-dynamic equations, the magnetic-elasticity dynamic buckling equation of a current plate under the action of a mechanical load in a magnetic field is derived. Then the buckling equation is changed into a standard form of the Mathieu equation using the Galerkin method. Thus, to solve the buckling problem is to solve the Mathieu equation. According to the eigenvalue relation of the coefficientsλandηin the Mathieu equation, the criterion equation for the buckling problem is also obtained here. As examples, the magnetic-elasticity buckling equation of a thin current-carrying plate applied four kinds of boundaries, simply supported, simply supported at three edges, simply supported and fixed opposite and fixed are obtained. The relation curves of the instability state and with variations in some parameters are also shown in this paper. The calculation results and the effects of the relative parameters are also discussed.
(2)The bifurcation characters of a long slender bar in an electromagnetic field were discussed applying Lagrange description from the perspective of nonlinear dynamics. The bifurcation characters were discussed at the situation state mechanical load, linear and nonlinear dynamic load compression model individual. The buckling bifurcation conditions, bifurcation point and the type of bifurcation were gotten when the long slender bar was under the action of mechanical load in a magnetic field. A hinged aluminum beam and a non-ferromagnetic steel beam were taken as computational example. Their critical load and magnetic intensity value of causing bifurcation were carried out. The relationship curves between these critical values and the length of the beam were shown in the paper.
(3)The bifurcation conditions, bifurcation point and bifurcation type of a thin current-carrying plate simply supported or fixed at all edges were obtained through discussed the stability of equilibrium when the plate applied the mechanical load in a electromagnetic field. The dynamic bifurcation conditions, bifurcation point and bifurcation type of the same plate were also obtained using LS method. At the same time, the critical load was carried out and the relation curves and the variation rules between critical load and the current density, the intensity of magnetic field and the geometry length of the plate were shown in the paper.
(4)The nonlinear equations of motion of a plate simply supported or fixed at all edges were established when the plate applied mechanical load in a transverse magnetic field. The balance points of the nonlinear dynamic system were gotten in the case of non-disturbance. The critical condition of chaos in the sense of Smale horseshoe of the system was also obtained using Melnikov function method when a disturbance happened.
(5)For a strip plate in a transverse electromagnetic field, the chaotic motion under the single-mode displacement was analyzed and obtained its critical chaotic conditions using Melnikov function method. More, the bifurcation point of the strip plate under the dual-mode displacement was obtained and the stability of bifurcation point was discussed using the averaging method. The differences of the simulation to the nonlinear behavior of a system using single-, dual-mode displacement mode were analyzed theoretically.
To sum up, it contains very rich and complex dynamic behavior when structural components are in electromagnetic, mechanical and many other physical fields at the same time. Therefore, whether it is through theoretical research or experimental study, it has the theoretical and practical application value. At the same time, the results obtained here can also be used as references to the reliability design for related electromagnetic structure.
引文
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46 Zhou,Youhe, Gao,Yuanwen, Zheng,Xiaojing. Buckling and post-buckling of a ferromagnetic beam-plate induced by magneto-elastic interactions. International Journal of Non-Linear Mechanics,2000,35(6):1059-1065
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49 Healey T.J. Large Rotating States of a Conducting Elastic Wires in a Magnetic Field: Subtle Symmetry and Multiparameter Bifurcation, J. of Elasticity, 1990,24:211-227
50 Wolfe P, Bifurcation Theory of an Elastic Conducting Rod in a Magnetic Field, Quart Mech.Appl.Math,1988,41(2):265-279
51 Peter Wolfe, Bifurcation Theory of an Elastic Conducting Rod Subjected to Magnetic Forces, Int. J. Nonlinear Mechanics,1990,25(5):597-604
52 Moon F C,Chattopadhyay S.Elastic Stability of a Thermonuclear Rector Coil, Princeton,(1973)
53 Moon F.C.Buckling of a Superconducting Ring in a Toroidal Magnetic Field, J. Appl. Mech, 1979, 46(1):151-155
54谢慧才,方葛丰,王满德.平面圆形线圈的磁弹性屈曲.力学学报,1991,23(6):706-711
55 Lenotovich M.A and Shafranov V.D.The Stability of a Flexible Conductor in a Magnetic Field, Plasms Physics and the Problem of Controlled Thermonucler Recations, Pergamon Press, 1961
56 Van Lieshout P H, Rongen P M J and Van de Ven A AFA. Variational Principle for Magneto-elastic Buckling. J. Eng. Math, 1987, 21:227-252
57 Van Lieshout P.H and Van de Ven A.A.F.A Variational Approach to the Magneto-elastic Buckling Problems of an Arbitrary Number of Superconducting Beams. J. Eng. Math, 1991, 25:353-374
58 Chattopadhyay S and Moon F.C.Magneto-elastic Buckling and Vibration of a Rod Carrying Electric Current.J. Appl. Mech, 1975,42:809-814
59 Van de Ven AAF and Couwenberg MJH.Magneto-elastic Stability of a Superconducting Ring in its Own Field.J. Eng. Math,1986,20:251-270
60王省哲,郑晓静,周又和.非圆Tokamak载流线圈的磁弹性分析.兰州大学学报(自然科学版),1999,35(1):34-43
61王海滨,周又和,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
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49 Healey T.J. Large Rotating States of a Conducting Elastic Wires in a Magnetic Field: Subtle Symmetry and Multiparameter Bifurcation, J. of Elasticity, 1990,24:211-227
50 Wolfe P, Bifurcation Theory of an Elastic Conducting Rod in a Magnetic Field, Quart Mech.Appl.Math,1988,41(2):265-279
51 Peter Wolfe, Bifurcation Theory of an Elastic Conducting Rod Subjected to Magnetic Forces, Int. J. Nonlinear Mechanics,1990,25(5):597-604
52 Moon F C,Chattopadhyay S.Elastic Stability of a Thermonuclear Rector Coil, Princeton,(1973)
53 Moon F.C.Buckling of a Superconducting Ring in a Toroidal Magnetic Field, J. Appl. Mech, 1979, 46(1):151-155
54谢慧才,方葛丰,王满德.平面圆形线圈的磁弹性屈曲.力学学报,1991,23(6):706-711
55 Lenotovich M.A and Shafranov V.D.The Stability of a Flexible Conductor in a Magnetic Field, Plasms Physics and the Problem of Controlled Thermonucler Recations, Pergamon Press, 1961
56 Van Lieshout P H, Rongen P M J and Van de Ven A AFA. Variational Principle for Magneto-elastic Buckling. J. Eng. Math, 1987, 21:227-252
57 Van Lieshout P.H and Van de Ven A.A.F.A Variational Approach to the Magneto-elastic Buckling Problems of an Arbitrary Number of Superconducting Beams. J. Eng. Math, 1991, 25:353-374
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59 Van de Ven AAF and Couwenberg MJH.Magneto-elastic Stability of a Superconducting Ring in its Own Field.J. Eng. Math,1986,20:251-270
60王省哲,郑晓静,周又和.非圆Tokamak载流线圈的磁弹性分析.兰州大学学报(自然科学版),1999,35(1):34-43
61王海滨,周又和,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
62马文麒,杨承辉.一类耦合非线性振子同步混沌Hopf分岔及其电路仿真.物理学报.2005,54(3):1064-1070
63 Lorenz E N. Deterministic nonperiodic flow. J. Atoms. Sci.20,1963
64吴祥兴,陈忠等编著.混沌学导论.上海:上海科学技术文献出版社,1996
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66陈予恕.非线性振动系统的分岔与混沌理论.北京:高等教育出版社,1993
67陈昌萍,傅衣铭.单轴转动截锥扁壳的全局分岔与混沌.非线性动力学报.2002,9(1-2):56-61
68徐耀寰,蔡宗熙.矩形屈曲板受微扰时的浑沌现象.固体力学学报.1997,18(1):65-69
69王新志,王钢,赵永刚,赵宏,宋曦.圆薄板非线性动力分岔及混沌问题.甘肃工业大学学报.2003,29(1):140-142
70王永岗,戴诗亮.具有初挠度的受热双层金属圆板的混沌运动.清华大学学报(自然科学版),2004,Vol.44,No.8:1134-1137
71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)
72李银山,陈予恕,李伟锋.各种板边条件下大挠度圆板的全局分岔和混沌.天津大学学报2001,Vol.34 No.6:718-722
73米晋生,孙晋兰,王京.非线性粘弹性板条的分叉和混沌.太原理工大学学报,2002, Vol.33, No.6:666-668
74高原文,周又和,郑晓静.横向磁场激励下铁磁梁式板的混沌运动分析.力学学报,2002:Vol.34, No.1:101-108
75吴晓.屈曲黏弹性矩形板的非线性振动分岔.力学与实践,2001:Vol.23,No.1:40-43
76陆启韶编著.常微分方程的定性方法和分叉.北京:北京航空航天大学出版社,1989
77王高雄,周之铭等编.常微分方程.北京:高等教育出版社,1982
78陆启韶.分岔与奇异性.上海:上海科技教育出版社,1995,1-46
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80高普云编著.非线性动力学.北京:国防科技大学出版社,2005
81 J Williamson. On an Algebraic Problem, Concerning the Normal Forms of Linear Dynamical Systems. Amer of Math, 1963,58:141-163
82АмбарцумянСА,БагдасарянГЕ,БелубекянМВ.Магнитоупругостьтонкихоболо-чекипластин.Москва:Наука,1977:150-160
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84ДресвянниковВИ.Построениеуравненийдинамикиэлектропроводящихупругопластическихоболочекпридействиинестационарныхипластичности.1982,вып20:10-23
85程昌钧,朱正右著.《结构的屈曲与分岔》.兰州:兰州大学出版社,1991年9月第1版
86王竹溪,郭敦仁.特殊函数概论.北京:北京大学出版社,2000:601-636
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