载流薄板的磁弹性稳定问题分析
摘要
电磁弹性系统中的杆、薄板、薄壳等，当没有机械约束时，系统至少存在一个不稳定的运动模态，而存在约束时，当电流或磁场达到某一临界值后，系统将发生屈曲。因此，杆、薄板、薄壳等的磁弹性稳定问题分析是一个重要的理论和应用问题。
以前，对磁弹性稳定性理论的研究主要集中以下两方面：一、软铁磁性薄板屈曲理论模型的建立及修正；二、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。而对工程中屡见不鲜的在强磁场作用下的载流薄板、薄壳等结构元件的稳定性研究则尚未见诸文献。
本文将载流构件磁弹性稳定问题由线圈、杆件拓展到载流薄板，并引入特殊函数判别磁弹性稳定问题的失稳临界状态，得到了相关参量的数值关系。首先通过理论推导给出了载流薄板的磁弹性动力学方程，几何与物理方程，洛仑兹力的表达式，电动力学方程。并在此基础上，得出了载流薄板在电磁场与机械荷载共同作用下的磁弹性动力稳定性方程。然后，应用Galerkin原理将稳定性方程整理为特殊函数马丢方程的标准形式，并利用马丢方程的稳定解区域与非稳定解区域的分界，即系数和的本征值关系得出了磁弹性最低失稳临界状态的判别方程。最后，通过具体算例，得出了四边简支、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矩形薄板的磁弹性动力稳定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。得出了外加磁感应强度的变化对载流薄板的失稳临界电流密度影响极大，减小外加磁感应强度可以有效地提高失稳临界电流密度等结论。
Pole , thin plate , thin shell in the electromagnetic elastic system ,etc. when there is no machinery to restrain, the system has an unstable movement mode at least, and while exists restrain, after up to a certain critical value in electric current or the magnetic fielding, buckling must take place in the system. So, the magnetoelastic stability analysis of the pole , thin plate , thin shell is an important theory and question of using.
In the past, two following respects that the study on magnetoelastic stability theory was main and centralized : First , setting-up and revision of the soft ferromagnetism plates buckling theory model; Second, represented by the Tokamak Thermonuclear current-carrying Rector Coil, the stability of current-carrying coils and poles under magnetic field has been studied. However, it has not been seen in all documents yet that the study of stability of thin current-carrying plates and shells, etc. structure under strong magnetic field which is common occurrence to project.
This text expanded the magnetoelastic stability issue of current-carrying component from coil , pole piece to thin current-carrying plates, and used special function to differentiate the critical state of losing magnetoelastic steady, the number value of getting the relevant parameter is solved .The text derives and provides the magnetoelastic movement equation of thin current-carrying plates, the geometry equations and the physics equations, the expression formula of Lorent’s force, electro dynamics equation through the theory at first. On this basis, draw magnetoelastic movement stability equation of thin current-carrying plates under the effect of electric magnetic field and machinery load. Then, use Galerkin principle to put the stability equation in order for the special function the Mathieu standard form, and utilize the boundary of the Mathieu solves’ steady area and unsteady area , namely special relationships of and,draw the discrimination equation of losing the magnetoelastic steady critical state. Finally, through regarding as the example concretely, relation curve between the relevant parameters while losing the magnetoelastic steady critical state were drawn in four kinds of different border conditions plate, and analyses about its change law. Have drawn the
following conclusion: the intensity of magnetic field outside has a great influence to the density of the electric current of lose steady of thin current-carrying plates, and reduce the intensity of magnetic field can effectively increase the density of the electric current of lose steady,etc.
以前，对磁弹性稳定性理论的研究主要集中以下两方面：一、软铁磁性薄板屈曲理论模型的建立及修正；二、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。而对工程中屡见不鲜的在强磁场作用下的载流薄板、薄壳等结构元件的稳定性研究则尚未见诸文献。
本文将载流构件磁弹性稳定问题由线圈、杆件拓展到载流薄板，并引入特殊函数判别磁弹性稳定问题的失稳临界状态，得到了相关参量的数值关系。首先通过理论推导给出了载流薄板的磁弹性动力学方程，几何与物理方程，洛仑兹力的表达式，电动力学方程。并在此基础上，得出了载流薄板在电磁场与机械荷载共同作用下的磁弹性动力稳定性方程。然后，应用Galerkin原理将稳定性方程整理为特殊函数马丢方程的标准形式，并利用马丢方程的稳定解区域与非稳定解区域的分界，即系数和的本征值关系得出了磁弹性最低失稳临界状态的判别方程。最后，通过具体算例，得出了四边简支、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矩形薄板的磁弹性动力稳定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。得出了外加磁感应强度的变化对载流薄板的失稳临界电流密度影响极大，减小外加磁感应强度可以有效地提高失稳临界电流密度等结论。
Pole , thin plate , thin shell in the electromagnetic elastic system ,etc. when there is no machinery to restrain, the system has an unstable movement mode at least, and while exists restrain, after up to a certain critical value in electric current or the magnetic fielding, buckling must take place in the system. So, the magnetoelastic stability analysis of the pole , thin plate , thin shell is an important theory and question of using.
In the past, two following respects that the study on magnetoelastic stability theory was main and centralized : First , setting-up and revision of the soft ferromagnetism plates buckling theory model; Second, represented by the Tokamak Thermonuclear current-carrying Rector Coil, the stability of current-carrying coils and poles under magnetic field has been studied. However, it has not been seen in all documents yet that the study of stability of thin current-carrying plates and shells, etc. structure under strong magnetic field which is common occurrence to project.
This text expanded the magnetoelastic stability issue of current-carrying component from coil , pole piece to thin current-carrying plates, and used special function to differentiate the critical state of losing magnetoelastic steady, the number value of getting the relevant parameter is solved .The text derives and provides the magnetoelastic movement equation of thin current-carrying plates, the geometry equations and the physics equations, the expression formula of Lorent’s force, electro dynamics equation through the theory at first. On this basis, draw magnetoelastic movement stability equation of thin current-carrying plates under the effect of electric magnetic field and machinery load. Then, use Galerkin principle to put the stability equation in order for the special function the Mathieu standard form, and utilize the boundary of the Mathieu solves’ steady area and unsteady area , namely special relationships of and,draw the discrimination equation of losing the magnetoelastic steady critical state. Finally, through regarding as the example concretely, relation curve between the relevant parameters while losing the magnetoelastic steady critical state were drawn in four kinds of different border conditions plate, and analyses about its change law. Have drawn the
following conclusion: the intensity of magnetic field outside has a great influence to the density of the electric current of lose steady of thin current-carrying plates, and reduce the intensity of magnetic field can effectively increase the density of the electric current of lose steady,etc.
引文
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Hasanyan, Davresh, Librescu, Liviu. Nonlinear vibration of finitely electro-conduct- ive plate-strips in a magnetic field. Collection of Technical Papers-AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference,2003,v 4:2827-2837
Gaganidze,E., Esquinazi,P., Ziese,M. Dynamical response of vibrating ferromagnets. Joural of Magnetism and Magnetic Materials,2000,210(3):49-62
胡宇达,白象忠.圆柱壳体的非线性磁弹性振动问题.工程力学,2000,17(1): 35-39
胡宇达,徐耀玲,白象忠.横向磁场中矩形金属薄板的振动问题.燕山大学学报,1999,23(1):18-19
马世麟,胡宇达,徐耀玲,等.广义洛仑兹力的磁弹性效应.?工程力学,1999, 16(2):79-84
马世麟,谭文锋,白象忠.洛仑兹力的应力应变效应.工程力学,2000,17(1):88-93
田振国,白象忠,马世麟.载流圆形薄板的磁弹性应力与变形分析.力学季刊,2002,23(1):109-114
梁伟,沈亚鹏,方岱宁.软铁磁材料平面裂纹问题的耦合场,力学学报,2001,33(6):758-767
Chun-Bo Lin, Chau-Shioung Yeh. The magnetoelastic problem of a crack in a soft ferromagnetic solid. International Journal of Solids and Structures,2002,(39):1-17
Chun-Bo Lin, Hsien-Mou Lin. The magnetoelastic problem of cracks in the bonded dissimilar materials. International Journal of Solids and Structures,2002,(39):2807- 2826
Chun-Bo Lin. On the bounded elliptic elastic inclusion in plane magnetoelasticity. International Journal of Solids and Structures,2003,(40):1547-1565
白象忠,胡宇达.电磁热裂纹止裂中的尺寸效应.燕山大学学报,1999,23(3):196-198
白象忠,胡宇达,脉冲电流在导电薄板裂纹止裂技术中的应用.固体力学学报,2000,21(4):335-340
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Takagi T, Tani J, Matsubara Y, et al. Dynamic behavior of fusion structural components under strong magnetic fields. Fusion Eng and Design, 1995, (27): 481-489
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van Lieshout P H, van de Ven A A F. A variational approach to the magnetoelastic buckling of an arbitrary number of superconducting beams. J Eng Math, 1991,(25): 353-374
Zhou Y H, Zheng X J. A theoretical model of magnetoelastic buckling for soft ferromagnetic thin plates. Acta Mechanica Sinica, 1996, 12(3):213-224
Zhou Y H, Miya K. A theoretical prediction of natural frequency of a ferromagnetic plate with low susceptibility in in-plane magnetic field. ASME J Appl Mech,1998,65(1):121-126
周又和,郑晓静.铁磁体磁弹性相互作用的广义变分原理与理论模型.中国科学A辑,1999,29(1):61-68
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
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,2002,34(1):101-108
Yang, W., Pan, H., Zheng, D. Energy method for analyzing magnetoelastic buckling and bending of ferromagnetic plates in static magnetic fields. Journal of Applied Mechanics, Transactions ASME,1999,66(4):913-917
Zhou,Youhe, Gao,Yuanwen, Zheng,Xiaojing. Buckling and post-buckling analysis for magneto-elastic–plastic ferromagnetic beam-plates with unmovable simple supports. Int. J. Solids and Structures,2003,(40):2875-2887
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
Woodson H.H and Melcher J.R, Electromechanical Dynamics, Part I, II, III, John Wiley and Sons, New York (1968)
Wolfe P, Bifurcation Theory of an Elastic Conducting Wire Subjected Magnetic Force. J. of Elasticity, (1990),23 (2-3):201-217
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
Wolfe P.Bifurcation Theory of an Elastic Conducting Rod in a Magnetic Field, Quart Mech.Appl.Math,1988,41(2):265-279
Peter Wolfe, Bifurcation Theory of an Elastic Conducting Rod Subjected to Magnetic Forces, Int. J. Nonlinear Mechanics,1990,25(5):597-604
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Moon F.C.Buckling of a Superconducting Ring in a Toroidal Magnetic Field, J. Appl. Mech, 1979, 46(1):151-155
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
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
Van Lieshout P.H, Rongen P.M.J and Van de Ven A.A.F.A Variational Principle for Magneto-elastic Buckling. J. Eng. Math, 1987, (21):227-252
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
Chattopadhyay S and Moon F.C.Magneto-elastic Buckling and Vibration of a Rod Carrying Electric Current.J. Appl. Mech, 1975,(42):809-814
Van de Ven A.A.F and Couwenberg M.J.H.Magneto-elastic Stability of a Superconducting Ring in its Own Field.J. Eng. Math,1986,(20):251-270
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
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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周又和,郑晓静.电磁固体结构力学.北京:科学出版社,1999:1-8
Hasanyan,D.J. Khachaturyan,G.M. Piliposyan,G.T. Mathematical modeling and inv- estigation of nonlinear vibration of perfectly conductive plates in an inclined magne- tic field.Thin-Walled Structures,2001,39(1):111-123
Hasanyan, Davresh, Librescu, Liviu. Nonlinear vibration of finitely electro-conduct- ive plate-strips in a magnetic field. Collection of Technical Papers-AIAA/ASME/ ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference,2003,v 4:2827-2837
Gaganidze,E., Esquinazi,P., Ziese,M. Dynamical response of vibrating ferromagnets. Joural of Magnetism and Magnetic Materials,2000,210(3):49-62
胡宇达,白象忠.圆柱壳体的非线性磁弹性振动问题.工程力学,2000,17(1): 35-39
胡宇达,徐耀玲,白象忠.横向磁场中矩形金属薄板的振动问题.燕山大学学报,1999,23(1):18-19
马世麟,胡宇达,徐耀玲,等.广义洛仑兹力的磁弹性效应.?工程力学,1999, 16(2):79-84
马世麟,谭文锋,白象忠.洛仑兹力的应力应变效应.工程力学,2000,17(1):88-93
田振国,白象忠,马世麟.载流圆形薄板的磁弹性应力与变形分析.力学季刊,2002,23(1):109-114
梁伟,沈亚鹏,方岱宁.软铁磁材料平面裂纹问题的耦合场,力学学报,2001,33(6):758-767
Chun-Bo Lin, Chau-Shioung Yeh. The magnetoelastic problem of a crack in a soft ferromagnetic solid. International Journal of Solids and Structures,2002,(39):1-17
Chun-Bo Lin, Hsien-Mou Lin. The magnetoelastic problem of cracks in the bonded dissimilar materials. International Journal of Solids and Structures,2002,(39):2807- 2826
Chun-Bo Lin. On the bounded elliptic elastic inclusion in plane magnetoelasticity. International Journal of Solids and Structures,2003,(40):1547-1565
白象忠,胡宇达.电磁热裂纹止裂中的尺寸效应.燕山大学学报,1999,23(3):196-198
白象忠,胡宇达,脉冲电流在导电薄板裂纹止裂技术中的应用.固体力学学报,2000,21(4):335-340
Moon F.C, Earnshaw’s Theorem and Magnetoelastic Buckling of Superconducting Structures, Proceedings of the IUTAM-IUPAP Symposium on the Mechanical Behaviors of Electromagnetic Solid Continua, Paris (1983):378-389
Brown W F. Magnetoelastic Interactions. New York: Springer－Verlag, 1966
Moon F C, Pao Y H. Magnetoelastic buckling of a thin plate. ASME J Appl Mech, 1968, 35(1): 53-58
Takagi T, Tani J, Matsubara Y, et al. Dynamic behavior of fusion structural components under strong magnetic fields. Fusion Eng and Design, 1995, (27): 481-489
Maugin G A. A continuum approach to magnonphonon couplings─I.Int J Eng Sci, 1981,(17):1073-1091
van Lieshout P H, van de Ven A A F. A variational approach to the magnetoelastic buckling of an arbitrary number of superconducting beams. J Eng Math, 1991,(25): 353-374
Zhou Y H, Zheng X J. A theoretical model of magnetoelastic buckling for soft ferromagnetic thin plates. Acta Mechanica Sinica, 1996, 12(3):213-224
Zhou Y H, Miya K. A theoretical prediction of natural frequency of a ferromagnetic plate with low susceptibility in in-plane magnetic field. ASME J Appl Mech,1998,65(1):121-126
周又和,郑晓静.铁磁体磁弹性相互作用的广义变分原理与理论模型.中国科学A辑,1999,29(1):61-68
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
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,2002,34(1):101-108
Yang, W., Pan, H., Zheng, D. Energy method for analyzing magnetoelastic buckling and bending of ferromagnetic plates in static magnetic fields. Journal of Applied Mechanics, Transactions ASME,1999,66(4):913-917
Zhou,Youhe, Gao,Yuanwen, Zheng,Xiaojing. Buckling and post-buckling analysis for magneto-elastic–plastic ferromagnetic beam-plates with unmovable simple supports. Int. J. Solids and Structures,2003,(40):2875-2887
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
Woodson H.H and Melcher J.R, Electromechanical Dynamics, Part I, II, III, John Wiley and Sons, New York (1968)
Wolfe P, Bifurcation Theory of an Elastic Conducting Wire Subjected Magnetic Force. J. of Elasticity, (1990),23 (2-3):201-217
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
Wolfe P.Bifurcation Theory of an Elastic Conducting Rod in a Magnetic Field, Quart Mech.Appl.Math,1988,41(2):265-279
Peter Wolfe, Bifurcation Theory of an Elastic Conducting Rod Subjected to Magnetic Forces, Int. J. Nonlinear Mechanics,1990,25(5):597-604
Moon F.C,Chattopadhyay S.Elastic Stability of a Thermonuclear Rector Coil, Princeton,(1973)
Moon F.C.Buckling of a Superconducting Ring in a Toroidal Magnetic Field, J. Appl. Mech, 1979, 46(1):151-155
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
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
Van Lieshout P.H, Rongen P.M.J and Van de Ven A.A.F.A Variational Principle for Magneto-elastic Buckling. J. Eng. Math, 1987, (21):227-252
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
Chattopadhyay S and Moon F.C.Magneto-elastic Buckling and Vibration of a Rod Carrying Electric Current.J. Appl. Mech, 1975,(42):809-814
Van de Ven A.A.F and Couwenberg M.J.H.Magneto-elastic Stability of a Superconducting Ring in its Own Field.J. Eng. Math,1986,(20):251-270
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
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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