不同边界条件下轴向运动矩形薄板的磁弹性振动
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摘要
随着现代科学技术的高速发展，利用铁磁、导体、压电等材料作为主要构件的电磁装置越来越多应用于工程实际中，使得磁弹性振动问题日益显现。而随着电磁固体力学的发展，电磁场和机械场共同作用下的轴向系统也受到更广泛的关注。因此，有必要进一步开展这方面的研究工作，如何对众多处于电磁场环境下的振动结构进行安全设计和可靠性评估，具有重要的理论和现实意义。
本文主要研究了不同边界条件下的轴向运动矩形薄板的磁弹性振动问题。首先，根据电磁理论和麦克斯韦方程组导出了电磁力和电磁力矩，根据薄板理论并运用哈密顿原理推导出了轴向运动矩形薄板的磁弹性振动方程。
其次，针对四种不同的边界条件，设定位移函数，并应用伽辽金方法对薄板磁弹性振动方程进行积分，分别导出了轴向运动矩形薄板的自由振动微分方程、受迫振动微分方程。通过对系统振动微分方程进行求解，得到了系统自由振动频率变化规律曲线图、响应图和相图，受迫振动的幅频曲线和相频曲线及响应图和相图；对薄板在四种不同边界条件下的曲线图进行了比较，讨论了板厚、速度、磁感应强度及边界条件变化对薄板振动特性的影响。
最后，针对轴向变速运动薄板的参数振动特性进行了分析。通过变化速度推导出了马蒂厄方程并利用平均法进行求解，得到了横向磁场中薄板振动的近似稳定性图，讨论了电磁和机械参数对薄板振动特性的影响，并对四种不同边界条件下的曲线图进行了分析比较。
With the rapid development of modern science and technology, using ferromagneticconductor, piezoelectric and other materials as a major component of the electromagneticdevices has widely used in various construction projects, which makes the magneto-elasticvibration problems become increasingly apparent. With the development of electro-magnetic solid mechanics, the axial system under the action of Electromagnetic andmechanical has more and more applications to the project construction. Therefore, it isNecessary to carry out further research in this area, how to perform the safety design andreliability assessment of many vibrations in the electromagnetic environment hasimportant theoretical and practical significance.
This paper mainly focuses on the axial movement of the magneto-elastic rectangularplate vibration problems under different boundary conditions. Firstly, electromagneticforce and electromagnetic torque were derived from the electromagnetic theory andMaxwell's equations, a transverse magnetic field in the axial movement of the rectangularthin plate vibration equationwas obtained using Hamilton's principle according to the thinplate theory and the axial movement of the motion of the system parameters.
Secondly, for four different boundary conditions,we set the displacement function,and apply plus Galerkinmethod for integral equations of the system vibration. Furthermore,we derive the differential equations of free vibration, forced vibration and parametricvibration in axial motion systems. Discussed the thickness, velocity and magnetic fluxdensity change the vibration characteristics of the system.
Finally, through calculating by matlab programming, axial movement of thetransverse magnetic field in the system vibration characteristics were analyzed. In the caseof given parameters, the system free vibration frequency plot,response plans and phasediagram of the forced vibration amplitude-frequency curves, phase-frequency curves andparametric vibration stability diagram were formed, diagramwereformed, By changing theparameters, the influence of electromagnetic mechanical parameters on the vibration characteristics of the systemwas was discussed, and the graph was analyzed and comparedunder four different boundary conditions.
本文主要研究了不同边界条件下的轴向运动矩形薄板的磁弹性振动问题。首先，根据电磁理论和麦克斯韦方程组导出了电磁力和电磁力矩，根据薄板理论并运用哈密顿原理推导出了轴向运动矩形薄板的磁弹性振动方程。
其次，针对四种不同的边界条件，设定位移函数，并应用伽辽金方法对薄板磁弹性振动方程进行积分，分别导出了轴向运动矩形薄板的自由振动微分方程、受迫振动微分方程。通过对系统振动微分方程进行求解，得到了系统自由振动频率变化规律曲线图、响应图和相图，受迫振动的幅频曲线和相频曲线及响应图和相图；对薄板在四种不同边界条件下的曲线图进行了比较，讨论了板厚、速度、磁感应强度及边界条件变化对薄板振动特性的影响。
最后，针对轴向变速运动薄板的参数振动特性进行了分析。通过变化速度推导出了马蒂厄方程并利用平均法进行求解，得到了横向磁场中薄板振动的近似稳定性图，讨论了电磁和机械参数对薄板振动特性的影响，并对四种不同边界条件下的曲线图进行了分析比较。
With the rapid development of modern science and technology, using ferromagneticconductor, piezoelectric and other materials as a major component of the electromagneticdevices has widely used in various construction projects, which makes the magneto-elasticvibration problems become increasingly apparent. With the development of electro-magnetic solid mechanics, the axial system under the action of Electromagnetic andmechanical has more and more applications to the project construction. Therefore, it isNecessary to carry out further research in this area, how to perform the safety design andreliability assessment of many vibrations in the electromagnetic environment hasimportant theoretical and practical significance.
This paper mainly focuses on the axial movement of the magneto-elastic rectangularplate vibration problems under different boundary conditions. Firstly, electromagneticforce and electromagnetic torque were derived from the electromagnetic theory andMaxwell's equations, a transverse magnetic field in the axial movement of the rectangularthin plate vibration equationwas obtained using Hamilton's principle according to the thinplate theory and the axial movement of the motion of the system parameters.
Secondly, for four different boundary conditions,we set the displacement function,and apply plus Galerkinmethod for integral equations of the system vibration. Furthermore,we derive the differential equations of free vibration, forced vibration and parametricvibration in axial motion systems. Discussed the thickness, velocity and magnetic fluxdensity change the vibration characteristics of the system.
Finally, through calculating by matlab programming, axial movement of thetransverse magnetic field in the system vibration characteristics were analyzed. In the caseof given parameters, the system free vibration frequency plot,response plans and phasediagram of the forced vibration amplitude-frequency curves, phase-frequency curves andparametric vibration stability diagram were formed, diagramwereformed, By changing theparameters, the influence of electromagnetic mechanical parameters on the vibration characteristics of the systemwas was discussed, and the graph was analyzed and comparedunder four different boundary conditions.
引文
[1]周又和,郑晓静.电磁固体力学[M].北京:科学出版社,1999:21-64.
[2]白象忠.板壳磁弹性理论基础[M].北京:机械工业出版社,1996:51-84.
[3]刘延柱,陈文良,陈立群.振动力学[M].北京:高等教育出版社,1998:2-6.
[4]刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001:83-95.
[5] Ambarcumian S A, Bagdasarian G E, Belubekian M V. Magneto-elastic of Thin Shell and Plate[J].Moscow: Science,1977,14(3):63-89.
[6] Moon F C. Magneto-solid Mechanics. New York: John[J]. Wiley and Sons,1984.11(4):55-68.
[7]周又和,郑晓静.软铁磁薄板磁弹性屈曲的理论模型[J].力学学报,1996,28(6):651-660.
[8] Lu Q S. Dynamic Stability and Bifurcation of An Alternating Load and Magnetic Field ExcitedMagneto-elastic Beam[J]. Journal of Sound and Vibration,1995,181(5):873-891.
[9]王知人,白象忠,边宇虹.对边简支载流薄板在电磁场中稳定性分析[J].机械工程学报,2007,43(10):155-160.
[10]胡宇达.传导薄板的非线性磁弹性振动问题[J].工程力学,2001,18(4):89-94.
[11]李佰洲.纵向磁场中导电条形薄板的非线性自由振动[D].河北:燕山大学硕士学位论文,2006:2-33.
[12]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[J].动力学与控制学报,2004,2(4):35-38.
[13]李晶.磁场中导电薄板的非线性振动与稳定性分析[D].河北:燕山大学硕士学位论文,2005:4-30.
[14]胡宇达,白象忠.圆柱壳体的非线性磁弹性振动[J].工程力学,2000,17(1):35-40.
[15]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[J].力学学报,2002,34(1):101-108.
[16]高原文,祈发强.磁脉冲作用下铁磁梁式板的磁弹塑性动力响应与失稳[J].振动工程学报,2007,20(1):61-65.
[17]周又和,高原文,郑晓静.铁磁矩形简支板的磁弹塑性弯曲行为分析[J].计算力学学报,2004,21(5):522-528.
[18] Wang X Z, Zhou Y H, Zheng X J. A Generalized Variational Model of Magneto-thermoelasticityfor Nonlinearly Magnetized Ferroelastic Bodies[J]. International Journal of Engineering Science,2002,30(17):1957-1973.
[19] Wang X Z, Zheng X J. Analyses on Non-linear Coupling of Magneto Thermo Elasticit ofFerromagnetic Thin Shell-I: Generalized Variational the Oretical Modeling[J], Acta MechanicaSolida Sinica,2009,22(3):189-196.
[20] Hu Y D, Li J. Magneto-elastic Combination Resonances Analysis of Current-conducting ThinPlate[J]. Applied Mathematics and Mechanics,2008,29(8):1053-1066.
[21] Hu Y D, Li J. The Magneto-elastic Subharmonic Resonance of Current-conducting Thin Plate inMagnetic Filed[J]. Journal of Sound and Vibration,2009,319(3-5):1107-1120.
[22]胡宇达,赵将.横向磁场中导电圆柱壳体的超谐波共振[J].机械强度,2011,33(1):11-14.
[23]胡宇达,赵将.杜国君横向磁场中导电壳体的亚谐波共振与稳定性[J].应用力学学报,2009,26(3):408-412.
[24]赵将,胡宇达.恒定磁场中简支圆柱壳的磁弹性振动分析[J].动力学与控制学报,2006,4(3):267-271.
[25]黄建亮,陈树辉.轴向运动体系非线性振动分析的多元L-P方法[J].中山大学学报,2004,43(4):115-117.
[26]黄建亮,陈树辉.轴向运动体系横向非线性振动的联合共振[J].振动工程学报,2005,18(1):18-23.
[27]黄建亮.轴向运动体系的非线性振动研究[D].广东:中山大学硕士学位论文,2004:4-9.
[28]殷振坤,陈树辉.轴向运动薄板非线性振动及其稳定性研究[J].动力学与控制学报,2007,5(4):314-318.
[29] Pellicano F. On the Dynamic Properties of Axially Moving Systems[J]. Journal of Sound andVibration,2005,3(22):593-609.
[30] Zhao H. On the Control of Axially Moving Material Systems[J]. Journal of Vibration andAcoustics,2006,8(128):527-531.
[31] Ghayesh M H, Khadem S E. Rotary Inertia and Temperature Effects on Non-linear Vibration,Steady-state Response and Stability of An Axially Moving Beam with Time-dependentVelocity[J]. International Journal of Mechanical Sciences,2008,3(50):389-404.
[32] Chena S H, Huanga J L, Szeb K Y. Multidimensional Lindstedt–poincaré Method for NonlinearVibration of Axially Moving Beams[J]. Journal of Sound and Vibration,2007,9(25):1-11.
[33] Wang X D. Numerical Analysis of Moving Orthotropic Thin Plates[J]. Computers and Structures,1999,2(4):467-486.
[34]钟阳,李锐,刘月梅.四边固支矩形板弹性薄板的精确解析解[J].力学季刊,2009,6(30):297-303.
[35] Hyde K, Chang J. Parameter Studies for Plane Stress In-plane Vibration of Rectangular Plates[J].Journal of Sound and Vibration,2001,10(25):471-487.
[36] Haoa X, Chen L H, Zhang W, et al. Nonlinear Oscillations, Bifurcations and Chaos ofFunctionally Graded Materials Plate[J]. Journal of Sound and Vibration,2007,5(20):862-892.
[37]陈伟球,叶贵如,蔡金标等.横观各项同性功能梯度材料矩形板的自由振动[J].振动工程学报,2001,14(3):1122-1131.
[38]胡宇达,杜国君.磁场环境下导电圆形薄板的磁弹性强迫振动[J].工程力学,2007,24(7):184-188.
[39]郝芹.轴向运动梁的横向振动[D].湖北:华中科技大学硕士学位论文,2006:35-50.
[40]杨志安,赵雪娟,席晓燕.非线性弹性地基上矩形薄板的主参数共振[J].岩土力学,2005,26(12):1921-1925.
[41]徐芝纶.弹性力学[M].北京:高等教育出版社,2003:142-150.
[42]彭旭麟,罗汝梅.变分法及其应用[M].武昌:华中工学院出版社,1983:21-52.
[43]钟益林,彭乐群,刘炳文.常微分方程及其Maple, MATLAB求解[M].北京:清华大学出版社,2007:16-63.
[2]白象忠.板壳磁弹性理论基础[M].北京:机械工业出版社,1996:51-84.
[3]刘延柱,陈文良,陈立群.振动力学[M].北京:高等教育出版社,1998:2-6.
[4]刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001:83-95.
[5] Ambarcumian S A, Bagdasarian G E, Belubekian M V. Magneto-elastic of Thin Shell and Plate[J].Moscow: Science,1977,14(3):63-89.
[6] Moon F C. Magneto-solid Mechanics. New York: John[J]. Wiley and Sons,1984.11(4):55-68.
[7]周又和,郑晓静.软铁磁薄板磁弹性屈曲的理论模型[J].力学学报,1996,28(6):651-660.
[8] Lu Q S. Dynamic Stability and Bifurcation of An Alternating Load and Magnetic Field ExcitedMagneto-elastic Beam[J]. Journal of Sound and Vibration,1995,181(5):873-891.
[9]王知人,白象忠,边宇虹.对边简支载流薄板在电磁场中稳定性分析[J].机械工程学报,2007,43(10):155-160.
[10]胡宇达.传导薄板的非线性磁弹性振动问题[J].工程力学,2001,18(4):89-94.
[11]李佰洲.纵向磁场中导电条形薄板的非线性自由振动[D].河北:燕山大学硕士学位论文,2006:2-33.
[12]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[J].动力学与控制学报,2004,2(4):35-38.
[13]李晶.磁场中导电薄板的非线性振动与稳定性分析[D].河北:燕山大学硕士学位论文,2005:4-30.
[14]胡宇达,白象忠.圆柱壳体的非线性磁弹性振动[J].工程力学,2000,17(1):35-40.
[15]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[J].力学学报,2002,34(1):101-108.
[16]高原文,祈发强.磁脉冲作用下铁磁梁式板的磁弹塑性动力响应与失稳[J].振动工程学报,2007,20(1):61-65.
[17]周又和,高原文,郑晓静.铁磁矩形简支板的磁弹塑性弯曲行为分析[J].计算力学学报,2004,21(5):522-528.
[18] Wang X Z, Zhou Y H, Zheng X J. A Generalized Variational Model of Magneto-thermoelasticityfor Nonlinearly Magnetized Ferroelastic Bodies[J]. International Journal of Engineering Science,2002,30(17):1957-1973.
[19] Wang X Z, Zheng X J. Analyses on Non-linear Coupling of Magneto Thermo Elasticit ofFerromagnetic Thin Shell-I: Generalized Variational the Oretical Modeling[J], Acta MechanicaSolida Sinica,2009,22(3):189-196.
[20] Hu Y D, Li J. Magneto-elastic Combination Resonances Analysis of Current-conducting ThinPlate[J]. Applied Mathematics and Mechanics,2008,29(8):1053-1066.
[21] Hu Y D, Li J. The Magneto-elastic Subharmonic Resonance of Current-conducting Thin Plate inMagnetic Filed[J]. Journal of Sound and Vibration,2009,319(3-5):1107-1120.
[22]胡宇达,赵将.横向磁场中导电圆柱壳体的超谐波共振[J].机械强度,2011,33(1):11-14.
[23]胡宇达,赵将.杜国君横向磁场中导电壳体的亚谐波共振与稳定性[J].应用力学学报,2009,26(3):408-412.
[24]赵将,胡宇达.恒定磁场中简支圆柱壳的磁弹性振动分析[J].动力学与控制学报,2006,4(3):267-271.
[25]黄建亮,陈树辉.轴向运动体系非线性振动分析的多元L-P方法[J].中山大学学报,2004,43(4):115-117.
[26]黄建亮,陈树辉.轴向运动体系横向非线性振动的联合共振[J].振动工程学报,2005,18(1):18-23.
[27]黄建亮.轴向运动体系的非线性振动研究[D].广东:中山大学硕士学位论文,2004:4-9.
[28]殷振坤,陈树辉.轴向运动薄板非线性振动及其稳定性研究[J].动力学与控制学报,2007,5(4):314-318.
[29] Pellicano F. On the Dynamic Properties of Axially Moving Systems[J]. Journal of Sound andVibration,2005,3(22):593-609.
[30] Zhao H. On the Control of Axially Moving Material Systems[J]. Journal of Vibration andAcoustics,2006,8(128):527-531.
[31] Ghayesh M H, Khadem S E. Rotary Inertia and Temperature Effects on Non-linear Vibration,Steady-state Response and Stability of An Axially Moving Beam with Time-dependentVelocity[J]. International Journal of Mechanical Sciences,2008,3(50):389-404.
[32] Chena S H, Huanga J L, Szeb K Y. Multidimensional Lindstedt–poincaré Method for NonlinearVibration of Axially Moving Beams[J]. Journal of Sound and Vibration,2007,9(25):1-11.
[33] Wang X D. Numerical Analysis of Moving Orthotropic Thin Plates[J]. Computers and Structures,1999,2(4):467-486.
[34]钟阳,李锐,刘月梅.四边固支矩形板弹性薄板的精确解析解[J].力学季刊,2009,6(30):297-303.
[35] Hyde K, Chang J. Parameter Studies for Plane Stress In-plane Vibration of Rectangular Plates[J].Journal of Sound and Vibration,2001,10(25):471-487.
[36] Haoa X, Chen L H, Zhang W, et al. Nonlinear Oscillations, Bifurcations and Chaos ofFunctionally Graded Materials Plate[J]. Journal of Sound and Vibration,2007,5(20):862-892.
[37]陈伟球,叶贵如,蔡金标等.横观各项同性功能梯度材料矩形板的自由振动[J].振动工程学报,2001,14(3):1122-1131.
[38]胡宇达,杜国君.磁场环境下导电圆形薄板的磁弹性强迫振动[J].工程力学,2007,24(7):184-188.
[39]郝芹.轴向运动梁的横向振动[D].湖北:华中科技大学硕士学位论文,2006:35-50.
[40]杨志安,赵雪娟,席晓燕.非线性弹性地基上矩形薄板的主参数共振[J].岩土力学,2005,26(12):1921-1925.
[41]徐芝纶.弹性力学[M].北京:高等教育出版社,2003:142-150.
[42]彭旭麟,罗汝梅.变分法及其应用[M].武昌:华中工学院出版社,1983:21-52.
[43]钟益林,彭乐群,刘炳文.常微分方程及其Maple, MATLAB求解[M].北京:清华大学出版社,2007:16-63.