磁场中轴向运动薄板的非线性振动
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摘要
在工程实际中，多种工程系统如动力传送带、磁带、带锯、纸张、航空航天工程中应用的复合材料层合板等都可以模型化为轴向运动板。因此，研究轴向运动板的振动问题具有重要的理论和实际意义。本文研究了在横向磁场环境中轴向运动条形薄板的非线性磁弹性振动问题。
首先，利用电磁场理论推导出了薄板的电动力学方程和电磁力表达式，再应用哈密顿变分原理导出了机械场和磁场共同作用下的轴向运动条形薄板的非线性磁弹性振动方程。将满足对边简支对边自由边界条件下的位移解代入到振动方程，运用伽辽金积分并进行无量纲化处理，推导出其达芬型振动微分方程。
其次，采用多尺度法分别对条形板的非线性自由振动、主共振、组合共振问题进行求解，得到了一次近似解析解。针对主共振和组合共振的情况，得到了幅频响应方程。基于李雅普诺夫稳定性理论，得到了解的稳定性判定条件，并对系统稳态运动下解的稳定性进行了分析。
最后，通过数值算例，分别给出了在不同参数情况下非线性自由振动、主共振、组合共振的时间历程响应图和相图。针对主共振和组合共振的情况，通过幅频响应方程，得到了幅频曲线图。针对组合共振情况，得到了频谱图。分析了磁感应强度、频率调谐参数、轴向速度、激励幅值、板厚等电磁参数和机械参数对系统振动特性的影响，并对系统存在的单周期运动、概周期运动和混沌等复杂动力学行为进行了探讨。
A variety of engineering systems with axially moving such as power transmissionbelts, tapes, band saw, paper and the composite material laminate of aerospace engineer-ing etc.can be simplified as axially moving plates in engineering practice application.Therefore, the investigation on vibration of the axially moving plates is significant notonly in thoeretical but also in engineering practice.In this paper, research works ofnonlinear magnetoelastic vibration of the axially moving strip thin plates subjected to antransverse magnetic field have done.
First, based on the electromagnetic theory,electrodynamics equation electromagne-tic forces are deduced. A sets of nonlinear magnetoelastic vibration equation of theaxially moving strip thin plates which in the mechanic filed and magnetic filed are givenby using Hamiltonian variational principle. Substitute the displacement solution into thevibration equation which meet the boundary condition of two opposite ends simplysupported and two opposite ends free, using the method of Galerkin to integrate it andmake it dimensionles, then the Duffing type vibration differential equation is deduced.
Second, solve the problems of nonlinear free vibration, main resonances and com-bination resonances by using the method of Multiple Scales, and the first approximateanalytic solutions is deduced, and amplitude-frequency fuctions of main resonances andcombination resonances are also obtained. based on the lyapunov stability theory, thediscriminant of stable solutions is obtained, and the stability of stable solution isanalyzed.
At last, by means of calculated example, the time history response plots and phasecharts of three vibration conditions: nonlinear free vibration, main resonances and com-bination resonances are obtained. In cases of mian resonances and combination resonan-ces, spectrum charts is obtained, and obtained the amplitude frequency curve by theamplitude frequency response equation. The influences of system’s vibration characteris-tics under the condition of magnetic induction intensity, speed, frequency tuning parame-ters, incentive amplitude, thickness of the electromagnetic parameters and mechanical parameters are analyzed, and the complex dynamic behaviors of axially moving systems:such as as single-frequency periodic motion, quasi-period motion and dou-bling-periodmotion are discussed.
首先，利用电磁场理论推导出了薄板的电动力学方程和电磁力表达式，再应用哈密顿变分原理导出了机械场和磁场共同作用下的轴向运动条形薄板的非线性磁弹性振动方程。将满足对边简支对边自由边界条件下的位移解代入到振动方程，运用伽辽金积分并进行无量纲化处理，推导出其达芬型振动微分方程。
其次，采用多尺度法分别对条形板的非线性自由振动、主共振、组合共振问题进行求解，得到了一次近似解析解。针对主共振和组合共振的情况，得到了幅频响应方程。基于李雅普诺夫稳定性理论，得到了解的稳定性判定条件，并对系统稳态运动下解的稳定性进行了分析。
最后，通过数值算例，分别给出了在不同参数情况下非线性自由振动、主共振、组合共振的时间历程响应图和相图。针对主共振和组合共振的情况，通过幅频响应方程，得到了幅频曲线图。针对组合共振情况，得到了频谱图。分析了磁感应强度、频率调谐参数、轴向速度、激励幅值、板厚等电磁参数和机械参数对系统振动特性的影响，并对系统存在的单周期运动、概周期运动和混沌等复杂动力学行为进行了探讨。
A variety of engineering systems with axially moving such as power transmissionbelts, tapes, band saw, paper and the composite material laminate of aerospace engineer-ing etc.can be simplified as axially moving plates in engineering practice application.Therefore, the investigation on vibration of the axially moving plates is significant notonly in thoeretical but also in engineering practice.In this paper, research works ofnonlinear magnetoelastic vibration of the axially moving strip thin plates subjected to antransverse magnetic field have done.
First, based on the electromagnetic theory,electrodynamics equation electromagne-tic forces are deduced. A sets of nonlinear magnetoelastic vibration equation of theaxially moving strip thin plates which in the mechanic filed and magnetic filed are givenby using Hamiltonian variational principle. Substitute the displacement solution into thevibration equation which meet the boundary condition of two opposite ends simplysupported and two opposite ends free, using the method of Galerkin to integrate it andmake it dimensionles, then the Duffing type vibration differential equation is deduced.
Second, solve the problems of nonlinear free vibration, main resonances and com-bination resonances by using the method of Multiple Scales, and the first approximateanalytic solutions is deduced, and amplitude-frequency fuctions of main resonances andcombination resonances are also obtained. based on the lyapunov stability theory, thediscriminant of stable solutions is obtained, and the stability of stable solution isanalyzed.
At last, by means of calculated example, the time history response plots and phasecharts of three vibration conditions: nonlinear free vibration, main resonances and com-bination resonances are obtained. In cases of mian resonances and combination resonan-ces, spectrum charts is obtained, and obtained the amplitude frequency curve by theamplitude frequency response equation. The influences of system’s vibration characteris-tics under the condition of magnetic induction intensity, speed, frequency tuning parame-ters, incentive amplitude, thickness of the electromagnetic parameters and mechanical parameters are analyzed, and the complex dynamic behaviors of axially moving systems:such as as single-frequency periodic motion, quasi-period motion and dou-bling-periodmotion are discussed.
引文
[1]刘延柱,陈立群.非线性振动[M].北京:高等教育出版社,2001:83-88.
[2] Nayfeh A H, Mook D T. Nonlinear Oscillations[M]. New York: John Wiley&Sons,1979:161-172.
[3]陈予恕.非线性振动[M].北京:高等教育出版社,1983:237-239.
[4]周又和,郑晓静.电磁固体结构力学[M].北京:科学出版社,1999:11-12.
[5]白象忠,田振国.板壳磁弹性力学基础[M].北京:科学出版社,2006:184-188.
[6]白象忠.磁弹性、热磁弹性理论及其应用[J].力学进展,1996,26(3):389-406.
[7]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.
[8]胡宇达,李佰洲.纵向磁场中导电薄板的亚谐共振[J].工程力学,2011,28(2):223-228.
[9] 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.
[10] Zhu L L, Zhang J P, Zheng X J. Multi-Field Coupling Behavior of Simply-Supported ConductivePlateunder the Condition of a Transverse Strong Impulsive Magnetic Field[J]. Acta MechanicaSolida Sinica,2006,19(3):203-211.
[11] Hu Y D, Li J. Magneto-Elastic Combination Resonances Analysis of Current-Conducting ThinPlate[J]. Applied Mathematic and Mechanics,2008,29(8):1053-1066.
[12]李晶,胡宇达.横向磁场中导电薄板的非线性自由振动[J].振动与冲击,2006,25(s):693-695.
[13]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.
[14]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.
[15]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.
[16]胡宇达,白象忠.磁场中圆柱壳体的轴对称振动[J].工程力学,1999,16(5):47-52.
[17]胡宇达,徐耀玲.白象忠.纵向磁场中条形板的非线性振动[J].非线性动力学学报,1998,5(1):183-189.
[18]胡宇达,白象忠.倾斜磁场中条形薄板的磁弹性振动[J].振动与冲击,2000,19(2):64-66.
[19]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.
[20] Zhou Y H, Zheng X J. A Generalized Variational Principle and Theoretical Moder forMagnetoelastic Interaction of Ferromagnetic Bodies[J]. Science In China,1999,42(6):618-626.
[21] 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.
[22] Wang X Z, Zheng X J. Analyses on Nonlinear Coupling of Magneto Thermo Elasticit ofFerromagnetic Thin Shell-I: Generalized Variational the Oretical Modeling[J], Acta MechanicaSolida Sinica,2009,22(3):189-196.
[23]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):522-527.
[24]王省哲,郑晓静.铁磁梁式板磁弹性初始后屈曲及缺陷敏感性分析[J].力学学报,2006,38(1):32-40.
[25]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.
[26] Pellicano F. On the Dynamic Properties of Axially Moving Systems[J]. Journal of Sound andVibration,2005,281(3-5):593-609.
[27]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(3):19-23.
[28] Zhao H. On the Control of Axially Moving Material Systems[J]. Journal of Vibration andAcoustics,2006,128(4):527-531.
[29]涂剑平.轴向运动系统动力学及若干应用[D].南京:南京航空航天大学硕士学位论文,2009:9-12.
[30] Chen L Q, Yang X D.Transverse Nonlinear Dynamics of Axially Accelerating ViscoelasticBeams Based on4-term Galerkin Truncation[J]. Chaos, Solitions and Fractals,2006,27:748-757.
[31]丁虎,陈立群.轴向运动黏弹性梁横向非线性受迫振动[J].振动与冲击,2009,28(12):128-131.
[32]丁虎.轴向运动梁横向非线性振动建模、分析和仿真[D].上海:上海大学博士学位论文,2008:2-5.
[33]丁虎,陈立群,戈新生.黏弹性轴向运动梁非线性受迫振动稳态幅频响应[J].力学季刊,2008,29(3):502-505.
[34]杨晓东,陈立群.粘弹性轴向运动梁的非线性动力学行为[J].力学季刊,2005,26(1):157-162.
[35]杨晓东,陈立群.变速度轴向运动粘弹性梁的动态稳定性[J].应用数学和力学,2005,26(8):905-910.
[36] 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,2007,25(9):1-11.
[37] 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,2008,50(3):389-404.
[38] Banichuk N, Jeronen J, Neittaanm ki P, et al. On the Instability of An Axially Moving ElasticPlate[J]. International Journal of Solids and Structures,2010,47(1):91-99.
[39]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):314-319.
[40]刘金堂,杨晓东,张宇飞,等.变速轴向运动正交各向异性板横向振动的直接多尺度分析[J].机械强度,2010,32(4):531-535.
[41] Hatami S, Azhari M, Saadatpour M M. Nonlinear Analysis of Axially Moving Plates UsingFEM[J]. International Journal of Structural Stability and Dynamics,2007,7(4):589-607.
[42] Lin C C. Stability and Vibration Characteristics of Axially Moving Plates[J]. InternationalJournal of Solids and Structures,1997,34(24):3179-3190.
[43] Zhou Y F, Wang Z M. Vibrations of Axially Moving Viscoelastic Plate with ParabolicallyVarying Thickness[J]. Journal of Sound and Vibration,2008,316(1-5):198-210.
[44] Hatami S, Ronagh H R, Azhari M. Exact Free Vibration Analysis of Axially Moving ViscoelasticPlates[J]. Computers and Structures,2008,86(17-18):1738-1746.
[45]徐芝纶.弹性力学[M].北京:高等教育出版社,2003:142-149.
[46] Tang Y Q, Chen L Q. Nonlinear Free Transverse Vibrations of In-Plane Moving Plates: Withoutand with Internal Resonances[J]. Journal of Sound and Vibration,2010,330(2011):110–126.
[2] Nayfeh A H, Mook D T. Nonlinear Oscillations[M]. New York: John Wiley&Sons,1979:161-172.
[3]陈予恕.非线性振动[M].北京:高等教育出版社,1983:237-239.
[4]周又和,郑晓静.电磁固体结构力学[M].北京:科学出版社,1999:11-12.
[5]白象忠,田振国.板壳磁弹性力学基础[M].北京:科学出版社,2006:184-188.
[6]白象忠.磁弹性、热磁弹性理论及其应用[J].力学进展,1996,26(3):389-406.
[7]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.
[8]胡宇达,李佰洲.纵向磁场中导电薄板的亚谐共振[J].工程力学,2011,28(2):223-228.
[9] 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.
[10] Zhu L L, Zhang J P, Zheng X J. Multi-Field Coupling Behavior of Simply-Supported ConductivePlateunder the Condition of a Transverse Strong Impulsive Magnetic Field[J]. Acta MechanicaSolida Sinica,2006,19(3):203-211.
[11] Hu Y D, Li J. Magneto-Elastic Combination Resonances Analysis of Current-Conducting ThinPlate[J]. Applied Mathematic and Mechanics,2008,29(8):1053-1066.
[12]李晶,胡宇达.横向磁场中导电薄板的非线性自由振动[J].振动与冲击,2006,25(s):693-695.
[13]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.
[14]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.
[15]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.
[16]胡宇达,白象忠.磁场中圆柱壳体的轴对称振动[J].工程力学,1999,16(5):47-52.
[17]胡宇达,徐耀玲.白象忠.纵向磁场中条形板的非线性振动[J].非线性动力学学报,1998,5(1):183-189.
[18]胡宇达,白象忠.倾斜磁场中条形薄板的磁弹性振动[J].振动与冲击,2000,19(2):64-66.
[19]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.
[20] Zhou Y H, Zheng X J. A Generalized Variational Principle and Theoretical Moder forMagnetoelastic Interaction of Ferromagnetic Bodies[J]. Science In China,1999,42(6):618-626.
[21] 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.
[22] Wang X Z, Zheng X J. Analyses on Nonlinear Coupling of Magneto Thermo Elasticit ofFerromagnetic Thin Shell-I: Generalized Variational the Oretical Modeling[J], Acta MechanicaSolida Sinica,2009,22(3):189-196.
[23]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):522-527.
[24]王省哲,郑晓静.铁磁梁式板磁弹性初始后屈曲及缺陷敏感性分析[J].力学学报,2006,38(1):32-40.
[25]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.
[26] Pellicano F. On the Dynamic Properties of Axially Moving Systems[J]. Journal of Sound andVibration,2005,281(3-5):593-609.
[27]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(3):19-23.
[28] Zhao H. On the Control of Axially Moving Material Systems[J]. Journal of Vibration andAcoustics,2006,128(4):527-531.
[29]涂剑平.轴向运动系统动力学及若干应用[D].南京:南京航空航天大学硕士学位论文,2009:9-12.
[30] Chen L Q, Yang X D.Transverse Nonlinear Dynamics of Axially Accelerating ViscoelasticBeams Based on4-term Galerkin Truncation[J]. Chaos, Solitions and Fractals,2006,27:748-757.
[31]丁虎,陈立群.轴向运动黏弹性梁横向非线性受迫振动[J].振动与冲击,2009,28(12):128-131.
[32]丁虎.轴向运动梁横向非线性振动建模、分析和仿真[D].上海:上海大学博士学位论文,2008:2-5.
[33]丁虎,陈立群,戈新生.黏弹性轴向运动梁非线性受迫振动稳态幅频响应[J].力学季刊,2008,29(3):502-505.
[34]杨晓东,陈立群.粘弹性轴向运动梁的非线性动力学行为[J].力学季刊,2005,26(1):157-162.
[35]杨晓东,陈立群.变速度轴向运动粘弹性梁的动态稳定性[J].应用数学和力学,2005,26(8):905-910.
[36] 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,2007,25(9):1-11.
[37] 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,2008,50(3):389-404.
[38] Banichuk N, Jeronen J, Neittaanm ki P, et al. On the Instability of An Axially Moving ElasticPlate[J]. International Journal of Solids and Structures,2010,47(1):91-99.
[39]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):314-319.
[40]刘金堂,杨晓东,张宇飞,等.变速轴向运动正交各向异性板横向振动的直接多尺度分析[J].机械强度,2010,32(4):531-535.
[41] Hatami S, Azhari M, Saadatpour M M. Nonlinear Analysis of Axially Moving Plates UsingFEM[J]. International Journal of Structural Stability and Dynamics,2007,7(4):589-607.
[42] Lin C C. Stability and Vibration Characteristics of Axially Moving Plates[J]. InternationalJournal of Solids and Structures,1997,34(24):3179-3190.
[43] Zhou Y F, Wang Z M. Vibrations of Axially Moving Viscoelastic Plate with ParabolicallyVarying Thickness[J]. Journal of Sound and Vibration,2008,316(1-5):198-210.
[44] Hatami S, Ronagh H R, Azhari M. Exact Free Vibration Analysis of Axially Moving ViscoelasticPlates[J]. Computers and Structures,2008,86(17-18):1738-1746.
[45]徐芝纶.弹性力学[M].北京:高等教育出版社,2003:142-149.
[46] Tang Y Q, Chen L Q. Nonlinear Free Transverse Vibrations of In-Plane Moving Plates: Withoutand with Internal Resonances[J]. Journal of Sound and Vibration,2010,330(2011):110–126.