轴向运动矩形板的非线性谐波共振研究
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摘要
在自然界和工程实际中,轴向运动体系的振动现象是十分普遍的。它广泛存在于军事、航空航天、机械、电子、土木工程中。如动力传送带、磁带、纺织纤维、带锯、纸张、航空航天工程中应用的复合材料层合板等。因此研究此类系统的振动特性具有重要的理论和实际意义。
本文致力于研究轴向运动矩形板在外界激励作用下的非线性强迫振动问题。重点研究在不同边界条件下的轴向运动矩形板的超、亚谐波共振等问题。首先,根据薄板理论和轴向运动系统的运动参数,应用伽辽金原理导出了轴向运动矩形板在横向简谐激励作用下的非线性振动方程。
其次,针对四边简支轴向运动矩形板的超谐波共振和亚谐波共振问题,应用多尺度法进行求解,分别导出了系统相应的幅频响应方程,并基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件;针对一对边简支一对边自由轴向运动矩形板的超谐波共振和亚谐波共振问题,基于参数变换的思想,进行了数值的分析计算。
最后,通过编程计算,对几种不同边界条件下的轴向运动矩形板进行了分析。分别给出了在不同参数情况下,四边简支轴向运动矩形板超谐波共振和亚谐波共振问题的幅频曲线及其时间历程图、相图、庞加莱映射图和频谱图;对于一对边简支一对边自由的轴向运动矩形板的超、亚谐波共振的分析,进行了数值分析计算,并给出其时间历程图、相图、庞加莱映射图和频谱图。讨论了载荷大小、阻尼系数、速度变化以及板厚等参数对系统非线性振动特性的影响。
In nature and engineering practice, the vibration of axially moving systems is a very common phenomenon. It is widespread in the military, aerospace, machinery, electronics, civil engineering, such as power transmission belts, tapes, textile fibers, band saw, paper, and the composite laminated panels of aerospace engineering. Therefore, the study on the vibration characteristics of axially moving systems has important theoretical and practical significance.
In this paper, the axially moving rectangular plate subjected to external excitation is studied. Super harmonic resonance and subharmonic resonance of the axially moving rectangular plates in the different boundary conditions are focused discussed. Firstly, according to the classical plate theory and the parameters of axially moving systems, the nonlinear forced vibration equations of axially moving rectangular plates respectively under the harmonic excitation in the different boundary conditions were deduced by using Galerkin method.
Secondly, for super harmonic resonance and subharmonic resonance of axially moving plate with four simply supported edges, the amplitude-frequency response equations were respectively derived using multi-scale method. Based on Lyapunov stability theory, the critical conditions of steady-state solutions’stability were gotten; for super harmonic resonance and subharmonic resonance of axially moving plate with two opposite ends simply supported and two opposite ends free, based on the parameters transformation, the analysis of numerical calculation was carried on.
Finally, the axially moving rectangular plates in different boundary conditions were analyzed by programming calculation. In the super harmonic resonance and subharmoic resonance, the amplitude-frequency curves, the time history plot, the phase chart, the Poincarémap and the spectrum plot of the axially moving rectangular plate with four simply supported edges were derived under different parameters. In the boundary condition of two opposite ends simply supported and two opposite ends free, the time history plot, the phase chart, the Poincarémap and the spectrum plot for the super harmonic resonance and subharmoic resonance the of the axially moving rectangular plate were also derived under different parameters. The influences of different parameters on nonlinear resonance properties of system,such as load, damping, speed and the width of plate , were discussed.
本文致力于研究轴向运动矩形板在外界激励作用下的非线性强迫振动问题。重点研究在不同边界条件下的轴向运动矩形板的超、亚谐波共振等问题。首先,根据薄板理论和轴向运动系统的运动参数,应用伽辽金原理导出了轴向运动矩形板在横向简谐激励作用下的非线性振动方程。
其次,针对四边简支轴向运动矩形板的超谐波共振和亚谐波共振问题,应用多尺度法进行求解,分别导出了系统相应的幅频响应方程,并基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件;针对一对边简支一对边自由轴向运动矩形板的超谐波共振和亚谐波共振问题,基于参数变换的思想,进行了数值的分析计算。
最后,通过编程计算,对几种不同边界条件下的轴向运动矩形板进行了分析。分别给出了在不同参数情况下,四边简支轴向运动矩形板超谐波共振和亚谐波共振问题的幅频曲线及其时间历程图、相图、庞加莱映射图和频谱图;对于一对边简支一对边自由的轴向运动矩形板的超、亚谐波共振的分析,进行了数值分析计算,并给出其时间历程图、相图、庞加莱映射图和频谱图。讨论了载荷大小、阻尼系数、速度变化以及板厚等参数对系统非线性振动特性的影响。
In nature and engineering practice, the vibration of axially moving systems is a very common phenomenon. It is widespread in the military, aerospace, machinery, electronics, civil engineering, such as power transmission belts, tapes, textile fibers, band saw, paper, and the composite laminated panels of aerospace engineering. Therefore, the study on the vibration characteristics of axially moving systems has important theoretical and practical significance.
In this paper, the axially moving rectangular plate subjected to external excitation is studied. Super harmonic resonance and subharmonic resonance of the axially moving rectangular plates in the different boundary conditions are focused discussed. Firstly, according to the classical plate theory and the parameters of axially moving systems, the nonlinear forced vibration equations of axially moving rectangular plates respectively under the harmonic excitation in the different boundary conditions were deduced by using Galerkin method.
Secondly, for super harmonic resonance and subharmonic resonance of axially moving plate with four simply supported edges, the amplitude-frequency response equations were respectively derived using multi-scale method. Based on Lyapunov stability theory, the critical conditions of steady-state solutions’stability were gotten; for super harmonic resonance and subharmonic resonance of axially moving plate with two opposite ends simply supported and two opposite ends free, based on the parameters transformation, the analysis of numerical calculation was carried on.
Finally, the axially moving rectangular plates in different boundary conditions were analyzed by programming calculation. In the super harmonic resonance and subharmoic resonance, the amplitude-frequency curves, the time history plot, the phase chart, the Poincarémap and the spectrum plot of the axially moving rectangular plate with four simply supported edges were derived under different parameters. In the boundary condition of two opposite ends simply supported and two opposite ends free, the time history plot, the phase chart, the Poincarémap and the spectrum plot for the super harmonic resonance and subharmoic resonance the of the axially moving rectangular plate were also derived under different parameters. The influences of different parameters on nonlinear resonance properties of system,such as load, damping, speed and the width of plate , were discussed.
引文
1刘延柱,陈立群.非线性振动.北京:高等教育出版社, 2001
2陈予恕.非线性振动.北京:高等教育出版社, 1983
3陈树辉,黄建亮,佘锦炎.轴向运动梁横向非线性振动研究.动力学与控制学报, 2004, 2(1):40-45
4陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报, 2005, 37(1):57-63
5黄建亮,陈树辉.轴向运动体系横向非线性振动的联合共振.振动工程学报, 2005, 18(3):19-23
6殷振坤,陈树辉.简支梁的组合谐波共振.华南理工大学学报(自然科学版), 2003, 31(7):23-25
7黄建亮,黄惠仪,陈恒,陈树辉.梁的强非线性超、次谐波共振.振动与冲击, 2004, 23(1):1-3
8丁虎,陈立群.轴向运动黏弹性梁横向非线性受迫振动.振动与冲击, 2009, 28(12):128-131
9丁虎,陈立群.轴向运动黏弹性梁平面耦合非线性受迫振动.上海大学学报(自然科学版), 2009, 15(6):649-658
10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.力学季刊, 2008, 29(3):502-505
11王波,陈立群,王洪伟,刘玉敬.非线性变速轴向运动黏弹性梁稳态响应.科技导报, 2009, 27(2):25-28
12杨晓东,陈立群.变速度轴向运动粘弹性梁的动态稳定性.应用数学和力学, 2005, 26(8):905-910
13 S. H. Chena, J. L. Huanga, K. Y. Szeb. Multidimensional Lindstedt–PoincaréMethod for Nonlinear Vibration of Axially Moving Beams. Journal of Sound and Vibration, 2007, 25(9):1-11
14 M. H. Ghayesh, S. E. Khadem. Rotary Inertia and Temperature Effects on Non-linear Vibration, Steady-state Response and Stability of An Axially Moving Beam with Time-dependent Velocity. International Journal of Mechanical Sciences, 2008,50(3):389-404
15 F. Pellicano. On the Dynamic Properties of Axially Moving Systems. Journal of Sound and Vibration, 2005, 281(3-5):593-609
16 H. Zhao. On the Control of Axially Moving Material Systems. Journal of Vibration and Acoustics, 2006, 128(4):527-531
17 N. Banichuk, J. Jeronen, P. Neittaanm?ki, T. Tuovinen. On the Instability of An Axially Moving Elastic Plate. International Journal of Solids and Structures, 2010, 47(1):91-99
18 C. Lin, J. Mote, D. Clayton. Wrinkling and Stress Sistribution in Rectangular Webs Under Nonlinearly Distributed Edge Loading. American Society of Mechanical Engineers, 1995, 11(1):239-245
19 S. Changho, C. Jintai, K. Wonsuk. Dynamic Characteristics of the Out-of-plane Vibration for an Axially Moving Membrane. Journal of Sound and Vibration, 2005, 286(4-5):1019-1031
20 Yinfeng Zhou, Zhongmin Wang. Vibrations of Axially Moving Viscoelastic Plate with Parabolically Varying Thickness. Journal of Sound and Vibration, 2008, 316(1-5):198-210
21高正,杨晓东.轴向运动粘弹性板横向振动分析的有限差分方法.沈阳航空工业学院学报, 2009, 26(3):14-18
22 S. Hatami, H. R. Ronagh, M. Azhari. Exact Free Vibration Analysis of Axially Moving Viscoelastic Plates. Computers and Structures, 2008, 86(17-18):1738-1746
23钟阳,李锐,刘月梅.四边固支矩形板弹性薄板的精确解析解.力学季刊, 2009, 30(6):297-303
24刘金堂,杨晓东,闻邦椿.基于微分求积法的轴向运动板横向振动分析.振动与冲击, 2009, 28(3):178-181
25王效民,刘新东,汤翔.重三角级数解一对边简支一对边自由的矩形板.山西建筑, 2007, 33(10):12-13
26许琪楼,张锋,姬鸿恩.谈一对边简支一对边自由矩形板自振频率解法.郑州工业大学学报, 2000, 21(4):1-3
27周银锋,王中民.轴向运动粘弹性板的横向振动特性.应用数学和力学, 2007,28(2):191-199
28 C. C. Lin. Stability and Vibration Characteristics of Axially Moving Plates. International Journal of Solids and Structures, 1997, 34(24):3179-3190
29 A. C. J. Luo, H. R. Hamidzadeh. Equilibrium and Buckling Stability for Axially Traveling Plates. Communications in Nonlinear Science and Numerical Simulation, 2004, 9(3):343-360
30 K. Joohong, C. Jooyong, L. Usik, P. Sangdeok. Modal Spectral Element Formulation for Axially Moving Plates Subjected to In-plane Axial Tension. Computers and Structures, 2003, 81(20):2011-2020
31 A. C. J. Luo. Resonant and Stationary Waves in Axially Travelling Thin Plates. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2003, 217(3):187-199
32 S. Hatami, M. Azhari, M. M. Saadatpour. Exact and Semi-analytical Finite Strip for Vibration and Dynamic Stability of Traveling Plates With Intermediate Supports. Advances in Structural Engineering, 2006, 9(5):639-651
33殷振坤,陈树辉.轴向运动薄板非线性振动及其稳定性研究.动力学与控制学报, 2007, 5(4):314-319
34刘金堂,杨晓东,张宇飞,闻邦椿.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.机械强度, 2010, 32(4):531-535
35 S. Hatami, M. Azhari, M. M. Saadatpour. Nonlinear Analysis of Axially Moving Plates Using FEM. International Journal of Structural Stability and Dynamics, 2007, 7(4):589-607
36 L. Tang, P. Michael. On the Instability and the Post-critical Behavior of Two-dimensional Cantilevered Flexible Plates in Axial Flow. Journal of Sound and Vibration, 2007, 305(1-2):97-115
37 L. Tang, P. Michael. The Influence of the Wake on the Stability of Cantilevered Flexible Plates in Axial Flow. Journal of Sound and Vibration, 2007, 310(3):512-526
38 L. Tang, P. Michael. A Fluidelastic Model for the Dynamics of Cantilevered Plates with An Additional Spring Support in Axial Flow. Computers & Structures, 2007,85(11-14):1089-1096
39 L. Tang, P. Michael, J. Jin. Cantilevered Flexible Plates in Axial Flow: Energy Transfer and the Concept of Flutter-mill. Journal of Sound and Vibration, 2009, 326(1-2):263-276
40 Y. X. Hao, L. H. Chen, W. Zhang, J. G. Lei. Nonlinear Oscillations, Bifurcations and Chaos of Functionally Graded Materials Plate. Journal of Sound and Vibration, 2007, 312(4-5):862-892
41胡宇达,李晶.导电薄板的磁弹性组合共振分析.应用数学和力学, 2008, 29(8):954-966
42吕书锋,胡宇达.正交各向异性叠层板的非线性主共振分析.动力学与控制学报, 2009, 7(1):35-38
43 K. Hyde, J. Y. Chang, C. Bacca, J. A. Wickert. Parameter Studies for Plane Stress In-Plane Vibration of Rectangular Plates. Journal of Sound and Vibration, 2001, 247(3):471-487
44 Xiaodong Wang. Numerical Analysis of Moving Orthotropic Thin Plates. Computers and Structures, 1999, 70(4):467-486
45杨伯源,张义同.工程弹塑性力学.北京:机械工业出版社, 2008
46徐芝纶.弹性力学(下).第四版.北京:高等教育出版社, 2006
47徐芝纶.弹性力学(上).第四版.北京:高等教育出版社, 2006
48彭旭麟,罗汝梅.变分法及其应用.武昌:华中工学院出版社, 1983
49曹国雄.弹性矩形薄板振动.北京:中国建筑工业出版社, 1983
50曹志远.板壳振动理论.北京:中国铁道出版社, 1989
51殷振坤.轴向运动薄板的非线性振动研究. [中山大学博士论文]. 2007:23-26
52 A. H. Nayfeh, Dean T Mook. Nonlinear Oscillations. New York: John Wiley & Sons, 1979
53钟益林,彭乐群,刘炳文.常微分方程及其Maple, MATLAB求解.北京:清华大学出版社, 2007
54周建兴. MATLAB从入门到精通.北京:人民邮电出版社, 2008
2陈予恕.非线性振动.北京:高等教育出版社, 1983
3陈树辉,黄建亮,佘锦炎.轴向运动梁横向非线性振动研究.动力学与控制学报, 2004, 2(1):40-45
4陈树辉,黄建亮.轴向运动梁非线性振动内共振研究.力学学报, 2005, 37(1):57-63
5黄建亮,陈树辉.轴向运动体系横向非线性振动的联合共振.振动工程学报, 2005, 18(3):19-23
6殷振坤,陈树辉.简支梁的组合谐波共振.华南理工大学学报(自然科学版), 2003, 31(7):23-25
7黄建亮,黄惠仪,陈恒,陈树辉.梁的强非线性超、次谐波共振.振动与冲击, 2004, 23(1):1-3
8丁虎,陈立群.轴向运动黏弹性梁横向非线性受迫振动.振动与冲击, 2009, 28(12):128-131
9丁虎,陈立群.轴向运动黏弹性梁平面耦合非线性受迫振动.上海大学学报(自然科学版), 2009, 15(6):649-658
10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.力学季刊, 2008, 29(3):502-505
11王波,陈立群,王洪伟,刘玉敬.非线性变速轴向运动黏弹性梁稳态响应.科技导报, 2009, 27(2):25-28
12杨晓东,陈立群.变速度轴向运动粘弹性梁的动态稳定性.应用数学和力学, 2005, 26(8):905-910
13 S. H. Chena, J. L. Huanga, K. Y. Szeb. Multidimensional Lindstedt–PoincaréMethod for Nonlinear Vibration of Axially Moving Beams. Journal of Sound and Vibration, 2007, 25(9):1-11
14 M. H. Ghayesh, S. E. Khadem. Rotary Inertia and Temperature Effects on Non-linear Vibration, Steady-state Response and Stability of An Axially Moving Beam with Time-dependent Velocity. International Journal of Mechanical Sciences, 2008,50(3):389-404
15 F. Pellicano. On the Dynamic Properties of Axially Moving Systems. Journal of Sound and Vibration, 2005, 281(3-5):593-609
16 H. Zhao. On the Control of Axially Moving Material Systems. Journal of Vibration and Acoustics, 2006, 128(4):527-531
17 N. Banichuk, J. Jeronen, P. Neittaanm?ki, T. Tuovinen. On the Instability of An Axially Moving Elastic Plate. International Journal of Solids and Structures, 2010, 47(1):91-99
18 C. Lin, J. Mote, D. Clayton. Wrinkling and Stress Sistribution in Rectangular Webs Under Nonlinearly Distributed Edge Loading. American Society of Mechanical Engineers, 1995, 11(1):239-245
19 S. Changho, C. Jintai, K. Wonsuk. Dynamic Characteristics of the Out-of-plane Vibration for an Axially Moving Membrane. Journal of Sound and Vibration, 2005, 286(4-5):1019-1031
20 Yinfeng Zhou, Zhongmin Wang. Vibrations of Axially Moving Viscoelastic Plate with Parabolically Varying Thickness. Journal of Sound and Vibration, 2008, 316(1-5):198-210
21高正,杨晓东.轴向运动粘弹性板横向振动分析的有限差分方法.沈阳航空工业学院学报, 2009, 26(3):14-18
22 S. Hatami, H. R. Ronagh, M. Azhari. Exact Free Vibration Analysis of Axially Moving Viscoelastic Plates. Computers and Structures, 2008, 86(17-18):1738-1746
23钟阳,李锐,刘月梅.四边固支矩形板弹性薄板的精确解析解.力学季刊, 2009, 30(6):297-303
24刘金堂,杨晓东,闻邦椿.基于微分求积法的轴向运动板横向振动分析.振动与冲击, 2009, 28(3):178-181
25王效民,刘新东,汤翔.重三角级数解一对边简支一对边自由的矩形板.山西建筑, 2007, 33(10):12-13
26许琪楼,张锋,姬鸿恩.谈一对边简支一对边自由矩形板自振频率解法.郑州工业大学学报, 2000, 21(4):1-3
27周银锋,王中民.轴向运动粘弹性板的横向振动特性.应用数学和力学, 2007,28(2):191-199
28 C. C. Lin. Stability and Vibration Characteristics of Axially Moving Plates. International Journal of Solids and Structures, 1997, 34(24):3179-3190
29 A. C. J. Luo, H. R. Hamidzadeh. Equilibrium and Buckling Stability for Axially Traveling Plates. Communications in Nonlinear Science and Numerical Simulation, 2004, 9(3):343-360
30 K. Joohong, C. Jooyong, L. Usik, P. Sangdeok. Modal Spectral Element Formulation for Axially Moving Plates Subjected to In-plane Axial Tension. Computers and Structures, 2003, 81(20):2011-2020
31 A. C. J. Luo. Resonant and Stationary Waves in Axially Travelling Thin Plates. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, 2003, 217(3):187-199
32 S. Hatami, M. Azhari, M. M. Saadatpour. Exact and Semi-analytical Finite Strip for Vibration and Dynamic Stability of Traveling Plates With Intermediate Supports. Advances in Structural Engineering, 2006, 9(5):639-651
33殷振坤,陈树辉.轴向运动薄板非线性振动及其稳定性研究.动力学与控制学报, 2007, 5(4):314-319
34刘金堂,杨晓东,张宇飞,闻邦椿.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.机械强度, 2010, 32(4):531-535
35 S. Hatami, M. Azhari, M. M. Saadatpour. Nonlinear Analysis of Axially Moving Plates Using FEM. International Journal of Structural Stability and Dynamics, 2007, 7(4):589-607
36 L. Tang, P. Michael. On the Instability and the Post-critical Behavior of Two-dimensional Cantilevered Flexible Plates in Axial Flow. Journal of Sound and Vibration, 2007, 305(1-2):97-115
37 L. Tang, P. Michael. The Influence of the Wake on the Stability of Cantilevered Flexible Plates in Axial Flow. Journal of Sound and Vibration, 2007, 310(3):512-526
38 L. Tang, P. Michael. A Fluidelastic Model for the Dynamics of Cantilevered Plates with An Additional Spring Support in Axial Flow. Computers & Structures, 2007,85(11-14):1089-1096
39 L. Tang, P. Michael, J. Jin. Cantilevered Flexible Plates in Axial Flow: Energy Transfer and the Concept of Flutter-mill. Journal of Sound and Vibration, 2009, 326(1-2):263-276
40 Y. X. Hao, L. H. Chen, W. Zhang, J. G. Lei. Nonlinear Oscillations, Bifurcations and Chaos of Functionally Graded Materials Plate. Journal of Sound and Vibration, 2007, 312(4-5):862-892
41胡宇达,李晶.导电薄板的磁弹性组合共振分析.应用数学和力学, 2008, 29(8):954-966
42吕书锋,胡宇达.正交各向异性叠层板的非线性主共振分析.动力学与控制学报, 2009, 7(1):35-38
43 K. Hyde, J. Y. Chang, C. Bacca, J. A. Wickert. Parameter Studies for Plane Stress In-Plane Vibration of Rectangular Plates. Journal of Sound and Vibration, 2001, 247(3):471-487
44 Xiaodong Wang. Numerical Analysis of Moving Orthotropic Thin Plates. Computers and Structures, 1999, 70(4):467-486
45杨伯源,张义同.工程弹塑性力学.北京:机械工业出版社, 2008
46徐芝纶.弹性力学(下).第四版.北京:高等教育出版社, 2006
47徐芝纶.弹性力学(上).第四版.北京:高等教育出版社, 2006
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