柔性梁和板结构非线性振动实验研究
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摘要
柔性梁和板结构无论是在大型空间站等航空、航天领域中,还是在船舶、汽车、建筑等民用工程领域中都是非常重要的组成部分。随着工程结构的大型化、柔性化和轻质化,以及对结构的高可靠性、高精度、高稳定性的要求越来越高,对柔性梁和薄板结构的动力学与振动控制的研究日益重要。非线性因素对机械柔性结构的动力学特性有非常重要的影响,因此,研究柔性梁和板结构的非线性振动具有非常重要的理论意义和实际应用价值,能够为结构的设计和控制提供重要的理论指导依据。
本论文主要对具有方形截面的柔性悬臂梁、金属材料矩形薄板和碳纤维复合材料层合矩形薄板进行系统的非线性振动实验研究。主要研究内容有如下几个方面:
1.研究了简谐参数激励作用下方形截面悬臂梁系统非线性振动响应,主要内容如下:
1)通过理论计算、Ansys有限元模拟给出了方形截面梁的固有频率并同实验观测到的共振频率对比,发现了超谐和亚谐共振等非线性现象;
2)实验研究了共振区域悬臂梁模态阶次和振动方向的转变过程;绘制了频响和力幅值曲线;讨论了系统受迫响应与初始振幅之间的关系;
3)研究了由于模态相互耦合作用,梁的响应在倍周期、概周期和混沌之间转化的非线性动力学行为;观察到了两类多模态组合共振的组合方式以及不同方式所对应的混沌脉冲跳跃现象;发现了梁的大幅回转运动和阵发混沌运动;
4)实验测定了梁对应不同模态的相对阻尼系数。
2.研究了几何尺寸、边界条件、激励等因素对金属材料矩形薄板和碳纤维复合材料层合矩形薄板系统振动响应的影响。具体内容如下:
1)通过理论计算、Ansys有限元模拟给出了矩形薄板的固有频率和模态;
2)通过锤击法、扫频等方法给出了板系统的共振频率,分别讨论了几何尺寸、边界条件、激励等因素对系统共振的影响;基于扫频结果,讨论了板的非线性频响特性;
3)实验发现了一类高阶激励频率激发起低阶非固有频率所对应的共振现象。
Flexible beam and plate structures have been increasingly used in fields of aerospace and civil engineering. Correspondingly, more and more requirements of reliability, precision and stability become considered due to the special characteristics of these structures such as light aromatics and flexibility. Research on the dynamics and vibration control of these structures, therefore, are of great importance. Nonlinear vibration, bifurcations and chaotic dynamics play an important role in nonlinear science. Specific researches on nonlinear dynamics of mechanical flexible structures can provide essential theoretical guidance for their design and control in engineering applications.
In this paper, nonlinear dynamic characteristics of flexible cantilever beam with square section, metallic plate and Composite Laminated Plate have been investigated in detail using experimental method. The main contents are as follow:
1. For cantilever beam, the case with harmonic parametric excitation has been maily discussed with the following details:
1) Natural frequencies obtained mathematically have been compared with those of experiments, by which sub-harmonic and super harmonic resonance has been found;
2) Translating processes between different mode orders and mode directions have been researched experimentally. Frequency and amplitude response curves have been drawn based on the observed points. Nonlinear character of sensitivity toward initial values has been studied and regareded to cause the curves;
3) Chaotic phenomena as a result of interaction of different modes have been studied. In resonant zones, two kinds of multi-mode combination have been found; two correspongding kinds of pulse jumping phenomena have been observed as well;
4) Relative damping coefficients have been determined experimentally.
2. With regard to rectangular thin plates, influences from geometric dimension, boundary conditons and kinds of excitations have been studied with the details as follow:
1) By means of theoretical calculation and finite element imitaiton, natural frequencies and modes have been obtained;
2) Resonant frequencies have been obtained using hammering method and frequency sweeping method. Effects from boundary conditons and excitions have been discussed;
3) A kind of low frequency mode resonance has been found.
本论文主要对具有方形截面的柔性悬臂梁、金属材料矩形薄板和碳纤维复合材料层合矩形薄板进行系统的非线性振动实验研究。主要研究内容有如下几个方面:
1.研究了简谐参数激励作用下方形截面悬臂梁系统非线性振动响应,主要内容如下:
1)通过理论计算、Ansys有限元模拟给出了方形截面梁的固有频率并同实验观测到的共振频率对比,发现了超谐和亚谐共振等非线性现象;
2)实验研究了共振区域悬臂梁模态阶次和振动方向的转变过程;绘制了频响和力幅值曲线;讨论了系统受迫响应与初始振幅之间的关系;
3)研究了由于模态相互耦合作用,梁的响应在倍周期、概周期和混沌之间转化的非线性动力学行为;观察到了两类多模态组合共振的组合方式以及不同方式所对应的混沌脉冲跳跃现象;发现了梁的大幅回转运动和阵发混沌运动;
4)实验测定了梁对应不同模态的相对阻尼系数。
2.研究了几何尺寸、边界条件、激励等因素对金属材料矩形薄板和碳纤维复合材料层合矩形薄板系统振动响应的影响。具体内容如下:
1)通过理论计算、Ansys有限元模拟给出了矩形薄板的固有频率和模态;
2)通过锤击法、扫频等方法给出了板系统的共振频率,分别讨论了几何尺寸、边界条件、激励等因素对系统共振的影响;基于扫频结果,讨论了板的非线性频响特性;
3)实验发现了一类高阶激励频率激发起低阶非固有频率所对应的共振现象。
Flexible beam and plate structures have been increasingly used in fields of aerospace and civil engineering. Correspondingly, more and more requirements of reliability, precision and stability become considered due to the special characteristics of these structures such as light aromatics and flexibility. Research on the dynamics and vibration control of these structures, therefore, are of great importance. Nonlinear vibration, bifurcations and chaotic dynamics play an important role in nonlinear science. Specific researches on nonlinear dynamics of mechanical flexible structures can provide essential theoretical guidance for their design and control in engineering applications.
In this paper, nonlinear dynamic characteristics of flexible cantilever beam with square section, metallic plate and Composite Laminated Plate have been investigated in detail using experimental method. The main contents are as follow:
1. For cantilever beam, the case with harmonic parametric excitation has been maily discussed with the following details:
1) Natural frequencies obtained mathematically have been compared with those of experiments, by which sub-harmonic and super harmonic resonance has been found;
2) Translating processes between different mode orders and mode directions have been researched experimentally. Frequency and amplitude response curves have been drawn based on the observed points. Nonlinear character of sensitivity toward initial values has been studied and regareded to cause the curves;
3) Chaotic phenomena as a result of interaction of different modes have been studied. In resonant zones, two kinds of multi-mode combination have been found; two correspongding kinds of pulse jumping phenomena have been observed as well;
4) Relative damping coefficients have been determined experimentally.
2. With regard to rectangular thin plates, influences from geometric dimension, boundary conditons and kinds of excitations have been studied with the details as follow:
1) By means of theoretical calculation and finite element imitaiton, natural frequencies and modes have been obtained;
2) Resonant frequencies have been obtained using hammering method and frequency sweeping method. Effects from boundary conditons and excitions have been discussed;
3) A kind of low frequency mode resonance has been found.
引文
1.陈予恕,非线性振动,天津科技技术出版社,天津, 1983.
2.陈予恕,工程非线性动力学的最新进展,中国力学学会一般力学学术会议报告,宜昌, 1994.
3. A. H. Nayfeh and D. T. Mock, Nonlinear Oscillation, Wiley-Interscience, New York, 1979.
4.陈予恕,非线性振动,分叉与混沌的最新进展与展望,黄文虎,陈斌,王照林, 一般力学最新进展,科学出版社,北京: p19-29, 1994.
5.陆启韶,黄克磊,非线性动力学,分叉和混沌,文虎,陈斌,王照林,一般力学最新进展,科学出版社,北京: p19-29, 1994.
6.张清杰,结构动力屈曲研究进展,力学进展23: p530-539,1994.
7. H. Hatwal, A. K. Mallik and A. Ghost, Nonlinear vibration of a harmonically excited autoparametric system, Journal of Sound and Vibration 81: p153-164, 1982.
8. H. Hatwal, Notes on an autoparametric vibration absorber, Journal of Sound and Vibration 83: p440-443, 1983.
9. K. L. Tuer, Vibration control of a flexible beam using a rotational internal resonance controller-part I, theoretical development and analysis, Journal of Sound and Vibration 167: p41-62, 1993.
10. B. Blazejczyk, Controlling chaos in mechanical systems, Applied Mechanics Reviews 48: p385-391, 1993.
11.卢文秀,转子系统碰摩故障的实验研究,清华大学学报45: p614-617, 2005.
12.俞翔,非线性隔振系统混沌特性实验研究4: p1006-1355, 2003.
13. L. Fortuna, A chaotic circuit with ferroelectric nonlinearity, Nonlinear Dynamics 44: p55–61, 2006.
14. F. C. Moon and P. J. Holmes, A magneto-elastic strange attractor, Journal of Sound and Vibration 65: p275-296, 1979.
15. F. C. Moon, Fractal boundary for chaos in a two-state mechanical oscillator, Physical Review Letters 53: p962-963, 1984.
16. J. P. Cusumano and F. C. Moon, Chaotic non-planar vibrations of the thin elastica-part I: experimental observation of planar instability, Journal of Sound and Vibration 179: p185-208, 1995.
17. W. Kreider and A. H. Nayfeh, Experimental investigation of single-moderesponses in a fixed-fixed buckled beam, Nonlinear Dynamics 15: p155-177, 1998.
18. W. Lacarbonara and A. H. Nayfeh, Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter system: analysis of buckled beam, Nonlinear Dynamics 17: p95-117, 1998.
19. K. Yasuda and K. Kamiya, Experimental identification technique of nonlinear beams in time domain, Nonlinear Dynamics 18: p185-202, 1999.
20. S. A. Emam and A. H. Nayfeh,Nonlinear responses of buckled beams to subharmonic-resonance excitations, Nonlinear Dynamics 35: p105–122, 2004.
21. P. Malatkar and A. H. Nayfeh, On the transfer of energy between widely spaced modes in structures, Nonlinear Dynamics 31: p225–242, 2003.
22. A. H. Nayfeh, A theoretical and experimental investigation of nonlinear vibrations of buckled beams, thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science In Engineering Mechanics, Blacksburg, Virginia, 1997.
23. T. A. Nayfeh, A. H. Nayfeh and D. T. Mook, A theoretical and experimental investigation of a three-degree-of-freedom structure, Nonlinear Dynamics 6: p353-374, 1994.
24. J. C. Ji and L. Yu, Bifurcation and amplitude modulated motions in a parametrically excited two-degree-of-freedom non-linear system, Journal of Sound and vibration 228: p1125-1144, 1999.
25.季进臣,陈予恕,参激非线性振子不稳定区域的实验研究, Journal of Vibration Engineering 10: p491-495, 1997.
26.季进臣,陈予恕,叶敏,参激屈曲梁的倍周期分岔和混沌运动的实验研究,实验力学12: p248-259, 1997.
27. W. Zhang and D. X. Cao, Studies on bifurcation and chaos of a string-beam coupled system with two degrees-of-freedom. Nonlinear Dynamics 45: p131-147, 2006.
28. D. X. Cao and W. Zhang, Analysis on nonlinear dynamics of a string-beam coupled system, International Journal of Nonlinear Sciences and Numerical Simulation 6: p47-54, 2005.
29. W. Zhang and D. X. Cao, Local and global bifurcations of L-mode to H-mode transition near plasma edge in Tokamak, Chaos, Solitons and Fractals 29: p223-232, 2006.
30. Maestrello, Nonlinear vibration and radiation from a panel with transition to chaos, AIAA Journal 30: p2632-2638, 1992.
31. Maestrello, Controling vibrational chaos of a curved structure, AIAA Journal 39: p581-588, 2001.
32. J.C. Roberts, Experimental, numerical and analytical results for bending and buckling of rectangular orthotropic plates, Composite Structures 43: p289-299, 1999.
33. G. J. Turvey, Experimental and computed natural frequencies of square pultruded, Composite Structures 50: p391-403, 2000.
34. M. Amabili, Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid-part iii: truncation effect without flow and experiments, Journal of Sound and Vibration 237: p617-640, 2000.
35. M. Amabili, Experiments on large-amplitude vibrations of a circular cylindrical panel, Journal of Sound and Vibration 260: p537–547, 2003.
36. M. Amabili, Nonlinear vibration of rectangular plates with different boundary conditions: theory and experiments, Computers and Structures 82: p2587-2605, 2004.
37. M. Amabili, Theory and experiments for large-amplitude vibrations of circular cylindrical panels with geometric imperfections, Journal of Sound and Vibration 298: p43–72, 2006.
38. M. Amabili, Effect of concentrated masses with rotary inertia on vibrations of rectangular plates,Journal of Sound and Vibration 295: p1–12, 2006.
39. M.Amabili, Theory and experiments for large-amplitude vibrations of rectangular plates with geometric imperfections, Journal of Sound and Vibration 291: p539–565, 2006.
40. K. Woznica, Experiments and numerical simulations on thin metallic plates subjected to an explosion, Journal of Engineering Materials and Technology 123: p203-209, 2001.
41. N. J. Kessissoglou, An analytical and experimental investigation on active control of the flexural wave transmission in a simply supported ribbed plate, Journal of sound and vibration 240: p73-85, 2001.
42. N. J. Kessissoglou, An analytical and experimental comparison of optimal actuator and error sensor location for vibration attenuation, Journal of Sound and Vibration 260: p671-691, 2003.
43. O. Thomas, Asymmetric non-linear forced vibrations of free-edge circular plates-part II: experiments, Journal of Sound and Vibration 265: p1075–1101, 2003.
44. G. K. Schleyer, Pulse pressure loading of clamped mild steel plates, International Journal of Impact Engineering 28: p223–247, 2003.
45. J. K. Paik, An experimental investigation on the dynamic ultimate compressive strength of ship plating,International Journal of Impact Engineering 28: p803–811, 2003.
46. J. Park, An investigation of the flow-induced sound and vibration of viscoelastically supported rectangular plates: experiments and model verification, Journal of Sound and Vibration 275: p249–265, 2004.
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55.袁尚平,张惠桥,王冰,参数激励下简支板的分叉与混沌,太原重型机械学院学报18: p133-139, 1997.
56.袁尚平,张惠侨,孙东,简支矩形屈曲薄板非线性振动特性及分叉分析,机械设计与研究2: p4-6, 1997.
57.张宇飞,弯曲薄板动态响应特性的实验研究,沈阳航空工业学院学报17: p7-10, 2000.
58.李银山,圆板振子超谐分岔和混沌运动的实验研究,实验力学16: p347-357, 2001.
59.李银山,陈予恕,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34: p718-722, 2001.
60.董卓敏,大型柔性结构振动主动控制的实验研究,实验力学18: p86-92, 2003.
61. G. Q. Li, Layer-wise closed-form theory for geometrically nonlinear rectangular composite plates subjected to local loads, Composite Structures 46: p91-101, 1999.
62. R. Erick, Nonlinearly asdic membrane model for heterogeneous plate: a formal asymptotic approach by using a new double scale variation, Formulation International Journal of Engineering Science 40: p2183-2202, 2002.
63. J. Cao, L. Liang and L. Lames, Chaotic and bifurcation for a simply-supported thermo-mechanical coupling circular plate with variable thickness, Chaos, Solitions and Fractals 22: p1013-1030, 2004.
64. S. Kitipornchai, J. Yang and K. M. Liew, Semi-analytical solution for nonlinear vibration of laminated FGM plates with geometric imperfections, International Journal of Solids and Structures 41: p2235-2257, 2004.
65. C. C. Ma and C. H. Huang, Experimental whole-field interferometer for transverse vibration of plates, Journal of Sound and Vibration 271: p493-506, 2004.
66. P. Malatkar and A. H. Nayfeh, On the transfer of energy between widely spaced modes in structures, Nonlinear Dynamics 31: p225-242, 2003.
2.陈予恕,工程非线性动力学的最新进展,中国力学学会一般力学学术会议报告,宜昌, 1994.
3. A. H. Nayfeh and D. T. Mock, Nonlinear Oscillation, Wiley-Interscience, New York, 1979.
4.陈予恕,非线性振动,分叉与混沌的最新进展与展望,黄文虎,陈斌,王照林, 一般力学最新进展,科学出版社,北京: p19-29, 1994.
5.陆启韶,黄克磊,非线性动力学,分叉和混沌,文虎,陈斌,王照林,一般力学最新进展,科学出版社,北京: p19-29, 1994.
6.张清杰,结构动力屈曲研究进展,力学进展23: p530-539,1994.
7. H. Hatwal, A. K. Mallik and A. Ghost, Nonlinear vibration of a harmonically excited autoparametric system, Journal of Sound and Vibration 81: p153-164, 1982.
8. H. Hatwal, Notes on an autoparametric vibration absorber, Journal of Sound and Vibration 83: p440-443, 1983.
9. K. L. Tuer, Vibration control of a flexible beam using a rotational internal resonance controller-part I, theoretical development and analysis, Journal of Sound and Vibration 167: p41-62, 1993.
10. B. Blazejczyk, Controlling chaos in mechanical systems, Applied Mechanics Reviews 48: p385-391, 1993.
11.卢文秀,转子系统碰摩故障的实验研究,清华大学学报45: p614-617, 2005.
12.俞翔,非线性隔振系统混沌特性实验研究4: p1006-1355, 2003.
13. L. Fortuna, A chaotic circuit with ferroelectric nonlinearity, Nonlinear Dynamics 44: p55–61, 2006.
14. F. C. Moon and P. J. Holmes, A magneto-elastic strange attractor, Journal of Sound and Vibration 65: p275-296, 1979.
15. F. C. Moon, Fractal boundary for chaos in a two-state mechanical oscillator, Physical Review Letters 53: p962-963, 1984.
16. J. P. Cusumano and F. C. Moon, Chaotic non-planar vibrations of the thin elastica-part I: experimental observation of planar instability, Journal of Sound and Vibration 179: p185-208, 1995.
17. W. Kreider and A. H. Nayfeh, Experimental investigation of single-moderesponses in a fixed-fixed buckled beam, Nonlinear Dynamics 15: p155-177, 1998.
18. W. Lacarbonara and A. H. Nayfeh, Experimental validation of reduction methods for nonlinear vibrations of distributed-parameter system: analysis of buckled beam, Nonlinear Dynamics 17: p95-117, 1998.
19. K. Yasuda and K. Kamiya, Experimental identification technique of nonlinear beams in time domain, Nonlinear Dynamics 18: p185-202, 1999.
20. S. A. Emam and A. H. Nayfeh,Nonlinear responses of buckled beams to subharmonic-resonance excitations, Nonlinear Dynamics 35: p105–122, 2004.
21. P. Malatkar and A. H. Nayfeh, On the transfer of energy between widely spaced modes in structures, Nonlinear Dynamics 31: p225–242, 2003.
22. A. H. Nayfeh, A theoretical and experimental investigation of nonlinear vibrations of buckled beams, thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Master of Science In Engineering Mechanics, Blacksburg, Virginia, 1997.
23. T. A. Nayfeh, A. H. Nayfeh and D. T. Mook, A theoretical and experimental investigation of a three-degree-of-freedom structure, Nonlinear Dynamics 6: p353-374, 1994.
24. J. C. Ji and L. Yu, Bifurcation and amplitude modulated motions in a parametrically excited two-degree-of-freedom non-linear system, Journal of Sound and vibration 228: p1125-1144, 1999.
25.季进臣,陈予恕,参激非线性振子不稳定区域的实验研究, Journal of Vibration Engineering 10: p491-495, 1997.
26.季进臣,陈予恕,叶敏,参激屈曲梁的倍周期分岔和混沌运动的实验研究,实验力学12: p248-259, 1997.
27. W. Zhang and D. X. Cao, Studies on bifurcation and chaos of a string-beam coupled system with two degrees-of-freedom. Nonlinear Dynamics 45: p131-147, 2006.
28. D. X. Cao and W. Zhang, Analysis on nonlinear dynamics of a string-beam coupled system, International Journal of Nonlinear Sciences and Numerical Simulation 6: p47-54, 2005.
29. W. Zhang and D. X. Cao, Local and global bifurcations of L-mode to H-mode transition near plasma edge in Tokamak, Chaos, Solitons and Fractals 29: p223-232, 2006.
30. Maestrello, Nonlinear vibration and radiation from a panel with transition to chaos, AIAA Journal 30: p2632-2638, 1992.
31. Maestrello, Controling vibrational chaos of a curved structure, AIAA Journal 39: p581-588, 2001.
32. J.C. Roberts, Experimental, numerical and analytical results for bending and buckling of rectangular orthotropic plates, Composite Structures 43: p289-299, 1999.
33. G. J. Turvey, Experimental and computed natural frequencies of square pultruded, Composite Structures 50: p391-403, 2000.
34. M. Amabili, Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid-part iii: truncation effect without flow and experiments, Journal of Sound and Vibration 237: p617-640, 2000.
35. M. Amabili, Experiments on large-amplitude vibrations of a circular cylindrical panel, Journal of Sound and Vibration 260: p537–547, 2003.
36. M. Amabili, Nonlinear vibration of rectangular plates with different boundary conditions: theory and experiments, Computers and Structures 82: p2587-2605, 2004.
37. M. Amabili, Theory and experiments for large-amplitude vibrations of circular cylindrical panels with geometric imperfections, Journal of Sound and Vibration 298: p43–72, 2006.
38. M. Amabili, Effect of concentrated masses with rotary inertia on vibrations of rectangular plates,Journal of Sound and Vibration 295: p1–12, 2006.
39. M.Amabili, Theory and experiments for large-amplitude vibrations of rectangular plates with geometric imperfections, Journal of Sound and Vibration 291: p539–565, 2006.
40. K. Woznica, Experiments and numerical simulations on thin metallic plates subjected to an explosion, Journal of Engineering Materials and Technology 123: p203-209, 2001.
41. N. J. Kessissoglou, An analytical and experimental investigation on active control of the flexural wave transmission in a simply supported ribbed plate, Journal of sound and vibration 240: p73-85, 2001.
42. N. J. Kessissoglou, An analytical and experimental comparison of optimal actuator and error sensor location for vibration attenuation, Journal of Sound and Vibration 260: p671-691, 2003.
43. O. Thomas, Asymmetric non-linear forced vibrations of free-edge circular plates-part II: experiments, Journal of Sound and Vibration 265: p1075–1101, 2003.
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