应用非线性Galerkin方法求解微梁的非线性动态响应
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摘要
微机电系统(MEMS)属于微尺度领域的范畴,具有一系列独特于宏观领域的性质。在力学方面,微尺度效应引发的粘附现象通常容易被忽略并导致设计上的失误。微梁正在发展成为各种微系统共振器,如共振加速度器、微机电滤波器、微振动电机等的重要部件。然而,由于装置的几何复杂性、物理条件以及微机械结构中不可忽略的微尺度效应等问题,使得难以整合到现有的计算程序中,设计工具远没有跟上MEMS发展的步伐。有效地仿真和预测MEMS的非线性行为的能力在估计系统性能和指导可靠性核查过程中是非常重要的。传统的Galerkin方法提供了一个简便的途径来建立较低维的非线性动力学模型,并借此预测微结构的机械性能。但是,就像在现有文献中所指出的一样,由于传统的Galerkin方法完全省去了高频部分,这一方法可能导致不精确的结果。非线性Galerkin方法则通过惯性流形(近似惯性流形)将高频部分向低频部分投影,考虑了高频分量对响应的贡献。本文首先简要介绍MEMS中相关背景知识,并补充文献中对非线性Galerkin方法的理论描述,使其可适用于一般的情形。其次,讨论粘附力在微共振器设计中的影响,计算在粘附力中起主要作用的毛细引力作用下微共振器的动力学响应。最后,分析电场力驱动下微共振器的非线性动力学特性。取前3阶模态,利用传统Garlekin方法得到3自由度的非线性动力学模型,再利用非线性Garlekin过程得到单自由度的降阶模型。用多尺度法计算降阶模型的动态响应,并得出了稳态响应的幅频特性曲线,与利用传统Garlekin方法直接取1阶模态所得的结果比较。以数值积分法求解3自由度模型得到的微共振器动力学响应为参考标准,验证了非线性Galerkin方法与传统Galerkin方法相比具有较高的精度。
Micro-electro-mechanical systems (MEMS) is a category of micro area with a series of unique nature distinguished from macro area. In aspect of mechanics, adhesion of micro-caused phenomenon is usually easy to be ignored and lead to the design of an error. Micro-beams are being developed as a key component in various resonator based micro-systems, such as resonant accelerometers, micro-electro- mechanical filters, and micro-vibromotors. The design tools, however, have not kept pace with this growth since the geometrical complexity and physical conditions of the devices are not easily incorporated into existing computational programs. The ability of effectively simulating and predicting nonlinear behaviour of MEMS is important in estimating system performance and guiding the reliability verification process. The traditional Galerkin method provides a simple approach to establish a nonlinear dynamic model with lower dimension and predicting the mechanical behaviour of microstructures, but it may lead inaccurate results since it neglect the parts of high frequencies completely as indicated in the literature. The nonlinear Galerkin method projects the parts of higher frequencies onto the parts of lower frequencies through an inertial manifold, takes the parts of higher frequencies into account. In this paper, a brief introduction of MEMS relevant background is addressed at first, and nonlinear Galerkin method is complemented in theoretical description to make it applicable to the general situation. Then, the influence of adhesion in the design of resonator is discussed. Calculate the response of micro-resonator driven by the capillary force, which is the major part of adhesion. Finally, analyze the dynamic behaviour of resonator driven by electric force. Consider the first three modes, use nonlinear Garlekin method to get a model of single degree of freedom. Use multiple scales method calculate the dynamic response of model got by the nonlinear Garlekin method. And draw the amplitude-frequency curve of steady state, compare it to the results obtained by traditional Garlekin method which only consider the first mode. Let the micro-beam dynamic response got by numerical integration of model with three degrees of freedom as a reference standard, validated that nonlinear Galerkin method with higher accuracy compared to traditional Galerkin method.
Micro-electro-mechanical systems (MEMS) is a category of micro area with a series of unique nature distinguished from macro area. In aspect of mechanics, adhesion of micro-caused phenomenon is usually easy to be ignored and lead to the design of an error. Micro-beams are being developed as a key component in various resonator based micro-systems, such as resonant accelerometers, micro-electro- mechanical filters, and micro-vibromotors. The design tools, however, have not kept pace with this growth since the geometrical complexity and physical conditions of the devices are not easily incorporated into existing computational programs. The ability of effectively simulating and predicting nonlinear behaviour of MEMS is important in estimating system performance and guiding the reliability verification process. The traditional Galerkin method provides a simple approach to establish a nonlinear dynamic model with lower dimension and predicting the mechanical behaviour of microstructures, but it may lead inaccurate results since it neglect the parts of high frequencies completely as indicated in the literature. The nonlinear Galerkin method projects the parts of higher frequencies onto the parts of lower frequencies through an inertial manifold, takes the parts of higher frequencies into account. In this paper, a brief introduction of MEMS relevant background is addressed at first, and nonlinear Galerkin method is complemented in theoretical description to make it applicable to the general situation. Then, the influence of adhesion in the design of resonator is discussed. Calculate the response of micro-resonator driven by the capillary force, which is the major part of adhesion. Finally, analyze the dynamic behaviour of resonator driven by electric force. Consider the first three modes, use nonlinear Garlekin method to get a model of single degree of freedom. Use multiple scales method calculate the dynamic response of model got by the nonlinear Garlekin method. And draw the amplitude-frequency curve of steady state, compare it to the results obtained by traditional Garlekin method which only consider the first mode. Let the micro-beam dynamic response got by numerical integration of model with three degrees of freedom as a reference standard, validated that nonlinear Galerkin method with higher accuracy compared to traditional Galerkin method.
引文
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2 L.S.Fan, Y.C.Tai and R.S.Muller. Tech. Digest IEEE Int. Electron Devices Meeting, San Fransisco. Dec.1988, 11-14:666.
3张威,张大成,王阳元. MEMS概况及发展趋势.微纳电子技术. 2002, (1):22~27
4孙立宁,周兆英,龚振邦. MEMS国内外发展状况及我国MEMS发展战略的思考.机器人技术与应用. 2002, (1):2~4
5赵晓峰,温殿忠. MEMS研究与发展前景.黑龙江大学自然科学学报. 2002, (1):64~69
6李志坚.微电子机械系统(MEMS)发展展望.电子科技导报. 1997, (1):2~9
7 R.Legtenberg, A.C.Tilmans, J.Elders, et al. Stiction of Surface Micromachined Structures after Rinsing and Drying:Model and Investigation of Adhesion Mechanisms .Sensors and Actuators, A. 1994, 43(11):230~238
8 N.Tas, T.Sonnenberg, H.Jansen, et al. Stiction in Surface Micromachining. Journal of Micromachining and Microengineering. 1996, (6):385~397
9 J.Ding, Y.Meng, S.Wen. Size Effect on Mechanical Properties of Microfabricated Polysilicon Thin Films. Journal of Materials Research. 2001, 16(8):2223~2228
10 Y.P.Zhao, L.S.Wang, T.X.Yu. Mechanics of Adhesion in MEMS-a Review. Journal of Adhesion Science and Technology. 2003, 17(4):519~546
11 W.M.V.Spengen, R.Puers, I.D.Wolf. On the Physics of Stiction and Its Impact on the Reliability of Microstructures[J]. Adhesion Science and Technology. 2003, 17(4):563~582
12 C.L.Britton, P.I.Jones Oden. Maltiinput Microcantilever Sensors[J]. Ultrascopy. 2000, 82(1-4):17~21
13 J.Fritz, M.K.Baller, H.P.Lang. Translating Biomolecular Recognition into Nanomechanics[J].Science. 2000, 288(5464):316~318
14 D.L.De Voe, A.P.Pisano. Modeling and Optimal Design of Pieozoelectric Cantilever Microactuators[J]. Journal of Microelectromechanical Systems. 1997,6(3):266~270
15 M.Koch, A.G.R.Evann, A.Brunnschweiler Coupled FEM simulation for the characterization of the fluid flow within a micromachined cantitler value. Journal of Micromachining and Microengineering. 1996, 6(1):1112~1124
16 M.Elwenspoek, R.J.Wiegerink. Mechanical Microsensors. Springer, Berlin. 2001
17 J.R.Clark. Parallel-Resonator HF Micromechanical Bandpass Filters. Proceedin-gs of the Conference on Solid State Sensors and Actuators. 1997:1161~1164
18 V.K.Varadan, K.J.Vinoy, K.A.Jose. RF MEMS and Their Applications. Wiley, New York. 2003
19 M.I.Younis, E.M.Abdel-Rahman, A.H.Nayfeh. Static and dynamic Behavior of an Electrically Excited Resonant Microbeam. Proceedings of the 43rd AIAA Structures, Structural Dynamics, and Materials Conference, Denver, CO. United States. 2002:22~25
20 E.M.Abdel-Rahman, M.I.Younis and A.H.Nayfeh. Characterization of The Mechanical Behavior of an Electrically Actuated Microbeam. Journal of Micromechanics and Microengineering. 2002, (12): 759~766
21 H.A.C.Tilmans, R.Legtenberg. Electrostatically Driven Vacuum-encapsulated Polysilicon Resonators. Part II. Theory and Performance. Sensors and Actuators 1994, 45(11):67~84
22 M.I.Younis, A.H.Nayfeh. A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuation. Nonlinear Dynamics. 2003, 31(1): 91~117
23 E.M.Abdel-Rahman, A.H.Nayfeh, M.I.Younis. Dynamics of an Electrically Actuated Resonant Microsensor. Proceedings of the International Conference of MEMS, NANO and Smart Systems. 2003:188~196
24 Chenggun Gui, R.Legtenberg, H.A.C.Tilmans, J.H.J.Fluitman, M.Elwenspoek. Nonlinearity and hysteresis of Resonant Strain Gauges. Journal of Microelectromechanical Systems. 1998, 7(1):122~127
25 A.H.Nayfeh, B.Balachandran. Applied Nonlinear Dynamics. John Wiley. 1995
26 A.H.Nayfeh, M.I.Younis, E.M.Abdel-Rahman. Dynamic Pull-in Phenomenon in MEMS Resonators. Nonlinear Dynamics. 2007, (48):153~163
27 A.H.Nayfeh and M.I.Younis. Dynamics of MEMS Resonators Under Super-harmonic and Subharmonic Excitations. Journal of Micromechanics and Micro-engineering. 2005, (15):1840~1847
28 R.Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer. l988
29 M.Marion and R.Temam, Nonlinear Galerkin Methods. SLAM J Numer. Ana1. 1989, 26(5):1139~1157
30 M.Chen and R.Temam, Incremental Unknows in Finite Differences: Condition Number of the Matrix. SLAM J Matrix Anal. 1993, 14(2):432~455
31 H.G.Matthies, M.Meyer. Nonlinear Galerkin Methods for the Model Reduction of Nonlinear Dynamical Systems. Computers and Structures. 2003, 81(12):1277~1286
32 S.C.Sinha, Sangram Redkar, Venkatesh Deshmukh, and E.A.Butcher. Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications. Nonlinear Dynamics. 2005, 41(8):237~273
33张新华,徐健学.非线性Galerkin方法在梁的非线性动力学性态分析中的应用.《工程力学》增刊. 1996:564~568
34彭云,李喜德.微梁与基底间毛细粘附作用.第十一届实验力学学术会议大连. 2005:1150~1156
35胡海岩,孙久厚,陈怀海.机械振动与冲击.航空工业出版社. 2002:169~170
36邹经湘.结构动力学.哈尔滨工业大学出版社. 1999:80~84
37黄祖洽,丁鄂江.表面浸润和浸润相变.上海科学技术出版社. 1994:1-3,38-45
38 M.I.Younis and A.H.Nayfeh. A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuation. Nonlinear Dynamics. 2003, 31(1):91~117.
39陈予恕.非线性振动.高等教育出版社, 2002:237~245
40闻邦椿,李以农,韩清凯.非线性振动理论中的解析方法及工程应用.东北大学出版社, 2001:59~75
41 A.H.Nayfeh, D.T.Mook. Nonlinear Oscillations. Wiley, New York. 1995: 444 ~467
2 L.S.Fan, Y.C.Tai and R.S.Muller. Tech. Digest IEEE Int. Electron Devices Meeting, San Fransisco. Dec.1988, 11-14:666.
3张威,张大成,王阳元. MEMS概况及发展趋势.微纳电子技术. 2002, (1):22~27
4孙立宁,周兆英,龚振邦. MEMS国内外发展状况及我国MEMS发展战略的思考.机器人技术与应用. 2002, (1):2~4
5赵晓峰,温殿忠. MEMS研究与发展前景.黑龙江大学自然科学学报. 2002, (1):64~69
6李志坚.微电子机械系统(MEMS)发展展望.电子科技导报. 1997, (1):2~9
7 R.Legtenberg, A.C.Tilmans, J.Elders, et al. Stiction of Surface Micromachined Structures after Rinsing and Drying:Model and Investigation of Adhesion Mechanisms .Sensors and Actuators, A. 1994, 43(11):230~238
8 N.Tas, T.Sonnenberg, H.Jansen, et al. Stiction in Surface Micromachining. Journal of Micromachining and Microengineering. 1996, (6):385~397
9 J.Ding, Y.Meng, S.Wen. Size Effect on Mechanical Properties of Microfabricated Polysilicon Thin Films. Journal of Materials Research. 2001, 16(8):2223~2228
10 Y.P.Zhao, L.S.Wang, T.X.Yu. Mechanics of Adhesion in MEMS-a Review. Journal of Adhesion Science and Technology. 2003, 17(4):519~546
11 W.M.V.Spengen, R.Puers, I.D.Wolf. On the Physics of Stiction and Its Impact on the Reliability of Microstructures[J]. Adhesion Science and Technology. 2003, 17(4):563~582
12 C.L.Britton, P.I.Jones Oden. Maltiinput Microcantilever Sensors[J]. Ultrascopy. 2000, 82(1-4):17~21
13 J.Fritz, M.K.Baller, H.P.Lang. Translating Biomolecular Recognition into Nanomechanics[J].Science. 2000, 288(5464):316~318
14 D.L.De Voe, A.P.Pisano. Modeling and Optimal Design of Pieozoelectric Cantilever Microactuators[J]. Journal of Microelectromechanical Systems. 1997,6(3):266~270
15 M.Koch, A.G.R.Evann, A.Brunnschweiler Coupled FEM simulation for the characterization of the fluid flow within a micromachined cantitler value. Journal of Micromachining and Microengineering. 1996, 6(1):1112~1124
16 M.Elwenspoek, R.J.Wiegerink. Mechanical Microsensors. Springer, Berlin. 2001
17 J.R.Clark. Parallel-Resonator HF Micromechanical Bandpass Filters. Proceedin-gs of the Conference on Solid State Sensors and Actuators. 1997:1161~1164
18 V.K.Varadan, K.J.Vinoy, K.A.Jose. RF MEMS and Their Applications. Wiley, New York. 2003
19 M.I.Younis, E.M.Abdel-Rahman, A.H.Nayfeh. Static and dynamic Behavior of an Electrically Excited Resonant Microbeam. Proceedings of the 43rd AIAA Structures, Structural Dynamics, and Materials Conference, Denver, CO. United States. 2002:22~25
20 E.M.Abdel-Rahman, M.I.Younis and A.H.Nayfeh. Characterization of The Mechanical Behavior of an Electrically Actuated Microbeam. Journal of Micromechanics and Microengineering. 2002, (12): 759~766
21 H.A.C.Tilmans, R.Legtenberg. Electrostatically Driven Vacuum-encapsulated Polysilicon Resonators. Part II. Theory and Performance. Sensors and Actuators 1994, 45(11):67~84
22 M.I.Younis, A.H.Nayfeh. A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuation. Nonlinear Dynamics. 2003, 31(1): 91~117
23 E.M.Abdel-Rahman, A.H.Nayfeh, M.I.Younis. Dynamics of an Electrically Actuated Resonant Microsensor. Proceedings of the International Conference of MEMS, NANO and Smart Systems. 2003:188~196
24 Chenggun Gui, R.Legtenberg, H.A.C.Tilmans, J.H.J.Fluitman, M.Elwenspoek. Nonlinearity and hysteresis of Resonant Strain Gauges. Journal of Microelectromechanical Systems. 1998, 7(1):122~127
25 A.H.Nayfeh, B.Balachandran. Applied Nonlinear Dynamics. John Wiley. 1995
26 A.H.Nayfeh, M.I.Younis, E.M.Abdel-Rahman. Dynamic Pull-in Phenomenon in MEMS Resonators. Nonlinear Dynamics. 2007, (48):153~163
27 A.H.Nayfeh and M.I.Younis. Dynamics of MEMS Resonators Under Super-harmonic and Subharmonic Excitations. Journal of Micromechanics and Micro-engineering. 2005, (15):1840~1847
28 R.Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer. l988
29 M.Marion and R.Temam, Nonlinear Galerkin Methods. SLAM J Numer. Ana1. 1989, 26(5):1139~1157
30 M.Chen and R.Temam, Incremental Unknows in Finite Differences: Condition Number of the Matrix. SLAM J Matrix Anal. 1993, 14(2):432~455
31 H.G.Matthies, M.Meyer. Nonlinear Galerkin Methods for the Model Reduction of Nonlinear Dynamical Systems. Computers and Structures. 2003, 81(12):1277~1286
32 S.C.Sinha, Sangram Redkar, Venkatesh Deshmukh, and E.A.Butcher. Order Reduction of Parametrically Excited Nonlinear Systems: Techniques and Applications. Nonlinear Dynamics. 2005, 41(8):237~273
33张新华,徐健学.非线性Galerkin方法在梁的非线性动力学性态分析中的应用.《工程力学》增刊. 1996:564~568
34彭云,李喜德.微梁与基底间毛细粘附作用.第十一届实验力学学术会议大连. 2005:1150~1156
35胡海岩,孙久厚,陈怀海.机械振动与冲击.航空工业出版社. 2002:169~170
36邹经湘.结构动力学.哈尔滨工业大学出版社. 1999:80~84
37黄祖洽,丁鄂江.表面浸润和浸润相变.上海科学技术出版社. 1994:1-3,38-45
38 M.I.Younis and A.H.Nayfeh. A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuation. Nonlinear Dynamics. 2003, 31(1):91~117.
39陈予恕.非线性振动.高等教育出版社, 2002:237~245
40闻邦椿,李以农,韩清凯.非线性振动理论中的解析方法及工程应用.东北大学出版社, 2001:59~75
41 A.H.Nayfeh, D.T.Mook. Nonlinear Oscillations. Wiley, New York. 1995: 444 ~467