微机电系统中微梁变形行为的尺度效应
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摘要
在微机电系统(MEMS)中,当微结构尺寸减小到微米量级后,其力学和物理性能表现出与块体结构不同的特性,具有一定的尺度效应。由于静电驱动微梁常作为驱动结构,因此对其尺度效应的研究具有重要意义。
本文主要分析了静电驱动器中常用机构悬臂梁和两端固支梁的静、动态中体现的尺度效应。首先,分析微梁的吸合效应现象,建立其临界电压和临界挠度的数学模型,并运用ANSYS中的直接耦合法—TRANS126降阶单元,建立微梁的机电耦合模型,得出了几何结构参数对微梁吸合效应的影响。
其次,基于微梁在变时载荷下的振动理论,采用TRANS126降阶单元,经过多次试算,找到电压与挠度的收敛点,得到微梁的振动固有频率和吸合时间,同时建立了直接嵌入微梁材料的特征长度的固有频率与厚度之间的定量关系分析模型。
最后,基于微梁外形尺寸进入微米量级后而导致力学行为的尺度效应,建立了微梁挠度变形的静态变形解析分析模型,将静电力作用下梁弯曲理论与幂级数展开式、欧拉降阶公式和微梁的边界条件进行结合,提出了一种考虑尺度效应的分析微梁挠度变形的近似算法,得到了微梁在低压驱动下的近似解析模型。
通过以上数学模型和ANSYS分析结果可知,当微梁的结构尺寸减小到可与材料的特征长度相比较时,材料和力学特性将体现出一定的尺寸效应。
In Micro Electro-Mechanical System (MEMS), when the size of the microstructure decreases to micrometer, its performance of the mechanical and physical properties will show the different characteristics with the bulk material, that is scale effect. The electrostatic driven micro-beam is often used in drivers, so it is significant to analyze the scale effects. This paper primarily analyzes the scale effect of the static analysis and the dynamic analysis of the micro-beam which is actuated by electrostatic force, details are as follows:
Firstly, base on the micro-beam pull-effect phenomenon, this paper establishes the mathematical model between the threshold voltage and the critical deflection, then uses the direct coupling method-the TRANS126 reduction unit of the finite element ANSYS to set up the electromechanical coupling model. Meanwhile, obtains how the geometry structure parameters impact upon the micro-beam pull-in effect.
Secondly, base on the vibration theory of micro-beam, under the variable load, the paper applies the TRANS126 reduction unit, after many times pilot calculation, gets the convergence points of the voltage and the deflection, and obtains the natural frequency and the pull time of micro-beam. Meanwhile, establishes the quantitative analysis model between the thickness and the natural frequency of intrinsic characteristics length of the micro-beam.
At last, this paper establishes a static deformation analysis model of micro-beam deflection based upon the scale effect of the mechanical behavior when the micro-beam dimensions range in micron dimension. Simultaneously, combines bending theory of beam under the action of static electricity with the power series expansion, the Euler's reduction formula and the boundary conditions of the micro-beam proposes a approximation analytical model for the scale effect on the micro-beam deformation, and approximate analytical model of micro-beam under the low-voltage driven is obtained.
Base on the results of the models and the ANSYS, it shows that the material and the mechanical properties will reflect the scale effect when the micro-beam structural dimension decreases to which it can compare with characteristic length.
本文主要分析了静电驱动器中常用机构悬臂梁和两端固支梁的静、动态中体现的尺度效应。首先,分析微梁的吸合效应现象,建立其临界电压和临界挠度的数学模型,并运用ANSYS中的直接耦合法—TRANS126降阶单元,建立微梁的机电耦合模型,得出了几何结构参数对微梁吸合效应的影响。
其次,基于微梁在变时载荷下的振动理论,采用TRANS126降阶单元,经过多次试算,找到电压与挠度的收敛点,得到微梁的振动固有频率和吸合时间,同时建立了直接嵌入微梁材料的特征长度的固有频率与厚度之间的定量关系分析模型。
最后,基于微梁外形尺寸进入微米量级后而导致力学行为的尺度效应,建立了微梁挠度变形的静态变形解析分析模型,将静电力作用下梁弯曲理论与幂级数展开式、欧拉降阶公式和微梁的边界条件进行结合,提出了一种考虑尺度效应的分析微梁挠度变形的近似算法,得到了微梁在低压驱动下的近似解析模型。
通过以上数学模型和ANSYS分析结果可知,当微梁的结构尺寸减小到可与材料的特征长度相比较时,材料和力学特性将体现出一定的尺寸效应。
In Micro Electro-Mechanical System (MEMS), when the size of the microstructure decreases to micrometer, its performance of the mechanical and physical properties will show the different characteristics with the bulk material, that is scale effect. The electrostatic driven micro-beam is often used in drivers, so it is significant to analyze the scale effects. This paper primarily analyzes the scale effect of the static analysis and the dynamic analysis of the micro-beam which is actuated by electrostatic force, details are as follows:
Firstly, base on the micro-beam pull-effect phenomenon, this paper establishes the mathematical model between the threshold voltage and the critical deflection, then uses the direct coupling method-the TRANS126 reduction unit of the finite element ANSYS to set up the electromechanical coupling model. Meanwhile, obtains how the geometry structure parameters impact upon the micro-beam pull-in effect.
Secondly, base on the vibration theory of micro-beam, under the variable load, the paper applies the TRANS126 reduction unit, after many times pilot calculation, gets the convergence points of the voltage and the deflection, and obtains the natural frequency and the pull time of micro-beam. Meanwhile, establishes the quantitative analysis model between the thickness and the natural frequency of intrinsic characteristics length of the micro-beam.
At last, this paper establishes a static deformation analysis model of micro-beam deflection based upon the scale effect of the mechanical behavior when the micro-beam dimensions range in micron dimension. Simultaneously, combines bending theory of beam under the action of static electricity with the power series expansion, the Euler's reduction formula and the boundary conditions of the micro-beam proposes a approximation analytical model for the scale effect on the micro-beam deformation, and approximate analytical model of micro-beam under the low-voltage driven is obtained.
Base on the results of the models and the ANSYS, it shows that the material and the mechanical properties will reflect the scale effect when the micro-beam structural dimension decreases to which it can compare with characteristic length.
引文
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[3] Haque M A, Saif M T A. Strain gradient effect in nanoscal thin films [J]. Acta materialia, 2003, 51:3053.
[4]苏继龙,庄哲峰等.MEMS微结构变形行为尺度效应的研究进展[J].传感器与微系统,2009,28(7):1-4.
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[7]徐泰然.MEMS和微系统设计与制造[M].北京:机械工业出版社,2004,1:294-295.
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[9]阎照文.ANSYS10.0工程电磁分析技术与实例详解[M].北京:中国水利水电出社,2006,11:374-378.
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[11] Haque M A,Saif M T A.Application of MEMS force sensors for in situ mechanical characterization of nano-scale thin films in SEM and TEM[J].Sensors and actuators,2002,A(97):239-245.
[12] L . Sun, X . E . Liu, X . J . Zheng, Numeriealsimulation of magnetostricitve film-substrate mieroeantilever.Chin.J.Appl.Mech, 2004, 21:41- 45.
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[18] S. Belaidi, et al. Electrostatic forces acting on the tip in atomic force microscopy: Modelization and comparision with analytic expressions [J].J.Appl.Phys.1997, 81(3):1023-1030.
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[20] S. Belaidi, F. Lebon, et al. Finite element simulation of the resolution in electrostatic force microscopy[J].Appl.Phys.A.1998,Vol.66:239-243.
[21] Daniel J Bums, Herbert F Helbig.A System for automatic electrical and optical characterization of microelectromechanical devices[J].Joumal of Microelectromechanical System, 1999, 8(4):473—82.
[22] Kevin E, Speller, Howard Goldberg, Jeff Gannon ,et a1.Upique MEMS characterization solutions enabled by laser Doppler vibrometer measurements[C].Proc SPIE,4827:478-485.
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[28] Sazzadur Chowdhury,M. Ahmadi, W. C.Miller. 1st International Conference on Sensing Technology [J].2005,112-117.
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[30]王洪喜,贾建援,樊康旗.求解静电驱动微梁变形的数值与解析方法[J].机械设计与研究,2005,21(2):35-38.
[31]张广平.微米尺寸不锈钢的形变与疲劳行为的尺寸效应[J].金属学报,2005,41(4):337-340.
[32]温诗铸,丁建宁.微型机械设计基础研究[J].机械工程学报,2000,36(7):39-42.
[33]郭香华,方岱宁.用电子散斑法对纯镍薄片弯曲变形的测量[J].力学与实践,2005,27(2):22-25.
[34]丁建宁,孟永刚,温诗铸等.多晶硅微电子机械构件材料强度尺度效应研究[J].机械强度,2001,23(4):385.
[35]李雷,谢水生,黄国杰,等.超薄板料微弯曲成型过程中尺度效应的数值研究[J].塑性工程学报,2005,12(12):93-97.
[36]陈广文.静电式开关硅悬臂梁的变形分析[J].传感器技术,2004,20(12):29-31.
[37]张健,李伟华.基于力学描述的MEMS悬臂梁系统级动态模拟方法[J].传感技术学报,2006,19(5):1376-1380.
[38]黄玉波.MEMS力学特性测试及可靠性分析中若干关键问题的研究[D].天津大学.2008,6:102-103.
[39]刘肖,陈旭远.RFMEMS开关的力电耦合仿真分析[J].青岛大学学报,2007,22(3):60-65.
[40]贾建援,王洪喜,樊康旗.静电力作用下的微梁失稳临界电压分析[J].应用力学学报.2005,22(1):95-98.
[41]陆静,韦笑梅,马小强.一种分析静电驱动微悬臂梁变形的高精度算法[J].广西工学院学报,2008,19(1):34-37.
[42]丁建宁.多晶硅微机械构件损伤的表面效应[J].机械强度,2005,27(2):211.
[43]董玮.体硅微机械光开关的设计与制作工艺的研究[D].吉林大学博士学位论文,2004.
[44] M. Lishchynska, N. Cordero, O. Slattery,et al.“Modelling electrostatic behaviour of microcantilevers incorporating residual stress gradient and non-ideal anchors”,J.Micromech.Microeng,2005,15,S10-S14.
[45]高世桥,刘海鹏.微机电系统力学[M].北京:国防工业出版社,2008:104-111.
[46]黄志洵,王小金.微波传输线理论与实用技术[M].北京:科学出版社,1996:102-106.
[47] Peter M.Osterberg and Stephen D.Senturia.M-TEST:A Test Chip for MEMS Material PropertyMeasurement Using Electrostatically Actuated Test Structures[J] . Journal of Microelectromechanical Systems,1997,7,6(2):107-118.
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[51] Chang Liu.Foundations of MEMS[D].Pearson Education,2006:79-80.
[52]刘庆玲.不同加载方式对微悬臂梁弹性系数的影响[J].机械设计与研究,2008,24(3):49-51.
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[55]刘晶波,杜修力.结构动力学[M].机械工业出版社,2005,1:152-154.
[56]康新,席占稳.基于Cosserat理论的微梁振动特性的尺度效应[J].机械强度,2007,29(1):1-4.
[2] Stolken J S, Evans A G. A micro-bend test method for measuring the plasticity length scale [J]. Acta materials,1998,46(14):5109.
[3] Haque M A, Saif M T A. Strain gradient effect in nanoscal thin films [J]. Acta materialia, 2003, 51:3053.
[4]苏继龙,庄哲峰等.MEMS微结构变形行为尺度效应的研究进展[J].传感器与微系统,2009,28(7):1-4.
[5]张向军,孟永钢,温诗铸等.微电子机械系统中的若干固体力学问题[J].力学与实践,2003,25(2):7.
[6]吕劲楠,唐洁影. MEMS微悬臂梁在冲击环境下的可靠性[J].功能材料与器件学报, 2008,14(1):103-106.
[7]徐泰然.MEMS和微系统设计与制造[M].北京:机械工业出版社,2004,1:294-295.
[8]梅涛.孔德义.张培强等.微电子机械系统的力学特性与尺度效应[J].机械强度,2001,23(4):373-379.
[9]阎照文.ANSYS10.0工程电磁分析技术与实例详解[M].北京:中国水利水电出社,2006,11:374-378.
[10] Pearson GL,Read W T, et al. Deformation and fracture of small silicon crystals [J].Acta Met, 1957, (5):181-191.
[11] Haque M A,Saif M T A.Application of MEMS force sensors for in situ mechanical characterization of nano-scale thin films in SEM and TEM[J].Sensors and actuators,2002,A(97):239-245.
[12] L . Sun, X . E . Liu, X . J . Zheng, Numeriealsimulation of magnetostricitve film-substrate mieroeantilever.Chin.J.Appl.Mech, 2004, 21:41- 45.
[13] EngelU,EcksteinR.Microforming-form basic research to its realization[J].Journal of Materials processing Technology, 2002, 125(1):35-44.
[14] Raulea L . Govaert L . Baaijens F . Grain and Specimen Size Effects in Processing Metal Sheets[J].Proceedings of the 6th ICTP.Berlin, 1999.
[15] Fleck NA, Muller GM, Ashby M F etal. Strain gradient plasticity: theory and experiment [J]. Acta Metall Mater ,1994, 42(2):475.
[16] Dominicus J. IJntema and Harrie A.C. Tilmans. Static and dynamic aspects of an air-gap capacitor[J].Sensors and Actuators A,1992, 35:121-128.
[17] Elmer S. Hung,et al. Generating Efficient Dynamical Models for Microelectromechanical Systems Systems from a Few Finite-Element Simulation Runs [J].Journal of Microelectromechanical System,1999, 8(3):280~289.
[18] S. Belaidi, et al. Electrostatic forces acting on the tip in atomic force microscopy: Modelization and comparision with analytic expressions [J].J.Appl.Phys.1997, 81(3):1023-1030.
[19] S. Watanabe, et al. Electrostatic force microscope imaging analyzed by the surface charge method[J].J.Vac.Sci.Technol.B.1993, 11(5):1774-1781.
[20] S. Belaidi, F. Lebon, et al. Finite element simulation of the resolution in electrostatic force microscopy[J].Appl.Phys.A.1998,Vol.66:239-243.
[21] Daniel J Bums, Herbert F Helbig.A System for automatic electrical and optical characterization of microelectromechanical devices[J].Joumal of Microelectromechanical System, 1999, 8(4):473—82.
[22] Kevin E, Speller, Howard Goldberg, Jeff Gannon ,et a1.Upique MEMS characterization solutions enabled by laser Doppler vibrometer measurements[C].Proc SPIE,4827:478-485.
[23] Christian Rembe, Lilac Muller, chard S Muller, et a1.Full three-dimensional motion characterization of a gimballed electmatatic microactuator[C].39th Annual Reliability Physics Symposium.Orlando, F1ofida, 200l, 9l-98.
[24] Kurth S.Interference microscopic techniques for dynamic testing of MEMS[M].Germany:Chemnitz University of Technology,2002.
[25] Sandia Report.Microscale Modeling and Simulation[R].Sandia National Laboratories,2001:11-16.
[26] Degani,O,et al.Pull-in Study of an Electrostatic Torsion Microactuator [J], Microelectromechanical System, 1998, 7:373-379.
[27] Hamed Sadeghian,Ghader Rezazadeh,Ehsan Malekpour et al. Pull-In Voltage of Fixed-Fixed End Type MEMS Switches with Variative Electrostatic Area[J].Sensors&transducers,2006,66:526-533.
[28] Sazzadur Chowdhury,M. Ahmadi, W. C.Miller. 1st International Conference on Sensing Technology [J].2005,112-117.
[29]刘凯,韩光平.微电子机械系统力学性能及尺寸效应[M].机械工业出版社,2009.2.
[30]王洪喜,贾建援,樊康旗.求解静电驱动微梁变形的数值与解析方法[J].机械设计与研究,2005,21(2):35-38.
[31]张广平.微米尺寸不锈钢的形变与疲劳行为的尺寸效应[J].金属学报,2005,41(4):337-340.
[32]温诗铸,丁建宁.微型机械设计基础研究[J].机械工程学报,2000,36(7):39-42.
[33]郭香华,方岱宁.用电子散斑法对纯镍薄片弯曲变形的测量[J].力学与实践,2005,27(2):22-25.
[34]丁建宁,孟永刚,温诗铸等.多晶硅微电子机械构件材料强度尺度效应研究[J].机械强度,2001,23(4):385.
[35]李雷,谢水生,黄国杰,等.超薄板料微弯曲成型过程中尺度效应的数值研究[J].塑性工程学报,2005,12(12):93-97.
[36]陈广文.静电式开关硅悬臂梁的变形分析[J].传感器技术,2004,20(12):29-31.
[37]张健,李伟华.基于力学描述的MEMS悬臂梁系统级动态模拟方法[J].传感技术学报,2006,19(5):1376-1380.
[38]黄玉波.MEMS力学特性测试及可靠性分析中若干关键问题的研究[D].天津大学.2008,6:102-103.
[39]刘肖,陈旭远.RFMEMS开关的力电耦合仿真分析[J].青岛大学学报,2007,22(3):60-65.
[40]贾建援,王洪喜,樊康旗.静电力作用下的微梁失稳临界电压分析[J].应用力学学报.2005,22(1):95-98.
[41]陆静,韦笑梅,马小强.一种分析静电驱动微悬臂梁变形的高精度算法[J].广西工学院学报,2008,19(1):34-37.
[42]丁建宁.多晶硅微机械构件损伤的表面效应[J].机械强度,2005,27(2):211.
[43]董玮.体硅微机械光开关的设计与制作工艺的研究[D].吉林大学博士学位论文,2004.
[44] M. Lishchynska, N. Cordero, O. Slattery,et al.“Modelling electrostatic behaviour of microcantilevers incorporating residual stress gradient and non-ideal anchors”,J.Micromech.Microeng,2005,15,S10-S14.
[45]高世桥,刘海鹏.微机电系统力学[M].北京:国防工业出版社,2008:104-111.
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