Scale effect on the natural frequency and vibration mode of a cantilever micro beam
XIE Xinji1,LIU Zhanfang1，2，3,DU Qiumei2
1. College of Aerospace Engineering, Chongqing University, Chongqing 400044, China;
2. State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chonqing University, Chongqing 400044, China;
3. Chongqing Key Laboratory of Heterogeneous Material Mechanics, Chongqing University, Chongqing 400044, China
Abstract：The first frequency of a cantilever microbeam predicted by classical elasticity is far lower than measured by experiments. Generalized elasticity is especially applicable to structure dynamics analysis with scale effect, where both the rotational deformation and couple stress is taken into account. The measurement of deformation was improved. Finite element dynamic equations of generalized elasticity were established through the principle of virtue work, and a numerical analysis method was used to study the natural frequency and vibration mode of the cantilever microbeam. The results show that the existent of scale effect of its natural frequency is related to its corresponding mode. The corresponding natural frequencies of bending and torsional modes have significant increment compared to classical elasticity. The torsional mode is taken into consideration. However, little change of natural frequency of tensile mode can be observed because deformation is not involved.
谢新吉1,刘占芳1,2,3,杜丘美1. 悬臂微梁固有频率和模态的尺寸效应[J]. 振动与冲击, 2018, 37(12): 187-192.
XIE Xinji1,LIU Zhanfang1，2，3,DU Qiumei2. Scale effect on the natural frequency and vibration mode of a cantilever micro beam. JOURNAL OF VIBRATION AND SHOCK, 2018, 37(12): 187-192.
[1] 王淑华. MEMS传感器现状及应用[J]. 微纳电子技术，2011, 48(8): 516-522.
Wang Shu-hua. Current Status and Applications of MEMS Sensors [J]. Micronanoelectronic technology, 2011,48(8):516-522.
[2] 刘林仙，张国军，许姣等. 双T型MEMS仿生矢量水听器的设计与测试[J]. 振动与冲击，2013, 32(2): 129-134.
LIU Lin-xian, ZHANG Guo-jun, XU Jiao. Design and test for a double T-shape MEMS bionic vector hydrophone [J]. Journal of vibration and shock, 2013, 32(2): 129-134.
[3] 韩雷，严国政. 底座激振下微型叠层芯片共振频率检测[J]. 振动与冲击，2012, 31(7): 153-157.
Han Lei, Yan Guo-zheng. Resonant frequency measurement for a micro stacked chip with base excitation [J]. Journal of vibration and shock, 2012, 31(7): 153-157.
[4] Fleck N A, Muller G M, Ashby M F, etal. Strain gradient plasticity: Theory and experiment[J]. Acta metallurgica et materialia, 1994, 42(2): 475-487.
[5] Stolken J S, Evans A G. A microbend test method for measuring the plasticity length scale[J]. Acta materialia, 1998, 46(14): 5109-5115.
[6] 郭香华，方岱宁，李喜德. 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.
Guo Xiang-hua, Fang Dai-ning, Li Xi-de. Measurement of deformation of pure Ni Foils by speckle pattern interferometry[J]. Mechanics in Engineering, 2005, 27(2): 22-25.
[7] 冯秀艳，郭香华，方岱宁，等. 微薄梁三点弯曲的尺度效应研究[J]. 力学学报，2007, 39(4): 479-485.
Feng Xiu-yan，Guo Xiang-hua, Fang Dai-ning, etal. Three-point microbend size effects for pure Ni foils[J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(4): 479-485.
[8] Lam D C C, Yang F, Chong A C M, etal. Experiments and theory in strain gradient elasticity [J]. Journal of the Mechanics & Physics of Solids, 2003, 51: 1477-1508.
[9] Cosserat E, Cosserat F. Theorie des Corps Deformables[M]. Paris: Herman et fils.1909.
[10] Toupin R A. Elastic materials with couple-stresses [J]. Archive for Rational Mechanics and Analysis, 1962, 11(1): 385-414.
[11] Mindlin R D, Tiersten H F. Effects of couple-stresses in linear elasticity [J]. Archive for Rational Mechanics and Analysis, 1962, 11(1): 415-448.
[12] Fleek N A, Hutehinson J W. A phenomenological theory for strain gradient effects in plasticity[J]. Journal of the Mechanics and Physics of Solids, 1993, 41: 1825-1857.
[13] Fleek N A, Hutehinson J W. Strain Gradient Plasticity [J]. Advances in Applied Mechanics, 1997, 33: 295-361.
[14] 黄克智，邱信明，姜汉卿. 应变梯度理论的新进展(一)—偶应力理论和SG理论[J]. 机械强度, 1999, 21(2): 81-87.
Huang Ke-zhi, Qiu Xin-ming, Jiang Han-qing. Recent advances in strain gradient plasticity-I—couple stress theory and SG theory[J]. Journal of Mechanical Strength, 1999, 21(2): 81-87. [15] 陈少华, 王自强. 应变梯度理论进展[J]. 力学进展, 2003, 33(2): 207-216.
Chen Shao-hua, Wang Zi-qiang. Advances in strain gradient theory[J]. Advances in Mechanics, 2003, 33(2):207-216.
[16] Yang F, Chong A C M, Lam D C C, etal. Couple stress based strain gradient theory for elasticity [J]. International Journal of Solids and Structures, 2002, 39: 2731-2743.
[17] Liu Z F, Fu Z, Scale Effects of the Stress Symmetry in Generalized Elasticity[J]. International Journal of Aerospace and Lightweight Structures, 2012, 2(4): 509-521.
[18] 颜世军, 刘占芳. 修正的偶应力线弹性理论及广义线弹性体的有限元方法[J]. 固体力学学报，2012, 33(3): 279-287.
Yan Shi-jun, Liu Zhan-fang. A modified couple stress linear elasticity and finite element method for generalized elastic bodies [J]. Chinese Journal of Solid Mechanics, 2012, 33(3): 279-287.
[19] 刘占芳, 颜世军, 冯晓伟. 离心场中含旋转梯度影响的弹性体动力特性分析[J]. 振动与冲击，2012, 31(16): 164-168.
Liu Zhan-fang, Yan Shi-jun, Feng Xiao-wei. Dynamic analysis of elastic body with rotation gradient effects in centrifugal field [J]. Journal of vibration and shock, 2012, 31(16): 164-168.
[20] Liu Z F, Sun X Y, Guo Y, et al. On elastic stress waves in an impacted plate [J]. International Journal of Applied Mechanics, 2014, 06(04): 1450047
[21] Park S K, Gao X L. Bernoulli–Euler beam model based on a modified couple stress theory [J]. Journal of Micromechanics and Microengineering, 2006, 16(11): 2355-2359.
[22] Ma H M, Gao X L, Reddy J N , A microstructure-dependent Timoshenko beam model based on a modified couple stress theory [J]. Journal of the Mechanics and Physics of Solids, 2008, 56(12): 3379-3391.
[23] 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(l): l-4.
Kang Xin, Xi Zhan-wen. Size effect on the dynamic characteristic of a micro beam based on Cosserat theory [J]. Journal of Mechanical Strength, 2007, 29(l): 1-4.
