Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

doi:10.1088/1674-1056/21/11/110401

中国物理B ›› 2012, Vol. 21 ›› Issue (11): 110401-110401.doi: 10.1088/1674-1056/21/11/110401

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

李晶^{a b c d}, 樊尚春^{a b c}, 李艳^{a b c}, 郭占社^{a b c}

a School of Instrument Science & Opto-electronics Engineering, Beihang University, Beijing 100191, China;
b Key Laboratory of Precision Opto-mechatronics Techonology, Ministry of Education Beijing 100191, China;
c Key Laboratory of Inertial Science and Technology for National Defence, Beijing 100191, China;
d School of Information and Communication Engineering, North University of China, Taiyuan 030051, China

收稿日期:2012-03-07 修回日期:2012-07-07 出版日期:2012-10-01 发布日期:2012-10-01
基金资助:
Project supported by the National Natural Science Foundation of China (Grant No. 60927005), the 2012 Innovation Foundation of BUAA for PhD Graduates, and the Fundamental Research Funds for the Central Universities, China (Grant No. YWF-10-01-A17).

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

Li Jing (李晶)^{a b c d}, Fan Shang-Chun (樊尚春)^{a b c}, Li Yan (李艳)^{a b c}, Guo Zhan-She (郭占社 )^{a b c}

a School of Instrument Science & Opto-electronics Engineering, Beihang University, Beijing 100191, China;
b Key Laboratory of Precision Opto-mechatronics Techonology, Ministry of Education Beijing 100191, China;
c Key Laboratory of Inertial Science and Technology for National Defence, Beijing 100191, China;
d School of Information and Communication Engineering, North University of China, Taiyuan 030051, China

Received:2012-03-07 Revised:2012-07-07 Online:2012-10-01 Published:2012-10-01
Contact: Li Yan E-mail:yanli.83@163.com
Supported by:
Project supported by the National Natural Science Foundation of China (Grant No. 60927005), the 2012 Innovation Foundation of BUAA for PhD Graduates, and the Fundamental Research Funds for the Central Universities, China (Grant No. YWF-10-01-A17).

摘要/Abstract

摘要： The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied. It is assumed that the theoretical formulations are based on Euler-Bernoulli beam theory. The governing equations of motion are derived by using Rayleigh-Ritz method and transformed into Mathieu equations, which is formed to determine the stability criterion and stability regions for parametricallyexcited linear resonant beams. An improved stability criterion is obtained using periodic Lyapunov functions. The boundary points on the stable regions are determined by using small parameter perturbation method. Numerical results and discussion are presented to highlight the effects of beam length, axial force and damped coefficient on the stability criterion and stability regions. While some stability rules are easy to anticipate, we draw some conclusions: with the increase of damped coefficient, stable regions arise; with the decrease of beam length, the conditions of the damped coefficient arise instead. These conclusions can provide a reference for the robust design of the parametricallyexcited linear resonant sensors.

关键词: resonant beams, dynamic stability, parametrically excitation, periodic axial force

Abstract: The parametric dynamic stability of resonant beams with various parameters under periodic axial force is studied. It is assumed that the theoretical formulations are based on Euler-Bernoulli beam theory. The governing equations of motion are derived by using Rayleigh-Ritz method and transformed into Mathieu equations, which is formed to determine the stability criterion and stability regions for parametrically-excited linear resonant beams. An improved stability criterion is obtained using periodic Lyapunov functions. The boundary points on the stable regions are determined by using small parameter perturbation method. Numerical results and discussion are presented to highlight the effects of beam length, axial force and damped coefficient on the stability criterion and stability regions. While some stability rules are easy to anticipate, we draw some conclusions: with the increase of damped coefficient, stable regions arise; with the decrease of beam length, the conditions of the damped coefficient arise instead. These conclusions can provide a reference for the robust design of the parametrically-excited linear resonant sensors.

Key words: resonant beams, dynamic stability, parametrically excitation, periodic axial force

中图分类号: (Relativistic stars: structure, stability, and oscillations)

04.40.Dg

02.30.Hq (Ordinary differential equations) 05.10.-a (Computational methods in statistical physics and nonlinear dynamics)

李晶, 樊尚春, 李艳, 郭占社 . Dynamic stability of parametrically-excited linear resonant beams under periodic axial force[J]. 中国物理B, 2012, 21(11): 110401-110401.

Li Jing (李晶), Fan Shang-Chun (樊尚春), Li Yan (李艳), Guo Zhan-She (郭占社 ). Dynamic stability of parametrically-excited linear resonant beams under periodic axial force[J]. Chin. Phys. B, 2012, 21(11): 110401-110401.

参考文献 35

[1]	Seshia A A 2002 Integrated Micromechanical Resonant Sensors for Inertial Measurement Systems (Ph.D. Thesis) (University of California, Berkeley)
[2]	Yang J B, Jiang L J and Chen D C 2004 J. Sound Vib. 274 863
[3]	Jeong C, Seok S, Lee B, Kim H and Chun K 2004 J. Micromech. Microeng 14 1530
[4]	Mahmoodi S N, Jalili N and Khadem S E 2008 J. Sound Vib. 311 1409
[5]	Zhang J H, Mao X L and Liu Q Q 2012 Chin. Phys. B 21 86101
[6]	Li S S, Lan J and Han B 2012 Chin. Phys. B 21 064601
[7]	Prak A1993 Silicon Resonant Sensors: Operation and Response (Ph.D. Thesis) (Netherlands: University of Twente)
[8]	Roessig T A 1998 Integrated MEMS Tuning Fork Oscillators for Sensor Applications (Ph.D. Thesis) (University of California, Berkeley)
[9]	Tabata O and Yamamoto T 1999 Sensor. Actuat. A-Phys. 75 53
[10]	Bilic D, Howe R T, Clark W A and Roessig T A 1999 The 10th International Conference on Solid-State Sensors and Actuators, Transducers Sendai, Japan, p. 1824
[11]	Recktenward G and Rand R 2005 Int. J. Nonlinear Mech. 40 1160
[12]	McLachlan N W 1964 Theory and Application of Mathieu Function (Dover Publicatons Inc)
[13]	Harris C and Crede C 1976 Shock and Vibration Handbook (2nd ed.) (Mc-Graw Hill) pp. 1-13
[14]	Liu S S and Liu S D 2002 Special Function (Weather Publicatons Inc) pp. 737-739 (in Chinese)
[15]	Wang Y Z and Zhou Y Z 2011 Chin. Phys. B 20 040501
[16]	Fan S C and Li Y 2011 Chin. Phys. B 21 050401
[17]	Francis J P and Glenn R F 2008 Proc. R. Soc. A 464 1885
[18]	Coisson R, Vernizzi G and Yang X K International Workshop on Open-source Software for Scientific Computation (Chap. 3), September 18-20 2009 Guiyang, China, p. 1
[19]	Rand R H, Sah S M and Suchorsky M K 2010 Commun. Nonlinear Sci. Numer. Simulat. 15 3254
[20]	Arrowsmith D Kand Mondragon R J 1999 Meccanica. 34 401
[21]	Younesian D, Esmailzadeh E and Sedaghati R 2005 Nonlinear Dyn. 39 335
[22]	Guennoun K, Houssni M and Belhaq M 2002 Nonlinear Dyn. 27 211
[23]	Susan, Wooden M and Sinha S C 2007 Nonlinear Dyn. 47 263
[24]	Thomas and Waters J 2010 Nonlinear Dyn. 60 341
[25]	Zounes R S and Rand R H 2002 Nonlinear Dyn. 27 87
[26]	Belhap M, Guennoun K and Houssni M 2002 Int. J. Nonlinear Mech. 37 445
[27]	Maharana P K and Shrivastava S K 1984 Acta Astronautica. 11 103
[28]	Taylor J H and Narendra K S 1969 SIAM J Appl. Math. 17 343
[29]	Gunderson H, Rigas H and Van Vleck F S 1974 SIAM J Appl. Math. 26 345
[30]	Pipes L A 1953 J. Appl. Phys. 24 902
[31]	Richards J A 1976 Appl. Math. 30 240
[32]	Richards J A 1977 Proc. IEEE 65 1549
[33]	Richards J A 1983 Analysis of Periodically Time Varying Systems (New York: Springer Verlag)
[34]	Michael G J 1967 IEEE Trans. Automat. Contr. 12 337
[35]	Ramarajan S and Rao S N 1971 IEEE Trans. Automat. Contr. 67 363

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

Dynamic stability of parametrically-excited linear resonant beams under periodic axial force

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