有界窄带激励下微梁系统1:2参数共振
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摘要
研究电路与微梁耦合系统在有界窄带激励下的参数共振问题。建立有界窄带激励下微梁系统的随机微分方程。应用多尺度法得到系统参数共振的频率响应方程,导出系统的Ito随机微分方程,采用矩法得到系统一阶矩和二阶矩的近似表达式,数值分析系统各个参数对微梁响应的影响。结果表明,参数共振稳态解稳定的充分必要条件与系统一阶矩和二阶矩稳定的充分必要条件是一样的;当微梁上极板的宽度、厚度和长度增加时,极板振幅的二阶矩减小;当上极板的阻尼系数、轴向力以及上下极板间的距离增加时,极板振幅的二阶矩增大。
To study the parametric resonance of series circuit and micro-beam coupling system subjected to the narrow-band random excitation, the stochastic differential equation of the micro-beam system subjected to narrow-band excitation is established. The frequency response equation of the system is obtained based on the method of multiple scales. Ito stochastic differential equation of system is derived. The first order and second order steady state response of the system are obtained and the corresponding to the amplitude-frequency curves are calculated. The influence of the system parameters on the response of the micro-beam is analyzed. The results show that the sufficient and necessary conditions for the stability of the parametric resonance are the same as the first order and second order moment stability of the system. The numerical simulation shows that with the increase of the width,thickness and length of the plate,the second order response of the plate amplitude is decreased. When the damping coefficient,the axial force and the distance between the two plates are increased,the second order response of the plate amplitude is increased.
To study the parametric resonance of series circuit and micro-beam coupling system subjected to the narrow-band random excitation, the stochastic differential equation of the micro-beam system subjected to narrow-band excitation is established. The frequency response equation of the system is obtained based on the method of multiple scales. Ito stochastic differential equation of system is derived. The first order and second order steady state response of the system are obtained and the corresponding to the amplitude-frequency curves are calculated. The influence of the system parameters on the response of the micro-beam is analyzed. The results show that the sufficient and necessary conditions for the stability of the parametric resonance are the same as the first order and second order moment stability of the system. The numerical simulation shows that with the increase of the width,thickness and length of the plate,the second order response of the plate amplitude is decreased. When the damping coefficient,the axial force and the distance between the two plates are increased,the second order response of the plate amplitude is increased.
引文
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[2]张冬至,胡国清.微机电系统关键技术及其研究进展[J].压电与声光,2010(03):513-520.ZHANG Dong Zhi,HU Guo Qing.Key technologies of microelectromechanical system and its recent progress[J].Piezoelectrics&Acoustooptics,2010(03):513-520(In Chinese).
[3]孔胜利.微梁力学性能尺寸效应的研究[D].济南:山东大学,2009:2-3.KONG Sheng Li,Size effect on mechanical properties if micro-beams[D].Jinan:Shangdong University,2009:2-3(In Chinese).
[4]杨志安,贾尚帅.RLC串联电路与微梁耦合系统的吸合电压与电振荡[J].应用力学学报,2010(04):721-726;2010(04):850.YANG Zhi An,JIA Shang Shuai.Pull-in voltage and electric oscillations of RLC series circuit and micro-beam coupled system[J].Chinese Journal of Applied Mechanics,2010(04):721-726;2010(04):850(In Chinese).
[5]杨志安,贾尚帅.RLC串联电路与微梁耦合系统1∶2内共振分析[J].应用力学学报,2010(01):80-85;2010(01):225.YANG Zhi An,JIA Shang Shuai.Analysis of the 1:2 internal resonance of coupled RLC circuit and microbeam system[J].Chinese Journal of Applied Mechanics,2010(01):80-85;2010(01):225(In Chinese).
[6]刘素娟,齐书浩,张文明.静电驱动裂纹微悬臂梁结构振动特性分析[J].振动与冲击,2013(17):41-45.LIU Su Juan,QI Shu Hao,ZHANG Wen Ming.Vibration behavior of cracked micro-cantilever beam under electrostaticexcitation[J].Journal of Vibration and Shock,2013(17):41-45(In Chinese).
[7]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(01):1-4.KANG Xin,Xi Zhan Wen.Size effrct on the dynamic characteristic of a micro beam based on cosserat theory[J].Journal of Mechanical Strength,2007(01):1-4(In Chinese).
[8]贾尚帅.微电子机械系统非线性动力学研究[D].唐山:河北理工大学,2008:6-9.JIA Shang Shuai.Study on nonlinear dynamics of micro-electromechanical system[D].Tangshang:Hebei Polytechnic University,2008:6-9(In Chinese).
[9]徐伟,方同,戎海武.有界窄带激励下具有黏弹项的Duffing振子[J].力学学报,2002(05):764-771.XU Wei,FANG Tong,RONG Hai Wu.Duffing oscillator with viscoelastic term under narrrow-band random excitation[J].Acta Mechanica Sinica,2002(05):764-771(In Chinese).
[10]何蕊,罗文波,王本利,孔宪仁.随机激励下振动系统非线性特性定性方法研究[J].工程力学,2010(04):24-29.HE Rui,LUO Wen Bo,WANG Ben Li,et al.Characterization of nonlinear vibrating systems under random excitation[J].Engineering Mechanics,2010(04):24-29(In Chinese).
[11]Zhu WQ.Stochastic Jump and Bifurcation of A Duffing Oscillator Under Narrow-Band Excitation[J].Acta Mechanica Sinica,1994(01):73-81.
[12]戎海武,王向东,孟光,等.谐和与随机噪声联合作用下Duffing振子的双峰稳态密度[J].应用数学和力学,2006(11):1373-1379.RONG Hai Wu,WANG Xiang Dong,MENG Guag,et al.On double peak probability density functions of a duffing oscillator to combined deterministic and random excitations[J].Applied Mathematics and Mechanics,2006(11):1373-1379(In Chinese).
[13]朱位秋.随机振动[M].北京:科学出版社,1992:137-134.ZHU Wei Qiu.Random vibration[M].Beijing:Science Press,1992:137-134(In Chinese).
[14]庄表中.非线性随机振动理论及其应用[M].杭州:浙江大学出版社,1986:169-177.ZHUANG Biao Zhong.Theory and application of nonlinear random vibration[M].Hangzhou:Zhejiang University Press,1986:169-177(In Chinese).