静电驱动阶梯型微悬臂梁吸合电压分析

Pull-in Voltage Analysis of Electrostatically Actuated Stepped Micro-cantilever Beam

针对静电驱动微机电系统(Micro-electro-mechanical system,MEMS)器件中常见的阶梯型微悬臂梁结构,提出一种吸合电压的计算方法。基于欧拉梁理论和修正的偶应力理论,运用能量法推导出吸合电压理论模型。采用试函数与待定系数的积来表示微悬臂梁位移,利用泰勒展开来简化求解过程。通过与有限元结果对比来验证模型的正确有效性,讨论试函数的选取以及泰勒展开阶数的确定,并与传统质量弹簧模型方法进行对比,最后研究其吸合特性。结果表明,泰勒展开阶数取8时截断误差可以忽略,试函数选择阶梯型微悬臂梁位移函数,理论模型预测误差小于5%,预测结果明显优于传统方法。吸合电压随宽度比增加而单调递增,随长度比增加出现先减小后增加的变化现象,可为低驱动电压MEMS器件设计提供参考。该理论模型中考虑了边缘场效应、尺度效应的影响,可应用于微纳米尺度的微悬臂梁的吸合电压预测。

Aiming at the stepped micro-cantilever beam commonly used in electrostatically actuated micro electromechanical systems(MEMS) devices, a pull-in voltage calculation method is proposed. Based on a modified couple stress theory and Euler theory, the pull-in voltage model is derived by the energy method. The beam deflection is expressed as the product of a trial function and an unknown coefficient, and the solving process is simplified by using Taylor’s expansion. The proposed method is validated by comparing with the finite element solutions. Further the selection of trial function and the determination of the Taylor’s expansion order are discussed, the pull-in behaviour of stepped beam is investigated, and the comparison with the traditional method is made. The results show that the truncation error can be neglected as setting the Taylor’s expansion order to be 8. The prediction error of the proposed model is less than 5% when using its deflection function, and the predicted results are obviously superior to the traditional method. The pull-in voltage monotonously increase with the increase of width ratio, and first decrease and then increase with the increase of length ratio, which can provide information for the design of low actuation voltage of MEMS devices. The proposed model considers the effect of the fringing field and the size-dependence, and can be used to predict the pull-in voltage in micro/nano scale.