常云霞 ,张琪昌
AuthorsHTML:常云霞 ^{1, 2}，张琪昌 ¹
AuthorsListE:Chang Yunxia ,Zhang Qichang
AuthorsHTMLE:Chang Yunxia ^{1, 2}，Zhang Qichang ¹
Unit:1. 天津大学机械工程学院，天津 300350；
2. 河北大学机械系，保定 071000

Unit_EngLish:1. School of Mechanical Engineering，Tianjin University，Tianjin 300350，China；
2. Department of Mechanical Engineering，Hebei University，Baoding 071000，China

Abstract_Chinese:在自顶向下制造纳米结构的过程中，热膨胀失配或边界支撑件发生底切刻蚀而引起的压缩松弛会产生应力 和应变，可导致纳机电系统静电驱动梁的初始曲率．因此，实际上很难构建无曲率的纳米结构．本文综合考虑尺度 效应、卡西米尔力和静电力的边缘效应等因素，应用 Euler Bernoulli 梁理论，建立了静电驱动纳米浅拱固支梁式谐 振器的分布质量模型；应用 Galerkin 方法研究了不同初始曲率幅值对系统静态分岔与谐振频率的影响；进而以静态 解为基础，引入五次非线性刚度的一阶离散动态模型，并应用多尺度方法研究了不同初始曲率幅值和五次非线性刚 度对于系统主共振频响特性的影响；提出了非线性软硬特性转变的判据．结果表明：与直梁相比，拱梁的静态吸合 电压增大，临界位移增大，随着初始曲率幅值的增加，导致系统出现突跳和吸合不稳定性；与直梁相比，初始曲率 改变了系统频率，降低了响应幅值，出现了复杂的非线性特性．因此，初始曲率缺陷不可忽略，反而可以加以利用 来扩展工作范围，提高系统性能．分析结果与已有报道结果相符．
Abstract_English:During top-down fabrication of nanostructures，stress and strain may develop due to thermal expansion mismatch or the compressive relaxation of the supports as they become undercut by plasma etching. Consequently， these make building of curvature-free nano structures difficult. In this paper，a distributed mass model of a nano shallow arch fixed beam resonator，driven by electrostatic force，is established using Euler Bernoulli beam theory. Here the scale effect，Casimir force and fringing effect of the electrostatic force are considered. The influence of different initial curvature amplitude on the static bifurcation and resonance frequency is studied through Galerkin method and a first-order discrete dynamic model of quintic nonlinear stiffness is introduced based on the static solution. The effect of different initial curvature amplitude and quintic nonlinear stiffness on the primary resonance is explored by the method of multiple scales. A criterion for the transformation of nonlinear softening-hardening characteristics is then proposed. Results reveal improved static pull-in voltage and increased critical displacement of the arch beam compared with that of the straight beam. In addition，the system is observed to experience snap-through and pull-in instability with increased initial curvature amplitude. Compared with the straight beam，the initial curvature reduces the system’s response amplitude，changes the system frequency，and presents complex nonlinear characteristics. This signifies that imperfections in the initial curvature can be used to expand the system’s working range and improve its performance. Analysis results of this study are consistent with those previously reported.
Keyword_Chinese:静电纳米拱；多尺度方法；主共振；非线性特性
Keywords_English:electrostatic nano arch；method of multiple scales；primary resonance；nonlinear characteristics

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