静电驱动微梁谐振器的厚度形态误差特性分析

刘成，冯晶晶，郝淑英，张伟; Liu Cheng; Feng Jingjing; Hao Shuying; Zhang Wei

引 en

静电驱动微梁谐振器的厚度形态误差特性分析^*

刘成^1,2
，冯晶晶^1,2,3
，郝淑英^1,2
，张伟³

1. 天津理工大学天津市先进机电系统设计与控制重点实验室机械工程学院, 天津 300384； 2. 天津理工大学机电工程国家级实验教学示范中心, 天津 300384； 3. 北京工业大学北京机械强度结构与非线性振动重点实验室机电学院, 北京 100124；

CHARACTERISTIC ANALYSIS OF THICKNESS SHAPE ERROR ON ELECTROSTATIC ACTUATION MICROBEAM RESONATOR^∗

Liu Cheng^1,2
， Feng Jingjing^1,2,3
， Hao Shuying^1,2
， Zhang Wei³

1. Tianjin University of Technology, Tianjin Key Laboratory for Advanced Mechatronic System Design and Intelligent Control, School of Mechanical Engineering, Tianjin 300384 , China； 2. Tianjin University of Technology, National Demonstration Center for Experimental Mechanical and Electrical Engineering Education, School of Mechanical Engineering, Tianjin 300384 , China； 3. Beijing University of Technology, Beijing Key Laboratory on Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing 100124 , China；

DOI:10.6052/1672⁃6553⁃2019⁃049

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出版信息

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目录contents

摘要Abstract
关键词Keywords
0 引言
1 数学模型
1.1 微梁谐振器模型
1.2 控制方程
2 方程的降阶与化简
3 静态分析
3.1 极板间电容仿真
3.2 分叉与稳定性研究
3.3 厚度形态误差对静态吸合电压的影响
4 动态分析
4.1 厚度形态误差对等效固有频率的影响
4.2 厚度形态误差对软硬特性的影响
5 结论
参考文献

摘要

本文研究了一类双极板静电驱动两端固支微梁谐振器.考虑中性面拉伸和厚度形态误差,建立了微谐振器连续体模型.厚度形态误差通过设置参数方程实现,方程中包含误差参数 λ,用于调整误差影响程度.利用伽辽金法和 Newton⁃Cotes 法将系统化简为单自由度模型.首先,通过 COMSOL Multiphysics 有限元软件对理论求得的静电力进行了仿真验证.然后,推导了不考虑厚度形态误差时系统的吸合情况,发现系统在特定情况下可以出现二次吸合现象.通过改变误差参数,分析了厚度形态误差对吸合情况的影响.最后,利用多尺度法求出系统小幅振动下的响应,研究了厚度形态误差对等效固有频率和系统软硬特性的影响

Abstract

A class of bipolar plate electrostatically actuation microbeam resonator with both clamped ends was studied. The continuum model was established by considering the influences of neutral surface tension and thick⁃ ness shape error. The thickness shape error was described by a parameter equation with an error parameter, which was used to adjust the degree of shape error. The Galerkin method and Newton⁃Cotes method were used to simpli⁃ fy the system into a single⁃degree⁃of⁃freedom model. The electrostatic force was simulated and verified by the mul⁃ tiphysics finite element software COMSOL. Then, the pull⁃in condition of the system was deduced without thick⁃ ness shape error. It was found that secondary pull⁃in appears in the system in certain situations. Furthermore, the influence of thickness shape error on pull⁃in condition was analyzed by adjusting the error parameter. The method of multiple scales was applied to obtain the response of small⁃amplitude vibrations. The influence of thickness shape error on equivalent natural frequencies as well as softening and hardening characteristics of the system were also studied.

关键词

微梁谐振器；厚度形态误差；静态吸合；多尺度法；软硬特性

Keywords

microbeam resonator ； thickness shape error ； static pull⁃in ； multiple scales method ； soften⁃ ing and hardening characteristics

0 引言
微谐振器作为MEMS器件中的一种典型动态结构而成为广大学者的研究对象^{^[1,2]}.对于微谐振器的研究,主要基于静态和动态两个方向进行分析^{^[3,4]}.由于微谐振器尺寸比较小,使得MEMS器件在加工制作方面产生了非常大的难度,在加工过程中常常会由于过度蚀刻或蚀刻不足导致器件形状发生变化,引起器件的力学性能改变^{^[5]}.它们不仅会引起微梁中性面弯曲,还可以改变微谐振器的宽度､厚度和极板与微梁间隙距离.因此,在研究谐振器时考虑表面加工误差对微梁力学性能影响,对于理解其全局动力学行为,开展动力学控制和优化振动设计是非常重要的.
产生误差的可能有几种,例如,残余应力^{^[6]} ､初始位移偏差^{^[7]} ､表面加工技术不完善^{^[8]} 等都会引起误差出现.残余应力会导致微梁发生弯曲形成微拱^{^[9]}.Krylov等^{^[10]}着重研究了静电驱动微梁的几何参数对系统动力学行为的影响,发现影响系统振动的主要因素为微梁厚度以及梁与极板之间的间隙距离.Wang等^{^[11]}研究了表面弹性和残余应力对微梁固有频率的影响,指出当梁的厚度达到微米或纳米级别时,表面弹性和残余表面张力不可被忽略.Hu等^{^[12]}考虑残余表面应力对含有裂纹的纳米梁进行了动力学分析.初始偏移缺陷可以破坏动态MEMS器件沿横向振动方向的对称性.在这种情况下,梁的形状仍为矩形梁.韩建鑫等^{^[13]} 考虑了初始偏移缺陷对微梁力学行为的影响.采用多尺度法和有限差分法对微谐振器进行全局静动态分析,并发现在中等和大幅度势阱内存在频响曲线从软化行为到硬化行为的转换现象.由于表面加工精度不足造成的误差会引起微梁宽度或厚度的变化.这种尺寸变化会影响结构刚度和静电力,所以,有必要研究此类误差.然而,这种误差对微梁形状的影响是随机的.因此, 学者一般通过设置参数方程进行光滑处理,并通过改变方程参数调节微梁形状.Herrera等^{^[14]}研究了单层可变微梁的共振行为.Joglekar和Trivedi ^{^[15,16]}提出了一个参数方程,此方程可以平滑的改变微梁不同位置的宽度.最后,方程的参数被优化,并例举几种情况验证了该方法的正确性.微梁厚度变化对系统力学性能的影响还很少有学者研究.Najar等^{^[17,18]} 模拟和分析MEMS器件中变截面梁的偏转和运动, 观察了几何参数的改变对静态分叉和频响曲线的影响.然而,在他的文献中,只有单边截面变化被研究.考虑到实际加工结果,微梁的各个部分都会发生改变.单截面变化仅适用于特殊情况.因此,需考虑两个微梁截面的同时变化,以确保所获得的研究模型更接近实际.
本文以静电驱动两端固支微梁谐振器为研究对象,在微梁上下截面同时设置参数方程,分别从静态和动态两个方面分析了厚度形态误差对微谐振器力学性能的影响.
1 数学模型
1.1 微梁谐振器模型
图1 为双极板静电驱动微梁谐振器模型,微梁两端固定.长度为L,宽度为b,弹性模量为E,密度为 ρ,固定端厚度､横截面积和极惯性矩分别为h, A₀=bh和I₀=bh ³/12.固定端处梁与电极板的间距为d.设置两个参数方程y₁(x)和y₂(x)用于反映加工误差导致的梁的厚度变化,方程中设有参数 λ 去调节厚度变化趋势,λ 主要分为三种情况:λ< 0, λ=0,λ>0.其中,λ<0 时,微梁沿固定端向中间逐渐变薄,如红色曲线;λ=0 时,为矩形梁,如黑色直线;λ>0 时,微梁沿固定端向中间变厚,如绿色曲线.微梁上下各放置一个电极板,上极板同时施加直流电压V_DC和交流电压V_AC sin(Ωt),下极板仅施加直流电压V_DC.微梁与极板之间形成电容器,产生静电力,梁在静电力的作用下发生位移和振动.
图1 静电驱动微梁谐振器模型
Fig.1 Electrostatic actuation microbeam resonator
1.2 控制方程
控制微梁谐振器厚度变化的方程表达式为:

y_{1} (x) = (h / 2 + λ h s i n (π x / L))

(1)

y_{2} (x) = - (h / 2 + λ h s i n (π x / L))

(2)

横截面积为A(x)=A₀(1+2λsin(πx/L))､转动惯量I(x)=I₀(1+2λsin(πx/L)) ³.
考虑中性面拉伸和静电力非线性,微梁谐振器的运动方程为^{^[19]}

\begin{matrix} \frac{\partial^{2}}{\partial x^{2}} (E I (x) \frac{\partial^{2} y}{\partial x^{2}}) + ρ A (x) \frac{\partial^{2} y}{\partial t^{2}} + c \frac{\partial y}{\partial t} \\ = \frac{E}{2 L} \int_{0}^{1} A (x) {(\frac{\partial y}{\partial x})}^{2} d x \frac{\partial^{2} y}{\partial x^{2}} + \\ \frac{ε_{0} ε_{r} b {[V_{D C} + V_{A C} s i n (Ω t)]}^{2}}{2 {(d - y_{1} (x) + \frac{h}{2} - y)}^{2}} - \\ \frac{ε_{0} ε_{r} b V_{D C}^{2}}{2 {(d - y_{1} (x) + \frac{h}{2} + y)}^{2}} \end{matrix}

(3)

其中,c为阻尼系数.ε₀ 为自由空间中的介电常数; ε_r 是间隙空间介质相对于自由空间的相对介电常数.
为了计算方便,考虑如下无量纲:
$\begin{matrix} \hat{x} = \frac{x}{L}, \hat{b} = \frac{b}{d}, \hat{y} = \frac{y}{d}, {\hat{y}}_{1} = \frac{y_{1}}{d}, {\hat{y}}_{2} = \frac{y_{2}}{d} \\ \hat{A} (\hat{x}) = \frac{A (x)}{A_{0}}, \hat{I} (\hat{x}) = \frac{I (x)}{I_{0}}, \hat{t} = \frac{t}{T}, ω = \frac{Ω t}{\hat{t}} \end{matrix}$
得到无量纲振动方程

\begin{matrix} \frac{\partial^{2}}{\partial {\hat{x}}^{2}} (I (\hat{x}) \frac{\partial^{2} \hat{y}}{\partial {\hat{x}}^{2}}) + A (\hat{x}) \frac{\partial^{2} \hat{y}}{\partial {\hat{t}}^{2}} + c^{'} \frac{\partial \hat{y}}{\partial \hat{t}} - \\ α_{2} \int_{0}^{1} A (\hat{x}) {(\frac{\partial \hat{y}}{\partial \hat{x}})}^{2} d x \frac{\partial^{2} \hat{y}}{\partial {\hat{x}}^{2}} \\ = α_{1} (\frac{1}{{(1 - {\hat{y}}_{1} (\hat{x}) + \frac{h}{2 d} - \hat{y})}^{2}} - \frac{1}{{(1 - {\hat{y}}_{1} (\hat{x}) + \frac{h}{2 d} + \hat{y})}^{2}}) + \\ \frac{2 α_{1} χ s i n (ω \hat{t})}{{(1 - {\hat{y}}_{1} (\hat{x}) + \frac{h}{2 d} - \hat{y})}^{2}} + \frac{α_{1} χ^{2} {s i n}^{2} (ω \hat{t})}{2 {(1 - {\hat{y}}_{1} (\hat{x}) + \frac{h}{2 d} - \hat{y})}^{2}} \end{matrix}

(4)

其中,
$T = \sqrt{\frac{L^{4} ρ A_{0}}{E I_{0}}}, c^{'} = \frac{c L^{4}}{E I_{0} T}, α_{1} = \frac{ε_{0} ε_{r} b L^{4} V_{D C}^{2}}{2 E I_{0} d^{3}}$ ,
在下文中为了书写方便去掉“ $^$ ”.
2 方程的降阶与化简
由于方程(4)为偏微分方程,只能对其进行数值求解,无法进行理论分析,因此,应用伽辽金离散将模型转换为由常微分方程组成的有限自由度系统.
定义微梁的挠度为:
$y (x, t) = \sum_{i = 1}^{\infty} u_{i} (t) φ_{i} (x)$ (5)
$u_{i} (t)$ 是第 $i$ 阶模态坐标幅值, $φ_{i} (x)$ 是第 $i$ 阶无阻尼线性正交模态函数.对于静电驱动微梁,单自由度模态足以捕捉Galerkin离散中所有关键的非线性性质^{^[20]}.然而,单模近似不能得到模态耦合效应和内共振现象.这些现象只有通过增加模态数来预测得到合理的结果.但是,相对的理论分析会变得更加复杂.由于本文的主要目的是探索非线性动力学问题中的主共振问题,一阶模态就足以获得良好的结果.假设y( x,t)=u( t) φ( x).将方程(5) 代入(4)中,在方程两端同时乘 φ(x),并在0 到1 上积分,获得如下方程式:

\begin{matrix} \ddot{u} + μ \dot{u} + k_{1} u - α_{2} k_{3} u^{3} \\ \begin{matrix} = α_{1} \int_{0}^{1} {\frac{φ}{{(1 - y_{1} + \frac{h}{2 d} - φ u)}^{2}} - \\ \begin{matrix} {(1 - y_{1} + \frac{h}{2 d} + φ u)}^{2}} d x + \\ 2 α_{r} χ s i n (ω t) \int_{0}^{1} \frac{φ}{{(1 - y_{1} + \frac{h}{2 d} - φ u)}^{2}} d x + \\ α_{r} χ^{2} {s i n}^{2} (ω t) \int_{0}^{1} \frac{φ (x)}{{(1 - y_{1} + \frac{h}{2 d} - φ u)}^{2}} d x \end{matrix} \end{matrix} \end{matrix}

(6)

其中, $\ddot{u}$ 代表u对t的导数,μ, $k_{1}$ , $k_{3}$ 的表达式如下:
利用Newton⁃Cotes法对方程(6)中的积分项进行曲线拟合^{^[17]},得到最后的数学模型:

\begin{matrix} \ddot{u} + μ \dot{u} + k_{1} u - α_{2} k_{3} u^{3} \\ = α_{1} (\frac{0.61}{(1 - δ λ - 1.48 u)^{2}} - \frac{0.61}{(1 - δ λ + 1.48 u)^{2}}) + \\ 2 α_{1} χ_{s i n (ω t)} \frac{0.61}{(1 - δ λ - 1.48 u)^{2}} + \\ α_{1} χ^{2} {s i n}^{2} (ω t) \frac{0.61}{(1 - δ λ - 1.48 u)^{2}} \end{matrix}

(7)

其中 δ=h/d.
不考虑阻尼及交流电的扰动,可以得到无量纲动力学方程所对应的的哈密顿系统如下:

{\begin{matrix} \dot{u} = v \\ \dot{v} = - k_{1} u + α_{2} k_{3} u^{3} + α_{1} (\frac{0.61}{(1 - δ λ - 1.48 u)^{2}} - \\ \frac{0.61}{(1 - δ λ + 1.48 u)^{2}}) \end{matrix}

(8)

此处应注意的是,微梁最大横向位移在中点处,无量纲化后y_max=φ(0.5) u∈[-1+λδ,1-λδ],由模态函数可知 φ(0.5)=1.59,所以,u∈[(-1+λδ)/1.59,(1-λδ)/1.59].
3 静态分析
首先,先对系统进行静态分析.通过静态分析可以得到微谐振器的安全区域,只有保证所取的参数能够使器件安全工作,才有研究的意义.
3.1 极板间电容仿真
先验证理论求出的静电力是否准确,选取了变厚梁,矩形梁和变薄梁3 种情况,利用COMSOL Multiphysics对极板间的电容进行仿真.设置微梁接地,在微梁固定端上下距离为d的位置施加电压.将整个模型置于空气中.为了可以直观地看到电势在模型中的分布情况,加入电场云图,如图2.假设物理参数:L=400μm,b=45μm,d=3μm,ρ=2.33× 10 ³ kg/m ³,E=165GPa,介电常数 ε₀=8.85×10-12 F/m, 固定端厚度h=2μm.整个模拟过程在稳态求解器中进行.
理论结果的对比见图3,通过对比可以发现两者比较吻合,尽管在电压较大的位置上有少许偏差,但影响不大.而且此时电压可能已经超过吸合电压,在实际工作中,取不到图中的高电压位置.对于吸合电压的具体数值,我们将在下面进行计算.总体来说,仿真结果验证了理论的正确性.
图2 电场分布云图
Fig.2 The contour of electric field distribution
图3 直流电压与电势能的关系
Fig.3 The relationship between DC voltage and potential energy
3.2 分叉与稳定性研究
微梁在静电力作用下产生位移,当移动到吸合位置时,微梁会讯速与极板发生碰撞,这种不稳定运动称为吸合不稳定,此时的临界电压为吸合电压.本节先算出等截面梁的吸合条件,再进一步分析方程参数对系统的影响.
令方程(8)中的 $\dot{v}$ = $\dot{u}$ =0,可以求得系统的平衡点为(u_D,0)(以下简称u_D),其中平衡位置u_D 可由静态方程确定,求得四个平衡点分别为0, $u_{D i}$ , $i$ =1, 2,3( $u_{D i}$ 由一元三次方程求根公式求出).不难看出,除(0,0) 点外,其余的平衡点均和方程参数 λ 有关,因此,研究方程变化对系统平衡点及静态吸合的影响是必要的.
首先,假设 λ=0,观察矩形梁时的吸合情况.通过对静态方程求根公式 Δ 的判断可知,系统的吸合情况如下(仅研究存在物理意义的点):

{\begin{matrix} 0 < δ < 0.643 {\begin{matrix} D : α_{1} < 138.6 \\ 2 : 138.6 < α_{1} < α_{1 s t} \end{matrix} \\ [\begin{matrix} 3 : α_{1} > α_{1 s t} \\ 0.643 < δ < 1 \end{matrix}} \\ \begin{matrix} (4) : α_{1} < 138.6 \\ (5) : α_{1} > 138.6 \end{matrix} \end{matrix}

(9)

其中,α_1st由对应的 δ 带入到 Δ=0 中求出.图4( a) 为每种情况对应的势能曲线,可以看到情况①和④ 在原点处有1 个稳定点,位于原点两侧对称的地方各有1 个鞍点.其对应的相空间流形见图4( b), 4(e).情况②在原点处为鞍点,位于原点两侧各有1 个稳定点和1 个鞍点,此时微谐振器发生了二次吸合现象.对应的相空间流形为图4(c),可以看到图中同时包含同宿轨道和异宿轨道,位于同宿轨道内的位置为小幅振动区域,位于同宿轨道和异宿轨道之间的位置为大幅振动区域,异宿轨道之外为不稳定区域.情况③和情况⑤仅有1 个鞍点,相空间流形为图4(d)和4(f),图中没有安全区域,即系统发生吸合现象.
图4 势能曲线和相位图,分别对应方程(9)中的5 种情况
Fig.4 Potential energy curves and phase diagrams corresponding to the five cases in equation(9)
3.3 厚度形态误差对静态吸合电压的影响
接下来详细研究不同截面形状下静态平衡点随 α₁ 的变化情况.首先取 δ=2/3 进行分析.从图5 中可以看出,λ 越小,发生吸合现象时 α₁ 越大,即所需的能量越大.当 λ=0 时,系统增加了一个平衡点,此时出现了新的稳定区域,二次吸合现象发生.且随着 λ 减小,此平衡点的位移和稳定区域均逐渐变大.因此,可得到结论:λ 越小,发生吸合所需的能量越大.另外,λ 变小,会促进微梁发生二次吸合现象.
图5 不同方程参数下无量纲参数 α1 与静态平衡点的关系
Fig.5 The relation between parameter α1 and static equilibrium point with different equation parameters
4 动态分析
由于在微谐振器中V_AC≪V_DC,因此,这里考虑V_DC=O(1),V_AC=O( ε ³),其中,ε 为小的无量纲参数,方程(7)可以改为:

u + ε^{2} μ \dot{u} + ω_{n}^{2} u + a_{q} u^{2} + a_{c} u^{3} = ε^{3} f s i n (ω t)

(10)

ω_n,a_q,a_c, $f$ 的表达式如下:
利用多尺度法对方程(10) 进行摄动分析,可得到微梁的频响方程:

a^{2} ({(\frac{μ}{2})}^{2} + {(σ + a^{2} κ)}^{2}) = {(\frac{f}{2 ω_{n}})}^{2}

(11)

其中, $κ = \frac{5 a_{q}^{2}}{12 ω_{n}^{3}} - \frac{3 a_{c}}{8 ω_{n}}$
4.1 厚度形态误差对等效固有频率的影响
图6 为 α₁ 与等效固有频率的关系.取 δ=2/3 进行分析.α₁=0 时,为不考虑施加外力时系统的固有频率,可以发现 λ 越大,微梁的固有频率越大.另外,随着 α₁ 的增大,等效固有频率降低直至为零, 体现了静电力明显的“弹簧软化”效应.若系统存在二次吸合现象,则等效固有频率第一次降到0 后, 会继续增大,然后再次迅速减小到0,且 λ 越小,等效固有频率增加的范围越大.
图6 不同方程参数下无量纲参数 α1 与等效固有频率的关系
Fig.6 The relation between dimensionless parameter α1 and equivalent natural frequency with different section parameters
4.2 厚度形态误差对软硬特性的影响
系统的软､硬特性由参数 $κ$ 决定.当 $κ$ <0 时,系统响应为硬特性;当 $κ$ >0 时,系统响应为软特性; 当 $κ$ =0 时,系统会经历单稳态振动,即线性振动, 这是微谐振器工作的理想状态.通过图7 可以发现,若系统不满足二次吸合产生,则随着参数 α₁ 的增大,响应由硬特性转化为软特性.λ 增大,促进软特性的产生.若系统满足二次吸合产生,在 α₁ 较小时,系统响应为硬特性,当发生二次吸合时,响应直接跳跃到软特性区域.
图7 不同方程参数下无量纲参数 α₁ 与 κ 的关系
Fig.7 The relation between dimensionless parameter α₁and κ with different section parameters
5 结论
本文研究了双极板静电驱动微梁谐振器,通过设置参数方程反映微梁厚度形态误差.伽辽金法与Newton⁃cotes法被应用去化简原数学模型.在静态分析中通过求根公式求出等截面时微谐振器会发生二次吸合需满足的条件.截面变薄会增大微谐振器的吸合电压,使谐振器安全区域增大,相同环境下工作更加稳定.同时,截面变薄会促进二次吸合现象的发生.然后,利用多尺度法求出系统在小幅振动下的频率响应.方程参数越大,微梁固有频率越大.二次吸合产生时,系统等效固有频率第一次降到0 后,会出现反弹现象,且 λ 越小,等效固有频率增加的范围越大.随激励电压的增大,微谐振器由软特性向硬特性过渡,同时,λ 增大促进软特性的产生.
参考文献
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- [2] Zhang W M,Hu K M,Peng Z K,et al.Tunable micro⁃ and nanomechanical resonators.Sensors,2015,15(10):26478~ 26566
- [3] Zhang W M,Tabata O,Tsuchiya T,et al.Noise⁃induced chaos in the electrostatically actuated MEMS resonator.Physics Letters A,2011,375:2903~ 2910
- [4] Li L,Zhang Q C,Wang W,et al.Dynamic analysis and design of electrically actuated viscoelastic microbeams considering the scale effect.International Journal of Non⁃ Linear Mechanics,2017,90:21~ 31
- [5] 郝淑英,齐成坤,罗昔柳.微机械陀螺的刻蚀误差特性分析.动力学与控制学报,2017,15(2):157~ 162(Hao S Y,Qi C K,Luo X L.Characteristic analysis of etching error on micro mechanical gyroscope.Journal of Dynamics and Control,2017,15(2):157~ 162(in Chinese))
- [6] Alkharabsheh S A,Younis M I.Statics and dynamics of MEMS arches under axial forces.Journal of Vibration and Acoustics⁃Transactions of the ASME,2013,135(2):021007
- [7] Krylov S,Harari I,Cohen Y.Stabilization of electrostati⁃ cally actuated microstructures using parametric excitation.Journal of Micromechanics and Microengineering,2005,15(6):1188~ 1204
- [8] Bataineh A M,Younis M I.Dynamics of a clamped⁃ clamped microbeam resonator considering fabrication im⁃perfections.Microsystem Technologies Micro⁃and⁃Nanosys⁃ tems Information Storage and Processing Systems,2015,21(11):2425~ 2434
- [9] Younis M I,Ouakad H M,Alsaleem F M,et al.Nonlin⁃ ear dynamics of MEMS arches under harmonic electrostatic actuation.Journal of Microelectromechanical Systems,2010,19(3):647~ 656
- [10] Krylov S,Ilic B R,Schreiber D.The pull⁃in behavior of electrostatically actuated bistable microstructures.Journal of Micromechanics and Microengineering,2008,18(5):055026
- [11] Wang G F,Xi Q F.Effects of surface elasticity and resid⁃ ual surface tension on the natural frequency of mi⁃ crobeams.Applied Physics Letters,2007,90(23):231904
- [12] Hu K M,Zhang W M,Peng Z K,et al.Transverse vibra⁃ tions of mixed⁃mode cracked nanobeams with surface effect.Journal of Vibration and Acoustics,2016,138(1):011020
- [13] Han J X,Qi H J,Jin G,et al.Mechanical behaviors of e⁃ lectrostatic microresonators with initial offset imperfection:qualitative analysis via time⁃varying capacitors.Nonlinear Dynamics,2018,91(1):269~ 295
- [14] Herrera⁃May A L,Aguilera⁃Cortés L A,García⁃Ramírez P J,et al.Modeling of the intrinsic stress effect on the resonant frequency of NEMS resonators integrated by beams with variable cross⁃section.Microsystem Technolo⁃ gies Micro⁃and⁃Nanosystems Information Storage and Pro⁃ cessing Systems,2010,16(12):2067~ 2074
- [15] Joglekar M M,Pawaskar D N.Shape optimization of elec⁃ trostatically actuated microbeams for extending static and dynamic operating ranges.Structural and Multidisciplinary Optimization,2012,46(6):871~ 890
- [16] Trivedi R R,Joglekar M M,Shimpi R R,et al.Shape optimization of electrostatically actuated micro cantilever beam with extended travel range using simulated annea⁃ ling.Lecture Notes in Engineering and Computer Science,2011,2192:2042~ 2047
- [17] Najar F,Choura S,Abdel⁃Rahman E M.Dynamic analy⁃ sis of variable⁃geometry electrostatic microactuators.Jour⁃ nal of Micromechanics and Microengineering,2006,16(11):2449~ 2457
- [18] Najar F,Choura S,El⁃Borgi S,et al.Modeling and de⁃ sign of variable⁃geometry electrostatic microactuators.Journal of Micromechanics and Microengineering,2005,15(3):419~ 429
- [19] 陈涛.静电驱动微桥非线性动力学特性研究[硕士学位论文].天津:天津大学,2016:24~ 26(Cheng T.The research of nonlinear dynamic characteristics of electro⁃ statically actuated micro⁃bridges[Master Thesis].Tianjin:Tianjin University,2016:24~ 26(in Chinese))
- [20] Feng J J,Liu C,Hao S Y,et al.Static and dynamic me⁃ chanical behaviors of electrostatic MEMS resonator with surface processing error.Micromachines,2018,9(1):34

基本信息

DOI: 10.6052/1672⁃6553⁃2019⁃049

基金信息

国家自然科学基金资助项目(11602169,11702192), 天津市自然科学基金资助项目(16JCQNJC04700),天津市高等学校自然科学研究计划一般项目(2017KJ262)；
The project supported by the National Natural Science Foundation of China(11602169,11702192), the Natural Science Foundation of Tianjin City (16JCQNJC04700) and the Scientific Research Project of Tianjin Municipal Education Commission(2017KJ262)；

引用信息

稿件历史

参考文献

[1] Younis M I,Nayfeh A H.A study of the nonlinear re⁃ sponse of a resonant microbeam to an electric actuation.Nonlinear Dynamics,2003,31(1):91~ 117
[2] Zhang W M,Hu K M,Peng Z K,et al.Tunable micro⁃ and nanomechanical resonators.Sensors,2015,15(10):26478~ 26566
[3] Zhang W M,Tabata O,Tsuchiya T,et al.Noise⁃induced chaos in the electrostatically actuated MEMS resonator.Physics Letters A,2011,375:2903~ 2910
[4] Li L,Zhang Q C,Wang W,et al.Dynamic analysis and design of electrically actuated viscoelastic microbeams considering the scale effect.International Journal of Non⁃ Linear Mechanics,2017,90:21~ 31
[5] 郝淑英,齐成坤,罗昔柳.微机械陀螺的刻蚀误差特性分析.动力学与控制学报,2017,15(2):157~ 162(Hao S Y,Qi C K,Luo X L.Characteristic analysis of etching error on micro mechanical gyroscope.Journal of Dynamics and Control,2017,15(2):157~ 162(in Chinese))
[6] Alkharabsheh S A,Younis M I.Statics and dynamics of MEMS arches under axial forces.Journal of Vibration and Acoustics⁃Transactions of the ASME,2013,135(2):021007
[7] Krylov S,Harari I,Cohen Y.Stabilization of electrostati⁃ cally actuated microstructures using parametric excitation.Journal of Micromechanics and Microengineering,2005,15(6):1188~ 1204
[8] Bataineh A M,Younis M I.Dynamics of a clamped⁃ clamped microbeam resonator considering fabrication im⁃perfections.Microsystem Technologies Micro⁃and⁃Nanosys⁃ tems Information Storage and Processing Systems,2015,21(11):2425~ 2434
[9] Younis M I,Ouakad H M,Alsaleem F M,et al.Nonlin⁃ ear dynamics of MEMS arches under harmonic electrostatic actuation.Journal of Microelectromechanical Systems,2010,19(3):647~ 656
[10] Krylov S,Ilic B R,Schreiber D.The pull⁃in behavior of electrostatically actuated bistable microstructures.Journal of Micromechanics and Microengineering,2008,18(5):055026
[11] Wang G F,Xi Q F.Effects of surface elasticity and resid⁃ ual surface tension on the natural frequency of mi⁃ crobeams.Applied Physics Letters,2007,90(23):231904
[12] Hu K M,Zhang W M,Peng Z K,et al.Transverse vibra⁃ tions of mixed⁃mode cracked nanobeams with surface effect.Journal of Vibration and Acoustics,2016,138(1):011020
[13] Han J X,Qi H J,Jin G,et al.Mechanical behaviors of e⁃ lectrostatic microresonators with initial offset imperfection:qualitative analysis via time⁃varying capacitors.Nonlinear Dynamics,2018,91(1):269~ 295
[14] Herrera⁃May A L,Aguilera⁃Cortés L A,García⁃Ramírez P J,et al.Modeling of the intrinsic stress effect on the resonant frequency of NEMS resonators integrated by beams with variable cross⁃section.Microsystem Technolo⁃ gies Micro⁃and⁃Nanosystems Information Storage and Pro⁃ cessing Systems,2010,16(12):2067~ 2074
[15] Joglekar M M,Pawaskar D N.Shape optimization of elec⁃ trostatically actuated microbeams for extending static and dynamic operating ranges.Structural and Multidisciplinary Optimization,2012,46(6):871~ 890
[16] Trivedi R R,Joglekar M M,Shimpi R R,et al.Shape optimization of electrostatically actuated micro cantilever beam with extended travel range using simulated annea⁃ ling.Lecture Notes in Engineering and Computer Science,2011,2192:2042~ 2047
[17] Najar F,Choura S,Abdel⁃Rahman E M.Dynamic analy⁃ sis of variable⁃geometry electrostatic microactuators.Jour⁃ nal of Micromechanics and Microengineering,2006,16(11):2449~ 2457
[18] Najar F,Choura S,El⁃Borgi S,et al.Modeling and de⁃ sign of variable⁃geometry electrostatic microactuators.Journal of Micromechanics and Microengineering,2005,15(3):419~ 429
[19] 陈涛.静电驱动微桥非线性动力学特性研究[硕士学位论文].天津:天津大学,2016:24~ 26(Cheng T.The research of nonlinear dynamic characteristics of electro⁃ statically actuated micro⁃bridges[Master Thesis].Tianjin:Tianjin University,2016:24~ 26(in Chinese))
[20] Feng J J,Liu C,Hao S Y,et al.Static and dynamic me⁃ chanical behaviors of electrostatic MEMS resonator with surface processing error.Micromachines,2018,9(1):34

静电驱动微梁谐振器的厚度形态误差特性分析^*

CHARACTERISTIC ANALYSIS OF THICKNESS SHAPE ERROR ON ELECTROSTATIC ACTUATION MICROBEAM RESONATOR^∗

摘要

Abstract

关键词

Keywords

0 引言

1 数学模型

1.1 微梁谐振器模型

1.2 控制方程

(1)

(2)

(3)

(4)

2 方程的降阶与化简

(6)

(7)

(8)

3 静态分析

3.1 极板间电容仿真

3.2 分叉与稳定性研究

(9)

3.3 厚度形态误差对静态吸合电压的影响

4 动态分析

(10)

(11)

4.1 厚度形态误差对等效固有频率的影响

4.2 厚度形态误差对软硬特性的影响

5 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

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静电驱动微梁谐振器的厚度形态误差特性分析*

CHARACTERISTIC ANALYSIS OF THICKNESS SHAPE ERROR ON ELECTROSTATIC ACTUATION MICROBEAM RESONATOR∗

摘要

Abstract

关键词

Keywords

0 引言

1 数学模型

1.1 微梁谐振器模型

1.2 控制方程

(1)

(2)

(3)

(4)

2 方程的降阶与化简

(6)

(7)

(8)

3 静态分析

3.1 极板间电容仿真

3.2 分叉与稳定性研究

(9)

3.3 厚度形态误差对静态吸合电压的影响

4 动态分析

(10)

(11)

4.1 厚度形态误差对等效固有频率的影响

4.2 厚度形态误差对软硬特性的影响

5 结论

参考文献

基本信息

基金信息

引用信息

稿件历史

参考文献

微信公众号二维码

手机版网站二维码

静电驱动微梁谐振器的厚度形态误差特性分析^*

CHARACTERISTIC ANALYSIS OF THICKNESS SHAPE ERROR ON ELECTROSTATIC ACTUATION MICROBEAM RESONATOR^∗