CN101402443A - 模态激励悬臂微梁及其模态形状电极的宽度确定方法 - Google Patents
模态激励悬臂微梁及其模态形状电极的宽度确定方法 Download PDFInfo
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Abstract
本发明公布了一种模态激励悬臂微梁及其模态形状电极的宽度确定方法,属微机电系统MEMS领域。所述模态激励悬臂微梁,包括弹性微梁、悬臂支承模块、模态形状电极。所述模态形状电极的宽度确定方法包括:确定悬臂微梁的尺寸,通过悬臂支承模块在弹性微梁和驱动电极间加电压,采用模态函数曲线作为驱动电极的基本形状,得到模态形状电极宽度。本发明结构简单,模态阶数可以按需求选择,不会引起模态截断误差,测试精度高。
Description
技术领域
本发明涉及一种模态激励悬臂微梁及其模态形状电极的宽度确定方法,属于微机电系统(MEMS)领域。
背景技术
静电激励的悬臂微梁是许多微机电器件的核心部分,例如:微梁式谐振器及滤波器。这类微梁通常按某阶固有频率振动,最常见的是按第一阶固有频率振动。静电激励频率就是这阶固有频率。目前,这类微梁的驱动电极都被设计成固定宽度(驱动电极形状都为矩形,例如:M.I.Younis,E.M.Abdel-Rahman,AliNayfeh.A Reduced-Oder Model for Electrically Actuated Microbeam-Based MEMS,Journal of Microelectromechanical Systems,vol.12,no.5,pp.672-680,2003;E.M.Abdel-Rahman,M.I.Younis,Ali Nayfeh.Characterization of the mechanicalbehavior of an electrically actuated microbeam.J.Micromech.Microeng,12,759-766,2002.)。
振动力学的模态理论及实验表明,无论激励频率是多少,微梁的振动都是由许多阶模态叠加而成的。但当前对这类微梁进行仿真计算时,为简化计算,都直接假设微梁按第一阶模态振动(C.Zhang,G.Xu,Q.Jiang.Characterization ofthe squeeze film damping effect on the quality factor of a microbeam resonator,Journal of Micromechanics and Microengineering,2004,14:1302-1306;M.I.Younis,E.M.Abdel-Rahman,Ali Nayfeh.A Reduced-Oder Model for Electrically ActuatedMicrobeam-Based MEMS,Journal of MicroelectromechanicalSystems,vol.12,no.5,pp.672-680,2003;Y.J.Yang,M.A.Gretillat,S.D.Senturia.Effect of air damping on the dynamics of nonuniform deformations ofmicrostructures,International conference on Solid-State Sensors and Actuators,Chicago,IL,1997,June 16-19,1093-1096),即仿真计算时只取第一阶模态,其余模态的振动都视为截断误差。显然,这种误差是无法克服的,在一些工况下直接影响仿真的可信度。如果能使微梁只按第一阶模态振动,那么仿真的模态截断误差为零。但是,目前这类微梁的驱动电极都被设计成固定宽度的矩形。按振动模态理论,这种电极能激励起微梁的所有模态。
发明内容
本发明要解决的技术问题是针对现有技术存在的缺陷提出一种模态激励悬臂微梁及其模态形状电极的宽度确定方法。
本发明模态激励悬臂微梁,包括弹性微梁、悬臂支承模块、模态形状电极,其中弹性微梁的一端固定于悬臂支承模块的上表面,悬臂支承模块的下表面与基底固定,模态形状电极位于弹性微梁正下方,并且模态形状电极固定于基底上。
所述的模态激励悬臂微梁的模态形状电极的宽度确定方法,包括如下步骤:
a)确定悬臂微梁的长度沿x方向尺寸为l、宽度沿z方向尺寸为b、厚度沿y方向尺寸为hδ;
b)通过悬臂支承模块在弹性微梁和模态形状电极间加电压V(t)=V0+v(t),其中V0是直流偏置分量,v(t)是交流分量,V0>>v(t);
c)采用模态函数曲线φi(x)作为驱动电极的基本形状,模态函数曲线φi(x)关于x轴对称,其表达式为:φi(x)=cosh(λix)-cos(λix)-γi[sinh(λix)-sin(λix)],0≤x≤l,其中 λil是梁振动方程的第i个特征根,i为模态阶数,取值范围为:i=1,2,3,…∞,模态形状电极宽度为:
本发明模态激励悬臂微梁及其模态形状电极的宽度确定方法结构简单,模态阶数可以按需求选择,不会引起模态截断误差,测试精度高。
附图说明
图1:本发明结构图;
图2:图1的A-A截面图。图中不包括固定支承部分的投影,只有模态电极投影。图中细虚线所围区域为现有的矩形电极,粗实线所围区域为本发明的模态形状电极。
具体实施方法
如图1所示,本发明模态激励悬臂微梁,包括弹性微梁1、悬臂支承模块2、模态形状电极3,其中弹性微梁1的一端固定于悬臂支承模块2的上表面,悬臂支承模块2的下表面与基底固定,模态形状电极3位于弹性微梁1正下方,并且模态形状电极3固定于基底上。
如图2所示,以构建第一阶模态函数为例。所述的模态激励悬臂微梁的模态形状电极的宽度确定方法,包括如下步骤:
a)确定悬臂微梁的长度沿x方向尺寸为l、宽度沿z方向尺寸为b、厚度沿y方向尺寸为hδ,模态形状电极是由两根曲线围成,最宽处等于微梁的宽度,两根曲线关于x轴对称。该曲线就是微梁的第一阶模态函数;
b)通过悬臂支承模块2在弹性微梁1和模态形状电极间加电压V(t)=V0+v(t),其中V0是数值大的直流偏置分量,v(t)是数值小的交流分量,V0>>v(t);g0是驱动电压为零时梁下底面与驱动电极的间距,当弹性微梁变形小于g0/20时,悬臂微梁的振动方程和边界条件分别为:
为了充分利用梁的宽度b,以获得尽可能大的驱动力,驱动电极的最宽处为b,采用模态函数曲线φ1(x)模拟模态形状电极,模态函数曲线φ1(x)关于x轴对称,其表达式为:φ1(x)=cosh(λ1x)-cos(λ1x)-γ1[sinh(λ1x)-sin(λ1x)],其中λ1l=1.875, λ1l是梁振动方程的第1个特征根,模态形状电极宽度为:
按振动理论,微梁的动态变形可以表示为:
式中,φi(x)是第i阶模态函数,qi(t)是第i阶模态坐标,i为模态阶数,取值范围为:i=1,2,3,…∞。则悬臂微梁的振动方程变为:
按振动理论,模态函数具有如下正交性
将上式即悬臂微梁的振动方程两边同时乘以φi(x),i=1,2,…∞,再在[0,]中积分,并利用模态函数正交性得:
qi(t)=0,i=2,3,…∞;
此时,微梁只有第一阶模态坐标q1(t)不为零,其余qi(t)都为零,即梁只按第一阶模态振动。
Claims (2)
1.一种模态激励悬臂微梁,其特征在于包括弹性微梁(1)、悬臂支承模块(2)、模态形状电极(3),其中弹性微梁(1)的一端固定于悬臂支承模块(2)的上表面,悬臂支承模块(2)的下表面与基底固定,模态形状电极(3)位于弹性微梁(1)正下方,并且模态形状电极(3)固定于基底上。
2.一种基于权利要求1所述的模态激励悬臂微梁的模态形状电极的宽度确定方法,其特征在于包括如下步骤:
a)确定弹性微梁(1)的长度沿x方向尺寸为l、宽度沿z方向尺寸为b、厚度沿y方向尺寸为hδ;
b)通过悬臂支承模块(2)在弹性微梁(1)和模态形状电极间加电压V(t)=V0+v(t),其中V0是直流偏置分量,v(t)是交流分量,V0>>v(t);
c)采用模态函数曲线φi(x)作为驱动电极的基本形状,模态函数曲线φi(x)关于x轴对称,其表达式为:φi(x)=cosh(λix)-cos(λix)-γi[sinh(λix)-sin(λix)],0≤x≤l,其中 λil是梁振动方程的第i个特征根,i为模态阶数,取值范围为:i=1,2,3,…∞,模态形状电极宽度为:
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CN107592089A (zh) * | 2017-09-14 | 2018-01-16 | 东南大学 | 一种具有通孔结构的低热弹性阻尼悬臂微梁谐振器 |
CN110598366A (zh) * | 2019-09-30 | 2019-12-20 | 清华大学 | 纵-扭复合振动式超声变幅杆基于频率耦合的设计方法 |
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CN107592089A (zh) * | 2017-09-14 | 2018-01-16 | 东南大学 | 一种具有通孔结构的低热弹性阻尼悬臂微梁谐振器 |
CN107592089B (zh) * | 2017-09-14 | 2020-04-21 | 东南大学 | 一种具有通孔结构的低热弹性阻尼悬臂微梁谐振器 |
CN110598366A (zh) * | 2019-09-30 | 2019-12-20 | 清华大学 | 纵-扭复合振动式超声变幅杆基于频率耦合的设计方法 |
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