基于应变梯度理论的准晶扭转微镜驱动器的稳定性分析

黄允祗; 冯淼林; 陈秀华

doi:10.1007/s10409-021-09031-x

Stability analysis of quasicrystal torsion micromirror actuator based on the strain gradient theory

doi: 10.1007/s10409-021-09031-x

Huang Yunzhi¹,
Feng Miaolin^{1,*
, ,},
Chen Xiuhua^{2,*
, ,}

1.
Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University,Shanghai 200240, China
2.
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai 200240, China

Funds:

the National Natural Science Foundation of China Grant

Corresponding authors: Corresponding authors. E-mail addresses: mlfeng@sjtu.edu.cn (M. L. Feng); chenxiuhua@sjtu.edu.cn (X. H. Chen); Corresponding authors. E-mail addresses: mlfeng@sjtu.edu.cn (M. L. Feng); chenxiuhua@sjtu.edu.cn (X. H. Chen)

摘要

摘要: 静电扭转微镜广泛应用于微米光开关、光衰减器、光学扫描仪和光学显示器等领域. 过去微镜镜面主要沿单轴或双轴方向偏转, 从而限制了入镜光线的反射范围. 本文通过建立圆形静电驱动微镜的动力学模型, 设计了一种可任意方向偏转的准晶微镜驱动器. 然后基于应变梯度理论进行数值求解, 分析了其静态和动态现象以及吸合失稳特性, 并比较了反射镜面在三种偏转方向下的结果. 研究表明, 镜面沿着不同方向偏转的结果存在显著差异. 当沿斜轴的偏转角度达到45°时, 吸合失稳电压值最小. 失稳电压还随着准晶的声子场-相位子场耦合弹性模量和相位子场弹性模量的增加而增加. 准晶的介电常数、应变梯度参数和空气阻尼都会影响到微镜动态系统的扭转. 此外, 微镜表面分布力的减小也会导致更大的吸合失稳电压.
Abstract: Electrostatic torsional micromirrors are widely applied in the fields of micro-optical switches, optical attenuators, optical scanners, and optical displays. In previous lectures, most of the micromirrors were twisted along the uniaxial or biaxial direction, which limited the range of light reflection. In this paper, a quasicrystal torsional micromirror that can be deflected in any direction is designed and the dynamic model of the electrostatically driven micromirror is established. The static and dynamic phenomena and pull-in characteristics are analyzed through the numerical solution of the strain gradient theory. The results of three kinds of mirror deflection directions are compared and analyzed. The results show the significant differences in the torsion models with different deflection axis directions. When the deflection angle along the oblique axis reaches 45°, the instability voltage is the smallest. The pull-in instability voltage increases with the increment of phonon-phason coupling elastic modulus and phason elastic modulus. The permittivity of quasicrystal, the strain gradient parameter, and the air damping influence the torsion of the micromirror dynamic system. A larger pull-in instability voltage generates with the decrease of surface distributed forces.
- Quasicrystals /
- Torsion micromirror /
- Instability /
- Strain gradient theory

HTML全文

1. A schematic view of the round, double-gimbaled system, showing the various relevant parameters: a the main view of the system; b the top view of the circular plate; c the bottom view of the circular plate; d the top view of the substrate.

下载: 全尺寸图片幻灯片

2. Schematic diagram of electrostatic torsional mirror with model considering coupled torsion and bending effects.

下载: 全尺寸图片幻灯片

3. A schematic view of the circular plate rotating along a the inner axis, b the outer axis, and c the oblique axis under the action of the electrostatic force in different areas.

下载: 全尺寸图片幻灯片

4. The static behaviors of macro reference model analyzed by finite element simulation: stress amplitude for rotating along a the inner axis, b the outer axis, and c the oblique axis; displacement amplitude for rotating along d the inner axis, e the outer axis, and f the oblique axis.

下载: 全尺寸图片幻灯片

5. Comparison of torsions in three directions (the inner, outer, and oblique axes) using FES and present model for macro mirror: a curve for torsion angle Θ; b curve for maximum displacement Δ.

下载: 全尺寸图片幻灯片

6. Influences of a phonon-phason coupling elastic modulus R₁ and b phason elastic modulus K₁, and influences of c horizontal deflection angle φ and d strain gradient parameter-induced additional stiffness J_s/J₀ on the dimensionless pull-in tilting angle Θ and voltage β, respectively.

下载: 全尺寸图片幻灯片

7. Comparison of time history curves of a the tilting angle Θ with different strain gradient parameter-induced additional stiffness J_s/J₀ and b the velocity of rotation ξ with different damping effects ${\bar{C}}_{vq}$ and ${\bar{C}}_{sq}$ , respectively.

下载: 全尺寸图片幻灯片

8. Influence of QC permittivity ε₁ on the maximum torsion angle Θ_m with different a damping effects ${\bar{C}}_{vq}$ ( = ${\bar{C}}_{sq}$ ) and b parameters of outer electrode position and width λ = b/a₁.

下载: 全尺寸图片幻灯片

9. Comparison of the phase portrait on the phase plane: a without damping effects ( ${\bar{C}}_{vq}$ = ${\bar{C}}_{sq}$ = 0 ) and b with damping effects ( ${\bar{C}}_{vq}$ = ${\bar{C}}_{sq}$ = 0.05).

下载: 全尺寸图片幻灯片

10. Influence of a van der Waals moment (α₃ = 0) and b thermal Casimir moment (α₄ = 0) on the pull-in voltage β_cp with different initial gap g₀.

下载: 全尺寸图片幻灯片

Table 1. Geometrical and material parameters of the combination structure of beam and plate

Geometrical parameters		QC material parameters
H = 20 μm, R = 200 μm, a = 200 μm,b = a/2, g₀ = H, a₁ = a, a₁ = 3a/2,L = b/2, L₁ = b/2, L₂ = 1.2R, L₃ = L₁, h₁ = 10 μm, h₂ = h₁		E = 197.5 × 10⁹ N/m²,K₁ = 122 × 10⁹ N/m²,R₁ = 17.7 × 10⁹ N/m²,ν = 0.25

下载: 导出CSV

参考文献(41)

[1]	M. E. Motamedi, MOEMS: Micro-Opto-Electro-Mechanical Systems (SPIE Press, Bellingham, 2005)
[2]	W. Sun, J. Lan, and J. T. W. Yeow, Constraint adaptive output regulation of output feedback systems with application to electrostatic torsional micromirror, Int. J. Robust Nonlin. Control 25, 504 (2015).
[3]	M. Taghizadeh, and H. Mobki, Bifurcation analysis of torsional micromirror actuated by electrostatic forces. Arch. Mech. 66, 95 (2014)
[4]	C. S. Harsha, C. S. Prasanth, and B. Pratiher, Prediction of pull-in phenomena and structural stability analysis of an electrostatically actuated microswitch, Acta Mech. 227, 2577 (2016).
[5]	X. M. Zhang, F. S. Chau, C. Quan, Y. L. Lam, and A. Q. Liu, A study of the static characteristics of a torsional micromirror, Sens. Actuat. A-Phys. 90, 73 (2001).
[6]	D. Sadhukhan, and G. P. Singh, Study of Electrostatic Actuated MEMS Biaxial Scanning Micro-Mirror with Comb Structure. Inter. Confer. Multi. Mater. 2019, doi: 10.1063/5.0019578
[7]	Y. Hua, S. Wang, B. Li, G. Bai, and P. Zhang, Dynamic modeling and anti-disturbing control of an electromagnetic mems torsional micromirror considering external vibrations in vehicular LiDAR, Micromachines 12, 69 33435401(2021).
[8]	W. M. Zhang, H. Yan, Z. K. Peng, and G. Meng, Electrostatic pull-in instability in MEMS/NEMS: A review, Sens. Actuat. A-Phys. 214, 187 (2014).
[9]	F. Khatami, and G. Rezazadeh, Dynamic response of a torsional micromirror to electrostatic force and mechanical shock, Microsyst. Technol. 15, 535 (2009).
[10]	Q. Li, J. Xi, and C. Hua, Bifurcations of a micro-electromechanical nonlinear coupling system, Commun. Nonlin. Sci. Numer. Simul. 16, 769 (2011).
[11]	S. Malihi, Y. T. Beni, and H. Golestanian, Dynamic pull-in stability of torsional nano/micromirrors with size-dependency, squeeze film damping and van der waals effect, Optik 128, 156 (2017).
[12]	F. M. M. Seyyed, A. Rastgoo, and M. Taghi Ahmadian, Size-dependent instability of carbon nanotubes under electrostatic actuation using nonlocal elasticity, Int. J. Mech. Sci. 80, 144 (2014).
[13]	H. Ren, X. Zhuang, and T. Rabczuk, A nonlocal operator method for solving partial differential equations, Comput. Methods Appl. Mech. Eng. 358, 112621 (2020).
[14]	H. Ren, X. Zhuang, and T. Rabczuk, A higher order nonlocal operator method for solving partial differential equations, Comput. Methods Appl. Mech. Eng. 367, 113132 (2020).
[15]	M. Shaat, F. F. Mahmoud, X. L. Gao, and A. F. Faheem, Size-dependent bending analysis of Kirchhoff nano-plates based on a modified couple-stress theory including surface effects, Int. J. Mech. Sci. 79, 31 (2014).
[16]	H. M. Sedighi, Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory, Acta Astronaut. 95, 111 (2014).
[17]	K. F. Wang, and B. L. Wang, A general model for nano-cantilever switches with consideration of surface effects and nonlinear curvature, Phys. E-Low-dimen. Syst. Nanostruct. 66, 197 (2015).
[18]	A. Zabihi, R. Ansari, J. Torabi, F. Samadani, and K. Hosseini, An analytical treatment for pull-in instability of circular nanoplates based on the nonlocal strain gradient theory with clamped boundary condition, Mater. Res. Express 6, 0950b3 (2019).
[19]	A. A. Daikh, M. S. A. Houari, and A. Tounsi, Corrigendum: Buckling analysis of porous FGM sandwich nanoplates due to heat conduction via nonlocal strain gradient theory (2019 Eng. Res. Express 1 015022), Eng. Res. Express 2, 049501 (2020).
[20]	A. Zabihi, J. Torabi, and R. Ansari, Effects of geometric nonlinearity on the pull-in instability of circular microplates based on modified strain gradient theory, Phys. Scr. 95, 115204 (2020).
[21]	S. M. J. Hosseini, J. Torabi, R. Ansari, and A. Zabihi, Geometrically nonlinear electromechanical instability of fg nanobeams by nonlocal strain gradient theory, Int. J. Str. Stab. Dyn. 21, 2150051 (2021).
[22]	B. Hu, J. Liu, Y. Wang, B. Zhang, and H. Shen, Wave propagation in graphene reinforced piezoelectric sandwich nanoplates via high-order nonlocal strain gradient theory, Acta Mech. Sin. doi: 10.1007/s10409-021-01113-y (2021).
[23]	S. M. J. Hosseini, R. Ansari, J. Torabi, K. Hosseini, and A. Zabihi, Nonlocal strain gradient pull-in study of nanobeams considering various boundary conditions, Iran J. Sci. Technol. Trans. Mech. Eng. 45, 891 (2021).
[24]	J. M. Dubois, Useful Quasicrystals (World Scientific, Singapore, 2005)
[25]	T. Y. Fan, The Mathematical Theory Elasticity of Quasicrystals and its Applications (Springer, Singapore, 2010)
[26]	V. Fournée, H. R. Sharma, M. Shimoda, A. P. Tsai, B. Unal, A. R. Ross, T. A. Lograsso, and P. A. Thiel, Quantum size effects in metal thin films grown on quasicrystalline substrates, Phys. Rev. Lett. 95, 155504 16241737(2005).
[27]	A. Inoue, F. Kong, S. Zhu, C. T. Liu, and F. Al-Marzouki, Development and applications of highly functional Al-based materials by use of metastable phases, Mat. Res. 18, 1414 (2015).
[28]	L. Zhang, J. Guo, and Y. Xing, Bending analysis of functionally graded one-dimensional hexagonal piezoelectric quasicrystal multilayered simply supported nanoplates based on nonlocal strain gradient theory, Acta Mech. Solid Sin. 34, 237 (2021).
[29]	X. Li, J. Guo, and T. Sun, Bending deformation of multilayered one-dimensional quasicrystal nanoplates based on the modified couple stress theory, Acta Mech. Solid Sin. 32, 785 (2019).
[30]	S. Malihi, Y. T. Beni, and H. Golestanian, Analytical modeling of dynamic pull-in instability behavior of torsional nano/micromirrors under the effect of Casimir force, Optik 127, 4426 (2016).
[31]	P. A. Thiel, Quasicrystal surfaces, Annu. Rev. Phys. Chem. 59, 129 17988201(2008).
[32]	I. Brevik, J. B. Aarseth, J. S. Høye, and K. A. Milton, Temperature dependence of the Casimir effect, Phys. Rev. E 71, 056101 16089596(2005).
[33]	F. Pan, J. Kubby, E. Peeters, A. T. Tran, and S. Mukherjee, Squeeze film damping effect on the dynamic response of a mems torsion mirror, J. Micromech. Microeng. 8, 200 (1999).
[34]	R. Atabak, H. M. Sedighi, A. Reza, and E. Mirshekari, Analytical investigation of air squeeze film damping for bi-axial micro-scanner using eigenfunction expansion method, Math. Meth. Appl. Sci. mma.6658 (2020).
[35]	J. Abdi, M. Keivani, and M. Abadyan, Microstructure-dependent dynamic stability analysis of torsional NEMS scanner in van der Waals regime, Int. J. Mod. Phys. B 30, 1650109 (2016).
[36]	P. Tong, F. Yang, D. C. C. Lam, and J. Wang, Size effects of hair-sized structures-torsion, Key Eng. Mater. 261-263, 11 (2004).
[37]	E. Samaniego, C. Anitescu, S. Goswami, V. M. Nguyen-Thanh, H. Guo, K. Hamdia, X. Zhuang, and T. Rabczuk, An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications, Comput. Methods Appl. Mech. Eng. 362, 112790 (2019).
[38]	C. Anitescu, E. Atroshchenko, N. Alajlan, and T. Rabczuk, Artificial neural network methods for the solution of second order boundary value problems, Comput. Mater. Continua 59, 345 (2019).
[39]	H. Guo, X. Zhuang, and T. Rabczuk, A deep collocation method for the bending analysis of Kirchhoff plate, Comput. Mater. Continua 59, 433 (2019).
[40]	T. Rabczuk, H. Ren, and X. Zhuang, A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem, Comput. Mater. Continua 59, 31 (2019).
[41]	N. Vu-Bac, T. Lahmer, X. Zhuang, T. Nguyen-Thoi, and T. Rabczuk, A software framework for probabilistic sensitivity analysis for computationally expensive models, Adv. Eng. Software 100, 19 (2016).

施引文献

资源附件(0)

访问统计

点击查看大图

图(10) / 表(1)

计量

文章访问数: 33
HTML全文浏览量: 26
PDF下载量: 0
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

留言板

Stability analysis of quasicrystal torsion micromirror actuator based on the strain gradient theory

doi: 10.1007/s10409-021-09031-x

Corresponding authors: Corresponding authors. E-mail addresses: mlfeng@sjtu.edu.cn (M. L. Feng); chenxiuhua@sjtu.edu.cn (X. H. Chen); Corresponding authors. E-mail addresses: mlfeng@sjtu.edu.cn (M. L. Feng); chenxiuhua@sjtu.edu.cn (X. H. Chen)

计量

目录

留言板

Stability analysis of quasicrystal torsion micromirror actuator based on the strain gradient theory

doi: 10.1007/s10409-021-09031-x

Corresponding authors: Corresponding authors. E-mail addresses: mlfeng@sjtu.edu.cn (M. L. Feng); chenxiuhua@sjtu.edu.cn (X. H. Chen); Corresponding authors. E-mail addresses: mlfeng@sjtu.edu.cn (M. L. Feng); chenxiuhua@sjtu.edu.cn (X. H. Chen)

计量

出版历程

目录