﻿ MEMS系统中微平板结构声振耦合性能研究
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 力学学报  2016, Vol. 48 Issue (4): 907-916  DOI: 10.6052/0459-1879-15-354 0

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Tang Yufan , Ren Shuwei , Xin Fengxian , Lu Tianjian . SCALE EFFECT ANALYSIS FOR THE VIBRO-ACOUSTIC PERFORMANCE OF A MICRO-PLATE[J]. Chinese Journal of Ship Research, 2016, 48(4): 907-916. DOI: 10.6052/0459-1879-15-354.
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文章历史

2015-09-21 收稿
2016-03-21网络版发表
MEMS系统中微平板结构声振耦合性能研究

1. 西安交通大学航天航空学院, 多功能材料与结构教育部重点实验室, 西安 710049 ;
2. 西安交通大学航天航空学院, 机械结构强度与振动国家重点实验室, 西安 710049

0 引言

1 考虑阻尼效应的微平板声振耦合理论模型

1.1 声振耦合模型的建立

 图 1 双板模型 Fig. 1 Double-micro-plates model

 图 2 声压、气膜力耦合模型 Fig. 2 Acoustic pressure and squeeze film coupling model

 $\omega_1 = \frac{\partial w}{\partial y},~~\omega_2 = \frac{\partial w}{\partial x}$ (1)

 $\left. \begin{matrix} {{\chi }_{11}}=\frac{{{\partial }^{2}}\omega }{\partial y\partial x},~~{{\chi }_{12}}=\frac{{{\partial }^{2}}\omega }{\partial {{y}^{2}}} \\ {{\chi }_{21}}=-\frac{{{\partial }^{2}}\omega }{\partial {{x}^{2}}},~~{{\chi }_{22}}=-\frac{{{\partial }^{2}}\omega }{\partial x\partial y} \\ \end{matrix} \right\}$ (2)

 $U = \int\left[{\mu _{\rm L} \left( {\varepsilon _{ij} \varepsilon _{ij} + l^2\chi _{ij} \chi _{ij} } \right) + \frac{\lambda _{\rm L} \varepsilon _{ii} \varepsilon _{jj} }{2}} \right]{\rm d}V$ (3)

 $\left. \begin{array}{*{35}{l}} T=\frac{1}{2}\int{\rho }h{{\left( \frac{\partial w}{\partial t} \right)}^{2}}\text{d}A \\ W=\int{\left( {{p}_{\text{i}}}+{{p}_{\text{r}}}-{{p}_{\text{t}}}+{{p}_{\text{sf}}} \right)}w\text{d}A \\ \end{array} \right\}$ (4)

 $\delta H = \int_{t_1 }^{t_2 } \left( {\delta U - \delta T - \delta W} \right){\rm d}t = 0$ (5)

 \begin{align} & \int_{{{t}_{1}}}^{{{t}_{2}}}{\int{\{}}\left[ \frac{E\left( 1+\text{j}\eta \right){{h}^{3}}\left( 1-\nu \right)}{12\left( 1+\nu \right)\left( 1-2\nu \right)}+\frac{E\left( 1+\text{j}\eta \right){{l}^{2}}h}{2\left( 1+\nu \right)} \right]{{\nabla }^{4}}w+ \\ & \rho h\frac{{{\partial }^{2}}w}{\partial {{t}^{2}}}-\left( {{p}_{\text{i}}}+{{p}_{\text{r}}}-{{p}_{\text{t}}}+{{p}_{\text{sf}}} \right)\}\delta w\text{d}A\text{d}t=0 \\ \end{align} (6)

 $D\nabla ^2\nabla ^2w + \rho h\frac{\partial ^2w}{\partial t^2} = p_{\rm i} + p_{\rm r} - p_{\rm t} + p_{\rm sf}$ (7)

 $D = \frac{E\left( {1 + {\rm j}\eta } \right)h^3\left( {1 - \nu } \right)}{12\left( {1 + \nu } \right)\left( {1 - 2\nu } \right)} + \frac{E\left( {1 + {\rm j}\eta } \right)l^2h}{2\left( {1 + \nu } \right)}$ (8)
1.2 耦合控制方程的求解

 $-\frac{\partial {{\Phi }_{1}}}{\partial \text{z}}=\text{j}{{\omega }_{\text{s}}}w$ (9)

 $-\frac{\partial {{\Phi }_{2}}}{\partial z}=\text{j}{{\omega }_{\text{s}}}w$ (10)

 ${{\Phi }_{\text{1},\text{i}}}\left( x,y,z;t \right)={{I}_{\text{i}}}{{\text{e}}^{\text{j}\left( {{\omega }_{\text{s}}}t-{{k}_{x}}x-{{k}_{y}}y-{{k}_{z}}z \right)}}$ (11)

 $\left.\begin{array}{l} k_0 = {\omega _{\rm s} }/ c \\ k_x = k_0 \sin \varphi \cos \theta \\ k_y = k_0 \sin \varphi \sin \theta \\ k_z = k_0 \cos \varphi \\ \end{array}\right\}$ (12)

 $\varPhi _{\rm 1,r} \left( {x,y,z;t} \right) = \beta _{\rm r} {\rm e}^{{\rm j}\left( {\omega _{\rm s} t - k_x x - k_y y + k_z z} \right)}$ (13)
 $\varPhi _{\rm 2,t} \left( {x,y,z;t} \right) = \varepsilon _{\rm t} {\rm e}^{{\rm j}\left( {\omega _{\rm s} t - k_x x - k_y y - k_z z} \right)}$ (14)

 $\varPhi _1 \left( {x,y,z;t} \right) = I_{\rm i} {\rm e}^{{\rm j}\left( {\omega _{\rm s} t - k_x x - k_y y - k_z z} \right)} + \beta _{\rm r} {\rm e}^{{\rm j}\left( {\omega _{\rm s} t - k_x x - k_y y + k_z z} \right)}$ (15)

 $p_{\rm r} + p_{\rm i} = {\rm j}\omega _{\rm s} \rho _0 \varPhi _1$ (16)

 $\frac{\partial \left( {{p}_{\text{i}}}+{{p}_{\text{r}}} \right)}{\partial \text{z}}={{\rho }_{0}}\omega _{\text{s}}^{2}w$ (17)

 $\varPhi _2 \left( {x,y,z;t} \right) = \varepsilon {\rm e}^{{\rm j}\left( {\omega _{\rm s} t - k_x x - k_y y - k_z z} \right)}$ (18)

2.3 振动微平板振动频率的影响

 图 6 不同振动频率对传声损失的影响 Fig. 6 Influence of oscillation frequency on STL

2.4 振动微平板振动幅度和板间距的影响

 图 7 振动幅度对传声损失的影响 Fig. 7 Influence of oscillation amplitude on STL

 图 8 微平板间距对传声损失的影响 Fig. 8 Influence of gap distance on STL

3 结论

(1) 本征长度的增大导致频谱曲线向高频移动，且本征长度越大，移动越明显；增大Knudsen数会使频谱曲线的波峰变陡，峰值变大；

(2) 控制并减小振动幅度可提高微平板的隔声性能，增大微平板间距也可提高微平板的隔声性能；

(3) 振动幅度越大，或板间距越小，传声损失曲线极大值所在的峰值越平坦.

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SCALE EFFECT ANALYSIS FOR THE VIBRO-ACOUSTIC PERFORMANCE OF A MICRO-PLATE
Tang Yufan1,2, Ren Shuwei1,2, Xin Fengxian1,2, Lu Tianjian1,2
1. MOE Key Laboratory for Multifunctional Materials and Structures, Xi'an Jiaotong University, Xi'an 710049, China ;
2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi'an Jiaotong University, Xi'an 710049, China
Abstract: Micro electromechanical system (MEMS) is an electromechanical device of microscale, in which micro-plate is the most typical structure. Its acoustical and mechanical properties influence the design of MEMS significantly. The vibroacoustic performance of simply supported micro-plate subjected to simultaneous stimulation of sound pressure and gas film (squeeze film) damping force is analyzed theoretically, the latter induced by the vibration of a micro-plate having similar size. By applying the Cosserat theory and the Hamilton principle, micro scale effects due to characteristic length and Knudsen number are taken into account. The governing equations are subsequently solved using the method of multiple Fourier transform to quantify sound transmission loss (STL) across the micro-plate. In the frequency domain, the effects of squeeze film under different circumstances (e.g., different characteristic lengths and Knudsen numbers, different vibration frequencies and amplitudes) on STL are investigated systematically. This work demonstrates the great influence of micro-scale effects, as well as vibration, on STL. Decreasing the vibration amplitude and increasing the distance between micro-plates lead to better performance of sound transmission. Results presented in this study provide useful theoretical guidance to the practical design of micro-plates in MEMS.
Key words: MEMS    micro-plate    squeeze film    sound transmission loss