CN101334415A

CN101334415A - Microfluidic drive control method for coupled analysis of electricity-solid-microflow in MEMS airtight cavity

Info

Publication number: CN101334415A
Application number: CNA2008100408231A
Authority: CN
Inventors: 李丽伟; 朱荣; 周兆英; 任建兴
Original assignee: Shanghai University of Electric Power
Current assignee: Shanghai University of Electric Power
Priority date: 2008-07-22
Filing date: 2008-07-22
Publication date: 2008-12-31

Abstract

The invention discloses a method for controlling the coupling effect of electricity-solid-micro-air flow in a MEMS microcavity. By comparing the undetermined coefficient λ′ of the displacement vibration shape affected by the air flow with the undetermined coefficient λ of the displacement vibration shape without the influence of the air flow, the method is found. The damping term of the micro-air squeeze film damping that affects the vibration displacement is obtained. The invention models and analyzes the coupling effect of static electricity-silicon film-micro-airflow in the MEMS airtight cavity, and provides relevant theoretical basis and control strategy for the driving and coordinated control of the micro-airflow.

Description

Microfluidic drive control method for coupled analysis of electricity-solid-microflow in MEMS airtight cavity

技术领域 technical field

本发明涉及MEMS微流体驱动技术领域，更具体的说是涉及一种针对MEMS密闭微腔内电-固-微气流耦合分析而获得的微流体驱动控制方法。The present invention relates to the technical field of MEMS microfluid drive, and more specifically relates to a microfluid drive control method obtained for the coupled analysis of electricity-solid-micro-air flow in a MEMS sealed microcavity.

背景技术 Background technique

MEMS器件是在硅等半导体基底材料上制作而成的微小结构腔体，可形成基于压电、静电、磁致伸缩及热电等效应工作的微传感器和微驱动器，它兼具信号处理功能、对外部世界的感知功能和作用功能，被广泛用作压力传感器、温度传感器、加速度计、微泵、微阀、微开关及药物微喷等多个领域。随着MEMS加工技术的发展，物体间距可以做得越来越小，微米级、甚至纳米级的间隙都已成为可能。当微间隙的厚度与该间隙中气体分子的平均自由程相当时，这种阻尼就称为挤压膜阻尼。气隙阻尼的微位移很小时，该阻尼结构极为有用。现代MEMS器件中，有些专门采用微间隙阻尼结构，如加速度计、微谐振器和谐振压力传感器等。微气隙挤压膜阻尼是适合加速度计等传感器的有效阻尼方式，该方式在同类及其他类似传感器中已得到应用。微间隙挤压膜阻尼的一般结构主要由两个相对运动的平行薄板组成，在两薄板之间存在一个很小的气体间隙。当薄板相对运动时，挤压两相对表面间的气体，从而产生阻尼力，气流将周期性地被压缩和膨胀。目前，涉及微间隙挤压膜阻尼效应的MEMS器件可采用开放式腔体和密闭式腔体等形式。以静电驱动密闭腔微气流阻尼结构为例，该类结构驱动膜的振动不仅包括与静电场之间的耦合，更重要的是受到密封腔内流动气体的作用，实际振动是静电-硅膜-微气流的耦合作用。事实上，微小密闭腔内受挤压的气流会使结构谐振频率以及振幅发生变化，随着尺寸的逐步减小将更加明显，这些都会影响到整个结构的谐振特性，而掌握MEMS微腔体内电-固-微气流的耦合作用是实现微流体驱动和协调控制的关键。MEMS devices are microstructure cavities made on semiconductor substrate materials such as silicon, which can form microsensors and microdrivers based on piezoelectric, electrostatic, magnetostrictive and pyroelectric effects. The perception and action functions of the external world are widely used in many fields such as pressure sensors, temperature sensors, accelerometers, micropumps, microvalves, microswitches, and drug microsprays. With the development of MEMS processing technology, the distance between objects can be made smaller and smaller, and micron-level or even nano-level gaps have become possible. When the thickness of the microgap is comparable to the mean free path of the gas molecules in the gap, the damping is called squeeze film damping. This damping structure is extremely useful when the micro-displacement of the air gap damping is small. Among modern MEMS devices, some specialize in micro-gap damping structures, such as accelerometers, micro-resonators, and resonant pressure sensors. Micro-air-gap squeeze film damping is an effective damping method for sensors such as accelerometers, and it has been used in this and other similar sensors. The general structure of the micro-gap squeezed film damper is mainly composed of two relatively moving parallel thin plates, and there is a small gas gap between the two thin plates. When the thin plates move relative to each other, the gas between the two opposite surfaces is squeezed, thereby generating a damping force, and the air flow will be compressed and expanded periodically. At present, MEMS devices involving the damping effect of squeezed film with micro-gap can be in the form of open cavity and closed cavity. Taking the micro-airflow damping structure driven by static electricity as an example, the vibration of the film driven by this type of structure not only includes the coupling with the electrostatic field, but more importantly, it is affected by the flowing gas in the sealed cavity. The actual vibration is electrostatic-silicon membrane- Coupling effect of micro air flow. In fact, the squeezed airflow in the tiny airtight cavity will change the resonance frequency and amplitude of the structure, which will become more obvious as the size gradually decreases, which will affect the resonance characteristics of the entire structure. The coupling effect of solid-microflow is the key to realize microfluidic drive and coordinated control.

发明内容 Contents of the invention

本发明所要解决的技术问题是针对MEMS密闭腔体内静电-硅膜-微气流的耦合作用进行建模分析，从而提供一种控制MEMS微腔体内电-固-微气流的耦合作用的方法。The technical problem to be solved by the present invention is to conduct modeling analysis for the coupling effect of static electricity-silicon film-micro-airflow in the MEMS airtight cavity, so as to provide a method for controlling the coupling effect of electricity-solid-micro-airflow in the MEMS microcavity.

本发明采用的技术方案：一种基于MEMS密闭腔内电-固-微气流耦合分析的微流体驱动控制方法，通过对比有气流影响的位移振形待定系数λ′与无气流影响的位移振形待定系数λ，找出对振动位移产生影响的微气流挤压膜阻尼的阻尼项，包括下列步骤：The technical scheme adopted in the present invention: a microfluidic drive control method based on the coupling analysis of electric-solid-micro-air flow in a MEMS airtight cavity, by comparing the undetermined coefficient λ′ of the displacement vibration shape affected by the air flow with the displacement vibration shape without the influence of the air flow Undetermined coefficient λ, to find out the damping term of the micro-flow extrusion film damping that affects the vibration displacement, including the following steps:

a.静电-硅膜的耦合振动分析a. Electrostatic-silicon membrane coupled vibration analysis

根据Rayleigh-Ritz能量法，硅膜的振动位移函数：w＝λ(1²-r²)²，使其满足周边固支边界条件：w(r)|_r＝l＝0，且 $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1为硅膜半径，选择适当待定系数λ，使得所述静电-硅膜耦合振动的总能量值I为最小，可获得所述静电-硅膜耦合振动的谐振频率ω，所述的待定系数λ被确定为：According to the Rayleigh-Ritz energy method, the vibration displacement function of the silicon membrane: w=λ(1 ² -r ² ) ² , so that it satisfies the peripheral fixed support boundary condition: w(r)| _r=l ＝0, and $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1 is the radius of the silicon film, select an appropriate undetermined coefficient λ, so that the total energy value I of the electrostatic-silicon film coupling vibration is the minimum, the resonant frequency ω of the electrostatic-silicon film coupling vibration can be obtained, and the undetermined coefficient λ It is determined as:

$λ λ = = \frac{{q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr}{M m}$

式中， $q_{3} = \frac{π ϵ_{0} U^{2}}{{D_{0}}^{2}};$ N₃(r)＝(r²-l²)²，In the formula, $q_{3} = \frac{π ϵ_{0} u^{2}}{{D.}_{0}^{2}};$ N ₃ (r)=(r ² -l ² ) ² ,

$M m = = {q q}_{11} {ω ω}^{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr - - 1616 {q q}_{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{22} ((r r)) rdr rdr,,$

并且：q₁＝πhρ； $q_{2} = \frac{πE h^{3}}{12 (1 - μ^{2})},$ And: q ₁ =πhρ; $q_{2} = \frac{πE h^{3}}{12 (1 - μ^{2})},$

N₁(r)＝(r²-l²)⁴，N ₁ (r)=(r ² -l ² ) ⁴ ,

N₂(r)＝(3r²-l²)²+2μ(3r²-l²)(r²-l²)+(r²-l²)²，N ₂ (r)=(3r ² -l ² ) ² +2μ(3r ² -l ² )(r ² -l ² )+(r ² -l ² ) ² ,

所述的总能量I包括振动硅膜的动能、势能及静电电场能，其中所述振动硅膜的动能为： $T_{\max} = q_{1} ω^{2} {&Integral;}_{0}^{l} w^{2} rdr,$ Described total energy I comprises kinetic energy, potential energy and electrostatic electric field energy of vibrating silicon membrane, and wherein the kinetic energy of described vibrating silicon membrane is: $T_{\max} = q_{1} ω^{2} {&Integral;}_{0}^{l} w^{2} rdr,$

其中所述振动硅膜的势能能基于硅膜的应力、应变分析获得，即：Wherein the potential energy of the vibrating silicon membrane is obtained based on the stress and strain analysis of the silicon membrane, namely:

${U u}_{S S max max} = = {q q}_{22} {&Integral; &Integral;}_{00}^{l l} [[{((\frac{{&PartialD; &PartialD;}^{22} w w}{&PartialD; &PartialD; {r r}^{22}}))}^{22} + + \frac{22 μ μ}{r r} \frac{{&PartialD; &PartialD;}^{22} w w}{&PartialD; &PartialD; {r r}^{22}} \frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; r r} + + \frac{11}{{r r}^{22}} {((\frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; r r}))}^{22}]] rdr rdr$

其中所述静电电场能可基于静电力做功获得，即： $U_{E \max} = q_{3} {&Integral;}_{0}^{l} wrdr;$ Wherein the electrostatic electric field energy can be obtained based on electrostatic force work, that is: $u_{E. \max} = q_{3} {&Integral;}_{0}^{l} wrdr;$

b.静电-硅膜-微气流的耦合振动分析b. Coupled vibration analysis of static electricity-silicon membrane-micro-air flow

首先确定微气流的压力分布，再根据所述微气流压力分布计算所述微气流挤压膜阻尼能，将所述微气流挤压膜阻尼能代入能量方程，进而实现所述的静电-硅膜-微气流的耦合分析，First determine the pressure distribution of the micro-airflow, then calculate the damping energy of the micro-airflow extrusion film according to the pressure distribution of the micro-airflow, and substitute the damping energy of the micro-airflow extrusion film into the energy equation, and then realize the electrostatic-silicon membrane -Coupled analysis of micro-gas flow,

微气流压力分布micro air flow pressure distribution

根据等温雷诺方程可求得微气流的压力分布，即：According to the isothermal Reynolds equation, the pressure distribution of the micro-airflow can be obtained, namely:

$P P ((R R,, T T)) = = - - (({Σ Σ}_{n no = = 11}^{\infty \infty} \frac{jσ jσ {B B}_{n no}}{{η η}_{00}^{n no} + + jσ jσ} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)))) {e e}^{jT J}$

微气流挤压膜阻尼能Squeeze film damping energy of micro air flow

根据微气流压力所做的功计算微气流挤压膜阻尼能，即：According to the work done by the pressure of the micro-airflow, the damping energy of the micro-airflow extrusion film is calculated, namely:

${U u}_{Fd Fd max max} = = \frac{{q q}_{44} {α α}^{22} {ω ω}^{' ' 22}}{{η η}_{00}^{11^{22}} + + {α α}^{22} {ω ω}^{' ' 22}} λ λ {&Integral; &Integral;}_{00}^{l l} {N N}_{44} ((r r)) rdr rdr$

式中，q₄＝2πb₁；In the formula, q ₄ =2πb ₁ ;

${N N}_{44} ((r r)) = = (({l l}^{22} - - \frac{{{η η}_{00}^{11^{22}} r r}^{22}}{44})) {(({l l}^{22} - - {r r}^{22}))}^{22};;$

${η η}_{00}^{11} = = {((3.8317 3.8317))}^{22} . .$

静电-硅膜-微气流的能量耦合Electrostatic-Silicon Membrane-Micro-airflow Energy Coupling

根据Rayleigh-Ritz能量法，选择适当待定系数λ′，使得所述静电-硅膜-微气流耦合振动的总能量值I′为最小，可获得所述静电-硅膜-微气流耦合振动的谐振频率ω′，所述的待定系数λ′被确定为：According to the Rayleigh-Ritz energy method, select an appropriate undetermined coefficient λ', so that the total energy value I' of the electrostatic-silicon membrane-micro-airflow coupling vibration is the smallest, and the resonance of the electrostatic-silicon membrane-microairflow coupling vibration can be obtained frequency ω′, the undetermined coefficient λ′ is determined as:

${λ λ}^{' '} = = \frac{{q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr}{M m - - \frac{{q q}_{44} {ω ω}^{' ' 22}}{{((\frac{{η η}_{00}^{11}}{α α}))}^{22} + + {ω ω}^{' ' 22}} {&Integral; &Integral;}_{00}^{l l} {N N}_{44} ((r r)) rdr rdr}$

所述的总能量I′包括前面所述的静电-硅膜耦合振动的总能量值I以及微气流挤压膜阻尼能；The total energy I' includes the total energy value I of the aforementioned electrostatic-silicon membrane coupling vibration and the damping energy of the micro-air extrusion membrane;

c.通过对比无微气流影响的振动位移函数待定系数 $λ = \frac{q_{3} {&Integral;}_{0}^{l} N_{2} rdr}{M}$ 和有微气流影响的位移振形待定系数 $λ^{'} = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M - \frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr}$ 可知，λ′的表达式比λ的表达式中增加了一项，即： $\frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr$ 该增加项即为对振动位移产生影响的微气流挤压膜阻尼的阻尼项。c. By comparing the undetermined coefficients of the vibration displacement function without the influence of micro air flow $λ = \frac{q_{3} {&Integral;}_{0}^{l} N_{2} rdr}{m}$ and the undetermined coefficient of the displacement mode shape affected by the micro-flow $λ^{'} = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{m - \frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr}$ It can be seen that the expression of λ′ has one more item than the expression of λ, namely: $\frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr$ The added term is the damping term of the micro-air extrusion film damping that affects the vibration displacement.

本发明的有益效果：在密闭式MEMS微腔体中，将微气流挤压膜阻尼能、静电电场能和静电-硅膜耦合结构的振动能量视为一个能量守恒的振动系统，因此可采用所述能量法进行分析。本发明针对MEMS密闭腔体内静电-硅膜-微气流的耦合作用进行建模分析，从而找出了对振动位移产生影响的微气流挤压膜阻尼的阻尼项，为微气流的驱动及协调控制提供相关理论基础及控制策略。Beneficial effects of the present invention: In the closed MEMS microcavity, the vibration energy of the micro-airflow extrusion film damping energy, electrostatic electric field energy and electrostatic-silicon membrane coupling structure is regarded as an energy conservation vibration system, so all analyzed using the energy method. The present invention conducts modeling analysis on the coupling effect of static electricity-silicon film-micro-airflow in the MEMS airtight cavity, thereby finding out the damping item of the micro-airflow extrusion film damping that affects the vibration displacement, which is the driving and coordinated control of the micro-airflow Provide relevant theoretical basis and control strategies.

附图说明 Description of drawings

图1是微间隙挤压膜阻尼结构简图；Fig. 1 is a schematic diagram of the micro-gap extrusion film damping structure;

图2是密封腔微气流阻尼结构简图；Fig. 2 is a schematic diagram of the micro-flow damping structure of the sealed cavity;

图3是硅膜的几何尺寸参数图。Fig. 3 is a parameter diagram of the geometric dimensions of the silicon film.

图2编号名称对应表Figure 2 Number and Name Correspondence Table

1 1 硅驱动圆膜 Silicon drive circular membrane 2 2 基体 Matrix

3 3 微气流间隙 Micro air gap 4 4 静电驱动电极 Electrostatic drive electrodes

具体实施方式 Detailed ways

下面通过附图和实施例对本发明进一步详细描述，如图1所示，它是微间隙挤压膜阻尼结构简图，(a)为矩形薄板阻尼结构，(b)为圆形薄板阻尼结构，微间隙挤压膜阻尼的一般结构主要由两个相对运动的平行薄板组成，在两薄板之间存在一个很小的气体间隙。当薄板相对运动时，挤压两相对表面间的气体，从而产生阻尼力，气流将周期性地被压缩和膨胀。如图2所示，本发明涉及的静电驱动密闭腔微气流阻尼结构主要包括驱动圆膜1和固定基体2，驱动圆膜周边固支且与基体间存在微气流间隙3。驱动圆膜1在外加静电场4作用下沿z向作受迫振动，同时受到微气流的阻尼作用。本发明建模及分析方法的特点是所述静电-硅膜-微气流耦合振动的理论分析是针对所述MEMS密闭微腔体进行的，在开放式MEMS微腔体中，微气流的挤压膜阻尼以耗散能的形式存在，而在密闭式MEMS微腔体中，可将微气流挤压膜阻尼能、静电电场能和静电-硅膜耦合结构的振动能量视为一个能量守恒的振动系统，因此可采用所述的Rayleigh-ritz能量法进行The present invention is described in further detail below by means of the accompanying drawings and examples. As shown in Figure 1, it is a schematic diagram of a micro-gap extruded film damping structure, (a) is a rectangular thin plate damping structure, (b) is a circular thin plate damping structure, The general structure of the micro-gap squeeze film damper is mainly composed of two relatively moving parallel thin plates, and there is a small gas gap between the two thin plates. When the thin plates move relative to each other, the air between the two opposite surfaces is squeezed, thereby generating a damping force, and the air flow will be compressed and expanded periodically. As shown in FIG. 2 , the micro-airflow damping structure of the electrostatically driven airtight chamber of the present invention mainly includes a driving circular membrane 1 and a fixed base 2 , and there is a micro-airflow gap 3 between the driving circular membrane and the base. The driving circular membrane 1 is forced to vibrate along the z direction under the action of the external electrostatic field 4, and is damped by the micro airflow at the same time. The characteristics of the modeling and analysis method of the present invention are that the theoretical analysis of the electrostatic-silicon film-micro-airflow coupling vibration is carried out for the MEMS airtight microcavity, and in the open MEMS microcavity, the extrusion of the microairflow Membrane damping exists in the form of dissipated energy, while in a closed MEMS microcavity, the micro-airflow extrusion membrane damping energy, electrostatic electric field energy and vibration energy of electrostatic-silicon membrane coupling structure can be regarded as an energy-conserving vibration system, so the Rayleigh-ritz energy method described can be used for

建模分析。本发明综合采用雷诺方程、微间隙挤压膜阻尼效应及Rayleigh-ritz能量法进行静电-硅膜-微气流耦合振动的理论分析。所涉及的密闭腔内微气流与静电-硅膜振动耦合的分析方案，考虑到在微气流挤压膜阻尼作用下，静电-硅膜耦合振动结构的振动特性会受到影响，为对比分析微气流挤压膜阻尼对静电-硅膜耦合结构谐振特性的影响，所述方案包括无微气流挤压膜阻尼影响的静电-硅膜耦合振动分析和受微气流挤压膜阻modeling analysis. The invention comprehensively adopts the Reynolds equation, the damping effect of micro-gap extruded film and the Rayleigh-ritz energy method to carry out the theoretical analysis of electrostatic-silicon film-micro-airflow coupling vibration. The analysis scheme for the coupling of micro-airflow and electrostatic-silicon membrane vibration in a closed cavity is involved. Considering that the vibration characteristics of the electrostatic-silicon membrane coupling vibration structure will be affected by the damping effect of the micro-airflow extrusion film, in order to compare and analyze the micro-airflow The impact of squeezed film damping on the resonance characteristics of electrostatic-silicon membrane coupled structures, the scheme includes the electrostatic-silicon membrane coupled vibration analysis without the influence of micro-airflow squeezed film damping and the impact of micro-airflow squeezed film damping

尼影响的静电-硅膜-微气流-耦合振动分析两步骤。Electrostatic-silicon membrane-micro-airflow-coupled vibration analysis of Neil effect in two steps.

一、静电-硅膜的耦合振动分析1. Electrostatic-silicon membrane coupled vibration analysis

根据所述的Rayleigh-Ritz能量法，假设硅膜的振动位移函数：w＝λ(1²-r²)²，使其满足周边固支边界条件：w(r)|_r＝l＝0，且 $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1为硅膜半径。According to the Rayleigh-Ritz energy method, it is assumed that the vibration displacement function of the silicon membrane is: w=λ(1 ² -r ² ) ² , so that it satisfies the peripheral fixed support boundary condition: w(r)| _r=l =0, and $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1 is the radius of the silicon film.

选择适当待定系数λ，使得所述静电-硅膜耦合振动的总能量值I为最小，可获得所述静电-硅膜耦合振动的谐振频率ω。所述的待定系数λ被确定为：An appropriate undetermined coefficient λ is selected so that the total energy value I of the electrostatic-silicon membrane coupling vibration is minimized, and the resonant frequency ω of the electrostatic-silicon membrane coupling vibration can be obtained. The undetermined coefficient λ is determined as:

式中， $q_{3} = \frac{π ϵ_{0} U^{2}}{{D_{0}}^{2}};$ N₃(r)＝(r²-l²)²；In the formula, $q_{3} = \frac{π ϵ_{0} u^{2}}{{D.}_{0}^{2}};$ N ₃ (r) = (r ² -l ² ) ² ;

$M m = = {q q}_{11} {ω ω}^{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr - - 1616 {q q}_{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{22} ((r r)) rdr rdr;;$

并且：q₁＝πhρ； $q_{2} = \frac{πE h^{3}}{12 (1 - μ^{2})};$ And: q ₁ =πhρ; $q_{2} = \frac{πE h^{3}}{12 (1 - μ^{2})};$

N₁(r)＝(r²-l²)⁴；N ₁ (r) = (r ² -l ² ) ⁴ ;

N₂(r)＝(3r²-l²)²+2μ(3r²-l²)(r²-l²)+(r²-l²)²；N ₂ (r)=(3r ² -l ² ) ² +2μ(3r ² -l ² )(r ² -l ² )+(r ² -l ² ) ² ;

所述的总能量I包括振动硅膜的动能、势能及静电电场能。The total energy I includes the kinetic energy, potential energy and electrostatic field energy of the vibrating silicon membrane.

其中所述振动硅膜的动能为： $T_{\max} = q_{1} ω^{2} {&Integral;}_{0}^{l} w^{2} rdr;$ Wherein the kinetic energy of the vibrating silicon membrane is: $T_{\max} = q_{1} ω^{2} {&Integral;}_{0}^{l} w^{2} rdr;$

其中所述振动硅膜的势能能可基于硅膜的应力、应变分析获得，即：Wherein the potential energy of the vibrating silicon membrane can be obtained based on the stress and strain analysis of the silicon membrane, namely:

其中所述静电电场能可基于静电力做功获得，即： $U_{E \max} = q_{3} {&Integral;}_{0}^{l} wrdr .$ Wherein the electrostatic electric field energy can be obtained based on electrostatic force work, that is: $u_{E. \max} = q_{3} {&Integral;}_{0}^{l} wrdr .$

二、静电-硅膜-微气流的耦合振动分析2. Coupled vibration analysis of static electricity-silicon membrane-micro-air flow

所述静电-硅膜-微气流的耦合振动分析，首先确定微气流的压力分布，再根据所述微气流压力分布计算所述微气流挤压膜阻尼能，将所述微气流挤压膜阻尼能代入能量方程，进而实现所述的静电-硅膜-微气流的耦合分析。The coupling vibration analysis of the electrostatic-silicon membrane-micro-airflow first determines the pressure distribution of the micro-airflow, and then calculates the damping energy of the micro-airflow extrusion film according to the pressure distribution of the micro-airflow, and damps the extrusion film of the micro-airflow. It can be substituted into the energy equation, and then realize the coupling analysis of the electrostatic-silicon film-micro-air flow.

(1)微气流压力分布(1) micro airflow pressure distribution

(2)微气流挤压膜阻尼能(2) The damping energy of micro-airflow extrusion film

根据微气流压力所做的功计算微气流挤压膜阻尼能，即：According to the work done by the micro-airflow pressure, the damping energy of the micro-airflow extrusion film is calculated, namely:

式中，q₄＝2πb₁；In the formula, q ₄ =2πb ₁ ;

${η η}_{00}^{l l} = = {((3.8317 3.8317))}^{22} . .$

(3)静电-硅膜-微气流的能量耦合(3) Electrostatic-silicon membrane-micro-airflow energy coupling

同理，根据Rayleigh-Ritz能量法，选择适当待定系数λ′，使得所述静电-硅膜-微气流耦合振动的总能量值I′为最小，可获得所述静电-硅膜-微气流耦合振动的谐振频率ω′。所述的待定系数λ′被确定为：Similarly, according to the Rayleigh-Ritz energy method, select an appropriate undetermined coefficient λ', so that the total energy value I' of the electrostatic-silicon membrane-micro-airflow coupling vibration is the smallest, and the electrostatic-silicon membrane-microairflow coupling can be obtained The resonant frequency ω' of the vibration. The undetermined coefficient λ' is determined as:

所述的总能量I′包括前面所述的静电-硅膜耦合振动的总能量值I以及微气流挤压膜阻尼能。The total energy I' includes the total energy value I of the electrostatic-silicon membrane coupled vibration mentioned above and the damping energy of the micro air flow squeezing the membrane.

为便于分析，将所获得的两个振动位移函数待定系数同时再列出：For the convenience of analysis, the obtained undetermined coefficients of the two vibration displacement functions are listed at the same time:

无微气流影响的振动位移函数待定系数： $λ = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M}$ Undetermined coefficients of the vibration displacement function without the influence of micro-airflow: $λ = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{m}$

有微气流影响的位移振形待定系数λ′： $λ^{'} = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M - \frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr}$ The undetermined coefficient λ′ of the displacement mode shape affected by micro-airflow: $λ^{'} = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{m - \frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr}$

经对比可知，λ′的表达式比λ的表达式中增加了一项，即： $\frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr$ It can be seen from the comparison that the expression of λ' has one more item than the expression of λ, namely: $\frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr$

本发明发现该增加项为一阻尼项，微气流挤压膜阻尼正是通过该阻尼项对振动位移产生影响的，通过改变和调整这一阻尼项可以有效调节和控制MEMS微腔体内的电-固-微气流耦合作用，从而为微气流的驱动及协调控制提供相关理论基础及控制策略。The present invention finds that the added item is a damping item, and the damping of the micro-airflow squeezed film affects the vibration displacement through this damping item. By changing and adjusting this damping item, the electric-power in the MEMS microcavity can be effectively adjusted and controlled. Solid-micro airflow coupling, thus providing relevant theoretical basis and control strategy for the driving and coordinated control of micro airflow.

实施例Example

图3用于说明静电驱动硅膜的几何尺寸参数，首先进行无微气流挤压膜阻尼影响的静电与硅膜耦合振动分析。Figure 3 is used to illustrate the geometric size parameters of the electrostatically driven silicon membrane. Firstly, the coupled vibration analysis of the static electricity and the silicon membrane without the influence of the micro-airflow squeezing the membrane damping is carried out.

一、静电-硅膜的振动分析1. Vibration analysis of electrostatic-silicon membrane

1、圆形硅膜的应力、应变：1. Stress and strain of circular silicon membrane:

应变分量： $S_{rr} = - z \frac{{&PartialD;}^{2} w}{&PartialD; r^{2}},$ $S_{θθ} = - \frac{z}{r} \frac{&PartialD; w}{&PartialD; r} - - - (1)$ Strain component: $S_{rr} = - z \frac{{&PartialD;}^{2} w}{&PartialD; r^{2}},$ $S_{θθ} = - \frac{z}{r} \frac{&PartialD; w}{&PartialD; r} - - - (1)$

式中，w为厚度方向(z向)的形变位移；S_rr，S_θθ为径向及环切应变分量。In the formula, w is the deformation displacement in the thickness direction (z direction); S _rr , S _θθ are the radial and circumferential shear strain components.

应力分量： $σ_{rr} = - \frac{Ez}{1 - μ^{2}} (\frac{{&PartialD;}^{2} w}{&PartialD; r^{2}} + \frac{μ}{r} \frac{&PartialD; w}{&PartialD; r})$ Stress components: $σ_{rr} = - \frac{Ez}{1 - μ^{2}} (\frac{{&PartialD;}^{2} w}{&PartialD; r^{2}} + \frac{μ}{r} \frac{&PartialD; w}{&PartialD; r})$

${σ σ}_{θθ θθ} = = - - \frac{Ez Ez}{11 - - {μ μ}^{22}} ((\frac{11}{r r} \frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; r r} + + μ μ \frac{{&PartialD; &PartialD;}^{22} w w}{&PartialD; &PartialD; {r r}^{22}})) - - - - - - ((22))$

式中，σ_rr，σ_θθ为硅膜径向及环向应力分量；E为材料杨氏弹性模量；μ为泊松比。In the formula, σ _rr , σ _θθ are the radial and hoop stress components of the silicon membrane; E is the Young's modulus of elasticity of the material; μ is Poisson's ratio.

2、静电力2. Electrostatic force

硅膜上所受的静电驱动力为：The electrostatic driving force on the silicon membrane is:

${F f}_{e e} = = \frac{11}{22} {ϵ ϵ}_{00} {((\frac{U u}{D D.}))}^{22} S S = = \frac{11}{22} {ϵ ϵ}_{00} {U u}^{22} {((\frac{11}{{D D.}_{00} - - w w}))}^{22} S S - - - - - - ((33))$

式中，ε₀-空气介电常数；U-上下电极之间的静电驱动电压；d-硅膜与下电极之间的距离；D₀-硅膜与下电极初始距离；S-硅膜与下电极间的正对面积。In the formula, ε ₀ - the dielectric constant of air; U - the electrostatic driving voltage between the upper and lower electrodes; d - the distance between the silicon film and the lower electrode; D ₀ - the initial distance between the silicon film and the lower electrode; The facing area between the lower electrodes.

硅膜进行的是微幅振动，w＜＜D₀，因此静电驱动力可近似为：The silicon membrane vibrates slightly, w<<D ₀ , so the electrostatic driving force can be approximated as:

${F f}_{e e} \approx \approx \frac{11}{22} {ϵ ϵ}_{00} {((\frac{U u}{{D D.}_{00}}))}^{22} S S - - - - - - ((44))$

于是单位面积上的静电力：Then the electrostatic force per unit area:

$d d {F f}_{e e} = = \frac{11}{22} {ϵ ϵ}_{00} {((\frac{U u}{{D D.}_{00}}))}^{22} dS wxya - - - - - - ((55))$

3、能量计算3. Energy calculation

(a.)硅膜振动的最大动能T_max：(a.) The maximum kinetic energy T _max of silicon membrane vibration:

振动硅膜的动能T_Si：The kinetic energy T _Si of the vibrating silicon membrane:

${T T}_{Si Si} = = \frac{11}{22} ρ ρ {&Integral; &Integral;}_{τ τ} [[{((\frac{&PartialD; &PartialD; u u}{&PartialD; &PartialD; t t}))}^{22} + + {((\frac{&PartialD; &PartialD; v v}{&PartialD; &PartialD; t t}))}^{22} + + {((\frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; t t}))}^{22}]] dτ dτ - - - - - - ((66))$

式中，ρ为硅膜的密度。In the formula, ρ is the density of the silicon film.

此处， $\frac{&PartialD; u}{&PartialD; t} = 0,$ $\frac{&PartialD; v}{&PartialD; t} = 0,$ 则：here, $\frac{&PartialD; u}{&PartialD; t} = 0,$ $\frac{&PartialD; v}{&PartialD; t} = 0,$ but:

${T T}_{Si Si} = = \frac{11}{22} ρ ρ {&Integral; &Integral;}_{- - h h / / 22}^{h h / / 22} {&Integral; &Integral;}_{00}^{l l} {&Integral; &Integral;}_{00}^{22 π π} {((\frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; t t}))}^{22} rdθdrdz rdθdrdz$

$= = πhρ πhρ {&Integral; &Integral;}_{00}^{l l} {((\frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; t t}))}^{22} rdr rdr - - - - - - ((77))$

式中，h为硅膜的厚度。In the formula, h is the thickness of the silicon film.

若驱动电极施加一正弦电压，硅膜的位移函数可表示为：w＝we^jωt，则其最大动能T_max为：If a sinusoidal voltage is applied to the driving electrode, the displacement function of the silicon film can be expressed as: w=we ^jωt , then its maximum kinetic energy T _max is:

${T T}_{max max} = = {q q}_{11} {ω ω}^{22} {&Integral; &Integral;}_{00}^{l l} {w w}^{22} rdr rdr - - - - - - ((88))$

其中，q₁为简化标记符，且q₁＝πhρ。Wherein, q ₁ is a simplified marker, and q ₁ =πhρ.

(b.)硅膜振动的最大势能U_Smax：(b.) The maximum potential energy U _Smax of silicon membrane vibration:

振动硅膜的势能U_S：The potential energy U _S of the vibrating silicon membrane:

${U u}_{S S} = = \frac{11}{22} {&Integral; &Integral;}_{τ τ} (({Σ Σ}_{i i = = 11}^{66} {σ σ}_{i i} {S S}_{i i})) dτ dτ = = \frac{11}{22} {&Integral; &Integral;}_{- - h h / / 22}^{h h / / 22} {&Integral; &Integral;}_{00}^{l l} {&Integral; &Integral;}_{00}^{22 π π} (({σ σ}_{rr rr} {S S}_{rr rr} + + {σ σ}_{θθ θθ} {S S}_{θθ θθ})) rdrdθdz rdrdθdz - - - - - - ((99))$

将式(1)及(2)代入式(8)，则最大势能U_Smax为：Substituting equations (1) and (2) into equation (8), the maximum potential energy U _Smax is:

${U u}_{S S max max} = = {q q}_{22} {&Integral; &Integral;}_{00}^{l l} [[{((\frac{{&PartialD; &PartialD;}^{22} w w}{&PartialD; &PartialD; {r r}^{22}}))}^{22} + + \frac{22 μ μ}{r r} \frac{{&PartialD; &PartialD;}^{22} w w}{&PartialD; &PartialD; {r r}^{22}} \frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; r r} + + \frac{11}{{r r}^{22}} {((\frac{&PartialD; &PartialD; w w}{&PartialD; &PartialD; r r}))}^{22}]] rdr rdr - - - - - - ((1010))$

其中，q₂为简化标记符，且 $q_{2} = \frac{πE h^{3}}{12 (1 - μ^{2})} .$ where q ₂ is a simplified marker, and $q_{2} = \frac{πE h^{3}}{12 (1 - μ^{2})} .$

(c.)最大静电场能U_Emax：(c.) Maximum electrostatic field energy U _Emax :

静电场能U_E：Electrostatic field energy U _E :

U_E＝∫_τwe^jωtdF_e (11)U _E ＝∫ _τ we ^jωt dF _e (11)

将式(5)代入式(11)，则最大静电场能U_Emax为：Substituting formula (5) into formula (11), the maximum electrostatic field energy U _Emax is:

${U u}_{E E. max max} = = {&Integral; &Integral;}_{00}^{l l} {&Integral; &Integral;}_{00}^{22 π π} ((\frac{11}{22} {ϵ ϵ}_{00} {((\frac{U u}{{D D.}_{00}}))}^{22})) wrdrdθ wrdrdθ = = {q q}_{33} {&Integral; &Integral;}_{00}^{l l} wrdr wrdr - - - - - - ((1212))$

其中，q₃为简化标记符，且 $q_{3} = \frac{π ϵ_{0} U^{2}}{{D_{0}}^{2}} .$ Among them, q ₃ is a simplified marker, and $q_{3} = \frac{π ϵ_{0} u^{2}}{{D.}_{0}^{2}} .$

4、静电-硅膜能量耦合求解4. Electrostatic-silicon membrane energy coupling solution

根据Rayleigh-Ritz能量法，设定硅膜振动的位移函数为：w＝λ(1²-r²)²，使其满足固支边界条件：w(r)|_r＝l＝0，且 $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1为硅膜半径。According to the Rayleigh-Ritz energy method, the displacement function of silicon membrane vibration is set as: w=λ(1 ² -r ² ) ² , so that it meets the fixed support boundary condition: w(r)| _r=l ＝0, and $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1 is the radius of the silicon film.

通过选择适当的待定系数λ，使得该耦合振动的总能量值I为最小，则可求得该耦合振动的谐振频率，即：By selecting an appropriate undetermined coefficient λ, so that the total energy value I of the coupled vibration is the smallest, the resonant frequency of the coupled vibration can be obtained, namely:

$\frac{&PartialD; &PartialD; I I}{&PartialD; &PartialD; λ λ} = = 00 - - - - - - ((1313))$

式中，λ为位移函数的待定系数；In the formula, λ is the undetermined coefficient of the displacement function;

I-静电与硅膜耦合振动的总能量，即：I-the total energy of electrostatic coupling vibration with silicon membrane, that is:

I＝T_max-U_Smax-U_Emax (14)I＝T _max -U _Smax -U _Emax (14)

将所推导的硅膜能量表达式即式(8)、(10)及(12)代入式(14)，可得总能量I，并将其代入式(13)，则可确定待定系数λ为：Substituting the derived silicon film energy expressions, namely formulas (8), (10) and (12) into formula (14), the total energy I can be obtained, and substituting it into formula (13), the undetermined coefficient λ can be determined as :

$λ λ = = \frac{{q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr}{M m} - - - - - - ((1515))$

其中，M，N₁(r)，N₂(r)，N₃(r)为简化标记符，M为：Among them, M, N ₁ (r), N ₂ (r), N ₃ (r) are simplified markers, and M is:

$M m = = {q q}_{11} {ω ω}^{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr - - 1616 {q q}_{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{22} ((r r)) rdr rdr - - - - - - ((1616))$

并且：N₁(r)＝(r²-l²)⁴；And: N ₁ (r)=(r ² −l ² ) ⁴ ;

N₃(r)＝(r²-l²)²。N ₃ (r)=(r ² −l ² ) ² .

当M＝0时，可得静电与硅膜耦合振动的谐振频率为：When M=0, the resonant frequency of electrostatic and silicon film coupling vibration can be obtained as:

${ω ω}^{22} = = \frac{1616 {q q}_{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{22} ((r r)) rdr rdr}{} - - - - - - ((1717))$

硅膜动态位移函数w为：The dynamic displacement function w of the silicon film is:

$w w = = \frac{{(({l l}^{22} - - {r r}^{22}))}^{22} {q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr}{M m} - - - - - - ((1818))$

二、密闭腔内静电-硅膜-微气流耦合建模2. Electrostatic-silicon membrane-micro-airflow coupling modeling in a closed cavity

静电-硅膜-微气流的耦合建模，首先要确定微气流的压力分布。再根据微气流压力分布计算微气流挤压膜阻尼能，进而利用Rayleigh-ritz能量法，对静电-硅膜-微气流进行振动耦合分析。The coupled modeling of electrostatics-silicon membrane-micro-airflow requires the first determination of the pressure distribution of the micro-airflow. Then calculate the damping energy of the micro-air extrusion film according to the micro-air pressure distribution, and then use the Rayleigh-ritz energy method to conduct a vibration coupling analysis of the static electricity-silicon membrane-micro-air flow.

1、流压力分布计算1. Flow pressure distribution calculation

微气流的压力分布是根据等温雷诺方程来求解的。The pressure distribution of the micro-gas flow is solved according to the isothermal Reynolds equation.

(a)微气流压力分布控制方程(a) Governing equation of micro-air pressure distribution

在进行挤压膜阻尼分析时，有如下假设：(1)气体作用为理想气体；(2)由于间隙很小，因而在气隙内气体的流动可以认为是粘性起主要作用的层流及等温流；(3)薄板为光滑表面，且气隙厚度远小于振动薄板的尺寸；(4)气隙平均厚度为D，薄板间相对位移为d，d＜＜D；(5)气隙中任一位置的气体压力与密度关系可表示为：p/ρⁿ＝const，n是取决于运动过程的常量，当气体为等温时，n≈1；(6)与法向速度梯度相比，径向及切向的速度梯度可以忽略。In the analysis of extrusion film damping, the following assumptions are made: (1) The gas acts as an ideal gas; (2) Since the gap is small, the flow of gas in the air gap can be considered as laminar flow and isothermal flow in which the viscosity plays a major role. (3) The thin plate has a smooth surface, and the thickness of the air gap is much smaller than the size of the vibrating thin plate; (4) The average thickness of the air gap is D, and the relative displacement between the thin plates is d, d<<D; (5) Any The relationship between gas pressure and density at a position can be expressed as: p/ρ ⁿ = const, n is a constant depending on the motion process, when the gas is isothermal, n≈1; (6) Compared with the normal velocity gradient, the radial The vertical and tangential velocity gradients can be ignored.

不考虑流体惯性及膨胀应力时，基于以上假设由Navier-Stokes方程及连续方程可得到等温气体挤压膜压力分布的控制方程，即等温非线性雷诺方程：When fluid inertia and expansion stress are not considered, based on the above assumptions, the Navier-Stokes equation and the continuity equation can be used to obtain the governing equation for the pressure distribution of the isothermal gas extrusion film, that is, the isothermal nonlinear Reynolds equation:

$\frac{&PartialD; &PartialD;}{&PartialD; &PartialD; X x} (({\overset{&OverBar; &OverBar;}{H h}}^{33} \overset{&OverBar; &OverBar;}{P P} \frac{&PartialD; &PartialD; \overset{&OverBar; &OverBar;}{P P}}{&PartialD; &PartialD; X x})) + + \frac{&PartialD; &PartialD;}{&PartialD; &PartialD; Y Y} (({\overset{&OverBar; &OverBar;}{H h}}^{33} \overset{&OverBar; &OverBar;}{P P} \frac{&PartialD; &PartialD; \overset{&OverBar; &OverBar;}{P P}}{&PartialD; &PartialD; Y Y})) = = σ σ \frac{&PartialD; &PartialD; ((\overset{&OverBar; &OverBar;}{P P} \overset{&OverBar; &OverBar;}{H h}))}{&PartialD; &PartialD; T T} - - - - - - ((1919))$

若假设驱动硅膜进行的是微幅振动，所引起的气体压力变化也很小，引入压力P和挤压膜厚度H两个摄动参数，且H＝1+H，P＝1+P，代入式(19)并忽略二阶及高阶项，则可得到线性化的雷诺方程：If it is assumed that the silicon membrane is driven by a slight vibration, the resulting change in gas pressure is also small, and two perturbation parameters, the pressure P and the extruded membrane thickness H, are introduced, and H=1+H, P=1+P, Substituting into equation (19) and ignoring the second-order and higher-order terms, the linearized Reynolds equation can be obtained:

${&dtri; &dtri;}^{22} P P - - σ σ \frac{&PartialD; &PartialD; P P}{&PartialD; &PartialD; T T} = = σ σ \frac{&PartialD; &PartialD; H h}{&PartialD; &PartialD; T T} - - - - - - ((2020))$

采用极平面坐标，拉普拉斯算子

Using polar plane coordinates, the Laplace operator

${&dtri; &dtri;}^{22} = = ((\frac{{&PartialD; &PartialD;}^{22}}{&PartialD; &PartialD; {R R}^{22}} + + \frac{11}{R R} \frac{&PartialD; &PartialD;}{&PartialD; &PartialD; R R} + + \frac{11}{{R R}^{22}} \frac{{&PartialD; &PartialD;}^{22}}{&PartialD; &PartialD; {θ θ}^{22}}))$

式中，P＝p/p_a，p为挤压膜内气体压力，p_a为大气压力；In the formula, P=p/p _a , p is the gas pressure in the extruded membrane, and p _a is the atmospheric pressure;

H＝h/h₀，h为挤压膜厚度，h₀为挤压膜初始厚度；H=h/h ₀ , h is the thickness of the extruded film, h ₀ is the initial thickness of the extruded film;

R＝r/1，1为振动圆膜半径；R=r/1, 1 is the radius of the vibrating circular membrane;

T＝ωt，ω为受迫振动频率；T=ωt, ω is the forced vibration frequency;

σ＝12μω1²/p_ah₀ ²，σ为薄膜挤压数，μ为空气粘度。σ=12μω1 ² /p _a h ₀ ² , σ is the film extrusion number, μ is the air viscosity.

考虑一阶轴对称模态，微流体压力分布与θ无关，于是只需求解如下无量纲形式的等温线性化雷诺方程：Considering the first-order axisymmetric mode, the microfluidic pressure distribution has nothing to do with θ, so it is only necessary to solve the isothermal linearized Reynolds equation in the following dimensionless form:

$\frac{{&PartialD; &PartialD;}^{22} P P}{&PartialD; &PartialD; {R R}^{22}} + + \frac{11}{R R} \frac{&PartialD; &PartialD; P P}{&PartialD; &PartialD; R R} = = σ σ ((\frac{&PartialD; &PartialD; P P}{&PartialD; &PartialD; T T} + + \frac{&PartialD; &PartialD; H h}{&PartialD; &PartialD; T T})) - - - - - - ((21 twenty one))$

由式(21)可知，其齐次方程为零阶Bessel方程。It can be seen from formula (21) that its homogeneous equation is the zero-order Bessel equation.

密封腔受迫振动空气挤压膜及圆膜的边界条件为：The boundary conditions of the sealed chamber forced vibrating air to extrude the membrane and the circular membrane are:

$\frac{&PartialD; P (R, T)}{&PartialD; R} |_{R = 1} = 0,$ H(R，T)|_R＝1＝0 (22) $\frac{&PartialD; P (R, T)}{&PartialD; R} |_{R = 1} = 0,$ H(R,T)| _R＝1 ＝0 (22)

(b)微气流压力控制方程求解(b) Solving the micro-flow pressure governing equation

假设P(R，T)＝Ψ(R)Φ(T)，H(R，T)＝H(R)Φ(T)，对式(21)进行分离变量：Assuming P(R, T)=Ψ(R)Φ(T), H(R, T)=H(R)Φ(T), separate variables for formula (21):

$\frac{{Ψ Ψ}^{' '' '} ((R R)) + + \frac{11}{R R} {Ψ Ψ}^{' '} ((R R))}{σ σ [[Ψ Ψ ((R R)) + + H h ((R R))]]} = = \frac{{Φ Φ}^{' '} ((T T))}{Φ Φ ((T T))} = = j j - - - - - - ((23 twenty three))$

进一步由式(23)可得：Further from formula (23), we can get:

${Ψ Ψ}^{' '' '} ((R R)) + + \frac{11}{R R} {Ψ Ψ}^{' '} ((R R)) + + jσΨ jσΨ ((R R)) = = - - jσH jσH ((R R)) - - - - - - ((24 twenty four))$

Φ′(T)-jΦ(T)＝0 (25)Φ′(T)-jΦ(T)＝0 (25)

由式(25)可解得时间变量：The time variable can be solved by equation (25):

Φ(T)＝e^jT (26)Φ(T)＝e ^jT (26)

根据空气压力边界条件：Ψ′(R)|_R＝1＝0，可得式(21)的齐次方程解：According to the air pressure boundary condition: Ψ′(R)| _R=1 ＝0, the homogeneous equation solution of formula (21) can be obtained:

$Ψ (R) = J_{0} (\sqrt{η_{0}^{n}} R),$ n＝1，2，3，... (27) $Ψ (R) = J_{0} (\sqrt{η_{0}^{no}} R),$ n=1, 2, 3, ... (27)

同时由边界条件Ψ′(R)|_R＝1＝0可推知： ${J_{0}}^{'} (\sqrt{η_{0}^{n}} L) = - J_{1} (\sqrt{η_{0}^{n}} L) = 0,$ 设J₁(x)＝0的根为：x₁ ⁿ，n＝1，2，...，则： $\sqrt{η_{0}^{n}} L = x_{1}^{n},$ 因此Ψ(R)的本征值为： $η_{0}^{n} = {(\frac{x_{1}^{n}}{L})}^{2}$ 此处L＝1。因 $x_{1}^{1} = 3.8317,$ 从而 $η_{0}^{1} = {(3.8317)}^{2} .$ At the same time, it can be deduced from the boundary condition Ψ′(R)| _R=1 =0: ${J_{0}}^{'} (\sqrt{η_{0}^{no}} L) = - J_{1} (\sqrt{η_{0}^{no}} L) = 0,$ Let the roots of J ₁ (x)=0 be: x ₁ ⁿ , n=1, 2,..., then: $\sqrt{η_{0}^{no}} L = x_{1}^{no},$ So the eigenvalues of Ψ(R) are: $η_{0}^{no} = {(\frac{x_{1}^{no}}{L})}^{2}$ Here L=1. because $x_{1}^{1} = 3.8317,$ thereby $η_{0}^{1} = {(3.8317)}^{2} .$

挤压空气膜与振动圆膜存在共同边界，因此空气膜的厚度变化H(R)可根据振动圆膜的位移函数来描述。There is a common boundary between the extruded air film and the vibrating circular film, so the thickness change H(R) of the air film can be described according to the displacement function of the vibrating circular film.

现假设挤压空气膜的厚度变化H(R)为：Now assume that the thickness change H(R) of the extruded air film is:

H(R)＝λ(1-R²)² (28)H(R)＝λ(1-R ² ) ² (28)

式中：λ-空气膜厚度变化函数的待定系数。In the formula: λ-the undetermined coefficient of the air film thickness variation function.

Ψ(R)的Bessel级数展开：Bessel series expansion of Ψ(R):

$Ψ Ψ ((R R)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} {C C}_{n no} {Ψ Ψ}_{n no} ((R R)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} {C C}_{n no} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) - - - - - - ((2929))$

于是空气压力P(R，T)为：Then the air pressure P(R, T) is:

$P P ((R R,, T T)) = = Ψ Ψ ((R R)) Φ Φ ((T T)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} {C C}_{n no} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) {e e}^{jT J} - - - - - - ((3030))$

将H(R)展开成关于Ψ(R)的Bessel级数：Expand H(R) into a Bessel series with respect to Ψ(R):

$H h ((R R)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} {B B}_{n no} {Ψ Ψ}_{n no} ((R R)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} {B B}_{n no} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) - - - - - - ((3131))$

${B B}_{n no} = = \frac{{&Integral; &Integral;}_{00}^{11} H h ((R R)) {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) RdR Rd}{{&Integral; &Integral;}_{00}^{11} {J J}_{00}^{22} ((\sqrt{{η η}_{00}^{n no}} R R)) RdR Rd} - - - - - - ((3232))$

于是挤压膜厚度变化H(R，T)为：Then the extruded film thickness change H(R, T) is:

$H h ((R R,, T T)) = = H h ((R R)) Φ Φ ((T T)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} {B B}_{n no} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) {e e}^{jT J} - - - - - - ((3333))$

将式(30)及式(33)代入式(21)，有：Substituting formula (30) and formula (33) into formula (21), we have:

${Σ Σ}_{n no = = 11}^{\infty \infty} {C C}_{n no} (({Ψ Ψ}_{n no}^{' '' '} ((R R)) + + \frac{11}{R R} {Ψ Ψ}_{n no}^{' '} ((R R)))) = = {Σ Σ}_{n no = = 11}^{\infty \infty} jσ jσ (({C C}_{n no} + + {B B}_{n no})) {Ψ Ψ}_{n no} ((R R)) - - - - - - ((3434))$

式(34)通过加减项变形为：Equation (34) is transformed into:

${Σ Σ}_{n no = = 11}^{\infty \infty} {C C}_{n no} (({Ψ Ψ}_{n no}^{' '' '} ((R R)) + + \frac{11}{R R} {Ψ Ψ}_{n no}^{' '} ((R R)) + + {η η}_{00}^{n no} {Ψ Ψ}_{n no} ((R R)) - - {η η}_{00}^{n no} {Ψ Ψ}_{n no} ((R R)))) = = {Σ Σ}_{n no = = 11}^{\infty \infty} jσ jσ (({C C}_{n no} + + {B B}_{n no})) {Ψ Ψ}_{n no} ((R R)) - - - - - - ((3535))$

根据式(27)即： $Ψ (R) = J_{0} (\sqrt{η_{0}^{n}} R),$ 可知 ${Ψ_{n}}^{''} (R) + \frac{1}{R} {Ψ_{n}}^{'} (R) + η_{0}^{n} Ψ_{n} (R) = 0,$ 则式(35)为：According to formula (27): $Ψ (R) = J_{0} (\sqrt{η_{0}^{no}} R),$ It can be seen ${Ψ_{no}}^{''} (R) + \frac{1}{R} {Ψ_{no}}^{'} (R) + η_{0}^{no} Ψ_{no} (R) = 0,$ Then formula (35) is:

$- - {Σ Σ}_{n no = = 11}^{\infty \infty} {C C}_{n no} {η η}_{00}^{n no} {Ψ Ψ}_{n no} ((R R)) = = {Σ Σ}_{n no = = 11}^{\infty \infty} jσ jσ (({C C}_{n no} + + {B B}_{n no})) {Ψ Ψ}_{n no} ((R R)) - - - - - - ((3636))$

比较式(36)两边关于Ψ_n(R)的系数可得：Comparing the coefficients of Ψ _n (R) on both sides of equation (36), we can get:

${C C}_{n no} = = - - \frac{jσ jσ {B B}_{n no}}{{η η}_{00}^{n no} + + jσ jσ} - - - - - - ((3737))$

于是空气压力为：Then the air pressure is:

$P P ((R R,, T T)) = = - - (({Σ Σ}_{n no = = 11}^{\infty \infty} \frac{jσ jσ {B B}_{n no}}{{η η}_{00}^{n no} + + jσ jσ} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)))) {e e}^{jT J} - - - - - - ((44 - - 3838))$

2、微气流挤压膜阻尼能计算2. Calculation of the damping energy of micro-air extrusion film

考虑推导出的微气流压力分布式(38)中的空间函数P(R)：Consider the spatial function P(R) in the derived microflow pressure distribution (38):

$P P ((R R)) = = - - {Σ Σ}_{n no = = 11}^{\infty \infty} \frac{jσ jσ {B B}_{n no}}{{η η}_{00}^{n no} + + jσ jσ} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) = = - - {Σ Σ}_{n no = = 11}^{\infty \infty} \frac{(({σ σ}^{22} + + j j {η η}_{00}^{n no} σ σ)) {B B}_{n no}}{{η η}_{00}^{{n no}^{22}} + + {σ σ}^{22}} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) - - - - - - ((3939))$

取式(39)中的实部，即：P(R)＝R_e(P)，则有微气流压力分布为：Get the real part in the formula (39), that is: P (R) = R _e (P), then there is the micro air flow pressure distribution as:

$P P ((R R)) = = - - {Σ Σ}_{n no = = 11}^{\infty \infty} \frac{{σ σ}^{22} {B B}_{n no}}{{η η}_{00}^{{n no}^{22}} + + {σ σ}^{22}} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} R R)) - - - - - - ((4040))$

将薄膜挤压数σ＝12μωl²/P_ah₀ ²＝αω以及B_n和J₀(η₀ ⁿR)的有量纲形式代入式(40)，于是有量纲形式的微气流压力分布p(r)为：Substituting the film extrusion number σ=12μωl ² /P _a h ₀ ² =αω and the dimensional form of B _n and J ₀ (η ₀ ⁿ R) into formula (40), then the micro-airflow pressure distribution in dimensional form p(r) is:

$p p ((r r)) = = - - {Σ Σ}_{n no = = 11}^{\infty \infty} \frac{{α α}^{22} {ω ω}^{22} {b b}_{n no}}{{η η}_{00}^{{n no}^{22}} + + {α α}^{22} {ω ω}^{22}} {J J}_{00} ((\sqrt{{η η}_{00}^{n no}} r r)) - - - - - - ((4141))$

其中，r_n为系数B_n的有量纲形式，且 $b_{n} = \frac{{&Integral;}_{0}^{l} h (r) J_{0} (\sqrt{η_{0}^{n}} r) rdr}{{&Integral;}_{0}^{l} {J_{0}}^{2} (\sqrt{η_{0}^{n}} r) rdr};$ where r _n is the dimensioned form of the coefficient B _n , and $b_{no} = \frac{{&Integral;}_{0}^{l} h (r) J_{0} (\sqrt{η_{0}^{no}} r) rdr}{{&Integral;}_{0}^{l} {J_{0}}^{2} (\sqrt{η_{0}^{no}} r) rdr};$

α为薄膜挤压数表达式中的简化标记符，α＝12μl²/P_ah₀ ²。α is a simplified symbol in the film extrusion number expression, α=12μl ² /P _a h ₀ ² .

对于式(41)所示的微气流压力分布级数解，当取n＝1时，可得到相应于截止频率ω_c的微气流压力

此处以

来近似微气流压力分布p(r)：For the micro-flow pressure distribution series solution shown in formula (41), when n=1, the micro-flow pressure corresponding to the cut-off frequency _ωc can be obtained

here to

To approximate the micro-flow pressure distribution p(r):

$p p ((r r)) \approx \approx \overset{^^}{p p} ((r r)) = = - - \frac{{α α}^{22} {ω ω}_{c c}^{22} {b b}_{11}}{{η η}_{00}^{11^{22}} + + {α α}^{22} {ω ω}_{c c}^{22}} {J J}_{00} ((\sqrt{{η η}_{00}^{11}} r r)) - - - - - - ((4242))$

其中， $b_{1} = \frac{{&Integral;}_{0}^{l} h (r) J_{0} (\sqrt{η_{0}^{1}} r) rdr}{{&Integral;}_{0}^{l} {J_{0}}^{2} (\sqrt{η_{0}^{1}} r) rdr},$ h(r)＝λ(l²-r²)²。in, $b_{1} = \frac{{&Integral;}_{0}^{l} h (r) J_{0} (\sqrt{η_{0}^{1}} r) rdr}{{&Integral;}_{0}^{l} {J_{0}}^{2} (\sqrt{η_{0}^{1}} r) rdr},$ h(r)=λ(l ² −r ² ) ² .

同时以截止频率ω_c近似静电-硅膜-微气流耦合谐振频率ω′，即ω′≈ω_c，则微气流压力分布p(r)为：At the same time, the electrostatic-silicon film-micro-airflow coupling resonant frequency ω _′ is approximated by the cut-off frequency ωc, that is, _ω′≈ωc , then the micro-airflow pressure distribution p(r) is:

$p p ((r r)) = = - - \frac{{α α}^{22} {ω ω}^{' ' 22} {b b}_{11}}{{η η}_{00}^{11^{22}} + + {α α}^{22} {ω ω}^{' ' 22}} {J J}_{00} ((\sqrt{{η η}_{00}^{11}} r r)) - - - - - - ((4343))$

根据微气流压力所做的功计算挤压膜阻尼能U_Fd为：According to the work done by the micro airflow pressure, the squeeze film damping energy U _Fd is calculated as:

${U u}_{Fd Fd} = = {&Integral; &Integral;}_{00}^{l l} {&Integral; &Integral;}_{00}^{22 π π} p p ((r r)) {we we}^{jωt jωt} rdrdθ rdrdθ - - - - - - ((4444))$

将式(43)代入式(44)，则最大微气流挤压膜阻尼能U_Fd max为：Substituting Equation (43) into Equation (44), the maximum micro-airflow extrusion film damping energy U _{Fd max} is:

${U u}_{Fd Fd max max} = = {&Integral; &Integral;}_{00}^{l l} {&Integral; &Integral;}_{00}^{22 π π} \frac{{α α}^{22} {ω ω}^{' ' 22} {b b}_{11}}{{η η}_{00}^{11^{22}} + + {α α}^{22} {ω ω}^{' ' 22}} {J J}_{00} ((\sqrt{{η η}_{00}^{11}} r r)) wrdrdθ wrdrdθ$

$= = {&Integral; &Integral;}_{00}^{l l} \frac{22 π π {α α}^{22} {ω ω}^{' ' 22} {b b}_{11}}{{η η}_{00}^{11^{22}} + + {α α}^{22} {ω ω}^{' ' 22}} {J J}_{00} ((\sqrt{{η η}_{00}^{11}} r r)) wrdr wrdr = = \frac{{q q}_{44} {α α}^{22} {ω ω}^{' ' 22}}{{η η}_{00}^{11^{22}} + + {α α}^{22} {ω ω}^{' ' 22}} λ λ {&Integral; &Integral;}_{00}^{l l} {N N}_{44} ((r r)) rdr rdr - - - - - - ((1515))$

其中，q₄、N₄(r)为简化标记符，且q₄＝2πb₁；Among them, q ₄ and N ₄ (r) are simplified markers, and q ₄ =2πb ₁ ;

${N N}_{44} ((r r)) = = {J J}_{00} (({η η}_{00}^{11} r r)) w w ((r r)) = = (({l l}^{22} - - \frac{{η η}_{00}^{11^{22}} {r r}^{22}}{44})) {(({l l}^{22} - - {r r}^{22}))}^{22},, {η η}_{00}^{11} = = {((3.8317 3.8317))}^{22} . .$

3、静电-硅膜-微气流耦合3. Electrostatic-silicon membrane-micro airflow coupling

下面根据Rayleigh-Ritz能量法，对静电-硅膜-微气流的耦合场进行分析。即为满足如下方程：Next, according to the Rayleigh-Ritz energy method, the coupling field of static electricity-silicon film-micro-air flow is analyzed. That is to satisfy the following equation:

$\frac{&PartialD; &PartialD; {I I}^{' '}}{&PartialD; &PartialD; {λ λ}^{' '}} = = 00 - - - - - - ((4646))$

式中，I′为静电、硅膜与微气流耦合振动的总能量，即：In the formula, I' is the total energy of static electricity, silicon membrane and micro-air coupling vibration, that is:

I′＝T_max-U_Smax-U_Emax-U_Fdmax (47)I'＝T _max -U _Smax -U _Emax -U _Fdmax (47)

将式(45)所示的最大微气流挤压膜阻尼能以及前面所推导的硅膜最大动能、势能及电场能表达式即式(8)、(10)及(12)代入式(47)，可得总能量I′，并将其代入式(46)，则可确定待定系数λ′为：Substitute the maximum damping energy of the micro-airflow extrusion film shown in Equation (45) and the expressions of the maximum kinetic energy, potential energy and electric field energy of the silicon membrane deduced above, namely Equations (8), (10) and (12) into Equation (47) , the total energy I' can be obtained, and if it is substituted into formula (46), the undetermined coefficient λ' can be determined as:

${λ λ}^{' '} = = \frac{{q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr}{M m - - \frac{{q q}_{44} {ω ω}^{' ' 22}}{{((\frac{{η η}_{00}^{11}}{α α}))}^{22} + + {ω ω}^{' ' 22}} {&Integral; &Integral;}_{00}^{l l} {N N}_{44} ((r r)) rdr rdr} - - - - - - ((4848))$

由式(16)可知M为简化标记符，且M表达式为：It can be seen from formula (16) that M is a simplified marker, and the expression of M is:

$M m = = {q q}_{11} {ω ω}^{' ' 22} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr - - 1616 {q q}_{22} {&Integral; &Integral;}_{00}^{l l} {N N}_{22} ((r r)) rdr rdr$

λ是位移振形函数的待定系数，λ的极大值即为位移的极大值。因此，若静电-硅膜-微气流耦合振动存在振动的最大能量，也应该是λ达到极大值，则对应的频率为谐振频率ω′，因此有：λ is the undetermined coefficient of the displacement mode shape function, and the maximum value of λ is the maximum value of the displacement. Therefore, if the electrostatic-silicon membrane-micro-airflow coupling vibration has the maximum vibration energy, λ should also reach the maximum value, and the corresponding frequency is the resonance frequency ω′, so:

$\frac{&PartialD; &PartialD; λ λ}{&PartialD; &PartialD; {ω ω}^{' '}} = = 00 - - - - - - ((4949))$

则有：Then there are:

Aω′⁴+Bω′²+C＝0 (50)Aω′ ⁴ +Bω′ ² +C＝0 (50)

其中，A、B、C为简化标记符，且：Among them, A, B, C are simplified markers, and:

$A A = = 22 {α α}^{22} {q q}_{11} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr [[{α α}^{22} {q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr + + {q q}_{44} {&Integral; &Integral;}_{00}^{l l} {N N}_{44} ((r r)) rdr rdr]];;$

$B B = = 44 {α α}^{22} {η η}_{00}^{11^{22}} {q q}_{11} {q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr;;$

$C C = = 3232 {η η}_{00}^{11^{22}} {q q}_{22} {q q}_{44} {&Integral; &Integral;}_{00}^{l l} {N N}_{22} ((r r)) rdr rdr {&Integral; &Integral;}_{00}^{l l} {N N}_{44} rdr rdr + + 22 {η η}_{00}^{11^{44}} {q q}_{11} {q q}_{33} {&Integral; &Integral;}_{00}^{l l} {N N}_{11} ((r r)) rdr rdr {&Integral; &Integral;}_{00}^{l l} {N N}_{33} rdr rdr . .$

由式(50)可获得静电-硅膜-微气流耦合振动的谐振频率ω′：The resonant frequency ω′ of electrostatic-silicon membrane-micro-airflow coupling vibration can be obtained from formula (50):

${ω ω}^{' ' 22} = = \frac{- - B B + + \sqrt{{B B}^{22} + + 44 AC AC}}{22 A A} - - - - - - ((5151))$

所述内容仅为本发明构思下的基本说明，而依据本发明的技术方案所作的任何等效变换，均应属于本发明的保护范围。The above content is only a basic description of the concept of the present invention, and any equivalent transformation made according to the technical solution of the present invention shall belong to the protection scope of the present invention.

Claims

Gu one kind based on the MEMS hermetic cavity electricity--microfluid drive and control method of micro-airflow coupling analysis, it is characterized in that the displacement of the airflow influence displacement of shape undetermined coefficient λ ' and the airless influence shape undetermined coefficient λ that shakes that shakes being arranged by contrast, find out the damping term of little air-flow squeeze-film damping that vibration displacement is exerted an influence, comprise the following steps:

A. the coupled vibration analysis of static-silicon fiml

According to the Rayleigh-Ritz energy method, the vibration displacement function of silicon fiml: w=λ (1 ²-r ²) ², make it satisfy periphery fixed boundary condition: w (r) | _R=l=0, and $\frac{dw (r)}{dr} |_{r = l} = 0,$ 1 is the silicon fiml radius, selects suitable undetermined coefficient λ, makes the total energy value I of described static-silicon fiml coupled vibrations be minimum, can obtain the resonance frequency omega of described static-silicon fiml coupled vibrations, and described undetermined coefficient λ is confirmed as:

$λ = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M}$

In the formula, $q_{3} = \frac{{πϵ}_{0} U^{2}}{{D_{0}}^{2}};$ N ₃(r)＝(r ²-l ²) ²，

$M = q_{1} ω^{2} {&Integral;}_{0}^{l} N_{1} (r) rdr - 16 q_{2} {&Integral;}_{0}^{l} N_{2} (r) rdr,$

And: q ₁=π h ρ; $q_{2} = \frac{{πEh}^{3}}{12 (1 - μ^{2})},$

N ₁(r)＝(r ²-l ²) ⁴，

N ₂(r)＝(3r ²-l ²) ²+2μ(3r ²-l ²)(r ²-l ²)+(r ²-l ²) ²，

Kinetic energy, potential energy and the static electric field energy of described gross energy I involving vibrations silicon fiml, the kinetic energy of wherein said vibration silicon fiml is: $T_{\max} = q_{1} ω^{2} {&Integral;}_{0}^{l} w^{2} rdr,$

The potential energy of wherein said vibration silicon fiml can obtain based on stress, the strain analysis of silicon fiml, that is:

$U_{S \max} = q_{2} {&Integral;}_{0}^{l} [{(\frac{{&PartialD;}^{2} w}{&PartialD; r^{2}})}^{2} + \frac{2 μ}{r} \frac{{&PartialD;}^{2} w}{&PartialD; r^{2}} \frac{&PartialD; w}{&PartialD; r} + \frac{1}{r^{2}} {(\frac{&PartialD; w}{&PartialD; r})}^{2}] rdr$

Wherein said static electric field can obtain based on the electrostatic force acting, that is: $U_{E \max} = q_{3} {&Integral;}_{0}^{l} wrdr;$

B. the coupled vibration analysis of static-silicon fiml-little air-flow

At first determine the pressure distribution of little air-flow, again according to the described little air-flow squeeze-film damping energy of described little stream pressure Distribution calculation, will described little air-flow squeeze-film damping energy substitution energy equation, and then realize the coupling analysis of described static-silicon fiml-little air-flow,

Little stream pressure distributes

Can try to achieve the pressure distribution of little air-flow according to the isothermal Reynolds equation, that is:

$P (R, T) = - (Σ_{n = 1}^{\infty} \frac{jσ B_{n}}{η_{0}^{n} + jσ} J_{0} (\sqrt{η_{0}^{n}} R)) e^{jT}$

Little air-flow squeeze-film damping energy

Calculate little air-flow squeeze-film damping energy according to little stream pressure institute work, that is:

$U_{Fd \max} = \frac{q_{4} α^{2} ω^{' 2}}{η_{0}^{1^{2}} + α^{2} ω^{' 2}} λ {&Integral;}_{0}^{l} N_{4} (r) rdr$

In the formula, q ₄=2 π b ₁

$N_{4} (r) = (l^{2} - \frac{η_{0}^{1^{2}} r^{2}}{4}) {(l^{2} - r^{2})}^{2};$

$η_{0}^{1} = {(3.8317)}^{2} .$

The energy coupling of static-silicon fiml-little air-flow

According to the Rayleigh-Ritz energy method, select suitable undetermined coefficient λ ', make the total energy value I ' of described static-silicon fiml-little air-flow coupled vibrations for minimum, can obtain the resonance frequency omega of described static-silicon fiml-little air-flow coupled vibrations ', described undetermined coefficient λ ' is confirmed as:

$λ^{'} = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M - \frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr}$

Described gross energy I ' comprises that the total energy value I of foregoing static-silicon fiml coupled vibrations and little air-flow squeeze-film damping can U _{Fd max}

C. by contrasting the vibration displacement function undetermined coefficient of no little airflow influence $λ = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M}$ With the displacement that little airflow influence the is arranged shape undetermined coefficient of shaking $λ^{'} = \frac{q_{3} {&Integral;}_{0}^{l} N_{3} rdr}{M - \frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr}$ As can be known, increased by one in the expression formula of the expression formula of λ ' than λ, that is: $\frac{q_{4} ω^{' 2}}{{(\frac{η_{0}^{1}}{α})}^{2} + ω^{' 2}} {&Integral;}_{0}^{l} N_{4} (r) rdr,$ This increase item is the damping term of little air-flow squeeze-film damping that vibration displacement is exerted an influence.