squeeze film air damping in mems

Sensors and Actuators A 136 (2007) 3–27

Review

Squeeze film air damping in MEMS

Minhang Bao a,∗, Heng Yang b

a State Key Lab of ASIC and System, Fudan University, Shanghai 200433, PR Chinab State Key Lab of Transducer Technologies, Shanghai Institute of Micro System and Information Technologies,

CAS, Shanghai 200050, PR China

Received 30 November 2006; accepted 2 January 2007Available online 16 January 2007

Abstract

The paper presents an overview and reports the recent progress of research on squeeze film air damping in MEMS. The review starts withthe governing equations of squeeze film air damping: the nonlinear isothermal Reynolds equation and various reduced forms of the equationfor different conditions. After the basic effects of squeeze film damping on the dynamic performances of micro-structures are discussed basedon the analytical solutions to parallel plate problems, recent research on various aspects of squeeze film air damping are reviewed, includingthe squeeze film air damping of perforated and slotted plate, the squeeze film air damping in rarefied air and the squeeze film air damping
of torsion mirrors. Finally, the simulation of squeeze film air damping is reviewed. For quick reference, important equations and curves areincluded.© 2007 Elsevier B.V. All rights reserved.
K

C

0d

eywords: Squeeze film air damping: Reynolds equation; Simulation

ontents

1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Reynolds equations for squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1. Nonlinear Reynolds equation for compressible film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2. Nonlinear Reynolds equation for squeeze film damping of parallel plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. Linearized Reynolds equation for compressible gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4. Linearized Reynolds equation for “incompressible gas” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Squeeze film air damping of parallel plate and basic damping effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1. Viscous damping force and elastic force of squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2. Squeeze film air damping of compressible gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2.1. The viscous damping force and the elastic damping force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2.2. The cut-off frequency of the squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2.3. The squeeze film air damping of rectangular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3. The effect of squeeze film damping on the dynamic behavior of vibration systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.1. Condition of ωo � ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3.2. Condition of ωo ≈ ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3.3. Condition of ωo � ωc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4. The verification of the effects of squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.5. The coefficients of viscous damping force for incompressible gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.5.1. Strip plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5.2. Circular plate (dish plate) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

∗ Corresponding author. Tel.: +86 21 55664527; fax: +86 21 65643449.E-mail address: [email protected] (M. Bao).

924-4247/$ – see front matter © 2007 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2007.01.008

mailto:[email protected]

dx.doi.org/10.1016/j.sna.2007.01.008

4 M. Bao, H. Yang / Sensors and Actuators A 136 (2007) 3–27

3.5.3. Annular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5.4. Rectangular plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.6. Amplitude effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.6.1. The amplitude effect on viscous damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6.2. The amplitude effect on elastic damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.7. Border effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7.1. Acoustic border conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.7.2. Extraction of elongation through FEM simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4. Squeeze film air damping of hole-plate and slotted plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.1. Squeeze film air damping of infinite, thin hole-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2. Squeeze film air damping of infinite, thick hole-plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.3. Squeeze film air damping of finite, thick hole-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3.1. Modified Reynolds’ equation for hole-plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.2. Solution to a strip hole-plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3.3. “Effective damping area” approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4. Squeeze film air damping of slotted plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165. Squeeze film air damping of rarefied air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.1. Effective coefficient of viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.2. Christian’s model for rarefied air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3. Energy transfer model for squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.3.1. Velocity change and energy transfer caused by collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.3.2. Quality factor for squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6. Squeeze film air damping of torsion micro-mirrors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.1. Damping of a strip mirror plate at balance position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2. Damping of a rectangular mirror plate at balance position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.3. Damping of a rectangular torsion mirror at a finite tilting angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.4. Damping of torsion mirror in rarefied air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7. Simulation of squeeze film air damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227.1. Physics level simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.2. System level simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247.3. Squeeze film damping in the free molecular region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

. . . . .

1

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apdrsrt

svfidMsawa

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. Introduction

As the volume forces such as gravity and inertia that workn a machine vary in direct proportion to the (length)3, whileurface forces such as viscous force vary in direct proportion tohe (length)2, the effect of surface forces on a micro-machines relatively greater than the effect of volume forces. There-ore, the effect of surface forces (most notably the dampingorce of the surrounding air), which can be neglected for aachine of conventional dimensions, may play an important roleith micro-machines and the significance of the effect becomesreater as micro-machined structures decrease in size. As a com-on result, the motion of small parts in a MEMS device can be

ffected by the surrounding air significantly. The air presents aounter reactive force on the moving of the plates.

Newell [1] observed that the ever-present damping effect ofhe surrounding air would be increased when a plate was oscil-ating near a second surface due to the squeeze film action ofhe gas between the surfaces. The squeeze film damping will be

ore important than the drag force damping of air if the thick-
ess of the gas film is smaller than one-third of the width ofhe plate. Therefore, for most MEMS devices for which frictionas been avoided, the squeeze film effect is the most importantamping effect on its dynamic behavior, as the most commonly
mca[

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

sed technologies are capacitive sensing and electric driving, forhich narrow air gaps often result.For MEMS devices with a plate (a proof mass) that moves

gainst a trapped film, squeeze film air damping has been aroblem of great importance as the mechanism dominates theamping and thus substantially affects the system frequencyesponse. The dynamic behavior of accelerometers, opticalwitches, micro-torsion mirrors, resonators, etc. is significantlyelated the squeeze film air damping of the mechanical struc-ures.

Though the resistive force of surrounding air is much lessignificant than the friction force in affecting the motion of con-entional machines, air film may play an important role in gaslm lubrication in addition to the well-known application inampers for pneumatic machines. Thus, prior to the advent ofEMS, an extensive literature had already been developed for

queeze film effect relating to air film lubrication, which hadpplication in air bearing and levitation systems. Most of theorks related to the parallel motion of the surfaces, with less

ttention being given to the squeeze film effect caused by nor-
al motion of the surfaces. The realization that squeeze films
ould have a significant effect on bearing stability promptedmore detailed analysis and experiments to test the theory

2,3].

nd Ac

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udsdIwfiTifitmtp

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vsig

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eetna

R

ww

“sttR

ncP

TiiPs

faot

2o

f

o

M. Bao, H. Yang / Sensors a

Blech [4] first refer the use of squeeze film air damping to theontrol of the frequency response of seismic accelerometers, anmportant technique widely used in MEMS devices thereafter.

ore recently the use of squeeze film air damping in tailoring theesponse of micro-accelerometers has been described by Allent al. [5], Starr [6] and Andrews et al. [7].

Though the basic nature of squeeze film air damping had beennderstood by 1980s, the research and development of MEMSevices and MEMS technologies have always stimulated inten-ive research activity in the area. Some examples are: (1) Theeep reactive ion etching (DRIE, or induction coupled plasma,CP) technologies developed in recent years produces devicesith thick, perforated plate. The adequate model for the squeezelm air damping of the perforated plates becomes desirable; (2)he encapsulation of MEMS devices in vacuum for high qual-

ty factor needs more accurate model for estimating the squeezelm air damping in rarefied air; (3) With the wide applications of

orsion mirrors in recent years, the squeeze film damping attractsore and more attention recently; (4) In addition, there is always

he need for numerically calculation of the damping effect for arecise design, once the basic model has been established.

For a review on the research on squeeze film air damping,his paper starts with the basic equations for squeeze film airamping in Section 2. The squeeze film air damping of parallellate and the effects of squeeze film air damping are reviewed inection 3. Then, the squeeze film air damping effects of perfo-ated plates, in rarefied air and for torsion mirrors are reviewedn Sections 4–6, respectively. Finally, the modeling and simula-ion are reviewed in Section 7. In this article, only the squeezelm damping problems for rigid plates are considered.

. Reynolds equations for squeeze film air damping

.1. Nonlinear Reynolds equation for compressible film

The behavior of squeeze film is in general governed by bothiscous and inertial effect within the fluid. However, for the verymall geometries encountered in MEMS devices, inertial effects often negligible. In such a case, the behavior of the fluid isoverned by the well-known Reynolds equation.

For the application of fluid lubrication, Osborne Reynoldsrst formulated the theory for the film between two surfaces inelative motion more than a century ago [2]. However, the mosteneral form of Reynolds equation for compressible gas is theonlinear differential equation first given by Tipei in 1954 [8]:

∂

∂x

(ρ

h3

μ

∂P

∂x

)+ ∂

∂y

(ρ

h3

μ

∂P

∂y

)

= 6

{2∂(hρ)

∂t+ ∂

∂x[ρh(u1 + u2)] + ∂

∂y[ρh(v1 + v2)]

}

(2.1)

here P is the pressure in the film, ρ the density, μ the coefficientf viscosity of the fluid, h the thickness of the film, u1 and u2 theelocities in the x-direction of the top plate and the bottom plate,espectively, and v1 and v2 are the velocities in the y-direction of P

tuators A 136 (2007) 3–27 5

he two plates. The conditions for the equation are that the gasow is steady (i.e., the time variation of velocity and the inertiaf the fluid can be ignored) and the thermal gradient through thelm thickness is negligible.

In 1962, Langlois [2] derived the general form of Reynoldsquation based on Navier–Stokes equations and the generalquations of viscous hydrodynamics [9]. The Reynolds equa-ion is obtained under the conditions that the modified Reynoldsumbers RS and RL are much smaller than unity. The RS and RLre defined as

S = ωρh2

μ, RL = VLρh2

wμ(2.2)

here VL is the relative velocity in lateral direction and w theidth of the plate.The condition of small RS is equivalent to the condition of

negligible inertia effect of fluid”. In reality, the condition ofmall RS is adequate for most MEMS devices and the rela-ive movement in lateral direction is not considered. Thus, forhe squeeze film damping problems encountered in MEMS, theeynolds equation is reduced to

∂

∂x

(ρ

h3

μ

∂P

∂x

)+ ∂

∂y

(ρ

h3

μ

∂P

∂y

)= 12

∂(hρ)

∂t(2.3)

For MEMS devices, the temperature variation is usuallyegligible due to the small dimensions. Under the isothermalondition, gas density ρ is directly proportional to its pressure. Thus, we have

∂

∂x

(Ph3

μ

∂P

∂x

)+ ∂

∂y

(Ph3

μ

∂P

∂y

)= 12

∂(hP)

∂t(2.4)

his nonlinear Reynolds equation has been commonly usedn MEMS for isothermal squeeze film damping of compress-ble gas. Note that, pressure P consists of two components:= Pa + p; Pa is the ambient pressure and p the deviatory pres-

ure caused by the squeeze film effect.A straightforward derivation of Reynolds equation can be

ound in [10,11]. By applying the conditions of force-balancend mass conservation to a volume element, Eq. (2.3) can bebtained directly. Readers are referred to the books [10,11] forhe derivation.

.2. Nonlinear Reynolds equation for squeeze film dampingf parallel plates

For normal motion of parallel plates, h as well as μ is not aunction of position. Eq. (2.4) can be simplified as

∂

∂x

(P

∂P

∂x

)+ ∂

∂y

(P

∂P

∂y

)= 12μ

h3

∂(hP)

∂t(2.5)

r

∂2 ∂2 24μ ∂(hP)

∂x2 P2 +∂y2 P2 =

h3 ∂t(2.6)

If the pressure is normalized by ambient pressure Pa, P =/Pa, by using the normalized variables, x = x/l, y = y/l, h =

6 nd Ac

h

oc

w

σ

wpc

2

t

p

tf(

AETσ

Twtq

2g

hitE

(

To

3d

3a

fidnoprfr

wtlo

fbTd

3

filp[op

3


/ho and τ = ωt, the nonlinear Reynolds equation for a platescillating with small amplitude around its balanced positionan be written in a non-dimensional form:

∂2

∂x2 P2 + ∂2

∂y2 P2 = 2σ∂(hP)

∂τ(2.7)

here σ is referred to as the squeeze number:

= 12μωl2

Pah2o

(2.8)

here ω is the radial frequency and l the typical length of thelate: the width or length of a rectangular plate, the radius of aircular plate, etc.

.3. Linearized Reynolds equation for compressible gas

For small displacement of the plate around its balance posi-ion (�h � ho and p � Pa), Eq. (2.5) can be linearized as [4]:

a

(∂2p

∂x2 + ∂2p

∂y2

)− 12μ

ho2

∂p

∂t= 12μpa

ho3

dh

dt(2.9)

By using normalized variables, the linearized Reynolds equa-ion for compressible gas can be written in the non-dimensionalorm:

∂2p

∂x2 + ∂2p

∂y2

)− σ

∂p

∂τ= σ

dh

dτ(2.10)

s the relaxation process of gas compression is considered inq. (2.9) or Eq. (2.10), the gas is considered as compressible.he degree of gas compression is related to the squeeze number.

For very large σ, from Eq. (2.9), we have Pa�h + ho�p = 0.his relation is equivalent to the Boyle’s law, PV = constant,hich means that the gas in the film is fully compressed, failing

o escape when it is compressed with high frequency or by auick squeeze action.

.4. Linearized Reynolds equation for “incompressibleas”

For very small σ (or slow squeeze action), the gas in the filmas enough time to “leak” so that �p/Pa � �h/ho, or, the gass not appreciably compressed. Thus, the condition is referredo as the “incompressible gas” condition. Under this condition,
q. (2.9) reduces to
∂2p

∂x2 + ∂2p

∂y2 = 12μ

h3o

dh

dt(2.11)

f

p

h

Fig. 1. Squeeze film air damping (a) cross-se

tuators A 136 (2007) 3–27

Or, in a non-dimensional form, we have

∂2p

∂x2 + ∂2p

∂y2

)= σ

dh

dτ(2.12)

hese are the equations used for MEMS devices at low frequencyr with slow movement.

. Squeeze film air damping of parallel plate and basicamping effects

.1. Viscous damping force and elastic force of squeeze filmir damping

For a pair of parallel plates as shown in Fig. 1(a), the resistiveorce to the plate moving normally against the stationary plates caused by the damping pressure between the two plates. Theamping pressure consists of two main components: the compo-ent to cause the viscous flow of air when the air is squeezed outf (or sucked into) the plate region and that to cause the com-ression of the air film. Here in this paper, the force componentelated to the viscous flow is referred to as the viscous dampingorce, and the force component related to the air compression iseferred to as the elastic damping force.

If the plate oscillates with a low frequency, or, the plate movesith a slow speed, the gas film is not compressed appreciably. In

his case, the viscous damping force dominates. It will be seenater that the viscous force is directly proportional to the speedf the plate.

On the other hand, if the plate oscillates with a very highrequency, or moves with a high speed, the gas film is compressedut fails to escape. In this case, the gas film works like a bellows.hus, the elastic force dominates. Obviously, the elastic force isirectly proportional to the displacement of the plate.

.2. Squeeze film air damping of compressible gas

According to Section 2, for small displacement, the squeezelm air damping of parallel plates is generally governed by the

inearized Reynolds equation, Eq. (2.9). The equation for com-ressible gas has been solved in different ways since the 1960s2–4,12]. For the convenience of comparison, only the problemf long rectangular plate (i.e., the strip plate) and/or rectangularlate (if available) will be reviewed in this section.

.2.1. The viscous damping force and the elastic damping
orce
Based on Eq. (2.4) and assuming that the distance of the twolates varies according to

= ho(1 + δ cos ωt) (3.1)

ctional view; (b) the damping pressure.

nd Ac

Laa(f

F

Td(

F

wf

c

f

f

Tsf

f

f

Tap

pσ

nif[

t

F

Fn

woyt

F

wfic

σ

3

pdtbttb

a

wfi

ω

o


anglois [2] solved the Reynolds equation for δ � 1 by usingperturbation method for the first order approximation (equiv-

lent to the solution to the linearized Reynolds equation, Eq.2.9)). The normalized damping force F for the strip plates isound to be

˜ = δ[−fe(σ) cos ωt + fd(σ) sin ωt] (3.2)

he first and the second terms in the equation are the elasticamping force and the viscous damping force, respectively. Eq.3.2) can be equivalently expressed as

˜ = δA(σ) cos[ωt + ϕ(σ)] (3.3)

here A(σ) =√

fe(σ)2 + fd(σ)2 is the amplitude of dampingorce and ϕ = arctan[fd(σ) + fe(σ)] is the phase lag.

Langlois found that, for small σ, the two force componentsould be approximated as

e(σ) = σ2

120+ · · · (3.4)

d(σ) = σ

12+ · · · (3.5)

hus, the amplitude of normalized viscous damping force formall σ is approximately equal to (σ/12)δ sin ωt, while the elasticorce is negligible.

For large σ, he obtained the approximations:

e(σ) = 1 −√

2/σ + · · · (3.6)

d(σ) =√

2/σ + · · · (3.7)

hus, the elastic damping force is approximately δ cos ωt withphase lag close to zero (i.e., the force is in phase with the dis-lacement δ cos ωt), while the viscous damping force vanishes.

Curves in Fig. 2 illustrate the dependence of the force com-onents on the squeeze number σ (note: ω is proportional to) for compressible gas film in the range where the most sig-ificant behavior is observed. The solution to Eq. (2.11) forncompressible gas approximation (the dash line) is includedor comparison. Similar results were found for dish plates in2].

If we now return to the real physics (non-normalized) quan-ities, the damping force is

= FPaA = PaAδ[−fe(σ) cos ωt + fd(σ) sin ωt]

ig. 2. The dependence of viscous damping force and elastic force on squeezeumber.

efop

c

A

k

ddtrf

tuators A 136 (2007) 3–27 7

here A is the area of the plate (i.e., A = Lw). If the displacementf the plate with reference to its balanced position is denoted as, we have y = h − ho = hoδ cos ωt and y = −hoδω sin ωt. Thus,he damping force takes the form of

= FPaA = −PaA

hofe(σ)y − PaA

hoωfd(σ)y = −key − cdy

(3.8)

here ke = PaAfe(σ/ho) is the coefficient of elastic dampingorce and cd = PaAfd(σ)/ho�) is the coefficient of viscous damp-ng force. According to Eqs. (3.4)–(3.8), we have ke � 0 andd � (μLw3/h3

o) for σ � σc, but ke = PaA/ho and cd = 0 for� σc.

.2.2. The cut-off frequency of the squeeze film air dampingGriffin et al. [3] treated the compressible squeeze film

roblem for a strip plate in a slightly different way. Theyetermined first the film response to a step change in the filmhickness and then used the principle of superposition com-ined with the convolution integral formulation to determinehe film response to any displacement function. The Laplaceransform of the damping force on the strip plate was giveny

F (s)

δ(s)= 96μLw3

π4h3o

∞∑m=1

1

(2m − 1)4

s

1 + s/ωc(3.9)

Then, the infinite series solution was truncated into one termpproximation:

F (s)

δ(s)= 96μLw3

π4h3o

s

1 + s/ωc(3.10)

here ωc is the approximate cut-off frequency of the squeezelm air damping:

c = π2h2opa

12μw2 (3.11)

Obviously, a cut-off frequency corresponds to a specific cut-ff squeeze number, σc = π2 = 9.8696. As pointed out by Griffint al., the exact squeeze number obtained from Langlois’ analysisor the strip plate is 10.1342. From the one term approximationf Griffin et al., the coefficient of viscous damping force for striplates is

d = 96μLw3

π4h3o

� μLw3

h3o

(3.12)

nd, the coefficient of elastic damping force is

e = 8LwPo

π2ho� 0.81

Apa

ho(3.13)

At the cut-off frequency (or the cut-off squeeze number)efined by Griffin et al., the elastic force equals the viscous
amping force. The cut-off frequency is an important parame-er in analyzing the damping effect on the amplitude–frequencyelation of a vibrating system. A similar discussion can also beound in [13].

8 nd Ac

awf

3

fMr

σ

wpn

t

f

a

f

Bno

Etwfit

Fa

ci

c

a

k

Td

3b

atdtsecuo

m

wq


Griffin et al. also verified the analytical results by experimentst low frequency. They found that the agreement is very goodhen the gap distance is smaller than the plate dimensions by a

actor of 10.

.2.3. The squeeze film air damping of rectangular plateIn 1983, Blech [4] treated the squeeze film damping problem

or rectangular plate, which has wide applications in modernEMS devices. He found the cut-off squeeze number for a

ectangular plate by a one-term approximation:

c = π2(

1 + 1

η2

)(3.14)

here η is the aspect ratio of the rectangular plate. For a striplate (η = ∞), he obtained the same result for cut-off squeezeumber by Griffin et al. [3].

The solutions to the rectangular damping problem were foundo be

d(σ) = 64σδ

π6

∑m,n odd

m2 + (n/η)2

(mn)2{[m2 + (n/η)2]2 + σ2/π4}

(3.15)

nd

e(σ) = 64σ2δ

π8

∑m,n odd

1

(mn)2{[m2 + (n/η)2]2 + σ2/π4}

(3.16)

y equating fd(σ) and fe(σ), the exact value of cut-off squeezeumber can be found via numerical calculation. The dependencef the cut-off squeeze number on 1/η is given in Fig. 3.

More recently, Darling et al. [12] derived the same results asqs. (3.15) and (3.16) by using Green function method. With

his method, they were able to treat various boundary conditionsith greater flexibility. For example, they treated the squeezelm problems with boundary conditions for one edge closed,

wo edges closed, etc.

ig. 3. The dependence of cut-off squeeze number on the aspect ratio of rect-ngular plate.

i

y

w

A

φ

wbrc

3

tcaca

tuators A 136 (2007) 3–27

By using the derivation process for Eqs. (3.12) and (3.13), theoefficient of viscous damping force of squeeze film air dampings

d(σ) = 64σpaA

π6ho

∑m,n odd

m2 + (n/η)2

(mn)2{[m2 + (n/η)2]2 + σ2/π4}

(3.17)

nd the coefficient of elastic damping force is

e(σ) = 64σ2paA

π8ho

∑m,n odd

1

(mn)2{[m2 + (n/η)2]2 + σ2/π4}

(3.18)

he equations are widely used for the analysis of squeeze filmamping in MEMS devices.

.3. The effect of squeeze film damping on the dynamicehavior of vibration systems

According to the analyses made by Langlois [2], Griffin etl. [3], Blech [4] and Darling et al. [12], the model is quite clearhat the squeeze film air damping provides additional viscousamping force and elastic force to the vibration system. Thus,he damping effect on the dynamic behavior of a spring-massystem might be significant. If the mass of the plate is m, thelastic constant of the springs supporting the plate is ko and theoefficient of damping force co for the vibration system in vac-um is negligible, the equation governing the dynamic behaviorf the vibration system in air is

y + cdy + (ko + ke)y = Fo sin ωt

here Fo sin ωt is an external driving force with a radial fre-uency ω. The steady response of the system to the driving forces

(ω) = Ao sin(ωt + φ) (3.19)

here Ao is the amplitude and φ is the phase lag:

o = Fo

m

√1

(ω2a − ω2)2 + c2

dω2/m2

(3.20)

= − arctancdω

m(ω2a − ω2)

(3.21)

here ω2a = (ko + ke)/m = ω2

o + ke/m. ωa is a function of ω

ecause ke is a function of ω. Thus, the damping effect is greatlyelated to the ratio of the natural vibration frequency ωo and theut-off frequency �c, as discussed as follows.

.3.1. Condition of ωo � ωc

For MEMS devices such as high sensitivity accelerometers,he free vibration frequency is low so that ωo � ωc. In this
ase, the frequency range of interest is ω ≤ ωo (for ω ≥ ωo, themplitude is very small). In this frequency range, the coeffi-ient of viscous damping force is approximately a constant cdond the elastic damping force is negligible (this means that the

M. Bao, H. Yang / Sensors and Actuators A 136 (2007) 3–27 9

Fω

gtlauo

3

aAtω

vpIr

3

tω

bwiki

Fω

Fω

ω

phcaii

3d

fitstsaag

ig. 4. The curves for amplitude–frequency relation under the condition of

o � ωc.

as is “incompressible”). Thus, the amplitude–frequency rela-ion is only determined by the damping ratio of the system atow frequency, ζ = cdo/2mωo. The curves in Fig. 4 show themplitude–frequency relations for three damping conditions:nder damping for ζ < 1, optimum damping for ζ ≈ 0.7 andver-damping for ζ > 1.

.3.2. Condition of ωo ≈ ωc

For MEMS devices with ωo ≈ ωc, the effect of squeeze filmir damping on the amplitude–frequency relation is significant.typical relationship is shown by the curve in Fig. 5. The ampli-

ude starts to decline appreciably when the frequency approachesc. The ratio of ke/k determines how far it may go. The value ofiscous damping force determines if the curve show a resonanteak at a frequency larger than ωo and how high the peak is.n this condition, both two damping components play importantoles on the dynamic performance of the system.

.3.3. Condition of ωo � ωc

For MEMS devices with ωo � ωc, the elastic constant ofhe structure ko is large so that ko � ke. At low frequency with

< ωc < ωo, the amplitude–frequency relation of the system isasically a horizontal line. The amplitude decreases slightly
hen ω approaches ωc due to the increase of the elastic damp-
ng force. However, the decrease is usually not significant, ase � ko. For ω > ωo, there will be a resonant peak at ωres, whichs larger than the free vibration frequency of the system, i.e.,

ig. 5. A typical curve of amplitude–frequency relation under the condition of

o ≈ ωc.

ttb

v

c

a

k

Tsc

9tt

ig. 6. A typical curve of amplitude–frequency relation under the condition of

o � ωc.

res > ωo, due to the effect of the elastic damping force. Theeak might be high, as the viscous damping force is small atigh frequency. A typical amplitude–frequency relation for thisondition is shown in Fig. 6. The effect of viscous damping forcend elastic damping force on the dynamic behavior of the systems usually slight, except for the condition that the film thicknesss extremely small.

.4. The verification of the effects of squeeze film airamping

To verify the effects of the two force components of squeezelm air damping on the dynamic performance of vibration sys-

ems, Andrews et al. [7] designed and fabricated a mass-loadedilicon plate supported by four beams by using micro-machiningechnology. The natural vibration frequency of the beam masstructure is about 10 kHz. The chip to which these beams arenchored is electro-statically bonded to a Pyrex glass on whichcavity has been etched. The air gap between the plate and thelass is about 2 �m, which provides squeeze film air damping forhe spring-mass system. The system is then used for the inves-igation of the squeeze film air damping effect on the dynamicehavior of the system.

As the plate is square in shape (η = 1), the coefficient ofiscous damping force is

d(σ) = 64σPaA

π6ho

∑m,n odd

m2 + n2

(mn)2{(m2 + n2)2 + σ2/π4}(3.22)

nd the coefficient of elastic force is

e(σ) = 64σ2PaA

π8ho

∑m,n odd

1

(mn)2{(m2 + n2)2 + σ2/π4}(3.23)

o change the cut-off frequency of the system, the spring-massystem was hermetically encapsulated in a chamber and thehamber was evacuated to different vacuum levels.

At a pressure of 760 Torr, the cut-off frequency is about.5 kHz, which is close to the natural vibration frequency. Ashe viscous damping force is very large and the damping ratio ofhe system is much larger than unity, the amplitude–frequency

1 nd Ac

rF

fiff3i

fifTF

wb

3i

teqvrag“

f

3

w

w

p

T

F

A

c

3

a

T

p

wt

p

T

F

wf

c

3

a

p

wrtca

c

w

g

3

w

f

p

gmis that the coefficient of the damping force can be written as

crec = μLw3

h3 β(η) (3.35)

0 M. Bao, H. Yang / Sensors a

elation is somewhat like the over-damping curve shown inig. 4.

At a pressure of 80 Torr, the cut-off frequency of the squeezelm damping is about 3.5 kHz, which is much smaller than theree vibration frequency of the structure. As the viscous dampingorce is still quite large, the damping ratio of the system is about. Thus, the amplitude–frequency relation is like the curve shownn Fig. 5, with an apparent resonant peak.

At a pressure of 10 Torr, the cut-off frequency of the squeezelm damping remains at about 3.5 kHz, but the viscous dampingorce is reduced appreciably so that the system is under-damped.he amplitude–frequency relation is like the curve shown inig. 6, with a high resonant peak.

Andrews et al. found that the experimental results agreed veryell with the prediction based on Blech’s equations. Thus, theasic theory of squeeze film damping was verified.

.5. The coefficients of viscous damping force forncompressible gas

In many practical conditions, only low frequency behavior ofhe vibration system is of interest, or, only the viscous dampingffect is important. These conditions include: (1) the cut-off fre-uency of the squeeze film damping is much larger than the freeibration frequency of the vibration structure; (2) the dampingatio of the vibration system is very large so that the vibrationmplitude is significant only in low frequency range. As theas film is not appreciably compressed in these cases, the gas isincompressible”. Thus, Reynolds equation (2.11) is applicable.

Based on Eq. (2.11), the coefficients of viscous damping forceor some typical plates have been derived in literatures [6,10,11].

.5.1. Strip plateFor a strip plate with its length L much larger than its width

= 2a, the damping pressure can be found by direct integrationith trivial boundary conditions. The result is

(x, t) = −6μ

h3 (a2 − x2)dh

dt(3.24)

he damping force on the plate is

strip =∫ w/2

−w/2p(x)L dx = −μw3L

h3 h (3.25)

nd the coefficient of damping force is

strip = μLw3

h3 (3.26)

.5.2. Circular plate (dish plate)For a circular plate, the Reynolds equation for squeeze film

ir damping can be written in a polar coordinate system as

1

r

∂

∂r

(r

∂

∂rp(r)

)= 12μ

h3 h (3.27)

he boundary conditions are

(a) = 0,dp

dr(0) = 0 (3.28)

tuators A 136 (2007) 3–27

here a is the radius of the plate. By integrating Eq. (3.27) withhe boundary conditions, the damping pressure is found to be

cir(r) = −3μ

h3 (a2 − r2)h (3.29)

he damping force on the circular plate is

cir =∫ a

0p(r)2πr dr = − 3π

2h3 μa4h = − 3

2π

μA2

h3 h (3.30)

here A = πa2 is the area of the plate. The coefficient of dampingorce is

cir = 3π

2h3 μa4 = 3

2πh3 μA2 (3.31)

.5.3. Annular plateFor an annular plate, the Reynolds equation for squeeze film

ir damping is also Eq. (3.27), but the boundary conditions are

(a) = 0, p(b) = 0

here a and b are the outer and inner radii of the annular plate,espectively. By solving Eq. (3.27) with the boundary conditions,he damping pressure and the damping force on the annular platean be found. Then, the coefficient of damping force for annnular plate is obtained:

ann = 3μa2A

2h3 g(η) (3.32)

here A = πa2, η = b/a and g(η) is

(η) = 1 − η4 + (1 − η2)2

ln η(3.33)

.5.4. Rectangular plateConsider a rectangular plate with comparable side lengths

= 2a and L = 2b as shown in Fig. 7. The boundary conditionsor squeeze film problem are

(±a, y) = 0, p(x, ±b) = 0 (3.34)

The solution to the Reynolds equation for incompressibleas, Eq. (2.11), is usually given in complicated series. There areany forms of solution in literatures. The common conclusion

Fig. 7. Rectangular plate with comparable edge lengths.

M. Bao, H. Yang / Sensors and Ac

woβ

3

pdfia

fito

∇

w

np

P

ae

O

O

Tidfttab

a

p

T

F

Noi

F

Trfnefi

F

d

F

U

d

F

T

Fig. 8. The dependence of factor β on the aspect ratio w/L.

here η = w/L and β(η) is a correction factor. The dependencef β(η) on η is shown by the curve in Fig. 8. For a strip plate,(0) = 1, and for a square plate (i.e., w = L), β(1) = 0.42.

.6. Amplitude effects

In the analyses given in above, the motion amplitude of thelate is considered small. However, in many applications, theisturbance amplitude may be comparable with the nominallm thickness. In this case, some amplitude effects should bepparent.

Sadd and Stiffer [14] analyzed the amplitude effect of squeezelm damping for small but finite squeeze number. By using

he normalized variables (with superimposed waves), for finitescillating amplitude, Eq. (2.7) can be written as

2P2 = 2σ

h3

∂

∂τ(P h) (3.36)

here τ = ωt and h = h/ho ≡ 1 + ε sin ωt.Since the concern is with amplitude effects at low squeeze

umber, Sadd and Stiffer [14] developed the pressure into aower series in σ, i.e.

˜ = 1 + p1(x, τ)σ + p2(x, τ)σ2 + O(σ3) (3.37)

Terms up to and including O(�2) are then retained in thenalysis. Using Eq. (3.37) in Eq. (3.36) gives the followingquations:

(σ) : ∇2p1 = 1

h3

∂

∂τh (3.38)

(σ2) : ∇2p2 = 1

h3

∂

∂τ(hp1) − 1

2∇2p2

1 (3.39)

he boundary conditions for the two equations are that the damp-ng pressure is zero on all outside boundary and the pressureistribution has zero slopes at the center of the plate. Note that,or small amplitude, the first order equation, Eq. (3.38) is in fact
he equation for incompressible gas film, Eq. (2.11). Based onhe equations, Sadd discussed four plates: strip, disk, annularnd rectangular, but only the results for strip and disk are givenelow.
f

k

tuators A 136 (2007) 3–27 11

For strip film, the normalized pressure distribution for finitemplitude is

˜ (x, τ) = 1 + 1

2h3(x2 − 1)σ ˙h + 1

24h6{h[x2(x2 − 6) + 5] ¨h

−[x2(5x2 − 18) + 13] ˙h2}σ2 (3.40)

he normalized damping force on the plate is

˜ = − 1

3h3˙hσ + 2

15

⎧⎨⎩ 1

h5¨h − 5

2

˙h2

h6

⎫⎬⎭ σ2 (3.41)

ote that the normalized damping force is not only a functionf ˙h but also a function of ¨h and σ. A noticeable phenomenons that the temporal average of the damping force is nonzero:

¯ = 1

2π

∫ 2π

0F dτ = ε2(4 + 3ε2)

24(1 − ε2)9/2 σ2

= 6ε2(4 + 3ε2)

(1 − ε2)9/2

μ2l4

P2a h4

oω2 (3.42)

he nonzero average of damping force means that there exists aectified force due to the non-linear behavior of the dampingorce for large amplitude. The vibration rectification phe-omenon is indeed observed in silicon accelerometer by Christelt al. [15]. However, it was not attributed to the effect of squeezelm air damping.

For disk plate, the temporal average of the damping force is

¯ = 5ε2(4 + 3ε2)

768(1 − ε2)9/2 σ2 = 15ε2(4 + 3ε2)

16(1 − ε2)9/2

μ2l4

P2a h4

oω2 (3.43)

From the highly nonlinear property, Sadd realized that theamping force should be developed in a more general form:

˜ = Ao + A1 cos τ + B1 sin τ + A2 cos 2τ + B2 sin 2τ

+A3 cos 3τ + B3 sin 3τ + · · · (3.44)

sing the orthogonal properties, the coefficient are given by

Ao = ¯F ; An = 1

π

∫ 2π

0F (τ) cos nτ dτ

and Bn = 1

π

∫ 2π

0F (τ) sin nτ dτ (3.45)

If the squeeze film is modeled by an equivalent linear spring-amper, we have

(t) = −kdy − cdy (3.46)

he equivalent elastic constant of the squeeze film air damping
or strip film is
e,strip = 2(1 + 3ε2 + 3ε4/8)

15ho(1 − ε2)9/2 PaLwσ2 (3.47)

1 nd Ac

a

c

w

t

k

a

c

w

vc

e

3

frtt

f

T

3

sm

A

w

f

T

3

ptcmdsaba

mpeihda

gItinsT


nd the coefficient of viscous damping force is

d,strip = μLw3

ho3(1 − ε2)3/2 (3.48)

here L is the length and w the width of the strip plate.Similarly, for a disk plate, the equivalent elastic constant of

he squeeze film air damping is

e,disk = (1 + 3ε2 + 3ε4/8)

48ho(1 − ε2)9/2 Paπa2σ2 (3.49)

nd the coefficient of viscous damping force is

d,disk = 3μπa4

2h3o(1 − ε2)3/2 (3.50)

here a is the radius of the disk.Obviously, the linear spring-damper approximation is only

alid when ε is not very large. Otherwise, higher harmonic forceomponents cannot be ignored.

Based on Sadd’s work [14], Starr [6] discussed the amplitudeffects on the viscous damping and the elastic damping.

.6.1. The amplitude effect on viscous dampingStarr [6] noticed from [14] that the viscous damping force

or small squeeze number and large amplitude for circular andectangular plates can be calculated by multiplying respectivelyhe expressions for small amplitude, Eqs. (3.31) and (3.35), byhe function:

d(ε) = 1

(1 − ε2)3/2 (3.51)

he dependence of fd on ε is shown in Fig. 9.

.6.2. The amplitude effect on elastic dampingFor strip plate, Starr found the ratio between the elastic con-

tant of squeeze film air damping and the elastic constant of the
echanical structure:
ke,strip

k= 0.8ζfk(ε)σ

ω

ωo(3.52)

Fig. 9. The dependence of fd on ε.

t

3

ncdo

wod

L

tuators A 136 (2007) 3–27

nd, for disk plate, he found:

ke,disk

k= 0.33ζfe(ε)σ

ω

ωo(3.53)

here ωo = √k/m, ζ = cd/2mωo and

e(ε) = 1 + 3ε2 + 3ε4/8

(1 − ε2)3 (3.54)

he dependence of fe(ε) on ε is also shown in Fig. 9.

.7. Border effect

The squeeze film damping of parallel plates presented in therevious subsections are derived with trivial boundary condi-ions, i.e., the gas at the borders is at ambient pressure. Thisondition is a good approximation if the plate dimensions areuch larger than the film thickness. However, in practical

evices the flow escaping from the borders might contributeignificantly to the damping force. Experimental measurementsnd simulation performed by Vemuri et al. [16] show that theorder effect considerably increases the damping force, even forratio of film width/thickness as high as 20.

In order to include the border effects on the analyticallyodel, an effective plate length Leff = L + �L and an effective

late width weff = w + �w are introduced. The values of theffective length and width are such that the coefficient of damp-ng force for the enlarged plate with trivial boundary conditionsas the same value as the coefficient of damping force of the realevice size with the border effects. That is, the elongations �Lnd �w are used to model the effect of the border effect.

Naturally, the border effect depends on the geometry of theas outlet. There are two typical geometries of the border outlets.f the top and bottom plates are identical and line up perfectly,he outlet is symmetric for the gas flow at the border. Otherwise,f the bottom plate is much larger than the top plate, the outlet ison-symmetric for the gas flow at the border. Obviously, the non-ymmetric border is more typical for practical MEMS devices.herefore, only the non-symmetric border is considered here.

Two methods have been proposed in literatures to computehe elongations �L and �w.

.7.1. Acoustic border conditionsVeijola and coworkers [17,18] considered that the flow chan-

el continues outside the gas film border and the border flowan be modeled in a simple way. Assuming that the pressurerops linearly in the extended length �L/2 of the channel, theybtained:

∂p

∂n

∣∣∣∣ = ±p|±L/2

�L/2(3.55)

here n is a direction normal to the border surface. Thus, a firstrder approximation for the effective length and effective width
ue to the end effects are derived
eff = L

√1 + 3AL(1 + 4Aw)3/8

√1 + 3Aw(1 + 4AL)1/8 (3.56)


ations

w

w

A

A

FnT

3

bedpec

�

�

Tm

4p

ofmei

4

bsharca

tia

ttpd“t

P

Bi[

F

wt

F

Fig. 10. Schematic drawing of perfor

eff = w

√1 + 3Aw(1 + 4AL)3/8

√1 + 3AL(1 + 4Aw)1/8 (3.57)

here AL and Aw are

L = 8

3π

1 + 2.676K0.659n

1 + 0.531K0.5n (h/L)0.238

h

L(3.58)

w = 8

3π

1 + 2.676K0.659n

1 + 0.531K0.5n (h/w)0.238

h

w(3.59)

or small Knudsen number (Kn � 1) and small gas film thick-ess (L, w � h), we have Leff ≈ L + 8h/3π and weff ≈ w + 8h/3π.he results agree with that of [19].

.7.2. Extraction of elongation through FEM simulationsAnother method to predict the elongation caused by the

order effect on parallel surfaces was proposed by Vemurit al. [16] and Veijola et al. [20,21]. A series of two-imensional and three-dimensional FEM simulations wereerformed and, based on the results, approximations for thelongations were extracted. Veijola et al. obtained a very simpleonclusion:

L = 1.3h, �w = 1.3h (3.60)

For high Knudsen number, the elongations is modeled as

a = 1.3(1 + 3.3Kn)h (3.61)

he simulations also show that this model can be used for torsionotion.

. Squeeze film air damping of hole-plate and slottedlate

Moving plates in microstructures are sometimes perforatedr slotted to reduce the damping effect to a certain levelor some applications such as accelerometers, microphones,icro-torsion mirrors, switches and relays, etc. Therefore, the

stimation of air damping force for a perforated or slotted plates important in designing the devices.

w

k

Tsr

(a) square array; (b) hexagon array.

.1. Squeeze film air damping of infinite, thin hole-plate

For a thin hole-plate, the squeeze film damping effect cane easily analyzed with some approximations [22,10,11]. Forimplicity, the plate is assumed to be perforated with circularoles with radius ro and the holes are typically distributed insquare or a hexagon array as shown in Fig. 10(a) and (b),

espectively. If the density of hole is n (in m−2), the area of theell containing a hole is A1 = 1/n. The cell can be approximateds an annulus with an outer radius of rc = 1/

√πn.

The damping force on the whole plate is the summation ofhe damping force of each cell. So the damping force on a cells considered first. As the squeeze number is usually small forn annular plate, Eq. (3.28) for an annulus is used for the cell.

Under the approximations: (1) the hole plate is much largerhan a cell and all the cells are identical, so the air-flow betweenhe cells is negligible (the “infinite plate” approximation); (2) thelate is thin when compared with the hole size so that the pressureifference causing the air flow through the hole is negligible (thethin plate” approximation). Thus, the boundary conditions forhe cell are

(ro) = 0,∂P

∂r(rc) = 0 (4.1)

y solving Eq. (3.28) with the boundary conditions, the damp-ng pressure is found and the damping force on a single cell is22,10,11]:

1 = −3μA12

2πh3 h(4η2 − η4 − 4 ln η − 3) (4.2)

here η is the ratio of ro/rc and A1 is the area of a cell. Thus,he total damping force on the perforated plate is

p = A

A1F1 = − 3μA

2πnh3 hk(η) (4.3)

here A is the total area of the hole-plate and k(η) is

(η) ≡ 4η2 − η4 − 4 ln η − 3 (4.4)

he dependence of k(η) on η is given by the curve in Fig. 11,howing that the perforation with large holes is very effective ineducing the damping effect.

14 M. Bao, H. Yang / Sensors and Ac

eh[dpefTd

aTfwp(ar

(

(

(

4

em

thbb

cltbcf

bezbfsd

mitt

oaa

ohdoi

ph

4

Vaeemtaps

ep

Fig. 11. The dependence of factor k on η.

For a hole-plate with a finite area, the damping force is overstimated by Eq. (4.3), as the air-flow through the borders of theole-plate has been neglected. To reduce the error, Davies et al.23] proposed an empirical correction scheme, assuming that theamping force given by Eq. (4.3) and the damping force of a solidlate of the same size work like two resistors “in parallel”. Forxample, for a rectangular hole-plate, the squeeze film dampingorce, Frec, of a solid rectangular plate is found using Eq. (3.35).hen the resultant damping force of the perforated plate, FR, isetermined by

1

FR= 1

Fp+ 1

Frec(4.5)

Though the empirical correction scheme is intuitive, it usu-lly provides an improved estimation for practical applications.he result is applicable for typical (but not all) thin hole-plates

abricated using surface micromachining technology. However,ith the development of MEMS technology, thick plate can beerforated with tiny holes by using deep reactive ion etchingdeep RIE, or ICP) technologies. For thick hole-plate, the abovenalytical results are no longer accurate. Improvement in someespects has to be considered:

a) The damping effect of holes in thick plate:For a thick hole-plate, the resistance to the air flow through

the holes might be large.b) The border effect:

For the above reason, an appreciable fraction of air mightflow out of the gap region through the borders instead ofthe holes. In this case, the border effect is not negligible.Theoretically, the problem should be treated by solving adifferential equation governing the damping pressure withappropriate boundary conditions instead of using the intu-itive equation, Eq. (4.5), for border effect.

c) The end effect of hole:For long and thin holes, the end effect is insignificant.

However, the end effect might be significant for short holesand has to be considered for many hole-plates in MEMS.

.2. Squeeze film air damping of infinite, thick hole-plates

Among the three problems mentioned above, the dampingffect of air flow through the holes of a thick plate receivedore attention than the others. A few studies were made early

vabw

tuators A 136 (2007) 3–27

his century on the problem. For the simplicity of analysis, theole-plate is considered infinite in lateral dimensions so that theorder effect of the hole-plate is neglected and the analysis cane made for a cell containing a hole at center.

Bao et al. [24] considered the damping effect of a circularell with a hole at the center. If the plate is large enough, theateral air flow between the neighboring cells is negligible. Theotal damping pressure in a cell consists of the pressure causedy the vertical Poiseuille flow through the hole and the pressureaused by lateral air flow to the hole. Then, the expression isound for minimum damping ratio of the structure.

Kwok et al. [25] treated the damping effect of air flow in holesy two methods. First, they established the modified Reynoldsquation in a square cell containing a square hole at center. Theero-flow boundary condition is used for the four outer cell wallsecause the air flow between the neighboring cells is negligibleor infinite hole-plate. The boundary condition for the four innerides was derived based on Poiseuille flow in the hole. Then theamping pressure was numerically solved.

The second method used by Kwok et al. is to find an approxi-ate close form equation for the pressure caused by lateral flow

n the square cell. The total damping pressure is found by addinghe pressure component to the pressure caused by the air flow inhe hole.

Homentcovschi and Miles [26] treated the damping in a cellf the hole plate based on Navier–Stokes equations in a strict waynd considered the design for minimum damping coefficient forspecific structure.

All the above methods assumed that the air flows in and outf the plate region only through the holes. This means that theole-plate has infinite lateral dimensions so that the pressureistribution is identical for all the cells. The total damping forcen the hole-plate is obtained by multiplying the damping forcen a cell with the number of cells in the hole-plate.

To take into account the border effect for a finite, thick hole-late, a method with modified Reynolds equation for the wholeole-plate must be considered.

.3. Squeeze film air damping of finite, thick hole-plate

Considering the impedance of the air flow through the holes,eijola et al. [27] added a term into the linear Reynolds equationnd gave out a modified Reynolds equation for hole-plates. Thequation is then solved for a rectangular plate and the resultingxpression is implemented using an electrical equivalent circuitodel. They found that the simulated results agreed well with

he experimental measurement qualitatively. The quantitativegreement is very good for small hole-size, but the simulationredicted larger damping coefficients than measurement for holeize larger than 3 �m.

Considering the damping effect of air flow in the holes, Baot al. [24,28] derived a modified Reynolds equation for dampingressure of hole-plates based on the principle of mass conser-
ation and force balance. The equation is linearized for smallmplitude to a modified Reynolds equation similar to that giveny Veijola et al. The Modified Reynolds equation can be solvedith boundary conditions of the hole-plate. As the derivation


le-pl

ar

4

iRa

wtnr

Π

wac

c

tcl

H

4

tc

√p

i

p

tripEptR

vc

p

iaip

Fd = 2aLRl2[

1 − l

atanh

(a

l

)](4.14)

Fig. 12. Schematic structure of a thick ho

pproach in [24,28] is more detailed and general in form, it iseviewed as follows.

.3.1. Modified Reynolds’ equation for hole-platesFor a plate perforated with holes of high density as shown

n Fig. 12, Bao et al. have established a nonlinear modifiedeynolds equation based on the principles of mass conservationnd force balance [24,28].

∂

∂x

(P

∂P

∂x

)+ ∂

∂y

(P

∂P

∂y

)− P

3η2ro2

2h3H

p

Π (η)

= 12μ

h3

∂(Ph)

∂t(4.6)

here p is the damping pressure in the gas film, P = Pa + p (Pa ishe ambient pressure), H the thickness of the plate, h the thick-ess of the gas film, ro the radius of the holes, η ro/rc, rc theadius of the cell. �(η) in Eq. (4.6) is defined as

(η) ≡ 1 + 3r4o

16Hh3 k(η)

here k(η) is defined by Eq. (4.4). For small squeeze numbernd small fluctuating amplitude, the modified Reynolds equationan be linearized as

∂2p

∂x2 + ∂2p

∂y2 − 3η2ro2

2h3H

1

Π(η)p = 12μ

h3

∂h

∂t+ 12μ

h2Pa

∂p

∂t(4.7)

For “incompressible gas”, the second term on the right sidean be neglected:

∂2p

∂x2 + ∂2p

∂y2 − 3η2ro2

2h3H

1

Π(η)p = 12μ

h3

∂h

∂t(4.8)

According to Sharipov and Seleznev [19], the end effect ofhe holes should be considered if the radius of the holes, ro, isomparable to the thickness of the plate, H. In this case, the pipeength H should be replaced by an effective length Heff:

eff = H + 3πro

8(4.9)

.3.2. Solution to a strip hole-plateTo demonstrate the nature of the squeeze film damping of

hick hole-plate, a strip hole-plate with a width of w = 2a is
onsidered. Thus, Eq. (4.8) becomes one-dimensional:
∂2p

∂x2 − 3η2r2o

2h3Heff

1

Π(η)p = 12μ

h3

∂h

∂t(4.10)

F

ate (a) top view (b) cross-sectional view.

By defining R = (12μ/h3)(∂h/∂t) and l =2h3HeffΠ(η)/3η2r2

o (l is the attenuation length of dampingressure), Eq. (4.10) can be simplified as

∂2p

∂x2 − p

l2+ R = 0 (4.11)

With boundary conditions p(±a) = 0, the damping pressures found as

(x) = Rl2(

1 − cosh(x/l)

cosh(a/l)

)(4.12)

If the holes are very thin and/or the plate is very thick sohat the attenuation length l is much larger than a, Eq. (4.12)educes to Eq. (3.24). This means that the gas flow in the holess negligible and the damping of the hole-plate is that of a solidlate. To another extreme, if the plate is very thin so that a � l,q. (4.12) reduces to Eq. (4.3); the damping force of the hole-late is that of an infinite, thin hole-plate. The consistency withhe proven results at the two extremes justifies the modifiedeynolds’ equation.

Generally, l is indeed much smaller than a, but has a finitealue so that its effect on the damping pressure in the border areasannot be neglected. Thus, Eq. (4.12) can be approximated as

(x) = Rl2(1 − e(a−x)/l − e(a+x)/l) (4.13)

This means that the damping pressure in the vast areas po = Rl2. However, the pressure drops exponentially whenpproaching the borders. For example, for a = 10·l, the damp-ng pressure across the plate is plotted in Fig. 13, where theressure has been normalized to po.

From Eq. (4.12), the damping force on the strip hole-plate is

ig. 13. Pressure distribution for a long rectangular hole-plate with a = 10·l.

1 nd Actuators A 136 (2007) 3–27

A

c

F

a

c

4

aFtrda

F

a

F

we

tl

f(ds

ab[

4

ufiis

F

Fc

edii

w

Π

cbht


nd, the coefficient of damping force is

= 2aL8μHeff

η2r2o

(1 + 3r4

ok(η)

16Heffh3

) [1 − l

atanh

(a

l

)](4.15)

For a � l, i.e., tanh(a/l) ∼= 1, we have

d ∼= 2(a − l)LRl2 (4.16)

nd

= 8μHeff

η2r2o

(1 + 3r4

ok(η)

16Heffh3

)2L · 2(a − l) (4.17)

.3.3. “Effective damping area” approximationAccording to Eq. (4.16), for a � l, the damping force is equiv-

lently caused by the pressure distribution shown by the curve inig. 14. Therefore, we may consider that the plate is only effec-

ive for damping in the width 2(a − l), excluding the two borderegions with a width of l each. This is referred to as the “effectiveamping area” approximation. Based on the approximation, for, b � l, the damping force of a rectangular hole-plate is

d ∼= 4Rl2(a − l)(b − l) (4.18)

Similarly, the damping force on a circular hole-plate can bepproximated as

d ∼= Rl2π(a − l)2 (4.19)

here a is the radius of the circular hole-plate. This scheme mayven be applicable to some hole-plate with irregular shape.

According to the definition of l, if the hole is large sohat r4

o � Heffh3, we have Π(η) ≈ (3ro

4k(η)/16Heffh3) and

≈ rc√

k(η)/8. Thus, we have l < rc if only β > 0.06. There-ore, the condition for effective damping area approximationa, b � l) is equivalent to a, b � rc or a, b > 3rc. The effectiveamping area approximation is reasonable if there are more thanix holes across the plate.

The analytical results by the modified Reynolds’ equationgree well with the numerical analyses using the model proposedy Mehner et al. [29] and the experimental data by Kim et al.30].

.4. Squeeze film air damping of slotted plate

Another scheme to reduce the squeeze film air damping is to
se a plate with parallel slots (the slotted plate). The squeezelm air damping of some MEMS devices using movable grid
s similar to that of slotted plate [31]. Typically, the shape oflotted plates is rectangular.

ig. 14. Equivalent pressure distribution for a strip hole-plate with a = 10·l.

H

wgd

c

wb2sa

ig. 15. Squeeze film air damping in a thick slotted plate (a) top view; (b)ross-sectional view.

With the method used in last section, a modified Reynoldsquation for squeezed film air damping of slotted plate has beenerived by Sun et al. [32]. For a structure schematically shownn Fig. 15, the equation for a slotted plate with finite thicknesss

∂2p

∂x2 + ∂2p

∂y2 − 4b3

ah3H

1

Π(η)p = 12μ

h3 h (4.20)

here

(β) =(

1 + 4ab3

3h3H(1 − η)3

)and η = b

a

For a slotted plate with a finite thickness (the length ofhannel), the flow effect caused by the channel ends has toe considered. According to Eq. (5.18) in [19], the geometriceight, H, should be replaced with an effective height, Heff, inhe equations:

eff = H + 16

3πb (4.21)

The modified Reynolds equation, Eq. (4.20), can be solvedith the boundary conditions of the slotted plate. For a rectan-ular slotted plate with its length much larger than its width, theamping coefficient of the plate is found to be:

= 24

h3 μLl2(w

2− l

)(4.22)

here l =√

ah3HΠ(η)/4b3. The accuracy of the result has
een verified by ANSYS simulation. For a slotted plate witha = 20 �m, 2b = 4 �m, H = 20 �m, h = 2 �m, L = 5 mm, and alot number of 19, the difference between the analytical resultnd that of simulation is about 3%.

nd Ac

5

pa

μ

wmitp

is1aaor

tittrppt

ptcamn[

5

Sbttngt

u

wcavT

fEam

lseF

ucμ

iv

μ

A

dfciaa

μ

s

μ

doe

igbfKovt


. Squeeze film air damping of rarefied air

As described in above sections, air damping is directly pro-ortional to the coefficient of viscosity of air. The viscosity ofir derived based on a simple model [33] is

= 1

3ρaλv (5.1)

here ρa is the mass density of gas, λ the mean free path of theolecules and v is the average velocity of the molecules. As ρa

s proportional to the pressure while λ is inversely proportionalo the pressure, the coefficient of viscosity, μ, is independent ofressure p.

Experimental data show that the effect of air damping isndeed almost constant when the air pressure is near the atmo-pheric pressure. For example, the viscosity of air is 1.79 ×0−5 Pa s at 1 atm and the viscosity reduces to 1.61 × 10−5 Pa st 0.5 atm [34]. However, the experimental results show that their damping is reduced appreciably when the air is well belowne atmospheric pressure. Thus, the damping effect shouldeduce accordingly.

There have been two basic approaches so far in consideringhe damping in rarefied air: the “effective coefficient of viscos-ty” and the free molecular model. The first approach suggestedhat the equations for squeeze film air damping remain effec-ive in rarefied air, but the “coefficient of viscosity” should beeplaced by an “effective” one, μeff, which is dependent on theressure via Knudsen number Kn = λ/d, where λ is the mean freeath of molecules and d the gap distance between the plate andhe substrate.

However, for a pressure much lower than an atmosphericressure, the collisions among the gas molecules are so reducedhat the gas can hardly be considered as a viscous fluid. In thisase, the concept of effective viscosity would become question-ble. For Kn � 1, i.e., the gap distance is much smaller than theean free path of the gas molecules, the viscous flow model is

o more valid and a free molecular model has to be considered35,36].

.1. Effective coefficient of viscosity

For small Knudsen number (say Kn > 0.01), the Navier–tokes equations can be used with a single modification: theoundary conditions at fluid–wall interface are changed fromhe standard non-slip condition to that of a slip-condition, i.e.,he flow velocity of gas at the border with a stationary wall isot zero. The flow velocity is related to the mean free path of theas and the gradient of the gas velocity in the direction normalo the wall [37,38]:

wall = 2 − σv

σvλ

∂u

∂y

∣∣∣∣wall

(5.2)

here σv is the tangential momentum accommodation coeffi-
ient (TMAC) and is defined as the fraction of molecules whichre diffusively reflected. As λ is a function of air pressure, theelocity at border uwall is an implicit function of the pressure.he accommodation coefficient depends on the gas, the sur-
tq

o

tuators A 136 (2007) 3–27 17

ace material and roughness and is determined by experiments.xperimental measurements [39] show that smooth silicon hasn accommodation coefficient of about 0.7 with several com-only used gases.Even if the slip-flow theory is only valid for the condition of

ow Knudsen number, it is often used at much higher Kn due to itsimplicity compared to other approaches (solving the Boltzmannquation or using direct simulation Monte Carlo computation).or simple geometries the usage provides adequate results.

In gas lubrication theory, the non-slip boundary condition issually included in the viscosity coefficient, i.e., the slip-flowondition is considered by replacing the coefficient of viscosityin Reynolds equation with the effective viscosityμeff that takes

nto account the rarefaction effects. The effective coefficient ofiscosity is commonly given in the form:

eff = μ

1 + f (Kn)(5.3)

nd the Reynolds equation becomes:

∂

∂x

([1 + f (Kn)]

Ph3

μ

∂P

∂x

)+ ∂

∂y

([1 + f (Kn)]

Ph3

μ

∂P

∂y

)

= 12∂(hP)

∂t(5.4)

There are quite a few expressions for f(Kn) based on differenterivation considerations [40–45]. The expressions are quite dif-erent in forms and even in values. Veijola et al. [46] presented aomprehensive review on the expressions. Based on the analyt-cal work using Boltzmann’s transportation equation by Fukuind Kaneko [42], Veijola et al. obtained a simple, empiricalpproximation for the effective coefficient of viscosity:

eff = μ

1 + 9.658Kn1.159 (5.5)

Based on Andrews’ experimental data [7], Li [47] gave aimilar approximation for the effective coefficient of viscosity:

eff = μ

1 + 6.8636K0.9906n

(5.6)

With the effective coefficient of viscosity, the squeeze filmamping effect in rarefied air is dependent on the dimensionsf the plate and the gap distance between the plates in the samequation as it is in an atmospheric pressure.

The concept of effective viscosity coefficient for rarefied airs reasonable when the air pressures is not very low so that theas can still be considered as a continuum though the non-slipoundary conditions have to be replaced by slip ones. However,or a pressure much lower than an atmospheric pressure (i.e.,n � 1, the gap distance is much smaller than the mean free pathf the gas molecules, so that the viscous flow model is no morealid), the collisions among the gas molecules are so reduced thathe gas can hardly be considered as a viscous fluid. In this case,
he concept of effective coefficient of viscosity would becomeuestionable and a free molecular model has to be considered.
For example, the mean free path of gas molecules is on therder of 0.1 �m in an atmospheric pressure. However, for a pres-

1 nd Ac

si

5

lmppppo

inod

f

wmrt

(

2p

P

o

P

Aog

f

F

widc

c

A

Q

wp

eitro

mddi

Cscta

5


ure of 1 Pa, the mean free path increases to about 1 cm, whichs much larger than the typical dimension of microstructures.

.2. Christian’s model for rarefied air damping

Christian proposed a free molecular model for damping inow vacuum [35,48]. In the model, the interaction between gas

olecules is neglected and the damping force on an oscillatinglate is found by the momentum transfer rate from the vibratinglate to the surrounding air through the collisions between thelate and the molecules. The quality factor of the oscillatinglate is then found directly without relying on the “coefficientf viscosity” in any form.

Consider the air damping force acting on a plate oscillatingn its normal direction (x-direction) as shown in Fig. 16. Theumber of molecules in a unit volume with velocity in the rangef vx to vx + �vx is dn = nf (vx)dvx, where f (vx) is Maxwellianistribution function:

(vx) =√

m

2πkTe−mv2

x/2kT

here k is the Boltzmann constant (k = 1.38 × 10−23 J/K). If theoving speed of the plate is x, for the molecules in the velocity

ange vx to vx + �vx, the number of head-on collision in a unitime on a unit area of front is

vx + x)dn × 1 × 1 = n(vx + x)f (vx)dvx

As the change in momentum for each collision molecule ism(vx + x), the pressure caused by the collisions on the frontlate surface is

f = 2mn

∫ ∞

−x

(vx + x)2f (vx)dvx (5.7)

Similarly, the pressure caused by the collisions on the backf the plate surface is

b = 2mn

∫ ∞

x

(vx − x)2f (vx)dvx (5.8)

The net damping force caused by collisions is Fr =(Pf − Pb), where A is the area of the plate. As the velocityf the plate, x, is much smaller than that of the majority of theas molecules, according to Eqs. (5.7) and (5.8), the damping

Fig. 16. Collisions of an oscillating plate with head-on molecules.

[sotmm

5c

rm

ooM

v

tuators A 136 (2007) 3–27

orce on the plate is

r ∼= 8mnA

∫ ∞

0vxxf (vx)dvx = 4

√2

π

√Mm

RTPAx (5.9)

here Mm is the molar mass of the gas and R = 8.31 kg m2/(s2 K)s the universal molar gas constant. Eq. (5.9) shows that theamping force decreases linearly with the pressure. The coeffi-ient of damping force in rarefied air by the model is

r = 4

√2

π

√Mm

RTPA (5.10)

nd, the Q factor of the system in low vacuum is

Chr = Hρpω

4

√π

2

√RT

Mm

1

P(5.11)

here H is the thickness of the plate, ρp the mass density of thelate and ω is the radial frequency of the plate.

Eq. (5.11) has been compared with experimental data by Zookt al. [49] and Guckel et al. [50]. The quality factor is indeednversely proportional to pressure P, but its value is overes-imated by an order of magnitude, i.e., the damping force inarefied air is underestimated by Christian’s model by an orderf magnitude.

To alleviate the problems, Kadar et al. [51] and Li et al. [52]odified the calculation for Christian’s model and reduced the

ifference between the theoretical results and the experimentalata appreciably. However, their modifications are consideredncorrect [36,53].

As a matter of fact, the discrepancy between the results byhristian’s model and the experimental data by Zook et al. [49]

tems from the fact that the effect of nearby substrate is notonsidered by Christian model; Christian model is a model forhe air damping of an isolated subject but not for squeeze filmir damping.

.3. Energy transfer model for squeeze film air damping

To overcome the difficulty of Christian’s model, Bao et al.36] proposed a new free molecular model. With the model, thequeeze film damping can be considered as well as the dampingf an isolated object. As the model calculates the energy losseshrough the collisions between the plate and the molecules, the

odel is also referred to as the Energy Transfer model. Theodel is introduced as follows.

.3.1. Velocity change and energy transfer caused byollisions

To calculate the energy transferred from the plate to the sur-ounding air, let us first consider the velocity change of a gasolecule after its collision with a moving plate.Consider that a plate moving on with a speed x makes a head-

n collusion with a gas molecule with a speed of v. As the massf the gas molecule m is much smaller than the mass of the plate, the gas molecule is bounced back with a speed:

+ = v + 2x (5.12)

nd Actuators A 136 (2007) 3–27 19

So

v

fd

�

qqt

5

sdt

ta

wv

v

titsc

�

co

v

Tp

�

FI

o

�

Wo

�

fi

Q

w

TQ

Q

4Qmt


imilarly, if a molecule makes a catch-up collusion on the rearf the plate, the resulting speed of the molecule would be

− = v − 2x (5.13)

Eqs. (5.12) and (5.13) show that the energy transferred by aront collusion is different from that by a back collusion. Theifference is

e = 12m[(v + 2x)2 − (v − 2x)2] = 4mvx

The energy loses caused by the collisions is related to theuality factor of the oscillating plate. It has been found that theuality factor found by using the energy transfer model is exactlyhe same as the result obtained by Christian model [36].

.3.2. Quality factor for squeeze film air dampingFor a plate oscillating with x = ao sin ωt near a stationary sub-

trate, as schematically shown in Fig. 17, if the nominal gapistance is ho and the displacement of the oscillating plate is x,he gap distance is h = ho − x.

If the peripheral length of the gap is S, the boundary area ofhe gas film is S(ho − x) and the number of molecules movingcross the boundary into the gap per unit time is

14nvS(ho − x) (5.14)

here n is the concentration of the molecules and v is the averageelocity of the molecules.

If a molecule enters into the gap with velocity components ofyzo in the y–z plane and vxo in the x-direction, it gains velocity inhe x-direction due to the collisions with the plate when travelingn the gap. If the lateral traveling distance in the gap is l, theime the molecule stays in the gap is �t = l/vyzo, which is muchmaller than an oscillating cycle of the plate and the times ofollision in the time period are

N = �t vxo

2(ho − x)= lvxo

2(ho − x)vyzo

(5.15)

As the molecule gains a speed increment of 2x each time itollides with the plate, the velocity in the x-direction at the endf the traveling in the gap is

x = vxo + �N × 2x = vxo + lvxo

(ho − x)vyzo

x (5.16)

he extra energy gained by the molecule via collisions with thelate is

ek = 1

2m

[2lv2

xo

(ho − x)vyzo

x + l2v2xo

(ho − x)2v2yzo

x2

](5.17)

Fig. 17. Squeeze film air damping in rarefied air.

6

bldm

wh

ig. 18. Comparison with experimental results of a beam (I: experimental data;I: Christian’s model; III: the Energy Transfer model).

According to Eqs. (5.16) and (5.17), the average energy lossf the plate in one vibration cycle is

Ecycle = 1

4nvS

1

ω

∫ 2π

0

ml2v2xo

2(ho − x)v2yzo

a2oω

2 cos2 ωt d(ωt)

ith some approximation in calculation, a close form result isbtained:

Ecycle ∼= Aa2oω

8ρav

S

do(5.18)

According to the definition, the quality factor for squeezelm air damping by the Energy Transfer model is

E,Sq = 2πEP

�Ecycle= (2π)3/2ρpHω

(ho

S

) √RT

Mm

1

p(5.19)

here ρp is the mass density of plate and S = 2L + 2w.From Eqs. (5.19) and (5.11), the quality factor by Energy

ransfer model, QE,Sq, is related to that by Christian’s model,Chr., by

E,Sq = 16πho

SQChr. (5.20)

According to [49,50], the beam is 200 �m long and0 �m wide, and the gap distance is 1.1 �m. Thus, we haveE,Sq = 0.115QChr.. As a result, the Energy Transfer modelatches the experimental data much better than that of Chris-

ian’s model as illustrated by Fig. 18.

. Squeeze film air damping of torsion micro-mirrors

In recent years, more and more torsion micro-mirrors haveeen used in a variety of MEMS devices, such as optical displays,ight modulator and optical switches. As the squeeze film airamping is the key factor to the dynamic performances of the
irror, it has been investigated extensively in recent years.A typical micro-mirror is schematically shown in Fig. 19,
hereϕ is the tilting angle of the mirror plate. As the gap distance= ho – xϕ and the moving speed of the plate xϕ are not uniform,


e of a

tϕ

m

tg

6

idE

Iia

Ws

p

F

T

Tt

c

E

6p

d

nB

wawv(

h

rtdss

T

wor

stdaed

tst

Fig. 19. Schematic structur

he coefficient of damping torque is a function of the tiling angle. Obviously, the analysis of squeeze film air damping of torsionirrors is more difficult than that of a parallel plate actuator.As the tilting angle of micro-mirror is generally small (≤10◦),

he squeeze film air damping of micro-mirror is approximatelyoverned by the Reynolds equation (2.3).

.1. Damping of a strip mirror plate at balance position

The simplest condition to analysis is a strip torsion mirror,.e., a torsion micro-mirror with its length (L = 2b, in the axialirection) much larger than its width (w = 2a). In this condition,q. (2.3) becomes one dimensional:

d2p

dx2 = − 12μ

(ho − xϕ)3 ϕx (6.1)

f the plate is moving around its balanced position (ϕ = 0) with annfinitesimal angular amplitude, the above equation is simplifieds

d2p

dx2 = −12μ

ho3 ϕx (6.2)

ith trivial boundary conditions, p(x ± a) = 0, the damping pres-ure is found to be

(x) = 2μ

h3oϕx(a2 − x2) (6.3)

rom Eq. (6.3), the torque of damping force is

d =∫ +a

−a

xp(x)L dx = 16μba5

15h3o

ϕ (6.4)

herefore, the coefficient of damping torque for infinitesimalilting angle is

ϕ(0) = μLw5

60h3o

= 16μba5

15h3o

(6.5)

This is the same result as Eq. (30) of Veijola et al. [21] andq. (15) of Wei et al. [54].

.2. Damping of a rectangular mirror plate at balance
osition
Pan et al. [34] made a general analysis of squeeze film airamping for a rectangular mirror plate. They started with the

tftt

rectangular torsion mirror.

onlinear isothermal Reynolds equation for compressible gas.y using the normalized variables, the equation takes the form:

∂

∂x

(h3p

∂p

∂x

)+ ∂

∂y

(h3p

∂p

∂y

)= σ

d(hp)

dτ(6.6)

here σ = 12μωL2/Pah2o. They considered a small, sinusoidal

ngular oscillation of the plate around the balanced positionith a radial frequency of ω. The normalized amplitude of theibration is δ. Thus, the normalized thickness of the air filmwith τ = ωt) is

˜ = 1 + δx sin(τ + φ) (6.7)

Under the assumption of small displacement and harmonicesponse of the mirror, Pan et al. linearized Eq. (6.6) and, with therivial boundary conditions, found the analytical expressions ofamping torque for a rectangular torsion mirror both in Fouriereries and in double sine series. The damping torque in doubleine series is

d = −16Lw3pa

π4ho

{ϕ

∑odd m

∑even n

σ2

m2n2

1

[(mπ)2 + (nπ/η)2]2 + σ2

+ 1

ωϕ

∑odd m

∑even n

σ

m2n2

(mπ)2 + (nπ/η)2

[(mπ)2 + (nπ/η)2]2 + σ2

}(6.8)

here η ( w/L is the aspect ratio of the rectangular plate. Basedn Green function method, Darling et al. have obtained the sameesult [12].

Pan et al. found that the Fourier series solution and the doubleine solution yield almost identical final results. As the dampingorque in Fourier series is a more complicated than the one inouble sine series, it is not given here. Pan et al. also verified thenalytical results by numerical calculation based on the nonlin-ar isothermal Reynolds equation and by the experiments on theynamic behavior of micro-mirrors.

The damping torque given in Eq. (6.8) consists of two terms:he term in direct proportion to the angular displacement corre-ponds to the elastic damping and the term in direct proportiono the angular speed corresponds to the viscous damping.

Pan et al. found that the effect of elastic damping force on
he damping torque is small relative to the viscous dampingorce in the range of general interest and thus is negligible whenhe squeeze number is small. This conclusion is consistent withhe results given by Langlois [2] and Starr [6]. Pan et al. also

nd Actuators A 136 (2007) 3–27 21

ca

taslatab

f

c

F

c

T

tg

T

T[

T

wa0i

erT

c

wΔ

Δ

Bi

ea

Fr

t

c

Taa

tmtt

c

c

wrwpc

6t

tesi

huc

Pa∂x2 +

∂y2 +∂x

+∂y


oncluded that, with the elastic damping force neglected, thenalytical results were valid for non-harmonic response.

Since the moving directions of plate on the two sides ofhe axis are opposite, the squeeze film effects on the two sidesre canceled out somewhat. Therefore, the cut-off frequency ofqueeze film damping of a torsion micro-mirror is apparentlyarger than that of the parallel plate actuator with the same platerea. Therefore, it is easier for the torsion micro-mirror to meethe “incompressible gas” condition of the same plate dimensionsnd working frequencies. Under this condition, the analysis cane simplified significantly.

When the spring effect is neglected, the damping coefficientor small squeeze number is

d(η) = 192μLw5

π6ho3

∑odd m

∑even n

1

m2n2[m2η2 + n2](6.9)

or a strip torsion mirror, η is zero. Thus, we have

d(0) = 192μLw5

π6h3o

∑odd m

∑even n

1

m2n4∼= μLw5

60h3o

(6.10)

his result is consistent with Eq. (6.5).As Eq. (6.9) converges very fast, Pan et al. proposed to use

he first term of Eq. (6.9) as a simple approximation. Thus, theyave

d = −ϕ48

π6(η2 + 4)

μLw5

h3o

(6.11)

his formula is very similar to the formula given by Kurth et al.55]:

d = −ϕKrotμLw5

h3o

(6.12)

here Krot is 0.01764 for the case of η = w/L = 1. However,ccording to Eq. (6.11), the correspondent coefficient Krot is.010 for η = 1, whereas the exact Krot for η = 1 from Eq. (6.9)s 0.0116. The value provided in [55] is a little too large.

Hao et al. [56] also treated the nonlinear isothermal Reynoldsquation for small rotation motion of a rectangular torsion mir-or, but the equation was solved with Green’s function method.he coefficient of damping torque is

d(η) = 48μL4w2

h3o

∞∑m=1

∞∑n=1

Π2mnΔmn(η) (6.13)

here �mn is a complicated function of m, n, L and w [56] butmn has a familiar look to us:

mn(η) = (mπ)2 + (nπ/η)2

[(mπ)2 + (nπ/η)2]2 + σ2

(6.14)

ased on the analytical expressions, Hao et al. discussed thenfluence of design parameters.

Minikes et al. [57] treated the nonlinear isothermal Reynolds
quation for small rotation motion of a rectangular torsion mirrorlso with Green’s function technique. The coefficient of damping
ig. 20. The dependence of geometric correction factor β(a/b) on the aspectatio a/b.

orque found takes the form:

(η) = 192μLw5

π6h3o

×∑

n=1,2,...

∑m=1,2,...

1

(2n)2(2m − 1)2

1

(2n)2 + (2m − 1)2η2

(6.15)

hough the summation looks quite different with that of Pan etl., it has been checked for η = 0 and η = 1 that Minikes’ resultgrees very well with that of Pan et al.

The common problem of the above-mentioned papers is thathe damping effect is only considered for small angular displace-

ent around the balanced position (ϕ = 0). Another problem ofhese works is that the results are given by complicated equa-ions; the application of the results is difficult.

For the ease of applications, the coefficient of damping torquean be written as

(η) ≡ c(0)β(a

b

)(6.16)

here β(a/b) is a correction factor for the aspect ratio of theectangular plate. β(a/b) is shown by the solid line in Fig. 20,here the correction factor for squeeze film air damping of aarallel actuator is also shown in the figure by dash line foromparison.

.3. Damping of a rectangular torsion mirror at a finiteilting angle

The squeeze film damping of a rectangular mirror at a finiteilting angle is analyzed by Bao et al. [58] based on the nonlin-ar Reynolds equation, Eq. (2.3), for a finite tilting angle. Forimplicity, only the dominating component, the viscous force,s considered.

For a finite tilting angle, the non-uniformity of gap distance,= ho − ϕx, has to be considered. As the damping pressure p issually much smaller than the ambient pressure Pa, Eq. (2.3)an be developed to(

∂2p ∂2p) [(

∂p)2 (

∂p)2

]

= 12μ

h2

∂p

∂t− 12μ

h3 Pa∂ϕ

∂tx + 3P

h

∂p

∂xϕ (6.17)

22 M. Bao, H. Yang / Sensors and Ac

4φ(

tt

c

Bbtrc

γ

T[prta

6

rqts

tppstp2ti

Q

wt

tftd

7

tg

Fig. 21. The dependence of angular correction factor γ(φ) on φ.

It has been found that, under the condition of φ �μa2φ/h2pmax (where φ is the normalized tilting angle,= ϕa/h), Eq. (6.17) is simplified to

∂2p

∂x2 + ∂2p

∂y2

)= − 12μ

(ho − ϕx)3 ϕx (6.18)

Therefore, the air damping of a rectangular mirror is relatedo the aspect ratio a/b as well as the tilting angle. As a final result,he coefficient of damping torque can be given as follows:

ϕ

(a

b, φ

)≡ cϕ(0, 0)γ(φ)β

(a

b, φ

)(6.19)

ased on the coefficient of damping torque for strip plate atalanced position, cϕ(0,0), γ(φ) gives the correction for finiteilting angle and β(a/b,φ) gives the correction for the finite aspectatio of the plate. The expression for γ(φ) has been found in alose form:

(φ) = 45

4φ3

{4

φ+ 2φ

3(1 − φ2)+

(1

3− 2

φ2

)ln

(1 + φ

1 − φ

)}

(6.20)

he expression for β(a/b,φ) is too complicated to be given here58]. As the expressions for γ(φ) and β(a/b,φ) are rather com-
licated, they are shown here by the curves in Figs. 21 and 22,espectively. Further discussion based on the above results showshat, to satisfy the condition for Eq. (6.18), the normalized tiltingngle φ should not exceed 0.7.
Fig. 22. The dependence of correction factor β(a/b,φ) on a/b and φ.

bioct

tS[bpTp

tsTbller

tuators A 136 (2007) 3–27

.4. Damping of torsion mirror in rarefied air

To reduce the squeeze film air damping effect, torsion mir-ors are sometime encapsulated in vacuum. Therefore, theuality factor of torsion mirrors in rarefied air is an impor-ant parameter of concern and needs to be evaluated at designtage.

Minikes et al. [57] utilized Bao’s energy transfer model [36]o the case of torsion mirrors for low vacuum. They used a sim-lified model, where the movements of the two portions of thelate on the two sides of the axis are considered as in oppo-ite normal directions. With this simplified model, they adaptedhe energy transfer model for a torsion mirror by modifying theeripheral length appropriately. The peripheral of a single plate,L + 2w, is now replaced by the sum of the peripherals of thewo portions, S = 4L + 2w. Thus, the quality factor of the systems

E,Sq = 2πEP

�Ecycle= (2π)3/2ρHω

(do

2w + 4L

) √RT

Mm

1

p

(6.21)

here L and w are length and width of the mirror plates, respec-ively.

Minikes et al. [57] compared the model with experimen-al measurement with two dedicated mirror devices. Theyound that the agreement of the theory with the experimen-al data is general good, but deteriorates as the gap distanceecreases.

. Simulation of squeeze film air damping

The analytical research has been reviewed in above sec-ions. As the analytical solutions are inherently limited to simpleeometries, the real MEMS structures are usually too complex toe treated with analytical model and, thus, numerical simulations often used to obtain a more accurate result at the final stagef device design. The simulation of the squeeze film dampingan be generally divided into two levels: the physics level andhe system level.

The physics level methods treat the damping on the con-inuous field by solving the Reynolds equation or the Navier–tokes equations with the finite element method (FEM)6,20,29,59–61]. Though, in principle very accurate results cane obtained, it might be prohibitive in many cases due to the com-lex geometries and the couplings to the other energy domains.he accuracy and the complexity of the model have to be com-romised.

The system level methods treat the damping effect withhe reduced order models, which are easily integrated into theystem level models of the whole electro mechanical devices.hough in the early stages the system models were considered toe less physics transparency and not fit for the predictive simu-
ation, the differences between the system level and the physicsevel methods become vague. Several equivalent circuit mod-ls treat the damping with Reynolds equation and give accurateesults with very little computation time [61–66].

nd Actuators A 136 (2007) 3–27 23

7

gdc

ctt

a

λ

rm−htTsp

tostaath

abtwtSsbpge

toascttig

tlA

soo0f

c

Ty

tsteaai

sAib[

mcsodpaAoRb[

Il


.1. Physics level simulation

It was only in recent years that Reynolds equation was inte-rated in the commercial simulation tools. Several methods wereeveloped to simulate the squeeze film air damping before theommercial simulation tools were available.

It was proposed by Starr that the squeeze film damping effectan be simulated with the thermal analog with the small ampli-ude assumption [6]. The linearized Reynolds equation and thehermal equation are [29]:

∂2p

∂x2 + ∂2p

∂y2 = 12μ

h30

dh

dt+ 12μ

h20p0

∂p

∂t(7.1)

nd

∂2T

∂x2 + λ∂2T

∂y2 = −Q + ρcp

∂T

∂t(7.2)

espectively. By replacing the temperature T with p, the ther-al conductivity λ with h3/12μ, the heat source density Q withdh/dt and letting the product of the density and the specific

eat ρcp be unity, both the viscous damping force and the elas-ic damping force are simulated with the thermal analysis [29].he viscous damping force results from the real part of the pres-ure integral while the elastic damping force from the imaginaryart.

When the tilting angle is small, the linearized Reynolds equa-ion, Eq. (7.1), is valid for a torsion structure. The dampingf the torsion structures can be simulated by varying the heatource density Q = −dh/dt on the plate surface according tohe speed distribution of the torsion plate [29]. The thermalnalogy takes much less computing time than the fluid flownalysis. It is possible to simulate the complex geometries andhe perforated structures without considering the effects of theoles.

The effect of the high Knudsen number in the gap is usu-lly treated with the effective coefficient of viscosity. The slipoundary conditions are included with the surface accommoda-ion coefficient. The amplitude effects are not able to be obtainedith the thermal analogy because the linearized Reynolds equa-

ion, Eq. (7.1), is valid for small amplitude, as discussed inection 2. As Reynolds equation is valid only when the pres-ure drop across the gap is negligible, the fringe effects causedy the edges are not included in the model and the widths oflates must be much larger than the gaps. However, the elon-ation models [16,20] may be employed to include the fringeffects caused by the edges.

The squeeze film air damping is able to be simulated withhe commercial simulation tools for fluid flow, which are basedn the Navier–Stokes equations [29,67,68]. When the vibrationmplitude is small, the damping of the parallel plates can beimulated with the channel flow model shown in Fig. 23. Thehannel has the same size as the gap between the moving and
he fixed plate. According to the non-slip boundary condition,he gas near the moving plate moves with the plate. Therefore,t is reasonable to model the moving plate as an inlet with theas flow vnormal = h and vparallel = 0, where h is the speed of
bshH

Fig. 23. Channel flow model of the parallel plates.

he moving plate. The squeeze film air damping can be simu-ated with fluid simulation programs such as CFD-ACE [67] andNSYS/FLOTRAN [59].The ANSYS/FLOTRAN model of the structure in Fig. 23 is

hown in Fig. 24(a), where only the gap is shown. The velocitiesn the moving plate are set as Vx = 0, Vy = h, while the velocitiesn the fixed plate are zero. The pressures at the edges are set to be(trivial boundary conditions). Then, the coefficient of damping

orce is obtained by [29]:

= Re

(∫p dA

h

)(7.3)

he elastic damping force can be simulated by a transient anal-sis for compressible gas [29].

The border effect at the edges can be considered by applyinghe ambient pressure in a distance from the moving plate, ashown in Fig. 24(b). The perforated structures are simulated withhe similar methods [28,32]. As the assumptions for Reynoldsquation, such as the small gap, uniform pressure distributioncross the gap, negligible fluid velocity normal to plate surface,re relaxed, the additional damping effects caused by the flown the edges and perforations are included.

The squeeze film air damping of torsion structures can beimulated by varying the flow rate on the moving plate surface.s Navier–Stokes equations are used, the damping at large tilt-

ng angles can be simulated. Large amplitude effects can alsoe simulated with the tools based on Navier–Stokes equations68].

Though the models based on Navier–Stokes equations areore accurate than the models based on Reynolds equation, a

omplete model of real structure is usually too complex to beolved. For example, it takes hours to simulate the damping forcef a pair of perforated parallel plates with trivial boundary con-itions with the incompressible gas option, in which the movinglate is 81 �m × 81 �m × 4.3 �m with a 15 �m × 15 �m rect-ngular hole in the center and the gap between the plates is 2 �m.s a comparison, the viscous damping force and the elastic forcef the same structure can be obtained within a few seconds witheynolds equation based models. The Navier–Stokes equationsased methods are better fit for studying the damping effects16,20] than for design purpose.

Reynolds equation has been included in commercial tools.n ANSYS [59], squeeze film damping is simulated with theinearized Reynolds equation, Eq. (7.1). Due to the similarity
etween Eqs. (7.1) and (7.2), the method in ANSYS has theimilar properties as the thermal analog method, except that theole effects of perforated structures have been included withagen–Poiseuille equation [59]. As Hagen–Poiseuille equation


RAN

mbKai[aet

RSata

7

rims

iaatd

ootVsocs

cfba

t

efr

wha[

q

q

Trlfs

R

I

wva

p

The damping force is obtained by

F =∑

i

piwL

2n + 1= 2n(n + 1)(4n − 1)

(2n + 1)3

μw3L

h3 h (7.10)

Fig. 24. Models of the parallel plates used in ANSYS/FLOT

odels the flow of incompressible gas, the spring effect causedy the gas flow in holes is neglected. The effects of the highnudsen number and the slip boundary condition are included

s real constants of the elements. The end effect of short holess treated with the method presented by Sharipov and Seleznev19]. The torsion structures can be simulated when the gaps arepproximately uniform. The amplitude effects are not consid-red because the linearized Reynolds equation is derived withhe small amplitude assumption.

In Coventerware, a hybrid approach called Navier–Stokes–eynolds (NSR) is developed [60,61], which uses the Navier–tokes equations to obtain the flow resistances of the criticalreas (such as holes and edges) and inputs the flow resistances ofhe border effects into Reynolds equation to obtain the dampingnd the elastic forces.

.2. System level simulation

System level methods treat the damping effects with theeduced order models. Equivalent circuit models are the dom-nant methods as they are easily integrated into the hierarchy

odels of the whole electro mechanical systems, which areolved with commercial EDA tools.

In the simplest models, the coefficients of the viscous damp-ng force and the elastic damping force are extracted by thenalytical models or the FEM tools and modeled with equiv-lent components. The method is less physics transparent andhe coupling between the fluid domain and the other physicsomains is difficult to simulate.

The equivalent circuits are also used to model the solutionsf the analytical models [17,18,20,21,27,46]. Parallel branchesf R–L components are used to model the series solutions ofhe analytical models. With the compact models developed byeijola et al., the viscous and the elastic effects of quite a fewtructures can be integrated into the system level models. Tobtain the accuracies and the flexibilities beyond the analyti-al models, researchers use the similar circuits to model theolutions of the FEM simulations [16,68].

Reynolds equation itself can be modeled with equivalent cir-uits as the balance equations of the gas flow have the similarorms as the circuit equations [62–66,13]. Accurate results can
e obtained efficiently and the coupling between damping effectsnd the effects in the other physics domains can be simulated.
A simple example is discussed to illustrate the principles ofhe equivalent circuit model of Reynolds equation. Reynolds

Fd

: (a) for trivial boundary conditions; (b) with border effects.

quation is derived with the balances of the mass flows and theorces. For incompressible gas, the balance of the mass flowequires:

∂qx

∂x+ ∂qy

∂x+ ∂h

∂t= 0 (7.4)

hich is equivalent to the conservation of the currents (Kirch-off’s current law) if the mass flows are equivalent to the currentsnd ∂h/∂t is equivalent to a current source. The mass flows are10,11]:

x = − h3

12μ

(∂P

∂x

)(7.5)

y = − h3

12μ

(∂P

∂y

)(7.6)

he above equations are equivalent to the I–V relations of theesistors dRx = (12μ/h3) dx and dRy = (12μ/h3) dy if P is equiva-ent to the voltage. A one-dimensional equivalent circuit modelor a strip plate is obtained by discretizing Eqs. (7.4)–(7.6), ashown in Fig. 25 [13]:

i = R = 12μw

(2n + 1)Lh3 (7.7)

i = I = wLh

2n + 1(7.8)

here w is the width and L is the length of the strip plate. Theoltage at each node is equal to the pressure, which can be solvednalytically in the case of Fig. 25:

i = Vi =n∑

m=n−i+1

mIR (7.9)

ig. 25. One-dimensional equivalent circuit model for the incompressible gasamping.

nd Ac

W

F

w

awuecademse

Mtaf

7

mTwm

dmattudc

A

sgF

R

[

[

[

[[

[


hen n → ∞, the damping force is

= μw3L

h3 h (7.11)

hich is equal to the analytical solution of strip plates.As Reynolds equation is modeled, the method is accurate

nd flexible. Complex geometries can be meshed and modeledith a corresponding net-list. Perforated structures are sim-lated by including the equivalent resistors representing theffects of holes. The effects of edges can also be included byonnecting additional equivalent resistors in series at appropri-te positions. Both the viscous damping effect and the elasticamping effect are simulated accurately and efficiently with thequivalent model [64,65]. Recently, Schrag et al. extend theirodel to torsion structures and set up a reduced order model to

imulate the damping for large scale, highly perforated structuresfficiently and yet accurately enough [66].

In the new releases of Coverterware, the Finite Elementethod is employed to discretize the Reynolds equation and

he model order reduction technique [69] is employed to gener-te system level models with very small number of degrees ofreedom [61,70], which are fast, yet accurate enough.

.3. Squeeze film damping in the free molecular region

All of the simulation tools above are based on the continuumodels, which are valid when the Knudsen number is small [53].he effective viscosity method extends the validity. However,hen the Knudsen number is larger than 10, the free molecularodel is more reasonable.Hutcherson et al. developed a one-dimensional molecular

ynamics simulation code [53] based on the energy transferodel proposed by Bao et al. [36]. However, some of the

ssumptions employed in Bao’s energy transfer model (Say,he constant particle velocity, the constant beam position and
he constant change in particle velocity) are relaxed. The sim-lation result showed better agreement with the experimentalata published by Zook et al. [49] than the results based on theontinuum models with effective viscosity, as shown in Fig. 26.
Fig. 26. Quality factor comparison of different results.

[

[

[

[

[

[

[

[

[

tuators A 136 (2007) 3–27 25

cknowledgements

The authors’ research on squeeze film air damping has beenupported by the Key Basic Research and Development Pro-ram of China (No. G1999033101) and by the National Scienceoundation of China (No. 69876009).

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HFudan University, Shanghai, China, in 1995 and 2001, respectively. He worked


iographies

inhang Bao graduated from Fudan University, Shanghai, China and joinedhe faculty of the Physics Department in 1961. He finished his graduate study
n solid-state physics in 1966. Since 1983 he has been with the Electronicngineering Department of Fudan University, where he is currently a professor.e is on the editorial boards of several International and Chinese Journals on
ensors and actuators. He is the author of “Micro Mechanical Transducers”Elsevier, 2000), “Analysis and Design Principles of MEMS Devices” (Elsevier,

awCs

tuators A 136 (2007) 3–27 27

005), and a number of technical papers. His current research interest is onicro-mechanical transducers and technologies.

eng Yang received the BS and PhD degrees in electronic engineering from

s a post doctor in Delft University of Technology from 2001 to 2003. He is nowith Shanghai Institute of Microsystem and Information Technology (SIMIT),hinese Academy of Sciences. His research interests include mechanical sen-

ors, simulation and modeling, and micro-/nano-fabrication technologies.

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