- Email: [email protected]

SenSOTS

PHYSiCAL

and Actuators A 70 (1998) 3241

Compact analytical modeling of squeeze film damping with arbitrary venting conditions using a Green’s function approach Robert B. Darfing *, Chris Hivick, Jianyang Xu University

of Washington,

Department

ofElecrrical

Ettgineering,

P.O. Box 352500,

Seattle, WA, 981952500,

USA

Abstract An analytical method based upon a Green’s function solution to the linearized Reynolds equation is presented which allows the resulting forces from compressible squeeze film damping to be rapidly calculated for arbitrary acoustic venting conditions along the edges of amoveable structure. The resulting models are computationally compact, and thus applicable for dynamic system simulation purposes. Arbitrary deflection profiles can also be treated, enabling the calculation of damping effects for cantilevers, diaphragms, tilting plates, and drum-head modes. The 0 1998 Elsevier Science method is theot-eticaily described and a catalog of several useful cases is then presented to illustrate its use. S.A. All righls reserved. Keywords: Compact models; Squeeze films; Damping; Green’s funcrions

1. Introduction Compressiblesqueezefilm damping has been a problem of great importance for microstructuresthat involve a proof massthat moves againsta trapped air film, asthis mechanism can dominate the damping and thus substantially affect the system frequency response.Existing modelsof squeezefilm damping solve a linearized Reynolds lubrication equation over a two-dimensional domain that representsthe compressedvolume [ 1,2]. All previous models have used the simplified boundary condition that the acousticpressurevanishesat the edgesof the structure, which is equivalent to a zero acoustic impedance[ 3-51. Analytic modelshave been reported by Andrews et al. [6], Veijola and Kuisma [7], Veijola et al. [ 7-91, and Bourgeois et al. [ 101, for mostly rectangular

microstructures.

Finite

difference

models

have

been developed by Veijola et al. [ 111, and finite element modelsby Yang and Senturia [ 121, and Yang et al. [ 131 to treat more complex structural shapes. Damping effects at low gaspressureshave alsobeenmodeledby Veijola and Kuisma [ 141. In practice, the suspensionparts of the moveablestructure impedethe air flow from the compressedvolume, creating a nonzero acousticimpedanceas the boundary condition. For example, short tethersto a proof mass can provide nearly free flow conditions which closely approximate the zero pressure * Corresponding author. E-mail: bdarling~ee.washington.edu 0924~4247/98/$ - xee front Pi~SO924-4737~98)QO109-5

(a) 09 cc> Fig. 1. Bulk micromachined structures illustrating a moveable proof mass with a variety of venting conditions. (a) Cantilevered suspension provides no venting along one edge. (b) Venting through a slotted passageway. (c) Venting into a closed, sun-ounding cavity.

boundary condition; however, suspensionby a continuous membraneproduces the oppositecase of zero flow on the boundaries. Any situation or combination in between the extremesof zero pressureor zero flow can be realized. If the venting involves channels,cavities, or other restrictive passageways, then the venting boundary condition becomesa complex acoustical impedance.Fig. 1 illustrates somecommon bulk micromachined situationsin which these effects occur.

2. Problem formlrlation The motion of a trapped gasfilm betweentwo moveable plates of spacingh(x,y,r) will have little inertial effect if the

mauer 0 1998 Elsevier Science S.A. AI1 rights reserved.

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plates have only small curvature and will be characterized by a nearly unidirectional, low Reynolds number flow. If the motion of the plates is only towards each other, Couette flow effects will be absent and the local pressure within the gap p(x,y,t) will be governed by Reynolds lubrication equation

The proper Green’s function can be constructed as an expansion of eigenfunctions over the domain of the compressed volume. Assuming that the pressure variation is approximately constant in the z direction (direction of compression), the eigenfunctions u,,, and eigenvalues k,,, are solutions to a two-dimensional scalar Helmholtz equation

where p is the viscosity. The equation of state for the gas film is taken to be p/p7 = constant, where p is the density. For an isothermal process 7 = 1, while for an adiabatic process q = K, where K = cP/c, is the ratio of the specific heats. If the plates have high thermal conductivity, then the isothermal limit is usually a valid approximation, whereas thermally nonconductive plates would be better described by the adiabatic limit [ 1,2] . For small variations in local pressure and gap spacing, p = P, + Sp,h = h, + Sh, the Reynolds lubrication equation can be linearized into the form

With nondegenerate eigenvalues, the eigenfunctions form a complete orthonormal set of expansion functions,

(2)

u,,*(r)4,tn, i “0

(r)dr=S,,,Snnr

(8) (9)

The Green’s function is constructed from these expansion functions with time-varying coefficients, G(r,tlr,,t,)=

$O(t-to)

where P = SplP, is the normalized local pressure variation, P, is the ambient pressure, H = S~Z//Z,is the normalized local

(10)

gap variation, and h, is the nominal gap spacing. Finally, a? = 12pl~$1,~P, is a constant. This equation now has the form of a linear diffusion equation with a source term of c&qaHli3t, which can be solved analytically by a variety of methods. To treat the widest possible range of boundary conditions and source terms, a Green’s function method is adopted.

where O(f) is the unit step function [ 151. The solution process in general consists of finding the proper eigenfunctions and eigenvalues for the domain of the moveable plate, constructing the Green’s function as per Eq. (lo), and then calculating the pressure profile via Eq. (6). Integrating the pressure over the area of the plate will then produce the net reaction force.

3. Green’s function solution 4. Normal motion of a rectangular

plate

Diffusion from a point source excitation is described by VQ-ol$

4.1. All edges vented

=-4d(r-r,)6(t-to)

(3)

where G(r,tlr,,f,) is the Green’s function which represents the response at an observation point (r,t) caused by an excitation at the source point (r,,&,) [ 151. All solutions to Eq. (3) satisfy a reciprocity relation G(r,tlr,,t,)=G(r,,-t,Ir,-r)

(4)

in that time reversal has the same effect as swapping the observation and source points. For an arbitrary source term, 3P

aH

To illustrate the details of the above method, the often cited result for normal motion of an ideally vented rectangular plate will be obtained using the Green’s function approach. The domain of the compressed volume is taken to be -a/2

(5)

(11)

the solution is expressible as an integral of the Green’s function over the source points,

for m,n = { 1,2,3,. . . }, where cos is used for odd indices and sin is used for even indices. The corresponding eigenvalues are

v,2P-c?$

P(r,t)=

0

=-4?Tp(ro,t,)=cY%/ato

G(r,tlr, ,t,Mr, 0 “0

,to)drodto

(6)

k;,=k;+k;=

m25r2 n2r2 a2 + b2

(12)

R.B. Darling

34

et al. /Sensors

The plate is taken to be a rigid body with sinusoidal motion perpendicular to its face. The source term in Eq. (6) is then (13) where H’ is a constant giving the normalized amplitude of the plate vibration. Performing the time integration in Eq. (6) yields

aizd Actuators

A 70 (1998132-41

relative to the x dimension of the plate, which can be viewed as a normalized frequency variable. Letting the plate aspect ratio be given by /3 = a/b allows the normalized force to be expressed as F(t) -=

-h-r,

abP,

j.jeM

jcr+m2rr2+~‘n’~2 (21)

The spring and damping components of the reaction force can be separated as the real and imaginary parts of the above, (14) e j”‘-exp(-ki,,th2)

=

jw-l-k~,h2

where the first term in the numerator is the sinusoidal steadystate term, which is retained, and the second term is the tumon transient, which is discarded for the present purposes of finding the frequency response. The integration over x,, in Eq. (6) gives, for m = odd,

F(t) t-1abP, spring 64

=Re

-h

c- 9i-4m2n2 ju+m2~2+/3’n2n-2 n=odd

1 H’

(22)

+a/2

23)‘“‘~‘“2

cos~,jxo=

(15)

I -a/2

64

=Im

cn=oddr4m2n2

-&-7 ja+m2~2+~2n2~2

and for m = even, +a/2

(16)

1 H’

(23)

These results agree with those of other workers for this benchmark case [ 3,461. 4.2. One edge closed

The integration over y0 proceeds identically. Collecting the above results into Eq. (6) yields the normalized pressure distribution of P(X>YJ)

Next, consider the same plate geometry as before, but with the edge at X= - al2 now closed to produce a zero flow condition, expressed by aP/ax = 0 along this one boundary. The eigenfunctions are now (24)

(17) Integrating this pressure profile over the area of the plate produces the net resultant force, +a/2

F(t)=P,

i

-a/2

-kbi2

i

P(x,y,f)dx

dy

(18)

k;,=k;+k+

+ 4a’

-b/2

Normalizing this force to the ambient pressure acting on the plate area yields

F(t) c 64 abP, m n=oddrr”m2n2

-=

where m = { 1,3,5,. . . } and n = { 1,2,3 ,... }, and in the second trigonometric factor cos is used for n = odd and sin is used for IZ= even. The corresponding eigenvalues are

16(-1)(“+“/2)

(19)

P(x,y,t> =

c m,n=odd

The negative sign in the numerator accounts for the reaction force being opposed to the direction of plate motion. Squeeze film damping is often expressed in terms of a dimensionless squeeze number (20)

(25)

Evaluating Eq. (6) using these produces a normalized pressure distribution of

-jwvHkj”’

jwi-k~,,lor2

n2r2 + b2

r2mn

jw, ,Hkj”’

jw+k&,h2 (267

The normalized reaction force is F(l) -= abP,

64 -jwqH’e’“’ cmn=oddr4m2n2 jwi-kk,,h’

(27)

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differing from the fully vented case only in the eigenvalues

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4.4. Three edges closed

kmn’

For boundary conditions of aPl&= 0 along x = - al 2 and xal2, aPlay = 0 along y = -b/2, and P= 0 along ybl2, the

4.3. Two edges closed

normalized pressure distribution becomes If two edges on opposite sides of the plate are closed, expressed as aPl&x=O on x=-a/2 and x= fal2, the eigenfunctions become

4(-1)(“-1)/2 fY-GY,t)=

c n=odd

-jwrlH~jwT

rrn

jw+kilcx2 (36)

where k,’ = n2$/4b2 and the normalized reaction force is for nz={0,1,2,3 ,... } and n={1,2,3 ,... }; cos is used for in= even and 11= odd; sin is used for nr = odd and n = even; and when III = 0 the normalization factor becomes J2/ab, corresponding to an eigenfunction which is constant with x. The associated eigenvalues are k2 =k2+k2=- rn2f12 n27T2 ~ mn m n a2 + b2

(29)

Evaluating the pressure integral of Eq. (6) retains only the m = 0 term in the x integration, yielding Rx,YJ)=

c

4(-l)(n-‘)‘2

-jwTH’ej*‘cosnay jwfk~lo12 b

rrn

n=odd

(30)

F(f) -=

abP,

This corresponds to effectively a one-dimensional problem with the pressure varying only in the y direction. If two adjacent edges of the plate are closed, expressed as aPI&= on X= --n/2 and aPlay=O on y= -b/2, with ideal venting P = 0 along the remaining two edges of x = + al2 and y= + b/2, the eigenfunctions are xs 2‘) cosz.,( y+ 2b,

(32)

for m,n = { 1,3,5,. . . } and eigenvalues of k&,=k;+k,T=

(37)

abP, 4.5. All edgesclosed

If VP = 0 is enforced along all edges of the plate, then the only nonzero eigenfunction will be a constant with x and y which produces a constant normalized pressure distribution of P(x,y,t) = - qH’&“’ and a normalized reaction force of F(t) /abP, = - qH’@“. The reaction force is completely in phase with the displacement for all frequencies, giving a pure spring force with no viscous damping losses. 4.6. Comparisonof cases

The normalized reaction force is then

zL,,(x,y)= p ZCOSG,,(

F(t) -=

in27r2 n297-’ 4a2 + 4b2

(33)

For a square plate (a = b) , the reaction forces from the above five fundamental venting cases are compared in Figs. 2 and 3 for a normalized plate oscillation amplitude of H’=O.lO, i.e., one-tenth of the gap. Isothermal conditions are assumed for which 77= 1. Real and imaginary components of each reaction force are separated to yield the spring and damping components, respectively, each plotted as function of the squeeze number CTdefined by Eq. (20). The overall trend is for the spring component to increase for all frequencies and the damping component to increase for low frequencies and decrease for high frequencies asthe venting becomes increasingly restricted. The critical frequency, defined as where the spring and damping components have equal ampli0.10 n

nC

U.UJ n no

I i

i

_/c--

__---.-

____-----

The resulting normalized pressure distribution is

m&t)= c

16(-1)(‘“+“)12

m,n=odd

xcos$x+

7r2rnn

jwTHkh’r

jw+ki,,lcx2

;)cos~(y+

;)

(34)

with a normalized reaction force of

F(t) c 64 mn=oddr4m2n2 jw+ki,lcx2 -jwrlHbh”

-= abP,

(35)

Squeeze

Number

G

Fig. 2. Springforce for normal motionof a square plate.

-4

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et al. /Sensors

and Actuators

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and normalizing this to the ambient pressure acting on the area of the plate yields

L -

“’-I-‘-‘. ”

3edgesven1ei 2 opposite edges venfed 2 adjacent edges vented 1 edge vented

2 0.07 2 ‘~0.06 E ,$ 0.05 -

---

(43)

This result agrees with that of Griffin et al. [3], who derived it using Laplace transforms. 6. Arbitrary

Squeeze

Fig. 3. Damping

Number

o

force for normal motion

of a square plate.

tude, also shifts to lower frequencies with increasingly more restrictive venting.

deflection profiles

The previous cases were easily integrated since the plate motion was normal to its surface, making the source term in the Green’s function integral of Eq. (6) constant over the area of the plate. Since this integral explicitly allows the source term to be an arbitrary function of position, arbitrary deflection profiles can be treated using the same methods. Two useful examples of this are discussed next. 6.1. Tilting plates

5. Normal motion of a circular plate Solutions to the scalar Helmholtz equation in cylindrical coordinates are Bessel functions which are applicable to constructing the Green’s function for circular plates. For a circular plate of radius c which is ideally vented along its edge, P = 0 for r = c, the normalized eigenfunctions are (38) form={1,2,3

,... }andn={1,2,3

,... },and (39)

for m = 0 and n = { 1,2,3,. , , }. The eigenvalues are k,, = y,J c, where ymnis the nth root of J,,( ymn) = 0, and J,(y) is the mth order Bessel function of the first kind. For uniform motion perpendicular to the surface of a rigid circular plate, there is no 0 variation in either the boundary conditions or the source term, so only the m = 0 eigenfunctions are retained. Using these eigenfunctions to evaluate the Green’s function and then the integral of Eq. (6) leads to a normalized pressure distribution of P(r,&t)=C-

2

-jwqHlejw’

~ ckOn jw+k&loL2

J,(k,,r) J,(ko,c)

(40)

where use has been made of the relation c

Jo(ko,r,)r,d~o= ~Jdkd i0 O??

P(r,t),t)d&

dr

H’ $ej-’

n=odd

m7i.x a

Xsin -cos-

nw

(45)

b

form={2,4,6 ,... }andn={1,3,5 ,... }. The restoring torque on the plate can be computed as 7(t)=Pa

(42)

(44)

for sinusoidal rocking about the y axis. The maximum angle of rotation is I!&, = tan-‘2&P/a, where It, is the nominal gap spacing, and the reaction force now takes the form of a restoring torque T. The plate is taken to be fully vented along each edge, so that the eigenfunctions and eigenvalues are given by Eqs. ( 11) and ( 12). The integration of the Green’s function in Eq. (6) proceeds as before, except for an additional factor of 2x/a. The resulting normalized pressure distribution is

(41)

c 2Tr

II 0 0

H(x,y,t)=

fal:!

Integrating the normalized pressure over the area of the plate yields the net resulting force, F(t)=P,

The compressed gap area is again taken to be -n/2

I

-a/2

+b/2

I

xf’W,Odx

dy

(46)

-b/2

Normalized to the ambient pressure acting on the plate area and the moment associated with rotation about they axis, the normalized restoring torque is found to be -= 7(t) a 2bP,/2

64 c- r4m2n2 jw+k$,/cx2 ??L=tTe* -jo$fkj”’

ri=odd

(47)

R.B. Darling

et al. / Senrors

Interestingly, when the torque is normalized as above, it takes on an identical form as the force for normal motion of the plate, except that the index m runs over even instead of odd values. Another interesting case of similar nature is a square plate that tilts about its diagonal. Such a structure is used in the Texas Instruments deformable micromirror display device, as an example. In this case the normalized displacement of the plate rocking sinusoidally about the axis y = --x can be expressed as

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parabolic deflection profile. The compressed air film gap is taken to occupy the domain 0

where H’ is the maximum z displacement which occurs at the tips of (a/2,a/2) and ( -n/2, -a/2). The maximum angle of rotation is 0,,, = tan -‘fih,H’la. For ideal venting along all edges, the eigenfunctions and eigenvalues are the same as those used previously, and the Green’s function integral for the pressure now involves a weighting factor of (X +y) ln, which results in a normalized pressure distribution of

m,y,o= c

8(_l),n/2-i(_l)(n-1)/2

Xsin-

n

cosy

+

c

8(-l)

m=odd

(VI-

1)/2(-p-

T’rnn

mrx sinz a b

m27T2 n27T2 -

~

4a2 +

fYx,y,f)=

(49)

16(-l)

dmn m7Tx nry cos2n b

cos -

7(f)=P, dy J J-$+y,Odx

(55)

fbl2

X+-Y

-a12

(50)

The net reaction force, referenced to the tip of the cantilever, is calculated as

-b/2

a +b/2

which evaluates to give a normalized restoring torque of fir(t) -= a3P,

b2

bz-1)/2(4)(n-1)/2

c m=odd n=odd

The net restoring torque is i-a/Z

(53)

Performing the Green’s function integration of Eq. (6) yields a normalized pressure distribution of

1

n=eVell x -jw~H’d”‘cOs jw+k&Ja2

nv b

for m,n = { 1,3,5,, . . }, and the corresponding eigenvalues are k:,,=k;,+k,T=

n=odd N1rr.x

cos-

-jwqHkiWf

T2mn

ni= e-Jell

where H’ is the maximum deflection of the cantilever tip at x = n. The eigenfunctions which satisfy the boundary conditions for the domain are

c

32 r4m2n2 m=eve*

-jwTH’ej”’ jufkJ$,la2

F(t)=P,

The normalized torque for rotation about the diagonal y = f x of a square plate is thus half of the normalized torque for rotation about an orthogonal x = 0 or y = 0 axis.

;

P(x,y,t)dx dy

(56)

0 -b/2

(51)

n=odd

2

JJO X

which yields a force normalized to the ambient pressure over the cantilever area of

F(t) c 64 abP, ,rL=oddT4m2n2

-=

rz=odd

6.2. Cantilevers

Cantilevers are another common micromachined structural element whose dynamics can be strongly impacted by squeeze film compressive damping. For electrostatic actuation, where the attractive force is inversely proportional to the square of the gap distance, the pull-down force is concentrated toward the free tip of the cantilever, and a simple approximation is to assume that the pull-down force is completely tip-loaded. A force applied to only the tip of a cantilever of uniform cross-section will bend it downward in a

The real and imaginary parts, which are the spring and damping components of the reaction force, are shown in Fig. 4 for a square (a = b) cantilever and a rigid, square, free plate in normal motion whose amplitude is equal to that of the cantilever tip. Both the spring and damping components are smaller for the cantilever, and the critical frequency for the cantilever is slightly lower. This result is largely because only the tip of the cantilever undergoes the full range of motion, whereas all points on the free plate move through the full amplitude.

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p,LIS. If the slotted passageway is sufficiently narrow, the viscous drag of the sidewalls will introduce a real valued loss term into the acoustic impedance. The acoustic impedance of a narrow slotted passageway is given as -c

/’

I

/

I

-.-0

5

10

15

20

Squeeze Number Q Fig. 4. Spring and damping forces of an ideally vented a square cantilever with a parabolic deflection profile.

7. Arbitrary

25

30

square plate and of

acoustic venting conditions

As demonstrated above, the two extreme cases of zero pressure (ideal venting) and zero flow (ideal closure) boundary conditions can be straightforwardly handled by choosing the eigenfunctions to satisfy P = 0 or VP = 0 on the appropriate part of the boundary. The ideal venting condition corresponds to a zero acoustic impedance, while the idealclosure condition corresponds to an infinite acoustic impedance. The specification of an arbitrary acoustic impedance along each of the bounding edges allows more general and physically relevant conditions to be treated. Acoustic propagation through an ideal gas is governed by the differential equations

where p is the local pressure, q is the flow velocity, pa is Ihe ambient gas density, y is the compressibility of the gas, and P, is the ambient pressure. An aperture of cross-sectional area S with a mean flow velocity of q through it is described by a volume velocity of Q=q.i;S, where ri is the unit normal to the aperture in the direction of flow. The acoustic impedance of an aperture relates the volume velocity through the aperture to the applied acoustic pressure that causes it, Z,=p/Q. For sinusoidal steady-state, the acoustic differential equations allow the acoustic impedance to be expressed as z = --+wa A s

P vp.ri

The structural elements which restrict the air flow around a moveable plate can often be represented by simple acoustic impedances. A closed cavity of volume V would represent an acoustic impedance of ZA = 1/j&C,, where the acoustic compliance is C, = Vl yP,. A slotted passageway of length L which houses a moveable slug of air has an acoustic impedance of ZA =joM,, where the acoustic inertia is MA=

where fi is the air viscosity, and the cross-sectional dimensions of the slot are W X H. As is commonly done in electroacoustics, combinations of these acoustic compliance, inertiance, and viscous loss elements can be synthesized from an equivalent circuit model of the physical structure [ 51. Including general acoustic boundary conditions such as these is accomplished by constructing the eigenfunctions of the compressed domain to satisfy the condition of Eq. (60). As an example, consider again a rectangular plate trapping an air film over a domain of -a/2

(63)

where - tan is used for odd indices, + cot for even indices, and similarly for the y direction eigenvalues k,. Note that if the acoustic impedance ZA is purely reactive, i.e., lossless and imaginary, then the eigenvalues and hence eigenfunctions will be purely real-valued. Proper normalization of these eigenfunctions yields coefficients of (64) with the + sign for odd and the - sign for even indices. For the one case of km = 0, the normalization is A ,n= 1I&. These eigenfunctions can then be directly integrated to construct the local pressure distribution using Eq. (6). For the case of a constant acoustic impedance Z, along the entire plate boundary, only the odd-valued indices will be retained, yielding a normalized pressure profile of P(*‘y7t)=,n~dd

16 (k,,a)(k,,b)

-jwqHkj"'

jw+(kz+k,‘)/or2

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39

where H’ is the normalized displacement amplitude. Integrating over the area of the plate, as in Eq. ( 18)) gives a net reaction force of F(t) -= abP,

64

c (k,na)2(k,b)z m=odd

-jqHb

j”’

jo+(k:+k,2)/ck”

n=odd

(66) o,ooL-rYc,

:/

0

For the case of ideal venting around all four edges in which 2, = 0, the eigenvalues from Eq. (63) become k,,,a = mrand k,b = nr for odd values of m and II. This reduces the above results to the previously discussed case of Eqs. (17) and (19). If the acoustic impedance is a pure inertiance, the eigenvalues are solutions to

k,,,a k,na -jw,n -2tan2=-=2sz,

-paa 2SM,

(67)

In this case, the eigenvalues are frequency independent and need only to be computed once for a given device model. If the acoustic impedance is a pure compliance, the eigenvalues are solutions to k,,,a -m k a 2 tan2=

-=-.iw,a 2sz*

-t dp,llC, 2s

(68)

in which the eigenvalues are frequency dependent and must be recomputed for each frequency of interest. As an example of this result, consider a rigid, square plate with motion normal to its surface and with slotted vents along all four edges through which the trapped gas film is vented. The acoustic mass of each slot vent is MA = p,L/S, where S is the cross-sectional area of the slot and L is its length. The right hand side of Eq. (67) then becomes simply -a/2L. Fig. 5 compares the resulting spring and damping forces for two square plates, one ideally vented and the other vented by slot vents for which a/2L = 1 and H’ = 0.10. As can be seen, the magnitude of the spring force is more than doubled by the inertiance of the slot vents, and the critical frequency is also lowered by nearly a factor of three, illustrating the substantial effect that restrictive venting conditions can produce.

8. Synthesis of complex plates The treatment of vent holes which are placed within the surface of the moveable plate, to either aid in etch release or to reduce squeeze film damping, has traditionally been difficult for both analytical and numerical methods. Constructing eigenfunctions to satisfy these boundary conditions is a rather unwieldy task, making the Green’s function method also dif-

ficult to employ. However, with somesimplifying assump-

5

I,,

I,,

I,

10

15

20

Squeeze

Fig. 5. Effect of the inertiance

Number

I,.

L,,

25

I

30

cr

of slot vents on a square plate

tions on the gas flow patterns, squeeze film damping effects for this type of complex plate can be synthesized from the library of cases that have been discussed above. If a complex rectangular plate is partitioned into several smaller rectangular plates, then a simplifying assumption is that there is ideal closure (no venting) at the partition line between two adjacent plates. Each of the constituent plates can then have its net squeeze film damping reaction forces calculated based upon the venting along each edge, and the overall reaction force on the complex plate approximated as the sum of these. If the partition line happens to be a line of symmetry through the plate, then this synthesis procedure incurs no error. As an example, multiplying by four the net reaction force for a square plate vented along only two adjacent edges yields the net reaction force for a square plate of four times the area that is vented along all edges, as if the larger plate were composed of four comer quadrants. In practical cases, however, lack of needed symmetry will incur some error if the partitioning lines are not well chosen. As a final example, consider a rigid, square plate with motion normal to its surface and which traps an air film over the domain -a/2

(69) where Fza and Fzo are the net reaction forces for two adjacent and two opposite edges being vented, Eqs. (35) and (3 1) .

Sincethe componentpieceshave edgesof a/3, thefrequency

40

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Acknowledgements This work was sponsored by the NSF Center for the Design of Analog-Digital Integrated Circuits [CDADIC], The authors wish to thank Steve Lewis of AnaIog Devices and Ian Getreu of Analogy for helpful discussions and encouragement. Thanks are also extended to Timo Veijola of Helsinki University of Technology for providing comments and preprints of his work. 0

20

40

60

80

100

120

Squeeze Number IS Fig. 6. Effect of a square vent hole in the center of a square plate.

as represented by the squeeze number g is scaled by a factor of l/9. The spring and damping forces for this case are compared against those of an equally sized square plate without the center vent hole in Fig. 6. This verifies some common rules of thumb that are used for placing vent holes, namely that the ultimate magnitude of the forces is not substantially changed by the reduction in plate area, but that the frequency dependence is scaled to a value that represents the distance between the holes, rather than that of the smaller overall plate dimension. In Fig. 6, the magnitude of spring and damping forces is reduced only slightly for high frequencies, corresponding to a loss of l/9 of the plate area, while the critical frequency is increased by a factor of 9. 9. Conclusions A broadly applicable Green’s function method has been developed for rapidIy computing the effects of compressible squeeze film damping with realistic venting conditions typical of those encountered in micromachined structures. This method also allows arbitrary displacement or deflection profiles to be directly treated, as in the case of tilting modes and deformable cantilevers or diaphragms. A library of useful cases has also been presented to illustrate the analytical technique. While the resulting expressions for the pressure distribution and net force involve infinite sums over the eigenfunctions, these sums are rapidly convergent because each term is weighted by factors of 1/m2n’, with the index taking&her even or odd values. Retaining only the first five terms in each sum typically gives accuracy to within l-2%. With truncation of the sums to retain only the first few terms, Veijola and Ryhanen [ 71 and Veijola et al. [ 1 1,8] have shown that the net reaction force can be compactly represented by an equivalent circuit composed of parallel R-C elements, making these models directly useful for circuit and system simulations. The above library of cases and the described synthesis method for complex plates suggests that automated layout extraction and model synthesis could be developed from these methods.

References I11

W.E. Langh~is, Isothermal squeeze films, Q. Appl. Math. 20 (1962) 131-150. I21 C. Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics, Oxford Univ. Press, New York, 1997, pp. 243-253. H.H. Richardson, S. Yamanami, A study of fluid [31 WS, Griffin, squeeze-film damping, Trans. ASME: J. Basic Eng. 88 (1966) 451: 456. J. Lubrication Tech. 105 [41 JJ. Blech, On isothermal squeeze f&s, (1983) 615-620. New York, 1954, pp. 1% 151 L.L. Beranek, Acouslics, McGraw-Hill, 143. I61 M. Andrews, I. Harris, G. Turner, A comparison of squeeze-film theory with measurements on a microstructure, Sensors and Actuators A 36 (1993) 79-87. accel171 T. Veijola, T. Ryhanen, Mode1 of capacitive micromechanical erometer including effect of squeezed gas film, Proc. ISCAS ‘95, Seattle, WA, USA, Apr. 30-May 3, 1995, pp. 664-667. 181 T. Veijola, H. Kuisma, J. Lahdenpera, T. Ryhanen, Equivalent-circuit model of the squeezed gas film in a silicon accelerometer, Sensors and Actuators A 48 (l995) 239-248. 191 T. Veijoia, H. Ku&la, J. Lahdenpera, Model for gas film damping in a silicon accelerometer, Dig. Tech. Papers, 1997 Int. Conf. Solid-State Sensors and Actuators (Transducers ‘97), Chicago, LL, USA, Vol. 2, June 16-19, 1997, pp. 1097-1100. F. Porret, A. Hoogerwerf, Analytical modeling of [lOI C. Bourgeois, squeeze-film damping in accelerometers, Dig. Tech. Papers, 1997 Int. Conf. Solid-State Sensors and Actuators (Transducers ‘97), Chicago, IL, USA, June 16-19, 1997, pp. 1117-1120. IllI T. Veijola, T. Ryhanen, H. Kuisma, J. Lahdenpera, Circuit simulation model of gas damping in microstructures with nontrivial geometries, Proc. Transducers ‘95 andEurosensors lX, Stockholm, Sweden, June 2529,1995, pp. 36-39. 1121 Y.-J. Yang, S.D. Senturia, Numerical simulation of compressible squeezed-film damping, Tech. Dig. 1996 Solid-State Sensor and Actuator Workshop (SSSAW-96), Hilton Head Island, SC, USA, June 26, 1996, pp. 76-79. 1131 Y.-J. Yang, M.-A. Grerillat, SD. Senturia, Effect of air damping on the dynamics of nonuniform deformations of microstructures, Dig. Tech. Papers, 1997 Int. Conf. Solid-State Sensors and Actuators (Transducers ‘97), Chicago, lL, US.4, June 16-19, 1997, pp. 10931096. [ 141 T. Veijola, H. Kuisma, J. Lahdenpera, T. Ryhanen, Simulation model for micromechanical angular rate sensor, Sensors and Actuators A 60 (1997) 113-121. [I51 P.M. Morse, H. Feshbach. Methods of Theoretical Physics, McGrawHill Book, New York, Vol. 1, 1953, pp. 857-869.

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et al. /Sensors

Biographies Robert Bruce Darling was born in Johnson City, TN on March 1.5, 1958. He received the BSEE (with highest honors), MSEE, and PhD in Electrical Engineering from the Georgia Institute of Technology in 1980, 1982, and 1985, respectively. He has held Summer positions with SperryUnivac, Bristol, TN, Texas Instruments, Johnson City, TN, and from 1982 to 1983, he was with the Physical Sciences Division of the Georgia Tech Research Institute, Atlanta, GA. Since 1985, he has been with the Department of Electrical Engineering, University of Washington, Seattle, where he is presently an Associate Professor. From 1995 to 1996, he was a Visiting Associate Professor at Stanford University, Stanford, CA. His research interests include electron device

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physics, device modeling, microfabrication, circuit design, optoelectronics, sensors, electrochemistry, and instrumentation electronics. Chris Hivick-biography

not available.

Jinnynng Xzr was born in Beijing, China, in 1973. She received the BS degree in Physics from Tsinghua University, Beijing, China, in 1996. She is currently pursuing the MSEE degree at the University of Washington, Seattle, WA. Her research interests include analysis and simulation of analog/ digital integrated circuits and device modeling for circuit design.