ANALYTIC MODEL FOR PERFORATED SQUEEZED-FILM DAMPERS
Timo Veijola
Helsinki University of Technology, P.O. Box 3000, FIN-02015 HUT, Finland,
email: timo.veijola@hut.fl, tel. +358 9 451 2293, fax. +358 9 451 4818.
ABSTRACT
The concept of a perforation cell to derive simple an-
alytic models for perforated squeezed-fllm dampers is
applied. Theperforationcellmodelsthecylindricalvol-
umearoundasingleperforation. AnextendedReynolds
equation is then used to model the damping due both
to the gas ow in the air gap and in the perforations.
The method is applied in a rectangular damper with
4...64 square holes to derive the damping coe?cient
analytically. 3D FEM simulations are used to verify
the model. The damping predicted by the model is
in good agreement with that obtained with 3D FEM
simulations.
Sinusoidal small-amplitude velocities are assumed,
and micromechanical dimensions are considered with
rare gas efiects in the slip ow regime (Kn < 0:1).
1. INTRODUCTION
Perforations are used in several MEMS devices includ-
ing microphones, capacitive switches resonators, and
accelerometers,sincetheyreducedampinginasqueezed-
fllm structure considerably.
Modelling of perforated structures is problematic
because of the complexity of their geometries: a perfo-
rated surface might consist of thousands of tiny holes.
Direct simulation of such structures with FEM tools
often fails since the simulation problem becomes too
large and slow. The small dimensions of the microme-
chanical structures (about 1?m) is another challenge
for modelling; the rare gas efiects become important
even at ambient pressures. The models that are valid
for the continuum ow regime are no longer su?cient.
Perforated dampers are usually made of a perfora-
tion grid. When this grid is uniform, and the holes
are large enough, a repetitive pressure pattern forms
around each hole. This will be utilized in modelling
perforated dampers: the ow problem is isolated in a
cylindrical region around a single perforation, a perfo-
ration cell. This simplifles the analysis, and also com-
pact,parameterizedmacromodelsfortheseperforation
cells can be derived.
Analytic modelsforsuchperforationcellshavebeen
presented in the literature [1], [2], [3], [4], and [5]. Un-
fortunately, none of these contains proper models for
the fringe ows, and the reported models are not prop-
erlyverifled. In [6]amodelforacylindricalperforation
cell is derived based on FEM simulations.
A model is often needed for the case when a consid-
erable amount of gas ows from the damper borders.
In this case, the pressure proflle is no longer repetitive,
and the perforation cell alone is not su?cient in mod-
elling the damping. The problem can be formulated as
an extended Reynolds equation that has an additional
\leakage" term due to perforation [2]. The great bene-
flthereisthattheproblemisreducednowto2D.There
aretwoalternatewaysto solvethisequationwith FEM
tools: a homogenization approach, where the leakage
due to perforation is homogenized uniformly on the
damper surface [2],[7],[8], and a \Perforation Proflle
Reynolds" (PPR) method [9], where the leakage is de-
scribed as a spatially variable perforation proflle. In
thispaper, thehomogenizationapproachhasbeenused
to derive an analytic compact model for perforated
dampers. The perforation cell model derived in [6] is
applied to calculate the \leakage" term needed in the
extended Reynolds equation.
2. PERFORATION CELL
Figure 1 shows the topology of the perforation cell. It
is assumed that the radius of this cell is rx (about half
of the perforation pitch) and the radius of the hole is
r0.
2.1. Perforation Cell Model
Figure 2 shows the lumped ow resistances, that are
used to model the ow resistance (mechanical resis-
tance) of the cell.
It is equivalent to the force F acting on the bottom
surface divided by the velocity of the bottom surface,
r
z
rX
r0
hc
h
Figure 1: Topology and dimensions of the axisymmet-
ric perforation cell.
RS
RC
A0 A2
A3
A4
RIS
A0
pI
RE
U3
U2
U1
U0
p0
A1
p = 0
p1
RIC
p2
RIB
p3
p4
Figure 2: Lumped ow resistances used in modelling
the ow in difierent regions of a perforation cell.
RP = F=vz, i. e.,
RP = RS +RIS +RIB + r
4x
r40 (RIC +RC +RE): (1)
The resistances RS and RC are derived analytically,
but the remaining resistances, that model the fringe
ows, are derived from FEM simulations in [6]. The
equations for the ow resistances in Eq. (1) are given
in Appendix.
The validity of the perforation cell model in [6] is
limited tocaseswhereKn < 0:1, r0=h < 4, andr0=rx <
0:95. If the intermediate ow resistances in Eq. (1) are
ignored, if reduces to the model presented in [3],[4].
2.2. Rare Gas Efiects
The Knudsen number Kn = ?=a is a measure of gas
rarefaction. It is the ratio between the mean free path
? and the nominal displacement a. Since ? is inversely
proportional to pressure, Kn increases when the pres-
sure drops below the ambient pressure. In this paper,
two difierent Knudsen numbers are used: one for the
channel ow in the air gap Kch, and one for the capil-
lary ow in the perforation Ktb. Here, we include also
the efiect of the surface condition into the Knudsen
number.
For the channel ow
Kch = P?h ; (2)
where h is the air gap height, and for the capillary ow
Ktb = P?r
0
; (3)
where r0 is the radius of the capillary. For the difiuse-
specular scattering model [10], P is specifled as
P = 2?fifi [1:016?0:1211(1?fi)]; (4)
where fi is the momentum accommodation coe?cient.
For difiuse scattering, fi = 1, and P reduces to 1:016.
The damping in the structure will change through
the relative ow rate coe?cients, that depend on the
Knudsen numbers. In the slip ow regime these are
Qch = 1+6Kch (5)
for the ow in the air gap and
Qtb = 1+4Ktb (6)
for the ow in perforations with circular cross-sections.
3. MODEL FOR PERFORATED DAMPER
3.1. Extended Reynolds Equation
The homogenization principle is used to combine the
vertical ow in the air gap escaping from the damper
borders with the ow through the perforations. The
model derived from the extended Reynolds equation
presented in [2] is used, and the edge efiect model is
adopted from [11]. This extension contains an ad-
ditional term p=(NRP) in the Reynolds equation that
models the ow through N perforations. RP is the
ow resistance of a single perforation, modelled here
with the perforation cell model given in Eq. (1). The
extendedReynoldsequationwithoutcompressibilityef-
fects can be written as
h3Qch
12?
@2p
@x2 +
@2p
@y2
?
? pNR
P
= vz; (7)
where p is the pressure variation (function of x and y)
and ? is the viscosity coe?cient.
Equation7canbegenerallysolvedwithFEMmeth-
ods, but for simple geometries it can be solved also
analytically.
3.2. Rectangular Damper
The perforation cell model is applied here in building
a simple model for a perforated damper with rectangu-
lar surfaces (a?b) in perpendicular motion. Uniform
perforation is assumed.
It is necessary to correct for the edge efiects if the
ratio a=h < 50 [11]. The efiective surface dimensions
modelling the edge efiects in the slip ow regime are
aefi = a+1:3(1+3:3Kch)h; (8)
befi = b+1:3(1+3:3Kch)h: (9)
The pressure function satisfying Eq. (7) is
p =
X
m;n
cm;n cos(m?x=aefi)cos(n?y=befi) (10)
where n and n and positive indices. The coe?cients
cm;n aredeterminedfromtheboundaryconditions. The
velocity of the surface is also expressed with a Fourier
series:
vz = vr
X
m;n
hm;n cos(m?x=aefi)cos(n?y=befi) (11)
where hm;n = 16=(mn?2) for perpendicular motion.
After inserting Eqs. (10) and (11) into Eq. (7), coe?-
cients cm;n are solved:
cm;n = ?hm;nvrh3Q
ch?2
12?
m2
a2efi +
n2
b2efi
?
+ 1NR
P
(12)
Integration over the surface gives the total pressure
ptot =
Z aefi=2
?aefi=2
Z befi=2
?befi=2
pdxdy (13)
The mechanical resistance of the damper is RD =
?ptot=vr
RD =
X
m;n
1
Gm;n + 1R
m;n
;
? m = 1;3;5;:::,
n = 1;3;5;:::, (14)
where
Gm;n =
m2
a2efi +
n2
b2efi
? m2n2?6h3Q
ch
768?aefibefi ; (15)
and
Rm;n = 64NRPm2n2?4: (16)
The indices m and n in Eq. (14) run to inflnity, but for
an accuracy better than 1%, a maximum value of 15
for m and n is su?cient (for a square surface).
4. VERIFICATION
Several perforated dampers in perpendicular motion
are characterized with full 3D Navier-Stokes simula-
tions. The surface is a square, and the perforation con-
sists of identical square holes. The number of holes N
varies from 4 to 64. The simulated ow resistance is
compared with the simple model given in Eq. (14).
The topology of one of the four difierent simulated
structures is shown in Fig. 3 (N = 16). Square hole di-
s
a
h
h
2s
6h
c
Figure 3: Structure of simulated dampers. Topology
with N = 16 holes is shown. The flgure also illustrates
the simulation space around the structure in the 3D
Navier-Stokes simulations.
mensionsandcellperipheriesareaccountedforbyspec-
ifying efiectiveradii rx and r0. Radius rx = 0:5sxp4=?
isselectedsuchthattheareasofthecircularandsquare
shapes match. Radius r0 results from equating the hy-
draulic resistances (acoustic resistances) of square and
circular capillaries:
r0 =
128Q
sq
28:454?Qtb
?1
4 s
2 ?1:096?
s
2; (17)
where s is the width of each hole and Qsq is the rel-
ative ow rate of square perforation. The approxima-
tion assumes that the difierence between Qtb and Qsq
is small.
4.1. 3D FEM Simulations
3DFEMsimulationswithNavier-Stokessolverareper-
formed for the structures and the free air around it.
The multiphysical FEM software Elmer [12] was used.
Thesimulationswereperformedhererelativelyreliably,
thanks to the small number of holes and the symme-
try in the structure (actually, one quarter of the struc-
ture was simulated). The simulated gas volume was
extended around the damper: free space around and
above the damper are 6h and 2s, respectively. A mesh
of 500000 elements is used, and slip boundary condi-
tions are used for the surfaces. The constant velocity
vz is set to 1m/s.
The comparison was made well below the cut-ofi
frequency, that is, the compressibility and inertia of
the gas were ignored. The dimensions and parameters
are summarized in Table 1.
Table 1: Dimensions and parameters for the simulated
perforated dampers. In all cases, the perforation pitch
is sx = 5?m. Altogether, 288 difierent topologies were
generated and simulated.
Description Value Unit
N number of holes 4, 16, 36, 64
a surface length 10, 20, 30, 40 10?6m
s hole width 0.5, 1, ..., 4.5 10?6m
hc thickness 0.5, 1, 2, 5 10?6m
h air gap height 1, 2 10?6m
? viscosity coefi. 20 10?6Ns/m2
? mean free path 69 10?9m
P slip coe?cient 1
4.2. Results
Figures 4 and 5 compare the FEM simulations and the
model response as a function of the perforation ratio
q = s2=s2x ?100% when N = 64 and N = 36, respec-
tively. The accuracy of the model is good, the
maximum relative error is less than 12 %. The other
simulations show similar behaviour, but the error in-
creases when the number of holes decreases. The max-
imum relative errorsare 10%, 12%, 20%, and 35% for
N of 64, 36, 16, and 4, respectively. The use of the
efiective surface dimensions, Eqs (8) and (9), to model
the edge efiects results in a small error even for ratios
of a=h as low as 20.
0 25 50 75 100
100n
300n
1?
3?
10?
30?
-15
-10
-5
0
5
10
15APLAC 8.10 User: HUT Circuit Theory Lab. Wed Feb 8 2006
RD
q/%
err/%
0.5?m
1?m
2?m
5?m
hc =
h=1?m
N=64
a)
0 25 50 75 100
100n
300n
1?
3?
-15
-10
-5
0
5
10
15APLAC 8.10 User: HUT Circuit Theory Lab. Wed Feb 8 2006
RD
q/%
err/%
0.5?m
1?m
2?m
5?m
hc =
h=2?m
N=64
b)
Figure 4: Simulated (2) and approximated (||) ow
resistanceofthe perforateddamperasafunction ofthe
perforation ratio q. The number of perforations is 64
and the air gap height is a) h = 1?m and b) h = 2?m.
The relative error is also shown ({ { {) on the right-
hand scale.
The maximum relativeerror was about 35% for the
damper with four holes. Such an error was expected,
because, in this case, the validity of the perforation cell
approach is questionable. The error is mostly due to
the fringe forces acting on the sidewalls and on the top
surfaceofthedamperthatareexcludedfromthemodel
presented. The error in the model was the greatest for
large perforation ratios, where the damping is at its
minimum, and the relative contribution of the fringe
efiects dominates. Also, approximatingthe rectangular
hole with a circular equivalent in Eq. (17) may be a
source of an error that increases with the perforation
ratio.
0 25 50 75 100
100n
300n
1?
3?
10?
-15
-10
-5
0
5
10
15APLAC 8.10 User: HUT Circuit Theory Lab. Wed Feb 8 2006
RD
q/%
err/%
0.5?m1?m
2?m
5?m
hc =
h=1?m
N=36
a)
0 25 50 75 100
30n
100n
300n
1?
-15
-10
-5
0
5
10
15APLAC 8.10 User: HUT Circuit Theory Lab. Wed Feb 8 2006
RD
q/%
err/%
0.5?m
1?m
2?m
5?m
hc =
h=2?m
N=36
b)
Figure 5: Simulated (2) and approximated (||) ow
resistanceof the perforateddamperasafunction ofthe
perforation ratio q. The number of perforations is 36
and the air gap height is a) h = 1?m and b) h = 2?m.
The relative error is also shown ({ { {) on the right-
hand scale.
5. CONCLUSIONS
A relatively simple model is derived here for uniformly
perforated, rectangular dampers. This model was veri-
fledagainst3DFEMsimulationsintheslip owregime.
The model shows good accuracy (better than 10%) for
a damper with 64holes, but the relativeerrorincreases
when the number of holes is reduced. The accuracy is
good due to the fringe resistances in the model, and
due to the included edge efiects. This simple model
could be improved by taking into account the efiect
of the ow past the holes in the air gap. Also, the
fringe ows around the damper should be included in
the damper model, especially if a=h is not large. It is
straightforward to implement this model to any soft-
ware tool, since the model consists of simple equations.
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Appendix
The owresistanceRP of the perforationcell in Eq.(1)
consists of lumped ow resistances and their efiective
elongations. The equations have been derived partly
analytically, partly by fltting the model to FEM sim-
ulations by varying the coe?cients in heuristic equa-
tions [6]:
RS = 12??r
4x
Qchh3
1
2 ln
rx
r0 ?
3
8 +
r20
2r2x ?
r40
8r4x;
?
RIS = 6??(r
2x ?r20)2
r0h2 ?S;
RIB = 8??r0?B;
RIC = 8??r0?C;
RC = 8??hcQ
tb
;
and
RE = 8???Er0:
The efiective elongations in the previous equations are:
?S =
0:56?0:32r0r
x
+0:86r
20
r2x
1+2:5Kch ;
?B = 1:33
1?0:812r
20
r2x
? 1+0:732K
tb
1+Kch fB
r
0
h ;
hc
h
?
;
where
fB(x;y) = 1+ x
4y3
7:11(43y3 +1);
?C = (1+0:6Ktb)
0:66?0:41r0r
x
?0:25r
20
r2x
?
;
?E = 0:944?3?(1+0:216Ktb)16
?
1+0:2r
20
r2x ?0:754
r40
r4x
?
fE
?r0
h
?
;
where
fE(x) = 1+ x
3:5
178(1+17:5Kch):