9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
A piecewise-linear reduced-order model of
squeeze-film damping for deformable structures
including large displacement effects
A.Missoffe
1
, J.Juillard
1
, D.Aubry
2
1
SUPELEC, Dpt. SSE, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, FRANCE.
E-mails: alexia.missoffe@supelec.fr, jerome.juillard@supelec.fr.
2
LMSS-MAT, ECP, CNRS UMR 8579, Grande Voie des Vignes, 92295 Ch?tenay-Malabry Cedex, FRANCE. Email :
denis.aubry@ecp.fr.
Abstract-. This paper presents a reduced-order model for the
Reynolds equation for deformable structure and large
displacements. It is based on the model established in [11]
which is piece-wise linearized using two different methods. The
advantages and drawbacks of each method are pointed out. The
pull-in time of a microswitch is determined and compared to
experimental and other simulation data.
I. INTRODUCTION
Correct modelling of damping is essential to capture the
dynamic behaviour of a MEMS device. Our interest is
squeeze-film damping which models the behaviour of a fluid
in small gaps between a fixed surface and a structure moving
perpendicular to this surface. The lateral dimensions of the
surfaces are large compared to the gap and the system is
considered isothermal. Squeeze film damping is then
governed by the Reynolds equation [1]:
3
()
12
GGP
PP
t?
???
??=
??
?
??
(1)
where ()tyxG ,, is the distance between the moving and the
fixed surface, ()tyxP ,, is the pressure, ? the effective
viscosity of the fluid [1]. For small excitation frequencies or
amplitudes the squeezed film behaves as a linear damper.
For larger amplitudes or frequencies, the gas has no time to
flow away and the pressure builds up creating a stiffening
effect coupled to a nonlinear damping. A complete review
on this equation and its different regimes can be found in [3].
Coupling the Reynolds equation to the equation governing
the mechanical behaviour of the microstructure leads to a
nonlinear system of partial differential equations (PDEs),
which has no analytical solution and must be simplified
thanks to some assumptions. The most commonly made
assumptions are the following:
- uniform displacements, i.e. 0=
?
?
=
?
?
y
G
x
G
- steady-state sinusoidal excitation, i.e. ()tGG
e
?sin=
[4]
- small displacements, i.e. gGG +=
0
and
0
Gg << ,
where
0
G is the nominal gap of the structure at rest or
close to a static equilibrium [5].
- small pressure variations, i.e. pPP +=
0
and
0
Pp << , where
0
P is the ambient pressure.
These hypotheses prove to be useful in a variety of
applications, if only for gaining insight of nonlinear damping
phenomena. However, in many cases, it is difficult to justify
their use: for example, it is clear to see that none of the first
three hypotheses holds when trying to estimate the switching
time of a micro-switch. Most micro-switches do not undergo
uniform displacements, nor can these displacements be
considered small, and the behaviour of a micro-switch is
fundamentally transient.
To date, the most notable attempts to tackle the problem of
reduced-order modelling (ROM) of squeeze-film damping
with large, non-uniform displacements have been made by
Younis et al. [5-7], Mehner et al. [4], Yang et al. [8], Hung
and Senturia [9] and Rewienski and White [10]. In [5-7], the
authors solve the linearized Reynolds equation for a
displacement of the beam around an operating point. In [4],
the authors use a modal projection method to calculate
modal frequency-dependent damping and stiffening
coefficients close to a determined operating point. To extend
this approach to large displacements, Mehner [4] gives an
analytical expression of these coefficients as a function of
mechanical modal coordinates established by fitting of
simulation data for different initial deformations. These
approaches are all based on several steady-state sinusoidal
calculations [5-7] or simulations [4], which increase the time
for setting up the reduced-order model. The most general
approaches may well be those developed in [8-10]. These
approaches, although very general, have a high
computational cost (because of the nonlinear /multiphysics
/transient simulation they require) and their accuracy
depends, to some degree, on the choice of the training
trajectory.
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
II. CONSTRUCTION OF THE REDUCED-ORDER MODEL
We work on the variable
0
PPp ?= , supposing
0
Pp << . (1) has then the following form:
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
=
?
?
?
?
?
?
?
?
??
0
3
1
12 P
p
G
t
p
G
?
(2)
The first reduction step is based on modal projection of (2)
which is first transformed via a change of variable on p (3).
)2/3( ?
= Gp ?
(3)
The aim of this change of variable is to obtain a spatial
operator on ? for which the Laplacian eigenmodes are more
relevant than for the operator in (2), conserving its self-
adjoint property thus guarantying convergence of the
solution. The reduced-order model may be written as:
sxHxfsxA )())()(( =?
dt
d
(4)
with
??
?
?
?= dGPf
ll
2/1
0
2 ? (5)
?
?
?
?
?
?
?
?
?
?
+?=
??
?
d
G
GP
H
lkklkkl
????
?
2/3
2/3
2
0
12
(6)
and
??
?
?
?= dGA
lkkl
??
2
(7)
where x and s are respectively the vectors of the mechanical
modal coordinates of the moving structure, and the modified
squeeze coordinates. For a structure under electrostatic
actuation, one may write the full coupled model as:
() ()
() ()
()
()
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
0
0
00
0
00
0
0
00
00
00
xp
s
x
x
xH
xBxK
I
xfs
x
x
xA
M
I
e
dt
d
(8)
where p
e
(x) is the electrostatic force. The Euler-Bernoulli
equation is used to model the mechanical part. Large
displacement effects can be taken into account. Let us write
(8) in a more compact way:
)())(())(())((( tuttt
dt
d
zBzfzg += (9)
where z is the state vector including all modal coordinates.
III. PIECEWISE LINEARIZATION BASED ON AN INPUT TRAJECTORY
The cost of evaluating the terms g(z(t)), f(z(t)), and B(z(t))
can be reduced using the piecewise linear approach
described in [10]. The fact that the coefficients in (3) only
depend on the mechanical modal coordinates reduces the
cost of construction of the piecewise linear model, which has
the following structure:
)())()()((
)))(()(
1
0
1
0
zFzzJFzfz
z(zJGzgz
iii
eleci
s
i
iii
s
i
i
w
w
dt
d
+?+
=
?
?
?
?
?
?
?+
?
?
?
=
?
=
(10)
where
i
JG and
i
JF are respectively the jacobians of the
functions g(z) and f(z) at the linearization points
i
z and
(z)F
elec
corresponds to the electrostatic force. Two
problems arise from the piecewise linearization: the choice
of the linearization points and the weighting procedure. We
choose the linearization points from a simulation of the fully
nonlinear reduced model (8). A new linearization point is
chosen when a point is ?far enough? from the already chosen
points. The state variables must be normalized to calculate
the distances in state-space. The weighting procedure is the
one described in [14].We work on the example of a
microswitch also treated in [8-10]. We use the piecewise
linear reduced order model to determine the switching time
of the device for a step voltage between 9 and 10.5 V at
atmospheric ambient pressure. The 21 linearization points
are chosen along a 9.5V input training trajectory. The time
integration scheme is not exactly implicit because the
weights are calculated from the last step point. Fig.1 shows
the response to a 10V input using a linear model, the modal
projection model, and the piecewise linearized model. Fig.2
shows the experimental and simulated data presented in [9]
and results of our piecewise linear reduced model of order 6
for the switching time.
0 1 2 3
x 10
-4
0
0.5
1
1.5
2
2.5
x 10
-6
time (s)
m
i
d
d
le
p
o
in
t
d
isp
la
ce
m
e
n
t
(m
)
ROM Projection + Piecewise linear
ROM Projection
ROM Linear SFD
9.5V input trajectory
Linearization points
Fig. 1- Middle point displacement for the 9.5 V step voltage training trajectory,
a 10 V full reduced model simulation, piecewise linearized model, and linear
model..
9-11 April 2008
?EDA Publishing/DTIP 2008 ISBN: 978-2-35500-006-5
Fig. 2- Pull-in time (s) versus applied voltage (V) for
0
P =1.013?10
5
Pa.
Comparison of the experimental and simulated results presented in [16] to the
simulated results obtained with our reduced-order model. The chosen squeeze
modes correspond to
x
k =0, 2 and
y
k =1, 3.
IV. TRAJECTORY-INDEPENDENT REDUCED ORDER MODEL
We can notice that the nonlinear terms in (8) depend only
on the mechanical modal coordinate and that the model is
linear with respect to the modified squeeze coordinates. It
can be written as:
PE(x)zxBxFPzxG +=? )())()((
dt
d
(11)
The piecewise linearization described in section II doesn?t
take advantage of this structure. This second piecewise
linearized model is based on the linearization of the terms G,
FP and B. As they only depend on the on mechanical
coordinate, there is no need for a training trajectory. It is
sufficient to discretize in an appropriate way the space
corresponding to the mechanical coordinates. This is a great
advantage as the resulting model does not depend on the
relevance of a training trajectory, as opposed to the model
presented in section II. On the other hand the resulting
equation is nonlinear which increases the resolution cost.
Number of
linearization
points
Middle point
displacement
error (in %)
Pull-in time
error (in %)
10 21 7
14 5 2.5
19 1 2
30 0.5 0.1
Table 1 Middle point displacement error and pull-in time error for different
number of linearization points compared to the full reduced model.
This equation is integrated in time using a fully implicit
Euler scheme. At each step the nonlinear equation is solved
using the Matlab ?fsolve? function. Table 1 shows results for
an input step voltage of 9.1 V. This model yields better
displacement error and correct results are obtained for a
minimum number of linearization points of around 15 which
is less than the model obtained using the approach described
in [10].
V. CONCLUSION
We have presented a piecewise linear model of squeeze-
film damping for flexible structure and large displacements
with the restriction of small pressure variations. The model
is based on a modal projection method and is then piecewise
linearized using two different methods. The advantages and
drawbacks of each method were pointed out. The first
method [10] is very general whereas the second takes
advantage of a specificity of the equation structure. This last
method appears to be more efficient, although the final
system that must be solved remains nonlinear. The pull-in
time of a microswitch was determined and compared to
experimental and other simulation data.
REFERENCES
[1] M.H. Bao, ?Micro mechanical transducers: pressure sensors,
accelerometers and gyroscopes?, Handbook of Sensors and Actuators, vol. 8,
Elsevier, Amsterdam, 2000
[2] T. Veijola et al., ?Extending the validity of squeezed-film damper models
with elongations of surface dimensions?, Journal of Micromechanics and
Microengineering, vol. 15, pp. 1624-1636, 2005
[3] M.H. Bao, H. Yang, ?Squeeze film air damping in MEMS?, Sensors and
Actuators A, vol. 136, pp. 3-27, 2007
[4] J. E. Mehner et al., ?Reduced-order modeling of fluid-structural
interactions in MEMS based on modal projection techniques?, 12
th
International Conference on Solid-State Sensors, Actuators and Microsystems,
2003, vol. 2, pp. 1840-1843
[5] M. I. Younis, A. H. Nayfeh, ?Microplate Modeling under Coupled
Structural-Fluidic-Electrostatic Forces?, 7
th
International Conference on
Modeling and Simulation of Microsystems, 2004, pp. 251-254
[6] M. I. Younis, A. H. Nayfeh, ?Simulation of squeeze-film damping of
microplates actuated by large electrostatic load?, Journal of Computational and
Nonlinear Dynamics, vol. 2, pp. 232-241, 2007
[7] M. I. Younis, A. H. Nayfeh, ?A new approach to the modeling and
simulation of flexible microstructures under the effect of squeeze-film
damping?, Journal of Micromechanics and Microengineering, vol. 14, pp. 170-
181, 2004
[8] Y. J. Yang et al., ?Macromodeling of coupled-domain MEMS devices with
electrostatic and electrothermal effects?, Journal of Micromechanics and
Microengineering, vol. 14, pp. 1190-1196, 2004
[9] E. S. Hung, S. D. Senturia, ?Generating efficient dynamical models for
microelectromechanical systems from a few finite-element simulation runs?,
Journal of Microelectromechanical Systems, vol. 8, pp. 280-289, 1999
[10] M. Rewienski, J. White, ?A trajectory piecewise-linear approach to model
order reduction and fast simulation of nonlinear circuits and micromachined
devices?, IEEE Transactions on Computer-Aided Design of Integrated Circuits
and Systems, vol. 22, pp. 155-170, 2003
[11] J.Juillard, A.Missoffe, D.Aubry ?A reduced-order model of squeeze-film
damping for deformable structures including large displacement effects?, to be
published in Journal of Micromechanics and Microengineering.
8. 6 8. 8 9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6
10
-4
10
-3
Voltage (V)
P
u
l
l-i
n
t
im
e
(s
)
experimental data
FEM (Hung-Senturia)
projection+piecewise linear model
order 6