Modelling of Squeeze Film Effects with FEM for Coupled Fluid
Structure Interaction
Anish Roychowdhury1,2*
Advisor: Prof. Rudra Pratap1,2
Dept. of Mechanical Engineering, 2Center for Nano Science and Technology
1
*anishroy@mecheng.iisc.ernet.in
Abstract—This work aims at modelling the squeeze film effects
in vibratory MEMS devices such as resonators and gyroscopes,
which often consist of a plate-like structure that vibrates normal
to a fixed substrate. They are generally not perfectly vacuum
packed, resulting in a thin air film trapped between the moving
structure and the fixed substrate which offers both stiffness and
damping. We model this phenomenon using finite element
methods to solve the elasticity equation coupled with the 2D
Reynolds equation for squeeze film to get the pressure
distribution on the vibrating plate and then calculate the squeeze
film damping and stiffness forces.
INTRODUCTION
Vibratory MEMS devices such as MEMS
resonators, microphones, etc., typically have a thin
air film trapped between a fixed substrate and a
vibrating structure. If the lateral dimensions of the
vibrating structure happen to be much larger than
the height of the air gap, the trapped air behaves
both like a spring and a viscous damper, a
phenomenon known as squeeze film effect. Squeeze
film damping is the dominant dissipation
mechanism in Si based MEMS devices operating in
the aforementioned conditions [1]. Accurate
modelling of such systems generally involves
coupling of three domains, electrostatics, structural
and fluid. Traditionally, the squeeze film effect is
modelled using the lubrication theory, via the
Reynolds equation [2]. With rigid plate assumption
the Reynolds equation can be decoupled from the
elasticity equation and, further, on linearization, can
be solved to obtain analytical expressions for
stiffness and damping. Blech [3] studied the effect
of squeeze film induced stiffness and damping for
rigid plates with trivial pressure boundary
conditions. Darling et al. [4] presented analytical
solutions to the linearized Reynolds equation for
various venting conditions, using a Greens function
approach. McCarthy et al. [5] studied cantilever
micro-switches using a transient finite difference
method approximating a parabolic pressure
distribution along the length and non variance along
I.
the width of the structure and obtained good
agreement with experimental measurements.
Pandey and Pratap [6] studied the effect of flexural
mode shapes on the squeeze film offered stiffness
and damping for a cantilever resonator. They used
Green’s function to solve the linearized
compressible Reynold’s equation and employed the
modal projection method available in ANSYS to
solve the coupled fluid structure problem for
several flexural modes of vibration. The analytical
and numerical values of damping obtained were in
good agreement with experimental results. Hannot
and Rixen [7] presented an approach to solve the
coupled elasticity equation and Reynolds's equation
for modelling a capacitive micro-switch. They
employed a non-linear Newmark time integration
scheme for the mechanical equations and a
trapezoidal rule for the fluid equations. The above
mentioned models attempt to solve the coupled
problem, though not in a single step. The geometry
modelled is also limited to 1D beam type structures.
In this study we attempt to model the squeeze
film effects for flexible plate structures with the
elasticity effect coupled with the fluid problem and
solved with coupled FEM in a single step method.
MODELLING
In order to have a better understanding of the
squeeze film forces, we initially studied the effect
of variable flow boundaries on squeeze film
parameters for an all sides fixed elastic micro-plate.
We used ANSYS 3D FLOTRAN modules to model
the fluid domain and imposed approximate first
mode shape to account for the plate flexibility. We
obtained the pressure distribution due to the
squeeze film on the moving plate. The squeeze film
forces were subsequently computed by integrating
the pressure over the plate surface. However for
accurate modelling one needs to solve the coupled
II.
TABLE I
fluid structure problem to get a more accurate result
Q FACTOR FOR THE FIRST THREE MODES FOR A CANTILEVER BEAM
for the pressure distribution. We have subsequently
Modes
Q factor comparison
developed a coupled FEM based methodology to
QEXP
QANSYS
QFEM
solve the Reynolds equation coupled with the 3D
1
1.20
1.11
1.095
2
7.58
6.94
5.849
elasticity equation. We present results from a one
3
18.52
20.0
20.379
step monolith [8] formulation where we solve the
fluid structure problem in a truly coupled sense as
IV. CONCLUSIONS
opposed to iterative and approximate methods
reported in prior works [5,7]. The numerical results We have discussed numerical modelling strategies
show good agreement with published experimental for vibratory MEMS devices with varying flow
boundaries and presented results from our FEM
data and existing analytical solutions.
model addressing the coupled fluid structure
problem due to squeeze film effects. The same
III. RESULTS
In order to benchmark our code, we compare our technique can be used further as a design tool for
numerical FEM based results for spring (Fs) and modelling and simulation of the dynamic response
damping (Fd) forces with available analytical of vibratory MEMS devices where accurate values
solutions for the all sides open "OOOO" of their Q factor may be required.
configuration (See Fig. 1). We see from Fig. 1 that
ACKNOWLEDGEMENTS
the numerical stiffness and damping forces are in
This work is partially supported by NPMASS
close agreement with the analytical results, at high
as well as low squeeze numbers (σ). The deviation grant for computational micro-systems. The author
between the numerical and analytical results have acknowledges the support from CoNE Lab of the
been found to be less than 2% for the entire Center for Nano Science and Engineering at IISC.
frequency range studied. In order to compare our and also wish to thank Arup Nandy and Prof. C.S.
results with experimental data, we have modelled a Jog from the Department of Mechanical
cantilever beam as per dimensions mentioned in the Engineering at IISC for their help and collaboration.
work of Pandey and Pratap [6]. We compare the
REFERENCES
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Mathematics Vol. 20, No. 2, pp. 131–150, 1962.
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Vol. 105, No. 4, pp. 615-620, 1983.
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[5]
[6]
[7]
[8]
Figure 1. Spring and damping forces vs squeeze number (σ ) for "OOOO"
configuration
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Anish holds BE and MS degrees in Mechanical
engineering from NIT Durgapur, and Louisiana
State University (LSU), USA, respectively, He
also has 6 years of experience as a mainframe
programmer. He is currently pursuing a PhD in the
field of micro-systems modelling.