Download coupling plate deformation, electrostatic actuation and squeeze film

Int. Conf. on Computational Methods for Coupled Problems in Science and Engineering
COUPLED PROBLEMS 2009
B. Schrefler, E. Oñate and M. Papadrakakis (Eds)
c
CIMNE,
Barcelona, 2009
COUPLING PLATE DEFORMATION, ELECTROSTATIC
ACTUATION AND SQUEEZE FILM DAMPING IN A FEM
MODEL OF A MICRO SWITCH
STEPHAN D.A. HANNOT∗ AND DANIEL J. RIXEN∗
∗ Delft
University of Technology
Department of Precision and Microsystems Engineering
Mekelweg 2, 2628 CD Delft, The Netherlands
e-mail: [email protected], [email protected]
Key words: FEM, Electromechanical coupling, MEMS, Fluid Film Damping, Microsystems
1
INTRODUCTION
A specific field of research that can be characterized by modeling coupled problems
is microsystems modeling. Microsystems of Micro-Electro-Mechanical-Systems (MEMS)
are sensors and actuators that have typically dimensions in the micrometer range and are
produced with the same lithographic technologies as integrated circuits.
This paper focuses on one specific type of microsystem: the microswitch. A simple
capacitive microswitch is a small microbridge that is suspended above a dielectric layer
as shown in figure 1. When a potential difference is applied between the ground electrode
and the microbridge it is attracted to the dielectric. This changes the overall capacitance
of the parallel electrodes, therefore such a device can be used as a switch in a RF circuit1 .
Top electrode
Ground
electrode
Dielectric layer
Figure 1: A microswitch and a model of a microswitch.
These switches typically do not operate in a vacuum, therefore the most dominant
form of damping is squeeze film damping between the two plates. Hence to accurately
model such a switch not only a proper electromechanicallly coupled model is crucial, also
the squeeze film problem should be modeled2 . This paper discusses the implementation
1
Stephan D.A. Hannot and Daniel J. Rixen
of a non-linear Reynolds equation to model this squeeze film damping into a coupled
electromechanical FEM model.
2
ELECTROMECHANICAL MODEL
The mechanical model is based on linear Mindlin-Reissner plate elements and the
electrostatic field is modeled using linear hexahedral elements to discretize the electric
potential3 . The forces on the mechanical model follow from the electrostatic solution.
Structural displacements cause a change in the configuration of the electrostatic domain,
changing the electrostatic solution. Therefore the electrostatic mesh has to deform to
follow the moving electrode. The implementation of the coupling including linearized
tangent matrices can be found in4 .
The resulting equations are for mechanics:
M uu ü + K uu u = f φ
(1)
where K uu is the linear stiffness matrix, M uu is the consistent mass matrix and u the vector containing the structural displacements and rotations. The applied force f φ contains
the forces due to the electric field.
The electric equations are:
K φφ (u)φ = Q(u, V )
(2)
where φ contains the nodal potentials, K φφ (u) is the stiffness matrix that is linear in φ,
but depends on u because it is computed with the deformed mesh and this mesh deformation depends on the structural displacement. The applied charge Q(u, V ) depends also
on the applied voltage V because the applied charge follows from the imposed potential
boundary condition on the top electrode.
3
SQUEEZE FILM DAMPING
The air gap is a three dimensional domain, however when the aspect ratio of plate
width over gap size is bigger than 3 it can be assumed that the pressure is constant in the
direction perpendicular to the plate5 . Also it is assumed that the flow in the direction
tangential to the plate can be described by Poiseuille flow. In that case the pressure field
can be described by the non linear Reynolds equation5,6 :
3
h ∂p
∂
∂ (hρ)
ρ
(3)
=
∂t
∂xi
12µ ∂xi
where h is the gap-height, ρ is the density, p the total pressure and µ the viscosity of the
fluid. Here we used Einstein’s summation convention for i = 1, 2.
It is assumed that given the small dimensions of MEMS the temperature variations are
negligible, hence that the density is proportional to the pressure. Therefore it is possible
to write the equations after finite element discretization as:
2
Stephan D.A. Hannot and Daniel J. Rixen
∂pj
+
Ni hNj dΩ
∂t
Ω
Z
∂hj
Ni pNj
dΩ = −
∂t
Ω
Z
Z
Ω
∂Ni ph3 ∂Nj
dΩpj
∂xk 12µ ∂xk
(4)
where the Ni are the nodal shape functions (assumed identical for p and h), the pj the
nodal pressures and the hj the nodal gap sizes. Note that in this equation the summation
over k is from 1 to 2 while the summation over i and j is from 1 to the number of nodes.
These equations can be written in matrix vector notation as:
Lṗ + Rḣ = −Dp
(5)
where L is a matrix function of h, R a matrix function of p and D a matrix function of
p and h3 .
The coupling between fluid and structural domain is pretty straightforward. First it is
known that h = h0 + u, secondly the nodal force on the structure is the fluid pressure
minus the ambient pressure (because the ambient pressure gives a force on the top side
of the movable electrode):
Z
fjp =
Nj (p − pambient )dΩ
(6)
Ω
which can be added to the right hand-side of equation (1).
4
TIME INTEGRATION
It is possible to integrate these full equations in a staggered manner. We chose a
Newmark non-linear time integration scheme for the mechanical equations3 and a simple
non-linear version of the trapezoidal rule for the fluid equations. Since the electric problem
is static, the electric forces are incorporated in the mechanical equations, which explains
the requirement of a non-linear time integration scheme for the mechanical equations.
However, when modeling with the Reynolds equation, the incompressible terms are
often neglected5 , reducing equation (5) to:
Rḣ = −Dp
(7)
This equation is still non-linear: R still depends on p and D still depends on p and h.
But it is now static equation for p, that can be included directly in the mechanical time
integration as was done with the electrostatic equations. This can greatly speed up the
time integration.
5
RESULTS
The equations were integrated for a plate with dimensions: length = 250µm, width
= 30µm, thickness = 1.4µm and an air gap of 3µm. The material properties were approximately those of poly silicon: a Young’s modulus of 160GP a, a density of 2330kg/m3
and a Poisson’s ratio of 0.28. The viscosity is 18.27µP a s.
Figure 2 shows the results of both compressible and incompressible time integration
for a step response of 20 Volts.
3
Stephan D.A. Hannot and Daniel J. Rixen
x 10
x 10
−7
displacement
−0.2
−0.5
−1
0
−0.4
−0.6
x 10
−0.8
−1
0
−7
0
Incompressible
Compressible
Z
0
−5
−2
Y
0.2
0.4
0.6
time
0.8
1
−4 0
X
1
2
x 10
−4
1.2
x 10
−5
Figure 2: Midnode displacement as function of time and deformed shape (color = pressure).
6
CONCLUSIONS
It was shown that it is relatively easy to implement a full non-linear Reynolds equation
for fluid film damping into a electromechanical FEM model. Furthermore it was shown
that for certain dimensions it is possible to neglect the incompressible terms from the
equation. Accounting for squeeze film damping shows that Microsystems can exhibit
special behavior close to pull-in (not shown here due to space limitations).
7
ACKNOLEDGEMENT
We want to acknowledge the financial support of the MicroNed program of the ministry
of economical affairs of The Netherlands.
REFERENCES
[1] X. Rottenberg, H. Jansen, P. Fiorini, W. De Raedt and H.A.C. Tilmans. Novel
RF-MEMS capacitive switching structures. 32nd European Microwave Conference,
Milano, Italy, (2002).
[2] J. Bielen, J. Stulemeijer, D. Ganjoo, D. Ostergaard and S. Noijen. Fluid-electrostaticmechanical modeling of the dynamic response of RF-MEMS capacitive switches. EuroSimE 2008, Freiburg, Germany, (2008).
[3] K.J. Bathe. Finite element procedures, Prentice Hall, (1996).
[4] V. Rochus, D.J. Rixen and J.C. Golinval. Monolithic Modeling of electro-mechanical
coupling in micro-structures. Int. J. Num. Meth. Engng., 65, 461–493, (2006).
[5] M. Bao and H. Yang. Squeeze film air damping in MEMS. Sens. Actuator. A. Phys.,
136, 3–27, (1999).
[6] A.H. Nayfeh and M.I. Younis. A new approach to the modeling an simulation of
flexible microstructures under the effect of squeeze film damping. J. Micromech.
Microeng., 14, 170–181, (2004).
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