Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
Reduced Order Modelling of MEMS Dynamics
Stefano Mariani 1 , Saeed Eftekhar Azam 1 , Aldo Ghisi 1 , Alberto Corigliano 1 , Barbara Simoni 2
1 Politecnico di Milano, Dipartimento di Ingegneria Strutturale
Piazza Leonardo da Vinci 32, 20133 - Milano (ITALY)
2
STMicroelectronics, MSH Division
Via Tolomeo 1, 20010 Cornaredo (ITALY)
Abstract- The dynamics of a uniaxial micro-accelerometer
subjected to accidental drop events is studied by means of a
reduced order model. A two degrees of freedom model is built,
which carefully reproduces the MEMS response under high
acceleration events. The results of the reduced order model are
compared to those obtained with a three-dimensional finite
element model, in terms of accuracy of the results and
simulation speed-up
I. INTRODUCTION
During accidental drop events, polysilicon MEMS
sensors are often exposed to high-g loadings because of
their extremely small mass and, therefore, inertia [1-3].
This fact can cause mechanical failure due to cracking in
high stressed regions. In a series of papers [4-8] we
recently proposed a numerical approach to accurately link
the features of the shock-inducing cause (like, e.g. a drop)
to the effects at the MEMS level. Because of the several
length-scales affecting the dynamics of the whole device
when subjected to such shocks, a multi-scale frame was
adopted. We then showed that macro-scale (at device level)
and meso-scale (at sensor level) analyses can be routinely
investigated making use of commercial finite element
codes, since the features of the polycrystalline film
constituting the movable parts of the MEMS have a
marginal impact. A different situation characterizes microscale
(at polysilicon film level) analyses, which turn out to
be extremely complicated and time demanding, in case high
accuracy of the results is mandatory.
A possible way to drastically reduce the computing time
is to make use of reduced order models for the whole
MEMS sensor, built in an accurate and micro-mechanically
informed way. Reduced models would allow to avoid
running analyses at the micro-scale, keeping a similar
accuracy in the results. This issue was partially addressed
in previous works [9-10].
In the present work we go further on in the use of reduced
models built on the basis of purely mechanical
considerations, routed by the investigated details of the
MEMS dynamics. A simple two degrees of freedom
reduced model is built for a commercial micro
accelerometer (Fig. 1) which measures the acceleration
orthogonal to the device substrate. The dynamics of
accidental drop events characterized by two acceleration
levels, is numerically studied by means of the reduce model
and compared with the outcome of a fully 3D finite element
model.
More sophisticated reduced order modelling techniques
could be used, e.g. based on the proper orthogonal
decomposition (POD) [11] and compared with the approach
presented in this paper; this issue will be the subject of
forthcoming works.
II. MODELLING IMPACTS IN MEMS ACCELEROMETERS
The accelerometer was assumed to be subjected to a low-g
acceleration input directed along the Z-axis (see Fig. 2),
whose maximum is about 90 g, and to a high-g input (see
Fig. 3), with a maximum acceleration peak of about 5,500 g.
As for the boundary conditions applied to the model, the
accelerometer is anchored at its center with two slender
suspension springs. As a reference solution a finite element
model of the accelerometer, featuring 34,000 nodes and
26,000 elements, i.e. about 100,000 degrees of freedom (dof)
has been considered. In parallel, a reduced, two-dof model
has been envisaged: if we impose a rigid behavior for the
plate, only dof reproducing the vibration modes #1 and #5
from Fig. 6 need to be considered. The spectral content of
the two input accelerations, as shown in Fig. 4-5 through
their energy spectral density, confirms that the relevant,
excited dynamics pertains the relative rotation of the plate
−=Δ
θθθ
and its relative translation −=Δ
www
at the
spring clamped end, as shown in Fig. 1.
The motion of the device in this two-dof framework can
be described as follows:
− aMuKu
=++ (1)
where the vector u=[Δw Δθ ] T collects the two dof,
superposed dots indicate time derivative. If we assume h to be
the plate thickness, L the plate length in the x direction, ρ the
polysilicon mass density (reduced to take into account for the
holed plate), l the spring length, and V the plate volume, then
the mass matrix collects the plate translational mass
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