Online proceedings - EDA Publishing Association

11-13
May 2011, Aix-en-Provence, France
input case, assuming the same damping coefficient as in the
previous damped case. The relative displacements at two
opposite (with respect to the spring axis) corner nodes for the
FE and the two-dof models are depicted in Fig. 11 and, again,
they well agree. It is worthwhile to point out that when one
Mode 1: 1,527 Hz
Mode 2: 10,047 Hz corner enters into contact with the die, since the plate is rigid
the opposite node is limited in its movement as well. This
confirms that for the low-g case the motion due to the
excitation is mainly rotational around the spring axis, i.e.
according to the degree of freedom Δθ . Again, the stresses
Mode 3: 17,737 Hz
Mode 4: 23,665 Hz
are underestimated by the two-dof model (Fig. 12), but their
absolute value is lower than the previous damped case when
no contact is considered. This means that, due to the small
lower gap, the die works as a stopper for the moving plate.
Finally, a calculation has been carried out for the high-g
acceleration input case, where both damping and contact have
Mode 5: 28,945 Hz
Mode 6: 164,090 Hz
to be considered. As in the previous low-g case with contact,
we present in Fig. 13 the relative displacements for both the
opposite corner nodes. Again, the dynamics of the sensor is
well captured by the two-dof model; in this case, however,
when one node enters into contact with the die, the opposite
corner evidences a larger displacement in the other direction.
Mode 7: 259,820 Hz
Mode 8: 359,310 Hz
This behavior is possible only if a contribution of the
translational degree of freedom Δw is also present; in other
words, the plate, still rigid, translates along the Z-axis and
rotates along the spring axis during high-g excitation. For the
high-g case, the stresses calculated by the two-dof model are
even more underestimated than the previous cases, as shown
Mode 9: 526,660 Hz
Mode 10: 847,540 Hz in Fig. 14.
In conclusion, the good performance of the two-dof model for
Fig. 6. Lowest vibration modes of the MEMS sensor.
what concerns the sensor dynamics is undermined by the lack
of accuracy for the stresses. Other methods like the one
recalled in the following Section IV could possibly solve this
drawback.
III. RESULTS
The advantages of the reduced order modelling in terms of
computational burden are evident: in our examples the FE
calculations required about 5 hours versus a few seconds for
the two-dof model. In order to appreciate, instead, the quality
of the reduced order model approximation Figs. 7-14 compare
the FE solution with the simplified, two-dof approach, in four
different cases.
First, for the low-g acceleration input case, an undamped and
damped response has been considered when the contact is
neglected for the moving plate. In Fig. 7 and Fig. 9 the
relative displacements along the Z-axis u Z at one corner node
of the plate are compared: in both the cases the sensor
dynamics is correctly captured (thus confirming the hypothesis
of a rigid plate), an almost negligible difference is visible only
in the peak of the damped case. In the Figs. 8 and 10 the
envelope of maximum principal stresses at the spring clamped
end section, calculated as in [9], are shown. The two-dof
stresses appear clearly underestimated with respect to the FE
solution; this discrepancy is due to the stress concentration
effect because of the rounded corners between the plate and
the spring end; this effect is captured by a refined FE mesh,
but is not reproduced by the two-dof model.
As a third case, we allow for contact in the low-g acceleration
Fig. 7. Time history of the relative vertical displacement at the plate corner
node for the low-g undamped case, neglecting contact.
©EDA Publishing/DTIP 2011
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