Page 270 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

Resonant Micromechanical Systems 269
2
M = ʌR BI (5.106)
Example: Determine the parameters defining the lumped-parameter
undamped model of the microcantilever illustrated in Fig. 5.44a when the
loop current is sinusoidal [Eq. (5.101)] and the center of the loop is placed at
a distance l 1 from the free end of the microcantilever having the length l.
The parameters defining the lumped-parameter model of the single-DOF
system of Fig. 5.1 are the mass m, stiffness k, and forcing amplitude F 0 . The
first two parameters are determined by Eqs. (2.66) and (2.61), respectively.
To find the force which is located at the microcantilever’s free end, the real
case where a bending moment acts at a distance l 1 is made equivalent to the
situation where a force acts at the free end, subject to the condition that both
systems produce the same rotation at the point where the bending moment
is applied. It can simply be shown that this requirement results in a force
2M
F = (5.107)
l + l
1
By combining now Eqs. (5.101), (5.106), and (5.107), the equivalent force can
be expressed whose amplitude is
2
2ʌR BI 0
F = (5.108)
0 l + l
1

5.4.3 Piezoelectric and piezomagnetic
transduction
Piezoelectric materials change dimension when exposed to a variation
in an external electric field, whereas piezomagnetic materials change
dimension when subjected to variations in an external magnetic field.
In such situations microdevices that are designed based on this effect
(also called reversed) behave as actuators. Conversely, when external
mechanical pressure is applied to piezoelectric/piezomagnetic materi-
als, they become electrically/magnetically polarized and therefore can
be used as deformation sensors. Most often, piezoelectric materials
(such as PZT, an alloy based on lead, zinc, and titanium; ZnO, zinc ox-
ide; or Al Ga 0.7 As, an alloy based on aluminum and gallium arsenide)
0.3
and piezomagnetic materials (such as Terfenol-D) are deposited on sub-
strate/structural layers and create sandwiched micro/nano cantilevers,
bridges, or membranes. Piezomagnetic materials are sensitive to the
relative position between the external magnetic field and their own po-
larization field. Positive magnetostrictive materials (see Jakubovics 13
for more details) do extend along the polarization direction when the
two fields are parallel and contract about the polarization direction
when the two fields are perpendicular.

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