Page 271 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

270 Chapter Five
compressed active layer

t1
t2

structural layer

(a) (b)
Figure 5.45 Bimorph with active piezoelectric layer in compression: (a) original
(undeformed) state; (b) deformed state.

Several MEMS/NEMS transducers are designed as bimorphs that
use piezoelectric or piezomagnetic materials. One component of the
bimorph is the substrate (or structural layer) whereas the piezo (active)
material is deposited over it. Variation of the electric/magnetic field
leads to compression/extension of the active layer, but due to bonding
to the substrate, the net result is upward/downward bowing of the
entire sandwich structure. Figure 5.45, for instance, shows a bimorph
which bends by prevented compression of an active piezoelectric layer.
An equivalent tip force F can be calculated which will produce the
same tip rotation as the bending moment M which is created by the
induced-strain actuation. It can simply be shown that the force is
related to the moment in the form:

2M
F = (5.109)
l
7
where l is the bimorph length. Lobontiu and Garcia gave the
relationship between the bending moment, the free induced strain İ ,
0
and the material/geometry properties of the bimorph, and therefore
Eq. (5.109) can be reformulated as
4
F = f İ (5.110)
l 0
E E A A (t + t )(E I + E I )
2 1
2
2 y2
1 y1
1
1 2
where f =
2
E A {4E I + E 4I y2 + A (t + t ) } (5.111)
2
2 1
1
1 1
2
2
+4E A (E I + E I + E I )
2 2 1 y 2 y1 2 y2
For a piezoelectric material, the free strain is expressed in terms of the
active layer thickness and applied voltage as
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