Page 254 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 241
It has also been shown in Chapter 1 that the bending of a sandwich beam
can be described by an equivalent bending rigidity which was defined
in Eq. (1.180) in terms of individual material and geometry properties of the
component layers. The bending moment that needs to be applied at the
cantilever’s tip in order to produce the curvature radius of Eq. (4.134) is
determined as:

Equation (4.134) or Eq. (4.135) can serve as a metric in comparing the
different possibilities of actuating a bimorph with a given geometry, but in
actuality only the peculiarities of the free strain will dictate the differences
in bending between two physically-identical bimorphs that are actuated by
means of different sources.
When used as an actuator, the bimorph needs to be characterized in terms
of its free displacement and bloc force capabilities, as mentioned in the
beginning of this section. For a fixed-free (cantilever) configuration, the free
displacement can be calculated as:

where the equivalent rigidity is given in Eq. (1.180) of Chapter 1.
Similarly, the force that will bloc the tip motion of a bimorph can be
calculated as:

For a bimorph with given cross-section, material and induced-strain
properties, the free displacement is proportional to the square of the length,
as indicated by Eq. (4.136), whereas the bloc force of Eq. (4.137) is inversely
proportional to the bimorph length.
The generic equations presented thus far can also be utilized as metric
tools in quantifying the mechanical motion or the environmental changes by
means of bimorph-based sensors. A variation of the tip bending moment or
deflection translates in an induced strain, and this latter amount can easily be
converted into an electrical signal for instance.

8.2 Thermal Bimorph

In a thermal bimorph, as the name indicates, the source strain is
induced thermally. In the hypothesis that the temperature of the upper layer
of Fig. 4.51’s bimorph decreases by (such that shrinking of this layer is
possible, according to the assumptions here), the induced strain is: