Page 235 - Mechanics of Microelectromechanical Systems
P. 235
222 Chapter 4
the interaction with an external electromagnetic field, as suggested by
Kruusing and Mikli [9] for instance. Figure 4.39 shows four different cases
of magnetization m of a microcantilever that is placed in an external
electromagnetic field B, whose direction is assumed to be constant. The
interaction between the magnetization vector and the external
electromagnetic field vector B results in a force and a moment that act on the
cantilever and which are figuratively shown in Figs. 4.39 (a) through (d). The
force and the moment are calculated as:
It can be seen that in the case where the electromagnetic field B varies about
the z-direction, a force will be produced about the same direction, and will
bend the microcantilever, irrespective of the direction of magnetization. The
moment, however, according to the definition of Eq. (4.91) will have
different directions, as a function of the magnetization direction. In the case
of Fig. 4.39 (b), the total moment will be a pure bending moment combining
to the bending effect produced by the force whereas Fig. 4.39 (c) depicts
the situation where the moment is a torsional one. When m has an arbitrary
direction, as shown in Fig. 4.39 (d), the resulting moment can be resolved
into a bending component and a torsion component.
Example 4.10
A microcantilever of length l and cross-sectional dimensions w and t is
magnetized about a direction as shown in Fig. 4.39 (d). The microdevice is
used to monitor a constant external field B, as sketched in the same figure.
An optical system can measure a maximum slope at the microcantilever
tip. What is the maximum value of the magnetic field that can be detected by
this sensing microsystem ?
Solution
There will be no force acting at the free end, because the external field is
assumed constant. The moment produced at that end can be resolved into a
torsional component and a bending one. The later one has the expression:
The tip slope can be found as:
By combining Eqs. (4.92) and (4.93) results in:
