Page 269 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

268 Chapter Five

bending axis
F
I torsion axis I
F F B
R
F
B
(a) (b)
Figure 5.44 Cantilever-based electromagnetic sensing: (a) bending mode; (b) torsional
mode.

d E
Ȧ =5.6 (5.103)
r 2 ȡ
l
When the forced vibrations are undamped, the amplitude ratio of Eq. (1.15)
can be expressed as
X 1
=
X 2 (5.104)
st 1 í (Ȧ Ȧ )
/ r
For an amplitude ratio of 2, Eq. (5.104) in combination with Eq. (5.103) pro-
duces the excitation frequency

d E
Ȧ =3.955 (5.105)
l 2 ȡ
which is equal to approximately 0.71Ȧ r .
Another possibility of electromagnetic transduction is illustrated in
Fig. 5.44 where a circular conducting wire is patterned to a micro-
cantilever. When the external magnetic field is parallel to the micro-
cantilever length, as in Fig. 5.44a, a couple is produced by the two
opposite forces F which act at two diametrically opposed points, and the
result is a bending moment which is applied about the axis shown in
the same figure. In the case where the magnetic field is perpendicular
to the cantilever length, as in Fig. 5.44b, the resulting Lorentz forces
F will generate a couple acting along the member’s longitudinal axis,
and therefore the cantilever will undergo torsion. For any other
direction of the magnetic field B situated in the microcantilever’s plane,
both bending and torsion will be produced. In both cases, the moment
resulting through the interaction between the external magnetic
field B and the current I yields a moment equal to

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