Page 176 - Mechanical design of microresonators _ modeling and applications

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Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 175
model are less than 2 percent; moreover, for bridges where the thickness-to-
length ratio is small, these errors are smaller and therefore almost negligible.

4.2.2 Torsion resonant frequency
The torsional resonant frequency can be determined by finding the
lumped-parameter stiffness and inertia for a half-length microbridge
and for the full-length microbridge, respectively.
For the half-length microbridge, according to the model sketched in
Fig. 4.4, a moment applied about the longitudinal (x) axis at the guided
end (which, as far as torsion is concerned, is considered free) produces
torsion of the bar, and it can simply be shown that the torsional stiffness
of that segment (the ratio of the applied moment to the resulting
rotation angle) is

2GI t
k = (4.24)
t,e l
The lumped-parameter mechanical moment of inertia of an
equivalent rigid body which is placed at the guided end is determined
by means of Rayleigh’s principle again. The torsion-related distribution
function is the ratio of the rotation angle at a generic point of
abscissa x (measured from point 1 toward the right in Fig. 4.4) to the
maximum rotation angle (at point 1 in the same figure) and is found
to be

2x
f (x) =1 í (4.25)
t l
As a consequence, the lumped-parameter mechanical moment of inertia
becomes

J t
J t,e = 6 (4.26)

where J t is the torsional mechanical moment of inertia of the full-length
microbridge. By combining Eqs. (4.24) and (4.26), the torsional resonant
frequency for a half-microbridge becomes

GI t
Ȧ t,e =3.46 lJ t (4.27)

The same result should be found when the calculus is performed for
the full-length microbridge, such as the one shown in Fig. 4.7.

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