Page 324 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

Microcantilever and Microbridge Systems for Mass Detection 323

6×10 -16
5×10 -16
4×10 -16
f m -16
3×10
2×10 -16
1×10 -16
0
0 0.0002 0.0004 0.0006 0.0008 0.001

f ω
Figure 6.25 Mass fraction in terms of the frequency shift ratio.

Example: Study the relationship between the quantity of added mass and
the resonant frequency shift in the case where the deposited mass is localized
3
on a microbridge at a = 2 “m. Known are: ȡ = 2200 kg/m , l = 20 “m,
w = 2 “m, t = 150 nm, and the resonant shift ǻȦ = 1000 rad/s.
Equation (6.62) is rewritten as

2
/
1 (1– f ) –1
Ȧ
ǻm =0.0016 m (6.63)
4
c (1– c ) 4
l l
ǻȦ
where f = (6.64)
Ȧ
Ȧ
b,0
By using the numerical values of this example, the plot of Fig. 6.25 is
obtained.
Figure 6.25 indicates that for a fixed position of the attached mass on the
microbridge, a higher resonant shift signifies that more mass has attached
to the structure (meaning a larger mass fraction f m ).
Layer-mass detection. Only the resonant approach is developed here for
layerlike mass deposition detection. Mass which attaches over a portion
of a microbridge can be treated by using the model developed for bi-
morph microbridges. If mass deposition is symmetrically localized with
respect to the microbridge midpoint, then Figs. 4.12 and 4.13 can be
utilized as well for the analysis that follows. As is the case with layerlike
mass deposition on microcantilevers, the similar phenomenon corre-
sponding to microbridges results in alteration of both the stiffness and
the mass of the structure, and therefore the change in the resonant
frequency results from the interplay between stiffness and mass
modifications. By using the half-length model of a dissimilar-length
bimorph microbridge, the original stiffness is

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