Page 328 - 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 327

1.03 0.1
ω b,0 / ω b
1
0 0
f m

c l 0
1 1
Figure 6.28 Bending resonant frequency ratio for a variable-cross-section microbridge in
terms of the nondimensional length parameter and mass fraction.

Conversely, the absolute frequency shift can be determined in the form:

1
ǻȦ = 1– Ȧ (6.76)
b 4 b,0
4
1+256c (1– c ) f m
l
l
Example: Study the quantity of deposited mass which can be detected res-
onantly by a paddle microbridge with step variable thickness, such as the
one pictured in Fig. 4.22, given the defining geometry and material param-
3
eters: l 1 = l 2 = 100 “m, w = 20 “m, t 1 = 0.2 “m, t 2 = 0.5 “m, ȡ = 2200 kg/m ,
and E = 160 GPa.
By using the numerical data of this example in conjunction with Eq. (4.134)
giving the stiffness and Eq. (4.136) giving the effective mass, both associated
with the midpoint of a paddle microbridge with step variable thickness, it is
found that the effective mass of the original structure is m = 2.216 × 10 í12 kg
and the bending resonant frequency is Ȧ b,0 = 1.377 MHz. By using Eq. (6.75)
which yields the deposited mass in terms of structural properties and the
nondimensional properties, the plot of Fig. 6.29 has been drawn showing the
variation of the added mass with the parameters f Ȧ and c l . It can be seen
that this microbridge design is capable of detecting masses on the order of
10 –14 kg (tens of femtograms) when the frequency shift ratio (the absolute
frequency shift to the original resonant frequency ratio) reaches values in the
vicinity of 0.0001. It has been assumed that the mass can attach anywhere
on the thickest middle portion.

6.5 Mass Detection by Means of Partially
Compliant, Partial-Inertia Microdevices

There are microcantilever- and microbridge-based designs where
certain structural segments can be considered rigid, generally because
their dimensions (particularly thickness) are larger than those of other

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