Page 304 - 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 303

0.003
0.0025
0.002
f m
0.0015
0.001
0.0005
0
0 0.002 0.004 0.006 0.008 0.01
f ω
Figure 6.10 Mass fraction in terms of the bending resonant frequency fraction for a
constant rectangular cross-section microbridge.

Example: Evaluate the mass fraction in terms of the resonant frequency
fraction [Eq. (6.12)] as well as the resonant frequency fraction in terms of the
mass fraction [Eq. (6.13)] by analyzing the free bending vibrations of a
constant rectangular cross-section microbridge having the material and
geometric properties given in the previous example.
The effective mass of this particular microbridge is 2.10×10 –13 kg
(0.21 ng), and the bending resonant frequency is 2.8 MHz. By using Eq. (6.12),
the mass fraction is plotted against the bending resonant frequency in
Fig. 6.10, and the converse relationship of Eq. (6.13) can be pictured in a
figure which, for the same range of the variable, is identical to Fig. 6.10 and
therefore is not plotted here.
As Fig. 6.10 indicates, a change of 0.001 in f m results in an approximate
change in f Ȧ of about 0.003, which in absolute figures means that a mass
addition of 0.001 × 0.21 ng = 0.21 pg can be realized through a change in the
resonant frequency of 2.8 MHz × 0.003 = 8.4 kHz.

6.3 Mass Detection by Means of
Microcantilevers
The presence of extraneous substances can be detected by means of
microcantilevers either statically (quasi-statically) when changes in
deflections and/or rotations are experimentally monitored or dynami-
cally when changes in the bending resonant frequency are, again,
experimentally determined after the monitored substances deposited
(attached) on a microcantilever by either physical means (adsorption,
for instance) or chemical reactions. Two situations are investigated in
this chapter, pertaining to either constant- or variable-cross-section
microcantilevers. The case, where the deposited mass can be considered
as a point mass, is analyzed first, whereas the second possibility, where
the deposited extraneous substance is a layer, is studied afterward. As
mentioned, experimental deformation or modal data are necessary in

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