Page 316 - 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 315
1.03
1.02
1.01
1
ω b,0 / ω b
0.99
0.98
0.97

0 0.2 0.4 0.6 0.8
c l1
Figure 6.19 Bending resonant frequency ratio in terms of patch position nondimensional
parameter.

6.3.2 Variable-cross-section
microcantilevers
Variable-cross-section microcantilevers can also be utilized for mass
deposition detection in a manner similar to the one presented for
constant-cross-section microcantilevers. Mass detection is treated
here similarly to mass detection by means of constant-cross-section
microcantilevers; therefore pointlike and layerlike mass detection is
analyzed by using both the static approach and the resonant one.

Point-mass detection
Static approach. As a reminder, when a point mass attaches to a
microcantilever, its position (as defined by the parameters l x and l y in
Fig. 3.37) and the mass quantity ǻm are the unknowns. By experimen-
tal evaluation of three elastic deformations at the free tip, for instance,
u 1z , ș 1y , ș 1x , the three equations that were valid for a constant-cross-
section configuration can be reformulated as

u = ǻmg(C – l C ) ș = ǻmg(C – l C )
1z 2,l x 2,c 1y 2,c x 2,r
(6.43)
ș = ǻmgl C
1x y 2,t
where the newly introduced compliances are calculated with respect
to point 2 in Fig. 3.37a (which explains the first subscript) and are
defined as

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