Page 313 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH06_312_08/30/05

Microcantilever and Microbridge Systems for Mass Detection

312 Chapter Six
This equation system involves solving of higher-degree algebraic
equations, which can be done numerically for specified geometry and
material parameters.
In the case where the microcantilever is functionalized over a length
l p (which is known a priori) and when this functionalized patch (which
is assumed to be covered entirely by the attached mass) starts from the
microcantilever’s free end (l 1 = 0), the deposited mass can be calculated
as either
24u 1z
ǻm = (6.35)
3
3
2
3
p /
g 3l p/ (EI ) +2(4l –3l l – l ) (E I )
p
1 y1
y e
6ș 1y
or ǻm = (6.36)
2
p /
g l p/ (EI ) +3l(l – l ) (E I )
1 y1
y e
depending on whether the tip deflection u 1z [as shown in Eq. (6.35)] or
the tip slope ș 1y [as in Eq. (6.36)] is available experimentally.
Resonant approach. Finding the attached mass by the resonant method
implies measuring the shift in the bending resonant frequency after
mass attachment, as a result of alterations in both the stiffness and the
mass of the microcantilever-based system. Again, after the extraneous
substance attaches to the original microcantilever, the compound sys-
tem behaves as a dissimilar-length sandwich, whose equivalent
lumped-parameter stiffness and inertia were discussed in Chap. 3. Fig-
ure 3.35, which has been utilized for dissimilar-length bimorph can-
tilever calculations, is also valid in the present case. By taking into
consideration the expressions determined for the equivalent stiffness
k b,e and the effective mass m b,e [Eqs. (3.172) and (3.174), respectively],
the added mass ǻm (which is identical to the patch mass in the dissim-
ilar-length sandwich microcantilever model) is determined as
l p 140 k b,e
ǻm = – m 1) (6.37)
l p '( 33 Ȧ 2

where m 1 is the original microcantilever mass and Ȧ is the altered
bending resonant frequency. Equation (6.37) is particularly useful
when the modified resonant frequency is available experimentally.
The bending frequency ratio can be expressed as

Ȧ k m
b,0 = b,0 b (6.38)
Ȧ b k b m b,0

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.