Page 321 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

320 Chapter Six
where higher resonant frequencies are expected than those produced
by constructively similar or identical microcantilevers. Constant- and
variable-cross-section bridge configurations are investigated in
relationship to their performance in static or resonant mass detection.

6.4.1 Constant-cross-section microbridges
The first part of this subsection analyzes the static and resonant re-
sponses of constant-cross-section microbridges when extraneous mass
attaches in either point- or layerlike fashion.

Point-mass detection. When the substance which is mobilized on a mi-
crobridge has small dimensions which warrant the pointlike mode to be
employed, the stiffness of the original sensing microdevice remains un-
altered, and the variations are either the deflected shape (in the static
approach) or the mass, which causes modification of the resonant fre-
quency (in the resonant approach), as discussed in the following.
Static approach. Figure 6.11b, which shows a constant-cross-section
microcantilever where a mass attaches, can also be utilized here to
find the quantity of deposited mass as well as its position on a half-
length microbridge (in other words, the originally free end of the
microcantilever is guided this time, and the length is only l/2). Three
experimental measurements are needed again to determine ǻm, l , and
x
l y . While measuring the deflection u 1z and torsional rotation ș 1x at the
guided end is feasible, another amount needs to be determined, except
for ș which is zero in this case. Another deflection, for instance, can
1y
be monitored such as that at a point situated at an abscissa a past the
location where the mass is deposited. The following equations are valid
in this case:

(l –2l ) 2 (l +4l ) (l –2a) (l +4a)
2
u = x x ǻmg u = ǻmg
1z 96EI 3z 96EI
y y
(6.54)
(l –2l )l
x y
ș 1x = ǻmg
2GI
t
The second of Eqs. (6.54) enables us to determine the quantity of
deposited mass as
96EI u
y 3z
ǻm = 2 (6.55)
(l –2a) (l +4a)g

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