Page 317 - Mechanical design of microresonators _ modeling and applications

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Microcantilever and Microbridge Systems for Mass Detection

316 Chapter Six

l l
1 dx 1 xdx
C = C =
2,r Eฒ I (x) 2,c Gฒ I (x)
l y l y
x x
(6.44)
l 2 l
1 x dx 1 dx
C 2,l = Eฒ I (x) C 2,t = Gฒ I (x)
l y l t
x x
The equation system (6.43), which includes the definition Eqs. (6.44),
is nonlinear as the unknown l x also intervenes in compliances, and
therefore its solution implies using numerical techniques.
Resonant approach. In using the resonant approach to determine the
quantity of pointlike deposited mass, the algorithm that has been de-
tailed for constant-cross-section microcantilevers can be extended to
variable-cross-section members. The effective mass which results after
mass deposition is

2
m = m b,0 + f (a) ǻm (6.45)
b
b
where f (a) is the distribution function corresponding to the position
b
where the mass ǻm has deposited. This distribution function is given
in Eq. (2.63) for a constant-cross-section microcantilever, and it was
shown to also be an accurate approximation for variable-cross-section
members. Equation (6.45) expresses the modified resonant frequency
in the form:
k b,0
Ȧ = (6.46)
b 2
m b,0 + ǻmf (a)
b
The direct use of Eq. (6.46) is to determine the amount of mass that
locally deposits on a variable-cross-section microcantilever in terms of
the altered bending resonant frequency (which can be determined
experimentally), namely,

2
k b,0/ Ȧ – m b,0
b
ǻm = (6.47)
2
f (a)
b
By also taking into account the original resonant frequency of a
microcantilever, the following frequency ratio can be formulated:
Ȧ 2
(
l)
b,0 = 1+ 1– 3 c + 1 3 f
c
Ȧ 2 l 2 m (6.48)
b
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