Page 314 - 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 313
where the subscript 0 indicates the original condition (no mass depo-
sited on the microcantilever). By using the equations corresponding to
the original stiffness and to the modified one [Eqs. (2.61) and (3.172),
respectively] the stiffness ratio of Eq. (6.38) can be expressed as
2
k b,0 c c c (3c 2 +3c c + c ){3+ c 6+ c (4+ c c ) }
l1
t E lp
l1 lp
lp
E t
t
t
=1– (6.39)
k b 1+ c c {4+ c 6+ c (4+ c c ) }
t
E t
t
E t
where the nondimensional parameters are defined as
t p l p l 1 E p
c = c = c = c = (6.40)
t t 1 lp l l1 l E E 1
Similarly, the mass ratio of Eq. (6.38) is expressed by means of
Eqs. (2.66), (3.174), (3.175), and (3.176) as
m b c c c 6 (c l1 + c ) i +1 – c i +1
lp
p t lp
l1
m =1 + 33 1+ ™ c i c (6.41)
b,0 i =1 lp
i5
with the coefficients c i being c 1 = í210, c 2 = 105, c 3 = 35, c 4 = –42,
c 6 = 5, and
ȡ ȡ
c = (6.42)
ȡ
ȡ
1
Example: Study the influence of the nondimensional parameters of
Eqs. (6.40) and (6.42) on the bending resonant frequency ratio in the case of
layerlike mass deposition on a cantilever whose parameters are ranging as
c l1 ĺ [0, 0.9], c lp ĺ [0, 1], c t ĺ [0.01, 1], c E ĺ [0.1, 1], c ȡ ĺ [0.5, 5].
By considering that c l1 = 0, c E = 0.2, and c ȡ = 0.5, the plot of Fig. 6.17
shows the variation of the resonant frequency ratio in terms of c lp and c t .
Figure 6.17 indicates that the patch length parameter c lp has a notable in-
fluence on the bending resonant frequency ratio in cases where the patch
thickness compares to the substrate thickness, when this ratio reaches a
maximum. The frequency ratio increases with the thickness ratio except for
the extreme cases where the patch length is almost equal to the substrate
length. Figure 6.18 is the three-dimensional plot of the same frequency ratio
in terms of nondimensional material parameters for c l1 = 0, c lp = 0.2, and
c t = 0.2. It can be seen that variation by a factor of 10 in the longitudinal
elasticity modulus leaves the resonant frequency almost unchanged.
Increasing the density of the patch relatively to the density of the substrate
increases the frequency ratio.

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