Page 318 - 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 317

1.03
0.1
ω b,0 / ω b
1
0 0
f m
c l
0
1 1
Figure 6.20 Bending resonant frequency ratio in terms of nondimensional length ratio
and mass fraction.

where c l and f m are length and mass fractions, respectively, and have
been defined in Eqs. (6.24). Equation (6.48) indicates that the altered
resonant frequency is always lower than the original (the ratio is
greater than 1). Another interesting peculiarity displayed by the reso-
nant frequency ratio of Eq. (6.48) is that it does not depend on the
particular properties (geometric and material) of a specific variable-
cross-section microcantilever. Figure 6.20 is the three-dimensional plot
of this frequency ratio as a function of the two nondimensional param-
eters. As the particle attaches farther away from the free tip, the
resonant frequency ratio diminishes because the altered frequency
increases relatively due to a decrease in the effective mass. Conversely,
an increase in the mass fraction will produce a corresponding increase
in the resonant frequency ratio.
The added mass can also be expressed in the following form:

2
/
1 (1– f ) –1
Ȧ
ǻm = m (6.49)
3 2
(1– 1.5c +0.5c )
l l
where f Ȧ is the resonant frequency fraction and was defined in Eq. (6.11)
in terms of the frequency shift and the original resonant frequency.
Conversely, the resonant frequency shift is
1
ǻȦ = 1– Ȧ b,0 (6.50)
3 2
1+ (1–1.5c +0.5c ) f m
l
l
Example: Compare the resonant frequency shift of a constant rectangular
cross-section microcantilever of dimensions l, w, and t to that of a rectangular
cross-section paddle microcantilever, as pictured in Fig. 6.21. Consider
that w 2 = 4w 1 , w 1 = w, and l 1 + l 2 = l, l 1 = 80 “m, w 2 = 10 “m, t = 0.4 “m,

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