Page 302 - 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 301
submultiples base unit

zepto atto femto pico nano micro milli

10 -21 10 -18 10 -15 10 -12 10 -9 10 -6 10 -3 1
Figure 6.8 Submultiples of a base unit of measure and their mutual relationships.

Ȧ = Ȧ – ǻȦ (6.7)
0
where ǻȦ is the resonant frequency variation. By analyzing Eqs. (6.5)
and (6.7) it follows that an increase of the mass variation ǻm leads to
a corresponding increase in the resonant frequency variation ǻȦ. Equa-
tions (6.5) and (6.7) enable us to define the mass sensitivity which
represents the frequency variation per unit of added mass and which
can be calculated as

ǻȦ Ȧ 2
0
= (6.8)
ǻm m(2Ȧ + ǻȦ)
0
A mass deposition sensing device will possess high sensitivity when its
resonant frequency is high and its mass is small, as Eq. (6.8) suggests.
It is also interesting to quantify the minimum mass quantity ǻm min
which can be experimentally detected through a resonant frequency
variation ǻȦ min . By using Eqs. (6.5) and (6.7), the following equation is
obtained:
ǻȦ min (2Ȧ + ǻȦ min )
0
ǻm min = m e (6.9)
Ȧ 2
0
This equation indicates that a decrease in the minimum detected mass
can be achieved by a sensing device whose equivalent mass is small and
whose resonant frequency is high, which, on aggregate, amounts to de-
creasing the mass of the vibrating micromember and increasing its
stiffness. The drive toward miniaturization is therefore clear, as a
smaller mass of the vibrating detector is possible only through size re-
duction, while keeping the relevant stiffness relatively large. Figure 6.8
illustrates the submultiples sequence for a generic unit of measure
(which can be meter or gram, for instance).
Example: Study the amount of deposited mass that can be reso-
nantly detected through bending by a constant rectangular cross-section

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