Page 301 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH06_300_08/30/05

Microcantilever and Microbridge Systems for Mass Detection

300 Chapter Six
experimentally employed to determine the specific mass, only the quan-
tity of deposited mass is of interest because the stiffness alteration is
negligible in terms of modifying a relevant resonant frequency. The
original resonant frequency of a certain micromember (be it cantilever,
bridge, or membrane) can be expressed generically by means of lumped-
parameter modeling as
k b,e
Ȧ = (6.1)
b,0 m b,e
where k b , e and m b , e are the bending-related effective stiffness and mass,
respectively. If torsion is monitored, Eq. (6.1) changes to

k t,e
Ȧ = (6.2)
t,0 J
t,e
where k t , e and J t , e are the torsion-related effective stiffness and
mechanical moment of inertia. By considering the free vibrations as
generic (and therefore they can be either bending- or torsion-gener-
ated), Eqs. (6.1) and (6.2) can be written in the single form:
k e
Ȧ = (6.3)
0 m
e
Mass deposition on a micromember changes its original mass and
therefore original resonant frequency such that the new resonant
frequency becomes

k e
Ȧ = (6.4)
m + ǻm
e
where ǻm is the deposited mass. By combining Eqs. (6.3) and (6.4), the
following relationship results between the original and modified
resonant frequencies:
Ȧ
0 = 1+ f (6.5)
Ȧ m
ǻm
where f =
m m (6.6)
e
is the mass fraction. Equation (6.5) indicates that the resonant
frequency decreases through mass addition, which means that

Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.