Page 200 - Mechanical design of microresonators _ modeling and applications
P. 200
0-07-145538-8_CH04_199_08/30/05
Microbridges: Lumped-Parameter Modeling and Design
Microbridges: Lumped-Parameter Modeling and Design 199
of the equivalent (effective) mass to the kinetic energy of the vibrating
microcantilever such that
l 1 l +l 2
1
f (x) w(x) dx + w f (x) dx
2
2
b
b
0 l 1
m b,e = ȡt (4.110)
2l +l 2
1
f (x) w(x) dx
2
+ b
l +l
1 2
Due to the transverse symmetry of the microbridge shown in Fig. 4.18,
Eq. (4.110) can also be expressed as
l 1 l +l 2
1
2
2
m = ȡt 2 f (x) w(x) dx + w f (x) dx (4.111)
b,e b b
0 l 1
A check is performed again to see whether the generic Eq. (4.111) re-
duces to the known mass equation in the case where the two end seg-
ments of the microbridge are identical to the middle one. Indeed, when
l 1 = l 2 = l/3 and w(x) = w, Eq. (4.111) reduces to Eq. (4.12), which gives
the effective mass of a constant rectangular cross-section microbridge
of length l and cross section defined by w and t.
By combining Eqs. (4.101) with Eq. (4.111), the resonant frequency
corresponding to bending vibrations of the microbridge sketched in
Fig. 4.18 is
1
Ȧ =
b,e l l + l
1 2 1 2 2
ȡ t 2. f (x) w(x) dx + w. f (x) dx
b
b
0 l 1 (4.112)
2 ƍ
2
ƍ
2 ƍ
ƍ
ƍ
ƍ
2
ƍ
ƍ
× c + c C í 2c c C + c C + c C í 2c c C + c C ƍ
4 l
3 4 c
3 r
2 r
1 2 c
1 l
For a microbridge formed of three identical segments (l 1 = l 2 = l/3) of
constant rectangular cross section, the generic Eq. (4.112) reduces to
Eq. (4.13), which defines the bending resonant frequency of a constant-
cross-section microbridge of length l.
Torsional resonant frequency. In torsion, the lumped-parameter stiffness
being associated with the midspan of the microbridge pictured in
Fig. 4.18 is determined by first calculating the endpoint reaction
moment produced by a torsional moment (about the x axis) applied at
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.
