Page 247 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_246_08/30/05

Resonant Micromechanical Systems

246 Chapter Five
out-of-the-plane translation about the z axis (which is achieved through
bending of the root flexible segments about the y axis).
The torsional stiffness is provided by the two root segments and is explic-
itly given in Eq. (5.28). The mechanical torsional moment of inertia is pro-
duced by only the middle segment. When we consider that this segment is a
relatively thin plate, its moment of inertia about its central (symmetry) axis
3
is, as indicated by Beer and Johnston for instance,
2
mw 2
J = (5.40)
t,C 12
Because of the fact that the bridge system of Fig. 5.20b will rotate about the
x axis, the moment of inertia about that axis will be, according to the parallel
axes theorem,

2 2 2
mw 2 mw 2 mw 2
J = + = (5.41)
t,e 12 4 3
The torsional resonant frequency of this microbridge becomes

t Gw 1
Ȧ = 2 (5.42)
t,e w ȡl l w
2 1 2 2
By using the nondimensional parameters

w 2 t
ƍ
c = c = (5.43)
w w t w
1 1
the following torsional frequency ratio can be formulated:
Ȧ *
t,e 2
Ȧ = 2 (5.44)
ƍ
t,e 1+ (c c )
t/ w
where Ȧ * t,e is the torsional resonant frequency of the symmetric paddle
microbridge shown in Fig. 4.19. Figure 5.21 is the three-dimensional plot of
the resonant frequency ratio expressed in Eq. (5.44). It can be seen that for
the selected parameter ranges this frequency ratio remains almost constant
and equal to 2, which indicates that the torsional resonant frequency of the
symmetric paddle bridge (Fig. 4.19) is twice the frequency of the asymmetric
bridge shown in Fig. 5.20a.

Example: Study the free response of the four-arm microbridge sketched in
Fig. 5.22, and compare it to the resonant behavior of a regular symmetric
microbridge such as the one shown in Fig. 4.19. Assume the central segment
is rigid and the legs are massless and compliant. Also assume the geometry

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.