Page 127 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_126_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

126 Chapter Three
The torsional resonant frequency can now be calculated by means of the
lumped-parameter stiffness [Eq. (3.56)] and mechanical moment of in-
ertia [Eq. (3.57)].
The bending stiffness is
2 3
Ewt t (t – t ) 3
2
1 2 1
k b,e =
2
3
2
/
2 { 2t (t – t ) l (3l (l + l ) + l ) (3.58)
1 1
2
1 1
2
2
2
2
3 3
t )
+3l t (3t – t )(t – t ) +2t ln(t 1/ 2 }
1 2
2
2
1
1
1
The bending effective mass is
6
7
6
ȡw 264l t +56l l (10t +23t ) +56l l (7t +26t )
1 2
1
2 2
1
2
1 2
2
2 5
3 4
+56l l (40t +59t ) +70l l (49t +83t )
1 2
1
2
2
1
1 2
4 3
5 2
+56l l (49t +116t ) +28l l (49t +149t ) (3.59)
1 2
1 2
2
1
1
2
7
+l (49t +215t )
1
2
1
m b,e =
1120(l + l ) 6
1
2
The bending resonant frequency is determined by combining Eqs. (3.58)
and (3.59) and is not included here.
The following particular conditions have been used: l = l = l/2,
1
2
t 1 = t 2 in Eqs. (3.53) through (3.59), which transformed the microcanti-
lever of Fig. 3.16 to a constant rectangular cross-section microcanti-
lever of length l, thickness t, and width w. Indeed, the corresponding
Eqs. (2.45), (2.49), (2.51), (2.55), (2.61), and (2.66) were used, which
define the constant rectangular cross-section microcantilever.
3.3.2 Filleted microcantilevers
Another possible combination of the basic units of Chap. 2 is studied
here, namely, by realizing compound cantilevers through the mixing of
circular and elliptic filleted units, either by themselves or with constant
rectangular cross-section segments.
Long, circularly filleted microcantilevers. Figure 3.17 shows the top view
with the main geometric parameters defining a long, circularly filleted
microcantilever which consists of a constant rectangular area at the
free end connected to a circularly filleted area at the fixed root. The
lumped-parameter stiffness and inertia will again be determined in
view of calculating the resonant frequencies of this micromember 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.