Page 76 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_75_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 75

y

w
w 2 1 w 1
x

l
Figure. 2.25 Trapezoid microcantilever geometry.

3
Et (w — w ) 3
k b,e = 2 1 (2.99)
2
3
6l (w — w )(3w — w ) +2w ln(w 2/ w )
1
2
1
2
1
1
When w 2 ĺ w 1 , Eq. (2.98) reduces to Eq. (2.61), which gives the bending
stiffness of a constant-cross-section microcantilever. The lumped mass
corresponding to bending vibrations can be calculated by means of
Eqs. (2.93), (2.94), and (2.98) and is found to be
215w +49w 2
1
m b,e = m (2.100)
560(w + w
1 2
Equation (2.100) reduces to Eq. (2.66)–which gives the bending mass
of a constant-cross-section microcantilever–when w 2 ĺ w 1 . The bend-
ing resonant frequency of the trapezoid microcantilever of Fig. 2.25 is,
by way of the stiffness of Eq. (2.99) and the mass of Eq. (2.100),
E(w í w )
1
2
Ȧ = 13.66
b,e 2
ȡ(215w +49w ) [ (w í w )(3w í w ) +2w ln(w w ) ]
1 2 1 2 1 2 1 2/ 1
(2.101)
t(w í w )
× 2 1
l 2
The free axial vibrations are solved similarly by following the cor-
responding generic-approach algorithm. By using the first of Eqs. (2.84)
coupled with Eqs. (2.87) and (2.98), the axial stiffness becomes

Et(w — w )
1
2
k a,e = (2.102)
l ln(w 2/ w )
1
For w ĺ w , Eq. (2.102) transforms to Eq. (2.45), which yields the axial
2
1
stiffness of a constant-cross-section microbar. The equivalent mass in

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.