Page 173 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_172_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

172 Chapter Four
l
l
u (0) = u (l) =0 ș (0) = ș y( ) = ș (l) =0 u z( ) = u (4.10)
z z y 2 y 2 z
the five unknown coefficients can be found, and the following ratio,
which is also the bending distribution function, can be formulated:
u (x) x 2 x 2
z
f (x) = =16 2 ( 1 ෹ ) (4.11)
b
u
z l l
The effective mass corresponding to the entire microbridge undergoing
free bending vibrations is therefore
l
ฒ
m = ȡA f (x) 2 dx = 128 (4.12)
b,e b 315m
0
where, again, m is the total mass of the microbridge. By using Eqs. (4.7)
and (4.12), the bending-related resonant frequency is
EI y
Ȧ = 21.737 (4.13)
b,e 3
ml
It can be seen that, compared to the exact value of the resonant fre-
quency in Eq. (4.6), the approximate value of Eq. (4.13) is lower, and
the error between the two values is approximately 2.8 percent, which
is still an acceptable figure.

Short microbridges. In the case of relatively short microbridges where
shearing effects are taken into account, the shear-dependent bending
Eqs. (2.72) defining the half-length model are applied again. It can be
shown that the angle ș is given by
y
F x(l / 2 í x)
ș (x) = (4.14)
y 2EI y
which resulted from the first of bending Eqs. (2.72). By substituting the
expression of ș y (x) of Eq. (4.14) into the second of bending Eqs. (2.72),
the following distribution function is obtained:

48țEI + GA(l í 2x)(l +4x)
y
f sh (x) = (l í 2x) (4.15)
b 2
l(48țEI + GAl )
y
which connects the deflection u z (x) to the tip deflection u z , according to
Eq. (2.62). The tip deflection is

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.