Page 74 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_73_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 73
l
2
= ȡt
m a,e ฒ f (x)w(x) dx (2.87)
a
0
where the distribution function is calculated as
C (x)
a
f (x) = (2.88)
a
C
a,e
l
1 dx
with C (x) = (2.89)
a Etฒ w(x)
x
Similarly, the mechanical moment of inertia which is dynamically
equivalent to the distributed inertia of a microcantilever during free
torsional vibrations is
l
ȡt
2
2
J = 12ฒ f (x)w(x) w(x) + t 2 dx (2.90)
t,e t
0
where the torsion-related distribution function is calculated as
C (x)
t
f (x) = (2.91)
t C t,e
l
3 dx 3E
with C (x) = = C (x) (2.92)
t
a
Gt 3ฒ w(x) Gt 2
x
It can easily be checked that the distribution functions in axial and
torsional vibrations of variable-cross-section microcantilevers are ac-
tually identical, and this conclusion extends to constant-cross-section
microcantilevers, too. For a constant-cross-section microcantilever,
the axial/torsional distribution function reduces to the one given in
Eq. (2.48).
In bending, the effective mass is calculated as
l
2
= ȡt
m b,e ฒ f (x)w(x) dx (2.93)
b
0
The bending-related distribution function of Eq. (2.93) is calculated in
2
Lobontiu under the assumption that both a force and a moment that
act at the microcantilever’s free end produce bending. However, in
defining the direct linear bending stiffness, only the effects of the end
force are taken into account, and therefore it can be shown that the
bending-related distribution function is calculated as

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.