Page 13 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH01_12_08/30/05

Design at Resonance of Mechanical Microsystems

12 Chapter One
4
3ʌȝR
c = (1.37)
2z 0 3

where R is the radius of the disk.
For a rectangular superposition area, the damping coefficient is also
5
based on evaluations by Starr and Andrews, Harris, and Turner, 6
namely,
( w w 2) w lȝ
3
2
c = 0.997í0.752 +0.163 3 (1.38)
l
z
l
0
where w is the plate width and l is the plate length. For a very narrow
strip, where w/l ඎ 0, Eq. (1.38) reduces to
3
w lȝ
c = 0.997 (1.39)
z 3
0
By knowing the damping coefficient c, it is possible to express the qual-
ity factor Q, which is connected to squeeze-film damping, according to
Eq. (1.26):
k
Q = (1.40)
cȦ
where k is the stiffness of the mobile structure and Ȧ is the relative
motion frequency.
All previous development was based on the condition that the flow be
a continuum. However, when the dimensions of the particles in the flow
approach the relevant dimensions of the channel they travel in, the
continuum property may no longer be valid. A quantifier that monitors
this aspect is Knudsen’s number Kn, which is defined as the ratio of the
free mean molecular path to the relevant dimension of the channel, in
this case:

Ȝ
Kn = (1.41)
z
When Kn < 0.01, the continuum property of the flow is preserved; but
when Kn < 10, the free mean path is comparable to (even larger than)
the relevant channel dimension and the flow is free molecular. For the
in-between range of 0.01 to 10 Kn, the flow is of a transition type where
slip is possible. The Knudsen number can also be utilized as a correc-
tion factor in expressing the dynamic viscosity Ș variability as indicated
7
by Veijola, Kuisma, and Lahdenpera, who gave an effective value of

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.