Page 185 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_184_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

184 Chapter Four
where the equivalent torsional rigidity is expressed as

3
3
w(t + t )
1
p
(GI ) = (4.57)
t e 3
In the end and by way of Eqs. (4.56) and (4.57), the torsional stiffness
becomes
3
3
3
2G t w(G t + G t )
p p
1 1
1 1
k = (4.58)
t,e 3 3
3 G lt + G (l í l )t p
p
p
1 1
When l p ඎ 0, the stiffness of Eq. (4.58) reduces to
3
2Gwt 1
k = 3l (4.59)
t
which is indeed the torsional stiffness of the half-length patched
microbridge.
The equivalent inertia fraction (mechanical moment of inertia) which
needs to be placed at the guided end of the half-model microbridge of
Fig. 4.13 is again calculated by means of Rayleigh’s principle and by
assuming the following velocity distribution:
(
˙
ș (x) = 1 í 2x ) ș ˙ x (4.60)
x
l
By using this procedure, the equivalent mechanical moment of inertia
becomes
2
2
2
2
w ȡ l t (w + t )(3l í 3ll + l ) 2 2
p
p
p p p
p
J t,e = 72 l 2 + ȡ lt (w + t ) (4.61)
1
1 1
When l p ඎ 0 and t p ඎ 0 (no patch, just the substrate), Eq. (4.61) reduces
to
2
2
J t m (w + t )
1
1
J = = (4.62)
t,e 6 72
which is the equation giving the equivalent inertia for the half-length
model of a single-component microbridge. The torsion-related resonant
frequency is obtained by combining Eqs. (4.58) and (4.61), namely,

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.