Page 162 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_161_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 161

6 (l + l ) i +1 í l i +1
ƍ 1 p 1
where l = 140l + ™ c i i / 33 (3.175)
p
p
i =1 l
i  5
with c 1 = í210, c 2 = 105, c 3 = 35, c 4 = í42, and c 6 = 5. Equation (3.174)
reduces to

33m 1
m = 140 (3.176)
b
in the case where there is no patch on the microcantilever (m p = 0), and
this is the known relationship, giving the equivalent mass of a homo-
geneous microcantilever.
The bending-related resonant frequency is now computed by means
of Eqs. (3.172) and (3.174) as

wl p
Ȧ b,e =1.03 ƍ (3.177)
(l m + l m )c b
p
p
p
1
2
3
2
l í l (3l +3l l + l )
p
1
1 p
p
with c =
b
E t 3
1 1
(3.178)
2 2
l 3l +3l l + l p)( 1 1 p p)
E t + E t
p( 1
1 p
+
2 4
2
2
2 4
E t + E t +2E E t t 2t +3t t +2t p)
1 1 p p 1 p 1 p( 1 1 p
Obviously, Eq. (3.177) reduces to that equation giving the bending res-
onant frequency of a homogeneous microcantilever when l p ĺ 0,
t p ĺ 0, and m p ĺ 0.
Note: Multimorph microcomponents, in either the equal- or dis-
similar-length configurations, are generally constructed of thin layers
(the substrate included), and therefore shearing effects (and the
corresponding model alterations imposed by the Timoshenko model)
can be ignored.
Example: Study how the patch position and length (quantified by l 1 and l p
in Fig. 3.35) of a bimorph microcantilever influence the bending resonant
3
frequency. Given: E 2 = 150 GPa, E 1 = 120 GPa, ȡ 1 = 2300 kg/m , ȡ 2 = 2500 kg/
3
m , l = 1500 “m, and w = 100 “m. Consider that the monitored parameters
range within the following intervals: l 1 ĺ [800 “m, 2000 “m], l p ĺ [50 “m,
100 “m].
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