Page 20 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 19
warmer compressed fibers

energy flow temperature gradient

colder extended fibers
Figure 1.18 Thermoelastic energy dissipation (TED) mechanism in a bent beam.

2 (
2 2 2 4
2 )
1.57ț 0.4c ȡ f t
p
r
Q TED = 1+ (1.55)
EĮ Tf t 2 ț
r
where f (f = 2ʌȦ ) is the resonant frequency and t is the cross-sectional
r
r
r
thickness. For a microcantilever (a fixed-free beam) Eq. (1.55) simplifies
to
2
2 0.42c E ȡt 6
ȡ
1.55țl E( 2 4 )
p
Q TED = 1+ (1.56)
2
EĮ Tt 2 ț l
where l is the cantilever length.
Example: Analyze the TED quality factor for a silicon nitride microcan-
3
–6
í1
tilever with E = 126 GPa, ȡ = 3440 kg/m , Į = 3 × 10 K , c p = 710 J/(kg • K),
and ț = 3.2 W/(m • K) for a temperature of 300 K.
Figure 1.19 highlights the fact that the quality factor owing to thermoe-
lastic losses registers a minimum for thicknesses in the range of 5 to 20 ȝm.
For thicknesses that are outside that range, the quality factor is not so sever-
ely affected by the TED mechanism. Figure 1.20 is another (two-dimensional)
plot showing the temperature influence on the quality factor for a silicon
nitride nanocantilever defined by a thickness t = 100 nm and a length
l = 60 ȝm.
The vast majority of microresonators are fabricated as monolithic systems
with virtually no friction losses due to the mating parts, losses which are
present in macroscale rotation bearings, for instance. However, losses are
still produced by energy dissipation through the anchors that attach the res-
16
onator to the substrate. Hosaka, Itao, and Kuroda, as well as Yasumara
15
et al., mention that the quality factor which expresses the clamping losses
for cantilevers can be determined by analyzing the vibrational energy which
is transmitted from the cantilever to the substrate (viewed as an infinite
elastic plate) and gives an evaluation of
l 3
Q § 2.17 ( ) (1.57)
cl t

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