Page 278 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_277_08/30/05

Resonant Micromechanical Systems

Resonant Micromechanical Systems 277
The dynamic equations of motion for the drive and sense directions
are
˙˙
˙
J ș + c ș + k ș = M sin(Ȧ t)
x x
d x
0
d
d x
˙
˙˙
˙
J (ș +2Ȧș ) + c ș + k ș =0 (5.132)
y y c s y s y
where k =2k 1t k =2k 2t (5.133)
s
d
with k and k being the torsional stiffnesses of the inner and outer
1t
2t
hinges, respectively.
By following an approach similar to the one presented for the linear
drive microgyroscope, it can be shown that the particular solution of the
drive direction is of the form:
ș = Ĭ sin(Ȧ t í ́ )
x x d d (5.134)
M 0 2ȟ ȕ
d d
with Ĭ = ́ = arctan (5.135)
d
x
2 2
k (1 íȕ ) + (2ȟ ȕ ) 2 1 íȕ 2
d d d d d
c d
and ȟ = (5.136)
d 2J Ȧ
x d,r
Similarly, the particular solution corresponding to the sense direction
is expressed as
ș = Ĭ sin(Ȧ t í ́ ) (5.137)
d
y
y
s
M 0,y
with Ĭ =
y 2 2 2 (5.138)
J y (Ȧ s,r íȦ ) + (2ȟ Ȧ Ȧ ) 2
s s,r d
d
and
2 2 2 2
(Ȧ íȦ ) + (2ȟ Ȧ Ȧ )
s,r d s s,r d
2ȟ Ȧ Ȧ + (5.139)
s s,r d 2
í [ M / ( J Ĭ )
0,y s y
́ = 2 arctan
s
2
Ȧ 2 íȦ + M 0,y/ (J Ĭ )
s,r d s y
In Eqs. (5.138) and (5.139),
M 0,y = í 2J ȦȦ d (5.140)
y
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