Page 275 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_274_08/30/05

Resonant Micromechanical Systems

274 Chapter Five
c s
where F = í 2mȦȦ X ȟ = (5.122)
0,y d s 2mȦ
s,r
and Ȧ s,r is the resonant frequency of the sense branch of the gyroscope. The par-
ticular solution to this equation is

y = Y sin(Ȧ t í ́ ) (5.123)
p
s
d
The amplitude Y is
F 0,y
Y = (5.124)
2 2
m (Ȧ 2 íȦ ) + (2ȟ Ȧ Ȧ ) 2
s,r d s s,r d
and the phase of the particular solution of Eq. (5.123) is

2ȟ Ȧ Ȧ
s s,r d
2 2
2
+ (Ȧ 2 íȦ ) + (2ȟ Ȧ Ȧ ) í F 0,y / (mY) 2
́ =2 arctan s,r d s s,r d
s 2 2 (5.125)
Ȧ íȦ + F 0,y / (mY)
s,r d
The sense amplitude can be reformulated by using Eqs. (5.119) and
(5.122) as
2 Ȧ FȦ d 0
Y = (5.126)
2 2
2 2
mȦ 2 Ȧ 2 (1 íȕ ) + (2ȟ ȕ ) 2 (1 íȕ ) + (2ȟ ȕ ) 2
d,r s,r d d d s s s
Ȧ d
where ȕ = (5.127)
s Ȧ
s,r
The following situations are possible:

1. Resonance in the drive branch: Ȧ d = Ȧ d,r . Because ȕ d = 1, the sense
amplitude of Eq. (5.126) becomes
ȦF 0
Y =
d 2 2 2 (5.128)
ȟ mȦ d,r s,r (1 íȕ ) + (2ȟ ȕ ) 2
Ȧ
s
d
s s
2. Resonance in the sense branch: Ȧ d = Ȧ s,r . In this situation ȕ s = 1, and
therefore the sense amplitude of Eq. (5.126) transforms to
ȦF 0
Y = (5.129)
s
2 2
ȟ mȦ 2 Ȧ (1 íȕ ) + (2ȟ ȕ ) 2
s d,r s,r d d d

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