Page 16 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 15

l x

v movable plate

y 0

fixed plate

Figure 1.13 Slide-film flow in the fluid surrounding a movable plate–fixed plate couple
(top view).

corresponding to the fluid damping generated through the relative mo-
tion on the side faces between two comb fingers is
ʌ 2 ȝ
Q C,s = Ȟ (1.47)
Ȧ
y
0
where the subscript s points to the side region of the conjugate plates.
Stokes-type flow is responsible for damping produced both above and
underneath the movable plate shown in Fig. 1.12. The turbulent Stokes
flow which extends over the penetration height į is characterized by the
quality factor
ʌ 2 ȝ
Q S,a = Ȧ Ȟ į (1.48)

10
According to Cho, Pisano, and Howe, the Stokes flow damping under-
neath the moving plate is
ʌ 2 ȝ
Q S,u = Ȧ Ȟ į f (ȕ) (1.49)

sinh(2ȕz ) +sin(2ȕz )
0
0
where f (ȕ) = (1.50)
cosh(2ȕz ) ෹ cos(2ȕz )
0 0
and
1
ȕ = (1.51)
į

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