Page 15 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

14 Chapter One
laminar flow

δ
Stokes flow
c
v m
k
t
z 0
Couette flow

Fixed plate (substrate)
Figure 1.12 Slide-film flow regimes in the fluid surrounding a movable plate–fixed plate
couple (side view).

distance į (called the penetration distance) and becomes stationary past
that distance. The penetration distance is calculated as
2ȝ
į = (1.44)
ȡȦ

and represents the distance corresponding to the position where the
velocity becomes
Ȟ
Ȟ = e (1.45)
į

where e is the natural logarithm base.
Figure 1.13 is the top view of the movable plate shown in relationship
with a similar fixed plate, which is located at the same height z 0 from
the substrate. The fixed and movable plates of Fig. 1.13 are the
corresponding fingers in a comb microdevice, which is utilized for
electrostatic transduction.
With reference to Fig. 1.12, the Couette flow damping can be
quantified by the following quality factor, as shown by Cho, Pisano, and
Howe: 10
ʌ 2 ȝ
Q C, u = Ȟ (1.46)
Ȧ
z
0
The subscript u indicates the zone underneath the movable plate. Sim-
ilarly, and with reference to Fig. 1.13, the Couette flow quality factor

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