Page 14 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 13
ȝ
ȝ = (1.42)
eff 1.159
1 + 9.638Kn
which was shown to be accurate for 0 < Kn < 880.
A concern that is directly related to squeeze-film damping is the
possibility that the fluid layer behaves as an elastic film when film
compressibility becomes a noticeable factor. The predictor that
indicates the possible spring behavior is the squeeze number, which, as
5
shown by Starr, is defined as
2
12ȝȦl
ı = (1.43)
2
z p
0 a
where l is the relevant in-plane dimension of the movable plate and
p is the ambient fluid pressure. For squeeze numbers smaller than 0.2,
a
5
as also indicated by Starr, the film is practically incompressible and
the spring behavior is negligible. For squeeze numbers larger than the
0.2 threshold value, the film behaves as a spring and therefore its
energy dissipation properties are diminished. An even more relaxed
8
condition is proposed by Blech who mentions that for ı < 3 the gas
escapes from the gap that is formed between the movable and fixed
members, and there is no sensible gas compression; whereas for ı > 3,
the gas is trapped in the gap and its compressibility generates the
5
spring behavior. Starr gave the stiffness of the film acting as a spring
and also presented correction functions that have to be applied when
the displacement of the mobile component is comparable to the film
thickness.
In slide film damping (schematically shown in Fig. 1.11b), as
10
9
indicated by Tang, Nguyen, and Howe ; Cho, Pisano, and Howe ; or
11
Zhang and Tang, there are a few flow regimes above and underneath
the moving plate. Figure 1.12 shows the side view of a plate of mass
m, which is attached by a spring of constant k. The mobile plate can be
the finger of a comb drive microtransducer case in which the fixed plate
underneath is the substrate. The distance between the two plates is
constant and equal to z . When the plate oscillates with a frequency Ȧ
0
in the direction shown in the figure, a damping force (of viscous nature)
resulting from the fluid-structure interaction will oppose the plate
motion. There are basically two types of damping forces being generated
by the following flows: a Couette-type flow, which is set between a
mobile plate and a fixed one and where the velocity decays from v (the
mobile plate velocity) at the mobile plate–fluid interface to zero at the
fixed plate–fluid interface; and a Stokes-type flow, which is set both
above and underneath the mobile plate. This flow is turbulent up to a

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