Page 12 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 11
motion
motion
direction
direction
moving plate moving plate
z
y
z 0
x fixed plate fixed plate

(a) (b)
Figure 1.11 Structure-fluid interaction damping: (a) squeeze-film; (b) slide-film.

2
2
˜ p ˜ p 12ȝ ˜z
+ = (1.35)
2 2 3 ˜t
˜x ˜y z 0
Equation (1.35), where p is the pressure in the fluid film and ȝ is the
dynamic viscosity coefficient, is a Poisson-type equation which can be
integrated (particularly when the contacting area of the two plates is
elementary, such as rectangular or circular) and the pressure is the
solution to it. Equation (1.35) is only valid, as mentioned by Starr, 5
when the Reynolds number Re satisfies the condition

ȡȦz 2
Re = <<1 (1.36)
ȝ
The Reynolds number is proportional to the ratio of inertia (turbulence)
to viscosity, and therefore Eq. (1.36) requires that viscosity dominate.
In addition to this Re number inequality, several other conditions need
to be complied with in order for the Poisson equation, Eq. (1.35), to be
valid, and these conditions and/assumptions are enumerated next.
The fluid film should be in isothermal condition (which ensures that
ȡ§ p — and this simplifies the original Navier-Stokes equations); the
pressure variations within the film are assumed to be small compared
to the average pressure; the film thickness is uniform; the displace-
ments by the movable plate are small compared to the film thickness;
and the fluid is incompressible.
As mentioned previously, by integrating Poisson’s equation for
pressure and then determining the damping force (as a viscous one
which is equal to the damping coefficient times the velocity), the
damping coefficient can be obtained for a specified surface geometry. In
5
the case of a circular conjugate surface, as mentioned by Starr and by
6
Andrews, Harris, and Turner, the damping coefficient is

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