Page 102 - Mechanical design of microresonators _ modeling and applications

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Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 101
which indicates that the bending and torsion-bending resonant frequencies
are almost identical for very thin rectangular cross-section rings.
A comparison between the torsional and bending resonant frequencies
[Eqs. (2.201) and (2.203)] leads to the relationship

Ȧ
t = 0.036 R
Ȧ w (2.209)
b
which shows that the torsional frequency is the smallest for designs where
R/w > 1/0.036 = 27.46.

2.3.3 Thin plates and membranes
Thin plates and membranes are utilized as microresonators in fluidic
applications, for instance, where bending resonant vibration of a thin
member (usually clamped along its contour) realizes pumping of the
fluid in or out of a cavity.
Thin plates (often referred to as Kirchhoff plates) are characterized
by their small thickness compared to the other two (in-plane) dimen-
sions; generally, a plate is considered thin when its thickness is less
than one-twentieth of the smallest in-plane dimension. Such a plate is
defined by its middle surface, and planes that are perpendicular to the
original (undeformed) middle surface remain planar and perpendicular
on the deformed middle surface. Another modeling feature/assumption
of thin plates is that the middle surface does not stretch or compress
under load. A thin plate can be defined by its deflection w (deformation
out of the plane about the z direction), and this is a function of both x
and y (see Fig. 2.38).
In characterizing the free bending response of a thin plate, the out-
of-the-plane deformation w is of the form:

w(x, y, t) = w (x, y)sin(Ȧt) (2.210)
0
5
It can be shown (as demonstrated by Timoshenko, for instance) that
the bending resonant frequency is given by the Rayleigh procedure

z

y

x

t
Figure 2.38 Thin plate of constant thickness.

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