Page 104 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_103_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 103

x 2 x 2 y 2 y 2
1 í
w (x, y) = a 2( ) ( ) (2.214)
1 í
0 l 2 l
l 1 1 l 2 2
By taking w 0 (x, y) of Eq. (2.214) and substituting it into Eqs. (2.211)
and (2.212), the following bending resonant frequency is obtained:
2 2
4
4
1 2
1
2
2
Ȧ = 6Et 2 7(l + l ) +4l l (2.215)
2
4 4
ȡ(1 íȝ ) l l
1 2
A similar expression can be determined for a circular thin plate by using
the generic formulation of Eqs. (2.211) and (2.212) and an appro-
5
priate w 0 function. Timoshenko indicates the following equation:
10.21 D
Ȧ = (2.216)
R 2 ȡt
While a thin plate is subjected to bending, a membrane (which can
be structurally identical to a plate) is acted upon by a uniform
stretching of its middle surface in a manner that would make negligible
5
the small deflections occurring during vibration (Timoshenko ). A thin
plate can be seen as the two-dimensional counterpart of a long beam,
whereas the membrane is the two-dimensional correspondent of a bar
subject to tension. We can find the first resonant frequency of a
membrane by applying Rayleigh’s procedure again and therefore by
equating the maximum potential energy to the maximum kinetic
5
energy. It can be shown (see Timoshenko, for instance) that the
resonant frequency of a generic membrane is calculated as
2
0/
0/
ฒ (˜w ˜x) + (˜w ˜y) 2 dA
2
Ȧ = s A (2.217)
ȡt ฒ w dA
2
A 0
where s represents a uniform tension per unit length of the boundary
(measured in newtons per meter, for instance), and w is the deflection
0
of the membrane–it was defined in Eq. (2.210)–and should con-
veniently chosen to satisfy the boundary conditions of the membrane.
5
Timoshenko shows that the lowest resonant frequency of the rectan-
gular membrane sketched in Fig. 2.39 is
s 1 1 2)
Ȧ =0.5 + (2.218)
ȡt( 2 l
l
1 2

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