Page 101 - Mechanical design of microresonators _ modeling and applications

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

100 Chapter Two
cross section; both have been previously discussed when we analyzed
straight-line members earlier in this chapter.
The circular ring can also undergo bending vibrations, as indicated
5
in Fig. 2.36b. Timoshenko suggests the following equation for resonant
frequency calculations:
2 2
1 i (i í 1) EI z
Ȧ b,i = 2 2 (2.202)
R i +1 ȡA
where i is the mode number. For i =1, the result is a rigid body motion
(no resonant frequency), and the first resonant frequency is obtained
for i = 2 as

EI z
Ȧ =6.726 (2.203)
b
mR 3
For mixed torsional-bending vibrations the modal frequencies change
to

2 2
1 i (i í 1) EI z
Ȧ = (2.204)
t í b,i 2 2 ȡA
R i +1+ ȝ
where Í is Poisson’s ratio. The first resonant frequency (i = 2) becomes

EI z
Ȧ = 15.04 (2.205)
t í b 3
(5+ ȝ )mR
Example: Compare the resonant frequencies corresponding to axial, tor-
sional, bending, and torsional-bending vibrations of a circular ring having
very thin rectangular cross section of thickness t and width w (t<<w).
It can be shown that the axial and torsional resonant frequencies are re-
lated as
Ȧ a
Ȧ = (2.206)
t 2
Also, by comparing Eqs. (2.203) and (2.205) it follows that

Ȧ
b
Ȧ =0.447 5+ ȝ (2.207)
t í b
For polysilicon where Í = 0.25, Eq. (2.207) results in
Ȧ =1.024Ȧ
b t í b (2.208)

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