Page 256 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

Resonant Micromechanical Systems 255

100 0.1
r ω
1-2
0
1
c t ´

c l 0.001
10
Figure 5.27 Resonant frequency ratio, the first of Eqs. (5.76).

Ȧ 1 l w E
2 1
rȦ = = 3
1í2 Ȧ t 2 2
2 1 16Gl +3El
1 2
(5.76)
Ȧ l 2
3 E( l )
rȦ = 2 = 1+ 16 G 1
2í4 Ȧ
4 2
By using the nondimensional variables
l 2 t 1
ƍ
c = c = (5.77)
l l t w
1 1
the three-dimensional plot of Fig. 5.27 and the two-dimensional plot of
Fig. 5.28 are obtained.
As Fig. 5.27 indicates, the resonant frequency Ȧ 1 can be 100 times larger
than the next resonant frequency Ȧ 2 , especially when the nondimensional
parameter cƍ t (to which the frequency ratio is particularly sensitive) is small.
The last resonant frequency is the smallest of the three, as indicated in
Fig. 5.28, which also shows that the resonant frequencies Ȧ 2 and Ȧ 4 tend to
be equal for large values of the plate dimension l 2 .

Example: Determine the resonant frequency of the system shown in Fig. 5.29
by considering only the two rotation DOFs (the ones produced through tor-
6
sion). Harrington, Mohanty, and Roukes utilized this design to study energy
dissipation issues in micromechanical resonators.
The microdevice of Fig. 5.29 can be modeled as a 2-DOF system, the gen-
eralized coordinates being the two rotation angles ș 1x and ș 2x . The kinetic
energy of the system is

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