Page 250 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_249_08/30/05

Resonant Micromechanical Systems

Resonant Micromechanical Systems 249
By applying Lagrange’s equations the following differential equations sys-
tem is obtained:
.. .. 2
mu +4k u =0 J ș + (4k + k w )ș =0 (5.51)
z l z C x t l 2 x
which indicates that the two motions are dynamically decoupled, according
to previous definitions introduced in Chap. 1. The mass and stiffness matri-
ces corresponding to Eqs. (5.51) are

4k
m 0 l 0
M = K = (5.52)
0 J C 0 4k + k w 2
r l 2
The dynamic matrix can now be formed as
4k
l 0
m
í1
A = M K = 2 (5.53)
4k + k w
0 r l 2
J
C
and its eigenvalues yield the following resonant frequencies:
2 2
2t 1 t w (16Gl +3Ew ) t 1 Et w
1 1
2
1
1 1
Ȧ = Ȧ = (5.54)
1 l w ȡl l t w 2 l ȡl l t w
1 2 1 2 2 2 2 1 2 2 2
In the particular case where the thickness of the middle plate is equal to that
of the four identical root legs, Eqs. (5.54) simplify to
2
2
w (16Gl +3Ew ) Ew
2t 1 1 2 t 1
Ȧ = Ȧ = (5.55)
1 l w ȡl l w 2 l 2 ȡl l w
1 2 2
1 2
1 2 2
Equations (5.54) and (5.55) indicate that the resonant frequency Ȧ 1 com-
bines both torsion and bending effects (through the simultaneous presence
of the elastic modulii E and G), whereas the other resonant frequency Ȧ 2
contains only the bending contribution. The combined effect of torsion and
bending in the resonant frequency Ȧ 1 is normal as rotation of the middle plate
around its central x axis results through the mixed torsion-bending defor-
mation of the four root legs.
Example: Compare the resonant frequencies of the four-leg microbridge
sketched in Fig. 5.22.
The following resonant frequency ratio can be formulated by using
Eqs. (5.54) and (5.55):

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