Page 206 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_205_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 205
applying the generic model developed in this section for a three-segment
microbridge.
The torsional stiffness associated with the midpoint of the micro-
bridge is
3
4Gt w w
1 2
k t,e = (4.128)
3(w l +2w l )
1 2 2 1
It can be seen that the stiffness expressed in Eq. (4.128), which corre-
sponds to the full-length paddle microbridge, is twice the torsional
stiffness of the half-microbridge [Eq. (4.110)].
The effective torsional mechanical moment of inertia is

2
2
2
2
3
2ȡt 2l w (16l +25l l +10l )(w + t )
1 1 1 1 2 2 1
4
2
2
2 2
4
3
3
+l w (w + t )(30l +60l l +40l l +10l l + l ) (4.129)
l 2
1
1 2
2
1 2
2
2 2
J t,e =
45(2l + l ) 4
2
1
When l 1 = l 2 = l/3 and w 1 = w 2 , Eq. (4.129) reduces to Eq. (4.33), which
provides the effective mechanical moment of inertia of a constant-cross-
section microbridge of length l. The resulting torsional resonant fre-
quency is, by way of the stiffness Eq. (4.128) and mechanical moment
of inertia Eq. (4.129),
1 /
5.48t(2l + l ) 2 Gw w 2 ȡ(w l +2w l )
2 1
1 2
2
1
Ȧ =
t,e 3 2 2 2 2
2l w (16l +25l l +10l )(w + t ) (4.130)
1 1 1 1 2 2 1
4
3
2 2
3
2
4
2
+l w (w + t )(30l +60l l +40l l +10l l + l )
2
2
1 2
1 2
2
2
1
l 2
Obviously, for the same limit conditions as above, namely l 1 = l 2 = l/3
and w 1 = w 2 , Eq. (4.130) reduces to Eq. (4.34), which yields the torsional
resonant frequency of a constant-cross-section microbridge of length l.
The lumped-parameter bending stiffness, which is associated with
the midpoint of the paddle microbridge, is
3
16Et w w (w l +2w l )
1 2
2 1
1 2
k b,e = (4.131)
2
2 4
2
2 4
w l +8l l (4l +3l l + l )w w +16w l
1 2 1 2 1 1 2 2 1 2 2 1
and this is twice that given in Eq. (4.123) for a half-length paddle
microbridge.
The bending-related effective mass is
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