Page 205 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_204_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

204 Chapter Four
principle. The distribution function which expresses the deflection at a
generic point x on the half-microbridge in terms of the maximum tip
deflection is, in this case,
2
2/
2/
(l + l 2 Ì 2x) (l + l 2+4x)
1
1
f (x) = (4.124)
b 3
2/
(l + l 2)
1
The lumped mass which is dynamically equivalent to the distributed
inertia of the half-microbridge undergoing free bending vibrations is
calculated as
2/
l 2 l +l 2
2/
1
2
2
m b,e = ȡt w 2 œ f (x) dx + w 1 œ f (x) dx (4.125)
b
2/
0 l 2 b
and its final equation is
5 2 2 6
ȡt 32l (52l +112l l +63l )w + l (2240l 1
1 2
1
1
1
2
2
5 4 2 3 3 2 4
+6720l l +7280l l + 3640l l + 1092l l
1 2
1 2
1 2
1 2
(4.126)
5 6
+182l l +13l )w 2
2
1 2
m =
b,e 6
70(2l + l )
1 2
This equation, again, reduces to Eq. (4.4)–expressing the lumped mass
for a constant-cross-section half-length microbridge–when w = w , l 1
2
1
= l/4 and l 2 = l/2. The lumped-parameter resonant frequency corre-
sponding to free bending vibrations is calculated with the aid of Eqs.
(4.123) and (4.126) as
2 4
2 1 /
Ew w (w l + w l ) {ȡ w l +8l l (4l 1 2
1 2
1 2
1 2
1 2
3
23.664(2l + l ) t
2
1
+3l l + l )w w +16w l }
2 4
2
2 1
1 2
1 2
2
Ȧ =
b 5 2 2
32l (52l + 112l l +63l )w 1 (4.127)
1
1 2
1
2
6 5 4 2
+ l (2240l + 6720l l + 7280l l
1
1 2
1 2
2
6
3 3
2 4
5
+3640l l +1092l l +182l l +13l )w 2
1 2
1 2
1 2
2
Generic-model approach. The torsion and bending resonant frequencies
can also be determined for the paddle microbridge of Fig. 4.19 by
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