Page 122 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_121_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 121
y

w2 w1
x

1 2 1 1
Figure 3.11 Top view of trapezoid paddle microcantilever.

3
2
2
2
2
2
2
ȡt{20l w (w + t ) +15l l (w + w )(2t + w + w )
2 2 2 1 2 1 2 1 2
2
3
3
2
3
2
+l 10w +6w w +3w w + w +5t (3w + w )
1 1 1 2 1 2 2 1 2
(3.46)
+2l l 10t (2w + w ) +3(4w +3w w +2w w + w ) }
3
2
2
2
3
2
1
1 2
1
2
1
2
1 2
2
J =
t,e 2
720(l + l )
1
2
The torsional resonant frequency is too complex and is not calculated
here, but it can simply be found by means of Eq. (3.46).
The bending stiffness is
3
Et w (w í w ) 3
1
2
2
k b,e =
2{ (w í w ) [2(w í w ) l (3l +3l l + l )
2
2
2
1 2 1 2 2 1 1 2 2 (3.47)
3
3
3
+3l w (w í 3w )]+ 6l w ln(w 1/ w )}
1 2 1 2 1 2 2
The effective bending mass is
7
6
6
ȡt 264l w +56l l (26w +7w ) +56l l (10w +23w )
2 2 1 2 1 2 1 2 1 2
3 4
2 5
+56l l (50w +49w ) +70l l (83w +49w )
1 2 1 2 1 2 1 2
4 3
5 2
+56l l (116w +49w ) +28l l (149w +49w ) (3.48)
1 2 1 2 1 2 1 2
7
+l (215w +49w )
1 1 2
m =
b,e 6
1120(l + l )
1
2
Again, the bending resonant frequency is found by means of Eqs. (3.47)
and (3.48) and is not included here. All the lumped-parameter stiff-
nesses and inertia fractions were checked by reformulating them for
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