Page 126 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_125_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 125
z

t2

x
t1

1 2 1 1
Figure 3.16 Side view of trapezoid paddle microcantilever.

3
2
2
ȡw 4l t +6l l (t + t ) +4l l (t +2t )
2 2 1 2 1 2 1 2 1 2
3
+l (t +3t ) (3.54)
1
1
2
m a,e =
12(l + l ) 2
1
2
The axial resonant frequency is
Et (t – t )
2 1
2
3.46(l + l )
1 2
t )]
ȡ [l (t – t ) + l t ln(t 1/ 2 (3.55)
2 1
2
1 2
Ȧ =
a,e 3 2 2 3
4l t +6l l (t + t ) +4l l (t +2t ) + l (t +3t )
1 2 1
1
1
2
2
2
1
2 2
1 2
The torsional stiffness is computed by inverting the torsional
compliance, which is calculated according to its definition in Eq. (2.26),
and it is
2 3
2Gt t w
1 2
k = (3.56)
t,e 2
3 2l t + l t (t + t )
1 2 1
2
2 1
The effective torsional mechanical moment of inertia is
2
2
2
2
2
3
2
ȡw{20l t (w + t ) +15l l (t + t )(2w + t + t )
2 2 2 1 2 1 2 1 2
3
3
2
3
2
2
+l 10t +6t t +3t t + t +5w (t +3t )
1 2 1 2 1 2 1 1 2 (3.57)
+2l l 10w (t +2t ) +3(4t +3t t +2t t + t ) }
2
3
3
2
2
2
1 2 1 2 2 1 2 1 2 1
J =
t,e 2
720(l + l )
1 2
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