Page 119 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

118 Chapter Three
z

t2

t1
x

1 2 1 1
Figure 3.9 Side view of a paddle microcantilever.

By applying the composition rules that have been derived for two-
segment microcantilevers, the axial stiffness is
Et t w
1 2
k = (3.34)
a,e l t + l t
2 1
1 2
The effective mass corresponding to free axial vibrations is
3
2
2
ȡ l (l +3l l +3l )t + l t w
2
1 1
1 2
1
2 2
m = (3.35)
a,e 2
3(l + l )
1 2
The resulting axial resonant frequency is
Et t
Ȧ =1.73(l + l ) 1 2 (3.36)
a,e 1 2 2 2 3
ȡw(l t + l t ) l (l +3l l +3l )t + l t
2 2
1 2
2 1
1 1
2
1 2
1
The torsional stiffness is determined by means of Eq. (3.20) from the
axial one. The effective torsional mechanical moment of inertia is
2
2
3
2
2
2
2
ȡw l t (l +3l l +3l )(t + w ) + l t (t + w )
1
2 2 2
2
1 1 1
1 2
J = (3.37)
t,e 2
36(l + l )
1 2
The torsional resonant frequency is
Gt t
1 2
3.46t t (l + l )
2
1 2 1
3
3
ȡ (l t + l t ) (3.38)
Ȧ = 1 2 2 1
t,e 2 2 2 2 3 2 2
l t (l +3l l +3l )(t + w ) + l t (t + w )
2 2 2
2
1
1 2
1 1 1
The bending stiffness is
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