Page 121 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

120 Chapter Three

40000
ω [rad/s] 0.9
0
0.5
ct
c1
0.1
1.5
Figure 3.10 Bending resonant frequency in terms of length and thickness parameters.

Another paddle-type microcantilever comprising a rectangular por-
tion at its root, connected to a trapezoid at its tip, is sketched in
Fig. 3.11, which also gives the geometric parameters defining this
configuration. The axial stiffness is
Etw (w – w )
k a,e = 2 1 2 (3.43)
l (w – w ) + l w ln(w 1/ w )
2
2
1
1 2
2
The effective mass associated to the free axial vibrations is
2
3
2
ȡt 4l w +6l l (w + w ) +4l l (2w + w )
1
2
2
1
2
1 2
1 2
2
3
+l (3w + w ) (3.44)
1
2
1
m a,e =
12(l + l ) 2
2
1
The corresponding resonant frequency is
Ew (w – w )
3.46(l + l ) 2 1 2
1 2
ȡ [l (w – w ) + l w ln(w 1/ 2
w )]
2
1 2
1
2
Ȧ =
a,e (3.45)
3
2
2
4l w +6l l (w + w ) +4l l (2w + w )
2 2 1 2 1 2 1 2 1 2
3
+l (3w + w )
1 1 2
The torsional stiffness, again, is connected to the axial one by means
of Eq. (3.20). The torsion-related effective mechanical moment of inertia
is
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