Page 148 - Mechanical design of microresonators _ modeling and applications

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Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 147
2 2 3
(4Gl +3El )w t
2
1
1
k t,e = (3.114)
6l 3
1
The effective mechanical moment of inertia, which is dynamically
equivalent to the distributed inertia of the hollow rectangular micro-
cantilever undergoing free torsional vibrations, is
2
2
2
2
2J t1 ȡt 2l w (w + t ) +3l w (w + t )
2
1
1 1
2 2
J t,e = 3 + J = 36 (3.115)
t2
The torsion resonant frequency is
2
2
2.45t (4Gl +3El )w 1
2
1
Ȧ = (3.116)
t,e l 2 2 2 2
1 ȡl 2l w (w + t ) +3l w (w + t )
2
1
1 1
1
2 2
The bending stiffness is
ew t 3
1
k b,e = (3.117)
2l 1 3
The effective mass in bending is
2 (
2 2)
33 33l w
1 1
m =2 m + m = ȡt + l w (3.118)
b,e 140 b1 70
and the bending-related resonant frequency is
5.92t Ew 1
Ȧ = (3.119)
b,e l 1 ȡl (33l w +70l w )
2 2
1 1
1
Example: Perform a comparison similar to that of the previous example, by
studying the bending resonant frequency of a solid constant rectangular
cross-section microcantilever in contrast to that of a hollow configuration. It
is assumed that both configurations have the same geometric envelope, and
identical thicknesses, and that they are built from the same material.
The sought frequency ratio is determined by using Eqs. (2.67) and (3.119),
which give the bending resonant frequencies of a solid constant rectangular
cross-section microcantilever and of a hollow rectangular one, respectively.
By using the nondimensional parameters:
w 1 w 2
c = c = (3.120)
1 l 2 2 l 1

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