Page 62 - Mechanical design of microresonators _ modeling and applications

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Basic Members: Lumped- and Distributed-Parameter Modeling and Design

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 61

t
w
Figure 2.13 Constant rectangular cross-section microbar.

Torsional vibrations. A similar path can be followed to determine the
resonant frequency of a fixed-free bar undergoing torsional vibrations.
The lumped stiffness at the free tip of the bar corresponding to torsional
deformation is
GI t
k t,e = l (2.51)

The torsional moment of inertia of a constant rectangular cross-section
with t << w (this configuration is referred to as very thin) can be ex-
7
pressed (see Boresi, Schmidt, and Sidebottom, for instance) as
wt 3
I = 3 (2.52)
t
There are designs where the cross section is relatively thick, with the
dimensions w and t being comparable, as sketched in Fig. 2.13.
1
2
For such designs, as shown in Young and Budynas or Lobontiu, the
torsional moment of inertia can be approximated to
(
3
I = wt 0.33 ෹ 0.21t ) (2.53)
t w
Circular cross-section microbars, such as those constructed of carbon
nanotubes, as shown in Fig. 2.14, are defined by the torsional moment
of inertia

ʌd 4
I = 32 (2.54)
t
where d is the diameter.

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