Page 58 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_57_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 57

m a,e

u x
Figure 2.8 Effective mass corresponding to free axial vibrations of a fixed-free microrod.

By equating Eqs. (2.37) and (2.38), the effective mass corresponding
to axial free vibrations is
l
2
= ȡ
m a,e ฒ f (x)A(x) dx (2.39)
a
0
and is physically represented by the body mass which is placed at the
cantilever’s free end, as shown in Fig. 2.8.
Torsional vibrations. In torsion, the effective inertia which corresponds
to the free vibrations is sketched in Fig. 2.9. An approach similar to the
one applied to axial free vibrations is used here, according to which the
kinetic energy of the distributed-parameter system is
l dș (x) 2
1 x
ȡ
T = 2 ฒ I (x) dx (2.40)
t t dt
0
The kinetic energy of the effective, lumped-parameter system is
1 dș x 2
T t,e = 2 J t( dt ) (2.41)

Based on Rayleigh’s assumption, the following relationships can be
written:

dș (x) dș x
x
ș (x) = f (x)ș x dt = f (x) dt (2.42)
x
t
t
By equating Eqs. (2.40) and (2.41), the lumped-parameter effective in-
ertia fraction corresponding to free torsional vibrations is found to be
l
2
= ȡ
J t,e ฒ f (x)I (x) dx (2.43)
t
t
0

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