Page 71 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_70_08/30/05

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

70 Chapter Two
2
EGI A(GAl +3țEI )
y
y
Ȧ sh = 11.832 (2.81)
b,e 2 2 2 2 2 2 4
ml(140ț E I +77țEGI Al +11G A l )
y
y
It can be shown that by neglecting the shear force S , the Timoshenko
z
model produces the equivalent stiffness, mass, and resonant frequency
that are given by the Euler-Bernoulli model in Eqs. (2.61), (2.66), and
(2.67), respectively.
Example: Compare the inertia distribution functions corresponding to
long- and short-beam models. Compare the bending-related resonant fre-
quencies of long and short microcantilevers of constant rectangular cross
section.
The ratio of the distribution function for a long member to that of a short
member can be expressed by Eqs. (2.77) and (2.63) as

sh
f b 2 (8+5Į )l í 4lx í 4x 2
3 2
r = = (2.82)
fb f 3
b (8+ 5Į )(l í x)(2l + x)
where t = Įl (2.83)
The usual relationship between the longitudinal and transverse elas-
ticity moduli [Eq. (2.70)] has been considered here with a Poisson’s ratio of
0.25 (the approximate value for polysilicon) and a shear factor of ൣ, as
1
indicated in Young and Budynas, for instance. Obviously, Į << 1 because,
for thin microcantilevers, t << 1. It is interesting to analyze the ratio of
Eq. (2.82) as a function of Į. This can be done in a graphical manner, and
Fig. 2.23 illustrates this ratio plotted in terms of Į and for x = 0.99l. It can
be seen that for very small values of Į (therefore very thin members), the
effects of shear are small (the ratio is slightly larger than 1) and often can be
neglected.
When we compare the bending resonant frequencies of long and short mi-
crocantilevers of constant rectangular cross section, we use Eqs. (2.67) and
(2.81), and Fig. 2.24 is the two-dimensional plot of the long-to-short bending
resonant frequency ratio.
For very small values of the parameter Į, the resonant frequency provided
by the long-beam model is slightly smaller than the one produced by the
short-beam model. With Į increasing, the roles reverse and the predictions
by the long-beam model exceed those provided by the short-beam model. The
differences between the two models are, however, small, as Fig. 2.24 indi-
cates. It is important to notice that the width w does not enter either of the
two models’ resonant frequency, as it cancels out. The only two parameters
that influence the resonant frequency of a constant rectangular cross section
are the length l and thickness t.

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