Page 82 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_81_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 81
To calculate the exact effective inertia corresponding to free bending
vibrations, the distribution function of Eq. (2.94) has to be utilized,
where

l 2
12
C (x) = Ewฒ x dx
l 3
x t(x)
l 2
12
C = Ewฒ x dx (2.122)
l 3
0 t(x)
l xdx
12
C (x) = Ewฒ 3
c
x t(x)
The effective mass is calculated as
l
2
m b,e = ȡw ฒ t(x) f (x)dx (2.123)
b
0
After we perform the calculations necessitated in Eq. (2.123), the
bending-related effective mass is
5 4 3 2 2 3
ȡlw{(t í t )(9t í 75t t + 197t t í 155t t
1
1 2
1 2
1
2
1 2
4 5 2 2 2 2 2 2
í44t t +8t ) í 12t t {6t (ln t í ln t ) + 12t lnt 2
2
1 2
2
1 2
2
1
2
t )}} (2.124)
í(2t í 7t )(2t í 3t ) ln(t
1 2 1 2 1/ 2
m b,e = 2 2
t )
36(t í t ) (3t í t )(t í t ) +2t ln(t 2/ 1
2
1
1
2
2
2
1
which is quite involved. The effective bending mass can also be calcu-
lated by using the distribution function of Eq (2.63), which corresponds
to a constant-cross-section microcantilever, instead of the exact distri-
bution function of Eq. (2.94). In doing so, the following effective mass is
obtained:
ȡlw(215t +49t )
1
2
*
m b,e = 1120 (2.125)
which is considerably simpler than what Eq. (2.124) yielded. Both
Eqs. (2.124) and (2.125) reduce to Eq. (2.66) which gives the effective
bending inertia of a constant-cross-section microcantilever when t 2 ĺ t 1 .
Again, a comparison has been performed between the effective masses
provided by the two distribution functions through analyzing the mass
ratio:
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