Page 78 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_77_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 77

z y
α
t t 1
t 2
w
x x
l

(a) (b)
Figure 2.26 Variable-thickness trapezoid microcantilever: (a) side view; (b) top view.

The lumped-parameter mass which is dynamically equivalent to the
distributed-parameter inertia of this microcantilever is calculated by
considering two possibilities of defining the distribution function,
namely, first, as provided by the generic Eq. (2.88), which takes into
consideration that the cross section is variable and, second, by using
Eq. (2.48), which assumes the cross section is constant. In the first
variant, the two compliances defining the distribution function of
Eq. (2.88) are
l l
1 dx 1 dx
C (x) = C = (2.108)
a Ewฒ t(x) a,e Ewฒ t(x)
x 0
The lumped-parameter mass is calculated as
l
2
m = ȡw ฒ t(x) f (x)dx (2.109)
a,e a
0
In both Eqs. (2.108) and (2.109), the variable thickness is determined
as

t — t 1
2
t(x) = t + x (2.110)
1 l
The effective mass is
2 2 2
t )
ȡlw{t — t —2t 1+ln(t 2/ 1 2/ 1 }
t ) ln(t
1
2
1
m a,e = (2.111)
2
4(t — t )ln (t 2/ 1
t )
1
2
This equation reduces to Eq. (2.49), which provides the effective mass
of a constant rectangular cross-section microcantilever, when t 2 ĺ t 1 .
As mentioned previously, the effective mass in axial vibrations can
also be calculated by using the distribution function of Eq. (2.48)–
which corresponds to a constant-cross-section member–instead of the
distribution function that has just been used. The objective here is to
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