Page 67 - Mechanical design of microresonators _ modeling and applications

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

66 Chapter Two
distribution law and function are different from those of Eqs. (2.16) and
(2.18), which dealt with the stiffness approach.
The kinetic energy of the distributed-parameter microcantilever
corresponding to bending about the z axis is

l . 2 l
ȡA . 2 ȡAu z 2
T = 2 ฒ u (x) dx = 2 ฒ f (x) dx (2.64)
z
b b
0 0
The kinetic energy of a mass m b,e which is placed at the free end of the
microcantilever is simply
u ˙
m b,e z 2
T = (2.65)
b,e 2
By equating Eqs. (2.64) and (2.65), the equivalent mass m b,e is found
to be

33
m b,e = 140 m (2.66)

where m is the total mass of the microcantilever. By combining
Eqs. (2.1), (2.61), and (2.66), the bending resonant frequency of the mi-
crocantilever becomes
EI y
Ȧ =3.567 (2.67)
b,e 3
ml
As will be demonstrated in Chap. 5, the exact value of the bending res-
onant frequency can be calculated by integrating a differential equa-
tion, and its value is
EI z
Ȧ =3.52 (2.68)
b 3
ml
In other words, the relative error in the first resonant frequency cal-
culated by the approximate Eq. (2.67) versus the exact Eq. (2.68) is
1.33 percent.

Example: Compare the axial, torsional, and bending resonant frequencies
of a long microcantilever having a constant and very thin rectangular cross
section (t << w).
If we use the following nondimensional parameters
t w
Į = ȕ = (2.69)
l l

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