Page 61 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_60_08/30/05

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

60 Chapter Two

u x
m a,e
F x
u x (x)
k a,e
x
l
u x
(a) (b)
Figure 2.12 (a) Distributed-parameter microrod undergoing axial vibration; (b) equiv-
alent lumped-parameter mass-spring system.

The deflection at an abscissa x measured from the free end of the
microrod of Fig. 2.12a, u x (x), is related to the free end’s deflection u x by
a distribution function f a (x) in the form:

u (x) = f (x)u (0) = f (x)u x (2.46)
x
a
x
a
where the distribution function is the one determined in Eq. (2.8)
The kinetic energy of the distributed-parameter microrod corre-
sponding to axial vibration about the x axis is
l ȡAu ˙ 2 l
ȡA 2 x 2
T = 2 ฒ u ˙ (x) dx = 2 ฒ f (x)dx (2.47)
x
a
a
0 0
The kinetic energy of a mass m a,e which is placed at the free end 1 is
. 2
m a,e u x
T = (2.48)
a,e 2
By equating Eq. (2.47) to Eq. (2.48), the equivalent mass is found
to be

m
m = (2.49)
a,e 3
where m is the total mass of the microrod. By combining Eqs. (2.1),
(2.45), and (2.49), the resonant frequency becomes

EA
Ȧ a,e =1.732 ml (2.50)

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