Page 97 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_96_08/30/05

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

96 Chapter Two
Equation (2.181) reduces to an expression similar to that provided by
Eq. (2.165) when F x = 0.
A better approximation of the natural frequency can be obtained
when we consider that the deflection curve is generated by a uniformly
distributed load q z acting over the entire length of the cantilever, which
is given as
4
3
4
q (x í 4l x +3l )
z
u = (2.182)
z 24EI y
In this case the natural frequency is
2
( 14EI ) EI y
5F l
x
Ȧ =3.53 1+ y ml 3 (2.183)
Rayleigh-Ritz procedure. Rayleigh’s procedure enabled us to find the first
resonant frequency (in bending, in this case, but torsion and axial load-
ing can also be dealt with). Ritz proposed an algorithm, based on
Rayleigh’s method, that enabled finding of the higher resonant fre-
quencies, in addition to the first one. According to the Rayleigh-Ritz
approximate procedure, the natural frequency can be found by enforc-
ing U = T and by choosing a deflection of the form:
u = a ́ (x) + a ́ (x) + 썳 + a ́ (x) + 썳 (2.184)
y 0 0 1 1 n n
where ij 0 (x), ij 1 (x), . . . , ij n (x) are shape functions that comply with the
(fixed-free) boundary conditions of the microcantilever, and the coeffi-
cients a , a , . . . , a are the unknowns of the problem. In truncating
n
0
1
the potentially infinite series of Eq. (2.184), one imposes limitations on
the deflection curve; as a consequence, the resonant frequencies will
always be higher than the real ones. To get smaller resonant frequen-
cies, the Rayleigh-Ritz procedure selects the coefficients a 0 , a 1 , . . . , a n
so as to make the function
l 2 2 2 l 2
EI ฒ (d u z/ dx ) dx + F ฒ (du z / dx) dx
1x 0
y 0
2
Ȧ = l 2 (2.185)
ȡA ฒ u dx
0
z
a minimum. This condition requires
˜Ȧ 2
˜a i =0 i = 1,2,. . . (2.186)
The generic Eq. (2.186) can be written in the form:

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