Page 98 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_97_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 97

™c a =0 j = 1,2,. . . (2.187)
i
ij
i
where c = c . This represents a set of homogeneous equations in a i
ji
ij
whose solution is nontrivial when the system’s determinant is zero,
namely,
c c ಸ c
11 12 1n
c 12 c 22 ಸ c 2n
det =0 (2.188)
ಸ ಸ ಸ ಸ
c 1n c 2n ಸ c nn

Equation (2.188), the characteristic equation, is an algebraic equa-
2
tion of the nth degree in Ȧ which will provide the first n resonant
frequencies (all bending-related) of the microcantilever.
The two shape functions

)
)
(
́ (x) = 1 í x 2 ́ (x) = x ( 1 Ì x 2 (2.189)
1 2
l l l
satisfy the boundary conditions of a fixed-free beam, which are zero
shearing force and zero bending moment at the free end, as well as zero
slope and zero deflection at the fixed end, namely,
3 2
dx |
d u 3 | =0 d u 2 | =0 du z =0 u (l) =0
z
z
dx x =0 dx x =0 x = l z
(2.190)
By using them in Eqs. (2.185) through (2.188), a second-degree charac-
teristic equation results whose roots are

EI y EI y
Ȧ =3.53 Ȧ = 34.81 (2.191)
1
2
ml 3 ml 3
While the first resonant frequency is very close to the exact one, the
second one is larger than the corresponding exact one. A better approx-
imation to this second resonant frequency can be obtained by using the
shape functions

x x 2 x 2 x 2
́ (x) = l ( 1 í l ) ́ (x) = l 2 ( 1 Ì l ) (2.192)
2
1
which produce:

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