Page 54 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_53_08/30/05

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 53
at a point along a specified direction and the corresponding load, by
using the partial derivative of the strain energy, namely,
˜U
d = (2.24)
i
˜L
i
In axial loading, the corresponding compliance is
l
1 dx
C = (2.25)
a Eฒ A(x)
0
Similarly, the torsion-related compliance is calculated as
l
1 dx
C = Gฒ I (x) (2.26)
t
0 t
For bending about the y axis, the direct linear, direct rotary, and cross
compliances are evaluated as

l 2 l l
1 x dx 1 dx 1 xdx
=
=
C l,y = Eฒ I (x) C r,y ฒ I (x) C c,y ฒ I (x) (2.27)
E
0 y E 0 y 0 y
The first of Eqs. 3. (2.27) is valid for microcantilevers that are
relatively long (where the length is at least 5 times larger than the
largest cross-sectional dimension, which is generally the width–see
1
Young and Budynas, for instance) and where the Euler-Bernoulli
assumptions and model do apply. For relatively short configurations,
shearing deformations add to the ones normally produced by bending
such that the linear compliance is expressed according to the Timo-
shenko model (which is presented in greater detail a bit later in this
2
chapter) as follows (see Lobontiu for more details):
sh
C l,y = C l,y + ț E C a (2.28)
G
where ț is a constant accounting for the micromember’s cross-sectional
shape.
The compliances corresponding to bending about the z axis are
expressed by using the y-z subscript interchange in Eqs. (2.27) and
(2.28), which define the bending compliances corresponding to the y
axis. In the end, the compliances corresponding to axial, torsion, and
two-axis bending can be arranged into the following compliance matrix:

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