Page 69 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_68_08/30/05

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

68 Chapter Two

400
0.1
ω a,e / ω t,e
1
0.0001
β
α
0.01
0.001
Figure 2.21 Axial-to-torsional resonant frequency ratio.

z qz
θy
Sz + d Sz
Sz normal to face

tangent to neutral axis
My + dMy
My du z /dx
dx
x
Figure 2.22 Portion of a short beam with external load and internal reactions.

resonant frequency is larger than the torsional one, by a factor of up to 400,
as pictured in Fig. 2.21.
Short microcantilevers. For relatively short beams (where the length is
less than 5 times the largest cross-sectional dimension, as mentioned
previously), the tangent to the neutral axis is no longer perpendicular
to the face, as indicated in Fig. 2.22.
This is due to the fact that shearing effects become important for
short beams, and they produce an additional angular deformation ș —
y
du z /dx, which is shown in the same figure. The equations describing the
combined effects of bending and shearing, according to the Timoshenko
model, are

dș (x)
y
M (x) = EI y dx
y
(2.72)
du (x) țS (x)
z
z
ș (x) í =
y dx AG
where ț is the coefficient accounting for the cross-sectional shape, which
is equal to ൣ for a rectangular shape. By taking into account that

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