Page 65 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_64_08/30/05

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

64 Chapter Two

uz (x) mb,e
uz uz

Fz
x kb,e
l

(a) (b)
Figure 2.17 (a) Distributed-parameter microcantilever; (b) equivalent lumped-parameter
microcantilever.

z
qz
θy
Sz Sz + d Sz
normal to face

My + dMy
My
dx
x
Figure 2.18 Portion of a long beam with external load and internal reactions.

Bending vibrations. The bending vibration of a microcantilever is il-
lustrated in Fig. 2.17. The original distributed-parameter member is
pictured in Fig. 2.17a whereas the corresponding lumped-parameter
(mass-spring) system is sketched in Fig. 2.17b. Two possibilities are
studied here, namely, the long configuration (following the Euler-
Bernoulli beam model) and the relatively short configuration, which is
described by the Timoshenko model.
Long microcantilevers. In the case of relatively long beam configurations
(where the length is at least 5 times larger than the largest cross-
sectional dimension), the tangent to the neutral axis is perpendicular
to the face, as sketched in Fig. 2.18, which pictures a portion removed
from a deformed (bent) beam.
The static equilibrium can be analyzed for this segment under the
action of the external distributed load q z , shearing forces S z , and
bending moments M y . As known from the mechanics of materials, the
bending moment can be expressed as

2
d u (x)
z
M (x) = EI y (2.59)
y
dx 2

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