Page 75 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_74_08/30/05

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

74 Chapter Two
C (x) — xC (x)
l
c
f (x) = C l (2.94)
b
where the newly introduced compliances in the numerator are calcu-
lated as
l 2 l
12 x dx 12 xdx
C (x) = C (x) = (2.95)
c
l
Et 3ฒ w(x) Et 3ฒ w(x)
x
x
It can easily be demonstrated that for a constant rectangular cross-sec-
tion microcantilever, the bending-related distribution function defined
in Eqs. (2.94) and (2.95) reduces to Eq. (2.63).
In the case of short microcantilevers, where shearing effects and the
corresponding deformations have to be accounted for, the bending-
related distribution function is calculated as
sh
C (x) — xC (x)
l
c
f b sh (x) = (2.96)
C sh
l
sh E
where C (x) = C (x) + ț G C (x) (2.97)
l
l
a
For a constant cross-section microcantilever, Eqs. (2.96) and (2.97) re-
sult in Eq. (2.77), which gives the bending-related distribution function
in a separate (and independent) derivation.
Several variable-cross-section microcantilever configurations such as
trapezoid or corner-filleted are analyzed next by providing lumped-
parameter stiffness and inertia fractions together with the resonant
frequencies corresponding to axial, torsional, and bending vibrations.

Trapezoid microcantilevers. A trapezoid configuration is shown in
Fig. 2.25 together with the defining geometry. It is assumed that the
microcantilever is fixed at its root and free at the opposite end, and that
its constant thickness t is small (thin configuration). The variable width
w, which is measured at a distance x from the free end, can be expressed
as

(w — w )x
2
1
w(x) = w + (2.98)
1 l
As previously mentioned, the bending stiffness corresponding to
point 1 in Fig. 2.25 and to deformation (rotation) about the y axis is
calculated with the aid of the generic Eq. (2.84) and the width definition
of Eq. (2.98). Thus, the bending stiffness is

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