Page 73 - Mechanical design of microresonators _ modeling and applications

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

72 Chapter Two
model which is presented first, and which allows other geometric curves
to be utilized for profile generation of different configurations. Similar
2
approaches to this subject can be found in Lobontiu and Lobontiu and
Garcia. 3

Generic formulation. As previously shown, the axial and torsional stiff-
nesses are found as the algebraic inverses of the corresponding compli-
ances, and this relationship is also valid for the so-called definition
bending stiffness (the one utilized in lumped-parameter resonant fre-
quency calculations). It therefore suffices to determine compliances,
which is a rather easier task for variable-cross-section members, com-
pared to directly finding stiffnesses. As a consequence, the axial, tor-
sional, and direct linear bending stiffnesses of a variable-cross-section
microcantilever are calculated as

1 Et
k a,e = =
l
C
0 /
a,e ฒ dx [w(x) ]
3
1 3Gt
k t,e = =
C
l
0 /
t,e ฒ dx [w(x) ] (2.84)
3
1 12Et
k b,e = =
C
l 2
/
l ฒ x dx [w(x) ]
0
Comparing the first two of Eqs. (2.84), we notice that
Gt 2
k t,e = 3E k a,e (2.85)
and, consequently, the torsional stiffness can be found from the axial one.
It has also been shown that the shearing effects are important for
short microcantilevers, and this translates to the direct linear bending
compliance being calculated according to Eq. (2.28). As a direct result,
the bending stiffness of a short microcantilever is
sh
k b,e = 1 = 1 (2.86)
/
C sh C + ț(E G)C
l l a,e
2
It can be shown, as also detailed in Lobontiu, that the lumped-
parameter mass which is dynamically equivalent to the distrib-
uted-parameter inertia of the microcantilever undergoing free axial
vibrations can be calculated as

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