Page 66 - Mechanical design of microresonators _ modeling and applications

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

Basic Members: Lumped- and Distributed-Parameter Modeling and Design 65
By integrating this equation twice and by applying the adequate bound-
ary conditions of the beam shown in Fig. 2.17a, which are zero slope
and zero deflection at the fixed end

du (x)
dx | =0 u (l) =0 (2.60)
z
z
x = l
the deflection equation can be found. It is known from the mechanics of
materials that the stiffness at the point is the ratio of a force that is
applied at that point about the z axis to the corresponding deflection,
and it is calculated as
3EI y
k b,e = 3 (2.61)
l
This is actually the inverse of the compliance C l,y of Eq. (2.27), which
has been calculated by means of the compliance approach. A different
value would be obtained for k if it were calculated by means of the
b,e
stiffness approach, as noted in that subsection of this chapter. That
value, however, is not the one needed for resonant frequency calcula-
tions, and therefore Eq. (2.61) is the one applicable for such purposes,
as demonstrated shortly.
The equivalent inertia fraction which has to be placed at the free
extremity of the microcantilever sketched in Fig. 2.17b is calculated
again by means of Rayleigh’s approximate method, according to which
the distribution of the velocity field of a bending vibrating component
is identical to the displacement (deflection here) distribution of the same
component. By equating the kinetic energy of the real, distributed-
parameter system to the kinetic energy of the equivalent, lumped-
parameter system, an equivalent (or effective) mass is produced. The
deflection at an abscissa x measured from the free end of the
microcantilever of Fig. 2.17a, u (x), is related to the free end’s deflection
z
u by means of a distribution function f (x) in the form:
z
b
u (x) = f (x)u (0) = f (x)u z (2.62)
b
b
z
z
2
where the distribution function is (see Lobontiu, for instance)
3x x 3
f (x) =1— + (2.63)
b 2l 2l 3
This distribution function can simply be determined by expressing the
deflection at a generic point, of abscissa x, as a function of the deflection
at the free end of a cantilever beam. It can be seen that the deflection

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