Page 194 - Mechanical design of microresonators _ modeling and applications

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Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 193
As a reminder, the prime in Eqs. (4.90) and (4.91) indicate compliances
that are calculated for the second segment in Fig. 4.15 with respect to
a switched reference frame (located at point 3), whereas the double
prime shows compliances calculated for the same second segment with
respect to a reference frame placed at point 1. Both compliance sets can
be expressed in terms of the bending-related compliances of the second
segment (regularly calculated with respect to the reference frame
placed at point 2) according to the compliance transformations of
Eqs. (3.2) and (3.10).
It should be emphasized that the generic Eq. (4.88) gives the lumped-
parameter stiffness of the entire two-segment microbridge by using
compliances that define one-half the microbridge, specifically one of the
two identical segments, and which can be denoted by C , C c , and C r . A
l
check has been performed of this generic calculation algorithm, by
considering the two identical segments of constant rectangular section.
By using the equations which define C l , C c , and C r of a constant
rectangular cross-section fixed-free segment of length l/2, Eq. (4.7) is
obtained, which defines the midpoint stiffness of a constant rectangular
cross-section microbridge of length l.
The lumped-parameter effective mass which needs to be placed at
midpoint 2, and is dynamically equivalent to the distributed mass of
the two-segment microbridge undergoing free out-of-the-plane bending
vibrations, is calculated by

l
= ȡt w(x) f (x) dx
m b,e ฒ b 2 (4.92)
0
where the bending distribution function f b (x) is expressed in Eq. (4.11)
and the variable width w(x) can be expressed as

l
w(x) 0 ” x <
2
w(x) = (4.93)
{ w( x í ) l ” x ” l
l
2 2
Equation (4.93) took into account the transverse symmetry of the
microbridge structure.
The bending-related resonant frequency can simply be calculated by
combining the lumped-parameter stiffness of Eq. (4.88) and effective
mass of Eq. (4.92), according to the definition.
A similar calculation procedure enables us to find the lumped-
parameter torsional resonant frequency of the entire structure,

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