Page 189 - Mechanical design of microresonators _ modeling and applications

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Microbridges: Lumped-Parameter Modeling and Design

188 Chapter Four
the distributed inertia of the half-length microbidge undergoing free
bending vibrations can be calculated by means of the corresponding
distribution function. This bending-related distribution function is
determined in the usual manner as the ratio of the deflection produced
at a generic point on the interval [0, l/2] to the maximum deflection (at
the guided end) under the action of a force F z applied at the guided end.
It can be shown that for a variable cross section, the bending-related
distribution function is

c/
c/
C (x) í (x + C C )C (x) + xC C C (x)
l
r
c
r
r
f (x) = 2 (4.71)
b
C í C c / C r
l
where the abscissa-dependent compliances in the numerator are de-
fined as
l 2
/
12 2 3
C (x) = E œ { x / [t w(x)] } dx
l x
l 2
/
12 3
/
C (x) = E œ x { x [t w(x)] } dx (4.72)
c
l 2
/
12 3
/
C (x) = E œ x dx [t w(x)]
r
The distribution function of Eq. (4.71) with the associated Eqs. (4.72)
results in
2
(l í 2x) (l +4x)
f (x) = (4.73)
b
l 3
when the rectangular cross section is constant. Equation (4.73) is ac-
tually identical to Eq. (4.2) which directly expresses the bending-related
distribution function for a half-length microbridge.
For short microbridge configurations, Eq. (4.71) becomes
sh
C (x) í (x + C / C )C (x) + xC / C C (x)
sh l c r c c r r
f b (x) = (4.74)
2
C sh í C / C
l c r
For a constant-cross-section microbridge, the generic Eq. (4.74) simpli-
fies to

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