Page 188 - Mechanical design of microresonators _ modeling and applications
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Microbridges: Lumped-Parameter Modeling and Design
Microbridges: Lumped-Parameter Modeling and Design 187
variable, whereas the thickness is constant. By applying a force F z at
the guided end in Fig. 4.4, it can be shown that the bending stiffness
related to that point is
F z 1
k = =
b,e u 2 (4.67)
C C
z
l c / C r
where the compliances defining these stiffness are calculated as
l 2
/
12 2 3
/
C = E { x [t w(x)]} dx
l
0
l 2
/
12 3
{ x [t w(x)]} dx
C = E / (4.68)
c
0
/
l 2
12 3
/
C = E dx [t w(x)]
r
0
A check has been performed on Eq. (4.67) to verify whether Eq. (4.1) is
obtained when the rectangular cross-section is constant, and this was
indeed the case.
For short microcantilevers, where shearing effects are taken into
consideration and the Timoshenko model is utilized, the bending
stiffness is expressed as
sh 1
k b,e = (4.69)
c /
C l sh í C 2 C r
where the direct-bending linear shear–dependent compliance is
expressed as in Eq. (2.28) with the aid of the axial compliance equation:
l 2
/
1
C = E 0 dx / tw(x) (4.70)
a
It can be checked again that for a constant rectangular cross-section
microbridge, Eq. (4.69) reduces to Eq. (4.17), and this proves the validity
of the generic formulation.
The lumped mass which needs to be placed at the guided end of the
half-microbridge in Fig. 4.4 and which is dynamically equivalent to
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