Page 240 - Mechanical design of microresonators _ modeling and applications

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Resonant Micromechanical Systems

Resonant Micromechanical Systems 239
k I Ew
I b,e t 1
Ȧ b,e = I = 2 (5.27)
m b,e l 1 ȡw l l
2 1 2
As a reminder, the assumptions are made in Eqs. (5.25), (5.26), and
(5.27) that the three segments have identical thicknesses t and that
their widths are different: w 1 for the end segments and w 2 for the middle
one (w 2 > w 1 ).
The torsional stiffness associated with the midpoint of the paddle
microbridge is again a combination of the two beam-springs, namely,
2GI t1 2Gw t 3
1
k I = = (5.28)
t,e l 1 3l 1
and the torsional moment of inertia of the paddle microbridge is
identical to that of the middle segment according to the assumptions of
model I, namely,

2
2
ȡl w t(w + t )
2 2
2
I
J t,e = 12 (5.29)
The torsional resonant frequency can now be calculated as
k I Gw
Ȧ I = t,e =2 2t 1 (5.30)
t,e I 2 2
J t,e ȡw l l (w + t )
2 1 2
2
Model II. Model I ignored the inertia contributions of the end compliant
segments, and this is acceptable as long as the mass of the middle seg-
ment is (considerably) larger than the masses of the end segments.
However, in cases where the two different segments comprising the
paddle bridge are comparable, the compliant end segments also con-
tribute to the total mass, and this additional mass can be calculated by
Rayleigh’s principle, which states that the deformation (displacement)
distribution of a beam in this case is identical to the velocity distribution
of the same member. By equating the kinetic energy of the real, dis-
tributed-parameter beam to that of the equivalent lumped-parameter
system, the additional mass is calculated as
l 1
2
m = ȡtw 1ฒ f (x) dx (5.31)
1,e b
0

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