Page 201 - Mechanical design of microresonators _ modeling and applications

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Microbridges: Lumped-Parameter Modeling and Design

200 Chapter Four
the midpoint, followed by calculation of the resulting angular deforma-
tion at the same midpoint, in the form:
M Cx
k = (4.113)
t,e ș Cx

After we perform the required calculations, the torsional stiffness of
Eq. (4.113) is expressed as

4GI t2
k t,e = (4.114)
l +2C GI
2 t t2
where C t is the torsional stiffness of one of the end segments and I t2 is
the torsional moment of inertia of the middle segment. When w(x) = w
and l 1 = l 2 = l/3, Eq. (4.114) simplifies to Eq. (4.28) which gives the tor-
sional stiffness of a constant-cross-section microbridge of length l.
The lumped-parameter torsional mechanical moment of inertia at the
midpoint, which is dynamically equivalent to the rotary inertia of the
distributed-parameter microbridge undergoing torsional vibrations, is
calculated by applying again Rayleigh’s principle and therefore by
equating the kinetic energy of the equivalent system to the kinetic
energy of the real one. The effective mechanical moment of inertia is

{ l 1 2 2 2
ȡt
2
J = 12 ฒ f (x) w(x) w(x) + t dx
t,e 0 t
(4.115)
l +l 2
1
2
2
2
+ w (w + t ) ฒ f (x) dx }
2 2 t
l
1
where the torsional distribution function is
4x(2l + l í x)
1
2
f (x) = (4.116)
t (2l + l ) 2
1 2
In the case where the two end segments are identical to the middle
one, and therefore have constant cross section and are of length l/3,
Eqs. (4.115) and (4.116) reduce to Eq. (4.33), which expresses the
effective moment of inertia corresponding to free torsional vibrations of
a constant-cross-section microbridge of length l.

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