Page 204 - Mechanical design of microresonators _ modeling and applications
P. 204
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Microbridges: Lumped-Parameter Modeling and Design
Microbridges: Lumped-Parameter Modeling and Design 203
[
2/
ȡt 2 2 l 2 2
J t,e = 12 w (w + t ) 0 f (x) dx
2
2
t
(4.120)
l t ]
l í l 2
2/
1
2
2
2
+ w (w + t ) f (x) dx
1 1 2/
After we perform the calculations involved in Eqs. (4.119) and (4.120),
the moment of inertia becomes
3 2 2 2
ȡt 8l w (w + t ) + l w (12l 1
2 2
1
1
1
2
2
2
+6l l + l )(w + t ) (4.121)
1 2
2
2
J =
t,e 2
72(2l + l )
1 2
When w 2 = w 1 , l 1 = l/4, and l 2 = l/2, Eq. (4.121) simplifies to Eq. (4.26),
which gives the effective torsional mechanical moment of inertia for a
constant-cross-section bar of length l/2. The lumped-parameter tor-
sional resonant frequency is calculated by combining the lumped-
parameter stiffness of Eq. (4.118) and the lumped-parameter inertia of
Eq. (4.121) as
1 2/
6.928(2l + l )t Gw w ȡ(w l +2w l )
1 2
2 1
1
2
Ȧ t,e = (4.122)
2
2
2
2
3
2
2
8l w (w + t ) + l w (12l +6l l + l )(w + t )
1 1 1 2 2 1 1 2 2 2
In bending, the stiffness of the half microbridge is found by applying
a force at the guided end in Fig. 4.20, perpendicularly on the beam’s
axis, and by determining the corresponding deflection. The lumped
stiffness is
3
8Et w w (w l +2w l )
2 1
1 2
1 2
k b,e = 2 4 2 2 2 4 (4.123)
w l +8l l (4l +3l l + l )w w +16w l
1 2 1 2 1 1 2 2 1 2 2 1
Equation (4.123) simplifies to Eq. (4.1), which calculates the bending
stiffness of a constant-cross-section half-microcantilever (of fixed-
guided boundary conditions) in the case where w = w , l = l/4, and
2
1
1
l 2 = l/2.
The lumped mass which needs to be placed at the guided end of the
half-microbridge in Fig. 4.20 is calculated again by means of Rayleigh’s
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