Page 212 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH04_211_08/30/05

Microbridges: Lumped-Parameter Modeling and Design

Microbridges: Lumped-Parameter Modeling and Design 211
Equation (4.146) simplifies to Eq. (4.28) which provides the torsional
stiffness of a constant-cross-section microbridge when w 2 = w 1 and
l 1 = l 2 = l/3.
The torsional mechanical moment of inertia is
ȡtD
J t,e = (4.147)
315(c +2) 4
l
where

2
D =14{22 + 36c +15c + (c +1) 10 + c (c +2)(17
1 l l l l l
2
+ c (c +7)) c } l t w + {5 37 + 4c (16 + 7c )
l l w 1 1 l l
(4.148)
+3 47 + 4c (18 + 7c ) c +3 29 + 2c (20 + 7c ) c 2
l l w l l w
3 3
+ 35 + 2c (232 + 7c (61 + c (40 + c (10 + c )))) c } l w
l l l l l w 1 1
Again, the mechanical moment of inertia of Eq. (4.147) reduces to
the one of a constant-cross-section microbridge of length l—when
w 2 = w 1 and l 1 = l 2 = l/3.
The lumped-parameter resonant frequency corresponding to free
torsional vibrations is calculated by means of Eqs. (4.146) and (4.147)
as
20.5t(2+ c ) 2 Gc (c í 1)
l
w
w
Ȧ t,e = (4.149)
l
l
1 ȡD c (c í 1) +2c ln c w
w
w
Figure 4.24 shows the design derived from Fig. 4.23 by eliminating
the constant rectangular middle portion.
The lumped-parameter stiffness, inertia, and resonant frequencies
corresponding to bending and torsion for this design are simply
obtained by taking into consideration that l = 0 and therefore c = 0—
2
l
according to the definition in Eq. (4.142)—in the equations defining
similar lumped-parameter properties for the more generic microbridge
of Fig. 4.23. The bending stiffness is therefore
3 2
Et w (c í 1) ln c w
1 w
k b,e = (4.150)
3
3l (c +1) ln c í 2(c í 1)
1 w w w
The effective mass which is dynamically equivalent to the bending-
vibrating distributed-parameter microbridge is

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