Page 77 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_76_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

76 Chapter Two
axial vibration is determined by means of Eqs. (2.87), (2.88), and (2.98)
as
3w + w 2
1
m = m (2.103)
a,e 6(w + w )
1 2
Again, Eq. (2.103) reduces to Eq. (2.49) when w 2 ĺ w 1 , as expected,
because the trapezoid configuration transforms into a constant rectan-
gular cross section to one for this limit condition. The axial vibration
resonant frequency is therefore

2 3 E(w — w )
2
1
Ȧ = (2.104)
a,e l ȡ(3w + w )ln(w 2/ w )
1
2
1
The torsional resonant frequency is determined similarly with
respect to the free end. The torsional stiffness is obtained from the axial
one, according to Eq. (2.85). The inertia fraction corresponding to
torsional vibrations is determined according to Eqs. (2.90), (2.91),
(2.92), and (2.98) as
2
3
2
2
3
ȡlt 10w +6w w +3w w + w +5t (3w + w )
2
1
1
2
1 2
2
1
J t,e = 720 (2.105)
When w 2 ĺ w 1 , Eq. (2.105) changes to Eq. (2.55), which defines the
torsional moment of inertia for a constant-cross-section microbar. The
torsional resonant frequency can be calculated as
4 15t G(w í w )
1
2
Ȧ =
t,e l 3 2 2
ȡ 10w +6w w +3w w
1 1 2 1 2 (2.106)
2
3
+w +5t (3w + w ) ln(w 2/ w )
2 1 2 1
Another trapezoid microcantilever is sketched in Fig. 2.26a and b
where the width w is constant and the thickness varies linearly from
t at the free end to t at the fixed root.
1
2
The axial stiffness of this microcantilever is
Ew(t — t )
1
2
k a,e = (2.107)
l ln(t 2/ 1
t )
When t 2 ĺ t 1 , Eq. (2.107) reduces to Eq. (2.45), which gives the axial
stiffness of a constant rectangular cross-section microcantilever.
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