Page 89 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_88_08/30/05

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

88 Chapter Two
The variable width w is expressed in terms of the minimum width
w 1 , the shorter semiaxis length b, and the variable angle ij as
w(x) = w +2b(1 í cos ́ ) (2.143)
1
The abscissa x is expressed as

x = a sin ́ (2.144)
o
o
and it spans the [0, a] interval when ij ranges between 0 and 90 .
The axial stiffness is
4Ebt
k a,e = (2.145)
1 /
a 4(2b + w ) w (4b + w ) arctan 1+4b w íʌ
/
1
1
1
When a ĺ R and b ĺ R, Eq. (2.145) changes to Eq. (2.130), which ex-
presses the lumped-parameter axial stiffness of a circularly filleted
microcantilever.
The lumped-parameter effective mass corresponding to free axial
vibrations is
a,e ( w 1
m = ȡat 0.036b + 3 ) (2.146)

When a ĺ R and b ĺ R, Eq. (2.146) transforms to Eq. (2.131), which
gives the effective mass of a circularly filleted microcantilever. The ax-
ially related resonant frequency is therefore

2 Eb
Ȧ =
a,e a
ȡ(0.036b + w 1/ 3) 4(2b + w ) (2.147)
1
/
/ w (4b + w ) arctan 1+4b w íʌ
1
1
1
The torsional stiffness is obtained from the axial one according to
Eq. (2.85). The effective mechanical moment of inertia corresponding to
free torsional vibrations is
2
2
2
3
ȡat 0.007b +0.038b w +0.333w (t + w )
1
1
1
2
2
+0.036b(t +3w ) (2.148)
1
J t,e = 12
For a ĺ R and b ĺ R, Eq. (2.148) changes to Eq. (2.133), which gives
the effective mechanical moment of inertia of a circularly filleted mi-
crocantilever. The torsional resonant frequency is
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