Page 161 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_160_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

160 Chapter Three

3
3
G t (G t + G t )
p p
1 1
1 1
Ȧ =6lt (3.170)
t,e 1 c
t
where
2
3
2
3
3
c =3 G lt + G (l í l )t {ȡ l t (t + w )
1 1
1 1
p
p
p
t
1
2
2
2
2
+3l l t 3ȡ t t + ȡ (3t + t + w )
p p
p
1 1 p
1
p
(3.171)
2
2
2
í 3ll t (2l + l ) 3ȡ t t + ȡ (3t + t + w )
p p 1 p 1 1 p p 1 p
2
2
2
2
+ l t (3l +3l l + l ) 3ȡ t t + ȡ (3t + t + w ) }
2
1 1 p
1 p
p
p
1
p p
1
p
The lumped-parameter stiffness in bending is determined by
calculating the tip displacement of the sandwiched microcantilever
which is produced by a tip force, by means of Castigliano’s displacement
5
theorem, for instance, as shown in Lobontiu and Garcia. This stiffness
is
w
k b,e =
3
2
3
2
p /
4{ l í l (3l +3l l + l ) (E t )
p 1 1 p 1 1
2
2
+ l (3l +3l l + l )(E t + E t ) (3.172)
p 1 1 p p 1 1 p p
2
2 4
2 4
2
/ E t + E t +2E E t t (2t +3t t +2t ) }
1 p
p
p p
1
1 1
p 1 p
1
Equation (3.172) simplifies to the known relationship
E wt 1 3 3EI y
1
k = = (3.173)
b
4l 3 l 3
when l ĺ 0 and t ĺ 0, a relationship that corresponds to a homoge-
p
p
neous (one-component) microcantilever.
In bending, the lumped-parameter mass which is placed at the free
end of the microcantilever is calculated by Rayleigh’s principle in a way
that has been described previously here. By equating the kinetic energy
of the distributed-parameter bimorph to that of an equivalent mass
placed at the free end of a massless cantilever, the inertia fraction is
ƍ
33(m + l p/ p p
l m )
1
m = (3.174)
b,e 140
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