Page 156 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH03_155_08/30/05

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design

Microhinges and Microcantilevers: Lumped-Parameter Modeling and Design 155
n
™ z E A i
i
i
i =1
z N = n (3.149)
™ E A i
i
i =1
The stiffness of this equivalent cantilever beam is
3(EI )
y e
k b,e = (3.150)
l 3
It can also be shown that the mass fraction that has to be placed at the
free end of the microcantilever and that is dynamically equivalent to
the distributed mass of the multimorph is
n
33l ™ (ȡ A )
i
i
i =1 33 (3.151)
m = = m
b,e 140 140
where m is the mass of the whole multimorph. The first bending reso-
nant frequency is calculated by means of Eqs. (3.150) and (3.151) as

k b,e (EI )
y e
Ȧ b,e = =3.567 (3.152)
m
b,e ml 3
In the case of a bimorph, Eq. (3.152) reduces to
2 4
2
2
2 4
1.03 E t + E t +2E E t t (2t +3t t +2t )
1 2 1 2
1 1
1 2
1
2 2
2
Ȧ = (3.153)
b,e 2
2 2 /
(E t + E t )(ȡ t + ȡ t ) l
1 1 2 2 1 1
When we express t 2 in terms of t 1 according to Eq. (3.132), then
Eq. (3.153) simplifies to
4
2
1.03t 1 E +2c 2+ c(3+2c) E E + c E 2 2
1 2
1
Ȧ = (3.154)
b,e 2 (E + cE )(ȡ + cȡ )
l 1 2 1 2
When c ĺ 0 (the bimorph becomes a unimorph formed of only material
1), Eq. (3.154) simplifies to
E I
1 y1
Ȧ = 3.567 (3.155)
b 3
m l
1
which is the known relationship.
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