The Functioning Principle Multimorph
154 generating a voltage of the same polarity as the poling voltage; whereas the compressive
forces on the lower piezoelectric elements generate a voltage of opposite polarity to that of the poling voltage.
Figure 7-3: Schematic diagram of a a multimorph structure and b transformed cross-section of a composite multimorph structure.
The output voltage of a piezoelectric cantilever can be estimated with the model developed by Roundy
et al
[12] and is rewritten here,
7-1
where
A
in
is the base input acceleration, is the dielectric constant of the piezoelectric material,
ζ
T
is the total damping ratio the sum of electrical and mechanical damping ratios,
C
p
is the capacitance of the piezoelectric material and
E
T
is the elastic modulus of the composite structure.
a h
l Rigidly Clamped
Centroid Plane of PZT Layer
Centroid Plane of PZT Layer
d h
N
b h
e1
h
p2
h
e4
h
e3
h
p4
h
p3
h
e2
h
p5
h
p1
p r
T r
r T
b in
p T
RC k
j l
dA h
d je
V
2 4
3
2 31
2 2
2 31
155 One of the controllable factors that can improve the output voltage is increasing the
distance between the PZT layer and the neutral axis of the multimorph structure,
d
. The neutral axis of the composite multimorph as shown in Figure 7-3 b can be determined
by the transformed-section method [93] as,
7-2
where
h
pi
is the thickness of PZT layer-
i
and
h
ej
is the thickness of electrode layer-
j
, while
H
E
is a parameter related to the elastic modular ratio,
n
ep
which is given by,
7-3
and the elastic modular ratio is,
7-4
where
e
e
and
e
p
is are the elastic modulus of electrode and PZT respectively. Taking ‘
’ as the reference point, the distance for the centroid of PZT of a particular section to the
neutral axis of a composite multimorph, as shown in Figure 7-3 a can be written as,
7-5
where is -1 for a layer above and +1 for a layer below the reference point as shown
in Figure 7-3 b. For simplification, the thickness of all the PZT sections and electrode sections are uniform with thickness
h
p
and
h
e
respectively as shown in Figure 7-4.
5 1
4 1
4 4
3 5
3 4
2 1
2 3
1 2
3 2
2 5
2 4
2 1
2 2
2 3
2 2
i j
ej ep
pi E
ep p
e p
e p
e e
p p
p e
p p
p p
p p
p N
h n
h H
n h
e h
h h
h h
h h
h h
h h
h h
h h
h h
h h
4 4
4 3
2 1
1 2
1 3
3 2
4 2
3 2
2 2
1
2
e p
e e
e e
e p
e e
p e
e e
e E
h h
h h
h h
h h
h h
h h
h h
h H
p e
ep
e e
n
N i
j ej
i pi
pi mm
h h
h h
d
1
1
2 1
156
Figure 7-4: Cross-sectional view of a composite multimorph with uniform thickness of PZT and electrode layers.
The bending modulus per unit width for a composite multimorph structure can be simplified as,
7-6
where
D
unimorph
is the unimorph bending modulus per unit width, which is derived from equation 3-13, and therefore the multimorph bending modulus can be written as,
7-7
The moment of inertia for the multimorph structure can be obtained by substituting equation 7-7 into equation 3-28,
7-8
The stress in each section of the PZT can be calculated in a manner similar to that derived for the unimorph structure as shown in equation 3-30.
½ h
p
-½ h
p
-h
e
+ ½ h
p
h
e
+ ½ h
p
2h
e
+ 1½ h
p
h
e
+ 1½ h
p
-h
e
+ 1½ h
p
-2h
e
+ 1½ h
p
p e
p e
p e
p e
h h
h h
e h
h h
h p
unimor ph multimor ph
dz z
E dz
z E
D D
2 3
2 2
3 2
2 3
2 1
2
2 2
2 2
3 2
2 3
15 2
15 8
3 6
8 27
3 2
e p
e p
e e
e p
e p
p p
multimor ph
h h
h h
h E
h h
h h
h E
D
2 2
3 2
2 3
15 2
15 8
3 6
8 27
3 2
p p
e p
e p
e e
p e
p p
multimor ph
h h
h h
h E
E h
h h
h h
w I
157 In order to be more precise when including the non-active PZT protective layers, the
total bending modulus is,
7-9
and the moment of inertia is,
7-10
a e
a p
e p
a e
p a
a p
multimor ph mm
h h
h h
h h
h h
h h
h E
D D
12 4
27 6
2 9
18 3
2
2 2
2 3
a e
a p
e p
a e
p a
a p
e e
p p
p
h h
h h
h h
h h
h h
h h
h h
h h
E 12
4 27
6 2
9 18
3 6
8 27
3 2
2 2
2 3
2 2
3
p e
e p
e e
h h
h h
h E
2 2
3
15 2
15 8
a e
a p
e p
a e
p a
a p
e e
p p
mm
h h
h h
h h
h h
h h
h h
h h
h h
w I
12 4
27 6
2 9
18 3
6 8
27 3
2
2 2
2 3
2 2
3
2 2
3
15 2
15 8
p p
e p
e p
e
h h
h h
h E
E
158