51 Therefore the distance of the neutral axis from a reference point as shown in Figure
3-3b can be derived as,
3-21
where
h
i
is the distance from the reference point to the centre of each layer of the material and
A
i
is the area of the
i
-th layer of the structure. The distance from the centroid of PZT layer to the neutral axis is therefore,
3-22
We can see from Equation 3-22 that, if the thickness of the upper electrode is similar to the lower electrode,
h
e1
=
h
e2
, the neutral axis is located at the centre of the PZT layer, therefore,
d
= 0. This will give a zero resultant stress, which will be discussed in the following section.
3.4.3 Maximum Allowed Stress
Figure 3-4: Bending beam of unimorph structure.
l h
d
Neutral axis Centroid plane of
PZT Layer Upper Electrode
Lower Electrode PZT layer
Rigidly Clamped
n i
i n
i i
i N
A A
h h
1 1
1 2
1 2
2 2
1 2
2 2
2 1
e e
ep p
e p
e e
ep p
p
h h
n h
h h
h h
n h
h d
52 The resultant stress on the clamped area of a beam for each layer of a unimorph is
proportional to the input moment divided by the inertia across the length of the beam as,
3-23
To find the exact value of stress of each layer of the material, the moment inertia of the beam,
I
b
has to be defined. The transformed cross-section of a unimorph is redrawn in Figure 3-5 with parallel-axis passing through the centroid of the beam.
Figure 3-5: Parallel-axis for a transformed cross-section of a unimorph.
From the parallel-axis theorem for the moment of inertia [93]
3-24
where
d
is the distance from the centroid of the layer to the neutral axis of the structure and
A
is the cross-section area of the layer. The integration of the second term at the right hand side of equation 3-24 is zero, therefore, the total moment of inertia for a
unimorph as shown in Figure 3-5 is,
3-25
dl I
d l
M l
b
l unimor ph
b l
1
½
h
e1
½
h
e2
h
2
h
3
h
N
Neutral axis
½h
p
h
w
h h
h h
h h
h h
dA d
dA h
d dA
h dA
d h
I
2 1
2 1
2 2
1 2
1 2
1 2
1 2
2 1
2 1
2
2
2 2
3 2
2 1
2 1
3 2
3 1
2 3
2 1
2 1
12 1
2 1
12 1
e e
e e
e ep
p N
p p
unimor ph
h h
h h
h he
h h
w n
h h
wh wh
I
