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etween x 2 to L (remember that x 2 is measured from the left
side of clamed end of the beam to electrode pair of sensor
part, and L is the length of beam).From figure 2, the electrode
pair can be used to covers from 0.77L to L and best output
voltage can be obtained from the first vibration mode. To
explain this, normalized mode shape for first vibration mode
is shown in Figure 3. Strain for piezoelectric film is
proportional to the curvature of beam. The strain distribution
between 0.77L to L and 0.22L to 0.77L will cancel each other.
Best output voltage will be obtained when the length of
output electrode pair is 0.22L.
Voltage (mv)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Nondimensional electrode length, x/L
Fig. 2 Effect of nondimensional electrode length on output voltage
Mass normalized mode
shape
2.00
1.00
-
-1.00
-2.00
output voltage
0 0.2 0.4 0.6 0.8 1
Fig.3 The mode shape and curvature of cantilever
4.2 Effect of cantilever length on output voltage and
frequency
deflection
curvature
Nondimensional beam coordinate, x/L
Figure 4 shows that voltage output and resonance
frequency for different lengths of cantilever lengths and the
transformer is also excited at first vibration mode. In this
study, material, geometric, and electromechanical parameters
of transformers are shown in Table 1 but cantilever lengths
vary from 5cm to 100cm. From figure 4, output is
proportional to the square of cantilever length and 1 st
resonance frequency is inversely proportional to the square of
cantilever length. Longer beam exceeds larger output but
lower excitation frequency.
4.3 Effect of the thickness on output voltage
Figure 5 shows that output for different thickness ratios
between PZT and substrate when the transformer is excited at
first resonance. In this study, material, geometric, and
electromechanical parameters of transformers are also shown
in Table 1 but thicknesses of substrate vary from 0.125mm to
1mm. The best output can be obtained when the thickness
11-13 May 2011, Aix-en-Provence, France
ratio between substrate and PZT is close to 2. Thickness ratio
between substrate and PZT will change the position of
neutral axis of beam. Figure 6(a) and 6(b) shows that neutral
axis lies within substrate and within PZT, respectively. To see
the stress distribution, tensile and compress stresses will
cancel each other when neutral axis lies with PZT layer. We
might think the best output will be obtained when the neutral
axis is located in the contact plane between substrate and
PZT. However, the maximum output voltage occurs when
neutral axis lies within substrate (Fig. 6(a)).
In this study, the thickness of PZT is 0.25 mm and
thickness of substrate varies. When the thickness of substrate
become larger, the distance from neutral axis to top of the
beam will increase but bending stiffness will also increase.
Larger bending stiffness will cause smaller deflection and of
beam when actuator part is excited at the same input voltage.
In the other word, maximum normal stress of beam will
decreases when bending stiffness increases. The output
voltage will be proportional to the summation of normal
stress of beam. The maximum output voltage will occurs
when neutral axis lies within substrate but not in the contact
plane between substrate and PZT. Figure 7 shows that output
voltages for different thickness ratio between PZT and
substrate when choosing different substrate. Young’s
modulus ratio between PZT and substrate varies from 1 to 2.8.
Different Young’s modulus ratio leads to different thickness
ratio to achieve best output of transformer. When the Young’s
modulus ratios are 1 and 2.8, the best output are 0.078V and
0.090V, respectively. The results are summarized in table 2.
frequency (Hz)
500
450
400
350
300
250
200
150
100
50
0
5 20 35 50 65 80 95
cantilever length (cm)
Fig. 4 voltage output and nature frequency with different cantilever lengths
Voltage (mV)
80
70
60
50
40
30
20
free vibration frequency
out put voltage
0.5 1 1.5 2 2.5 3 3.5 4
thinkness ratio, substrate/PZT
Fig. 5 output voltage vs. thickness ratio between substrate and PZT layers
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
voltage (V)
output voltage
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