133 significantly contribute to the total mass. Furthermore, the experiments are conducted at
a relatively low frequency ≤ 500 Hz and low acceleration level ≤ 10 ms
2
, therefore the damping effect of the tape could be ignored.
Figure 6-3: a Diagram of a sequence test system and b a shaker table where the device is being tested.
Figure 6-4: Schematic diagram of four different proof masses M1 – M4 shaded with the same
thickness of 1 mm distributed on the tip of a cantilever.
Planar view of cantilevers M1
M3 9 mm
2.5 mm 5 mm
M2 9 mm
4.5 mm
M4 2.5 mm
2.5 mm 4.5 mm
Device under test
Shaker PC
Function Generator
Analog-Digital
Shaker DUT
Sensor Signal Conditioner
Amplifier Programmable
Resistance Substituter
a b
134
6.4 Mechanical Characterisation
The mechanical properties of the composite free-standing cantilever samples were investigated and compared with the model developed in Chapter 3. The samples were
tested under two conditions; unloaded and loaded with proof mass. The experimental results will be used to calculate the coupling factor and the energy conversion efficiency
of the device.
6.4.1 Excitation without Proof Mass
Figure 6-5 shows a typical frequency response for the cantilevers. Those with a length of 18 mm sample D6 have a resonant frequency of around 230 Hz, while shorter
cantilevers, with a length of 4.5 mm, have a resonant frequency of about 2.3 kHz. Sample A1 and B1 were the initial batch of fabrication for series A and B respectively.
Other samples in the same series could not be measured because of fabrication defects. C and D series were the improved version of the samples. Sample C series are printed
with an additional layer of non-electro-active PZT layer compared with sample D series, as described in Table 4-, therefore the fundamental resonant frequency for sample C
series is higher than sample D series. The natural frequency of sample A1 with length 13.5 mm and sample B1 with length 11.25 mm are within the natural frequency range
of sample C and D series. This shows that the fabrication process was reasonably repeatable in producing uniform cantilever structures.
Figure 6-5: Experimental results in agreement with theoretical calculation for resonant frequency as a function of cantilever length.
Sample B1 Sample A1
135 The composite structure of sample D, with width 9 mm, length 1 cm, and total thickness
of 196 m was weighed at 0.11g. This gives an average density of 6240 kgm
3
. Sample C and D series were found to have a thickness of 208
m and 192 m respectively. Both samples D and C series were assumed to have the same density. Since the natural
frequency structure is inverse proportional to the length of the cantilever structure, therefore the Youn
g’s modulus of the structure can be estimated by using equation 3-8
. The calculated Young’s modulus of sample C and D series are 3.78 10
10
Nm
2
and 1.17 10
10
Nm
2
respectively.
The total
Q
-factor,
Q
T
of the structure can be determined experimentally by exciting the free-standing structures over a range of frequencies close to the fundamental resonant
frequency to determine the value of the full bandwidth at half maximum electrical output power, then substituting this value into equation 6-7. Figure 6-6 shows that the
calculated values for
Q
T
of the samples lie in the range 120 to 215, with the largest value associated with sample D3, which is roughly a square shape. Shorter or longer
cantilever lengths do not appear to exhibit the same
Q
-factor as those having a square structure. This is because shorter or longer cantilever structures suffer losses at different
rates and with different dominant factors. The energy dissipation losses at the support are dominant for a shorter structure [108], while air damping losses become dominant
for longer cantilever structure [114]. With the measured Q-factor value, the total damping ratio for the samples was calculated to be in the range of 0.002 to 0.005.
Figure 6-6:
Q
T
as a function of cantilever length.
