The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
A parametric study of wind-induced flutter of piezoelectric
patches for energy harvesting
J. M. McCarthy a, A. Deivasigamani a, S. Watkins a, S. J. John a, F. Coman b
a
RMIT University, School of Aerospace, Mechanical & Manufacturing Engineering,
71, Bundoora, Victoria 3083, Australia
b
FCST Pty., PO Box 122, South Carlton, Victoria 3053, Australia
PO Box
ABSTRACT: A study of key parameters affecting flutter characteristics and power output levels
from a thin piezoelectric patch immersed in a parallel, smooth flow was examined. The piezoelectric patch vibration amplitudes were augmented through the use of a polymeric “leaf”, freely
hinged to the trailing edge of the patch. The leading edge of the patch was rigidly clamped to a
relatively narrow, rectangular support that extended across the wind-tunnel test domain, giving
effectively a two-dimensional testing scenario. The influence of clamping base geometry on the
flutter characteristics was evaluated. It was found that the stream-wise dimension of the rectangular clamp had little effect, whilst an increasing cross-stream dimension tends to delay the onset
of flutter. Stream-wise proximity experiments were conducted with two leaf-stalk systems, and it
was found that there exists a point tandem and downstream of a fluttering leaf-stalk system,
whereby a second leaf-stalk can output more than 40% more power than when alone.
KEYWORDS: piezoelectric, energy harvesting, flow-induced flutter, vortex shedding
1 INTRODUCTION
A fluttering leaf on a tree, a vibrating automobile antenna, or blinds flapping in the wind; all of
these seemingly trivial events actually reveal a complex engineering and scientific problem: fluid-flow induced flutter. Traditionally, flutter has been considered a detriment to structural integrity, and engineers would design the structure to obviate flutter occurrence. In fact, much research has been conducted to comprehend the underlying mechanisms of an object fluttering in a
fluid flow (e.g. Lord Rayleigh 1879; Theodorsen 1935; Kornecki et al. 1976; Fredericks et al.
1986; Huang 1995; Paidoüssis 1998; Zhang et al. 2000; Argentina & Mahadevan 2005;
Paidoüssis et al. 2011).
Recently, however, there has been a realization that flutter could be exploited for the purposes
of energy harvesting, using thin, flexible piezoelectric materials immersed in a fluid flow. Flutter
is related to resonance of the system, and thus the amplitudes of an object in flutter become
large. In order to extract large amounts of energy from a piezoelectric material large mechanical
strain must be applied, and if this can be induced by flutter then energy can be extracted from a
fluid flow. One of the first known attempts in the laboratory environment to realize this concept
was by Allen & Smits (2001); where flutter of a thin, polyvinylidene-fluoride1-laden membrane
was induced by introducing an upstream vortex-shedding, bluff body. The vortices would impinge on the flexible membrane, and due to the imposed pressure gradients, would cause timevarying membrane deformation according to the vortex shedding frequency of the bluff body.
This method of flutter excitation is classed by Naudasher & Rockwell (1980) as Extraneously
Induced Excitation, or EIE; flutter is induced from external pressure gradients caused by turbulence or vortices. Taylor et al. (2001) carried out a similar study, whereby a bluff body was used
1
Polyvinylidene-fluoride (PVDF) is a flexible, isotropic piezoelectric material.
1401
to excite undulations of the so-called piezoelectric “eel” within a water flow; however, they focused their attention on optimizing the electrical subsystem. Pobering & Schwesinger (2004)
suggest two designs that exploit vortex shedding in a water flow; along with a simplified analysis, they state that such a piezoelectric system, when scaled up, could have a higher power density than that of a conventional wind turbine.
Dickson (2008) envisioned a piezoelectric flutter system that would not only be efficient, but
aesthetically-pleasing and safe; a tree-like construct with multiple piezoelectric “stalks” with
polymeric “leaves” attached the stalks – figure 1 shows this concept.
Figure 1. The piezoelectric “tree” concept proposed by Dickson (2008).
Li & Lipson (2009) initiated research on this concept, by starting with a single piezoelectric,
cantilevered leaf and stalk system in parallel, laminar flow. There was no vortex-shedding bluff
body in this study, as flutter in this type of arrangement is meant to be self-induced. Naudasher
& Rockwell (1980) classified this type of flutter excitation as Movement Induced Excitation, or
MIE. Here, flutter is caused by a small perturbation of the membrane in the flow and further
growth of this instability. Most theoretical and numerical studies in the past have focused on this
type of flutter (see: references 1-9), because the instability growth and mechanisms by which the
flutter initiates are still not quite fully understood. An important parameter to consider with this
type of flutter is the critical flutter speed (Uc), which is the flow speed at which flutter of the
membrane starts. In their studies, Li & Lipson (2009) examined different piezoelectric-stalk
lengths, as well as different leaf geometries. They found that a triangular-shaped leaf coupled
with a piezoelectric short stalk in the vertical configuration (Fig. 2b), gave the highest power
output when properly load-matched, at a wind speed of 8.0m/s.
Figure 2. The two different piezoelectric leaf and stalk arrangements used by Li & Lipson (2009, Fig. 5).
1402
The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
They concluded that the hinge might play a significant role in amplifying the vibrations of the
piezoelectric stalk, though no analysis was carried out.
Bryant et al. (2011) looked at the aerodynamic interaction between two piezoelectric/leaf
beams, and the subsequent influence on the power output. The stalks were constructed of thin
steel with a piezoelectric patch bonded along their length. In their experiments, one piezoelectric/leaf beam was placed at a fixed point in a laminar-flow wind tunnel, and another identical
replica was placed downstream at varying stream-wise and cross-stream locations. It was found
that when the second beam was placed aligned and downstream of the first beam, it could produce up to 30% more power in tandem than when alone. At all times, the frequencies of the
beams were monitored, and the resistive load adjusted accordingly in order that the maximum
power from each beam was always attained. The reason this was done is because there exists, for
a given flutter frequency and piezoelectric capacitance value, an optimum load resistance value
that will permit maximum power to be extracted (see Li and Lipson 2009; Roundy et al. 2003;
Sodano et al. 2004). Nevertheless, real-world deployment of these piezoelectric/leaf devices
would demand that minimal power is consumed by the device itself (i.e. would operate with a
high efficiency), and it is unknown if the benefits of an active resistance tuning circuit would
outweigh the performance degradation in terms of power output; Roundy & Zhang (2005) investigated active-tuning of a piezoelectric device via active mechanical resonance tuning and it was
found that active tuning never resulted in a net power output increase, but active electrical tuning
was not examined.
Such devices are likely to be “driven” by the atmospheric wind which is inherently turbulent
and may approach from a wide range of angles, thus simulations or laboratory experiments in
(usually) smooth flow may not give correct predictions of power outputs. Wind conditions vary
widely with geographical location, notwithstanding the prevalence of local wind effects within
urban settings. For instance, in Melbourne, Australia, the average wind speed in the Greater Metropolitan area as modeled by the Sustainable Energy Authority Victoria (2003) is 3.0m/s at a geometric height of 65 meters. However there are wind over- and under-speed effects, that occur
due to topography and/or building proximity effects, hence the average wind speed may be far
removed from the local speed.
Here, we aim to investigate the effect of the leading-edge geometry on Uc, as this has not been
quantified in previous analyses. We also, for a given leaf area and aspect ratio, examine the power output effects of two of these piezoelectric beams in proximity with each other. Unlike Bryant
et al. (2011), this will be done with a predetermined, load-matched circuit resistance that will
remain unchanged during the course of the experiments.
2 METHOD
2.1 Wind Tunnel Calibration and Setup
The wind tunnel utilized in the experiments is a subsonic, closed-circuit design with an octagonal
test section, measuring 1320mm high by 1070mm wide. A honeycomb mesh and an antiturbulence screen, plus a 4:1 contraction ratio inure the test-section incoming airflow. A 134horsepower DC motor powers a six-bladed fan, giving a maximum tunnel flow speed of approximately 45m/s. According to previous calibration work, the free-stream longitudinal turbulence
intensity was much less than 3% (Creazzo 1999). A pitot-static tube installed that linked to an
MKS Baratron® was used for velocity measurements, according to a standardized air density
value of 1.23kg/m3. A simple error analysis was carried out, in order to determine the impact of
standardizing the density across a large range of tests, and a maximum error of 0.5% in velocity
1403
measurements was obtained. Tunnel blockage due to the experimental setup was also found to be
negligible.
Due to the electrical nature of the experiments, the wind-tunnel electromagnetic noise (EMI)
was evaluated so that a signal-to-noise ratio could be estimated and any destructive interference
identified and attenuated. First, the coaxial, shielded cable for the experiments was connected to
a 20MHz oscilloscope, and the other end left free near the wind tunnel (but not contacting the
wind tunnel, so as to avoid an ‘antenna’ effect). The wind tunnel was ramped up to a wind speed
of 14.6m/s and voltage curve observed, with no noticeable peaks occurring throughout the frequency sweep range. Next, a piezoelectric element was mounted in the wind tunnel, and the
shielded cable connected to the electrodes. The voltage curve was then observed with no wind
speed, and a small peak of the order 2.5-10mV was appearing at a frequency of 40Hz, presumably stemming from the fluorescent lights in the room. Finally, the voltage curve was evaluated
for a piezoelectric element in flutter, initiated by the airflow in the wind tunnel traveling at
around 5.1m/s. The voltage peaked at 3-3.2V, at a frequency of around 7.1Hz, the flutter frequency of the piezoelectric element. Therefore, the signal-to-noise ratio was deemed large
enough that no further action was required.
2.2 Base Clamping Geometry Experimental Setup
In order to assess the influence that the clamping geometry could potentially have on the critical
flutter speed, a quick parametric study that included varying the length and width of the rectangular clamp was carried out. Figure 3 shows a diagrammatic of the clamp cross-section characteristic dimensions.
’
Figure 3. The base clamping setup, showing the clamping length, Lclamp, and the clamping width, Wclamp. The dashed
line illustrates the test specimen, and the free-stream uniform flow is denoted by U∞.
The test specimen was chosen based on tests performed elsewhere, and consisted of a polypropylene membrane that was 248mm long by 60mm wide by 0.35mm thick, placed well outside
of the tunnel boundary layer. The membrane was clamped at its leading edge so that the effective
length was 236mm. The membrane material density is 995kg/m3 and the elastic modulus is
1261MPa. A stand was installed in the wind-tunnel test section to fix the top of the clamping
base, and the bottom was fastened to the wind-tunnel floor. Guying wire was used so that there
would be no transverse vibrations of the clamp in the flow, as shown in figure 4.
1404
The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
Figure 4. A photograph of the clamping geometry sensitivity tests wind-tunnel setup.
The clamping base was fabricated out of two lengths of 1.25mm-thick steel; one 600mm-long
strip that extended from the stand to the wind-tunnel floor, and a 100mm strip that would be fastened to the other strip with the specimen clamped in between (Fig. 4). The baseline width of the
clamping base was 3.5mm, and the baseline length was 12mm.
The base clamping length was varied first, while keeping the clamping width at the baseline
dimension. The length was varied from the baseline value to 70mm. Then, the clamping width
was altered while keeping the baseline length dimension; the width was varied from 3.5mm to
12.5mm. For each dimension, the wind speed was ramped up slow enough so as to ensure
steady-state flow, and the membrane start-up flutter wind speed was recorded. A Reynolds number range of 129,000-214,000 was observed in the testing, based on the membrane effective
length.
2.3 Proximity Testing Experimental Setup
Due to their flexibility, durability and relatively low cost, PVDF piezoelectric patches were utilized (Measurement Specialties, Inc., LDT1-028K/L type) in these experiments. The length,
width and thickness of the piezoelectric patch are 72mm, 16mm and 205!m respectively. These
patches were the same ones used by Li & Lipson (2009) in their work. Bryant et al. (2011) utilized Lead-Titanium-Zirconate (PZT) patches bonded to their steel beams, which are capable of
outputting higher power but are generally less durable than PVDF.
The leaves used for the experiments were fabricated from 0.35mm-thick polypropylene
(whose properties are mentioned in section 2.2). The shape of the leaf is an isosceles triangle
with dimensions of 80mm by 80mm (base by height), as this is the shape and area that caused the
PVDF’s to output the highest power in previous work (Li and Lipson 2009). The leaf and piezoelectric stalk were coupled with a plastic revolute hinge, which would allow free rotation of leaf
about the vertical axis. The mass of the hinge was 0.009kg. The leading edge of the piezoelectric
stalk was securely clamped and the leaf end was free. The clamping length utilized for these experiments was 12mm. This length was chosen in order to securely hold the leads and wires attached to the PVDF stalk. The clamping strips were bolted to the wind-tunnel floor and ceiling,
and guyed to the sidewalls to prevent any transverse oscillations. The clamping base was also
taped, so as to prevent the piezoelectric electrodes from contacting the metal. The overall length
(L) of the system with the piezoelectric stalk, leaf, hinge and the clamping length were measured
to be 180mm.
In order to determine the optimum placement of these leaf-stalk systems in a given volume,
two such systems were placed co-linearly in the wind tunnel and the stream-wise distance be-
1405
tween them (d) varied, see figure 5. The separation distance was normalized by a single leafstalk length such that it is expressed as (d/L). At first, the two devices were positioned with a
normalized separation distance of 1.0. Subsequently, the separation distance was increased to 2.0
and 3.0. This was done to examine the aerodynamic effects of the upstream leaf-stalk system on
the downstream one. The wind speed range considered here was 3-8m/s. Figure 6 shows a photograph of the wind-tunnel setup.
’
Figure 5. A cross-section schematic of the co-linear proximity experiments, showing the relevant parameters.
Figure 6. A photograph of the proximity experimental setup in the wind tunnel.
The piezoelectric stalks were connected to a simple parallel electrical circuit. The load resistance used for this experiment was 5.6MΩ. This value was experimentally obtained by recording the power output from a single leaf-stalk system across various load resistances in parallel
(RL); from 1.0MΩ to 60MΩ, at a constant wind speed of 5.0m/s. The voltages from the piezoelectric stalks were measured using a differential probe (Elditest, GE8115) which has a high internal resistance (60MΩ) compared to the load resistance. The data from the differential probe
was sent to a DAQ board (National Instruments, BNC2110) and the RMS voltage (VRMS) of the
AC waveform was calculated in one-second intervals using LabView®.
The electrical power generated in one second of leaf-stalk flutter is given by equation 1 (Bryant et al. 2011):
Pi =
2
VRMS
i
(1)
RL
Then, the total average power generated over the 30-second data acquisition window was calculated as:
1 30
Pave = ∑ Pi
30 i=1
(2)
1406
The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
3 RESULTS AND DISCUSSIONS
3.1 Clamping Base Geometry Sensitivity
Figures 8 and 9 show the experimental results for the sensitivity of the critical flutter speed to the
clamping length and width dimensions, respectively. The critical flutter speed is normalized by
the critical flutter speed observed with the baseline clamp dimensions. The ordinate axis is plotted as such, for clarity.
Normalised Critical Flutter Speed
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
10
20
30
40
50
60
70
Clamping Length (mm)
Normalised Critical Flutter Speed
Figure 8. Clamping length sensitivity results.
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
14
Clamping Width (mm)
Figure 9. Clamping width sensitivity results.
Altering the length of the clamping base does not seem to affect the critical flutter speed of
the membrane. This could mean that the stream-wise boundary layer manifesting on the clamp
topology is not causal to membrane flutter, at least for the Reynolds number range experienced.
However, if the length of the clamping base were to greatly increase, assuming a constant width,
the membrane root would then by definition encounter a thicker boundary layer, since the
boundary layer thickness varies as the square root of distance from the leading edge (e.g. see
Anderson 2007, chpt. 18). Additionally, depending on the state of the boundary layer at the root
location of the membrane, whether laminar or turbulent, the critical flutter speed may change.
The clamping width does seem to affect the membrane critical flutter speed. Considering previous theoretical treatments of flutter (e.g. Kornecki et al. 1976; Argentina and Mahadevan
2005), one major assumption made is that the clamped leading edge of the membrane is infinitely stiff, yet infinitesimally thin. Of course, this is very difficult to achieve in practice, and it can
1407
be seen that as the width of the clamp deviates upwards from the infinitesimal value, there comes
an increase in critical flutter speed. This could be attributed to vortex shedding off of the clamp
at a frequency other than the membrane natural frequencies, thereby causing destructive interference and henceforth a damping effect. However, once a clamping width that generates shed vortices at a frequency equal to the membrane natural frequency is deployed, it is anticipated that
the critical flutter speed would decrease dramatically.
3.2 Proximity Experimental Results
Figure 10 presents the proximity testing results for the downstream leaf-stalk. The output power
is normalized by the power extracted with no upstream leaf-stalk (Pbaseline) present, and this quantity is plotted against the wind speed.
Normalised Power, P/Pbaseline
1.5
1.4
1.3
1.2
1.1
1
d/L = 1
0.9
d/L = 2
0.8
d/L = 3
0.7
3
4
5
6
7
8
Wind Speed (m/s)
Figure 10. The downstream leaf-stalk power output as a function of wind speed for multiple separation distances.
A maximum normalized power of more than 1.4 is being obtained at 8.0m/s, even though the
load resistance was optimized for a wind speed of 5.0m/s; this suggests that there is potential for
even higher power outputs from the downstream leaf-stalk, should an actively-tuned load resistance be used. Bryant et al. (2011) obtained a trailing leaf-stalk normalized power of 1.3, for a
wind speed of around 6-7m/s, so the results obtained here agree quite well. This correlation is
despite a smaller cantilever length, different piezoelectric material and configuration, a different
leaf shape, and a leaf area less than half of what Bryant et al. (2011) use. This strongly suggests
that the increase in output power is purely an aerodynamic phenomenon, and not a structural dynamic one. Also, the flutter type occurring in this wind speed range is steady state, limit-cycle,
sinusoidal; elsewhere it was determined by power spectral analysis that this flutter type contains
the most bending energy at the discrete membrane natural frequencies (Alben and Shelley 2008).
Once the flutter transitions to chaotic, stochastic motion with random snap-through events (Connell and Yue 2007), the maximum attainable power decreases and response becomes more
broadband. It can be seen in figure 10 that for d/L = 1 and 2, the normalized power approaches a
maximum. However, for d/L = 3 the normalized power growth rate is instead increasing. It could
be that for separation distances of 1 and 2, transition to the chaotic flutter regime is starting and
thus the power output has reached a peak. It is also interesting to note the local minimum transpiring at a wind speed of 4.0m/s – this is most likely due to anti-resonance occurring between
the upstream leaf-stalk vortex shedding frequency and the flutter frequency of the downstream
leaf-stalk.
1408
The Seventh International Colloquium on Bluff Body Aerodynamics and Applications (BBAA7)
Shanghai, China; September 2-6, 2012
Figure 11 shows the normalized power plotted against the normalized separation distance, for
wind speeds of 3.0 and 8.0m/s. Clearly, there is a maximum occurring at a separation distance of
2 for a wind speed of 3.0m/s, but there is no global maximum occurring with a wind speed of
8.0m/s.
Normalised Power, P/Pbaseline
1.5
1.4
1.3
1.2
1.1
1
3 m/s
0.9
8 m/s
0.8
1
1.5
2
2.5
3
Normalised Streamwise Separation Distance, d/L
Figure 11. The normalized power plotted against the normalized separation distance for the minimum and maximum
wind speeds tested.
Evidently, optimum increased power output is occurring for the downstream leaf-stalk at a
normalized separation distance of 2, as this is also what Bryant et al. (2011) found. However, the
coalescence of the normalized power output across the separation distances for a wind speed of
8.0m/s is intriguing. Perhaps the higher kinetic energy in the flow sustains the advection of the
shed vortices from the upstream leaf-stalk flutter and delays transition to chaotic flutter, which
was found to occur at wind speeds slightly greater than 8.0m/s for a single leaf-stalk. Further
work would entail flow visualization using a dual smoke wire approach.
4 CONCLUSIONS
The sensitivity of the critical flutter speed to the clamping geometry was first examined. The results suggested that the clamping length has little or no effect on the critical flutter speed, and
that an increasing clamping width tends to delay the onset of flutter. These results only hold for
parallel, laminar flow; it would be erroneous to extrapolate to turbulent flow, and also the effect
of the wind approaching from any direction hasn’t been quantified. That said, any future work
concerning piezoelectric fluttering systems could benefit from the findings, especially if there is
a need to control the critical flutter speed, or perhaps minimize it.
Proximity testing, investigating exclusively the effect of a tandem configuration in the streamwise direction, was carried out. Results agree well with findings reported elsewhere for similar
wind speeds tested. It is found that for wind speeds of 3-7m/s, a normalized separation distance
of 2 causes the greatest increase in power output from the downstream leaf-stalk system, over a
40% increase. A wind speed of 8.0m/s caused a coalescence of the normalized power across the
range of separation distances, and it is speculated that the increased kinetic energy in the flow
1409
more readily sustains the advection of the vortices shed from the upstream leaf-stalk system,
such that the transition of the downstream leaf-stalk system to chaotic flutter is impeded.
5 REFERENCES
1 M. Paidoüssis, Fluid-Structure Interactions - Slender Structures and Axial Flow, Elsevier Academic Press, 1998,
Vol. 1.
2 M. Paidoüssis, S.J. Price and E. de Langre, Fluid-Structure Interactions - Cross-Flow-Induced Instabilities,
Cambridge University Press, 2011.
3 Lord Rayleigh, Proc. Lond. Math. Soc., 1879, Vol. X, pp. 4-13.
4 T. Theodorsen, General Theory of Aerodynamic Instability and the Mechanism of Flutter, NACA Tech. Rep.
No. 496, 1935.
5 A. Kornecki, E.H. Dowell and J. O'Brien, On the aerodynamic instability of two-dimensional panels in uniform
incompressible flow, Journal of Sound and Vibration, 2 (1976), 163-178.
6 W. Frederiks, H.C.J. Hilberink and J.A. Sparenberg, On the Kutta condition for the flow along a semi-infinite
elastic plate, Journal of Engineering Mathematics, 20 (1986), 27-50.
7 L. Huang, Flutter of Cantilevered Plates in Axial Flow, Journal of Fluids and Structures, 9 (1995), 127-147.
8 J. Zhang, S. Childress, A. Libchaber and M. Shelley, Flexible filaments in a flowing soap film as a model for
one-dimensional flags in a two-dimensional wind, Nature, 408 (2000), 835-839.
9 M. Argentina and L. Mahadevan, Fluid-flow-induced flutter of a flag, Proceedings of the National Academy of
Sciences of the United States of America, 6 (2005), 1829-1834.
10 J.J. Allen and A.J. Smits, Energy Harvesting Eel, Journal of Fluids and Structures, 3-4 (2001), 629-640.
11 E. Naudascher and D. Rockwell, Oscillator-model approach to the identification and assessment of flow-induced
vibrations in the system, Journal of Hydraulic Research, 18 (1980), 59-82.
12 G.W. Taylor, J.R. Burns, S.M. Kammann, W.B. Powers and T.R. Welsh, The energy harvesting Eel: A small
subsurface ocean/river power generator, IEEE Journal of Oceanic Engineering, 4 (2001), 539-547.
13 S. Pobering and N. Schwesinger, A novel hydropower harvesting device, International Conference on MEMS,
NANO and Smart Systems, ICMENS 2004, August 25 - August 27, pp. 480-485.
14 R.M. Dickson, New Concepts in Renewable Energy, Lulu Enterprises Inc., 2008.
15 S. Li and H. Lipson, Vertical-stalk flapping-leaf generator for wind energy harvesting, ASME Conference on
Smart Materials, Adaptive Structures and Intelligent Systems, SMASIS2009, September 21 - September 23, pp.
611-619.
16 M. Bryant, R. Mahtani and E. Garcia, Synergistic Wake Interactions in Aeroelastic Flutter Vibration Energy
Harvester Arrays, ASME Conference on Smart Materials, Adaptive Structures and Intelligent Systems,
SMASIS2011, September 18 - September 21, pp. 1-7.
17 S. Roundy, P.K. Wright and J. Rabaey, A study of low level vibrations as a power source for wireless sensor
nodes, Computer Communications, 26 (2003), 1131-1144.
18 H.A. Sodano, G. Park and D.J. Inman, Estimation of electric charge output for piezoelectric energy harvesting,
Strain, 40 (2004), 49-58.
19 S. Roundy and Y. Zhang, Toward self-tuning adaptive vibration based micro-generators, Proceedings of SPIE,
Smart Structures, Devices, and Systems II, Vol. 5649 (2005), pp. 373-348.
20 Sustainable Energy Authority Victoria, Victorian Wind Atlas (Cartographic Material), 2003.
21 J. Creazzo, The interaction between a non-imbedded longitudinal vortex and turbulent boundary layer under the
influence of a stream-wise pressure gradient, PhD thesis, RMIT University, 1999.
22 J.D. Anderson Jr., Fundamentals of Aerodynamics, 4th edn., McGraw-Hill, New York, 2007
23 S. Alben and M.J. Shelley, Flapping states of a flag in an inviscid fluid: Bistability and the transition to chaos,
Physical Review Letters, 100 (2008), 074301-1 – 4.
24 B.S.H. Connell and D.K.P. Yue, Flapping dynamics of a flag in a uniform stream, Journal of Fluid Mechanics,
581 (2007), 33-67.
1410