Article citation information:
Haniszewski,
T., Gąska, D., Margielewicz, J., Opasiak, T. Numerical
investigations on the dynamic behaviour of a 2-DOF airfoil with
application in energy harvesting system. Scientific
Journal of Silesian University of Technology. Series Transport. 2023, 118, 77-91. ISSN: 0209-3324. DOI: https://doi.org/10.20858/sjsutst.2023.118.6.
Tomasz HANISZEWSKI[1], Damian GĄSKA[2], Jerzy MARGIELEWICZ[3], Tadeusz OPASIAK[4]
NUMERICAL INVESTIGATIONS ON THE DYNAMIC BEHAVIOUR OF A 2-DOF AIRFOIL
WITH APPLICATION IN ENERGY HARVESTING SYSTEM
Summary. This article
presents the basic airfoil model with two degrees of freedom - the semi-rigid
model, where its forced vibrations were considered, and the exciting force is
the aerodynamic force, including its periodic changes, that is, gusts. Since
the phenomenological model under study has a coupled form, its versions after
decoupling are presented, which has an impact on the results of the final
research. The airfoil model presented in this way was shown from the
application side in the system of a simple energy harvester based on a
deformable beam with piezoelectric elements. The result of the simulation tests
is a preliminary analysis of the possibility of using the airfoil as a
vibration generator for the energy harvesting system. Along with the
application of the mechanical part, a numerical simulation of the electrical
part was also implemented, related to the transformation of the voltage
generated by piezoelectric elements into a constant voltage signal with a
connected receiver with power consumption similar to the Atmega microcontroller
with battery charging.
Keywords: airfoil,
simulation, energy harvesting, piezo element, microcontroller
1.
INTRODUCTION
For several
decades, the development of various technologies for obtaining energy not based
on fossil fuels has been observed. Since the beginning of this century, energy
harvesting has been developing very dynamically, that is, a method of
recovering small amounts of energy, but significant for the technique, from
sources that normally waste it, for example, vibrations [1, 2], temperature
fluctuations [3], air flow [4, 5], solar, etc., which is beneficial in social
and ecological aspects. The most frequently developed and tested type of energy
harvesting is the recovery of energy from vibrating machine elements; however,
in recent years, constantly increasing attention has been given to the air
flow, especially considering the flow in ventilation, air conditioning, subway
tunnels, etc. The target of energy harvesting is to operate autonomous powered
electronic devices, for instance, sensors, but without external power sources.
Wind energy
commonly exists in the environment. There is a possibility to use small wind
turbines, but in the case of small sensors, it is pointless as traditional wind
turbine has a relatively large size and a high cut-in wind speed. The
wind-induced vibration energy harvesting is a microenvironmental
energy-capturing device [6] designed to harvest beam vibration energy imposed
by the air flow by different kinds of energy converters [7] – mostly
piezoelectric but also magnetic and others [8]. Piezoelectric transducers have
attracted significant interest because they can be used over a wide range of
frequencies and are easy to apply [9].
Considering
only the energy harvesters whose task is to recover energy from the air flow,
we usually deal with a beam at the end of which there is an active element that
causes the beam to vibrate [10]. This bluff-body is being placed in a flow
field and excited to undergo large oscillations absorbing energy from the air
that can be converted into electrical energy using piezoelectric and/or
electromagnetic transducers [11]. Aeroelastic phenomena are classified
according to their causes and main characteristics. Therefore, we are dealing
with flutter [12] when the bluff-body is an airfoil section, vortex-induced
vibrations [13, 14], in the case of circular cylinders, galloping [15, 16] for
prismatic structures and wake galloping [17, 18] when considering parallel
cylinders.
It is
important in the design that the energy harvesting takes place for different
air flow conditions, including flow velocity variation in analogy to realistic
environmental conditions. An airfoil may undergo stall-induced oscillations
beyond the critical flutter speed. Such oscillations yield intense periodical
motions that can be used to convert the airflow energy into electrical power
[19]. The harvester consisted of a rigid airfoil supported by nonlinear
flexural and torsional springs under the combination of vibratory base
excitations and aerodynamic loadings was considered in [20]. The effects of
system parameters on the performance of an energy harvester with three-to-one
internal resonance were analyzed in [21]. The research proved that the
harvested voltage could be significantly improved in the presence of internal
resonance. A modified airfoil-based piezo aeroelastic energy harvester with
double plunge degrees of freedom was proposed in [22]. In such systems, there
is also the possibility of chaotic movement, which was under consideration in
[23]. The investigated airfoil model with higher-order nonlinearities showed
multistability with changing airspeed and an infinitely countable number of
coexisting attractors. The airfoil as a bluff-body in energy harvesters was
also considered in [24, 25]. The current wind-induced vibration energy
harvesters usually work in a relatively narrow wind speed range or have a high
cut-in wind speed. In a real application, airflow often has random and unstable
characteristics and fluctuation in the wide velocity range.
The aim of
this paper is a preliminary analysis of the nonlinear airfoil as a vibration
generator for the energy harvesting system together with a numerical simulation
of the electrical part related to the transformation of the voltage generated
by piezoelectric elements into a constant voltage signal. The most important
part of the preliminary research presented in this article is the presentation
of the electrical subsystem responsible for the transformation of voltage from
alternating AC to constant DC. Also, the relationship of how important it is to
achieve a stable waveform of sinusoidal AC voltage generated by the
piezoelectric during the operation of the energy harvester was shown.
2.
PHENOMENOLOGICAL MODEL OF THE CONSIDERED NONLINEAR AIRFOIL SYSTEM WITH SEHS
As shown in the diagram, the system with the energy harvester includes
the rigid airfoil model with pitching and plunging degrees of freedom. The
piezoelectric coupling is included in the plunging degree of freedom. The
equations of motion of the 2 DOF airfoil and energy harvester with wind flow
excitation can be represented by equations (1). Symbols
|
|
(1) |
|
|
|
The forcing of vibrations in the system presented in
(2) is a combination of the lift force and the moment acting on the airfoil
with the change of wind speed and its gusts (4) with selected parameters.
Fig. 1.
Physical model of an airfoil with two degrees of freedom
|
|
(2) |
The nonlinearity of the model is implemented through a
nonlinear torsional stiffness, which is assumed to be described as a polynomial
function (3).
|
|
(3) |
Relationship 4 shows
the time course of the wind speed, which is a factor that forces the airfoil
structure to vibrate, where
|
|
(4) |
As the presented
system of differential equations (1) is in coupled form, the system was
decoupled and presented in form (6). The obtained equations (1) of the motion
of the system are inertia-coupled. The model built directly on their basis will
contain algebraic loops on signal lines representing the second derivatives of
the generalized coordinates. The standard way to solve the problem of algebraic
loops is to break them with a signal delay element. However, this may affect
the accuracy of the numerical calculations. The signal delay is a kind of
additional damping in the system, which in this case, is difficult to justify
physically. Due to the accuracy of numerical calculations, it is advisable to
inertia decoupling the system of equations of motion. These equations can be
treated as a system of algebraic equations due to the second derivatives of the
generalized coordinates. It can be checked (to simplify the work, a symbolic
calculator was used) that the determinant of the aforementioned system of
algebraic equations will be equal to:
|
|
(5) |
Since this determinant is always positive, the system
of equations can be solved where finally the uncoupled equations are obtained
in the form:
|
|
(6) |
2.1. Initial
identification of zones that are favourable from the energy harvesting approach
The bifurcation diagrams are
aimed at presenting the relationship between the value of the voltage generated
by the piezoelectric element and the spectrum of changes in the amplitude of
the excitation, which in this case, is the amplitude of changes in wind speed.
Fig.
2. Bifurcation diagram for wind speed 12, 14 m/s as a function of
wind gust amplitude in the range of 0-20 m/s
The diagrams presented in Figures 2, 3, and 4 show
2400 simulations for changes in the input amplitude
Fig. 3.
Bifurcation diagram for wind speed 8, 10 m/s as a function of
wind gust amplitude in the range of 0-20 m/s
Fig.
4. Bifurcation diagram for wind speed 4, 6 m/s as a function of
wind gust amplitude in the range of 0-20 m/s
Tab. 1
Table with calculation cases
for selected values based on the bifurcation diagrams (Figures 2-4)
|
1 |
|
|
|
|
||
|
2 |
|
|
|
|
||
|
3 |
|
|
|
|
The speed of vertical
vibrations is correspondingly -0.6 to 0.6 m/s for wind speed 12 m/s and for 14
m/s appropriately -0.9 to 0.95 m/s. In the case of torsional vibrations, the
values of airfoil deflection (torsional vibrations around the beam axis) for wind
speed are 12 m/s from -0.92 to 0.56 rad and for 14 m/s appropriately -0.94 to
0.68 rad. The speed of torsional vibrations is within the range -13.12 for
11.55 rad/s for 12 m/s, and wind speed of 14 m/s takes the values of -0.9 to
0.95 m/s. Noticeably, increasing the wind speed by 2 m/s causes an increase in
the amplitude of vertical vibrations responsible for the beam deflection with
the piezoelectric element, thus, generating higher voltage values, where the
peak-to-peak voltage for the wind speed of 12 m/s is 30.61 V and for 14 m/s, it
is the value of 47.12 V.
Fig. 5.
Results of the simulation tests for two wind speeds, 12 and 14 m/s, no gusts
According to the adopted plan of excitations applied
to the airfoil model (Table 1), the second option is to additionally apply the
effect of a gust of wind with an amplitude of 1 and 3 m/s and a frequency
of 10 rad/s. As can be seen in Figures 6 and 7, the amplitude of vertical
vibrations reaches its maximum values in the range from -0.1 to 0.1 m for a
wind speed of 14 m/s and a gust with an amplitude of 3 m/s, thus, giving the
highest peak voltage value of 51 V.
Fig. 6.
Results of the simulation tests for two wind speeds, 12 and 14 m/s, a gust of
wind with an amplitude of 1 m/s and the excitation frequency of 10 rad/s
Fig. 7.
Results of the simulation tests for two wind speeds, 12 and 14 m/s, a gust of
wind with an amplitude of 3 m/s and the excitation frequency of 10 rad/s
For the model under consideration, the negative effect
of the gusts is a significant increase in the amplitude of torsional vibrations
to the value of about 1 rad, which will contribute to the rapid destruction of
the system. It should also be noted that the proposed model does not consider
the deformations resulting from torsional vibrations, and the voltage values
are obtained only based on the vertical vibrations of the airfoil, that is,
they are related to the resulting deflected form of the beam. The speed of the end of the piezoelectric beam is not
without significance here, as it, together with the deflection, is responsible
for the value of the obtained voltage. High values of the vertical
vibration velocities of the airfoil can be observed when the wind gust is used;
these velocities range from -1.73 to 1.48 m/s, and thus, contribute to the
generation of the highest peak-to-peak voltage values.
2.2. Energy
acquired - the efficiency of the solution for recovering energy from mechanical
vibrations
In Figure 8, the
proposed electronic system is presented, enabling the conversion of the
alternating voltage generated by the piezoelectric to a constant value of a
specific level. The circuit consists of a full-wave rectifier based on
rectifying diodes and a filter capacitor to smooth the DC voltage. Then, to
regulate the voltage and ensure the situation of one-way power flow, a
buck-converter controlled by the PWM signal and a diode were used to obtain a
voltage in the range of 3-4 V at the output, needed to power the
microcontroller.
Fig.
8. Voltage transformation system
The pulse generator controls
the buck-converter in an open loop with a fixed switching frequency and a
defined duty cycle. The energy harvester system initially charges the battery,
where after 5 seconds, a receiver with a power of 20 mW and a voltage of 3.3 V
is connected to the circuit.
Fig. 9.
Results of the simulation tests for a constant wind speed of 12 and 14 m/s
At the input of the system
(Figure 8), the voltage generated based on numerical simulations carried out on
the tested object was applied. As a result of the conducted conversion of the
voltage generated by the energy harvester system into DC voltage, a series of
time waveforms was obtained, in the form of voltages, obtained power and the
battery charging curve.
Fig.
10. Time course of battery charging for constant wind speed, 12 and 14 m/s
Figures 9, 12, and 15 show the
time history of the voltage generated by the piezoelectric, after rectification
by the full-wave rectifier, and the voltage decreased to the desired value for
the batteries. Observably, depending on the applied excitation, the values of
the voltage generated by the piezoelectric differ given the vibration amplitude
and the speed of the vibrating beam, changing from 14.8 to 23.1 V for no gust
of wind, 15.8 to 24.6 V for gusts by the amplitude of 1 m/s and 27.5 to 46.3 V
for gusts of the amplitude of 3 m/s.
Fig. 11.
Results of simulation tests, for constant wind speed, 12 m/s, where output
power at final time = 27.86 mW, and 14 m/s, where output power at final time =
72.03 mW
Fig. 12.
Results of the simulation tests for wind speed 12 m/s,
a gust of wind with an amplitude of: a) 1 m/s,
b) 3 m/s for the frequency of changes in the gust amplitude 10 rad/s
Fig. 13. Time
course of battery charging for wind speed 12 m/s, a gust of wind with
amplitude: a) 1 m/s, b) 3 m/s for the frequency of changes in the gust
amplitude 10 rad/s
Fig. 14. Results of the simulation tests, for wind speed, 12 m/s, a gust
of wind with amplitude: a) 1 m/s where output power at final time = 33.6063 mW,
b) 3 m/s where output power at final time = 911.2049 mW for frequency variation
of the blast amplitude 10 rad/s
Figures 10, 13, and 16 show
the battery charging curves, where for almost every case of extortion, the
curve breaking point is characteristic, related to the inclusion of a 20 mW
load in the system. The highest charging capacity was obtained for the case of
forced wind gusts with an amplitude of 3 m/s, where the charging value closed
at 2 mAh in 10 s, giving theoretically 720 mAh charging in one hour, assuming
simultaneous operation of the loading device. Charts 11, 14, and 17 present the
time history of the power obtained for the tested energy harvester. In the case
of no gust, for both 12 and 14 m/s wind speed, constant power values in the
range of 27-72 mW were obtained. The use of gusts of wind increases the power
generated by the piezoelectric; however, these are only moments reaching the
peaks of 15.000 mW.
Fig. 15.
Results of the simulation tests, for a wind speed of 14 m/s, a gust of wind
with an amplitude of: a) 1 m/s, b) 3 m/s, and the excitation frequency of
10 rad/s
Fig. 16. Time
course of battery charging for a wind speed of 14 m/s,
a gust of wind with an amplitude of: a) 1 m/s,
b) 3 m/s for the frequency of changes in the amplitude of the gust 10 rad/s
Fig.
17. Results of the simulation tests for a wind speed of 14 m/s, a gust of wind
with an amplitude of: a) 1 m/s, b) 3 m/s and the excitation frequency of
10 rad/s,
output power at final time = 641.0324 mW and output power at final time =
3639.8759
The average maximum value for the obtained waveforms
with an amplitude of 3 m/s is 7800 mW, with the indication that these
values are obtained for large deformations of the beam, which should not be the
case with the actual system. According to the data obtained from bifurcation
diagrams, voltages of the order of 1000 V can be obtained; however, the
vibration amplitudes of the beam with the piezoelectric element reach values
only possible in theory due to the exceeding of the achievable limit stresses
and deformations already beyond the elastic range, thus, leading to the
immediate destruction of the device.
3. CONCLUSIONS
The presented
phenomenological model of a simple energy harvester based on airfoil
vibrations, with two degrees of freedom, shows features that allow its use in
energy harvesting systems. Nevertheless, it requires a special design due to
the high vibration amplitudes. According to the adopted plan of excitations
applied to the airfoil model (Table 1), with the addition of the influence of
the wind gust, it can be observed that the amplitude of vertical vibrations
reaches its maximum values in the range from -0.1 to 0.1 m for a wind speed of
14 m/s and a gust amplitude of 3 m/s, thus giving the highest peak-to-peak
voltage of 51V. For the model under consideration, the negative effect of the
gusts is a significant increase in the amplitude of torsional vibrations to the
value of about 1 rad, which will contribute to the rapid destruction of the
system. Further, it should be noted that the proposed model does consider the
deformations resulting from torsional vibrations, and the voltage values are
obtained only based on the vertical vibrations of the airfoil, that is, they
are related to the resulting deflected form of the beam, which should be
considered in future research also on a real object. The speed of the end of
the piezoelectric beams is not without significance here, as it, together with
the deflection, is responsible for the value of the obtained voltage. High
values of the vertical vibration velocities of the airfoil can be observed in
the case of blasts. It should also be remembered that it is necessary to obtain
a stable sinusoidal waveform of voltage changes, otherwise, the energy gain of
the solution will be lower, which is associated with the transformation of the
alternating signal into a constant one, where there are losses when passing
through the rectifier system and regulating the voltage supplied to the
receiver in the form of a battery that is charged by mechanical vibration. An
important fact is also to carry out the necessary tests related to the fatigue
strength of the proposed solution, aimed at estimating the lifetime of the
energy harvesting system presented in this way.
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Received 15.10.2022; accepted in
revised form 02.12.2022
Scientific Journal of Silesian University of Technology. Series
Transport is licensed under a Creative Commons Attribution 4.0
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[1] Faculty of Transport and Aviation
Engineering, The Silesian University of Technology, Krasińskiego 8 Street,
40-019 Katowice, Poland. Email: tomasz.haniszewski@polsl.pl. ORCID:
https://orcid.org/ 0000-0002-4241-6974
[2] Faculty of Transport and Aviation
Engineering, The Silesian University of Technology, Krasińskiego 8 Street,
40-019 Katowice, Poland. Email: damian.gaska@polsl.pl. ORCID:
https://orcid.org/ 0000-0002-2968-1626
[3] Faculty of Transport and Aviation
Engineering, The Silesian University of Technology, Krasińskiego 8 Street,
40-019 Katowice, Poland. Email: jerzy.margielewicz@polsl.pl. ORCID:
https://orcid.org/ 0000-0003-2249-4059
[4] Faculty of Transport and Aviation
Engineering, The Silesian University of Technology, Krasińskiego 8 Street,
40-019 Katowice, Poland. Email: tadeusz.opasiak@polsl.pl. ORCID:
https://orcid.org/ 0000-0002-0777-2316