Piezoelectric Energy Harvesting from Flow-Induced Vibrations in Micro Aerial Vehicles
Eduardo Piteira Pereira
Thesis to obtain the Master of Science Degree in Aerospace Engineering
Supervisor: Prof. João Manuel Melo de Sousa
Examination Committee Chairperson: Prof. Fernando José Parracho Lau Supervisor: Prof. João Manuel Melo de Sousa Member of the Committee: Prof. Filipe Szolnoky Ramos Pinto Cunha
November 2019 ii To Fatima,´ Carlos, David, Mar´ılia
”All it takes to make a smooth and easy flight is to fly loose and carefree.” - Richard Bach
iii iv Acknowledgments
First and foremost, I would like to express my gratitude to Professor Dr. Joao˜ M.M. Sousa for all the endless hours he spent teaching me about the subject and guiding me to accomplish the goals of this work in time. Even when the work seemed to be going nowhere, Professor Dr. Joao˜ M.M. Sousa kept me going and without his support this thesis would not have been possible. I would also like to thank Mar´ılia Matos for her help in making the MATLAB codes work and output the beautiful pictures it did. Additionally, I would like to thank Francisco Formigao˜ and Antonio´ Ramalho for helping me put the electric circuit working in order to obtain the useful data. Without a lot of help coming from outside of the university, I would not have been able to finish this thesis and so I would like to thank my parents Fatima´ and Carlos, my brother David and my girlfriend Mar´ılia for their unconditional support and motivation to keep me going. Lastly, I would also like to thank all my friends and colleagues with whom I shared some thoughts about my thesis.
v vi Resumo
Os maiores desafios que Micro Ve´ıculos Aereos´ (MVA) enfrentam atualmente sao˜ os limites opera- cionais devido as` restric¸oes˜ de peso das baterias e o desempenho menos bom devido ao efeito de perda a altos angulosˆ de ataque. De modo a superar este ultimo´ desafio um sistema de controlo de perda h´ıbrido ativo-passivo foi previamente desenvolvido no Instituto Superior Tecnico.´ O controlo ativo de perda e a ampliac¸ao˜ do alcance do MVA requerem de potenciaˆ adicional, potenciaˆ essa que nao˜ deve ser obtida a partir de um aumento do tamanho das baterias devido ao aumento excessivo do ve´ıculo. Consequentemente, este trabalho foca-se no estudo da viabilidade de um dispositivo piezoeletrico´ mon- tado no bordo de fuga da asa que oscila com a libertac¸ao˜ de vortices´ da asa de modo a gerar a en- ergia necessaria´ para o sistema ativo de controlo de perda entre outros. Um mo-delo em MATLAB foi desenvolvido com o proposito´ de encontrar o dispositivo otimo´ cuja frequenciaˆ de resonanciaˆ que cor- respondesse a` frequenciaˆ de libertac¸ao˜ de vortices´ da asa. Um segundo modelo foi desenvolvido com a capacidade de prever a voltagem que o dispositivo produz aquela` frequencia.ˆ Baseado no primeiro modelo, os dispositivos adquiridos foram sujeitos a testes no tunel´ de vento com o dispositivo A a ser capaz de gerar potenciasˆ entre 1 e 3 nW com um peso total de 0.05 g. As voltagens medidas em circuito fechado e aberto foram de 0.06 V e 0.3 V, respetivamente. Adicionalmente, mostrou-se que a asa possui um racio´ de sustentac¸ao˜ por resistenciaˆ superior com o dispositivo montado na asa face a` asa sem dispositivo. O modelo aero-electro-mecanicoˆ desenvolvido mostrou ter uma boa coerenciaˆ com os resultados experimentais apesar de algumas previsoes˜ darem valores inferiores aos resultados experimentais.
Palavras-chave: Micro ve´ıculos aereos,´ Controlo de perda, Gerac¸ao˜ de energia piezoeletrica,´ Vibrac¸oes˜ induzidas por vortices´
vii viii Abstract
The biggest challenges Micro Aerial Vehicles (MAV) face at present time are the limited operation windows due to weight restrictions on the batteries and the decreased performance due to early onset of stall at high angles of attack. To overcome this problem a hybrid active-passive stall control system was previously developed at Instituto Superior Tecnico.´ The active control system and the extension of the MAV’s range require additional power, which is not feasible with current batteries due to their excessive weight. Hence, this work aimed to study the feasibility of a piezoelectric device mounted at the trailing edge of a wing which oscillates with the shedding of the vortices from the wing to power the active stall control system among others. A model was developed to find the device whose resonant frequency matched best the vortex shedding frequency of the wing. Another model was developed with ability to predict the voltage output of the device operating at this frequency. Based on the first model, the acquired devices were put to the test in the wind tunnel with the most promising device (A) harvesting powers between 1 and 3 nW for its light weight of 0.05 g. The voltages in closed circuit are observed to be of the magnitude of 0.06 V with the open circuit voltages reaching 0.3 V. Additionally, the wing is shown to generally have a higher lift to drag ratio with the mounted piezoelectric flag relative to the wing without the device. The aero-electro-mechanical shows to be in good agreement with the obtained results despite some underprediction being conclusive.
Keywords: Micro Aerial Vehicle, Stall control, Piezoelectric Energy Harvesting, Vortex-Induced Vibrations
ix x Contents
Acknowledgments...... v Resumo...... vii Abstract...... ix List of Tables...... xiii List of Figures...... xv Nomenclature...... xvii Glossary...... xxi
1 Introduction 1 1.1 Motivation...... 1 1.2 Objectives...... 1 1.3 State-of-the-Art...... 2 1.4 Thesis Outline...... 7
2 Background 9 2.1 Theoretical Overview...... 9 2.1.1 Renewable Energy Sources...... 9 2.1.2 Piezoelectric Materials...... 15 2.1.3 Piezoelectric Modes...... 16 2.1.4 Piezoelectric Device Types...... 18 2.1.5 Polarisation of Piezoelectric Layers...... 18 2.1.6 Piezoelectric Constitutive Equations...... 19
3 Prediction Models 23 3.1 Frequency Model...... 23 3.2 Aero-Mechanical-Electrical Model...... 31 3.2.1 Electro-Mechanical Model of the Piezoelectric Beam...... 32 3.2.2 Aerodynamic forcing term...... 36 3.2.3 Solution of the system...... 38
4 Experimental Test Setup 41 4.1 Aerodynamic Forces and Moments...... 42
xi 4.2 Wind Tunnel Instrumentation...... 43 4.3 Multimeter...... 43
5 Results 45 5.1 Experiments...... 45 5.1.1 Impedance dependency for device A...... 45 5.1.2 Velocity dependency for device A...... 46 5.1.3 Comparison of device A with devices B and C...... 49 5.1.4 Series connection of multiple devices A...... 50 5.1.5 Effect of Lift and Drag Coefficient...... 51 5.1.6 Test using a standard wing...... 51 5.1.7 Power Density Ratio...... 52
6 Conclusions 57 6.1 Achievements...... 57 6.2 Future Work...... 58
Bibliography 59
A Piezoelectric Device Catalogue 65
B Technical Datasheets 67
xii List of Tables
2.1 Energy densities for the three types of converters...... 14 2.2 Charge coefficients of the primary piezoelectric materials...... 15 2.3 Comparison between the two modes [38]...... 17
3.1 Frequency results for the minimum substrate thickness...... 30
5.1 Device Masses...... 53
xiii xiv List of Figures
1.1 Power as a function of the lifetime of batteries, solar cells and vibration harvesters....3 1.2 Experimental setup of a beam subjected to axial flow...... 4 1.3 Possible configurations of energy harvesters...... 4 1.4 Oscillating ’eel’ behind a flat plate...... 5 1.5 Section types of the comparative study...... 5 1.6 Galloping effect of a triangular-shaped bluff body...... 6 1.7 Geometries of the tested cantilever beams...... 6 1.8 Comparison of extracted power for four beams at angles 0◦ and 90◦ ...... 6 1.9 Wing design and piezoelectric device by Esmaeili...... 7
2.1 Duck Curve - The total energy demand versus the energy produced from solar harvesting in California on a given day...... 10 2.2 Photo of Sunrise I’s maiden flight...... 10 2.3 Thin film solar panels on wing top of UAV...... 11 2.4 Solar Impulse 2...... 11 2.5 Airbus A320 Family Ram Air Turbine...... 12 2.6 Vibrational energy harvesters...... 13 2.7 Mechanical-to-electrical converters...... 14 2.8 Transverse mode configuration...... 16 2.9 Longitudinal mode configuration...... 17 2.10 Device Types...... 19 2.11 Connection types of piezoelectric beams: (a) series connection and (b) parallel connection 19
3.1 Two installations of the piezoelectric device on the wing...... 23 3.2 Frequency analysis of the two installations of the piezoelectric device on the wing..... 24 3.3 Flow fields showcasing the passing of the vortices along the airfoil and the movement of the piezoelectric plate relative to its stationary state...... 25 3.4 Harvested power as a function of the frequency from a piezoelectric flag...... 25 3.5 Sensitivity study of a bimorph plate with a Brass substrate...... 27 3.6 Sensitivity study of a bimorph plate with an Aluminium substrate...... 27 3.7 Sensitivity study of a bimorph plate with a Steel substrate...... 27
xv 3.8 Sensitivity study of a bimorph plate with a FR4 substrate...... 28 3.9 Sensitivity study of a bimorph plate with a Polyester substrate...... 28 3.10 Sensitivity study of an unimorph plate with a Brass substrate...... 29 3.11 Sensitivity study of an unimorph plate with an Aluminium substrate...... 29 3.12 Sensitivity study of an unimorph plate with a Steel substrate...... 29 3.13 Sensitivity study of an unimorph plate with a FR4 substrate...... 30 3.14 Sensitivity study of an unimorph plate with a Polyester substrate...... 30 3.15 Oscillation regimes as a function of the parameter β ...... 31 3.16 Theodorsen model of a harmonically oscillated thin airfoil...... 36 3.17 Results of the aeroelectromechanical model...... 39 3.18 Results of the aeroelectromechanical model...... 39
4.1 Wind tunnel diagram with engine and data acquisition system...... 41 4.2 Graphic interface of AeroIST software...... 42 4.3 Forces and Moments Balance...... 42 4.4 Diagram with the various balance components...... 43 4.5 Yokogawa 7551...... 43
5.1 Mouser Electronic Devices...... 45 5.2 Power as a function of the circuit’s impedance...... 46 5.3 Harvested power vs. Flow velocity...... 47 5.4 Tested positions (y/c) on the wing...... 48 5.5 Voltage vs AoA for device A in open circuit...... 48 5.6 Power vs AoA for device A in closed circuit...... 48 5.7 Device B tests in open and closed circuit at Re = 140000...... 50 5.8 Device C tests in open and closed circuit at Re = 140000...... 50 5.9 Results of series connection of multiple devices A at Re = 140000...... 51
5.10 CL/CD ratio for the wing with and without piezoelectric device...... 52 5.11 Results for simple wing for R = 1.55 MΩ ...... 52 5.12 Power per Mass vs AoA of device A...... 53 5.13 Power per Mass vs AoA of devices B and C...... 53 5.14 Power as function of the impedance for Re = 140000 and α = 20◦ ...... 54 5.15 Power density ratio optimisation...... 54
B.1 Material Datasheets...... 67
xvi Nomenclature
Greek symbols
α Angle of attack.
β Non-dimensional structural stiffness.
χr Modal coupling term.
S 33 Piezoelectric permittivity.
th ηr Modal coordiante of the beam for the r mode.
λ Root of the characteristic equation.
µ Mass per length unit.
ν Poisson coefficient.
νem Electro-mechanical coupling term.
th ωr Undamped natural frequency of the r mode.
th ωrd Damped natural frequency of the r mode.
th φr Mass normalized of the beam for the r mode.
ρ Density.
th ξr Mechanical damping ratio of the r mode.
Roman symbols
B Dimensional flexural rigidity. c Wing chord. ca Damping due to air viscosity.
CD Drag Coefficient. cs Damping due to structural viscoelasticity.
D Electric Displacement.
xvii d, g, i Piezoelectric constants. d31 Piezoelectric strain coefficient.
E Electric Field.
E Young’s Modulus. e Permittivity constant.
F Fluid force. f Frequency.
H Width. h Thickness.
I Second moment of area.
L Length.
Mbm Internal bending moment. p Pressure.
R Electrical resistance.
Re Reynolds Number.
S Strain. s, b Elastic compliance.
T Stress. t Time.
U Air speed. v Voltage. x, y, z Cartesian components.
YI Bending stiffness of the composite cross section. q Electric charge
Subscripts
∞ Free-stream condition. f Fluid. p Piezoelectric characteristics.
xviii s Substrate characteristics. ref Reference condition.
Superscripts t Transpose.
xix xx Glossary
APU Auxiliary Power Unit is the secondary power source of modern aircraft
AoA Angle of Attack is the angle of a wing relative to the horizontal plane being positive for a nose-up
CFD Computational Fluid Dynamics is a branch of fluid mechanics that utilises numerical methods and algorithms to solve problems that involve fluid flows
ICE Internal Combustion Engine is the motor used in most cars and airplanes at the moment
LCO Limit-cycle oscillations
MAV Micro Aerial Vehicle
MEMS Micro-Electro-Mechanical System
NA Neutral Axis
NM Nautical Miles
PVDF Polyvinylidene fluoride is the piezoelectric polymer used in this work
PZT Lead Zirconate Titanate is the most widely used piezoelectric material
Re Reynolds Number relates the viscous forces and inertial forces in a flow
TOW Take-Off Weight is the weight of an airplane or drone when its fully loaded for the mission
UAV Unmanned Aerial Vehicle
xxi xxii Chapter 1
Introduction
1.1 Motivation
Being wary about the effects of fossil fuels on the climate and human health, finding alternative ways to re-use ambient energy is vital to power aircraft in the future. Internal combustion engines (ICE) powered by fossil fuels are steadily becoming obsolete in the majority of the transport sector, with aviation being the one sector lagging behind. Invariably this change is already hitting the aviation industry, with new all-electric light aircraft being launched in the upcoming years. However, the utilisation of an electric system, such as the ones found in current electric vehicles, is only feasible on its own for low autonomy missions, due to the weight restrictions on the battery packs. The Eviation Alice, an Israeli electric airplane, is able to carry 9 passengers for 540 nautical miles (NM) with the batteries accounting for nearly two thirds of the aircraft’s take-off weight (TOW) [1]. In order to increase the range of such aircraft, one would have to increase the battery capacity as well, which in turn reduces the performance of the aircraft, thus reducing its range. Hence, harvesting ambient energy such as solar, thermal, wind, acoustic waves or mechanical vi- brations is key to enhancing the range of aircraft, with each one of these types having certain pros and cons in different mission types. Herein comes the idea of using the vortices created by the flow on the wing to harvest energy to power certain aircraft systems and in the best case scenario to power the aircraft’s engines. However, before jumping straight away into large aircraft, one will start by analysing smaller vehicles, MAVs.
1.2 Objectives
As will be shown in section 1.3, previous work performed at Tecnico´ had the aim of designing a hybrid passive-active stall control for MAVs [2]. However, the piezoelectric device vibrating inside the wing did not show promising results to justify the inclusion of the device in the MAV. The obtained power was not compensating the added mass of the piezoelectric device. This was due to the aeroelastic filter that attenuates the mechanical vibrations of the wing. In fact, since all wings possess this type of filter to
1 prevent flutter and other aeroelastic behaviours, the option of placing the piezoelectric device inside the wing is not feasible. Tuning the device to match the frequency of the vibrations was a considered option. This adds more mass to the device with no substantial benefits, though [2]. As such, the attention was turned from the mechanical energy of the wing to the ambient energy induced by the flow in the wing. The characteristics of the leading edge of the wing make it so vortices are being shed along the wing at a certain frequency. This is due to the shape of the leading edge of the wing, with the vortices being constantly shed from the leading edge of the wing, particularly where the valleys are placed. By matching the piezoelectric device to that same frequency, the piezoelectric device can harvest the vortex-induced vibrational energy. As seen in section 1.3, there are two possible configurations: piezoelectric flag connected at the trailing edge of the airfoil or inverted flag mounted on a support away from the upper surface of the airfoil [2]. The main goal of this work is to test the first configuration with the wing with the leading edge hump- back whale modification in a wind tunnel provided by Tecnico.´ The piezoelectric device will be tested at air speeds ranging from 0 to 10 m/s. The tests will be done in the following order:
1. Evaluation of the effect of the impedance of the electrical circuit on the generated power.
2. Evaluation of the effect of the air speed on the generated power.
3. Evaluation of the position of the piezoelectric device and the wing AoA at fixed speeds on the generated power.
4. Testing of similar piezoelectric devices with some minor differences in characteristics.
5. Evaluation of the effect of the device on the CL and CD of the wing
1.3 State-of-the-Art
In the last decade much thought and research has been done in the area of piezoelectric energy harvesting particularly because of the limited available energy for aircraft in flight, which compromises the operation and mission profiles of Unmanned Aerial Vehicles (UAV) and Micro Aerial Vehicles (MAV). An increase in available energy through harvesting would lead to enhanced range and extended mission profiles. Currently, electrical energy is stored in batteries because of their high energy density and low self discharge. The problem lies in batteries losing storage performance over time and needing replacement once the operating cycle is complete, which can be observed in figure 1.1, where the power over time of batteries is compared to renewable energy sources. Additionally, batteries might heat up and ultimately ignite due to poor design, charge and discharge cycles [3]. Since most UAV missions are performed in hard-to-access environments, replacing batteries is not only cost-intensive but also extremely difficult. Hence, an energy harvester can be introduced to operate low power consumption devices and to store and recharge batteries. These harvesters are beneficial relative to the current designs because
2 Figure 1.1: Power as a function of the lifetime of batteries, solar cells and vibration harvesters [2] there is no need to replace batteries, no need for extra cabling and reduced maintenance costs of such a device [3].
Vibration energy harvesters have been thoroughly studied in recent years. The most common studies involve the configuration of a flapping flag, fixed at the leading edge of the flag and free to flap at the trailing edge, subject to axial flow. These harvesters oscillate due to the destabilising fluid forces being attenuated by the stabilising elastic force of the beam (structural stiffness). The oscillations can then be self-sustained if a critical velocity value is reached. [4][5][6] develop numerical models to predict the behaviour of the flag in flutter boundary and post-critical conditions. [6] shows that when the length of the rigid wing is short, the flow over it is affected by the oscillations of the piezoelectric plate. The longer the wing is, the less it is affected by the movement of the plate. As expected, an increase in material damping and a higher drag coefficient of the moving plate, have a stabilising effect on the flutter critical point. Tang and Dowell [7] concluded that as flow velocity increases, the amplitude of limit-cycle oscillations (LCO) increases as well while their frequency decreases. LCO’s prevent structural damage from occurring by limiting the amplitude of the oscillations. They impact the wing’s performance negatively and contribute to structural fatigue. Tang et al. [8] studied the energy transfer between a cantilevered flexible plate (various locations) and the fluid under flutter conditions and axial flow. The conclusion was that once flutter is reached, the vibrations would be self-sustained and energy would be consistently be pumped to the plate. An experimental study on the effects of the fluid-structure interaction on piezoelectric flag at the root of a cantilever beam, see Figure 1.2, concluded that 17% of the flow’s energy could be harvested [9].
Doare´ and Michelin [10] derives a fully linear model for the fluid-solid interaction and the electrical circuit of a flexible plate undergoing flutter. Additionally, a stability analysis was performed yielding higher rigidity and damping due to the piezoelectric energy harvester. Also, the interaction and subsequent flow- induced vibration between two cantilever beams was studied [11]. Bryant et al. [12] studied the effects of stacking several energy harvesters and how the wake of each one affected the energy harvesting
3 Figure 1.2: Experimental setup of a beam subjected to axial flow [9] potential of the subsequent harvesters. Energy harvesters can work in various configurations with each one being studied for purposes of maximum extracted energy [13]. It was concluded that mode I had the highest power output out of all the modes due to the high deflection of the piezoelectric energy harvester across its whole area, see Figure 1.3.
Figure 1.3: Possible configurations of energy harvesters [13]
Bluff bodies have also been subject to extensive studies with two types of oscillations being observed: galloping oscillations and vortex-induced vibrations. More interesting is to study the effects of the wake of bluff bodies on a structure, like a cantilever beam, placed downstream of the bluff body. Most studies revolve around using a cylinder to induce these wake vortices that vibrate the cantilever beam. Akaydın et al. [14] studied the power output of an energy harvesting device in the wake of a cylinder subjected to a high Re flow. The vortices in the turbulent wake flow induce vibrations on the device. Their experiments resulted in a maximum power output of 2 µW measured at the centerline at a distance of x/D=2. It was concluded that as the distance of the piezoelectric beam to the cylinder is increased along the centerline, the power output decreases because of viscous vortex dissipation. Allen and Smits [15] examined the
4 effects of vortex-induced vibration of a membrane in the wake of a flat plate, shown in Figure 1.4. It was concluded that the von Karm´ an´ vortex sheet forming behind the bluff body induces vibrations on the membrane, which shows lock-in behaviour, the membrane vibrating at the same frequency as the vortex-shedding frequency of the bluff body.
Figure 1.4: Oscillating ’eel’ behind a flat plate [15]
More studies come in all shapes and forms of the bluff bodies like T-shaped cantilevers [16] and H-shaped beams [17]. Yang et al. [18] compared several shapes of bluff bodies, see Figure 1.5 and arrived at the conclusion that square-shaped bluff bodies present power output advantages over other shapes of bluff bodies with enough energy harvested to power low-consumption sensors.
Figure 1.5: Section types of the comparative study [18]
Sirohi and Mahadik [19] analysed the galloping effect of a tringular-shaped bluff body on the har- vested power of piezoelectric beams in its wake, Figure 1.6. Moreover, a quasi-steady aerodynamics model was derived to couple to the electro-mechanical model in order to predict the obtained power. It was concluded that the model was not in total agreement with the experimental results. The authors believe that the apparent mass effect, disregarded by them, had a significant impact in the aerodynamics at those test speeds. Another interesting topic studied is the effect of turbulent air flow on the energy harvesting capabilities of a piezoelectric beam. Goushcha et al. [20] tested various flexible beams (Figure 1.7) with several angles of incidence to understand the electrical outputs generated by the interaction. The graph from Figure 1.8 shows the obtained results for every beam that was tested. It was con- cluded that the power outputs are higher for increasing velocities and closeness to the wall due to increased turbulence intensity. All these studies are focused on the effects on the piezoelectric beam by the wake of a bluff body or
5 Figure 1.6: Galloping effect of a triangular-shaped bluff body [19]
Figure 1.7: Geometries of the tested cantilever beams [20]
Figure 1.8: Comparison of extracted power for four beams at angles 0◦ and 90◦ [20] a galloping effects. In the literature there is not much about the wake of an airfoil, though. Most studies that include airfoils are numerical studies [21] or morphing wing studies [22][23].
6 More recently, [2] studied the energy harvesting of a piezoelectric beam inside a MAV wing being oscillated by the mechanical vibrations of the wing. His goal was to harness enough energy to power an active stall control system to add to his passive stall control which was based on a leading edge change of a NASA LS(1)-0417 wing inspired by the flipper of humpback whales.
(a) Leading edge change of the NASA LS(1)-0417 wing (b) Piezoelectric device inside the wing
Figure 1.9: Wing design and piezoelectric device by Esmaeili [2]
Gunasekaran and Ross [24] also studied the vortex-induced vibrations on a piezoelectric flag in an inverted configuration in the wake of a NASA LS(1)-0417 airfoil and a SD7003 airfoil. They concluded that for the Re tested, the voltage was highest at a certain position above and downstream of the trailing edge of the wing. They also concluded that using such a device for wake sensing is viable, while not disturbing the wing itself.
1.4 Thesis Outline
This thesis is divided into 5 chapters with the following structure: In Chapter 2 the necessary theory to understand this work will be presented. The focus will be on the piezoelectric effect and how its fundamental principles enable energy harvesting. Chapter 3 will introduce the reader to the relevant models that devise the problem and the respec- tive constitutive equations. These are the models that enable the prediction of the frequency and power values the experiment will produce. The three underlying concepts of the problem (aerodynamic, me- chanical and electrical) will be explained along with how they mesh together to formulate the problem. The integral parts that compose the experimental test setup will be shown in Chapter 4. The reader will understand how the wind tunnel works and how the data is measured. Chapter 5 will include all the results from the run tests and the discussion of the obtained results. Comparisons between devices and other results available in the literature will be made. Finally, Chapter 6 will present the conclusions made from the work and possible future work that can be done on the back of this work.
7 8 Chapter 2
Background
2.1 Theoretical Overview
With the growing demand for not just energy but clean and sustainable energy, many everyday thoughts focus on finding new ways to use renewable energy sources in everything that revolves around our everyday life. For this, the human being is primarily harvesting solar, wind and water energy to shift from coal and gas towards a more greener future.
2.1.1 Renewable Energy Sources
Solar Energy
The development of photovoltaic cells to harvest energy from solar rays has been continuously de- veloped since the middle of the last century. The practical applications of these cells are enormous and in the last few decades their implementation in our lives has been widespread. They power homes, offices, cars and even aircraft and can be used for heating water. However, solar energy has a fundamental problem in that energy can only be harvested during daytime. This phenomenon of harvesting energy during the day and then relying on batteries or grid energy for the night is called the duck curve, Figure 2.1, and is the reason why solar farms are an intelligent way to move to green energy but can only function on liaison with other types of energy sources that suppress the energy demand during the night. In addition to this, current photovoltaic cells are made of crystalline silicon with a maximum achieved efficiency of 26,7% and a reachable limit of 29% [26]. Most studies revolve around trying to reach higher efficiencies of energy production. Along with improving manufacturing processes, installation and research and development, these make the growth of solar systems a costly venture [2]. Moreover, crystalline silicon photovoltaic cells are considerably heavy presenting a major obstacle in UAVs because it limits their operational range. In spite of these limitations, solar-powered aircraft have received a lot of attention since the maiden flight of the solar-powered aircraft Sunrise I-unmanned aircraft in California in 1974, Figure 2.2[27].
9 Figure 2.1: Duck Curve - The total energy demand versus the energy produced from solar harvesting in California on a given day [25]
Figure 2.2: Photo of Sunrise I’s maiden flight [27]
To overcome the problem of weight restrictions on UAVs and MAVs a different solution was developed by Anton and Inman [28], in which amorphous silicon is painted or rolled onto a thin layer of substrate in order to reduce the weight in comparison with ordinary solar panels.
10 Figure 2.3: Thin film solar panels on wing top of UAV [2]
Their results showed good energy harvesting capabilities allowing them to extend the range of the UAV or power some aircraft’s systems. In the last decade, the Solar Impulse 2 (Figure 2.4) aircraft was developed and completed the first solar-powered round-the-world trip in 2015 after 23 days and 550 hours of total flight time which included the crossing of the Atlantic and Pacific Oceans [29]. With this flight, the team also achieved the goal of flying various nights without stopping, something never done before. This presents the greatest challenge of solar-powered UAVs or MAVs, harvesting enough energy during the day to power it at night or in the presence of environmental phenomena such as clouds or storms. As such, most current projects are focused on high altitude missions where the aerodynamic drag is lower, environmental phenomena are less likely to occur. Solar-powered UAV applications include surveillance missions, fire fighting and spacecraft applications for space exploration [2].
Figure 2.4: Solar Impulse 2 [30]
Fixed-wing UAVs can definitely benefit from solar cells to power their systems. However, MAVs pose another set of challenges that did not appear when analysing UAVs. MAVs’ propulsion systems are
11 smaller and cannot propel the aircraft to as high altitudes as UAVs’ propulsion systems. Additionally, MAVs typically operate small cameras that are not capable of capturing images in high altitudes. Thus, MAVs fly at low altitudes, hence being more subject to encounter adverse environmental phenomena (winds, storms, clouds). Another problem rises with the smaller batteries incorporated in MAVs in com- parison to the ones used in UAVs that do not possess the capabilities to store as much energy, making it difficult to operate during nighttime. However, endurance can still be boosted during daytime by the use of solar panels. Large scale solar panels, which require larger wing surface areas, are needed to achieve typical endurance times of MAVs, which in turn means larger aircraft leading to conflicting with the definition of MAV.
Wind Energy
The use of wind energy is also widespread in the world at this time, with the number of wind turbines in coastal and mountainous areas rising steadily since its introduction. In view of the more continuous energy production by wind turbines, many developments have been made in this area including the research and development of offshore wind turbines. Offshore wind turbines benefit from high and consistent winds in high seas due to massive air currents that flow between the sea’s surface and the atmosphere as opposed to onshore wind turbines that suffer from physical barriers and other constraints. Offshore wind energy resources can be up to twice as much as onshore ones and do not need to be wary of noise pollution leading to increased output capacities [31].
In aviation wind turbines seldom see some application in large airplanes and its only use comes in case of an emergency when primary (engines) and secondary (APU) power sources fail [32]. They are either used in case of loss of primary electric generation or in case of loss of hydraulic systems. In the event this occurs, the ram air turbine will power flight control systems and instrumentation or power an electrical generator. A picture of an Airbus A320 Family Ram Air Turbine is shown in Figure 2.5.
Figure 2.5: Airbus A320 Family Ram Air Turbine [33]
Although useful in emergencies, these turbines are not a viable option when it comes to harvesting ambient energy efficiently and with a small overall increase in weight.
12 Thermal Energy
The thermoelectric effect, transformation of temperature differences into electric voltage, was first discovered in 1834 [34]. A conducting material subjected to a temperature gradient gives rise to heat fluxes creating a flow of charge carriers between the cold and hot zones thus generating a voltage difference [2]. In the world there are many processes where heat is created as a byproduct and ex- hausted without having any goal. This wasted heat can otherwise be utilised to generate electricity via the thermoelectric effect for instance in nuclear and coal power plants [34]. However, for thermal energy to originate usable electric energy high temperature gradients are needed. In 1997, it was demonstrated how a 10 ◦ C temperature gradient was capable of generating 15 µW/cm3 of electric energy in a microdevice [35]. Therefore, such a device does not produce significant amounts of energy for the typical temperature gradients found and as such is not viable option for MAV applications.
Vibration Energy
Structural vibration is present everywhere with the structures themselves being built in a way that prevents those vibrations from destroying the structure. The idea of vibrational energy harvesting is to convert these vibrations into electrical energy. With the rise of micro-electro-mechanical system (MEMS) devices, this type of energy harvesting developed due to the amounts of energy that could be harvested versus the cost and weight of such a device [2]. In reality this process is done in two steps: a mechanical-to-mechanical conversion first, usually through a mass-spring system, followed by a conver- sion to electricity by a mechanical-to-electrical converter. The use of a mass-spring system is primarily due to the fact that ambient vibrations are comparatively low, which are then boosted by the mass-spring system inducing resonance in the mass, which is were the harvested power is largest, Figure 2.6. This resonance phenomenon amplifies the amplitude of the mobile mass relative to amplitude of the ambient vibrations [36].
(a) Concept (b) Resonance (c) Model
Figure 2.6: Vibrational energy harvesters [36]
There are three types of mechanical-to-electrical converters: piezoelectric converters, electromag- netic converters and electrostatic converters. Figure 2.7 shows the principles on how they operate [36].
13 Figure 2.7: Mechanical-to-electrical converters [36]
Piezoelectric devices operate in two ways: mechanical stresses generate electrical voltage or elec- trical voltage generates mechanical stresses. Thus, there is a wide application for these materials. The harvested power of piezoelectric devices is typically of the µW magnitude, sufficient for most micro-scale devices [2]. Electromagnetic converters generate an electromotive force from the relative movement between a magnet and a coil. In depth, mechanical vibrations induce a movement of the magnets that produce small currents by moving in relation to conductors. Still it is difficult for these devices to be built in MEMS scale and their outputs are not large enough to justify their implementation. Moreover, their efficiency decreases with decreasing frequencies [2]. In electrostatic converters the plates of a charged capacitor are separated by the continuous me- chanical vibrations, thus converting the mechanical energy into electrical. To be viable, this type of harvester requires very high voltages, in excess of 100 V , and a polarisation source, which poses sev- eral hindrances, though [2]. Additionally, their output is seriously affected by parasitic capacitances and by low overall capacitances [36]. An in-depth review of the three types of converters was done by Wei and Jing [37]. From these three options, the one that shows the most promise and is possible to integrate in current MEMS-scale devices are piezoelectric devices. Piezoelectric devices benefit from their simple structures and ease of miniaturisation, whereas electromagnetic and electrostatic converters are complex and difficult to build in MEMS-scale. Although, piezoelectric devices may suffer from depolarisation, brittleness in bulk layer and difficulties in connecting the piezo-film to the electrode, they do not have any moving parts which increases their lifetime massively and they present a higher energy density than the other options, as is shown in table 2.1.
Table 2.1: Energy densities for the three types of converters [2] Converter Energy Density Assumption Piezoelectric 35.4 PZT 5H Electromagnetic 24.8 0.25 Tesla Electrostatic 4 3 × 107 V m−1
For these reasons, piezoelectric devices have been the primary choice in recent studies and devel- opments made in this field of study.
14 2.1.2 Piezoelectric Materials
Piezoelectricity is defined as the mechanism through which a mechanical stress generates an elec- tric charge in a material. The materials that possess this capability are called dielectric materials. Con- versely, when an external electric field is applied on the same material, the material deforms and vi- brates. The former is called the direct piezoelectric effect or generator effect and the latter is called the indirect piezoelectric effect. Hence, piezoelectric materials are transduction systems capable of converting mechanical energy into electrical energy or vice-versa [2]. The piezoelectric effect was first elaborated by Pierre and Jacques Curie in 1880 with the materials used at the time showcasing low piezoelectric outputs, though. With the development of ceramics capa- ble of much higher power levels, the practicality of these materials for implementation in electromechani- cal devices rose significantly. The first ceramic that kick-started this field was Barium Titanate (BaTiO3). Recently, many other ceramics have been developed that can be divided into following groups based on their chemical composition [38]:
• Lithium Niobate (LiNbO3)
• Lithium Tantalate (LiTaO3)
• Sodium Tungstate (Na2WO3)
• Lead Zirconate Titanate (PZT) (Pb(Zr,Ti)O3)
• Potassium Niobate (KNbO3)
Additionally, piezoelectric polymers such as Polyvinylidene Fluoride (PVDF) have been discovered. It is known that natural crystals, for instance quartz, have a high output voltage. However, natural materials are not a viable solution since they require large forces (magnitude of kN) to induce vibration which penalises the weight of the MAV. Therefore, synthetic crystals are the best solution at the moment for this application. For the presented types of piezoelectric ceramics and polymers, table 2.2 shows the charge coefficients of the two operating modes. These modes will be explained in section 2.1.3.
Table 2.2: Charge coefficients of the primary piezoelectric materials [38][39] Operating mode PZT PVDF Ba-Titanate Li-Niobate Li-Tantalate
d31 -250 -18 -30 -0.85 -3
d33 700 30 145 6 5.7
From the values, clearly PZT based devices have the highest charge coefficients of the tested ma- terials. Hence, it is the most widely used piezoelectric material in all sorts of applications. Besides its piezoelectric properties, PZT also boasts good pyroelectric properties and a large dielectric constant enabling its use in other areas of science. Typically, PZT is doped with acceptor dopants, which give rise to oxygen (anion) vacancies, creating hard PZT or with donor dopants, which induce metal (cation) vacancies, creating soft PZT. These two types of PZT also differ in their piezoelectric constants, a value
15 that is proportional to the polarisation or the mechanical stress produced per unit of electric field ap- plied. Soft PZT exhibits a higher piezoelectric constant compared to hard PZT. However, due to being more maleable, soft PZT also has higher internal losses due to friction than hard PZT [2][38]. Recently, PVDF has also seen its use rise in many areas like biomedical and aerospace due to its simplicity and high piezoelectric constant. It is much less rigid and heavy than PZT, making it much more viable in applications where weight is a constraint.
2.1.3 Piezoelectric Modes
There are two modes in which the electrodes can be configured in: mode [3-1] and mode [3-3]. The configuration is dependent on the direction of the applied strain and the direction of the electric field. These modes have some fundamental differences, which will be presented and explained.
Mode [3-1]
Figure 2.8 shows a typical representation of a piezoelectric device in [3-1] mode. This mode is defined by the strain vector being perpendicular to the electric field vector, meaning that the applied force and poling have the same direction. In Figure 2.8, one can see the substrate represented in yellow surrounded by two piezoelectric layers in grey and four electrodes (dark lines). This is also called a bimorph configuration due to the fact that it includes two piezoelectric layers [2].
Figure 2.8: Transverse mode configuration [2]
Mode [3-3]
Figure 2.9 shows the schematic of a harvester operating in [3-3] mode. This mode is characterised by an applied force being perpendicular to the poling direction, which leads to the electric field and strain vectors being parallel. As can be seen in the schematic, the electric field does not function in a straight line along the piezoelectric layer, though. Having differently poled electrodes on each piezoelectric layer, one can observe how the electric field needs to bend to reach from one electrode to the other. As such, the electromechanical modelling of this mode is much harder than for mode [3-1] [2].
16 Figure 2.9: Longitudinal mode configuration [2]
Comparison of modes
When choosing the design of the piezoelectric device, one has to take into account that only the structural stiffness and the density influence the generated power. While the density is constant in both operating modes, the structural stiffness is not due to the way the piezoelectric layers are constructed. Knowing that the bending stiffness is theoretically higher in [3-1] mode than in [3-3] mode and that the power is inversely proportional to the structural stiffness, mode [3-3] has theoretically higher power. Moreover, the latter mode allows for control of the voltage via the spacing of the electrodes due to the relation of the voltage with the electric field, while the former only allows for the electrode spacing to be determined by the thickness of the piezoelectric layer [2]. However, mode [3-3] poses some challenges in the manufacturing process due to its complexity for little added benefit in terms of generated power. Therefore, if the power and voltage output requirements can be satisfied using mode [3-1], then there is no point in using the more costly and complex mode [3-3]. Additionally, some studies have shown that even though mode [3-3] is theoretically beneficial in raw power outputs, experimentally that might not be the case, as is shown by Wu and Lee [40]. They compared the two modes in two piezoelectric harvesters, which resulted in the following values:
Table 2.3: Comparison between the two modes [38]
Mode fres [Hz] Optimal Load [kΩ] Power [µW] Voltage (open) [V] Voltage (closed) [V] 3-1 255.6 150 2.099 2.415 1.58 3-3 214 510 1.288 4.127 2.292
As can be seen from the values of table 2.3, in this experiment, mode [3-3] has higher output voltages in open and closed circuits than mode [3-1]. However, since the optimal load of the former is three times higher than of the latter, the power output is higher for the latter. The reason mode [3-1]’s optimal load is lower than that of mode [3-3] is due to the higher capacitance of the former. The optimal load and capacitance follow an inversely proportionality rule. In addition, the power output of the [3-3] mode was lower because of the non-uniform poling direc- tion of the PZT material being poled by interdigitated electrodes. It meant that the material under the
17 electrodes was not being poled correctly and thus not used. Moreover, the poling electric field strength becomes less effective, the farther away the surface of the PZT is. This all leads to a performance drop of mode [3-3], even though the measured voltages were higher due to distance adjustments of the electrodes [2][40].
2.1.4 Piezoelectric Device Types
Bimorph and unimorph cantilever beam devices have attracted the most attention for energy harvest- ing purposes. While the bimorph configuration utilises two piezoelectric layers patched onto a substrate layer, the unimorph configuration only utilises one piezoelectric layer. To overcome the brittleness of the piezoelectric materials and increase the elasticity of the device, an elastic metallic layer is used as a substrate. The efficiency of these configurations is dependent on the subjected conditions. Unimorph devices produce more energy when the frequency and load resistance are low, while bimorph devices in series connection produce the most energy when the frequency and load resistance are high. In the middle come the bimorph devices with a parallel connection [2]. Bimorph devices present several advantages relative to unimorph devices in terms of mechanical strength, voltage output and manufacturing cost. However, since the frequency at which the device will operate is low, unimorphs are the optimal choice. In the market, there are a lot of options of piezoelectric device in both bimorph and unimorph configuration. Available devices, Figure 2.10, are thin-layer uni- morphs, for instance THUNDER devices with a bottom layer of stainless steel, middle layer of PZT and top layer of aluminium, active fiber composites, where piezoelectric fibers are surrounded by a polymer matrix, radial field diaphragms, which comprise of a thin circular piezoelectric ceramic disk surrounded by two dielectric films and bimorphs with two piezoelectric layers glued onto a substrate layer and an intermediate layer for higher rigidity [2]. The problem encountered with the available devices in the market was that most had increased rigidity due to the substrate layer(s) used. In chapter3, a frequency model is developed that gives appropriate layer thicknesses for the typical frequencies encountered in this work. As such, the choice fell upon acquiring the piezoelectric layer that works better alone or with a thin and not very rigid substrate layer to achieve the optimal frequency.
2.1.5 Polarisation of Piezoelectric Layers
When constructing a bimorph piezoelectric harvester, there are two ways the two piezoelectric layers can be polarised, as is shown in Figure 2.11. As a result of the polarisation and wiring, the layers can either operate in serial or parallel connection [2]. In serial connection, the two piezoelectric layers are poled in the opposite direction achieving hereby a higher voltage and lower current. This means that the applied strains generate electric fields that occur in the same direction above and below the neutral axis (NA). In parallel connection, the two piezoelectric layers are polarised in the same direction which leads to a higher current and lower voltage being generated.
18 (a) Thin layer unimorph (b) Active Fiber composite
(c) Radial Field Diaphragm (d) Bimorph
Figure 2.10: Device Types [41][42]
Figure 2.11: Connection types of piezoelectric beams: (a) serial connection and (b) parallel connection [2]
2.1.6 Piezoelectric Constitutive Equations
As stated before in section 2.1.2, piezoelectric materials are dielectric materials. This makes the poled piezoceramics transversely isotropic materials. According to the IEEE Standard on Piezoelectricity
19 [43], the plane of isotropy for these materials is defined as a 12-plane. Besides displaying symmetry about the 3-axis, the poling axis of the material, piezoelectric materials exhibit a linear behaviour within a certain range. Thus, the constitutive relations show a total strain as a sum of the mechanical strain induced by the stress apllied and the electrical strain caused by the electric voltages [2]. [43] gives these relations in four sets with each set having two of the four variables as independent variables. The four variables are given by (S) and (T), the strain and stress components respectively, and by (E) and (D), the electric field and electric displacement components respectively. The four sets of equations are:
• T and E are independent
S = [sE]T + [dt]E, (2.1) D = [d]T + [T ]E,
• T and D are independent
S = [sD]T + [gt]D, (2.2) E = −[g]T + [eT ]D,
• S and D are independent
T = [bD]S + [it]D, (2.3) E = −[i]S + [es]D,
• S and E are independent
T = [cE]S + [et]E, (2.4) D = [e]S + [S]E, with d, g and i being alternative forms of the piezoelectric constants and s and b, e and being the elastic compliance and permittivity constant respectively. Note that in [43] h and c are respectively used instead of i and b. However, h and c are already used as other variables in this work, so a change had to be made. These can be transformed under different mechanical and electrical conditions, which is found in [43]. Furthermore, the superscripts D and T indicate that the constants are calculated at constant electric displacement and stress, respectively. While superscript E implies strain applied at zero or constant electric field, superscript S implies zero or constant strain. As usual, t stands for the transpose. From the presented sets of equations, 2.1 is the most widely used because of the dependence of the stress T with the geometry and the dependence of the electric field with the placement of electrodes. The resulting relation becomes easier to use once this dependence is eliminated, which can be done. The set of equations, also knows as the (S-D) form of the coupled equations, is written in matrix form as:
20 S sE dt T = , (2.5) D d T E
and in expanded form:
E E E S1 s11 s12 s13 0 0 0 0 0 d31 T1 E E E S2 s12 s11 s13 0 0 0 0 0 d31 T2 E E E S3 s13 s11 s33 0 0 0 0 0 d33 T3 E S4 0 0 0 s55 0 0 0 d15 0 T4 = E . (2.6) S5 0 0 0 0 s55 0 d15 0 0 T5 E S6 0 0 0 0 0 s66 0 0 0 T6 T D1 0 0 0 0 d15 0 11 0 0 E1 T D2 0 0 0 d15 0 0 0 11 0 E2 T D3 d31 d31 d33 0 0 0 0 0 33 E3
The transversely isotropic material behaviour simplifications are directly applied in the expanded form E E of the equations (e.g s11 = s22 etc.). This represents the general form of linear constitutive equations in use in engineering applications for piezoelectric materials [2]. Regarding the MEMS manufacturing process, cantilever piezoelectric energy harvesters are the most recommended configuration due to their simplicity and low structural stiffness. By assuming that the cantilever beam is thin, one can utilise Euler-Bernoulli beam theory, in which only one-dimensional bending stress is not zero, to reduce the equations to a more simplified form. Therefore, equation 2.5, when considering as well that an electrode pair covers the faces perpendicular to the 3-direction consistent with mode [3-1], becomes [2]:
E S1 s11 d31 T1 = . (2.7) T D3 d31 33 E3
In order for the reaction on the stress and strain fields remain in agreement with all the terms, the poling direction of the material must be taken into account. As such some of the terms will need to be negative. Additionally, one should consider that for positive poling positive piezoelectric constants and vice-versa. Complementing these equations with a mechanical model will present a coupled electro- mechanical system, where the spatial displacements are related to the voltage and power [2].
21 22 Chapter 3
Prediction Models
3.1 Frequency Model
While the previous work performed at Tecnico´ focused on scavenging the energy from the mechan- ical vibrations inside the wing, the nature of the problem changed when trying to predict the behaviour of the piezoelectric device to harvest the flow’s energy outside of the wing. In the first instance, the frequency at which the device operates is largely independent of the wing incidence at a certain Re, equal to 140,000 in this case. As such the device could be fine-tuned to match the fundamental frequen- cies produced by the wing’s mechanical vibrations. This involved changing the device itself with a steel tip plate and a tip mass so as to reduce the natural frequency of the device from 110 Hz down to the fundamental frequency measured by the modal analysis of 17,6 Hz [2]. For the underlying problem, predicting how the device will vibrate with the oncoming flow is key to choosing the device with a design that matches the vortex shedding frequency. Fine-tuning a device to match this frequency is not efficient since the added mass involved in this process does not bring any benefits for the whole system. As such, a model to predict the frequencies involved, was formulated. Before proceeding to the model, there are two installation procedures that need to be considered for the experiment that show the most promising energy harvesting results. The first configuration is the piezoelectric flag mounted at the trailing edge and in the wake of the airfoil (shown in Figure 3.1(a)) and the second is the inverted piezoelectric flag above and downstream of the upper surface of the airfoil (seen in Figure 3.1(b)):
(a) Piezoelectric flag mounted at the trailing edge of the airfoil (b) Inverted piezoelectric flag mounted above and downstream of the upper surface of the airfoil
Figure 3.1: Two installations of the piezoelectric device on the wing [2]
23 It is relevant at this time to show that the two possible configurations have distinct fundamental frequencies, as can be observed in Figure 3.2. As such, the characteristics of the piezoelectric flag for the two configurations will be different as well so as to match the desired frequencies.
(a) Frequency analysis of the piezoelectric flag mounted at the (b) Frequency analysis of the inverted piezoelectric flag trailing edge of the airfoil mounted above and downstream of the upper surface of the airfoil
Figure 3.2: Frequency analysis of the two installations of the piezoelectric device on the wing [2]
The second installation shows more promise in terms of harvesting the energy from the shed vortices of the leading edge of the wing while also increasing the pre-stall and post-stall lift-to-drag ratio of the airfoil. In [2], a numerical study with the aid of CFD was performed to assess the effects of this configuration on the lift and drag coefficients of the airfoil. The conclusion was that there is a decrease in lift accompanied by a larger decrease in drag, which resulted in an overall increase of the lift-to-drag ratio for the airfoil. The decrease in drag is related to the resizing of the shed vortices by the piezoelectric flag by turning large vortices into smaller ones, thus reducing the overall drag.
The interaction of the piezoelectric plate with the vortices is relevant to understand the drag reduction effect. As a vortex is shed from the leading edge of the airfoil and along the chord the airfoil, the low pressure core of the vortex passing below the piezoelectric plate causes it to bend towards this low pressure zone. The circulation of the vortex causes another vortex to be shed from the leading edge which this time will be shed along the airfoil and above the piezoelectric plate. This, in turn causes the beam to bend upward towards the low pressure zone. This means that the plate’s surface pressures are dependent on the pressure of the vortices passing by [2]. Hence, the movement of the plate and its interaction with the oncoming flow is explained. This movement is presented in Figure 3.3.
While elaborating the frequency model, the focus was on the inverted flag configuration. It can be observed that the vortex shedding frequency lies at 18.8 Hz. The process of harvesting energy from the vortices is not linear, though, see Figure 3.4, with the highest amount of energy being harvested at the natural frequency ωn of the device. The goal is not to find the device which can harvest the most energy at natural frequencies because the MAV will not be operated at these conditions solely. There is a whole spectrum of flight conditions that need to be analysed. By choosing a device with a slightly lower natural
24 (a) Downward bending of the piezoelectric plate as the vortex (b) Upward bending of the piezoelectric plate as another vortex passes below passes above
Figure 3.3: Flow fields showcasing the passing of the vortices along the airfoil and the movement of the piezoelectric plate relative to its stationary state [2] frequency, energy can be harvested more efficiently at lower air speeds.
Figure 3.4: Harvested power as a function of the frequency from a piezoelectric flag
As such, the goal of this model was to obtain the material characteristics of the plate that would vibrate at frequencies in the range of 10 to 15 Hz. The device used for the tests of the mechanical vibrations inside the wing has a natural frequency of 110 Hz and had been fine-tuned to match the fun- damental frequencies of the mechanical vibrations. This device was not a viable option for the current experiment, because adding mass to the MAV is not optimal since the performance gained from harvest- ing vibration energy is lost by having a heavier MAV. An extensive Web search for other piezoelectric plates showed no such plates with a natural frequency below 49 Hz. Thus, another option had to be
25 considered. Recently, other studies were performed using a PVDF sheet glued to a substrate layer. This model will also evaluate the viability of such a configuration.
In order to obtain the vibration frequencies of the piezoelectric plate, one has to analyse what the vibration is depending on. The plate is subject to a destabilising motion by the flow instabilities and a stabilising motion by the plate’s structure [44]. The non-dimensional bending stiffness β represents the magnitude of the bending force relative to the fluid inertial force [44][45] and is given by:
B β = 2 3 , (3.1) ρf U∞L where ρf the density of the fluid, U∞ the freestream velocity, L is the length of the beam and B is the dimensional flexural rigidity and for a uniform plate, given by:
Et3 B = , (3.2) 12(1 − ν2) where E is Young’s Modulus, t the thickness and ν the Poisson ratio of the plate.
In the event of a multilayer plate with different materials this equation can be changed to account for the different material properties. For a 3-layer sandwich plate with the top and bottom corresponding to the piezoelectric materials and the material in the middle being the substrate, also called a bimorph plate, B is given by [46]:
3 2 3 ! EshsH 2EphpH hs hshp hp B = 2 + 2 + + , (3.3) 12(1 − νs ) 1 − νp 4 2 3 where the subsript s and p are related to the substrate layer and piezoelectric layers, respectively, H is the total plate width and h is the thickness of the layers. While a bimorph plate is option to be considered, the unimorph plate certainly is viable as well. Efforts to find a similar relation as above were unsuccessful in the literature, though. Hence, an approximation was to compute the values of B for the unimorph case by dividing the term representing the piezoelectric layers by 2 to represent an unimorph layer. After having the parameter B computed, the frequency can then be obtained from the following relation [46]:
s ω 3.515 B f = n = , (3.4) 2π 2πL2 µ where µ is the mass per unit length.
With these equations, a sensitivity study was performed in order to evaluate how the frequency changed in relation to the use of different subtrate materials as well as different sizes of plate length, PVDF thickness and substrate thickness. Figures 3.5 to 3.9 show the sensitivity study for a bimorph plate with a substrate layer composed of different materials ranging from metals to polymers with subfigures (a) constisting of a thinner PVDF sheet than subfigures (b). The datasheet for most materials can be found in AnnexB[47]. For the aluminium, the properties were found in [48], since it is not of common use in such applications.
The study shows that, independently of the substrate material, the natural frequency of the plate
26 (a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.5: Sensitivity study of a bimorph plate with a Brass substrate
(a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.6: Sensitivity study of a bimorph plate with an Aluminium substrate
(a) Case 1: Two PVDF sheets with 28 µm thickness (b) Case 2: Two PVDF sheets with 52 µm thickness
Figure 3.7: Sensitivity study of a bimorph plate with a Steel substrate is largely impacted by the substrate thickness. Fabricating these substrate materials in thicknesses smaller than 0.1 mm is not feasible since these are hard to find on offer, so this was the minimum
27 (a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.8: Sensitivity study of a bimorph plate with a FR4 substrate
(a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.9: Sensitivity study of a bimorph plate with a Polyester substrate thickness considered. This thickness has proven to be the ideal case, since the frequency values of all tests neared the frequency range aimed for. While the length of the plate has some importance with the difference in natural frequency between a short and a long plate being more accentuated for thicker substrates, but not insignificant for the minimum substrate thickness, the thickness of the PVDF is not shown to be of great importance with the values being similar except for the polyester. More on this subject after the results of table 3.1. The unimorph plates’ sensitivity studies are presented in Figures 3.10 to 3.14. The study of the unimorph plates shows that the frequency has a similar behaviour to the bimorph plate study. What is interesting to see here, is how the two cases fare out against each other for the minimum substrate thickness and for the different PVDF thicknesses. The data used to do this comparison is presented on table 3.1. From the results, it can be concluded that both steel and aluminium are not viable substrates since both have larger natural frequencies than the vortex shedding frequency. FR4 and brass possess similar natural frequencies, brass being a better candidate because of widespread use and hence ease of
28 (a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.10: Sensitivity study of an unimorph plate with a Brass substrate
(a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.11: Sensitivity study of an unimorph plate with an Aluminium substrate
(a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.12: Sensitivity study of an unimorph plate with a Steel substrate availability. Polyester would be a viable option as well bar the fact that it is not common and as such not easy to find and order online. For polyester it is also noticeable that it boasts the largest difference in
29 (a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.13: Sensitivity study of an unimorph plate with a FR4 substrate
(a) Case 1: PVDF sheets with 28 µm thickness (b) Case 2: PVDF sheets with 52 µm thickness
Figure 3.14: Sensitivity study of an unimorph plate with a Polyester substrate
Unimorph Bimorph E[Pa] ν ρ [kg m−3] 28 µm 52 µm 28 µm 52 µm Brass 16.4247 16.5969 16.4148 16.7413 1.00 × 1011 0.32 8300 Steel 22.8459 22.8370 22.6779 22.6688 1.93 × 1011 0.29 8000 Aluminium 23.3799 23.3212 22.8990 22.8512 6.90 × 1010 0.32 2700 FR4 16.7530 17.3440 16.6839 17.6764 2.60 × 1010 0.17 1900 Polyester 9.8896 12.0282 10.9047 13.7871 3.65 × 109 0.48 1380
Table 3.1: Frequency results for the minimum substrate thickness natural frequency between the thinner and thicker PVDF sheets. If the option of polyester as a substrate had been pursued, the choice of PVDF thickness would impact significantly the obtained results. Brass was the chosen substrate material for the experiments. The inverted flag configuration relative to flag at the trailing edge configuration is more complex to build as an additional structure is needed to support the inverted flag and a specific substrate layer is required to achieve the natural frequencies. As such and due to the fact that Gunasekaran and Ross
30 [44] had already made wind tunnel testing in the inverted flag configuration albeit with another airfoil, the decision was made to move away from this configuration to the piezoelectric flag mounted at the trailing edge configuration. As previously stated and seen in Figure 3.2, the natural frequency of this installation is considerably smaller at around 8 Hz. The sensitivity study did not present any substrates able to achieve such small frequencies and as such the option to do the experiment without substrate was considered. The MATLAB results showed this to be a viable option with the natural frequency of the PVDF sheet being 7,4 Hz. This configuration also presents an easier assembly since no additional support for the flag is needed as opposed to the inverted flag configuration. The plate can just be glued to the trailing edge and connected to the electrical circuit.
Non-dimensional bending Stiffness
The parameter β did not appear in any form in the frequency sensitivity study because it is a non- dimensional parameter that represents in which regime the plate is oscillating. Following are the four possible regimes based on β:
Figure 3.15: Oscillation regimes as a function of the parameter β [44][49]
In the MATLAB code, β was calculated to evaluate whether the flag would be in the intended os- cillating regimes for the range of Re capable with the given wind tunnel facility. This is an important parameter for the inverted flag configuration because the free end of the plate is subject to the oncoming flow directly. Despite the fact the model shows good results in this matter, the results are not presented since the inverted flag configuration was not pursued.
3.2 Aero-Mechanical-Electrical Model
The frequency analysis was of major importance to determine which piezoelectric devices and sub- strate (for future work with the inverted flag configuration), that are optimal for the experiments at hand, should be acquired. However, predicting the energy harvesting capabilities of these devices is of paramount importance so as to have a comparison term between the theoretical and experimental results obtained. The prediction model has a large complexity since there is a need to incorporate three different kinds of differential equations, each representing a physical concept (aerodynamical, mechanical and
31 electrical).
3.2.1 Electro-Mechanical Model of the Piezoelectric Beam
Since these models can quickly become overwhelmed in complexity with little gains in accuracy, several assumptions were made to facilitate the equations without harming the overall accuracy of the model:
• The harvester is considered to be an uniform Euler-Bernoulli beam of a PVDF sheet (for the in- verted flag a composite structure of substrate plus PVDF sheet would be considered).
• The electrical circuit has a resistive load only with the harvester connected to it through the elec- trodes.
• The electrodes are perfectly conductive and cover the entire bottom and top surfaces of the PVDF sheet so that the electric field may be assumed uniform across the entire beam.
• The beam is continuously oscillated by the fluid flow, hence enabling continuous harvesting of the electrical energy.
• The leakage resistance of the PVDF is considered to be negligible, as the PVDF is connected parallel to the resistive load.
• The piezoelectric constitutive equations generate an electrical capacitance term and will be ac- counted for in the equations, albeit not being shown in the circuit as parallel to the resistive load - it is considered as an internal parameter.
The passing vortices exert the force to excite the harvester beam with its vertical displacement y(x,t) being a function of the horizontal position x and time t. The equation of motion of the beam is then given by [2][50]:
∂2M (x, t) ∂5y(x, t) ∂y(x, t) ∂2y(x, t) bm + c I + c + m = F (t) , (3.5) ∂x2 s ∂x4∂t a ∂t ∂t2 where Mbm(x,t) represents the internal bending moment, csI the equivalent damping term due to struc- tural viscoelasticity with cs being the equivalent coefficient of strain rate damping and I the second moment of area of the cross-section, ca the viscous air damping coefficient, m the mass per unit length and F the fluid force exerted on the beam by the flow. Both damping coefficients satisfy the proportional damping criterion, thus being mathematically ap- propriate for the modal analysis solution [2][50][51]. For the case of a composite flag with two or more layers, the internal bending moment can be computed via integration of the first moment of the stress distribution over the cross sectional area of each layer. For the substrate layer the stress-strain relationship is trivial and given by Hooke’s Law.
32 For the piezoelectric layer, the stress-strain-electric field relationships are given by the piezoelectric equations:
T = cE S − et E, (3.6a)
D = [e]S − [S]E, (3.6b) where E represents the electric field, D the electric displacement, c the elastic compliance, e the piezo- electric coupling coefficient and the permittivity constant. The superscript E relates to the zero or constant electric field strain and superscript S the zero or constant strain, while t represents the trans- pose. The internal bending moment is then obtained with Hooke’s Law from equation 3.6a and utilizing the terms of radius of curvature one can express the bending strain relation [2][50][52]. By writing the uniform electric field as
v(t) E3(t) = − , (3.7) hp with v(t) representing the voltage and hp the thickness of the piezoelectric layer, the internal bending moment follows:
∂2y(x, t) M (x, t) = YI + ν v(t) , (3.8) bm ∂x2 em where YI corresponds to the bending stiffness of the composite cross section given by:
E (h3 − h3) + E (h3 − h3) YI = b s b a p c b . (3.9) 3
Here, E stands for Young’s Modulus and b for the beam width, while the subscripts p and s stand for the piezoelectric and substrate layers, respectively. The distance of the bottom of the substrate layer to the neutral axis is given by ha, while hb denotes the distance of the bottom of the piezoelectric layer to the neutral axis. hc is the distance from the top of the piezoelectric layer to the neutral axis.
In equation 3.8, νem stands for the electro-mechanical coupling term, which can be written as:
Epd31b 2 2 νem = − (hc − hb ) , (3.10) 2hp with d being the piezoelectric coupling constant and the subscripts 1 and 3 indicating the axial strain and polarization directions, respectively. By substituting equation 3.8 in equation 3.5, one gets the mechanical equation of motion with the electrical coupling term:
∂4y(x, t) ∂5y(x, t) ∂y(x, t) ∂2y(x, t) dδ(x) dδ(x − L) YI + c I + c + m + ν v(t) − = F (t) , (3.11) ∂x4 s ∂x4∂t a ∂t ∂t2 em dx dx where δ(x) corresponds to the Dirac delta function.
33 By looking now at equation 3.6b, the relationship between the electrical displacement and field and the strain is established leading one to the second equation needed to model the problem, the electrical equation with the mechanical coupling term.
At this moment it is necessary to assume that the cantilever harvester is thin (most common config- uration of piezoelectric harvesters) in order to simplify the equation. In Euler-Bernoulli beam theory, the only stress components of a thin beam that is not negligible is the axial bending stress. Additionally, by assuming that the electrode pair is working in mode 3-1 and as such covering the faces perpendicular to the 3-direction, equation 3.6b becomes:
S S v(t) D3(x, t) = e31S1(x, t) + 33E3(t) = d31EpS1(x, t) − 33 . (3.12) hp
There is a relationship between the average bending strain S and the beam displacement y at any given time t and horizontal position x, given by:
∂2y(x, t) S (x, t) = −h , (3.13) 1 pc ∂x2 which substituted into equation 3.12, gives:
2 ∂ y(x, t) S v(t) D3(x, t) = −d31Ephpc 2 − 33 . (3.14) ∂x hp
From [43][50], the electric charge q(t) is given by:
Z q(t) = D · ndA , (3.15) A where D is the vector of electric displacement and n is the unit normal. Inserting 3.14 into 3.15, one obtains:
Z L 2 ∂ y(x, t) S v(t) q(t) = − d31Ephpcb 2 + 33b dx . (3.16) x=0 ∂x hp
The electric current is given by the time derivative of the electric charge as follows:
Z L 3 S dq(t) ∂ y(x, t) 33bL dv(t) i(t) = = − d31Ephpcb 2 dx + . (3.17) dt x=0 ∂x ∂t hp dt
This equation shows that the generated current has two components: the generation by oscillatory motion and the generation by the voltage differece in the piezoelectric layer. This second term relates to the capacitance of the piezoelectric layer that was previously explained as being internal to the PVDF. As such, no external capacitance is needed in the circuit. The voltage is then given by:
"Z L 3 S # ∂ y(x, t) 33bL dv(t) v(t) = Ri(t) = −R d31Ephpcb 2 dx + . (3.18) x=0 ∂x ∂t hp dt
34 Rearranging, the second differential equation of the system is:
S Z L 3 33bL dv(t) v(t) ∂ y(x, t) + = − d31Ephpcb 2 dx . (3.19) hp dt R x=0 ∂x ∂t
These equations can be solved with the modal solution approach, by substituting the transverse beam displacement by an absolutely and uniformly convergent series of eigenfunctions like so:
∞ X y(x, t) = φr(x)ηr(t) , (3.20) r=1 where φr(x) stands for the mass normalised eigenfunction and ηr(t) is the modal coordinate of the beam for the rth mode. As the system is proportionally damped, φr(x) corresponds to the eigenfunctions of the undamped free vibration problem and is thus given by [2][50][51]:
r 1 λrx λrx sinhλr − sinλr λrx λrx φr(x) = cosh − cos − sinh − sin , (3.21) mL L L coshλr + cosλr L L where λr is the root of the characteristic equation
1 + cosλ × coshλ = 0 . (3.22)
Through some mathematical manipulation, equation 3.20 inserted in the partial differential equation of motion 3.11 results in:
∂2η (t) ∂η(t) r + 2ξ ω + ω2η (t) + χ v(t) = f(t) , (3.23) ∂t2 r r ∂t r r r where ωr stands for the undamped natural frequency of the rth mode given by:
r YI ω = λ2 , (3.24) r r mL4
ξr is the mechanical damping ratio of the rth mode, which includes both the structural damping and the viscous air damping:
csIωr ca ξr = + , (3.25) 2YI 2mωr and χr is the modal coupling term given by:
dφ (x) χ = r . (3.26) r dx x=L
3.23 is thus the first differential equation of the problem. The second is obtained from substituting 3.20 in the second differential equation of the system 3.19, which is shown to be:
∞ dv(t) hp X dηr(t) + = ϕ , (3.27) dt RS bL r dt 33 r=1
35 where d Y h Z L d2φ (x) d Y h dφ (x) ϕ = − 31 p pc r dx = − 31 p pc r . (3.28) r S 2 S L x=0 dx L dx 33 33 x=L Having the two equations needed to model the problem, the only term still to be determined is the aerodynamic forcing term, which will be modelled in subsection 3.2.2.
3.2.2 Aerodynamic forcing term
When considering the forcing term, the two configurations have distinct behaviours that need to be accounted for. Even though, the tests will be done in the piezoelectric flag mounted at the trailing edge configuration, an aerodynamic forcing term will be explored for the inverted flag configuration as well since it can be useful for later works in this field of study. The inverted flag configuration will be explained first (in case future work on this configuration is pursued) followed by the normal flag configuration ensued by the solution of the system and the results of the aero-electro-mechanical model. Modelling the forces on an inverted flag is difficult and usually relies on numerical simulations as can be observed in the literature. As such, a more simplistic approach was taken, that relies on modelling the forces via unsteady aerodynamics. Theodorsen was one of the first to study the flapping wing dynamics of birds and draw a parallel to be used for the unsteady aerodynamics of helicopter analysis [53]. Theodorsen assumes inviscid, incompressible flow subjected to small disturbances to arrive at a solution for the unsteady forces on a 2-d harmonically oscillated airfoil. The basic Theodorsen model represents the airfoil and shed wake by a vortex, as can be seen in Figure 3.16, with the shed wake extending as a planar surface from the trailing edge to infinity. The planar surface can only be assumed with regard to the small angle of attack disturbances.
Figure 3.16: Theodorsen model of a harmonically oscillated thin airfoil [53]
The equation that models the system is:
1 Z c γ (x, t) 1 Z ∞ γ (x, t) w(x, t) = b dx + c dx , (3.29) 2π 0 (x − x0) 2π c (x − x0) where γb is the bound vorticity, γw represents the wake vorticity and c is the chord of the airfoil. A
36 pressure difference can be sustained by γb which results in a lift force, while γc has a net pressure jump of zero over the vortex sheet. Through some assumptions and manipulations, Theodorsen arrives at the solution for the simple harmonic oscillation, which is given by:
" # b b b2 h˙ bα˙ 1 L = πρV 2b h¨ + α˙ − aα¨ + 2πρV 2b + α + − α C(k) , (3.30) V 2 V V 2 V V 2 and in coefficient form:
" # b b b2 h˙ bα˙ 1 C = πb h¨ + α˙ − aα¨ + 2πC(k) + α + − α , (3.31) l V 2 V V 2 V V 2 where α and h represent the pitching and plunging motions, respectively, a is the pitch axis location relative to the mid-chord of the airfoil and V is the steady velocity. The equation is composed of two distinguishable terms, the first representing the non-circulatory contribution (apparent mass contribution) and the second representing the circulatory contribution (quasi-steady contribution) with C(k) being the circulatory term also known as Theodorsen’s function, which includes the effects of the shed wake on the unsteady airloads and is given by the complex transfer function:
C(k) = F (k) + iG(k) . (3.32)
This equation is a function of the reduced frequency k, a non-dimensional variable that describes the unsteadiness of the problem and is given by:
wc k = . (3.33) 2V
A further study by Von Karm´ an´ [54] added a contribution to the lift that Theodorsen did not include, the wake-induced lift . Hence, the total lift has three contributions and is as follows:
C0 = C0 + C0 + C0 . L Lqs Lam Lw (3.34)
The wake-induced lift contribution can be written in terms of the quasi-steady contribution like so:
C0 = C0 [C(k) − 1] , Lw Lqs (3.35) yielding a lift coefficient of the form:
C0 = C(k)C0 + C0 . L Lqs Lam (3.36)
Although this equation enables a quite simply approach to the wake-induced lift contribution, a more accurate equation was derived by Tchieu and Leonard [55] and is as follows:
N 0 X Γj CL = − . (3.37) w q 2 1 j=1 xj − 4
37 For the normal flag configuration, the derived equation of force is different. As the vortices that hit the piezoelectric flag at the trailing edge are very difficult to model with an empiric equation, in the literature studies are done entirely numerically, and the vortical structures are different at different positions along the wingspan, another method had not be researched in order to accomplish the objective of predicting the output power for a given velocity and impedance. So, the model was calibrated for a reference forcing term obtained from the experiments with an impedance of 1.55 MΩ at Re = 140000.
3.2.3 Solution of the system
There are two ways to solve the system of equations obtained. The first way is to consider that the modal response, which is the solution of the ordinary differential equation 3.23, can be simplified with the Duhamel Integral as shown:
1 Z t −ξr ωr (t−τ) ηr(t) = [f(t) − χrv(τ)]e sin(wrd(t − τ))dτ , (3.38) wrd τ=0 with ωrd representing the damped natural frequency given by
p 2 ωrd = ωr 1 − ξr . (3.39)
Then, the second equation of the system 3.27 can be solved using an integrating factor:
Ψ(t) = et/τc , (3.40)
with τc equal to S R33bL τc = . (3.41) hp By multiplying both side of equation 3.27 with the integrating factor and subsequently manipulating the equation, the following is obtained:
Z ∞ ! X dηr(t) v(t) = e−t/τc et/τc ϕ dt + c , (3.42) r dt r=1 where c stands for an arbitrary constant that is dependant on the initial voltage in the circuit and the velocity of the beam. Since equation 3.38 assumes the form of zero initial displacement and velocity, c is only dependant on the voltage. As such, for simplicity, initial voltage is considered to be zero and hence c is also equal to zero. The second does not include any use of mathematical simplifications to solve the system. This makes use of the MATLAB code Ode45 to solve the partial differential equations numerically and give the solution when it has converged. Although this method requires more processing power due to the extensive calculations, it was chosen due to simpler implementation because of previous experience with this kind of code. As stated before, the forcing term was calibrated for a reference output. The results of the model are
38 shown in Figures 3.17 to 3.18. The first shows the voltage and power outputs of the model as a function of the impedance. It is clear that the voltage rises quickly for lower resistances, while rising more slowly and asymptotically for very high impedances. This behaviour is expected to continue until an infinite resistance where the open circuit voltage would be achieved. The power clearly follows the same trend up until 20 MΩ, where it starts to decrease due to small increase in voltage output being counteracted by high impedance.
(a) Voltage as a function of the impedance (b) Power as a function of the impedance
Figure 3.17: Results of the aeroelectromechanical model
Figure 3.18 shows an almost linear increase in voltage with increasing air speed. The increase in voltage is higher for lower Re becoming more linear after Re = 100000.
(a) Voltage as a function of the Reynolds Number (b) Power as a function of the Reynolds Number
Figure 3.18: Results of the aeroelectromechanical model
39 40 Chapter 4
Experimental Test Setup
The Mechanical Engineering Department (DEM) from Instituto Superior Tecnico´ has a low speed open circuit blow wind tunnel, capable of performing aerodynamic tests up to velocities of 10 m s−1. Its test section dimensions are 1.35 m × 0.80 m, as depicted in Figure 4.1.
Figure 4.1: Wind tunnel diagram with engine and data acquisition system [56].
The instrumentation of this wind tunnel consists of an external aerodynamic forces and moments balance, which is installed in the test section. This balance was developed by the German company Schenck and installed in the early 1990’s and is capable of measuring the forces and moments in the three reference axes and the angles of attack and side-slip, which determine the attitude of the model being tested. The data acquisition system is done in a personal computer, by using the AeroIST software developed by Roque [56] and it runs on a Linux operating system. The wind tunnel operator is able to change the attitude and sidelip angles of the model manually or by using the AeroIST software. The graphic interface of the software AeroIST is as depicted in Figure 4.2, where all data can be read and exported to other files for further analysis.
41 Figure 4.2: Graphic interface of AeroIST software [56].
4.1 Aerodynamic Forces and Moments
The balance coordinate system is defined with the XY plane in a horizontal position and with the Z axis aligned with the vertical support, as depicted in Figure 4.3. The angle of attack α is defined positive for a nose-up attitude (positive as according to the right-hand rule about the Y axis) and the angle of sideslip is defined positive for a nose-left attitude (negative as according to the right-hand rule about the Z axis). The positive X axis direction opposes the flow direction.
The software used requires calibration before a measurement is started, with a reference being measured before the experiment is started.
Figure 4.3: Forces and Moments Balance
42 4.2 Wind Tunnel Instrumentation
The tunnel is equipped with a DC Thrige-Titan Lak 160 LA motor with nominal power of 21.5 kW and nominal angular velocity of 2280 rpm controlled by a Siemens’ SIMOREG DC-Master 6RA70. The reference motor velocity can also be controlled with the AeroIST software.
A schematic representation of the various components of the wind tunnel instrumentation is pre- sented in Figure 4.4.
Figure 4.4: Diagram with the various balance components [56]
4.3 Multimeter
The Yokogawa Model 7551 Multimeter was used to measure the voltages obtained in the experiment. In order to obtain the voltage at certain moments in time, the Trigger mode was utilised. Figure 4.5 shows the multimeter.
Figure 4.5: Yokogawa 7551
43 44 Chapter 5
Results
5.1 Experiments
For this work, three types of piezoelectric devices (unimorph and without substrate) were tested, with all being PVDF devices with a protective coating differing only in thickness of the piezoelectric layer and length. Device A has a piezoelectric layer thickness of 52 µm and a length of 30 mm. Device B has the same thickness but double the length and device C is 28 µm thick and 30 mm long. These can be viewed in Figure 5.1. The catalogue with relevant information regarding these devices can be seen in AnnexA.
(a) Device A (b) Device B (c) Device C
Figure 5.1: Mouser Electronic Devices
5.1.1 Impedance dependency for device A
In order to find the optimal impedance of the electric circuit to be used in the baseline experiments, the piezoelectric device A was positioned at the trailing edge and subjected to an oncoming flow of Re = 140,000 at α = 15◦ at the mid-point of the wing’s span (y/c = 3/4). The impedance was parametrically varied from 25 kΩ to 2300 kΩ with the power being measured for each impedance. Since only a mul- timeter was available to measure the voltage at each instant, 30 measurements were made in order to then calculate the root-mean-square (rms) voltage and subsequently the peak-to-peak (p-p) voltage, as follows:
r 1 V = (v2 + v2 + ... + v2 + v2 ) , (5.1) rms n 1 2 n−1 n
45 √ Vp−p = 2 2Vrms . (5.2)
The power was then computed from: V 2 P = p−p , (5.3) R with R being the electric resistance. Figure 5.2 shows the evolution of the harvested power as a function of the circuit impedance. It can be observed that the power rises until reaching a peak at 1550 kΩ (red dot on Figure 5.2). One can see that this value is overshooting the expected trend due to using the trigger function to obtain 30 voltage values. In fact a continuous rise of the power is to be expected until 10 MΩ. At this impedance the highest power value was measured. However, an impedance this high is impracticable in a real-world application. As such, the impedance where the highest power output was measured in the range of 25 to 2300 kΩ was taken instead. The maximum obtained power value for 1550 kΩ is equal to 2.474 nW. There are noticeable peaks and valleys in the graph and in all subsequent graphs shown in this chapter. These peaks and valleys are due to the way the measurements were made, where measuring at an instant where the voltage was very high in relation to the rest of the measurement and the other measurements, induced these peaks and valleys. The important aspect of the measurement is to obtain the trend of the power, though, which is accomplished.
Figure 5.2: Power as a function of the circuit’s impedance
5.1.2 Velocity dependency for device A
With the impedance at 1550 kΩ, device A was tested at different air speeds to evaluate the effect of the flow velocity on the power output. From Figure 5.3, one can infer that the higher the flow velocity the higher the power output as well.
46 (a) Open circuit (b) Closed circuit
Figure 5.3: Harvested power vs. Flow velocity
The two graphs show different behaviours at the measured speeds, though. The open circuit graph has a steady increase from Re = 40000 to Re = 70000, where it remains steady until reaching Re = 120000, after which it rapidly increases again until reaching its maximum value at Re = 140000. The maximum measured voltage was 0.315 V. In the closed circuit graph one can observe a slow increase of the power output from Re = 40000 to Re = 70000 followed by a large increase until Re = 100000. Again, the maximum is reached at Re = 140000, with a power output of 1.109 nW, after a steady increase observed from Re = 100000. Albeit the differences in the graphs, the overall trends of the overall power outputs increasing with the flow velocity remain the same. This is due to the more energised flow being captured by the oscillations of the piezoelectric device.
Afterwards, device A was tested at angles of attack ranging from 0◦ to 20◦ , at 5 different positions of the wing, which can be seen on Figure 5.4 (the two positions not shown in this figure are symmetrical to positions 1 and 2), and for Re = 70,000 and Re = 140,000. The device was also tested with an open circuit to measure the theoretical maximum voltage that can be obtained in the set configuration. The results are shown as the power as a function of the angle of attack with each line representing a position of the device on the wing (y/c), where y corresponds to the span of the wing and c is the chord of the wing.
In Figure 5.5, one can see that, except for the outlier in (a), the voltage trends follow a similar pattern with the maximum voltage measured at high AoA (15◦ , 20◦ ) which correspond to post-stall conditions. As the tests were running, it could be observed that at high AoA, the device operated in the large- deflection deformed flapping oscillation mode, while at low AoA the observed oscillation mode was the large-amplitude flapping for Re = 140,000 and y/c = 3/4 and small deflection deformed flapping at Re = 70,000 and y/c = [1/4,1/2]. At high AoA, the device enters the large-deflection deformed flapping mode due to entering post-stall conditions, where the flow is starting to separate and reaching the piezoelectric device with another vortical structure. The large-deflection deformed flapping corresponds to the oscillation mode that releases the most energy and as such the mode one should strive for maximum energy harvesting. In the last test, shown later on, coefficients of lift and drag are shown as
47 Figure 5.4: Tested positions (y/c) on the wing
(a) Re = 70000 (b) Re = 140000
Figure 5.5: Voltage vs AoA for device A in open circuit
(a) Re = 70000 (b) Re = 140000
Figure 5.6: Power vs AoA for device A in closed circuit well to conclude if the drag penalisation is large enough to offset the energy gains from this oscillation mode.
48 The closed circuit tests, presented in Figure 5.6, are coherent with the open circuit results, showing higher power outputs for high AoA. It is interesting to see that for the outer positions, y/c = [1/4,5/4], the highest power outputs come at an AoA of 20◦ , while for the inner positions they are measured at 15◦ . Additionally, it is noticeable that at low AoA at the outer positions the harvested power is 1 or 2 orders of order of magnitude smaller than for the inner positions. This was apparent, while running the test, in the way the device oscillated at each AoA. At low AoA almost no oscillation was seen, while at high AoA the device oscillated vehemently. This trend makes the device at the outer positions worthy for another type of application: wake sensing. By having a device that does not have significant power outputs at lower AoA and increases significantly at high AoA, one can utilise it as a sensor to inform the on-board computer that it is reaching stall conditions. Such a device at the inner positions would not be so useful since the differences in power are not large enough to detect such a behaviour. It is also noteworthy to mention the fact that the power outputs are highly symmetrical between the positions at the same distance from the mid-point of the wing’s span. For instance, the trend for the power output at position y/c = 1/4 is largely similar to the power outputs at y/c = 5/4. The same follows for positions y/c = 1/2 and y/c = 1. As such, from here on, experiments were conducted for the first three span positions only. Lastly, as expected and seen previously in the velocity experiment, one can conclude that larger power outputs come with higher flow speeds.
5.1.3 Comparison of device A with devices B and C
The first observation made when testing device B was the fact it started already from a bent position with the wind tunnel at standstill. Also, when running the wind tunnel, this device oscillated much more than devices A and C due to its length being double that of the latter. As can be seen in Figure 5.7, this leads to a higher energy output at high AoA when compared to device A, despite the fact of the open circuit voltage being lower than for device A. When comparing the values of both devices at low AoA, device A has generally higher power outputs at the inner positions, while having lower output in the outer position. The power trends remain largely similar with almost steady values from 0◦ to 10◦ and then reaching maximum values at 15◦ or 20◦ for the inner positions and staying 1 order of magnitude lower at low AoA and then increasing rapidly for high AoA for the outer position. Device C, being less thick than the other two devices, has some fundamental differences in behaviour as per Figure 5.8. One can deduce that unlike the previous devices, device C is more susceptible to oscillate more and thus produce more energy at low AoA, with the maximum open circuit voltage measured at 0◦ and y/c = 3/4. Additionally, the closed circuit graph shows mostly higher power values at y/c = 1/2 than y/c = 3/4, which has not been the case for the first two devices. Again, this device seems to work better in this position because of its inferior thickness. From observation, this result was expected, due to smaller oscillations noticed when running this device, even at high AoA. Its worth remembering at this time that devices A and B oscillated vehemently at high AoA and high Reynolds number, unlike
49 (a) Open circuit (b) Closed circuit
Figure 5.7: Device B tests in open and closed circuit at Re = 140000
(a) Open circuit (b) Closed circuit
Figure 5.8: Device C tests in open and closed circuit at Re = 140000 device C, hence the smaller energy outputs. Interestingly, device C has its maximum power output at 20◦ at y/c = 1/4, with P = 1.736 nW. This high value can be related to power spikes again, but as seen before with the other devices, position y/c = 1/4 shows higher oscillations at high AoA and as such this trend is not surprising.
5.1.4 Series connection of multiple devices A
By connecting 5 devices in series, one at each position y/c, in an effort to add their voltages, one should accomplish higher power outputs since the impedance remained the same. From Figure 5.6, it is possible to realise that adding these voltage sources in series would grant optimally power values in the order of magnitude of 10−8 W. However, this was not verifiable from the experiment, as can be seen in Figure 5.9. In fact, voltage levels in open circuit and power outputs in closed circuit are down from the values recorded in the single tests. This result was unexpected and so another test was run which showed similar results. The only plausible explanation is that having multiple devices at the trailing edge creates
50 (a) Open circuit (b) Closed circuit
Figure 5.9: Results of series connection of multiple devices A at Re = 140000 a sort of extension of the wing meaning that the vortices were mostly being shed downstream from the piezoelectric devices than at the trailing edge. When observing the oscillations while the test was running, the only device that was really oscillating similar to the single-device test was the one at y/c = 3/4, with the others not showing significant oscillations even at high AoA. Based on these observations the results obtained should have been expected, however not what was idealised, which would be an increase in power levels due to the coupling of multiple voltage sources.
5.1.5 Effect of Lift and Drag Coefficient
Analysing the effects of the device on the overall lift and drag coefficient of the wing is interesting because, as stated before, other configurations might be more advantageous in terms of power output, penalising the overall efficiency through higher drag coefficients, though. The lift and drag forces on the wing were tested with and without the piezoelectric device for the typical range of AoA at Re = 140000. The ratio of these coefficients is shown in Figure 5.10. Given the uncertainty of the forces balance at for low values of drag and lift, the results for the first few angles of attack are not very coherent. However, from 15 ◦ on the overall trend is already noticeable.
Theoretically, the CL/CD ratio is higher for the wing with the piezoelectric device than for the wing without piezoelectric device.
5.1.6 Test using a standard wing
The obtained results for a standard wing without changes to the leading edge are shown in Figure 5.11. The other tests were performed with the leading change of this wing. It can be observed that the harvested power in this wing follows a much different trend than in the wing with the leading edge change. Here, the voltage and power are maximum at 0◦ and keep decreasing as the AoA increases. Even though the maximum power output is not much lower than that of the wing with the leading edge change, for the other AoA the harvested power is generally much lower. These
51 Figure 5.10: CL/CD ratio for the wing with and without piezoelectric device
(a) Voltage output as function of AoA (b) Power output as a function of AoA
Figure 5.11: Results for simple wing for R = 1.55 MΩ values are due to the fact there is an absence of the streamwise vortices generated by the leading edge modification of the other tested wing, which give rise to more oscillations of the piezoelectric device in the other tests. The device oscillates due to the difference in pressure between the underside and upperside of the wing.
5.1.7 Power Density Ratio
The obtained results across all tests do not boast significantly high enough power outputs when considering raw values. However, in comparison with other solutions viewed in recent studies, these devices are significantly lighter than those. As stated before, lighter solutions are favoured because they do not penalise the range as much due to their added mass. Thus, an analysis on the harvested power per device mass is interesting in order to draw some conclusions. Knowing the components and the measurements of each device, their mass followed trivially and is shown in table 5.1. Figures 5.12 and 5.13 show the obtained results for this case.
52 Device A B C Mass [kg] 5.2234 × 10−5 1.0337 × 10−5 3.2058 × 10−5
Table 5.1: Device Masses
(a) Re = 70000 (b) Re = 140000
Figure 5.12: Power per Mass vs AoA of device A
(a) Device B (b) Device C
Figure 5.13: Power per Mass vs AoA of devices B and C
From the graphs, one can see that when the mass is considered in the calculations and since all devices are considerably light, the results are of the order of magnitude 10−5 W kg−1. Despite having the highest raw power value, Device B has the lowest value of power per mass because its heavier than devices A and C. Devices A and C show relatively similar results in this analysis because both have a mass of the same order of magnitude. Conclusively, device A has the best Power/Mass ratio.
At this point, in order to conclude if the operation of a piezoelectric device at the trailing edge is viable, one has to compare the obtained results with the previous work of having the device inside the wing. As seen in section 1.3,[2] obtained peak power values of the order of magnitude 20 to 30 µW for a device weighing approximately 10 g, like shown in Figure 5.14. This means [2] was able to achieve a power density ratio of 3 µW g−1 or 3 × 10−3 W kg−1, which translates to 2 orders of magnitude higher than the
53 results obtained from the device placed at the trailing edge. Additionally, [2] performed a multidisciplinary design optimisation to find the optimal device and electrical circuit characteristics to achieve the highest power density ratio. By doing so, a power density ratio of 39 µW g−1 was reached, meaning 3 orders of magnitude higher than the results obtained in this work, see Figure 5.15.
Figure 5.14: Power as function of the impedance for Re = 140000 and α = 20◦ [2]
Figure 5.15: Power density ratio optimisation [2]
While comparison with results from Esmaeili [2] is relevant, comparison with other studies with piezo- ceramic devices are much more interesting. In fact Song et al. [57] was able to achieve 15 mW cm−3 with a tuned device. Although this value seems much better than the ones achieved in this work, the results did not take into account the added mass of the tip masses that were used to optimise the har- vested energy. In raw numbers, Song et al. [57] was able to harvest 112.8 µW of power with a device that weighed 1.872 g. This means, Song et al. [57] was able to achieve a power density ratio of 60.25 µW g−1. Despite these results being several orders of magnitude higher than the ones obtained in this
54 work, one must not forget the device was tuned by tip masses to a fixed vibration acceleration to op- timise the harvested energy. In a real world application such tunings are hard to accomplish because the vibration acceleration is not constant and well-defined. A similar study to this work was performed by Rahman et al. [58], in which the dielectric properties of a PVDF was tuned with nanocomposites to achieve better energy harvesting. The maximum power harvested was 36 nW. Since the weight of the devices used by Rahman et al. [58] was similar, the power density ratio is also of the same order of magnitude. The obtained results are consistent to the ones obtained in this work.
55 56 Chapter 6
Conclusions
6.1 Achievements
With this work, it was possible to quantify the harvested energy from a piezoelectric device oscillating as a flapping flag at the trailing edge of a wing. The predictive aerodynamic-electric-mechanical model developed enabled to predict how the device would operate in the given conditions. Despite the fact that the model assumed some simplifications in order to achieve results in more rapid fashion, there was a good agreement between the model and the experimental results obtained in the wind tunnel. In the impedance test, the model generally underpredicts the voltage outputs relative to the wind tunnel testing. This in fact was to be expected due to the calibration of the model to a reference force. A numerical model for the force would certainly achieve a better predictive result in detriment of a more rapid solution. Such numerical models are very costly in terms of computational power and time. The model also predicted the power output of the piezoelectric device. Here, there are two primary observations to be made when comparing the model with the experimental results. The model under- predicts the power outputs up until 2.3 MΩ, while overpredicting the power outputs from 10 MΩ forward. Additionally, one can also observe that the model shows a maximum power output for 20 MΩ and a decline in power outputs from then on, while the experimental results show this maximum power output happening at 10 MΩ. This has to do with the fact that the model predicts a more pronounced increase in voltage between 10 and 20 MΩ, which was not verified in the experiments. In the air speed test, the model shows good agreement with the obtained results in the wind tunnel, with the same trends and approximate values being seen in both the model and the experiments. The major difference why the model is more effective here than in the impedance test is the fact that the calibrated force term behaves linearly in the equations, while the impedance appears in various shapes and forms in the equations, resulting in a worse prediction. When comparing the three types of devices used, there is a general advantage in using the smaller devices A and C in detriment of the longer device B. Despite having a higher maximum power output, device B also weighs more than A and C, thus having a lower power density ratio. Between devices A and C, A is the more viable option due to higher power outputs and a not significant increase in weight
57 resulting in an overall better power density ratio. In terms of energy harvesting, the devices did not perform as one would have hoped for with the results showing power outputs in the magnitude of nW. With the widespread of microelectronics, small amounts of energy are required to power micro-sensors, for instance. With some tuning and optimised conditions for energy harvesting, one should be able to power such micro electronic devices. Addition- ally, this configuration allows for energy gains without penalising the overall weight of the MAV, since the devices are lightweight. Other studies in the literature harvested more power, but at the cost of added weight to the MAV. If not for power harvesting purposes, these devices showed good results in managing to identify the onset of stall while operating at the outer positions of the wing. This wake sensing is an important tool to inform the pilot or the onboard computer of imminent stall situation so that it can be dealt with.
6.2 Future Work
In the future, one should study the inverted flag configuration. It is a known fact that the harvested energy quantities in this configuration will be larger than in the flapping flag at the trailing edge configu- ration with the same device weight. However, the support structure weight has to be taken into account as well as the additional drag created by said structure and the flapping of the flag in large-deflection deformed flapping. Therefore, in order to obtain a clear view of which configuration is best, these tests need to be done. While these devices show great promise in harvesting energy, the fact that they only boast significant voltage outputs at high resistive loads harms the overall energy gains. As such, there should be focus in developing these devices to work at lower resistive loads in order to make them more viable, since having a device operating in 10 MΩ of impedance is not feasible. Moreover, there are huge margins to work with in terms of optimising the piezoelectric devices for energy harvesting purposes. It is known that they are manufactured for other purposes mainly. However, with some focus on harvesting vortex-induced vibrations, the devices’ energy harvesting potential could be boosted. The predictive model can also be enhanced to obtain a more realistic result in some of the areas where it was lacking. The wing was created as a passive control of stall, delaying the onset of stall for the typical operating conditions a MAV encounters. It is possible to optimise its leading edge further in order to enhance its induced vortices and thus increase the energy harvested from the piezoelectric device, though.
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63 64 Appendix A
Piezoelectric Device Catalogue
The catalogue of the acquired piezoelectric devices from Mouser Electronics.
65 LDT1-028K PIEZO SENSOR w/ Lead Attachment
INSTRUCTIONS Direct Impact Sensing: Using an adhesive (such as double sided tape), adhere the sensor area to a pliable pad to absorb impact with the full length protective laminate on the impact face. Apply the force (such a finger LDT1-028K PIEZO SENSOR touch or hammer blow) to the sensor area.
w/ Leads Attached Vibration and Motion Sensing: Mount the element in a cantilever arrangement, allowing the sensing area to vibrate up and down. Add a small weight to the end of the sensor if greater sensitivity to lower frequencies is desired.
The direct adherence of the LDT1-028K to the vibrating body can detect vibration, but another piezo film sensor SPECIFICATIONS configuration (SDT1-028K) is available and designed for this application. The SDT1-028K (part number 1- Piezoelectric Polymer 1000288-0) is a fully shielded sensor that is related to the LDT1-028K. Multi-purpose A cantilever arrangement will allow the piezo element to be deflected and this can be used to detect a Vibration Sensing Bending: striking object when the element is flexed. It is essential that the film not be in the neutral axis of the beam. Impact Sensing Otherwise signal cancellation can result, minimizing signal. Laminated
Dual Wire Lead Attached The LDT1-028K device is unshielded by design. If shielding is required, the sensor should be enclosed in a proper environment. Metallized tapes can be used to cover the sensor but these may impede motion and subsequent output. Wire leads can be twisted or covered. Other piezo film devices such as SDT1-028K are available and are The LDT1-028K is a multi-purpose, piezoelectric shielded. sensor for detecting physical phenomena such as DIMENSIONS vibration or impact. The piezo film element is laminate to a The LDT1-028K is designed to cover a wide range of sensing applications. Specific sensors requiring wider Dimensions in mm [inches] sheet of polyester (Mylar), and produces a useable dynamic range, more or less sensitivity, different area coverage, different shapes, extended life, resistance to electrical signal output when forces are applied to the sharp objects, and higher temperature range, etc., can be constructed to fit the applications as special orders. sensing area. The dual wire lead attached to the sensor allows easy connection to a circuit or monitoring device to ORDERING INFORMATION process the signal. Total Film Cap Model Number Part Number Film Electrode Film Electrode Thickness Thickness (nF) FEATURES (μm)
LDT1-028K/L w/rivets 1-1002910-0 28 μm .64 (16) .484 (12) 1.63 (41) 1.19 (30) 157 1.38
Minimum Impedance: 1 MΩ Preferred Impedance: 10 MΩ and higher LDT1-052K/L w/rivets 2-1002910-0 52 μm .64 (16) .484 (12) 1.63 (41) 1.19 (30) 181 .740 Output Voltage: 10 mV-100 V depending on force and LDT2-028K/L w/rivets 1-1003745-0 28 μm .64 (16) .484 (12) 2.86 (73) 2.42 (62) 157 2.85 circuit impedance LDT2-052K/L w/rivets 2-1003745-0 52 μm .64 (16) .484 (12) 2.86 (73) 2.42 (62) 181 1.55 Storage Temperature: -40°C to +70°C [-40°F to 60°F] Operating Temperature: 0°C to +70°C [32°F to 160°F] LDT4-028K/L w/rivets 1-1002405-0 28 μm .86 (22) .740 (19) 6.72 (171) 6.13 (156) 157 11.00 LDT4-052K/L w/rivets 2-1002405-0 52 μm .86 (22) .740 (19) 6.72 (171) 6.13 (156) 181 5.70
APPLICATIONS NORTH AMERICA EUROPE ASIA Measurement Specialties, Inc., MEAS Deutschland GmbH Measurement Specialties (China), Ltd., Sensing direct contact force a TE Connectivity Company a TE Connectivity Company a TE Connectivity Company Recording time of an event Tel: +1-800-522-6752 Tel: +49-800-440-5100 Tel: +86 0400-820-6015 Counting number of impact events Email: [email protected] Email: [email protected] Email: [email protected] Measuring impact related events Sensing vibration using a cantilevered beam Wakeup switch Motion detection
TE.com/sensorsolutions
Measurement Specialties, Inc., a TE Connectivity company. Measurement Specialties, TE Connectivity, TE Connectivity (logo) and EVERY CONNECTION COUNTS are trademarks. All other logos, products and/or company names referred to herein might be trademarks of their respective owners. The information given herein, including drawings, illustrations and schematics which are intended for illustration purposes only, is believed to be reliable. However, TE Connectivity makes no warranties as to its accuracy or completeness and disclaims any liability in connection with its use. TE Connectivity‘s obligations shall only be as set forth in TE Connectivity‘s Standard Terms and Conditions of Sale for this product and in no case will TE Connectivity be liable for any incidental, indirect or consequential damages arising out of the sale, resale, use or misuse of the product. Users of TE Connectivity products should make their own evaluation to determine the suitability of each such product for the specific application. © 2015 TE Connectivity Ltd. family of companies All Rights Reserved.
SENSOR SOLUTIONS /// LDT1-028K Piezo Sensor Rev 1 07/2017 Page 1 SENSOR SOLUTIONS /// LDT1-028K Piezo Sensor Rev 1 07/2017 Page 2 Appendix B
Technical Datasheets
Figure B.1: Material Datasheets
67 68