Piezoelectric energy harvesting by aeroelastic means

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Piezoelectric Energy Harvesting by Aeroelastic

Means

Department of Mechanical and Aerospace Engineering

Dottorato di Ricerca in Ingegneria Aeronoutica e Spaziale – XXXII Ciclo

Candidate Hassan Elahi ID number 1740782

Thesis Advisor Prof. Paolo Gaudenzi

Co-Advisor

Prof. Luca Lampani Dr. Marco Eugeni

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Prof. Roberto Camussi (chairman) Prof. Michele Ferlauto

Prof. Riccardo Vescovini

Piezoelectric Energy Harvesting by Aeroelastic Means

This thesis has been typeset by LA_{TEX and the Sapthesis class.}

Version: March 4, 2020

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Dedicated to Mr. and Mrs. Sheikh Muhammad Noor Elahi

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Abstract

For the last few decades, piezoelectric (PZT) materials have been widely used in the field of micro/nano-electromechanical systems. One of the most important applications of the PZT material is energy harvesting by absorbing ambient energy from the operational conditions and converting it into electrical energy. This energy can be used to operate sensors and actuators. Moreover, it can be stored in batteries for later tasks. In this thesis, the harvester absorbs energy from the airflow, thanks fluid-structure interaction (FSI), and converts it into useful electrical energy. To analyze FSI, it is important to consider the whole dynamics of the system formed by the structure and the flow i.e., aeroelastic system rather than considering them as two different systems.

This coupling, from the mathematical point of view, occurs because the natural boundary condition of the structure is defined by the flow pressure which is mutually influenced by the structure. This leads to a very complex phenomenon that is intrinsically non-stationary and it is no longer possible to study it by considering the structure and the flow separately. The aeroelastic system remains stable up to a critical velocity of the flow known as flutter velocity which depends on the following media and the mechanical properties of the surrounding system. After this particular velocity, the aeroelastic system is no longer stable in its unperturbed condition. The system can no longer be considered as linear and stable oscillations arise, the so-called Limit Cycle. Indeed, the interaction of the fluid in the form of airflow with structure i.e., airfoil will transfer oscillations to the PZT which will result in energy harvesting.

In the present work, the possibility of extracting energy by means of PZT transduction from an aeroelastic behavior, known as the Limit Cycle Oscillation (LCO), is investigated analytically, numerically and experimentally. A suitably designed aeroelastic device which is based on the use of PZT components is presented thanks to the flag-flutter phenomenon. The presented harvester is studied from the analytical, numerical and experimental point of view. A nonlinear piezoelectric aeroelastic energy harvester (PAEH) is modeled based on the FSI that represents

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an important area of research for the development of innovative energy harvesting solution. This PAEH operates on LCOs that arise after the flutter velocity. The aim of this research is to study and design a nonlinear aeroelastic energy harvester. The PZT transduction from the Limit Cycle is investigated. Particular emphasis is placed on demonstrating a correct model of unsteadiness of aerodynamics. The unsteady aerodynamic model is a critical ingredient for a sound prediction of the nonlinear behavior of an aeroelastic system. Thus, it plays a vital role for the correct evaluation of the performance of an energy harvester based on the flutter phenomenon. Moreover, it is shown that if the unsteady nature of aerodynamics is not taken into account, the evaluation of the system stability margins is totally incorrect, even if a quasi-steady hypothesis is considered. Therefore, it is emphasized that the determination of aerodynamic model is necessary for the correct prediction of PAEH performance. Indeed, harvesting performances, flutter boundaries, aeroelastic modes, and LCOs amplitude predicted by different models, are compared with the experimental data provided by wind tunnel tests.

The present harvester has various applications in the field of aerospace engineer-ing. As a result, it is shown that the overall system is suitable for energy harvesting and can be utilized to drive microelectronics i.e., wireless sensors in sub-orbital missions, launchers, space vehicles and in various aerospace applications.

Keywords: Piezoelectric, Fluid-structure Interaction, Aeroelasticity, Energy Har-vesting, Nonlinear, Electromechanical, Flag-Flutter, Harvester

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Acknowledgments

I would like to thank my supervisor, Prof. Paolo Gaudenzi, and express my deepest gratitude for the patient guidance, encouragement and valuable advice provided throughout my time as a PhD student. I would like to thanks to my cosupervisor, Prof. Luca Lampani, he always encourages me and helped a lot in numerical and experimental part of the thesis. It will be totally unfair if I do not mention the name of Dr. Marco Eugeni, he always supported me in each and everything, from technical point of view to bureaucratic work. I had a perfect time with my lab mates specially Dr. Valerio Cardini, Dr. Luciano Pollice, Dr. Guo-Yan Zhao and Mr. Fune Federico. I would also like to thank Sapienza University and the Department of Aeronautic and Space Engineering for this opportunity. Moreover, thanks to Rome for hosting me in a superb way during my stay here and not letting me crazy.

I would like to acknowledge my all family members, Mr. and Mrs. Sheikh Muhammad Noor Elahi, Ms. Sadaf, Dr. Saba, Ms. Maryam, Engr. Raza, Dr. Umer and Dr. Aqsa for encouraging me. At the end, special thanks to my dear friends, Dr. Bashir, Dr. Bharti, Dr. Munir and Dr. Thakker for their moral support.

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List of Figures

2.1 Piezoelectric effect in Quartz. . . 9 2.2 Cubic structure of BaTiO3. Where, the red spheres represent the

oxide centres, blue spheres represent the Ti4+ _{cations, and the green}

spheres represent the Ba2+. . . 9 2.3 Overview of PEH mechanism; (a) piezoelectric effect with 33 and 31

strain-charge coupling; (b) polarization process; (c) bi-morph PEH mechanism . . . 10 2.4 Electromechanical coupling in PZT effect. . . 10 2.5 PZT for sensing and actuating. (a) top: hysteresis plot of P-E; bottom:

S-E field plot, (b) PZT before and after poling (c) applied voltage and polarity directions voltage are in same direction, (d) applied voltage and polarity voltage directions are in different direction, (e) generated voltage and polarity voltage are in different direction during compression, (f) generated voltage and polarity voltage are in different directions during tension. . . 11 2.6 Undeformed and deformed configuration under the effect of σ₃. . . . 11 2.7 non-deformed and deformed configuration under the action of E₃. . . 12 2.8 Open- and closed-circuit configurations of a PZT under compression.

Where, D is electric flux and E is electric field. . . . 15 2.9 The path of mechanical loading considering different circuit

configu-rations. . . 16 2.10 The path of electrical loading considering different circuit configurations. 16

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2.11 Typical cantilever piezoelectric harvesters; where V is the voltage

that can be harvest from this PEH. . . 19

2.12 Overall mechanism for piezoelectric energy harvesting; where K is the modal stiffness, C is the modal damping, Ce is the external capacitance and V is the output voltage generated. . . . 20

2.13 Application of PEH in wireless sensors. . . 20

3.1 Collar’s triangle for Aeroelasticity. . . 22

3.2 Polar diagram for divergence and flutter condition. . . 23

3.3 Subcritical Hopf bifurcation. . . 24

3.4 Supercritical Hopf bifurcation. . . 25

3.5 Sample arrangement of doublet lattice aerodynamic boxes. . . 29

3.6 Sample of aerodynamic and structural model for a wing. . . 30

4.1 Overall scheme for FSI based PEH (For more details see Ref. [98]). . 32

4.2 Mechanism of VIV based PEH. . . 33

4.3 Bifurcation diagram. . . 33

4.4 Cylinder-based piezoaeroelastic energy harvester; subjected to free stream velocity U∞; R is the electrical load resistance; C and K are the structural damping and stiffness respectively. . . 34

4.5 Flutter-based piezoaeroelastic energy harvester schematic. . . 35

4.6 Bifurcation diagram [7]. . . 35

4.7 Piezoaeroelastic system schematic [7] (A 3-DOF aiirfoil supported elastically by linear plunge and linear torsional spring). . . 36

4.8 Transverse galloping-based PEH schematic (A D-shaped prismatic structure is subjected to airflow; attached at the end of elastic can-tilever beam) . . . 38

4.9 Wake galloping-based PEH schematic (Two bluff bodies are subjected to airflow; One is attached at the end of elastic cantilever beam and another bluff body is placed infront of it) . . . 38

4.10 Galoping based PEH in test section of subsonic wind tunnel [70] . . 39

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5.2 Bifurcation graph [68] . . . 44

5.3 Basin of attraction [68] . . . 44

5.4 arctan approximation; dashed line represents arctan approximation; solid line represents the piecewise linear system . . . 45

5.5 Typical section of 3-dof aeroelastic system . . . 46

5.6 Root locus with steady state aerodynamics . . . 51

5.7 Root locus with quasi-steady aerodynamics. . . 52

5.8 Root locus with unsteady aerodynamics. . . 53

5.9 Output power at variable resistance for; (a) P ZT − 5A, (b) BaT iO₃. 54 5.10 Comparison of output power in terms of resistance for PZT-5A and BaTiO3. . . 55

5.11 Output power at variable length of piezoelectric patch. . . 55

5.12 Bifurcation diagram. - Normal Form; Numerical Integration. . . . 55

5.13 Limit cycle oscillation at U=3.0250. . . 56

5.14 Limit cycle oscillation at U=3.030. . . 57

6.1 Flag geometry. . . 59

6.2 Flag mesh. . . 61

6.3 Flutter velocity trend for flag 1 convergence analysis. . . 64

6.4 Flutter velocity trend for flag 2 convergence analysis. . . 65

6.5 Flutter quantities for unpatched flags. . . 69

6.6 Vibration mode of the unpatched flags. . . 70

6.7 Extended non-dimensional stability diagrams for unpatched models. 70 6.8 Flutter analysis of unpatched flags. . . 71

6.9 Vibration mode of the Al patched flags. . . 72

6.10 Extended non-dimensional stability diagrams of Al patched models. 72 6.11 Curvature and area results of Al patched flags. . . 73

6.12 Flutter analysis for piezoelectric patched flags. . . 74

6.13 Vibration mode of the Piezoelectric patched flags. . . 75

6.14 Extended non-dimensional stability diagrams for Piezoelectric patched models. . . 75

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6.16 Comparison of velocity stability diagrams. . . 77

6.17 Comparison of extended non-dimensional stability diagrams. . . 78

6.18 Comparison of frequency stability diagrams. . . 78

6.19 Non-dimensional stability diagrams of unpatched models. . . 79

7.1 PAEH experimental mechanism. . . 81

7.2 Overall experimental setup. . . 82

7.3 PAEH flag flutter. . . 83

7.4 Initial manufacturing of flag. . . 84

7.5 Unwanted deformation in flag. . . 85

7.6 Final stripes. . . 85

7.7 Steel bracket used for clamping the edge. . . 86

7.8 Patches of Al and Plastic. . . 87

7.9 Al patched flag. . . 87

7.10 Flow data acquisition system. . . 88

7.11 Fast camera, optics and software. . . 88

7.12 Blocking system setup. . . 89

7.13 First tests experimental setup. . . 89

7.14 Experimental flutter analysis of Al patched flags. . . 91

7.15 Experimental mode oscillation for different flags. . . 91

7.16 P-876 transducer . . . 92

7.17 Electrical instruments for EH measures. . . 92

7.18 Circuit for harvesting. . . 93

7.19 Circuital elements (zoomed). . . 93

7.20 Wind tunnel setup. . . 95

7.21 Energy harvesting output. . . 95

7.22 Experimental output voltage for different resistances. . . 97

7.23 FFT results. . . 98

7.24 Damping for different resistances. . . 98

7.25 Capacitor measures. . . 98

7.26 Instant and average power. . . 99

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7.28 Bifurcation diagram for open circuit flag. . . 101

7.29 Bifurcation diagram for optimal resistance flag. . . 101

7.30 Critical mode shape for lower and higher velocity than critical one. . 102

7.31 Dimensional critical values comparison. . . 103

7.32 Not dimensional critical values comparison. . . 103

7.33 Non dimensional stability diagrams. . . 104

7.34 Deformation comparison for Al patched flags. . . 107

7.35 Deformation comparison for PZT patched flag. . . 108

7.36 Tip motion locus comparison for Al patched flag. . . 108

7.37 Voltage output comparison with correlation value. . . 109

7.38 Power output comparison with correlation value. . . 109

7.39 Not dimensional stability diagram for velocity. . . 111

7.40 Not dimensional stability diagram for frequency. . . 111

7.41 Adimensional stability diagrams in literature. . . 112

7.42 Numerical deformation comparison. . . 112

7.43 Numerical and experimental deformation comparison with literature for lower and higher velocities than critical one. . . 113

7.44 Bifurcation diagrams comparison with literature. . . 114

7.45 Experimental and literature comparison for flag tip behaviour. . . . 116 7.46 Voltage and power comparison with experimental data in literature. 118

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List of Tables

1.1 Comparison of various flutter based PAEHs [59]. . . 3

4.1 Comparison of various FSI based PEH . . . 40

4.2 Other mechanisms for FSI based PEH . . . 41

5.1 Airfoil parameters [110] . . . 49

5.2 PZT coefficients for harvesting [60, 61]. . . 49

6.1 Geometric parameters of flag. . . 59

6.2 Nodes and elements for discretization. . . 61

6.3 Convergence analysis of flags dimensions. . . 62

6.4 Glass fiber characteristics. . . 63

6.5 Flag 1 preliminary analysis results. . . 63

6.6 Flag 1 convergence analysis results. . . 64

6.7 Flag 2 preliminary analysis results. . . 64

6.8 Flag 2 convergence analysis results. . . 65

6.9 Polystyrene characteristics. . . 66

6.10 Validation analysis results for flag 1. . . 66

6.11 Validation analysis percentage errors for flag 1. . . 66

6.12 Validation analysis results for flag 2 . . . 66

6.13 Percentage errors for flag 2 during validation. . . 67

6.14 Flag models geometric features. . . 68

6.15 Flutter analysis results of aluminium flags. . . 71

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6.17 Normalised average curvature and area values of Al patched flags. . 73 6.18 Flutter analysis results of piezoelectric patched flags. . . 74 6.19 Absolute average curvature and area values of PZT patched flags. . . 75 6.20 Normalised average curvature and area values of PZT patched flags. 76

7.1 Material properties of PIC 255. . . 81 7.2 Flag and piezoelectric element geometry. . . 82

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List of Symbols

b: half chord length

cα: pitch structural damping coefficient

ch: plunge structural damping coefficient

Cp: capacitance of the piezoelectric layer

d: width of the piezoelectric patch

dd_im: piezoelectric coefficients for direct piezoelectricity dc_jk: piezoelectric coefficients for converse piezoelectricity DA: the aerodynamic damping

Di: dielectric displacement vector

DOF : degree of freedom

e31: piezoelectric constant in 31 coupling direction

eσ_ij: dielectric permittivity at constant stress E: electrical energy generated

Ej: applied electric field vector

F SI: fluid structure interaction h: plunge deflection

Iα: mass moment of inertia about the elastic axis

K: the structural stiffness matrix

KA: the aerodynamic contribution to stiffness

kh: structural stiffness for the plunge motion

KM E: electromechanical coupling

kα: structural stiffness for the pitch motion

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LCO: limit cycle oscillations M : moment about the elastic axis Mα(α): torsional spring moment

mT: total mass of the wing with its support structure

mW: wing mass alone

M EM S: micro electromechanical systems p: lift aerodynamic moment for the wing

P AEH: piezoelectric aeroelastic energy harvester P EH: piezoelectric energy harvester

P ZT : piezoelectric

q: electrical charge on electrodes R: load resistance

Re: Reynolds numbers

r: pitch aerodynamic moment for the wing rα: reduced radius of gyration about elastic axis

r_β: reduced radius of gyration about flap hinge s: pitching moment acting on the flap

S_kmE elastic compliance matrix at constant electric field U : flow velocity

U∞: free stream velocity

v: electric potential between two electrodes V : voltage across this load resistance V IV : vortex induced vibrations

xα: dimensionless distance between the center of mass and the elastic axis

α: pitch motion β: flap motion

δ: thickness of piezoelectric layer ∆P_M: change in metabolic power

δφ: change in the flexibility of the patch k: strain vector

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ηDevice: efficiency of device

ωξ: uncoupled plunging frequency

ωα: uncoupled pitching frequency

ωβ: uncoupled flapping frequency

σm: stress vector

τ : dimensionless time θ: angular displacement

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1

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Introduction

The phenomenon of energy harvesting based on PZT transducers can be defined as, the transformation of energy absorbed by the transducer from operating environment into electric voltage that can be used on spot for actuation or stored in batteries for future usage [77]. The world is shifting to low powered electronic devices due to compact engineering designs, which leads to micro and nano powered electronic circuits [91]. Numerous researchers focused on the usage of piezoelectric energy harvester (PEH) as a self-powered source over the usage of batteries [20]. In recent technological advancement, these harvesters are ideal to be used in micro electromechanical systems (MEMS), smart structures, structural health monitoring and as wireless sensors for suborbital mission [57].

From the last few decades, PZT materials are widely used in the field of micro-electromechanical systems and nano-micro-electromechanical systems [77]. They have various applications in the field of aerospace engineering [57, 74, 75] i.e., as a smart structure and/or smart embedded composite structure; in the field of structural health monitoring [78] i.e., for the detection and propagation of cracks in the host structure; in the field of sensors and actuators [40], because of their voltage dependent actuation i.e., accelerometers [60]. Moreover, they play a vital role in the field of energy harvesting due to direct effect of piezoelectricity i.e., to operate wireless sensors in aerospace industry.

With recent advancements in compact and portable electronic technology, power sources have also been evolved. For such portable electronic devices, it is necessary to have their own power source rather than relying on external batteries that have limited lifespan and in some cases their replacement is problematic as well i.e.,

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wireless sensors for suborbital missions. In these conditions, PZT materials play a vital role as an alternative source of power supply to drive microelectronics [121, 138]. Many researchers have emphasized the importance of modeling of PZT systems and many techniques have been developed, both in the linear and nonlinear cases i.e., for 3D solids [58], and for structural elements like plates and shells [78]. They can harvest useful electrical energy by acquiring energy from the surrounding which can be utilized to drive circuits or can be stored in battery for later tasks. In this research work, PAEH absorbs energy from airflow (which is an example of FSI) and converts it into useful electrical energy.

The effect of electromechanical and fluid-surface interaction is an important phenomenon in piezoelectricity for characterizing smart structures, energy harvesters, integration of sensors and actuators for structural health monitoring especially in aerospace industry [81, 98]. To analyze FSI, it is important to consider the whole dynamics of the system formed by the structure and the flow i.e., aeroelastic system rather than considering them as two different systems. This coupling, from the mathematical point of view, occurs because the natural boundary condition of the structure is defined by the flow pressure which is mutually influenced by the structure. This leads to a very complex phenomenon that is intrinsically non-stationary and it is no longer possible to study it by considering the structure and the flow separately [50, 52]. The aeroelastic system remains stable up to a critical velocity of the flow known as flutter velocity which depends on the flowing media and the mechanical properties of the surrounding system. After this particular velocity, the aeroelastic system is no longer stable in its unperturbed condition. The system can no longer be considered as linear and stable oscillations arise, the so called Limit Cycle [52]. From the mathematical point of view, this means that the aeroelastic system is experiencing a bifurcation of its unperturbed solution [85]. Once the flutter phenomenon occurs, the self excited oscillations exhibited by the aeroelastic system, are very rich from the dynamical point of view. It has been widely studied in the literature [69, 85] and is of great interest for the phenomenon of PZT energy harvesting. Indeed, the interaction of the fluid in the form of airflow with structure i.e., airfoil will transfer oscillations to the PZT which will result in energy harvesting [1, 2].

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PZT materials are attractive source for energy harvesting. They can be used to recharge the batteries and to power other electronic accessories like communication devices and wireless sensors [77]. In particular, the possibility to extract energy from the operational environment where the component is placed to operate, is of absolute interest in the present advanced industrial applications, especially in the aerospace field where energy saving is an absolute task and a network of wireless sensors for health management is a very promising option. For energy harvesting via PZT material, direct effect of piezoelectricity is applicable, these PZT actuators can be used as active and semi-passive controllers for some modification in the behavior of aeroelastic wing [16, 35].

The use of PZT transducers as energy harvesters by means of FSI is of great interest for aerospace applications. Many theoretical and experimental studies in the dynamic aerodynamic profile with nonlinear freeplay were conducted by many researchers. Woolston et al. [162] and Shen et al. [134] investigated the effects of structural nonlinearity on aeroelastic systems. Yang et al. studied the effects of freeplay nonlinearity on the typical aeroelastic system [168]. Later on, this response was analyzed experimentally and numerically by Tang et al. [145] and Conner at al. [38]. Followed by Tang and Dowell, who analyzed the freeplay nonlinearity of different aeroelastic models [51] such as, wing-store model [146], wing section with control surface [147] and horizontal-tail model [148]. Investigations performed by them elobrated the importance of system parameters, nonlinearity and initial conditions as they affected the nonlinear aeroelastic system significantly. Comparison of various flutter based PAEHs from literature is represented in Table 1.1.

Table 1.1. Comparison of various flutter based PAEHs [59].

Design Material used Layer(s) Power (mW)

NACA0014 PZT 1 0.003

Symmetric PSI-5A4E 1 0.2

NACA 0012 QP 10N 2 2.2

Nonlinear PAEH based on flutter mechanism was introduced by Bibo and Daqaq [30, 31], in which they considered a lumped-parameter model under a combination of harmonic base excitations and quasi-steady aerodynamic loadings. Bae et al., [25]

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and Eugeni et al. [66], studied the in-detail performance of 2-DOF PAEH with cubic nonlinearities, and particular importance is given to the correct modeling of FSI aerodynamics. Abdelkefi et al., analyzed the effect of controlled freeplay nonlinearity and the performance of 2-DOF PAEH based on flutter mechanism experimentally [5]. Moreover, post critical aeroelastic behavior is stressed for the energy harvesting mechanism [3]. Selection of optimal load resistance plays a vital role in energy harvesting mechanism, as it can increase the order of harvested energy magnitude [8].

The project is overall supervised by Prof. Paolo Gaudenzi. Energy harvesting from fluid-structure interaction is one of the major projects ongoing in the research group headed by Prof. Gaudenzi. For this project, he collaborated with different re-search groups in the Department of Mechanical and Aerospace Engineering, Sapienza University of Rome, Rome, Italy. For setup of the experiments related to energy harvesting via PZT transduction, he collaborated with Prof. Luca Lampani. For the aeroelastic system design, he collaborated with Prof. Franco Mastroddi. For experimental fluid dynamics activities (the activities carried out in subsonic wind tunnel), he collaborated with Prof. Giovanni Paolo Romano. Mr. Hassan Elahi and Dr. Marco Eugeni collaborated as a bridge between all these activities. These collaborations can also be seen in the published articles of this project.

Prof. Paolo Gaudenzi is also the author of the book entitled “Smart structures: physical behavior, mathematical modeling, and applications. John Wiley & Sons, 2009”. This book offered valuable insight into both how smart structures behave, how and at what cost they could be designed and produced for real-life applications in cutting edge fields such as vibration control, shape morphing, structural health monitoring, and energy transduction. Moreover, this book offered a basic and fundamental description of smart structures from the physical, mathematical and engineering viewpoint. It explains the basic physics relating to the behavior of active materials, gives the mathematical background behind the phenomena and provides tools for numerical simulation.

The aim of this research is to study and design a PZT energy harvester based on aeroelastic means. The piezoelectric transduction from aeroelastic behavior, as the

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1.1 Outline of the Thesis 5

Limit Cycle, is investigated. Particular emphasis is placed on demonstrating correct modeling for the unsteadiness of aerodynamics. The unsteady aerodynamic model is a critical ingredient for a sound prediction of the nonlinear behavior of an aeroelastic system. Thus, it plays a vital role in the correct evaluation of the performances of an energy harvester based on the flutter phenomenon. Moreover, it is shown that if the unsteady nature of aerodynamics is not taken into account, the evaluation of the system stability margins is totally incorrect, even if a quasi-steady hypothesis is considered. With the perspective of using the flutter phenomenon for energy harvesting purposes, deeper comprehension of the dynamics of the fluid-structure interaction of a cantilevered flag is required. This thesis aims towards the numerical and experimental evaluation of a piezoelectric energy harvester based on flag-flutter. The LCOs are the cause of energy harvesting in the designed harvester. Further, it is shown that the designed harvester is effective and can be utilized in many aerospace industrial applications to drive micro/nano-electronic devices i.e., wireless sensors in sub-orbital missions, launchers, space vehicles, high altitude platforms, stratospheric probe balloons, and unmanned aerial vehicles.

1.1 Outline of the Thesis

The outcomes of three-year period of research and study carried out at the De-partment of Mechanical and Aerospace Engineering, Sapienza University of Rome, Rome-00186, Italy are reported in this thesis. This thesis is composed of journal publications, conference publications and congresses attended by the author under the supervision of Prof. Paolo Gaudenzi and Prof. Luca Lampani. The core of the work is to evaluate PAEH analytically, numerically and experimentally. The thesis is divided into 7 Chapters as follow:

1.1.1 Chapter 1: Introduction

A general presentation of the current challenge of energy harvesting by aeroelastic means is presented in this chapter. Moreover, motivation, research objectives and outline of the thesis is also elaborated.

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1.1.2 Chapter 2: Piezoelectric Materials

The physics physics of piezoelectric effect and the mathematical formulation of the behavior is presented in this chapter. The phenomenon of piezoelectric materials for energy harvesting applications under mechanical, thermal and fluid loading is also presented.

1.1.3 Chapter 3: Aeroelastic Problems

The general issues related to aeroelasticity, sub-critical and post-critical bifurcation are elaborated. Moreover, the mechanisms for fluid-structure interaction are pre-sented in this chapter. At the end of chapter, p-k method of aeroelastic problem formulation is also presented.

1.1.4 Chapter 4: Piezoelectric Energy Harvesting by Aeroelastic Means

The energy harvested by the integration of fluid-structure interaction and piezo-electricity is presented in this chapter. Piezoelectric energy harvesting mechansims based on vortex induced vibrations, flutter and galloping are also elaborated in this chapter.

1.1.5 Chapter 5: Analytical Model

The analytical model of PAEH is presented in this chapter. The mathematical model of the designed piezoelectric energy harvester along with the mechanism is elaborated. Moreover, it is emphasized that unsteady state of aerodynamics is important for the true evaluation of the harvester.

1.1.6 Chapter 6: Numerical Model

The numerical model of PAEH is presented in this chapter. The modeling of the desgined harvester is elaborated with model validation. Moreover, flutter analysis of Al patched, PZT patched and no patch attached flags is also studied numerically in this chapter.

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1.1 Outline of the Thesis 7

1.1.7 Chapter 7: Experimental Campaign

The experimental campaign of PAEH is presented in this chapter. The manufacturing of flags to the real time experimentation of the flag in a wind-tunnel carried out at DIMA is elaborated. Moreover, bifurcation graph for Al patched, PZT patched and no patch attached flags is also studied experimentally in this chapter.

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Piezoelectric Materials

This chapter is based on the results published as a chapter in the book by the author after peer review:

• Elahi, H., Eugeni M., Gaudenzi P. (2018) Electromechanical Degradation of Piezoelectric Patches. In: Altenbach H., Carrera E., Kulikov G. (eds) Analysis and Modelling of Advanced Structures and Smart Systems. Advanced Structured Materials, vol 81. Springer, Singapore.

In this chapter physcics behind the piezoelctric material is elaborated. There exits certain materials which have the ability to generate electrical voltage by application of mechanical stress on them (i.e., direct effect of piezoelectricity) and vice versa (i.e., converse effect of piezoelectricity). Pierre and Jacques Curie discovered direct PZT effect whereas, Gabriel Lippmann discovered the converse effect of piezoelectricity which was then experimentally verified by "the Curie brothers". The PZT generated electric voltage is proportional to the input stress and vice versa [77]. The PZT materials belong to the crystalline group. The topic of energy harvesting has become of significant importance over the last few decades and development of this field has been responsible for revolutionizing micro-electromechanical systems i.e., wireless sensors [24]. The PZT sensors and transducers that absorb ambient energy from vibrations are very much in demand because of their energy harvesting capability [132]. Many researchers are working on numerical simulations of PZT based sensors, actuators and structural health monitoring systems based on finite element method (FEM) [141]. The PZT effect is considered to be a couple between electrical and mechanical properties of a material [154]. The overall effect of piezoelectricity in Quartz is represented in Fig. 2.1. A typical 3D crystal structure of barium titanate

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2.1 The piezoelectric effect 9

(BaTiO₃) ceramic is shown in Fig. 2.2. For enhancing the output electric potential it is important to have a poling effect on the PZT materials so that all the poles are in approximately the same direction. The overall mechanism of bimorph PEH is presented in Fig. 2.3 with elaboration of polarization process.

Figure 2.1. Piezoelectric effect in Quartz.

Figure 2.2. Cubic structure of BaTiO3. Where, the red spheres represent the oxide centres,

blue spheres represent the Ti4+ _{cations, and the green spheres represent the Ba}2+_.

2.1 The piezoelectric effect

The piezoelectric effect is defined as the generation of the non-mechanical output (electric potential) in response to the mechanical stimulus or mechanical output (mechanical strain). The collar’s triangle for piezoelectricity is presented in Fig. 2.4. The effect of poling is elaborated in Fig. 2.5 for PZT in sensors and actuators. Moreover, the hysteresis plot of polarization vs. electrical field and plot of strain vs. electric field are also shown in Fig. 2.5.

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Figure 2.3. Overview of PEH mechanism; (a) piezoelectric effect with 33 and 31

strain-charge coupling; (b) polarization process; (c) bi-morph PEH mechanism

Figure 2.4. Electromechanical coupling in PZT effect.

2.1.1 Direct effect of piezoelectricity

Piezoelectricity is defined as the mechanical stress along the polarization direction to generate electric voltage. By the application of this mechanical stress σ₃ along the x3 axis it undergoes a cell deformation, an elongation along the stress application

axis as represented in Fig. 2.6. The titanium atom by changing its position increases polarization which becomes Pr+ ∆P . For small values of σ3:

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Figure 2.5. PZT for sensing and actuating. (a) top: hysteresis plot of P-E; bottom: S-E

field plot, (b) PZT before and after poling (c) applied voltage and polarity directions voltage are in same direction, (d) applied voltage and polarity voltage directions are in different direction, (e) generated voltage and polarity voltage are in different direction during compression, (f) generated voltage and polarity voltage are in different directions during tension.

Figure 2.6. Undeformed and deformed configuration under the effect of σ3.

Where d₃₃1 is a positive constant. Similarly it is possible to define all the other constants. Considering all the applied stresses the direct piezoelectric coupling

1

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matrix d is defined as:            ∆P₁ ∆P2 ∆P₃            =       0 0 0 0 d15 0 0 0 0 d15 0 0 d31 d31 d33 0 0 0                                      σ1 σ2 σ3 σ4 σ5 σ6                                (2.2)

The Equation 2.2 can be written as:

∆P = dσ (2.3)

2.1.2 Converse effect of piezoelectricity

Piezoelectricity is defined as application of an electric field along polarization axis which determines the displacement of positive charges in the same direction for a positive electric field applied. There is an elongation along the x3 axis and a

compression along x₁ and x₂ axes. For converse effect of piezoelectricity:

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2.1 The piezoelectric effect 13                                ε1 ε2 ε3 ε4 ε5 ε6                                =              0 0 d13 0 0 d13 0 0 d33 0 d42 0 d51 0 0                         E1 E2 E3            (2.4) ε = dE (2.5)

2.1.3 Constitutive equations of piezoelectricity

By definition, the induced polarization in a dielectric material from the electric field vector is

Pi= χijEj (2.6)

In which χ_ij(F m−1) is the tensor of the dielectric susceptibility of the material2_.

The charge density induced in the material is obtained by introducing the dielectric displacement vector D_i[Cm−2], which is defined as

Di= 0Ei+ Pi (2.7)

Where, 0 represents vacuum dielectric permeability. Introducing the dielectric

material permeability

ij = 0δij+ χij (2.8)

Substituting (2.6) and (2.8) in (2.7):

Di= 0Ei+ Pi = (0δij + χij)Ej = ijEj (2.9)

The presence of an additional polarization ∆P due to the direct piezoelectric effect can be seen as an additional term of D which can therefore be written as,

D = (σ)E + d σ (2.10)

2

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Where, 

σ is the dielectric matrix with constant or zero stress. It is also possible

to write the deformation vector by adding a term due to piezoelectric effect of the material:

ε = Q(E)σ + dTE (2.11)

Where, the material compliance matrix is indicated by Q.

Unifying (2.10) and (2.11) the constitutive matrix of a continuous piezoelectric is:

     D      =    ε(σ) _d dT Q(E)         E σ      (2.12) Or in a tensorial way: Di = εσijEj+ dijkσjk ij = F_ijhkE σhk+ dtijkEi (2.13)

It is important to introduce a coefficient that links the mechanical stress applied to the electric field produced [76, 79, 80]. This together with d allows to characterize the piezoelectric material applicability in order to take full advantage of its properties. Defining the coefficient g as:

gij = dijij (2.14) Follows: Ei = gijσj = dij ij σj (2.15)

Once the available electric field is established the best use of material in question can be identified according to d or g values. An high d is optimal for the actuation applications whereas a high g maximizes sensor performance of the material.

2.1.4 Open- and closed-circuit condition under compression

For compression in PZT, two types of configurations can be used for the phenomenon of energy harvesting i.e., Open- and closed-circuit configurations. In the open-circuit condition, the charges are disposed and are not free to move towards electrodes which results in no electric flux. Whereas in closed-circuit configuration, both the electrodes have the same electric potential resulting in no electric field. The phenomenon of

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Open- and closed-circuit conditions of PZT harvesters under compression is presented in Fig. 2.8.

Figure 2.8. Open- and closed-circuit configurations of a PZT under compression. Where,

D is electric flux and E is electric field.

2.1.5 Energy coupling coefficients

It is important to evaluate the efficiency of the PEH. It can be calculated by evaluating the ratio of electrical work W₁ to the mechanical work W₂ with respect to electrical work i.e., W1+ W2. The path of mechanical loading considering different

circuit configurations is expressed in Fig. 2.9 and the path of electrical loading considering different circuit configurations is shown in Fig. 2.10. The coupling coefficient K₃₃ for open circuit condition can be expressed as:

K33= s

W1

W1+ W2

(2.16)

For mechanical loading, by taking in to account Fig. 2.9 the energy coupling coefficient can be expressed as:

K33= s FE 33− F33D F₃₃E = s d2 33 F₃₃Eσ₃₃ (2.17)

K31, the energy coupling coefficient can be expressed as:

K31= s

d2 31

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Figure 2.9. The path of mechanical loading considering different circuit configurations.

Figure 2.10. The path of electrical loading considering different circuit configurations.

2.2 PZT for Energy Harvesting

PEH can be a very suitable alternative source of energy as compared to traditional one and they have vital applications in the area of automotive, aerospace and defense sector due to micro scale devices [32]. They can operate various sensors and actuators. Moreover they have very compact structure and become an environmental friendly source of energy generation. That is why piezoelectric patches are widely used in various optical devices [103], space missions [57], biomedical devices [130],

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2.2 PZT for Energy Harvesting 17

mechanical/civil structures [14] and precise measurement tools [123]. PEH are preferred because of their flexibility, low electromagnetic interference, high positioning and high torque to volume ratio [64, 115]. PEH is composed of three major steps. Step one, the piezoelectric patch which converts environmental input i.e., FSI, bio-mechanic vibrations, etc into alternating current. Step two, the storage unit i.e., a super capacitor or a battery to store the charge generated by PEH. Step three, the modulating circuit responsible for the conversion of AC into DC [131]. Step two can be ignored in order to utilize energy directly from PEH [60, 131].

2.2.1 Need of PZT

Design robustness of self-powered electronic devices has made themselves quite demanding these days [77]. As batteries are heavy in weight and are expensive to maintain so there is a need of mechanism that can power the nano or micro-electronics by absorbing structural energy [32]. There are many mechanisms for energy transformation i.e., electromagnetic, electromechanical, and fluid-structure interaction system [94]. Among these mechanisms, electromechanical system plays a vital role because of its voltage dependent actuation [61]. Many researchers are working on such techniques that drive electronic circuits in electromechanical systems [35, 60]. The piezoelectric material is mostly used to harvest electrical energy by absorbing mechanical energy (mechanical vibrations) from the surrounding [62]. The need for PEH arises because batteries have less operational life as compared to the circuit. In many conditions, replacement or maintenance of battery is impossible and cost of battery is very much costly for the operation. There occurs maintenance issues because of the battery usage while their operation in harsh environment i.e., high-altitude places, cold or hot climate, icy or snowy regions, as these conditions lead to damage battery life. In such cases, even recycling of the batteries is a problem as well as an environmental hazard (specially lead ion batteries) too. As per these conditions, one of the solutions is to use different external energy sources i.e., PEH so that durability of the battery can be increased [116]. These issues lead to utilization of PEH as an additional energy source which can assist electronic devices directly or indirectly via batteries by increasing their operational life time [32].

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2.2.2 FEA for PZT

The finite element method for piezoelectric materials can be accessed in many commercially available softwares to perform the static and dynamic analysis. In these software packages, structural elements of the material are coupled with electrical properties of the piezoelectric material. FEA is a very attractive tool for modeling and simulation of electromagnetic and electromechanical sensors and acquisitions [37]. AH Allik introduced the first numerical simulation of piezoelectric material in the 1970 [18]. The output power of PEH plates was also predicted with an electromechanical coupled FE model. The goal was to power small electronic devices by transforming the waste vibration energy available in atmosphere into electrical energy. The advantage of FEA over analytical results is that mechanical stress variations and electrical field calculations of complex geometries of the material can be more readily calculated. FEA was performed to calculate the stress and electric field distributions under static loads and under any electrical frequency. Thus the impact of material geometry can be evaluated and improved without the need to make and test various materials [36]. In addition, FEA can likely predict lifetime expectations without any need to conduct time consuming tests if the significant electrical and mechanical parameters are obtained.

2.2.3 Energy generation

The energy generated by the piezoelectric harvesters can be utilized either to power microelectronic systems or can be stored in batteries. Many researchers have emphasized on self-powered sensors and actuators rather than relying on batteries. As batteries are heavy in weight and their maintenance is expensive and sometimes impossible i.e., suborbital missions. So this gap opens up a chance for piezoelectric harvesters to develop self-powered portable electronic devices. The piezoelectric based devices are light in weight and are capable of self dependent actuation [122]. The efficiency of harvested energy for direct effect of piezoelectricity can be analyzed by calculating difference of mechanical energy converted in to electrical energy and loss in energy conversion [17, 104, 105, 113]. This harvesting mechanism is dependent on the medium of interaction [44, 71, 99, 117] i.e., transformation of kinetic energy

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2.2 PZT for Energy Harvesting 19

Figure 2.11. Typical cantilever piezoelectric harvesters; where V is the voltage that can

be harvest from this PEH.

to PZT transducer [100, 163, 164]. These medium may be mechanical vibrations [54, 158, 159], fluid structure interaction [82, 125] and thermal interaction [12, 72]. Cantilever beam is the most widely used configuration for energy harvesting technique in electromechanical systems. This configuration has wide applications in the field of sensors, actuators, structural health monitoring systems and energy harvesters [57]. In this configuration piezoelectric material is attached to the beam, which is fixed from one end. If the number of PZTp is one then it is known as unimorph cantilever harvester. The number of PZTp defines the configuration of piezoelectricity i.e., bi-morph or tri-morph. When these harvesters are exposed to vibrations, mechanical energy is absorbed by PZTp and transformed in to useful electrical energy [58]. The mechanism of typical cantilever piezoelectric harvesters is represented in Fig. 2.11.

The piezoelectric cantilever harvesters follow the direct effect of piezoelectricity i.e., mechanical input results in electrical output. When these harvesters are subjected to mechanical loading i.e., vibrations, the charge is collected on the surface of PZTp causing the voltage difference between the thickness of the patch. There is a need to improve sensitivity or energy efficiency of PEH mechanism in order to get higher voltage for same excitation [95]. The overall mechanism for piezoelectric energy harvesting is represented in Fig. 2.12. Moreover, the application of PEH in wireless sensors is elaborated in Fig. 2.13.

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Figure 2.12. Overall mechanism for piezoelectric energy harvesting; where K is the modal

stiffness, C is the modal damping, Ceis the external capacitance and V is the output voltage generated.

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21

3 |

Aeroelastic Problems

A system where interaction between fluid flow and a structure is considered, it is important to monitor overall fluid dynamics built by the whole system, this type of system is named as an aeroelastic system. This chapter discusses the issues inclined by such aeroelastic systems.

3.1 General issues on aeroelasticity

To have a full comprehension of the aeroelastic problem, it is compulsory to under-stand what aeroelasticity actually means. It is a field which studies the interaction of elastic structures deformation due to flow and their corresponding aerodynamic forces. Observing the Collar’s triangle we come to know the multidisciplinary nature of aeroelasticity as shown is Fig. 3.1. Generally there are two types of phenomenon Static aeroelastic phenomenon: which is outside the triangle and Dynamic aeroelastic phenomenon: which lies within the triangle, involving all the forces.

The problem arises by the mutual interaction between an elastic structure and a flow. It is basically governed by variation of parameters which determine flow conditions. Some of them are unperturbed speed or the Mach number. These parameters influence the system solution stability and are responsible for bringing the equilibrium to any unstable condition. Before proceeding with the aeroelastic analysis, some basic issues of aeroelasticity are addressed in this chapter.

3.1.1 Basic definitions on stability

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• A system is asymptotically stable if and only if, all the eigenvalues (or poles considered in the transfer function) are such that Re(λi) < 0.

• A system is stable if and only if, all the eigenvalues (or poles considered in the transfer function) are such that Re(λ_i)_{6 0. If there is an eigenvalue (or a} pole) with a null real part, this necessarily has geometric multiplicity equal to one.

It is also necessary to introduce a simple characterization of aeroelastic instability types:

• Static instability: divergence. It involves only steady aerodynamic and elastic forces. Mathematically it occurs when eigenvalue crosses the imaginary axis with its only real part, which in turn becomes positive as shown in Fig. 3.2(a).

• Dynamical instability: flutter. It involves inertial, aerodynamical and elastic forces. Mathematically it is described as a phenomenon when a couple of complex conjugate eigenvalues cross the imaginary axis acquiring positive real part as presented in Fig. 3.2(b).

For the better understanding about the aeroelastic stability analysis, it is impor-tant to highlight issues related to bifurcation problems.

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3.2 Bifurcation problems 23

(a) Sample of divergence condition. (b) Sample of flutter condition.

Figure 3.2. Polar diagram for divergence and flutter condition.

3.2 Bifurcation problems

Starting from an ordinary differential equation system depending on a k-dimensional parameter µ:

˙

x = F_µ_{(x) = F (µ, x); x ∈ R}n, µ ∈ Rk (3.1)

For the equilibrium state:

Fµ= 0 (3.2)

Is possible to describe the equilibria varying the parameters µ with a smooth function using Implicit Function Theorem for as long as the relation xeq(µ) remains

bi-univocal. When its no more bi-univocal, different solutions coincide for different parameter sets. A lot of different mechanisms can characterize a bifurcation process but in this work, it will only be considered the dynamical one called Hopf bifurcation. It is also clear that the statical or dynamical nature of this process depends on the spectrum behaviour of the linearised system.

3.2.1 Hopf bifurcation

A Hopf bifurcation occurs when a fixed-point solution becomes linearly unstable beyond the critical value (µ_c) of a parameter µ. It could be of two types depending on the parameter value:

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equilib-rium stability is conditioned with an unstable LCO that defines the stability margin, and after the bifurcation point only an unstable equilibrium exists as represented in Fig. 3.3.

(a) Subcritical Hopf bifurcation diagram.

(b) Subcritical Hopf bifurcation with turning point. , equi-librium solution; •, initial condition.

Figure 3.3. Subcritical Hopf bifurcation.

• It is called supercritical when µ > µ_c; after the bifurcation point there is an unstable equilibrium point and a stable LCO as represented in Fig. 3.4.

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3.3 Aeroelastic problem formulation 25

(a) Supercritical Hopf bifurcation diagram.

(b) Supercritical Hopf bifurcation with turning point. , equilibrium solution; •, initial condition.

Figure 3.4. Supercritical Hopf bifurcation.

3.3 Aeroelastic problem formulation

Considering what has been said so far, we obtain the fundamental equation that describes the aeroelastic system, written in the domain of Laplace.

(s2M + K)˜q = qDE(s; U∞, M∞)˜q + ˜f (3.3)

Where, M , K and E are the mass, stiffness and GAF matrices respectively. ˜q and ˜f are the Laplace transformed vectors of displacement and external force respectively.

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q_D is the dynamic pressure, U∞ the flow velocity, M∞ is the Mach number, these

parameters describe the flow. The stability problem can be solved using iterative methods, like the p-k method to the associated equation:

[s2M + K − qDE] ˜w = 0 (3.4)

3.3.1 p-k method

The aeroelastic analysis carried out using SOL 145 implemented in MSC Nastran is based on the p-k method. Assuming that the aerodynamics calculated for the pure harmonic motion is a good approximation for weakly damped ones. It is possible to affirm for a complex frequency p = α + jk ' jk: neglecting the real part, i.e. the damping whereas transfer aerodynamic matrix is calculable. In this method furthermore aerodynamic forces are considered as additional terms of stiffness and damping. Under this hypothesis the Equation (3.4) could be written as:

[s2M + sD + K − qD[ER(k; M∞) + jEI(k; M∞)]] ˜w = 0 (3.5)

The GAF matrix is evaluated as a function defined in the Laplace’s adimensional sub-domain k := =(s)l/U∞ (i.e. Fourier’s domain): the approximation is as better

as closer to the imaginary axis.

Introducing the velocity vector ˜v := s ˜w and defining:

A%∞,U∞,M∞;k :=    0 I −M−1hK − 1₂%∞U∞2ER(k; M∞) i −M−1hD − 1₂%∞U∞bEI(k;M_k ∞) i    (3.6) and ˜ u :=      ˜ w ˜ v      (3.7)

We obtain the following first order problem:

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3.3 Aeroelastic problem formulation 27

The implicit dependence on the complex eigenvalue s through k makes it necessary to recur to an iterative method.

Iterative method

1. Flight physics parameters U∞, %∞ and M∞ are fixed in this way A matrix

depends only on k

2. Divergence condition

A is evaluated for k = 0 and the divergence condition occurs if the calculated eigenvalue is zero.

3. Different from zero poles calculation

First structural mode frequency ω1 is used in order to estimate a first attempt

value of k

k₁(0):= ω1b U∞

It is possible to calculate the 2N problem eigenvalues s(0)_i . Among this eigenvalues s(0)₁ , the one with imaginary part closest to ω1 is the new estimation

value of the reduced frequency

k(1)₁ := s

(0) 1 b

U∞

Defining ε the required accuracy of the solution, if the condition

k (1) 1 − k (0) 1 < ε

is satisfied, s(0)₁ is the first aeroelastic pole, otherwise a new evaluation of the E matrix is required to calculate a new group of eigenvalues, iterating the proceedings until convergence condition is satisfied.

Once the first pole is detected, the others could be calculated by the same iteration method using the associated structural frequency as first attempt value. This method could also be used for varying the other parameters for the stability scenario.

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3.4 Critical modes

Once the eigen problem is resolved, it is also possible to represent the critical response of the system in time in terms of Lagrangian variables.

qcr(t) = Nm X

n=1

cnesntw(n) (3.9)

Where Nm is the number of modes used in the stability analysis and cn is

constant, which depends on the initial condition of problem. Due to the critic condition, the system steady response is dominated by critical modes.

q_cr(t) ' ceiωcrt_w

cr+ C.C. ≡ 2c [wRcrcos (ωcrt) − wIcrsin (ωcrt)] (3.10)

In terms of physical displacements:

ucr(ξα, t) ' N X i=1 qcri(t)φi(ξ α₎ =2c N X i=1

[w_R_cricos(ω_crt) − wI_crisin(ωcrt)]φi(ξα)

=2c N X i=1 |wcri| cos(ωcrt + w ∠ cri)φi(ξ α₎ (3.11)

The (3.11) shows how free aeroelastic response is in critical conditions, it is a combination of φn(ψα) functions (i.e. the normal modes of the structures) multiplied

simply by harmonic functions with frequency equal to ω_cr.

3.5 Finite Elements Method for flag-flutter

This section is devoted to numerical aspects. The numerical model analysed for the aeroelastic stability of the system is implemented in SOL 145, solver of software MSC Nastran, in which it is considered as a linear unsteady behaviour for both the structure and aerodynamics.

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3.5 Finite Elements Method for flag-flutter 29

3.5.1 Numerical model

The aeroelastic analysis carried out is based on the use of both structural and aerodynamic finite elements. Aeroelastic analysis of the composite swept wings for the aircraft has been carried out in the literature by Polli et al. [126, 127]. A finite element is defined by its geometry, while its physical displacement, u(x; t) is described by the functions of form φ(n)(x), associated with the related nodes or grids: u(x, t) ∼= N X n φ(n)(x)qn(t) (3.12)

In particular, these are composed of panels, aligned in the direction of steady (or harmonically unsteady) flow, on which act the aerodynamic forces. The mathematical model used also known as Doublets lattice method (or simply DLM), is based on the linearised theory of aerodynamic potential. On each panel there is a distribution of doublets equivalent to the pressure drop between the upper and lower surfaces of the panel itself. Each doublet has an unknown intensity and is placed on the line 1/4 of the panel chord. Every element has a collocation point in the middle of the line placed at 3/4 of the chord in which the normal wash, w = ∇φ · n is calculated. This is equal to the elastic oscillation of the structure itself as shown in Fig. 3.5.

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Writing the equations for all the panels we obtain a system of algebraic equations, which allows the calculation of doublets intensity and consequently the pressure jump across the lifting surface. It is possible to calculate the aerodynamic forces acting on the surface itself. Since the structural grids do not usually coincide with the aerodynamic ones, it is necessary to make an interpolation between them as presented in Fig. 3.6.

Figure 3.6. Sample of aerodynamic and structural model for a wing.

This step is fundamental in order to generate the same deformation of structure by two systems of distinct forces acting on the aerodynamic and structural grid points. This is achieved majorly by equal amount of virtual work done by two force systems. Unsteady aerodynamic forces are described by the GAF (Generalized Aerodynamic Force) matrix which relates the forces acting on the structure to its deformations. This GAF matrix is namely the [E] matrix, which was introduced in Equation. (3.3). In the next chapter, methodologies on piezoelectric energy harvesting by aeroelastic means are discussed in detail.

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31

4 |

Methodologies on

Piezoelec-tric Energy Harvesting by

Aeroelastic Means

This chapter is based on the results published by the author in a peer reviewed journal:

• Elahi, H., Eugeni, M., Gaudenzi, P. (2018). A review on mechanisms for piezoelectric-based energy harvesters. Energies, 11(7), 1850.

In this chapter various methodologies on piezoelectric energy harvesting by aeroelastic means are discussed in detail. From last few decades, fluid-surface interaction based PEH is of great interest for many researchers [46, 144, 172].The PZT harvester is exposed to a fluid flow for energy harvesting to drive electronic devices or for storage [169]. In mega structures, aerodynamic phenomena i.e., vortex induced vibrations (VIV) [73], flutter [46, 155] and galloping [166] may result in excessive vibrations that can damage or even destroy the structures [2]. The overall scheme for FSI based PEH is expressed in Fig. 4.1 [98]. These aeroelastic mechanisms of energy harvesting are described in this chapter and it is highlighted that how energy can be generated from the occurance of these phenomenon.

4.1 Vortex Induced Vibrations based PEH

From last few years, many researchers are working on development of piezoelectric energy harvester that can absorb energy from environment i.e., VIV and transform

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Figure 4.1. Overall scheme for FSI based PEH (For more details see Ref. [98]).

them in to useful electrical energy [42, 43, 112, 117]. The most commonly used process is, attach the PZT patch to the the fixed end of elastic cantilever beam and attach the circular cylinder to the free end of beam as shown in Fig. 4.2 [42, 43]. The PZT patch is shocked with external resistance as it has vital effect on the amplitude of oscillations, coefficient of lift and output energy generated by the harvester [2]. Mehmood et al. performed numerical simulations by attaching a PZT patch to the transverse DOF by taking into account the Reynolds numbers (Re) ranging from 96-118 and resistance from 500Ω to 5MΩ [112]. Re are selected on the basis that it can accumulate pre-synchronization, synchronization, and post-synchronization regimes as shown in Fig. 4.3 [112]. Schematics of this proposed mechanism is expressed in Fig. 4.4 [112]; where, U∞ is the free stream velocity, C is the structural

damping, K is the structural stiffness and R is the electric resistance applied. Dai et al. demonstrated based on Galerkin discretization that the first four modes are necessary to evaluate the performance of harvester correctly [43]. Moreover, both linear and non-linear analysis have been performed in order to analyze the efficiency of the system and it was observed that when the flow is at synchronization region, electromechanical damping associated to it decreases causing increase in harvested energy [42]. Franzini et al. carried out the sensitivity study that can influence the dimensionless quantities characterizing the PZT harvesters and proposes 50% increase in the efficiency for particular reduced frequency [73]. The enhancement of

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4.1 Vortex Induced Vibrations based PEH 33

Figure 4.2. Mechanism of VIV based PEH.

Figure 4.3. Bifurcation diagram.

the voltage generated is carried out by optimizing the parameters based on genetic algorithm [23]. The efficiency of energy harvested by VIV can be analyzed for 1 DOF as represented in Equ. 4.1 [73].

ηel,x= 4π2 U3 r (θ∗)2 f∗ σ2,x σ1,x (m∗+ Cα)v2x (4.1)

Where, θ∗ = θx/θy, f∗ = wn,x/wn,x, ηel,xis the dimensionless electric power harvested

at cross-wise and in-line harvesters, U_r is the reduced velocity, σ_2,x, σ_1,x is the dimensionless quantities related to the piezoelectric harvesters, θx, θy are the

Electro-mechanical coupling constants, w_n,x,w_n,y are the cylinder’s natural frequencies, m∗ and Cα are the mass parameter and potential added mass coefficient respectively.

Franzini et al. carried out numerical solution of PEH from VIV via dynamics of a rigid cylinder, that is mounted at the end of elastic beam attached with piezoelctric patch. They considered wake oscillator model as hydrodynamic load, coupled the solid and electric oscillators by linear constitutive equations. The dynamics of the FSI based PEH system were investigated for the influence of an additional structural DOF i.e., in-line oscillations. Moreover, they demonstrated that the efficiency of PEH can be increased upto 50% at any particular reduced frequency [73].