school of industrial and information engineering master of … · 2018-11-13 · figure 2.1 gpeh...

POLITECNICO DI MILANO

School of Industrial and Information Engineering Master of Science in Mechanical Engineering

Comparison between different configurations of piezoelectric energy

harvesting from galloping instability

Supervisor: Prof. Gisella TOMASINI

Co-Supervisor: Ing. Stefano GIAPPINO

Master’s Thesis of:

Giancarlo MARIANI Id. Number 863526

Academic Year 2017 – 2018

CONTENTS

i

CONTENTS

CONTENTS ........................................................................................................................ i

FIGURES ......................................................................................................................... iv

TABLES ......................................................................................................................... viii

LIST OF SYMBOLS ............................................................................................................ x

ABSTRACT .......................................................................................................................2

SOMMARIO .....................................................................................................................3

INTRODUCTION ...............................................................................................................4

1 STATE OF THE ART ...................................................................................................6

1.1 Wind energy harvesting ..................................................................................6

1.1.1 Wind micro-turbine .................................................................................6

1.1.2 Aerodynamic instability ...........................................................................8

1.2 Galloping piezoelectric energy harvester ......................................................12

1.2.1 GPEH configurations ..............................................................................15

2 MATHEMATICAL MODEL .......................................................................................17

2.1 GPEH reference layout ..................................................................................17

2.2 Piezoelectric beam characteristics equations................................................19

2.3 Galloping force method.................................................................................20

2.3.1 Influence of bluff body shape ................................................................22

2.4 Distributed parameter model of a GPEH .......................................................23

2.4.1 Assumptions ..........................................................................................23

2.4.2 Boundary conditions and stationary solutions ......................................24

2.4.3 Kinetic energy ........................................................................................33

2.4.4 Elastic energy ........................................................................................35

2.4.5 Charge force work .................................................................................38

2.4.6 Aerodynamic force work .......................................................................39

2.4.7 Dissipative Term ....................................................................................43

CONTENTS

ii

2.4.8 Conclusive equations .............................................................................45

2.4.9 Numerical simulation ............................................................................46

3 SENSITIVITY ANALISYS OF GPEH ............................................................................47

3.1 Effect on onset velocity and vortex shedding ................................................47

3.2 Reference case ..............................................................................................50

3.3 Variation of parameters ................................................................................53

3.3.1 Variation second region length .............................................................54

3.3.2 Variation beam width ............................................................................57

3.3.3 Variation beam thickness ......................................................................60

3.3.4 Variation beam material ........................................................................62

3.3.5 Variation damping ratio.........................................................................65

3.3.6 Variation bluff body density ..................................................................68

3.3.7 Variation bluff body length ....................................................................70

3.3.8 Variation bluff body side .......................................................................73

3.3.9 Variation piezo patch.............................................................................75

3.3.10 Variation number of beam ....................................................................78

3.3.11 Resistance effect ...................................................................................79

3.3.12 Longitudinal vs transversal ....................................................................82

3.4 Final conclusions and prototype design ........................................................83

3.4.1 Longitudinal prototype ..........................................................................85

3.4.2 Transversal prototype ...........................................................................86

4 EXPERIMENTAL VALIDATIONS ...............................................................................89

4.1 Experimental setup .......................................................................................89

4.1.1 Motion-imposed tests components ......................................................89

4.1.2 Wind tunnel setup .................................................................................93

4.2 Identification of the modal parameters ........................................................94

4.2.1 Transfer function ...................................................................................95

4.2.2 Estimation of modal parameters .........................................................103

4.2.3 Damping ratio ......................................................................................110

4.2.4 Resistance effect .................................................................................113

4.3 Validation of aero-electromechanical model ..............................................114

CONTENTS

iii

4.3.1 Longitudinal configuration ..................................................................115

4.3.2 Transversal configuration ....................................................................121

4.3.3 Comparison between longitudinal and transversal model ..................125

CONCLUSION ...............................................................................................................129

BIBLIOGRAPHY ............................................................................................................132

FIGURES

iv

FIGURES

Figure 1.1 Example of micro-turbine. ..............................................................................7

Figure 1.2 A prototype of micro-turbine proposed by Pray. ...........................................7

Figure 1.3 Representation of the vortex shedding phenomenon....................................8

Figure 1.4 (a) the scheme of the prototype provides by Pobering [5] and (b) the

prototype of Akaydin [6] ...............................................................................................10

Figure 1.5 Wind tunnel experiment of the flutter energy harvesting prototype provide

by Chawdhury and Morgental [11] ...............................................................................10

Figure 1.6 Schematic of wake galloping phenomenon [12] ..........................................11

Figure 1.7 Scheme of galloping piezoelectric energy harvesting ...................................12

Figure 1.8GPEH prototypes. (a) transversal model [15], (b) 2DOF system [29] and (c)

longitudinal model [14] .................................................................................................14

Figure 1.9 GPEH configurations scheme. (a) transversal one beam, (b) transversal two

beams and (c) longitudinal ............................................................................................16

Figure 2.1 GPEH layout for a longitudinal configuration .................................17

Figure 2.2GPEH layout for a transversal configuration with two cantilever beams ......17

Figure 2.3 Convention for the positive direction of stress, strain and bending moment

for a beam with two PZT attached to a metal beam .....................................................19

Figure 2.4 Representation of aerodynamic forces per unit length acting on a square-

section bluff body .........................................................................................................20

Figure 2.5 Transversal Model ........................................................................................25

Figure 2.6 Longitudinal Model ......................................................................................25

Figure 2.7 Angle Attack .................................................................................................39

Figure 2.8 Angle α transversal Model ...........................................................................40

Figure 2.9 Example of decay .........................................................................................43

Figure 2.10 Exact Vs Approximate Formulation ............................................................44

Figure 3.1 Reference case CAD......................................................................................51

Figure 3.2 Modal shape reference case ........................................................................52

Figure 3.3 Modal shape variation second region ..........................................................55

Figure 3.4 Effect of variation length second region.......................................................56

Figure 3.5 Time response variation second region. (a) Voltage Vs wind, (b) tip

displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind ..............57

Figure 3.6 Modal shape width beam variation ..............................................................58

Figure 3.7 Effect of variation width beam .....................................................................59

Figure 3.8 Time response variation width beam. (a) Voltage Vs wind, (b) tip


FIGURES

v

Figure 3.9 Modal shape thickness beam variation ........................................................61

Figure 3.10 Effect of thickness beam variation .............................................................61

Figure 3.11 Time response thickness beam variation. (a) Voltage Vs wind, (b) tip


Figure 3.12 Modal shape material beam variation .......................................................63

Figure 3.13 Time response material beam variation. (a) Voltage Vs wind, (b) tip


Figure 3.14 Modal shape damping ratio variation ........................................................66

Figure 3.15 Effect of damping ratio variation ................................................................66

Figure 3.16 Time response material beam variation. (a) Voltage Vs wind, (b) tip


Figure 3.17 Modal shape bluff body density variation ..................................................68

Figure 3.18 Effect of bluff body density variation .........................................................69

Figure 3.19 Time response bluff body density variation. (a) Voltage Vs wind, (b) tip


Figure 3.20 Modal shape bluff body length variation ...................................................71

Figure 3.21 Effect of bluff body length variation ...........................................................71

Figure 3.22 Time response bluff body length variation. (a) Voltage Vs wind, (b) tip


Figure 3.23 Modal shape bluff body side variation .......................................................73

Figure 3.24 Effect of bluff body side variation ..............................................................74

Figure 3.25 Time response bluff body side variation. (a) Voltage Vs wind, (b) tip


Figure 3.26 Modal shape piezo patch variation ............................................................76

Figure 3.27 Time response piezo patch variation. (a) Voltage Vs wind, (b) tip


Figure 3.28 Modal shape number of beam variation ....................................................78

Figure 3.29 Time response number of beam variation. (a) Voltage Vs wind, (b) tip


Figure 3.30 Modal shape resistance variation...............................................................80

Figure 3.31 Effect resistance variation ..........................................................................80

Figure 3.32 Time response number of beam variation. (a) Voltage Vs wind, (b) tip


Figure 3.33 Time response number of beam variation. (a) tip displacement Vs

resistance, (b) power Vs resistance ...............................................................................82

Figure 3.34 Modal shape different prototype ...............................................................83

Figure 3.35 Dimension of piezo patch PPA1011 ............................................................85

Figure 3.36 Different longitudinal prototypes. (a) A40, (b) A30, (c) A20, (d) B20, (e) B30

and (f) B40.....................................................................................................................87

Figure 3.37 Different transversal prototypes. (a) C5 and (b) D9 ...................................88

Figure 4.1 (a) experimental setup scheme of a shaker and (b) shaker setup for

prototype B20 ...............................................................................................................90

FIGURES

vi

Figure 4.2 Experimental devices. (a) c-DAQ module and (b) electromechanical shaker

......................................................................................................................................92

Figure 4.3 (a) wind tunnel with prototype, (b) instrumentation used in wind tunnel ...94

Figure 4.4 Scheme of measurement acquisition points for longitudinal prototype ......96

Figure 4.5 Prototype scheme and shaker force .............................................................97

Figure 4.6 Comparison between laser-accelerometer sensors for prototype A30 ........99

Figure 4.7 Motion-imposed tests: comparison between experimental and numerical

results in term of FRF prototype A20 with R = 100 kΩ. (a) A Tip/A Constraint, (b) V

PPA/A Constraint and (c) V PPA/L Tip .........................................................................101


results in term of FRF prototype A20 with R = 10 kΩ. (a) A Tip/A Constraint, (b) V



results in term of FRF prototype A20 with R = 1 MΩ. (a) A Tip/A Constraint, (b) V PPA/A

Constraint and (c) V PPA/L Tip ....................................................................................102


results in term of FRF prototype C5 with R = 100 kΩ. (a) A Tip/A Constraint, (b) V PPA/A

Constraint and (c) V PPA/L Tip ....................................................................................102


results in term of FRF prototype D9 with R = 100 kΩ. (a) A Tip/A Constraint, (b) V


Figure 4.12 Example of concentrated mass for (a) prototype A20 and (b) prototype A40

....................................................................................................................................106

Figure 4.13 Analytical modal shape prototype A30 ....................................................107

Figure 4.14 Piezo patch deformation for models B .....................................................108

Figure 4.15 Modal shape comparison between analytical and experimental results for

(a) A20, ........................................................................................................................110

Figure 4.16 Example of decay on prototype A30 ........................................................111

Figure 4.17 Interpolation for damping ratio (a) A20, (b) B30, (c) A40, (d) B40, (e) C5 and

(f) D9 ...........................................................................................................................112

Figure 4.18 Numerical simulation to show the effect of the resistance load on the

power ..........................................................................................................................113

Figure 4.19 Power response prototype A20 ................................................................114

Figure 4.20 Wind tunnel tests: comparison between analytical and experimental

results for prototype A40 (a) tip displacement vs wind, (b) PPA displacement vs wind

and (c) power vs wind .................................................................................................116







FIGURES

vii


results for prototype B40 (a) tip displacement vs wind, (b) PPA displacement vs wind








Figure 4.26 Different transient velocities for prototype B30 ......................................120


results for prototype C5 (a) tip displacement vs wind, (b) PPA displacement vs wind



results for prototype C5 (a) tip displacement vs wind, (b) PPA displacement vs wind



results for prototype D9 (a) tip displacement vs wind, (b) PPA displacement vs wind


Figure 4.30 Moment study on prototype D9 ...............................................................125

Figure 4.31 Comparison between prototypes A40, D9 and B40 in terms of power

recovered in function of wind speed ..........................................................................127

Figure 4.32 Focused on PZT patch deformation from analytical modal shape for

prototypes C5 and D5 .................................................................................................127

Figure 5.0.1 A non-perfect interaction between galloping and vortex shedding for

prototype B40 in term of power in function of the wind speed..................................130

TABLES

viii

TABLES

Table 1.1 Summary of different small-scale windmills ....................................................8

Table 1.2 Effect of the shape section on Strouhal number .............................................9

Table 1.3 Comparison between different energy harvesting technology from

aerodynamic instabilities ..............................................................................................12

Table 1.4 Cross sections of Yang's work [16] .................................................................13

Table 1.5 Comparison of GPEH prototypes ...................................................................15

Table 2.1 List of variables used in the modal approach ................................................18

Table 2.2 Aerodynamic coefficients for different bluff body shape ..............................23

Table 3.1 Parameters Reference Case ...........................................................................50

Table 3.2 Natural frequency reference case .................................................................51

Table 3.3 Modal mass reference case ...........................................................................51

Table 3.4 Aerodynamic result reference case ...............................................................52

Table 3.5 Results variation second region .....................................................................55

Table 3.6 Results variation width beam ........................................................................58

Table 3.7 Results thickness beam variation ..................................................................61

Table 3.8 Results material beam variation ....................................................................63

Table 3.9 Results damping ratio variation .....................................................................66

Table 3.10 Results bluff body density variation ............................................................69

Table 3.11 Results bluff body length variation ..............................................................71

Table 3.12 Results bluff body side variation..................................................................74

Table 3.13 Results piezo patch variation .......................................................................76

Table 3.14 Results number of beam variation ..............................................................78

Table 3.15 Results resistance variation .........................................................................80

Table 3.16 Results different prototype .........................................................................82

Table 3.17 Constant parameters ...................................................................................84

Table 3.18 Parameters piezo patch PPA1011 ................................................................85

Table 4.1 Characteristics of lasers .................................................................................91

Table 4.2 Characteristics of accelerometer ...................................................................92

Table 4.3 List of channels used during the experimental tests .....................................92

Table 4.4 Properties of wind tunnel ..............................................................................93

Table 4.5 Characteristics of the prototypes used ..........................................................95

Table 4.6 Comparison between numerical and experimental natural frequency .......104

Table 4.7 Comparison between numerical and experimental modal mass .................106

Table 4.8 Experimental piezo patch deformation ratio longitudinal prototypes ........110

Table 4.9 Damping ratio value for prototypes at 0.5 mm displacement of PPA ..........113

Table 4.10 Power obtained at the same amplitude for model A20 .............................113

TABLES

ix

Table 4.11 Analytical velocities and speed ratio for longitudinal prototypes..............115

Table 4.12 Analytical velocities and speed ratio for transversal prototypes ...............121

Table 4.13 Natural frequency variation due to wind contribution for prototype D9 ..124

Table 4.14 Comparison between experimental results for prototypes with the same

bluff body volume in both configurations ...................................................................126

Table 5.0.1 Efficiency for prototypes with the same bluff body volume and density .130

Table 5.0.2 Power recovered for prototypes with the same bluff body volume and

density ........................................................................................................................131

Table 5.0.3 cut-in speed for prototypes with the same bluff body volume ................131

LIST OF SYMBOLS

x

LIST OF SYMBOLS

𝑋𝑖,𝑗

For the quantity named “X” the first subscript indicates the material used (“p”: piezoelectric layer, “s”: supporting beam, “b”: bluff body, “e”: extension for the bluff body). The second subscript, if present, indicates the region in which the quantity “X” is considered

𝜙𝑗(𝑖) i-th mode of vibration for the region j

𝑉𝑖,𝑗 Volume of the material I for the region j

𝑤𝑖 Width of layer i

𝜌𝑖 Density of layer i

𝑡𝑖 Thickness of layer i

𝑚𝑖 Mass per unit length of layer i

𝐸𝑖 Young modulus of layer i

𝑆𝑖 Strain of layer i

𝐿𝑗 Length of layer i

𝑑𝑖 Distance of the layer i from the neutral axis of the region j

𝐷 Side with of the bluff body

𝑓𝑛 Natural frequency of the system in Hz

𝜔𝑛 Natural frequency of the system in rad/s

h Non-dimensional damping of the system

𝑟𝑎𝑒𝑟𝑜 Aerodynamic damping

LIST OF SYMBOLS

xi

𝑟𝑚𝑒𝑐ℎ Structural damping

𝑛𝑠 Strouhal number

𝑑𝑖 Distance of the layer I from the neutral axis of the region j

𝜌𝑏𝑙𝑢𝑓𝑓 Bluff body density

𝜌𝑎𝑖𝑟 Air density


𝑈𝑔 Galloping speed

𝑈𝑣 Vortex shedding speed

U Wind speed

𝑉𝑠 Voltage output

𝐶𝑠 Capacitance of the PZT patch

𝜒𝑠 Electromechanical coupling

ABSTRACT

2

ABSTRACT

The purpose of this thesis is to compare two different configurations (longitudinal and

transversal) of energy harvesting systems using piezoelectric patches, based on the

vibrations induced by galloping aerodynamic instability. Through the piezoelectric

effect, the prototype is studied to convert the electric energy from mechanical energy

of the vibrations induced by galloping aerodynamic instability.

Both energy harvesting configurations systems were designed using a numerical

model and experimentally tested with wind tunnel tests.

The first part of the work is focused on the development of a non-linear one degree of

freedom model, which describes the electromechanical system, consisting of the

deformable beam, where the piezoelectric patch is mounted, connected to the bluff

body, and its interaction with the aerodynamic force. The model is able to calculate

the amplitude of oscillation reached by the bluff body and the power recovered, in

function of the wind speed and to the load resistance.

In the second part of the research, the developed model is used to perform a

sensitivity analysis to the main parameters of the system and for the design of the

prototypes to be tested.

The last part of the work shows the experimental results obtained, useful to validate

the numerical model and to evaluate the obtainable performance, in term of the

power recovered, for each prototype studied in function of wind speed.

For longitudinal model the maximum power obtained at the velocity 6.14 m/s is 1.69

mW, with an efficiency, compared to the ideal power introduced by the wind, of

0.07%. Transversal model, with the same volume, recovers 7.1 mW at V = 7.92 m/s

which it corresponds to an efficiency of 0.15%.

SOMMARIO

3

SOMMARIO

L’obiettivo di questo lavoro di tesi è il confronto tra due configurazioni diverse

(longitudinale e trasversale) di sistemi a recupero di energia mediante lamine

piezoelettriche, basati sulle vibrazioni indotte dall’instabilità aerodinamica da galoppo.

Tramite l’effetto piezoelettrico, il prototipo è studiato per convertire in energia

elettrica l’energia meccanica delle vibrazioni indotte dall’instabilità aerodinamica del

galoppo.

Le due configurazioni di sistemi energy harvesting sono state progettate mediante un

modello numerico e testate sperimentalmente con prove in galleria del vento.

La prima parte del lavoro è focalizzata sullo sviluppo di un modello non lineare a un

grado di libertà, che descrive il sistema elettromeccanico, costituito dalla trave

deformabile collegata al corpo tozzo sulla quale è montata la lamina piezoelettrica, e

la sua interazione con la forza aerodinamica. Il modello è in grado di calcolare

l’ampiezza di oscillazione raggiunta dal corpo tozzo e la potenza recuperata, in

funzione della velocità del vento e della resistenza elettrica applicata.

Nella seconda parte della ricerca, il modello sviluppato è utilizzato al fine di eseguire

un’analisi di sensibilità ai principali parametri del sistema e per la progettazione dei

prototipi da testare.

L’ultima parte del lavoro mostra i risultati sperimentali ottenuti, utili sia per validare il

modello numerico sia per valutare le prestazioni effettivamente ottenibili, in termini di

potenza recuperata, dai diversi prototipi studiati al variare della velocità.

Con il modello longitudinale, la massima potenza introdotta alla velocità di 6.14 m/s è

1.69 mW, corrispondente ad un’efficienza, rispetto alla potenza ideale introdotta dal

vento, pari a 0.07%. Il modello in configurazione trasversale, a parità di volume,

recupera invece 7.1 mW a V = 7.92 m/s che corrisponde al un’efficienza di 0.15%.

INTRODUCTION

4

INTRODUCTION

The research on energy harvesters from different energy sources is becoming more

and more important during these last years thanks to the possibility to power

miniaturised devices, as wireless sensor nodes, able to measure, elaborate and

wirelessly communicate data for diagnostic/monitoring purposes in a variety of fields.

Sensor nodes, according to the different applications, can work continuously but with

very low consumption. The target of this work is to study an energy harvesting system

able to provide energy for a long time, without interruption to a sensor node.

In particular, within the different energy sources adopted for energy harvester

(mechanical energy, solar energy, etc.) the purpose of this work is to recover energy

from the wind.

There are two main research fields that explore energy conversion method from wind:

• centimetre-scale wind turbine: this solution has high efficiency and power

density for a relatively high range of flow speeds.;

• aerodynamic instabilities: in this case the instabilities due to the wind action

induce vibrations on a bluff body and allow to recover energy mainly by the

piezoelectric effect. This solution allows to recover lower energy with respect

to the cm-scale turbines, but it is possible to use these devices starting from a

lower set of wind speeds. Instabilities used for this purpose are: flutter

instability, galloping and vortex shedding.

In particular, this work will focus only on the aerodynamic instabilities generated by

galloping.

Studies about galloping instability started many years ago to understand the unstable

motion of the transmission lines, due to the presence of ice on cables.

In recent years some researches have highlighted the possibility to use this instability

to generate vibrations on a structure.

In this work we are interested in comparing two different types of Galloping Piezo

Electrical Harvesters (GPEH) prototypes, with longitudinal and transversal

configurations.

The goal of this work is to study which is the best configuration in terms of generated

power, starting galloping velocity and maximum displacement of the bluff body.

A numerical model of both the configurations was developed and used to design

different prototypes. The sensitivity analysis performed by the numerical model is

INTRODUCTION

5

then verified, for the most important parameters, by means of wind tunnel

experimental tests. Moreover, the interaction between vortex shedding and galloping

phenomena was analysed and the parameters influencing the coupling between them

are highlighted.

In detail, the present work is divided into four chapters and a conclusive section.

Chapter 1 is the state of the art about harvesting systems using wind energy source.

We focus our attention on energy harvesting from galloping instability, where an

overview is proposed about experimental results and mathematical models used in

literature.

In Chapter 2 the mathematical model is presented. It is composed by an

electromechanical system forced by the aerodynamic force, . The electro-mechanical

model is represented by a distributed parameter model and a modal approach while

the aerodynamic force is modelled by using the quasi-steady theory.

Once described the mathematical model, the sensitivity analysis is presented in

Chapter 3. This chapter is an explanation about all effects we have to consider and at

the end we provide different solutions to use in the experimental tests.

All prototypes designed are tested in wind tunnel and the results are provided in the

chapter 4.

The experimental setup of the different tests carried out as well as the results are

presented. A comparison between the experimental and numerical results is

performed.

The last section is about the conclusions that we can draw from the studies and

experimental results obtained. Here it is possible to establish if there are real

advantages to use one configuration respect than the other. Furthermore it is

provided a comparison between our results with literature results. We can also

consider the accuracy of the mathematical model.

STATE OF THE ART

6

1 STATE OF THE ART

In this chapter we want to provide all information about energy harvesting from wind,

in particular harvesting systems from galloping aerodynamic instability.

At the start we want to give some information on different type of methods exist for

wind energy harvesting, with micro turbines and aerodynamic instability.

At the end we analyse the existing research, to study which type of configurations

exist and what we can do to increase knowledge about this phenomenon, in particular

analysing two different configurations of GPEH.

1.1 Wind energy harvesting

In this paragraph it is our interest to see which the two main fields for wind energy

harvesting are:

• Wind micro-turbines;

• Vibration provide aerodynamic instability.

1.1.1 Wind micro-turbine

The concept of this device it is to create a rotational motion of a fan, that is connected

to an electrical motor, in order to convert mechanical energy, the rotation induced by

the wind, to electrical energy, that it is possible to restore on a battery.

A microturbine works in the same way to a turbine, the same concept for wind or

water application, but the big different is the dimension of the device, because in this

case we work with a little device, for sensor node.

This application it is the most famous for wind energy harvesting, because we can

reach high value of density power, respect than others possible applications.

The maximum power generated from a micro-turbine can be expressed using the Betz

formula:

𝑃 =

1

2𝜌𝑎𝑖𝑟𝐴𝐶𝐵𝑈3 (1.1)

Where A is the sectional turbine area, U the wind speed and 𝐶𝐵 is equal to 0.59.

We can find two disadvantages for this application. The first is the high cut-in velocity.

In fact, respect than aerodynamic instabilities, the start velocity is higher, and this is a

problem if we want to adopt this solution for all cases where we want to work with

low velocities.

STATE OF THE ART

7

The second problem is about the critical velocity, in fact if we reach high velocity,

there is the possibility to have some mechanical damages on the structure of the

turbine, with the risk of the break after some cycles.

We also have to consider that, like for turbine, also for micro-scale device we need to

provide a good maintenance, like lubrification, and this can be a problem if we want to

adopt this solution for autonomous system.

Of course, we can also find some advantages for this application. The first is the

maximum power, as we just said before, and the second is about the dimension of the

device, because we can obtain a really compact design, that is require for micro

system sensorial node.

Figure 1.1 Example of micro-turbine.

In recent years some researches proposed some solution in order to reduce the cut-in

wind speed. One example was the work conducted by Pray [1], that proposed a

piezoelectric windmill to harvest energy from low speed wind velocity.

With his second work [2], he obtained a prototype with a cut-in wind speed of 2.1 m/s

and a cut-off wind speed of 5.4 m/s. It is possible to see that in this case the cut-in

velocity is really low, but there is a big limitation on the range velocities, because the

device can work only until 5.4 m/s, after this we can have damages.

Figure 1.2 A prototype of micro-turbine proposed by Pray.

STATE OF THE ART

8

Author Transduction Wind speed at

max power (m/s)

Power density per

volume (𝐦𝐖

𝐜𝐦𝟐)

Howey [3] Electromagnetic 10 0.535

Priya [2] Piezoelectric 4.5 0.0663

Chen [4] Piezoelectric 5.4 0.0134 Table 1.1 Summary of different small-scale windmills

1.1.2 Aerodynamic instability

The second method to harvester energy from wind is to use the instabilities create

from aerodynamic force. These phenomena are studied in precedent for them

destructive effect on the structure, but in the last years some research studied the

possibility to use them to obtain available energy.

To extract energy it is possible to use different method:

• Magnetostrictive conversion;

• Electrostatic conversion;

• Electromagnetic conversion;

• Piezoelectric conversion.

In our work we’re going to use the piezoelectric technology in order to harvester

energy from vibrations.

1.1.2.1 Vortex shedding based energy harvesters

Vortex shedding is created to an alternative separation of the boundary layer from

opposite parts of the body that generate a force, whit an alternate positive and

negative pressure. Typically, this phenomenon is present in the smooth flow.

Figure 1.3 Representation of the vortex shedding phenomenon

The frequency of vortex shedding is given by:

𝑓𝑠 = 𝑆𝑡

𝑈

𝐷 (1.2)

STATE OF THE ART

9

Where 𝑆𝑡 is the Strouhal number and it depend from the geometrical characteristics of

the body, U is the mean flow speed and D is the body dimension. In Table 1.2 it is

possible to see the effect of the shape section on the Strouhal number.

Table 1.2 Effect of the shape section on Strouhal number

The importance of the phenomenon of vortex shedding is the lock-in effect. In fact

exist a certain flow speed where shedding frequency becomes equal to natural

frequency of the body. In this case the system is forced in resonant and we can

observe an increment of oscillation of the body.

The vibration oscillation of the system can reach value near the dimension of the body

and is limited by the natural damping of the system itself.

The flow speed at which this happen is:

𝑈𝑣 = 𝑓𝑛

𝐷

𝑛𝑠 (1.3)

Where 𝑓𝑛 is the natural frequency of the structure. The lock-n condition doesn’t occur

only at that velocity, but in a range of speeds, called lock-in range. For this reason it is

possible to see why the vortex shedding instability can be used only in a small range of

velocity in order to obtain energy.

Exist two different strategies in literature:

• A bluff body go in instability by the effect of vortex shedding, near to the

natural frequency of the body itself, and this creates vibration, VIV (Vortex

Induced Vibration);

• The instability is generated from a body to a first body that is clamp to a

second system, and so this is forced to oscillate. It is clear that in this case the

vibration is created from the first body, and this effect is called WIV (Wake

Induced Vibration).

First researches were addressed for water flow application instead air flow. With the

work of Pobering and Schwesinger [5] there was the first prototype tested in both

flow, wind and water. This prototype was a VIV model, so it was composed with a bluff

body fixed on a piezoelectric bimorph cantilever.

A second example of VIV model was proposed by Akaydin et al. [6], and the interesting

part of them works was the study on the mutual coupling behaviors between

aerodynamics, structural vibration and electrical response.

STATE OF THE ART

10

(a) (b)

Figure 1.4 (a) the scheme of the prototype provides by Pobering [5] and (b) the prototype of Akaydin [6]

1.1.2.2 Flutter based energy harvesters

Flutter instability is found in two degree of freedom system, when the natural

frequency of translational mode and rotational natural frequency become equal and

so we have a coupling of motions. This instability depends from the system stiffness

and in order to study the stability of the system we have to determine the sign of the

extra-diagonal terms of the stiffness matrix.

It is possible to observe this instability also on some bridges, an example of destructive

case is the Takoma Bridge, in Tacoma.

In literature there are a lot of works about flutter instability, with different possible

solutions of design:

• The first case is the prototype where the flutter is obtained from an axial flow

conditions, as the prototype proposed by Dunnmon [7] and Doarè [8];

• A prototype designed in cross-flow conditions, an example is the work of De

Marqui Jr [9];

• With flapping airfoil designs, like the model of Erturk [10].

Figure 1.5 Wind tunnel experiment of the flutter energy harvesting prototype provide by Chawdhury and

Morgental [11]

https://www.researchgate.net/profile/Samir_Chawdhury

STATE OF THE ART

11

1.1.2.3 Wake galloping based energy harvesters

In this case the instability in the result of the interaction between two structures, in

fact the motion on the one can create an instability on the others. For example, in

parallel cylinders the wake galloping instability occur depends on the position of the

first cylinder and it is related with the distance between the two bodies.

Figure 1.6 Schematic of wake galloping phenomenon [12]

Jung and Lee [12] studied this effect in order to extract power from this instability.

They developed a system with two paralleled cylinders, where the power is extracted

from the leeward cylinder, that it oscillates thanks to the wake generate from the

wind ward cylinder. In order to generate power, they used an electromagnetic

transduction, but also a piezoelectric energy harvesting could be use. They concluded

that exist a proper distance to optimize the generation of wake galloping.

A second prototype was presented by Abdelkefi et al. [13]. They studied the effect to

use a device with a circular cylinder in the windward and a square galloping cylinder

leeward. They found that besides the dimension and the distance between the two

bodies, also resistance load should be taken in account to design a wake galloping

energy harvesting device.

1.1.2.4 Galloping based energy harvesters

Galloping is a single degree of freedom instability in which the fluid-flow force

generates an oscillation of the structure, in the perpendicular direction respect to the

flow. Often this instability is the cause of the transmission line break, in presence of

ice on the cable, because in normal case, with a cylindrical shape, galloping doesn’t

occur. This instability happens after a certain velocity, onset speed, and it depends

from the stiffness component of the system. Above the onset velocity, the system

starts to increase the amplitude of the oscillation, with the same frequency of the

structure. Generally, these prototypes are composed by a cantilever beam with a

transducer, for example a piezoelectric patch, and a bluff body, and of course at the

galloping velocity the bluff body starts to oscillate and this create a deformation on

the beam, with the consequent generation of power from the transducer.

STATE OF THE ART

12

Figure 1.7 Scheme of galloping piezoelectric energy harvesting

Author Transduction Wind speed at

max power (m/s)

Power density per

volume (𝐦𝐖

𝐜𝐦𝟑)

Galloping

Sirohi and Mahadik [14]

4.7 0.0134

Zhao et al. [15] Yang et al. [16]

8.0 0.0346

Zhao [17] 5.0 0.0162

Zhao and Yang [18] 8.0 0.0455

Zhao [19] 5.0 0.0274

Ewere and Wang [20]

8.0 0.0512

Flutter

Bryant and Garcia [21]

7.9 0.0072

Erturk [10] 9.3 0.0023

Kwon [22] 4 0.333

Wake Galloping Jung and Lee [12] 4.5 0.111

Abdelkefi [13] 3.05 0.00057

VIV Song et al. [23] 0.35 0.004

Weinstein et al [24]

5.5 0.011

Table 1.3 Comparison between different energy harvesting technology from aerodynamic instabilities

1.2 Galloping piezoelectric energy harvester

In literature it is possible to find researches about the galloping energy harvesting

because, respect than vortex shedding, this source presents interesting advantages

due to the self-excited and self-limiting characteristics. A first important aspect of this

instability is the work range, in fact we can obtain a large wind speeds range.

A first research about galloping, using for energy harvesting, was provided by Barrero-

Gil [25] modelling the system, a 1DOF model, as a simple mass-spring-damper system.

In this work only a numerical solution was given, without a real design for a prototype.

In order to represent the aerodynamic force, it was used a cube polynomial in

according whit the quasi-steady hypothesis.

STATE OF THE ART

13

A continuum of the work was given by Sorribes-Palmer and Sanz-Andreas [26]. They

obtained the aerodynamic coefficient curve directly from experimental, avoid using

coefficients by literature. In this way It is possible to avoid some problems with the

fitting curve, and also to obtain a correct value of the curve 𝐶𝑧 for a specific prototype.

The first study on the structure was provide by Sirohi and Mahadik. In fact in them

work we can find a study on:

• A first use of two cantilever beams, instead only one, with a bluff body at the

end, free to move. For this system was adopted the Rayleigh-Ritz method in

order to model the coupled electromechanical effect [27];

• A piezoelectric composite beam with a bluff body, D-shape cross section,

connected in parallel with the beam. In this case we can see two articular

things, the first is the configuration of the prototype, because a longitudinal

model, and the second is the new shape of the bluff body. An interesting

result obtained in this research was that the natural frequency of the

oscillation was equal to the natural frequency of the cantilever, with a second

observation about the power, because it was observed that it increased with

the wind speed [14].

The influence of the bluff body shape was studied by Zhao et al [15] and Yang et al

[16]. In particular they studied different type of shape, comparing experimental results

for all prototypes. They studied also the effect of the aspect ratio for the rectangular

shape. In Table 1.4 it is possible to see all shapes.

Section Shape

Dimension h x d (mm)

40 x 40 40 x 60 40 x 26.7 40 (side) 40 (dia.)

Table 1.4 Cross sections of Yang's work [16]

From these researches was established that the best solution, in term of power, was

the square section.

Using a 1DOF model it was possible to predict the power response of the system,

where the aerodynamic force was modelled with a seventh order polynomial.

Another study about different bluff body shape was conducted by Abdelkefi et al. [28].

They used different shape including square, two isosceles triangles (with different

base angle) and a D-section and they concluded that D-section was recommended for

high wind speeds and isosceles triangle for small wind speed. They also discovered

that aerodynamic coefficients are sensitive to the flow condition, laminar or turbulent.

STATE OF THE ART

14

In order to optimize a galloping energy harvesting device, Zhao et al. [29] provided a

comparison of different modelling methods, trying to find advantages and

disadvantages for all.

In particular they studied a 1 DOF model, a single mode Euler-Bernoulli distributed

parameter model and a multimode Euler-Bernoulli distributed parameter.

At the end they concluded that the best way in order to represent the aerodynamic

force is the parameter model, instead in order to give a prediction of the cut.in wind

speed, it was better to a 1 DOF model.

A problem about energy harvesting based on galloping instability is that the amplitude

of oscillation can increase too much during the operational wind speeds range, with

the risk of failure of the device.

In order to avoid this problem Zhao et al. [17] tried a new model that it combined an

electromechanical system with a magnetic effect. This is a 2DOF model and using the

electricity It is possible to change the intensity of the magnetic field, with a reduction

of the amplitude oscillation at high speed.

A second method for this problem was provided by Ewere et al. [30]. They introduced

a bump stop on the system in order to reduce the fatigue problem and also to reduce

the amplitude oscillation with the galloping instability.

(a) (b)

(c)

Figure 1.8GPEH prototypes. (a) transversal model [15], (b) 2DOF system [29] and (c) longitudinal model [14]

STATE OF THE ART

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Author Configuration Bluff body shape

Cut-in wind speed [m/s]

Bluff body dimensions

[cm]

Cantilever dimensions

[cm]

Sorohi and Mahadik

[14] Longitudinal D-shape 2.5

3 in dia., 23.5 in length

9 x 3.8 x 0.0635

Zhao et al. [15]

Yang et al. [16]

Transversal Square 2.5 4 x 4 x 15 15 x 3.8 x

0.06

Zhao et al. [17]

Transversal Square 1.0 4 x 4 x 15 17.2 x 6.6 x

0.06

Zhao and Yang [18]

Transversal Square 2.0 4 x 4 x 15 8 x 4.5 x 0.5

Zhao et al. [19]

Transversal Square 5.0 2 x 2 x 10 8.5 x 2 x

0.03

Ewere and Wang [20]

Transversal Square 8.0 5 x 5 x10 22.8 x 4 x

0.04

Table 1.5 Comparison of GPEH prototypes

1.2.1 GPEH configurations

As we just said in the previous paragraph, a galloping energy harvesting is composed

by three main elements: a structure of support that is the beam, and it possible to

have different material for this element, a bluff body, with a precise dimension and

shape, and at the end a transducer, in our case we are interested only in piezoelectric

patch.

Of course, the prototype is also composed by a clamp mechanism and an electrical

circuity, and both these parameters can change the response of the system.

The PZT patches are mounted on the cantilever beam, one per side, in this way we

have two patches per each beam. These two patches can be connected in parallel or in

series. In our case we use the series connection. The output voltage is dissipated on

the resistive load, another parameter that can influence the result in term of power.

STATE OF THE ART

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We just saw that exist two different configurations for galloping energy harvesting in

literature:

• The first configuration is the transversal model, called also T-shape design. In

this case the bluff body is mounted on the tip of the beam orthogonally to the

beam. In this configuration we can divide in two cases, with one or two

support beams. In both these models all parameters remain the same but

using two beams we can avoid the moment acting on bluff body axis. Another

possible advantage is that using this second model we use four PZT patches,

and for this reason It is reasonable to think that the power is higher than with

one beam;

• The second configuration is the longitudinal model. Here the bluff body is

mounted following the beam direction. In this configuration there is only one

beam and there is the possibility to reduce the length of the beam, the region

without piezoelectric patch, in order to reduce the dimension of the total

prototype.

(a) (b)

(c)

Figure 1.9 GPEH configurations scheme. (a) transversal one beam, (b) transversal two beams and (c) longitudinal

MATHEMATICAL MODEL

17

2 MATHEMATICAL MODEL

In this second our aim is to provide a complete mathematical model for both

porotypes we want to compare at the end of this work.

We start from models we just described in the first chapter, using a general dimension

for the system, because correct values we’re going to be obtain in the third chapter,

after the sensitivity analysis.

Before the modelling of prototypes, we want to provide a little explanation of quasi-

steady theory and assumptions of distributed parameter model, because they need

for continuation.

2.1 GPEH reference layout

For application we want to study, we need to consider both configurations of energy

harvester, longitudinal one and transversal. In literature there isn’t a comparison

between longitudinal and transversal prototypes and with this research our purpose is

to provide this comparison.

Figure 2.1 GPEH layout for a longitudinal configuration

Figure 2.2GPEH layout for a transversal configuration with two cantilever beams

MATHEMATICAL MODEL

18

Longitudinal layout is composed by a supporting beam structured with two

piezoelectric patches attached at each side of the beam ad a bluff body attached at

the end of the beam. Bluff body can be designed with different material, in order to

modify the weight, and also different dimensions, to modify the aerodynamic forces.

The origin of the structure is placed on the clamped end, with the X is along the

longitudinal direction and Y axis is in the oscillation direction. In Table 2.1 it is possible

to find every variable used in the modal approach.

For transversal prototype we have to consider two supporting beams, instead one,

with four piezoelectric patches attached. In this second configuration the bluff body is

attached perpendicular to the end of both beams.

𝑋𝑖,𝑗

For the quantity named “X” the first subscript indicates the material used (“p”: piezoelectric layer, “s”: supporting beam, “b”: bluff body, “e”: extension for the bluff body). The second subscript, if present, indicates the region in which the quantity “X” is considered

𝜙𝑗(𝑖) i-th mode of vibration for the region j

𝑉𝑖,𝑗 Volume of the material I for the region j

𝑤𝑖 Width of layer i

𝜌𝑖 Density of layer i

𝑡𝑖 Thickness of layer i

𝑚𝑖 Mass per unit length of layer i

𝐸𝑖 Young modulus of layer i


𝐿𝑗 Length of layer i

𝑑𝑖 Distance of the layer i from the neutral axis of the region j

𝐷 Side with of the bluff body

Table 2.1 List of variables used in the modal approach

MATHEMATICAL MODEL

19

2.2 Piezoelectric beam characteristics equations

For our layout we consider a unimorph piezoelectric patch. This patch consists of a

single piezoelectric layer and it is combined with different substrate able to guarantee

suitable elastic properties for the patch. As we just said before, at the final prototype

we use one PZT patch per each side of the beam, or beams in the transversal

configuration. In our configuration we consider the piezo patch attached at the middle

of the beam.

When the beam considered is subjected to deformation, In Y direction, stress and

strain for each layer vary according to the bending moment direction.

Figure 2.3 Convention for the positive direction of stress, strain and bending moment for a beam with two

PZT attached to a metal beam

Piezoelectric layer produces a charge when mechanically strained. This effect is used

to convert mechanical energy to electrical energy. In our case we use this effect to

convert the energy introduce by aerodynamic forces, galloping instability, to electrical

power. It is possible to describe this process with the constitutive equations for

piezoelectric materials:

𝑆𝑖𝑗 = 𝑆𝐸𝑖𝑗𝑘𝑙𝑇𝑘𝑙 + 𝑑𝑘𝑖𝑗𝐸𝑘 (2.4)

𝐷𝑗 = 𝑑𝑖𝑘𝑙𝑇𝑘𝑙 + 휀𝑇𝑖𝑘𝐸𝑘 (2.5)

Where the mechanical strain S and stress T tensors are introduced, as well as the

electric displacement D and the electric field E (the spatial directions are represented

with the subscripts i, j, k, l). The other coefficients are the mechanical compliance at

constant electric field 𝑆𝐸𝑖𝑗𝑘𝑙, the permittivity at constant stress 휀𝑇

𝑖𝑘 and the coupling

matrix 𝑑𝑘𝑖𝑗 . These equations compose a 3x6 matrix, made up with all piezoelectric

coefficients along different directions.

For our application the PZT patch operates only in the 3-1 mode, bending mode,

meaning that the deformation in applied only at the direction 1, while the voltage is

harvested along direction 3. Formulas (2.4) and (2.5) can be simply as follow:

MATHEMATICAL MODEL

20

𝑆1 = 𝑆𝐸11𝑇1 + 𝑑31𝐸3 (2.6)

𝐷1 = 𝑑31𝑇1 + 휀𝑇33𝐸3 (2.7)

And we can rearrange (2.6) and (2.7) in matrix form:

[𝑆1

𝐷3] = [

𝑆𝐸11 𝑑31

𝑑13 휀𝑇33

] [𝑇1

𝐸3] (2.8)

2.3 Galloping force method

The GPEH system can be seen as a single degree of freedom body, where the whole

body is modelled as a concentrated mass, a spring and a damper, that represent the

structural stiffness and damping capability. The aerodynamic force can be modelled as

a constant flow, that hits the body surface in a normal direction. In Figure 2.4 is

showed a square section bluff body subjected to aerodynamic forces. We need to

consider relative velocity that is generated by the interaction between wind velocity

and bluff body velocity in vertical direction.

Figure 2.4 Representation of aerodynamic forces per unit length acting on a square-section bluff body

The total aerodynamic load results along vertical direction is equal to:

𝐹𝑦 = 𝐹𝐿 cos 𝛼 − 𝐹𝐷 sin 𝛼 (2.9)

where 𝛼 = tan−1 (�̇�

𝑈) .

For small displacements of the angle of attack 𝛼, around the zero value and under the

quasi-steady aerodynamic hypothesis, we can adopt the following simplifications:

• The relative fluid speed can be approximated as 𝑉𝑟 = √𝑈2 + �̇�2 ≈ 𝑈2

• The angle of attack may be simplified as 𝛼 ≈ (�̇�

𝑈)

MATHEMATICAL MODEL

21

• It is possible to use a first-degree Maclaurin approximation for sine and cosine:

sin 𝛼 ≈ 𝛼 and cos 𝛼 = 1

• Aerodynamic coefficients depend upon the angle of attack, so it is possible to

us the Taylor’s formula as:

𝐶𝐿 = 𝐶𝐿|𝛼=0 +

𝜕𝐶𝐿

𝜕𝛼|𝛼=0

𝛼 (2.10)

𝐶𝐷 = 𝐶𝐷|𝛼=0 +

𝜕𝐶𝐷

𝜕𝛼|𝛼=0

𝛼 (2.11)

It can be observed that for a bluff body we have:

• 𝐶𝐿|𝛼=0 = 0;

• 𝜕𝐶𝐷

𝜕𝛼|𝛼=0

= 0.

Aerodynamic force along y direction can be rewrite as:

𝐹𝑦 =

1

2𝜌𝐷𝐿𝑈2(𝐶𝐿 cos 𝛼 − 𝐶𝐷 sin 𝛼) (2.12)

Considering the previous simplifications, it is possible to rewrite the force as follow:

𝐹𝑦 =

1

2𝜌𝐷𝐿𝑈2 (

𝜕𝐶𝐿

𝜕𝛼|𝛼=0

− 𝐶𝐷|𝛼=0)�̇�

𝑈 (2.13)

In this linearized expression for the aerodynamic force, it is possible to express this

value as an equivalent aerodynamic damping action. It can be expressed as:

𝑟𝑎𝑒𝑟𝑜 = −

1

2𝜌𝐷𝐿𝑈 (

𝜕𝐶𝐿

𝜕𝛼|𝛼=0

− 𝐶𝐷|𝛼=0) (2.14)

Note that this contribution term can also assume a negative value and in this case if

the pair of complex conjugate poles, that represent the solution of the motion

equation, are moved on the right-hand side of the imaginary plane, the equilibrium of

the system will become unstable and the system will start to increase the oscillation

amplitude.

In order to evaluate the condition of galloping instability acting on the system, we can

observe when the total damping is negative:

𝑟𝑎𝑒𝑟𝑜 + 𝑟𝑚𝑒𝑐ℎ < 0 (2.15)

MATHEMATICAL MODEL

22

If we introduce (2.14) inside the formula (2.15) it is possible to obtaing the galloping

velocity:

𝑈𝑔 >𝑟𝑚𝑒𝑐ℎ

−12

𝜌𝐷𝐿𝑈 (𝜕𝐶𝐿

𝜕𝛼|𝛼=0

− 𝐶𝐷|𝛼=0)

(2.16)

At the end it is possible to show another way to express the aerodynamic force acting

on the vertical direction. In fact we can interpolate the aerodynamic coefficient with a

higher order polynomial in tan (α):

𝐶𝑦 = ∑𝛼𝑖 (

�̇�

𝑈)𝑖𝑁

𝑖=1

(2.17)

Where coefficients 𝛼𝑖 are estimated from experiments, and 𝐶𝑦 takes in account drag

and lift effects. Replacing (2.17) inside (2.13) we obtain:

𝐹𝑦 =

1

2𝜌𝐷𝐿𝑈2 (𝑎1

�̇�

𝑈+ 𝑎3 (

�̇�

𝑈)3

) (2.18)

This final expression is really important for us because we will use it inside our

mathematical model, whit aerodynamic coefficients from literature.

2.3.1 Influence of bluff body shape

As we have just seen, aerodynamic coefficients 𝑎1 and 𝑎3 are obtained from

experimental tests. In literature it is possible to find different values of coefficients,

depending from the cross-section geometry of the bluff body.

It is important to remember that these values are valid only for small angle of attack,

and them won’t still valid in big angles range. Reynolds number influences these

coefficients, and for this reason in literature it is possible to find different value for

different conditions. In Table 2.2 are showed different aerodynamic coefficients.

Cross-section 𝒂𝟏 𝒂𝟑 Re Source

Square 2.3 -18 33000 - 66000 Parkinson and

Smith [31]

Isosceles triangle (delta = 30°)

2.9 -6.2 105 Alonso and

Meseguer [32]

Isosceles triangle (delta = 30°)

0.79 -0.19 105 Luo et al. [33]

MATHEMATICAL MODEL

23

D-section 1.9 6.7 104 Novak and Tanak

[34] Table 2.2 Aerodynamic coefficients for different bluff body shape

2.4 Distributed parameter model of a GPEH

Distributed parameter model is derived by the Hamilton’s principle using Eulero-

Lagrange formulation, which starts with the definition of various energy forms. The

formulation is the same as the one presented in the works by Du Toit [35] and

Preumont [36] and is given by the following integral:

∫ [𝛿(𝐸𝑘 − 𝐸𝑝 + 𝑊𝑒) + 𝛿𝐿𝑎𝑒𝑟𝑜 + 𝛿𝐿𝑒𝑙]𝑑𝑡 = 0

𝑡2

𝑡1

(2.19)

where 𝑡1 and 𝑡2 are the initial and final times, 𝐸𝑘 is the kinetic energy, 𝐸𝑝 the elastic

energy, 𝑊𝑒 is the electrical energy, 𝛿𝐿𝑎𝑒𝑟𝑜 is the work done by aerodynamic force and

𝛿𝐿𝑒𝑙 is the work done by the electric charge.

2.4.1 Assumptions

We can describe the dynamic of the GPEH using the bending vibrations of beam of

continuous systems.

This model is valid under the following assumptions:

1) Small displacements;

2) Linear elastic constitutive law. Under this assumption we can consider

isotropic relationship between strain and stress. This is valid in particular for

metallic materials and we can consider true until the plastic yield limit is

achieved;

3) Constant section and homogeneous material. For this reason, if we have

different sections, we can split all of them in different regions and use the

beam theory for each section. Another important concept is that all the

physical properties don’t depend from the position or orientation inside the

beam;

4) Damping effects are neglected;

5) No forces are applied, except at the boundaries;

6) The beam isn’t subject to tension/compression;

7) The centre of gravity of the beam is on the principal axis, for this reason the

bending motion is decoupled from torsional vibrations;

8) Plane bending of the beam is studied, assuming that the plane where the bending motion occurs contains one of the principal axes of the beam section. It is easy to verify that, under this assumption, the plane bending motion

MATHEMATICAL MODEL

24

studied is totally de-coupled from a second component of bending, occurring in an orthogonal plane which contains the other principal axis of the beam section;

9) The beam is thin, the ratio between the height and length is really small, less than 1:

10) If h is the high of the section and l is the length, we must have: ℎ

𝑙≪ 1 (2.20)

2.4.2 Boundary conditions and stationary solutions

It is possible to define the modal equation, for every j-mode, as a combination of time

and displacement. The equation is defined for each i-region of the structure:

𝑦𝑖(𝑥, 𝑡) = 𝛼𝑖𝑗(𝑥)𝛽𝑗(𝑡) (2.21)

t is referred to the time and x is referred to the distance along the structure direction,

as it is possible to see in Figure 2.5 and Figure 2.6. Instead α and β are two coefficients

that we can define as:

𝛼𝑖𝑗(𝑥) = 𝐴𝑖𝑗 cos 𝛾𝑥 + 𝐵𝑖𝑗 sin 𝛾𝑥 + 𝐶𝑖𝑗 cosh 𝛾𝑥 + 𝐷𝑖𝑗 sinh 𝛾𝑥 (2.22)

𝛽𝑗(𝑡) = 𝐸𝑗 cos𝜔𝑗𝑡 + 𝐹𝑗 sin𝜔𝑗𝑡 (2.23)

Total displacement is calculated using every single displacement for each mode, so It is

necessary to sum every contribute:

𝑦𝑖(𝑥, 𝑡) = ∑𝛼𝑖𝑗(𝑥)𝛽𝑗(𝑡)

∞

𝑗=1

(2.24)

For the present problem it is reasonable to accept an approximation and limit the

summation of infinite modes to the first n modes. Depending on the range of

frequencies of interest for the problem, the model may be reduced to a three-modes

or even single-mode model. For a galloping energy harvester, many authors have

reported experimentally that the only significant mode is the first one [29] [27] [18].

Each domain can be considered, according to its properties, as deformable or rigid

elements. For deformable sections we can use the following expression:

𝑦𝑖(𝑥, 𝑡) = (𝐴𝑖 cos 𝛾𝑥 + 𝐵𝑖 sin 𝛾𝑥 + 𝐶𝑖 cosh 𝛾𝑥 + 𝐷𝑖 sinh 𝛾𝑥)𝑒𝑖𝜔𝑛𝑡 (2.25)

MATHEMATICAL MODEL

25

Instead rigid sections can be described as:

𝑦𝑖(𝑥, 𝑡) = (𝐴𝑖𝑥 + 𝐵𝑖)𝑒𝑖𝜔𝑛𝑡 (2.26)

In this project we are going to study two different layouts of prototype and for both of

them we’re going to define the mathematical model. In Figure 2.5 and Figure 2.6 are

showed both scheme configurations.

Figure 2.5 Transversal Model

Figure 2.6 Longitudinal Model

For longitudinal model the third region is composed by bluff body and cantilever

beam, because this is the part where the bluff body attaches on the beam.

In both of models there are regions whit more the one layer, with different stiffness

and density, I can use an approximation in order to compute an equivalent inertia of

that region:

MATHEMATICAL MODEL

26

𝐽𝑒𝑞 =

1

𝐸𝑝∑𝐸𝑖 (

𝑤𝑖𝑡𝑖3

12+ 𝑤𝑖𝑡𝑖𝑑𝑖

2)

𝑛

𝑖=1

(2.27)

Where 𝐸𝑝 is the Young’s modulus of the equivalent material used for the inertia

computation. In our case 𝐸𝑝 is referred to beam elastic modulus.

2.4.2.1 Transversal Model

Now It is possible to define all the formulas for each region, in order to obtain the

displacement associate to the first natural mode.

𝑦1(𝑥, 𝑡) = (𝐴1 cos 𝛾𝑥 + 𝐵1 sin 𝛾𝑥 + 𝐶1 cosh 𝛾𝑥 + 𝐷1 sinh 𝛾𝑥)𝑒𝑖𝜔1𝑡 (2.28)


𝑦3(𝑥, 𝑡) = (𝐴3𝑥 + 𝐵3)𝑒𝑖𝜔1𝑡 (2.30)

From this system of equations, I can define a vector of all coefficient:

𝑥 =

[ 𝐴1

𝐵1

𝐶1

𝐷1

𝐴2

𝐵2

𝐶2

𝐷2

𝐴3

𝐵3]

(2.31)

It is possible to compute these coefficients using the boundary conditions. To compute

these coefficients I need to obtain all the natural frequencies that I want to analyse, in

our case only the first.

We have a linear system, compose by ten equations, to solve:

[𝐻]𝑥 = 0 (2.32)

Matrix H is composed by all the boundary conditions on the equilibria of

displacements and speed continuities, momentum and shear force balances between

each region and considering that the first region is clamped, while the tip is a free end:

MATHEMATICAL MODEL

27

𝑦1(0, 𝑡) = 0 (2.33)

𝛿𝑦1(0, 𝑡)

𝛿𝑥= 0 (2.34)

𝑦1(𝐿1, 𝑡) = 𝑦2(0, 𝑡) (2.35)

𝛿𝑦1(𝐿1, 𝑡)

𝛿𝑥=

𝛿𝑦2(0, 𝑡)

𝛿𝑥 (2.36)

𝐸1𝐽1

𝛿2𝑦1(𝐿1, 𝑡)

𝛿𝑥2 = 𝐸2𝐽2𝛿2𝑦2(0, 𝑡)

𝛿𝑥2 (2.37)

𝐸1𝐽1

𝛿3𝑦1(𝐿1, 𝑡)

𝛿𝑥3 = 𝐸2𝐽2𝛿3𝑦2(0, 𝑡)

𝛿𝑥3 (2.38)

𝑦2(𝐿2, 𝑡) = 𝑦3(0, 𝑡) (2.39)


𝛿𝑥=

𝛿𝑦3(0, 𝑡)

𝛿𝑥 (2.40)

𝐸2𝐽2𝛿2𝑦2(𝐿2, 𝑡)

𝛿𝑥2 + 𝑀3

𝛿2𝑦3 (𝐿32

, 𝑡)

𝛿𝑥2 + 𝐽3𝛿3𝑦3 (

𝐿32

, 𝑡)

𝛿𝑡2𝛿𝑥= 0

(2.41)


𝛿𝑥23 − 𝑀3

𝛿3𝑦3 (𝐿32

, 𝑡)


(2.42)

Now if I substitute equations (2.28), (2.29) and (2.30):

𝐴1 + 𝐶1 = 0 (2.43)

𝐵1𝛾1 + 𝐷1𝛾1 = 0 (2.44)

𝐴1 cos(𝛾1𝐿1) + 𝐵1 sin(𝛾1𝐿1) + 𝐶1 cosh(𝛾1𝐿1) +𝐷1 sinh(𝛾1𝐿1)= 𝐴2 + 𝐶2

(2.45)

−𝐴1𝛾1 sin(𝛾1𝐿1) + 𝐵1𝛾1 cos(𝛾1𝐿1) + 𝐶1𝛾1 sinh(𝛾1𝐿1)+ 𝛾1𝐷1 cosh(𝛾1𝐿1) = 𝐵2𝛾2 + 𝐷2𝛾2

(2.46)

MATHEMATICAL MODEL

28

𝐸1𝐽1(−𝐴1𝛾12 cos(𝛾1𝐿1)

− 𝐵1𝛾12 sin(𝛾1𝐿1) + 𝐶1𝛾1

2 cosh(𝛾1𝐿1)+𝐷1𝛾1

2 sinh(𝛾1𝐿1)) = 𝐸2𝐽2(−𝐴2𝛾22 + 𝐶2𝛾2

2) (2.47)

𝐸1𝐽1(𝐴1𝛾13 sin(𝛾1𝐿1) − 𝐵1𝛾1

3 cos(𝛾1𝐿1) + 𝐶1𝛾13 sinh(𝛾1𝐿1)

+ 𝐷1𝛾13 cosh(𝛾1𝐿1)) = 𝐸2𝐽2(−𝐵2𝛾2

3 + 𝐷2𝛾23)

(2.48)

𝐴2 cos(𝛾2𝐿2) + 𝐵2 sin(𝛾2𝐿2) + 𝐶2 cosh(𝛾2𝐿2) +𝐷2 sinh(𝛾2𝐿2)= 𝐵3

(2.49)

−𝐴2𝛾2 sin(𝛾2𝐿2) + 𝐵2𝛾2 cos(𝛾2𝐿2) + 𝐶2𝛾2 sinh(𝛾2𝐿2)+ 𝐷2 𝛾2 cosh(𝛾2𝐿2) = 𝐴3

(2.50)

𝐸2𝐽2(−𝐴2𝛾22 cos(𝛾2𝐿2)

− 𝐵2 𝛾22sin(𝛾2𝐿2) + 𝐶2 𝛾2

2 cosh(𝛾2𝐿2)

+ 𝐷2 𝛾22 sinh(𝛾2𝐿2)) + 𝑀3 [−𝜔𝑛

2𝐿3

2(𝐴3

𝐿3

2+ 𝐵3)]

− 𝐽3𝜔𝑛2𝐴3 = 0

(2.51)

𝐸2𝐽2(𝐴2𝛾23 sin(𝛾2𝐿2) − 𝐵2𝛾2

3 cos(𝛾2𝐿2) + 𝐶2𝛾23 sinh(𝛾2𝐿2)

+ 𝐷2𝛾23 cosh(𝛾2𝐿2)) + 𝑀3𝜔𝑛

2𝐴3

𝐿3

2+ 𝑀3𝜔𝑛

2𝐵3

= 0

(2.52)

In the next page It is possible to see the matrix H that I can obtain from this system of

equations.

MATHEMATICAL MODEL

29

MATHEMATICAL MODEL

30


For the longitudinal prototype we saw that there are to consider four regions instead

three:




𝑦4(𝑥, 𝑡) = (𝐴4𝑥 + 𝐵4)𝑒𝑖𝜔1𝑡 (2.56)

Like for the transversal model we’re going to define a vector with all the coefficients:

𝑥 =

[ 𝐴1

𝐵1

𝐶1

𝐷1

𝐴2

𝐵2

𝐶2

𝐷2

𝐴3

𝐵3

𝐶3

𝐷3

𝐴4

𝐵4]

(2.57)

As for the transversal prototype we have to rewrite the boundary conditions:

𝑦1(0, 𝑡) = 0 (2.58)

𝛿𝑦1(0, 𝑡)

𝛿𝑥= 0 (2.59)

𝑦1(𝐿1, 𝑡) = 𝑦2(0, 𝑡) (2.60)


𝛿𝑥=

𝛿𝑦2(0, 𝑡)

𝛿𝑥 (2.61)

𝐸1𝐽1

𝛿2𝑦1(𝐿1, 𝑡)

𝛿𝑥2= 𝐸2𝐽2

𝛿2𝑦2(0, 𝑡)

𝛿𝑥2 (2.62)

MATHEMATICAL MODEL

31

𝐸1𝐽1

𝛿3𝑦1(𝐿1, 𝑡)

𝛿𝑥3 = 𝐸2𝐽2𝛿3𝑦2(0, 𝑡)

𝛿𝑥3 (2.63)

𝑦2(𝐿2, 𝑡) = 𝑦3(0, 𝑡) (2.64)


𝛿𝑥=

𝛿𝑦3(0, 𝑡)

𝛿𝑥 (2.65)

𝐸2𝐽2

𝛿2𝑦2(𝐿2, 𝑡)

𝛿𝑥2 = 𝐸3𝐽3𝛿2𝑦3(0, 𝑡)

𝛿𝑥2 (2.66)

𝐸2𝐽2

𝛿3𝑦2(𝐿2, 𝑡)

𝛿𝑥3 = 𝐸3𝐽3𝛿3𝑦3(0, 𝑡)

𝛿𝑥3 (2.67)

𝑦3(𝐿3, 𝑡) = 𝑦4(0, 𝑡) (2.68)


𝛿𝑥=

𝛿𝑦4(0, 𝑡)

𝛿𝑥 (2.69)


𝛿𝑥2 + 𝑀4

𝛿2𝑦4 (𝐿42

, 𝑡)

𝛿𝑥2 + 𝐽4𝛿3𝑦4 (

𝐿42

, 𝑡)


(2.70)


𝛿𝑥23 − 𝑀4

𝛿3𝑦4 (𝐿42

, 𝑡)


(2.71)

Without substitute equations inside the boundary conditions, it is possible to write the

matrix H.

MATHEMATICAL MODEL

32

MATHEMATICAL MODEL

33

2.4.3 Kinetic energy

In order to define the modal mass, we can use the kinetic energy 𝐸𝑘. This is written as

the sum of every mass contribution from different regions. We must define modal

mass for both prototypes.

2.4.3.1 Transversal model

From the previous consideration, we just know that GPEH is composed by three

regions:

𝐸𝑘 =

1

2∫ �̇�1

𝑇𝜌𝑠�̇�1𝑑𝑉

𝑉𝑠1

+ 21

2∫ �̇�1

𝑇𝜌𝑝�̇�1𝑑𝑉

𝑉𝑝1

+1

2∫ �̇�2


𝑉𝑠2

+1

2∫ �̇�3

𝑇𝜌𝑏�̇�3𝑑𝑉

𝑉𝑏3

(2.72)

With subscripts s, p and b we refer to beam, piezo and bluff body.

Because the structure is symmetric, we can double the contribution. Now it is possible

to introduce the modal approach, using only the first natural mode, according to the

previous assumption:

𝑦𝑖 = 𝜙𝑖(1)𝑞(1) where 𝜙𝑖

(1) = 𝜙𝑖(1)(𝑥𝑖) (2.73)

And deriving:

𝑦�̇� = 𝜙𝑖(1)�̇�(1) (2.74)

So now we are going to substitute this equation inside the kinetic energy.

We can also simplify the volume integrals because the quantities have to be integrated

only along the axial coordinate, and we obtain the following form:

𝐸𝑘 =

1

2[𝑤𝑠𝑡𝑠𝜌𝑠 ∫ 𝜙1

2 𝑑𝑥1

𝐿1

0

+ 2𝑤𝑝𝑡𝑝𝜌𝑝 ∫ 𝜙12 𝑑𝑥1

𝐿1

0

+ 𝑤𝑠𝑡𝑠𝜌𝑠 ∫ 𝜙22 𝑑𝑥2

𝐿2

0

+ 𝑤𝑏𝑡𝑏𝜌𝑏 ∫ 𝜙32 𝑑𝑥3

𝐿3

0

] �̇�2

=1

2𝑀∗�̇�2

(2.75)

MATHEMATICAL MODEL

34

At the end from the last equation we can define the modal mass:

𝑀∗ = 𝑤𝑠𝑡𝑠𝜌𝑠 ∫ 𝜙1

2 𝑑𝑥1

𝐿1

0


𝐿1

0


𝐿2

0


𝐿3

0

(2.76)

The overall kinetic energy is therefore:

𝐸𝑘 =

1

2𝑀∗�̇�2 (2.77)

By differentiating this equation with respect to the derivative of the modal coordinate

q it is obtained:

𝐸𝑘 = 𝛿�̇�𝑀∗�̇� (2.78)

2.4.3.2 Longitudinal model

Now for this second prototype the only different is about the number of the regions

that I’ve to consider. Indeed, as already discussed, now we’ve four regions, instead

three:

𝐸𝑘 =

1

2∫ �̇�1


𝑉𝑠1

+ 21

2∫ �̇�1


𝑉𝑝1

+1

2∫ �̇�2


𝑉𝑠2

+1

2∫ �̇�3


𝑉𝑠3

+1

2∫ �̇�3


𝑉𝑏3

+1

2∫ �̇�4


𝑉𝑏4

(2.79)

Without rewrite every step, the modal mass is defined as follow:

𝑀∗ = 𝑤𝑠𝑡𝑠𝜌𝑠 ∫ 𝜙1

2 𝑑𝑥1

𝐿1

0


𝐿1

0


𝐿2

0


𝐿3

0


𝐿3

0


𝐿4

0

(2.80)

MATHEMATICAL MODEL

35

By differentiating this equation with respect to the derivative of the modal coordinate

q it is obtained:

𝐸𝑘 = 𝛿�̇�𝑀∗�̇� (2.81)

2.4.4 Elastic energy

The elastic energy is determined by the sum of strain and stress deformations that the GPEH experiences. For this reason, since the fourth region is not deformable it will not be included in the following definition of the total elastic energy:

𝐸𝑝 =

1

2∫ 𝑆𝑇

𝑠1𝑇𝑠1𝑑𝑉𝑠1

𝑉𝑠1

+ 2 ∫𝑆𝑇𝑝1𝑇𝑝1𝑑𝑉𝑝1

𝑉𝑠1

+1

2∫𝑆𝑇


𝑉𝑠2

+1

2∫𝑆𝑇


𝑉𝑠3

+1

2∫𝑆𝑇

𝑏3𝑇𝑏3𝑑𝑉𝑏3

𝑉𝑠1

(2.82)

Then the formula is re-arranged by substituting the following definitions that apply respectively for stress and strain:

𝑆 = −𝑧

𝛿2𝑦

𝛿𝑥2 (2.83)

𝑇 = 𝑐𝑆 (2.84)

According to the piezoelectric constitutive law it is possible to state:

𝑇𝑝1 = 𝑐𝐸11𝑆𝑝1 − 𝑒31𝐸3 (2.85)

The overall elastic energy can be split in the electrical contribution, it is given by the piezoelectric patch, and contribution to the other layers.

𝐸𝑝𝑝𝑧𝑡

= ∫𝑧2 (𝛿2𝑦

𝛿𝑥2)

2

𝑐𝐸11𝑑𝑉𝑝

𝑉𝑝

+ ∫𝑧2𝛿2𝑦

𝛿𝑥2 𝑒31𝐸3𝑑𝑉𝑝𝑉𝑝

(2.86)

The connection for our prototypes is the series connection. The value 𝑒31 has opposite sign for the top and bottom PZT layers, and electric fields are:

𝐸3 = −

𝑉𝑠2𝑡𝑝

(2.87)

where 𝑉𝑠 is the voltage across the output terminals.

MATHEMATICAL MODEL

36

Considering that in equation (2.86) the quantities width of cross-section, elastic modulus and electric field are constant for any location of the volume, integrals can be simplified as:


= 𝑤𝑝𝑐𝐸

11 ∫ (𝛿2𝑦

𝛿𝑥2)

2

∫ 𝑧2𝑑𝑧𝑑𝑥1

ℎ𝑝+𝑡𝑝

ℎ𝑝

𝐿1

0

+ 𝑒31

𝑉𝑠2𝑡𝑝

𝑤𝑝 ∫𝛿2𝑦

𝛿𝑥2 ∫ 𝑧𝑑𝑧𝑑𝑥1

ℎ𝑝+𝑡𝑝

ℎ𝑝

𝐿1

0

(2.88)

The integrals on the z axis are then solved:


= 𝑤𝑝𝑐𝐸

11 ∫ (𝛿2𝑦

𝛿𝑥2)

21

3[(ℎ𝑝 + 𝑡𝑝)

3− (ℎ𝑝)

3] 𝑑𝑥1

𝐿1

0

+ 𝑒31

𝑉𝑠2𝑡𝑝

𝑤𝑝 ∫𝛿2𝑦

𝛿𝑥2

𝐿1

0

1

2[(ℎ𝑝 + 𝑡𝑝)

2− (ℎ𝑝)

2]

(2.89)

Solving the axial integration, it is found:


= 𝑤𝑝𝑐𝐸

11

1

3[(ℎ𝑝 + 𝑡𝑝)

3− (ℎ𝑝)

3] ∫ 𝜙1

′′(1)2𝑑𝑥 𝑞2𝐿1

0

+1

2𝑒31

𝑉𝑠2𝑡𝑝

𝑤𝑝[𝑡𝑝2 + 2𝑡𝑝ℎ𝑝][𝜙1

′(1)(𝐿1)

− 𝜙1′(1)(0)]𝑞

(2.90)

The variable 𝜃𝑠 is then introduced as:

𝜃𝑠 = 𝑒31

𝑤𝑝

2(𝑡𝑝 + 2ℎ𝑝) (2.91)

Come back to the definition of elastic energy, the rest of integrals can be solved in a

similar way:

MATHEMATICAL MODEL

37

𝐸𝑝𝑚𝑒𝑐

= 𝑤𝑠1𝑐𝑠1

1

2∫ (

𝛿2𝑦1

𝛿𝑥12)

2

∫ 𝑧2𝑑𝑧 𝑑𝑥1

𝑡𝑠2

−𝑡𝑠2

𝐿1

0

+ 𝑤𝑠2𝑐𝑠2

1

2∫ (

𝛿2𝑦2

𝛿𝑥22)

2


𝑡𝑠2

−𝑡𝑠2

𝐿2

0


1

2∫ (

𝛿2𝑦3

𝛿𝑥32)

2


𝑡𝑠2

−𝑡𝑠2

𝐿3

0

+ 2𝑤𝑏𝑐𝑏

1

2∫ (

𝛿2𝑦3

𝛿𝑥32)

2


𝑠2+𝑡𝑏

−𝑡𝑠2

3

0

(2.92)

The integrals in the z-direction are solved as:

𝐸𝑝𝑚𝑒𝑐


1

2∫ (

𝛿2𝑦1

𝛿𝑥12)

2

[1

3((

𝑡𝑠2)3

+ (𝑡𝑠2)3

)]𝐿1

0

𝑑𝑥1


1

2∫ (

𝛿2𝑦2

𝛿𝑥22)

2

[1

3((

𝑡𝑠2)3

+ (𝑡𝑠2)3

)]𝐿2

0

𝑑𝑥2


1

2∫ (

𝛿2𝑦3

𝛿𝑥32)

2

[1

3((

𝑡𝑠2)3

+ (𝑡𝑠2)3

)]𝐿3

0

𝑑𝑥3

+ 𝑤𝑏𝑐𝑏

1

2∫ (

𝛿2𝑦3

𝛿𝑥32)

2

[1

3((

𝑡𝑠2

+ 𝑡𝑏)3𝐿1

0

− (𝑡𝑠2)3

)] 𝑑𝑥3

(2.93)

Considering modal approach formulation:

𝐸𝑝


1

2[1

12𝑡𝑠

3]∫ 𝜙′′12𝑑𝑥1

𝐿1

0


1

2[1

12𝑡𝑠

3]∫ 𝜙′′2

2𝑑𝑥2

𝐿2

0

+𝑤𝑠3𝑐𝑠3

1

2[1

12𝑡𝑠

3]∫ 𝜙′′3

2𝑑𝑥3

𝐿3

0

+ 𝑤𝑏𝑐𝑏

1

3[((

𝑡𝑠2

+ 𝑡𝑏)3

− (𝑡𝑠2)3

)]∫ 𝜙′′3

2𝑑𝑥3

𝐿3

0

(2.94)

At the end the overall elastic energy can be expressed as:

𝐸𝑝 =

1

2𝐾∗𝑞2 +

1

2𝜒𝑠𝑉𝑒𝑙𝑞 (2.95)

MATHEMATICAL MODEL

38

Where 𝐾∗ is:

𝐾∗


1

2[1

12𝑡𝑠

3]∫ 𝜙′′1

2𝑑𝑥1

𝐿1

0


1

2[1

12𝑡𝑠

3]∫ 𝜙′′22𝑑𝑥2

𝐿2

0

+𝑤𝑠3𝑐𝑠3

1

2[1

12𝑡𝑠

3]∫ 𝜙′′32𝑑𝑥3

𝐿3

0

+ 𝑤𝑏𝑐𝑏

1

3[((

𝑡𝑠2

+ 𝑡𝑏)3

− (𝑡𝑠2)3

)]∫ 𝜙′′32𝑤𝑝

𝐿3

0

𝑐𝐸11

1

3[(ℎ𝑝 + 𝑡𝑝)

3

− (ℎ𝑝)3] ∫ 𝜙1

′′(1)2𝑑𝑥 𝑞2𝐿1

0

(2.96)

And the electromechanical coupling term is expressed as:

𝜒𝑠 = 𝜃𝑠[𝜙′1(𝐿1) − 𝜙′

1(0)] (2.97)

Differentiating the elastic energy respect to 𝛿𝑞:

𝛿𝐸𝑝 =

1

2𝛿𝑞𝐾∗𝑞 +

1

2𝛿𝑞𝜒𝑠𝑉𝑠 +

1

2𝛿𝑉𝑒𝑙𝜒𝑠𝑞 (2.98)

2.4.5 Charge force work

The electric energy of the two piezoelectric layers, per each beam, can be defined,

considering the symmetry of the structure, as:

𝑊𝑒 =

1

22 ∫𝐸𝑇𝐷𝑑𝑉

𝑉𝑝

(2.99)

And:

𝑊𝑒 = ∫(−

𝑉𝑒𝑙

2𝑡𝑝)

𝑇

(𝑒31(−𝑤𝑆1) − 휀𝑠31

𝑉𝑒𝑙

2𝑡𝑝)𝑑𝑉

𝑉𝑝

(2.100)

Introducing modal approach, it is possible to obtain:

𝑊𝑒 = ∫(−

𝑉𝑒𝑙

2𝑡𝑝)

𝑇

(𝑒31(−𝑤𝜙′′1𝑞) − 휀𝑠

31

𝑉𝑒𝑙

2𝑡𝑝)𝑑𝑉

𝑉𝑝

(2.101)

MATHEMATICAL MODEL

39

And constant 𝐶𝑠 is introduced as:

𝐶𝑠 = 휀𝑠

33

𝑤𝑝𝐿1

𝑡𝑝 (2.102)

Charge force equation can be expressed as:

𝑊𝑒 = −

1

2𝑉𝑒𝑙𝜃𝑞 −

1

2𝛿𝑞𝜃𝑉𝑒𝑙 + 𝛿𝑉𝑒𝑙𝐶𝑠𝑉𝑒𝑙 (2.103)

2.4.6 Aerodynamic force work


We need to write the virtual work of the force that act on the bluff body. In fact in our

case we assume that the aerodynamic force is acting only on the bluff body and not on

the beam, we can assume that his contribution is negligible.

𝛿∗𝐿 = �⃗� ∗ 𝛿�⃗�𝐹 (2.104)

For our model the force in only on the longitudinal axis, for this reason 𝐹 = 𝐹𝑦.

Figure 2.7 Angle Attack

This force is defined as:

𝐹𝑦 = 𝐹𝐿 cos 𝛼 + 𝐹𝐷 sin 𝛼 =

1

2𝜌𝑉𝑟𝑒𝑙

2𝐿𝑡𝑖𝑝𝐷𝑡𝑖𝑝𝐶𝑦 (2.105)

In this case 𝑉𝑟𝑒𝑙 ≅ 𝑈 because we are in the case of little displacement.

MATHEMATICAL MODEL

40

Where the angle of attack and the rotation of the bluff body is considered inside the

𝐶𝑦 value:

𝐶𝑦 = 𝐶𝐿 cos 𝛼 + 𝐶𝐷 sin 𝛼 (2.106)

It is possible to change the coefficient, using 𝑎1and 𝑎3 from previous experimental

results:

𝐶𝑦 = 𝑎1𝛼 + 𝑎3𝛼3 (2.107)

In our case we decide to stop the interpolation to the third order.

The virtual displacement can be rewrite using the modal approach:

𝛿�⃗�𝐹 = 𝜙3

1 (𝐿3

2) 𝛿𝑞 (2.108)

Remember the previous equations, we can substitute all inside the virtual work, until

to obtain:

𝛿∗𝐿 = 𝐹𝑦𝜙3

1 (𝐿3

2) 𝛿𝑞 (2.109)

At the end we have to define the angle 𝛼 for this prototype, because there is the

contribution of angle attack of the wind and also the rotation of the body. This effect

is showed in the Figure 2.8.

Figure 2.8 Angle α transversal Model

MATHEMATICAL MODEL

41

In order to calculate the angle of the body, we can consider the rotation at the starter

point of the bluff body:

𝜃 =

𝜕𝑦3

𝜕𝑥3|𝑥3=0

𝑞 (2.110)

And remember that region three is a concentrate body, the equation is 𝑦3(𝑥, 𝑡) =

(𝐴3𝑥 + 𝐵3)𝑒𝑖𝜔1𝑡, for this reason the angle is the coefficient 𝐴3.

Instead the angle of attack is calculated from the triangle of the velocity, using the

trigonometric relations:

tan 𝛾 =

�̇�

𝑈 (2.111)

Considering little displacement and also to consider the passage from continuous to

modal approach, It is possible to obtain:

𝛾 ≅�̇�

𝑈=

𝜙3 (𝐿32

) �̇�

𝑈

(2.112)

From all these considerations we’ll rearrangement the previous equation of the total

angle:

𝛼 =𝜕𝑦3

𝜕𝑥3|𝑥3=0

𝑞 +𝜙3 (

𝐿32

) �̇�

𝑈 (2.113)

Now there all the element for obtaining the final equation of the aerodynamic force:

𝐹𝑎 =1

2𝜌𝑎𝑖𝑟𝑈

2𝐿𝑡𝑖𝑝𝐷𝑡𝑖𝑝 (𝑎1 (𝜕𝑦3

𝜕𝑥3|𝑥3=0

𝑞 +𝜙3 (

𝐿32

) �̇�

𝑈)

+ 𝑎3 (𝜕𝑦3

𝜕𝑥3|𝑥3=0

𝑞 +𝜙3 (

𝐿32

) �̇�

𝑈)

3

) 𝜙3 (𝐿3

2)

(2.114)

The differentiation of the aerodynamic force work with respect to the variable q is:

𝛿𝐿𝑎𝑒𝑟𝑜 = 𝐹𝑎 (2.115)

MATHEMATICAL MODEL

42

2.4.6.2 Longitudinal Model

In longitudinal prototype the force is expressed as the integral contribution of the

aerodynamics forces on the third and fourth regions.

It is important to consider that the angle is only composed by the angle of attack:

𝛿𝐿𝑎𝑒𝑟𝑜 = ∫ 𝐹𝑦3

(𝑥)𝑇𝛿𝑦3 𝑑𝑥3

𝐿3

0

+ ∫ 𝐹𝑦4(𝑥)𝑇𝛿𝑦4 𝑑𝑥4

𝐿4

0

(2.116)

Passing now to the modal approach:

𝛿𝐿𝑎𝑒𝑟𝑜 = ∫ 𝐹𝑦3

(𝑥)𝑇𝜙3 𝑑𝑥3 𝛿𝑞𝐿3

0

+ ∫ 𝐹𝑦4(𝑥)𝑇𝜙4 𝑑𝑥4 𝛿𝑞

𝐿4

0

(2.117)

Rewriting now the aerodynamic force:

𝛿𝐿𝑎𝑒𝑟𝑜 = ∫

1


2𝐷𝑡𝑖𝑝[𝑎1𝛼 + 𝑎3𝛼3]𝑇𝜙3 𝑑𝑥3 𝛿𝑞

𝐿3

0

+ ∫1


2𝐷𝑡𝑖𝑝[𝑎1𝛼 + 𝑎3𝛼3]𝑇𝜙4 𝑑𝑥4 𝛿𝑞

𝐿4

0

(2.118)

Where the angle is expressed as in the transversal model, but only considering the

angle of attack and not the body rotation:

𝛼 = tan−1

�̇�

𝑈≅

�̇�

𝑈 (2.119)

And at the end we have to consider also the transformation to the modal approach:

𝛼 ≅

𝜙𝑖�̇�

𝑈 (2.120)

It is possible to substitute in order to obtain the final form of the virtual work:

𝛿𝐿𝑎𝑒𝑟𝑜 =

1


2𝐷𝑡𝑖𝑝 {∫ [𝑎1 [𝜙3�̇�

𝑈] + 𝑎3 [

𝜙3�̇�

𝑈]3

]

𝑇

𝜙3 𝑑𝑥3

𝐿3

0

+ ∫ [𝑎1 [𝜙4�̇�

𝑈] + 𝑎3 [

𝜙4�̇�

𝑈]3

]

𝑇

𝜙4 𝑑𝑥4

𝐿4

0

} 𝛿𝑞

(2.121)

MATHEMATICAL MODEL

43

The final equation of the aerodynamic force, that we’ve to insert to the system, is:

𝐹𝑎 =

1

2𝜌𝑎𝑖𝑟𝑈𝐷𝑡𝑖𝑝 {𝑎1

�̇�

𝑈∫ 𝜙3

2 𝑑𝑥3

𝐿3

0

+ 𝑎3 (�̇�

𝑈)3

∫ 𝜙34 𝑑𝑥3

𝐿3

0

+ 𝑎1

�̇�

𝑈∫ 𝜙4

2 𝑑𝑥4

𝐿4

0

+ 𝑎3 (�̇�

𝑈)3

∫ 𝜙44 𝑑𝑥4

𝐿4

0

}

(2.122)

The differentiation of the aerodynamic force work with respect to the variable q is:

𝛿𝐿𝑎𝑒𝑟𝑜 = 𝐹𝑎 (2.123)

2.4.7 Dissipative Term

The dissipative term is a component that can be obtained from experimental tests,

because it is really difficult to estimate.

To estimate the value, we can use the decay method. The structure is subjected to a

force and it is left free to oscillate, and the only effect acting on system is the damping

ratio.

The free damped motion is express by the following equation:

𝑥(𝑡) = 𝑋𝑒−ℎ𝜔0𝑡 cos(𝜔𝑡 + 𝜑) (2.124)

Where the factor ℎ is the non-dimensional damping, defined as:

ℎ =

𝑅

2𝑀∗𝜔0 (2.125)

Here there is an example of a logarithm decay:

Figure 2.9 Example of decay

MATHEMATICAL MODEL

44

Using the logarithmic decrement it is possible to compute the value of the non-

dimensional damping, taking two peaks of the temporal history 𝑥𝑖(𝑡) and 𝑥𝑖+1(𝑡 + 𝑇),

where 𝑇 is the period between two peaks:

𝛿 = ln

𝑥𝑖(𝑡)

𝑥𝑖+1(𝑡 + 𝑇)= ln

𝑋𝑒−ℎ𝜔0𝑡 cos(𝜔𝑡 + 𝜑)

𝑋𝑒−ℎ𝜔0(𝑡+𝑇) cos(𝜔(𝑡 + 𝑇) + 𝜑) (2.126)

At the end we can obtain:

𝛿 = ℎ

2𝜋

√1 − ℎ2 (2.127)

From the Figure 2.10 it is possible to demonstrate that for little value of ℎ, in particular

under the assumption ℎ ≪ 1, the formulation can change, and it can be simplified as

follow:

𝛿 = 2𝜋ℎ (2.128)

Figure 2.10 Exact Vs Approximate Formulation

Having ℎ value, we can compute the value of the dissipative term as:

𝑅 = 2𝑀∗𝜔0ℎ (2.129)

MATHEMATICAL MODEL

45

2.4.8 Conclusive equations

Taking the Hamilton principle it is now possible to write:

∫ 𝛿�̇�𝑀∗�̇� −

𝑡2

𝑡1

𝛿𝑞𝐾∗𝑞 − 𝛿𝑞𝜃𝑉𝑠 − 𝛿𝑉𝑒𝑙𝜃𝑉𝑠 + 𝛿𝑉𝑠𝐶𝑝𝑉𝑠 + 𝐹∗𝛿𝑞

+ 𝑄𝑒𝑙𝛿𝑉𝑠𝑑𝑡 = 0

(2.130)

Which brings to:

𝛿𝑞𝑀∗�̈� + 𝛿𝑞𝐾∗𝑞 + 𝛿𝑉𝑒𝑙𝜃𝑞 + 𝛿𝑞𝜃𝑉𝑠 − 𝛿𝑉𝑠𝐶𝑝𝑉𝑠 − 𝐹∗𝛿𝑞 − 𝑄𝑒𝑙𝛿𝑉𝑠= 0

(2.131)

It is possible to express the superimposition system between the two system variables

q and 𝑉𝑒𝑙 as:

{

𝑀∗�̈� + 𝑅�̇� + 𝐾𝑞 + 𝜒𝑠𝑉𝑝 = 𝐹𝑎

𝑉𝑝

𝑅+ 𝐶𝑝�̇�𝑝 − 𝜒𝑠�̇� = 0

(2.132)

The system can be rewrite as a state space representation, defining by the following

system variables:

[

𝑥1

𝑥2

𝑥3

] = [

𝑞�̇�𝑉𝑠

] (2.133)

The final model is:

[

𝑥1̇

𝑥2̇

𝑥3̇

] =

[ −

𝑅∗ −12

𝜌𝑎𝑖𝑟𝑈𝐷𝑎1𝑘1

𝑀∗ −𝐾∗

𝑀∗ −𝜒

𝑀∗

1 0 0𝜒

𝐶𝑝0 −

1

𝐶𝑝𝑅]

[

𝑥1

𝑥2

𝑥3

]

+ (1

2𝜌𝑎𝑖𝑟𝐷𝑎3

𝑘3

𝑈) [

𝑥1

00

]

3

(2.134)

MATHEMATICAL MODEL

46

2.4.9 Numerical simulation

The system of equation is implemented in MATBLAB. In order to solve this system we

use a numerical integration method, in our case we decide to use ODE45 function. This

function is just implemented in MATLAB.

It is necessary to set initial conditions with a non-zero displacement in order to find

the velocity that makes negative the total damping. Maximum amplitude reaches by

the system is obtained when the transient response is completed. It is necessary to

implement a method to find when the transient response is finished, because per each

speed, the transient time response can be different.

SENSITIVITY ANALISYS OF GPEH

47

3 SENSITIVITY ANALISYS OF GPEH

In this chapter the aim is to analyse all the effect of parameters, to study which is the

best way to optimize the system.

We are interested on investigate how velocities ratio can be reduced, how to improve

the interaction between galloping and vortex shedding and at the same time if there is

a way to increase the galloping velocity. At the end of this chapter we want to provide

final conclusion with a set of prototypes designed.

3.1 Effect on onset velocity and vortex shedding

It is possible to notice that when we change a parameter, there is a connection with

vortex shedding velocity and galloping velocity.

To give a first approximation of the problem about the sensitivity, we can take the

galloping formula showed in literature [37], considering only the mechanical problem.

We want to have a negative value of 𝑅𝑡𝑜𝑡 and considering mechanical and electrical

component:

𝑅𝑡𝑜𝑡 = 𝑅𝑚 + 𝑅𝑒𝑙 (3.135)

Where:

𝑅𝑚 = 2ℎ𝑀∗√𝑘

𝑀∗ (3.136)

𝑅𝑒𝑙 = −

1

2𝜌𝑎𝑖𝑟𝐷𝑡𝑖𝑝𝑈𝑎1 (3.137)

On first place, just for this approximation I can consider that the mass is associate to

the bluff body, because it gives the most important contribution, for this reason I

have:

𝑀∗ = 𝑉𝜌𝑡𝑖𝑝 = 𝐿𝑡𝑖𝑝𝐷𝑡𝑖𝑝𝜌𝑡𝑖𝑝 (3.138)


48

Considering all these equations, I can write the final form of the galloping velocity in

this approximation:

𝑈𝑔 ≅ 4

ℎ√𝜌𝑡𝑖𝑝√𝑘

𝜌𝑎𝑖𝑟𝑎1√𝐿𝑡𝑖𝑝

(3.139)

In some cases it is important to reduce this velocity and from this formula we can see

that 𝑈𝑔 is reduced when:

• Reducing damping ratio can reduce also the onset velocity, the relation is

proportional. To achieve this result, we must change some parameters of the

structure, for example, the material or dimensions of the beam;

• Decrease the value of the bluff body density is another way to reduce the

onset velocity. The only way is by chancing the material of the bluff body;

• Changing the shape of bluff body is an important thing if we want to reduce

the value of 𝑈𝑔 through 𝑎1 value. For this reason, we must study the effect of

different shape to select the best one;

• The last parameter is 𝐿𝑡𝑖𝑝. When we increase this value, we can increase the

aerodynamic force acting on the bluff body and for that is the reason why It is

possible to obtain before the instability.

Of course, all these parameters aren’t independent between them and it is possible

that if we change one value, it will affect another one.

The second velocity that we consider is the vortex shedding velocity, took again all the

consideration I mentioned before, and the result is:

𝑈𝑣 =

𝜔0𝐷𝑡𝑖𝑝

𝑛𝑆𝑡𝑟𝑜𝑢ℎ𝑎𝑙≅

√𝑘

𝑛𝑠√𝐿𝑡𝑖𝑝√𝜌𝑡𝑖𝑝

(3.140)

Now It is important to consider the ratio between these two velocities, because this

parameter can be important to obtain a correct response to the galloping instability.

If there is a variation of 𝐿𝑡𝑖𝑝 of course we increase also the mass of the bluff body (if

the density remains the same obviously) and this create an effect also on the natural

frequency and in the end there will be a variation also on the vortex shedding.

For this It is important to consider, before moving the parameters, what can happen if

we move each of them.


49

In fact, from literature [38] it just studied this factor of the coupled between these two

parameters and the importance of the ratio:

1. Full interaction: In this case there is a completely interaction between the two

velocities, vortex shedding can help the galloping to establish, without a big

oscillation, but a controlled increase of vibrations.

𝑈𝑔

𝑈𝑣≤ 1.8 (3.141)

2. Partial interaction: In this second case we have more distance between

galloping and vortex shedding, for this reason it is possible that the oscillation

starts to decrease after vortex shedding, because the wind goes out from the

locked frequency, and from galloping start to increase again.

𝑈𝑔

𝑈𝑣≈ 1.8 ÷ 2.5 (3.142)

3. Low interaction: Now galloping isn’t affected too much from the vortex

shedding, in fact the oscillation goes until almost zero before it restart to

increase, with the galloping instability. However, also in this case there is still a

little influence of the vortex shedding on the second instability.

There is the possibility that in this region the effect of the interaction is

influence by the time to reach the steady state.

𝑈𝑔

𝑈𝑣≈ 2.5 ÷ 3 (3.143)

4. In this last case there isn’t the influence of the first instability on the second,

for this reason when the structure enters on the galloping velocity there isn’t a

smooth increment, but we can see a big oscillation on the structure.

𝑈𝑔

𝑈𝑣≥ 3 (3.144)

Using previous equations of velocity, it is possible to write the ratio between them in

order to understand which parameters we need to change in order to have a good

value of the ratio:

𝑈𝑔

𝑈𝑣= 4

ℎ𝑛𝑠𝜌𝑡𝑖𝑝

𝜌𝑎𝑖𝑟𝑎1 (3.145)

From this first approximation we conclude that in order to reduce the ratio we have to

work only on:


50

• Damping ratio;

• Number of Strouhal;

• Aerodynamic coefficient;

• Density of bluff body.

In our study we will analyse only a square bluff body, that is why Strouhal and 𝑎1 can’t

change, so we can move only the last two parameters.

3.2 Reference case

At first, we need to define is to define a reference case where we start the study. We

just spoke about this model and we have to remember that in this prototype there are

three regions to consider. Table 3.1 shows characteristics of the reference case.

Parameter Reference value

Piezo - PPA1011

Number of piezo - 2

Connection - Series

Length of second region 𝑳𝟐 mm 50

Beam width 𝒘𝒔 mm 28

Beam thickness 𝒕𝒔 mm 0.5

Damping ratio % 1

Bluff body density 𝝆𝑩 𝑘𝑔/𝑚3 240

Beam material - Steel

Bluff body shape - Square

Dimension bluff body side 𝑫𝒕𝒊𝒑 mm 40

Length bluff body 𝑳𝒕𝒊𝒑 mm 200

Number of beam - 1

Resistance - 100 𝑘𝛺

Type of prototype - Transversal

Table 3.1 Parameters Reference Case


51

Figure 3.1 Reference case CAD

For this reference case we can see the result and compare it with the results when we

change the parameter.

We will take first the mechanical point of view, starting with the natural frequency and

modal mass:

Mode Frequency [Hz]

First Mode 11.78

Second Mode 57.24

Third Mode 397.12

Table 3.2 Natural frequency reference case

Parameter Reference case

Modal mass Kg 0.0174 Table 3.3 Modal mass reference case

We said that for our case only the first mode will be considered, so for this reason we

won’t take another mode.

In Figure 3.2 the modal shape associate with the first natural mode.


52

Figure 3.2 Modal shape reference case

From the modal shape it is possible to compute da value of the electro-mechanical

coupled:

𝜒 = 𝜃𝑠𝑒𝑟𝑖𝑒𝑠 (�̇�1(𝐿1) − �̇�1(0)) = 3.94𝐸 − 04 (3.146)

Now we can obtain the aerodynamic result, onset velocity and vortex shedding, before

to integrate on the time in order to compute the response:


Galloping m/s 8.30

Vortex Shedding m/s 3.86

Speed ratio - 2.15 Table 3.4 Aerodynamic result reference case

Now It is possible to show the response of the prototype with a wind velocity of

35 𝑚/𝑠 or less.

It is important to give four different responses:

• Tip displacement: maximum displacement of the bluff body. In this case we

consider the centre of the bluff body;

• PPA displacement: It is important to estimate the correct maximum deflection

of the piezo because exist a maximum value before the break. In this

reference case this value is 20 mm;

• Voltage: Also, this value is critical because exist a limit value and we must

check to remain inside the range;

• Power: In this way It is possible to compute the value of power that we can

recover from the wind the power. In this plot is considered only the effect of

one PZT patch.


53

(a) (b)

(c) (d)

Figure 3.2.2 Time response reference case. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

3.3 Variation of parameters

In this paragraph we’ll analyse different variation respect the reference case.

Parameters that we want to study are:

• Mechanical parameters:

o Length of the second region 𝐿2;

o Beam width;

o Beam thickness;

o Beam material;

o Damping ratio;

o Density of the bluff body;

• Aerodynamic parameters:

o Length of the bluff body 𝐿𝑡𝑖𝑝;

o Bluff body side 𝐷3;


54

• Electric parameters:

o Resistance;

o Piezoelectric patch;

• Type of prototype:

o Number of beam;

o Longitudinal model.

In this section we’re going to check if the simplify equations that we used at the

beginning of the chapter is correct or there are some effects that it is impossible to

consider using them.

Of course, for this study we considered all parameters independent, for this reason all

the other parameters remain constant when we move the value of one. Obviously, this

is an approximation only to understand the behaviour of that parameter, but in the

real prototype if we change a parameter we create some effects in line the others.

3.3.1 Variation second region length

Taking previous equations, we can see that both velocities are dependent from this

value, in the same way, so the ratio should be remaining equal even if I change this

value.

𝑈𝑔 ≅

1

√𝛿𝐿𝑡𝑖𝑝

(3.147)

𝑈𝑣 ≅

1


(3.148)

Looking at these formulas we expect to have the ratio that remain constant, but we

are able to reduce to velocity of galloping and vortex shedding, when we increase the

length of second region.

We said we won’t consider the second effect of this variation, and for this we will keep

the damping ratio value as a constant, but in the real case we expect to reduce this

value when we increase the length of second region.


55

Figure 3.3 Modal shape variation second region

From the numerical simulation we obtained the follow results:

Variation: 𝐋𝟐 mm 50 75 100

Frequency Hz 11.78 8.71 6.74

Galloping m/s 8.30 5.60 4.20

Vortex shedding m/s 3.86 2.85 2.21

Speed ratio - 2.15 1.96 1.90

Modal Mass Kg 0.0174 0.0198 0.0220

Chi - 3.96E-4 2.50E-4 1.73E-4 Table 3.5 Results variation second region


56

Figure 3.4 Effect of variation length second region

From these results it is possible to notice that the trend that we expected is correct, so

for this It is appropriate to use the simply formula.

We have to remember that in the simply formula we did a simplification on the modal

mass, because we as said before, it is completely associate with the bluff body, instead

there is also the effect of the beam, and this can be the explanation of the little

decrement of the speed ratio.

The difference is around 10% and for this reason can be consider completely

negligible.

At the end It is possible to conclude that changing this parameter is only in order to

modify the velocities but there isn’t big effect on the speed ratio, and for this we can’t

use it to reduce the distance between the two velocities. In Figure 3.5 are showed all

plots for this parameter variation.

(a) (b)


57

(c) (d)

Figure 3.5 Time response variation second region. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

3.3.2 Variation beam width

The variation of this parameter should give only the effect of increase or decrease the

stiffness of the prototype.

From formulas we can see the relation that exist between the velocities and these

parameters:

𝑈𝑔 ≅ √𝛿𝑘 (3.149)

𝑈𝑣 ≅ √𝛿𝑘 (3.150)

The first consideration is that in theory, from this approximation, we shouldn’t have

effect on the speed ratio, moving this value. This is because both velocities are linked

in the same way to the stiffness.


58

Figure 3.6 Modal shape width beam variation

From the numerical simulation we obtained the follow results:

Variation: 𝐰𝐛 mm 28 30 32

Frequency Hz 11.78 12.06 12.33

Galloping m/s 8.30 8.70 9.00


Speed ratio - 2.15 2.19 2.23

Modal Mass Kg 0.0174 0.0176 0.0179

Chi - 3.96E-4 4.16E-4 4.36E-4 Table 3.6 Results variation width beam


59

Figure 3.7 Effect of variation width beam

The effect of this parameter is almost equal to what we expected, in fact when we

increase the width of the beam there is an increment of the stiffness and for this

reason the value of both velocities must increase.

The only difference is again the speed ratio, because it should remain constant,

instead there is a little difference when we increase the value of the parameter.

Anyway, the increment is really low, and for this we can conclude that the effect of the

parameter is good using the simplify formula.

In conclusion we can use this parameter only for changing the value of velocities, but

we don’t have good effect on the speed ratio.

Obviously, we expect by changing the width of the beam, there will be a second effect

on the damping ratio which should increase when we increase the width, but here we

won’t consider this effect.

(a) (b)


60

(c) (d)

Figure 3.8 Time response variation width beam. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

3.3.3 Variation beam thickness

This parameter is really similar to the previous one, also in this case we have that

associated with the variation of the thickness there is the variation on the stiffness of

prototype.

𝑈𝑔 ≅ √𝛿𝑘 (3.151)

𝑈𝑣 ≅ √𝛿𝑘 (3.152)

Again, we have the same formulation for this case and the same consideration that we

did before.

We expect again that the prototype has constant speed ratio when we change the

value of the thickness, because both velocities should change in same way and with

the same ratio.

Incrementing the value of the thickness there is an important growing of the stiffness,

with a consequence on the velocities which increment as well.


61

Figure 3.9 Modal shape thickness beam variation

Variation: 𝐭𝐛 mm 0.5 0.6 0.7

Frequency Hz 11.78 14.71 17.63

Galloping m/s 8.30 11.30 14.40


Speed ratio - 2.15 2.34 2.49

Modal Mass Kg 0.0174 0.0184 0.0194

Chi - 3.96E-3 5.58E-3 7.22E-3 Table 3.7 Results thickness beam variation

Figure 3.10 Effect of thickness beam variation


62

We just saw the effect of beam parameters on the prototype and again in this case we

can have an increment of the galloping velocity but having an almost constant value of

the speed ratio.

In this way It is possible to set the desired velocity without changing the distance

between two velocities.

(a) (b)

(c) (d)

Figure 3.11 Time response thickness beam variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

3.3.4 Variation beam material

One parameter that we can change in order to modify the structure of the prototype is

the material of the beam, because in this way it is possible to change elastic modulus

and the stiffness of the beam.

In order to do this, we can use two different materials:

• Steel;

• Aluminium.


63

Of course, we must take all the others constant parameters between the two

prototypes, in this way we can explore only the effect of the material.

Characteristics of materials are:

• Steel:

o 𝜌𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 = 7850 𝑘𝑔

𝑚3

o 𝐸 = 209 𝐺𝑃𝑎

• Aluminium:

o 𝜌𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙 = 2700 𝑘𝑔

𝑚3

o 𝐸 = 64 𝐺𝑃𝑎

First, it is possible to compare the different modal shape:

Figure 3.12 Modal shape material beam variation

Below in Table 3.8 we can see the results of numerical simulation.

Variation: Material Beam Steel Aluminium

Frequency Hz 11.78 7.01

Galloping m/s 8.30 4.20

Vortex shedding m/s 3.86 2.30

Speed ratio - 2.15 1.82

Modal mass Kg 0.0174 0.0164

Chi - 3.96E-4 1.51E-4 Table 3.8 Results material beam variation


64

From results we can see that material has a big effect on the results, in fact using

aluminium we have a low value of elastic modulus and density, and for this reason we

obtain a low value of the frequency respect to the steel.

The effect on the velocities is that we can decrease both value and at the end we have

a little decrement of the speed ratio.

For the last effect we can say that we could study different material in order to obtain

a suitable value of speed ratio.

Of course, we must consider the effect on the galloping velocity, because we have to

deal with a low speed ratio but also to a high value of the galloping, in this way we can

reach high velocity without problems.

(a) (b)

(c) (d)

Figure 3.13 Time response material beam variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind


65

3.3.5 Variation damping ratio

Now we are studying the parameter that is linked to one velocity but is independent

from the second.

In fact, damping ratio modify only galloping velocity, leaving constant vortex shedding:

𝑈𝑔 ≅ 𝛿ℎ (3.153)

This aspect is really important for our purpose because we just knew that speed ratio

is defined as:

𝐼𝑛𝑑𝑒𝑥 𝑟𝑎𝑡𝑖𝑜 =

𝑈𝑔

𝑈𝑣 (3.154)

So, it is clear that if we can modify only one velocity, becomes simply to set the speed

ratio moving only one parameter.

In this particular case It is obvious that damping ratio must decrease to obtain a low

value of speed ratio, and in our study, this is the effect that we want.

Of course, this parameter is linked to the other parameters that we just studied and

for this reason the difficult part is to estimate correctly the value of damping ratio.

In fact, this value is estimate from experimental and after we can use it inside the

numerical simulation.

It is possible to consider some effect of other parameters on damping ratio, and from

a structural point of view we can decrease ℎ with:

• Change material beam;

• Increase the length of the beam;

• Reduce the thickness of the beam;

• Reduce the width of the beam.

All these parameters change the mechanical damping of the beam.

In this study we don’t consider the effect of the drag coefficient on the bluff body, that

of course create an aerodynamic damping on the structure.

For this we found a value of damping ratio that consider all the effect inside and is

proportional with the oscillation of the body, so it isn’t a constant value.


66

Figure 3.14 Modal shape damping ratio variation

Variation: 𝐡 % 1.5 2 2.5

Frequency Hz 11.78 11.78 11.78

Galloping m/s 8.30 10.50 12.70


Speed ratio - 2.15 2.72 3.29

Modal Mass Kg 0.0174 0.0174 0.0174

Chi - 3.96E-4 3.96E-4 3.96E-4 Table 3.9 Results damping ratio variation

Figure 3.15 Effect of damping ratio variation


67

As we expected, damping ratio can move only galloping velocity keeping constant

vortex shedding velocity.

From this point of view, we want to reduce damping ratio to obtain an improvement

on the speed ratio, for this our purpose is to reach the lowest possible value, using

parameters that we wrote before.

Obviously when there is a variation of damping ratio, modal shape remains the same,

because damping ratio hasn’t an effect on the modal simulation, only on the

numerical simulation of the system.

Another important thing is the big effect that damping ratio has on galloping, in fact

with a small variation of this value, there is a high increment (or decrement) in the

galloping.

For example, between 1.5% and 2% there is an increment of 26.5% of the velocity, so

the effect is really big.

(a) (b)

(c) (d)

Figure 3.16 Time response material beam variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind


68

3.3.6 Variation bluff body density

This parameter is the second value that we can use in order to reduce the speed ratio,

because density has an opposite effect on the two velocities:

𝑈𝑔 ≅ 𝛿𝜌𝑏𝑙𝑢𝑓𝑓 (3.155)

𝑈𝑣 ≅

1

𝛿𝜌𝑏𝑙𝑢𝑓𝑓 (3.156)

Changing the bluff body density, we can approach the two speeds, in order to reach

our target of speed ratio.

To decrease the density, we have two different approaches:

• To modify bluff body material, with a different value of density;

• To carve the bluff body, in this way It is possible reduce the equivalent density

of the body.

Both of these methods give the same effect on the prototype.

Figure 3.17 Modal shape bluff body density variation


69

Variation: 𝛒𝐛𝐥𝐮𝐟𝐟 𝐛𝐨𝐝𝐲 𝐤𝐠/𝐦𝟑 160 200 240

Frequency Hz 13.94 12.72 11.78

Galloping m/s 7.10 7.70 8.30


Speed ratio - 1.55 1.84 2.15

Modal Mass Kg 0.0215 0.0149 0.0174

Chi - 3.98E-4 3.97E-4 3.96E-4 Table 3.10 Results bluff body density variation

Figure 3.18 Effect of bluff body density variation

From results is really clear which is the effect of the density, in fact we are able to

modify the ratio between the two velocities using this parameter.

It is important to reduce a lot the bluff body density in order to reach our target, and

in this way, we should modify only this parameter without creating some second

effect to the others parameters, because in this case nothing else change when we

move the density.


70

(a) (b)

(c) (d)

Figure 3.19 Time response bluff body density variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

3.3.7 Variation bluff body length

Taking formulas of velocities, we can see that the dimension of bluff body shouldn’t

have effect on speed ratio:

𝑈𝑣 ≅

1


(3.157)

𝑈𝑔 ≅

1


(3.158)

From these we expect to have constant value of speed ratio because the length of

bluff body gives the same effect on both velocities, for this reason we should be able

to increase or reduce the galloping velocity, without having some risk on modify the

value that we have as target.


71

Figure 3.20 Modal shape bluff body length variation

Variation: 𝐋𝐛𝐥𝐮𝐟𝐟 𝐛𝐨𝐝𝐲 𝐦𝐦 200 300 400

Frequency Hz 11.78 8.25 6.06

Galloping m/s 8.30 4.90 3.30


Speed ratio - 2.15 1.81 1.66

Modal Mass Kg 0.0174 0.0208 0.0244

Chi - 3.96E-4 3.53E-4 3.30E-4 Table 3.11 Results bluff body length variation

Figure 3.21 Effect of bluff body length variation


72

Despite we shouldn’t have effect on speed ratio, we can see that this isn’t true

because when we have an increment of the length we can obtain a small decrement of

the speed ratio.

This can be explained considering that modifying the dimension of the bluff body It is

possible to modify also the stiffness of the structure.

In theory only, the beam parameters should modify this value, instead also the bluff

body can create some effect and in order to verify this aspect we will see the case of

section bluff body after.

Anyway, at the end we conclude that this parameter can be helpful in order to

decrease the speed ratio, because if we increase the length, there is a decrement of

that value.

Again, also in this case we can notice that to decrease the speed ratio we must deal

the decrement of the galloping velocity and this could be a negative aspect if we want

to maintain a high value.

(a) (b)

(c) (d)

Figure 3.22 Time response bluff body length variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind


73

3.3.8 Variation bluff body side

Before we analysed the effect of the bluff body length on the prototype, and we found

that dimension of the bluff body can interfere with the stiffness of the structure.

So now we can explore the second dimension of the tip, the side of the section.

From formulas we can find that there isn’t the effect of the side on both velocity:

𝑈𝑣 ≅

√𝑘

𝑛𝑠√𝐿𝑡𝑖𝑝√𝜌𝑡𝑖𝑝

(3.159)

𝑈𝑔 ≅ 4

ℎ√𝜌𝑡𝑖𝑝√𝑘

𝜌𝑎𝑖𝑟𝑎1√𝐿𝑡𝑖𝑝

(3.160)

For this reason we expect to find constant values when we move this parameter,

because in theory all of them are independent from this one.

But if the conclusion that we saw before is true, we should find a variation of stiffness

also in this case and we can confirm that also dimensions of bluff body have effect on

mechanical structure.

Figure 3.23 Modal shape bluff body side variation


74

Variation: 𝐃𝐛𝐥𝐮𝐟𝐟 𝐛𝐨𝐝𝐲 𝐦𝐦 20 30 40

Frequency Hz 18.61 14.46 11.78

Galloping m/s 13.90 10.30 8.30


Speed ratio - 4.55 2.89 2.15

Modal Mass Kg 0.0107 0.0141 0.0174

Chi - 5.49E-4 4.61E-4 3.96E-4 Table 3.12 Results bluff body side variation

Figure 3.24 Effect of bluff body side variation

With these results we can confirm what we said previously, the characteristic of the

bluff body can change the stiffness of the prototype.

In conclusion we can see that in order to decrease the value of speed ratio we need to

increase the value of the bluff body side.

We have to consider also this parameter in order to design a prototype, the effect that

this can have on the structure.

(a) (b)


75

(c) (d)

Figure 3.25 Time response bluff body side variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

3.3.9 Variation piezo patch

Now It is interesting to see the effect of the piezo parch on the structure, because this

component of course can change the stiffness of the beam and can increase also the

damping ratio with the electrical component.

Of course, we are interested also to how much power can store this device, and for

this reason we can compare two different components:

• PPA1011;

• PPA1021.

Both of these are the same length, for this reason the first domain doesn’t change the

dimension, but different width.

In fact, PPA1011 has 25.4 mm and PPA1021 has 10.33 mm.


76

Figure 3.26 Modal shape piezo patch variation

Variation: Piezo Patch PPA 1011 PPA 1021





Modal Mass Kg 0.0174 0.0181

Chi - 3.96E-4 2.02E-4 Table 3.13 Results piezo patch variation

By changing the piezo patch of course we can change the mechanical structure and

characteristics, because in this particular case we have two patches with different

width.

Another important effect is that using the PPA1021 we can obtain a different modal

shape, increasing the deformation of the piezo patch, so in theory in this way we can

increase the coupling between the mechanical and electrical domain.

But we have to remember that for this piezo patch we have also a low width, for this

reason there is less part that can store the energy and in real the coupling effect is

reduced.


77

(a) (b)

(c) (d)

Figure 3.27 Time response piezo patch variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

As we expected, we can see that there is a positive effect of the PPA1011 because,

despite we have less deformation of piezo patch, we can store more energy and of

course we are more interested to this aspect.

Of course, there is also a positive aspect for the PPA1021, because if we want to

reduce the dimension of the prototype, we should prefer this second piezo patch

respect to previous one, because in this way we could also reduce the dimension of

the beam thus obtaining the possibility to reduce all the structure.

In fact if there is a reduction of the width, we can also reduce the length of the beam

leaving constant the aspect ratio of the beam with an almost constant damping ratio.


78

3.3.10 Variation number of beam

We have to study which is the behaviour of the structure if we use two beams instead

that one.

In transversal prototype it can be useful to use two cantilever beams in order to

reduce the risk to have a moment on the bluff body axis.

If we use two beams, we must have on both two piezo patches.

Figure 3.28 Modal shape number of beam variation

Variation: Number Beam 1 2






Chi - 3.96E-4 3.99E-4 Table 3.14 Results number of beam variation

Modifying the number of beam has the effect to change the frequency of the structure

because we increase the total stiffness, but the modal shape remains the same.

Here we don’t consider the effect of this parameter on the damping ratio, but

obviously we have associate with this also a change of that value.

We can expect that if there is an increment of beam, damping ratio can only increase,

because is the same if we increase the width of one beam.


79

(a) (b)

(c) (d)

Figure 3.29 Time response number of beam variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind

At the end we can conclude that the second beam is used only in order to protect the

structure from the moment, also if we have a lot of disadvantages for this

configuration.

3.3.11 Resistance effect

Until now we saw the response in time always with the same resistance. In fact we’ll

see that exist an optimal resistance that can improve the power and reduce the tip

displacement.

To do that we have to simulate the numerical case for each resistance at the same

velocity, in this way it is possible to obtain an optimal case.

This study has been done to understand which resistance is useful to use in our

prototype, to improve the characteristic of piezo patch.


80

Figure 3.30 Modal shape resistance variation

Variation: 𝑹 𝐤𝛀 10 100 1000

Frequency Hz 11.78 11.78 11.78

Galloping m/s 6.80 8.30 7.00


Speed ratio - 1.76 2.15 1.81

Modal Mass Kg 0.0174 0.0174 0.0174

Chi - 3.96E-4 3.96E-4 3.96E-4 Table 3.15 Results resistance variation

Figure 3.31 Effect resistance variation


81

From numerical point of view It is possible to notice that also galloping velocity is

linked to the resistance.

Of course, this effect isn’t present inside the simplify formula, because in that case we

consider only the mechanical structure, without the effect of piezo patch, instead in

numerical simulation this effect is taking in account.

Over this effect we have to consider also which is the efficiency of the piezo, so we

decide to use the resistance to increase the energy stored respect to use it for

changing the velocities.

Anyway, this parameter can be used to modify the speed ratio.

(a) (b)

(c) (d)

Figure 3.32 Time response number of beam variation. (a) Voltage Vs wind, (b) tip displacement Vs wind, (c) power Vs wind and (d) PPA displacement Vs wind


82

In Figure 3.33 is showed power and tip displacement at constant velocity, for different

resistance values.

(a) (b)

Figure 3.33 Time response number of beam variation. (a) tip displacement Vs resistance, (b) power Vs resistance

From the last two plots It is possible to define which is the optimal resistance for the

reference case, and this is 150 kΩ.

If fact in that point I can minimize the tip displacement maximize the power that I can

store.

3.3.12 Longitudinal vs transversal

In this last part we want to see which is the different between two prototypes, in

particular we’ll focus on the parameters that It is possible to obtain in the two cases.

We just saw in the mathematical chapter which are difference between the two

prototypes, because there is an important difference to the aerodynamic force that

there is on the structure.

If we consider the same length of the bluff body, there is a problem of space about the

longitudinal prototype, because it extends more than the transversal one.

Variation: Prototype Transversal Longitudinal






Chi - 3.96E-4 1.19E-4 Table 3.16 Results different prototype


83

Figure 3.34 Modal shape different prototype

From results we can notice that with the longitudinal prototype It is possible to obtain

a structure with a small speed ratio respect to the transversal structure, but we have a

big disadvantage on the galloping velocity, because is really small, and our target is to

increase this velocity.

At the end we can conclude that transversal prototype is better than longitudinal for

two important things:

• Transversal design is more compact than longitudinal, for this reason is more

suitable for a real application;

• With transversal we have a high value of galloping velocity, so is better if our

target is to increase as much as possible the limit value of the range work.

3.4 Final conclusions and prototype design

After all these simulation, we can give some consideration in order to build a suitable

prototype:

• In order to obtain a low value of speed ratio:

o Decrease bluff body density;

o Decrease damping ratio;

• Bluff body side is a parameter that we have to consider, because this can

modify the final results;

• We expect transversal prototype is better than longitudinal prototype

• There is an optimal resistance and we must find this in order to obtain the

best result as possible;


84

• If there is the interest to reduce the dimension of the prototype, It is possible

to consider a variation of the piezo patch, using a smaller patch;

• It is important to consider if we must use one or two beams;

• In order to modify damping ratio, we’ll use only one parameter, and this is the

length of the second region.

After all these considerations we decided to build both type of prototype, in order to

see from an experimental point of view which is the best type between longitudinal

and transversal.

We decide to design all the prototypes with bigger bluff body, more than 20 cm,

because we could have some technological problems with smaller prototype.

In fact, we design a prototype with a carved bluff body, in order to reduce the

equivalent density, in this way It is possible to use better material without the risk of

break.

If we use for example polystyrene we could have a really small density, but it isn’t

possible to guarantee the solidity of the prototype.


Piezo - PPA1011

Number of piezo for beam - 2

Clamp position - 0

Connection - Series

Beam width 𝒘𝒔 mm 28

Beam thickness 𝒕𝒔 mm 0.5

Beam material - Steel

Bluff body shape - Square

Dimension bluff body side 𝑫𝒕𝒊𝒑 mm 40

Resistance - 100 𝑘𝛺

Table 3.17 Constant parameters


85

In Table 3.18 Parameters piezo patch PPA1011 there are all parameters for piezo

patch PPA1011 at clamp position zero.

Clamp 0

Capacitance nF 100

Mass G 3

Full Scale Voltage Range V 120

Thickness mm 0.71

Effective stiffness N/m 446.28

Effective mass g 0.614

Deflection mm 20.5 Table 3.18 Parameters piezo patch PPA1011

Figure 3.35 Dimension of piezo patch PPA1011

3.4.1 Longitudinal prototype

For longitudinal case we realised a prototype with six different configurations. The

prototypes present a density of 250 𝑘𝑔

𝑚3, and this parameter is always constant for all

six configurations.

The first classification is about the second region length as we can see:

• A: in this first case the length of the second region is 2 cm;

• B: in this second case we don’t have the second region, because we remove

the space between the piezo patch and bluff body.

The sub classification is about the bluff body length:

• 20 cm;

• 30 cm;

• 40 cm.


86

At the end the six different configurations are A20, A30, A40, B20, B30 and B40.

We decided to study the effect of the second region to see the effect of this

parameter on damping ratio. Instead using different length of bluff body we can study

the effect of this parameters on the speed ratio, to see if the prototype is able to start

the oscillation if it becomes more compact.

3.4.2 Transversal prototype

In this second case we decided to design a prototype like the longitudinal one, to

compare the behaviour of both in the same way.

For this reason, we maintained the same piezo patch with the same characteristic of

the beam.

The only difference in this case is that we must put two beams instead of one because

the risk of moment with big bluff body is really high. Of course, in this case we have

four piezo patches, two for every beam.

For transversal layout we designed three different prototypes but leaving the same

length of bluff body.

We chose a dimension of 40 cm, in this way we can reduce the interaction of beams

on the wake, this should improve aerodynamic effect of the structure.

Two prototypes differ only for the density of bluff body, in this way we can study from

an experimental point of view the parameter, and we chose to use a density that can

give to us a high value of speed ratio and the second density that can reduce that

value.

In this case the name for transversal prototype is indicated with:

• C: This prototype has a density of 𝜌𝑏𝑙𝑢𝑓𝑓 𝑏𝑜𝑑𝑦 = 90𝑘𝑔

𝑚3, in this way we should

obtain a light structure;

• D: of course, in this case the procedure is the same as before and

𝜌𝑏𝑙𝑢𝑓𝑓 𝑏𝑜𝑑𝑦 = 300 𝑘𝑔

𝑚3.

The second region is important because also in this case we can use this value in order

to modify the damping ratio.

Because we can’t know the damping ratio from a numerical point of view, we need to

design a prototype with the possibility of change the second region:

• 5 cm: C5 and D5 prototypes;

• 9 cm: D9 prototype.


87

(a) (b)

(c) (d)

(e) (f)

Figure 3.36 Different longitudinal prototypes. (a) A40, (b) A30, (c) A20, (d) B20, (e) B30 and (f) B40


88

(a) (b)

Figure 3.37 Different transversal prototypes. (a) C5 and (b) D9

EXPERIMENTAL VALIDATIONS

89

4 EXPERIMENTAL VALIDATIONS

This chapter explains how the experimental tests are done and it presents all results

obtained, with shaker and wind tunnel tests.

Before to show the results, we show all the materials used to do this experimental

work.

The results are showed for each prototype, and it is possible to compare the real

results with numerical values evaluated with the analytical model described in the

chapter 2.

In the chapter are described two types of tests:

• motion-imposed tests with electromechanical shaker for the identification of

the modal parameters of the analytical model;

• wind tunnel tests at different wind speeds for the evaluation of amplitude of

oscillation and power recovered.

In particular, in section 4.1 the experimental setup is presented for both typology of

tests. Instead in section 4.2 experimental results are provided.

4.1 Experimental setup

4.1.1 Motion-imposed tests components

Motion-imposed tests are based on the vibrating table used to study the

electromechanical behaviour of the prototype. Respect than aerodynamic forces,

acting on the bluff body, in this experiment the input force is introduce from the

clamp position, at the starting point of the beam. In Figure 4.1 is showed a scheme of

this experiment configuration.


90

(a) (b)

Figure 4.1 (a) experimental setup scheme of a shaker and (b) shaker setup for prototype B20

Prototype is mounted in vertical, in this way it is possible to avoid gravitational

contribution on the response.

The shaker used is an electromechanical model and it is able to generate sinusoidal

forcing. We use sinusoidal forcing to excite a range of frequencies. This is a method

used to find the natural frequency.

In this configuration we have an open-loop controlled, this meaning there aren’t

feedback information from the shaker about the oscillation.

In our experiment we are interested to investigate the behaviour of the system, only

for the first natural frequency.

From the personal computer we can set the frequency range, setting the maximum

and the minimum values and the acquisition time.

The shaker is controlled by:

• Personal computer;

• Amplifier;

• Portable sensor measurement system.

From the personal computer we can set the frequency range, setting the maximum

and the minimum values and the acquisition time.

Furthermore the PC must collect all the measurements from sensors, like lasers or

accelerometers measures. Through MATLAB it is possible to acquire information from

National Instrument’s devices.

Every signal, input or output, is collected by a portable sensor measurement system,

this is a device able to have different modulus. For example, It is possible to have an

input and an output module in the same device, in this way we can link only this to the

PC.


91

In our case we use a c-DAQ-9178, with analog I/O or digital I/O measurements, using

the following modulus:

• NI-9234: four channels with the range of voltage ±5 Volt. This module is used

for accelerometers. It is able to acquire IEPE signal without any further

conditioning equipment;

• NI-9239: analog input module, with a range of ±10 Volt with four channels.

This is used to acquire laser signals;

• NI-9229: this module is used to acquire piezoelectric voltage, and it can work

in range ±60 Volt. Also in this case is an analog input module with four

channels;

• NI-9269: analog output module, this is the last module we use in our

experiment and is the module we need in order to control the shaker. Using

this we can send the signal to the amplifier.

To increase the voltage output from the c-DAQ we need to insert an amplifier.

Using the amplifier, we can manually set the level of amplification, and it is important

to leave constant this value to have the same set of measurements without some

different from oscillations shaker.

Now we can focus on signal acquisition devices. We use different sensors in order to

check the validation of measurements:

• Laser sensor;

• Accelerometer sensor.

We decide to measure three different point using lasers. The first is fixed to the base,

because we need the oscillation of the shaker, in this way we can have the real

displacement of the other points of the structure.

The second is on the end of the piezoelectric patch, in this way it is possible to check

which is the deformation of the piezo, and we can see how much we can still deform

before the brake point.

The last point is at the end of the bluff body, in order to study which is the maximum

amplitude that the body can reach.

In Table 4.1 it is possible to see which lasers we use in the test.

M7L/10 M7L/20 M7L/100

Manufacturer MIKROELEKTRONIK© MIKROELEKTRONIK© MIKROELEKTRONIK©

Operational range 10 mm 20 mm 100 mm

Voltage range ±10 V ±10 V ±10 V

Resolution 0.5 µm 0.9 µm 6 µm

Maximum gap 23.8-33.8 mm 55-75 mm 170-270 mm Table 4.1 Characteristics of lasers


92

We have only two accelerometers, and we have to put the first on the base, in this

way It is possible to have the accelerometer of the shaker, and the second we can put

at the end of the bluff body.

In Table 4.2 below It is possible to see which the type of accelerometer we use. It is a

really light device, in this way we don’t risk to change in the mass of the bluff body,

and then we can obtain the real value without a high error.

In Table 4.3 there is a summary of channels used in experimental tests.

352A24

Manufacturer PCB-PIEZOTRONICS©

Operational range ±50 g

Voltage range ±5 V

Resolution 0.0002 g

Sensitivity 100 mV/g

Sensing element Ceramic Table 4.2 Characteristics of accelerometer

L Constraint Laser placed at the clamp position

L PPA Laser placed at the end of PPA

L Tip Laser placed at the free end of the bluff body

A Constraint Accelerometer placed at the clamp position

A Tip Accelerometer placed at the free end of the bluff body

V PPA Voltage output from PPA terminal

Table 4.3 List of channels used during the experimental tests

(a) (b)

Figure 4.2 Experimental devices. (a) c-DAQ module and (b) electromechanical shaker


93

4.1.2 Wind tunnel setup

Wind tunnel is used to reproduce the real effect of aerodynamic forces acting on the

bluff body.

We have to study different wind velocities, for this reason the wind tunnel must be

controllable to set the desired speed.

To measure the velocity, a differential pressure sensor is used to measure the wind

kinetic pressure at the inlet and so the velocity of the wind.

During our experiments we use the wind tunnel located at the Department of

Aerospace Science and Technology Aerodynamics laboratories of the Politecnico di

Milano.

Inside the wind tunnel we use only the accelerometer sensors and not lasers because

we need to avoid interactions with the wind flow. This because uniform flow is

required during our tests. In Table 4.4 are showed the properties of the wind tunnel

we used.

Wind tunnel

Type Closed-jet

Section width x height 1.5 x 1 m

Max wind speed [m/s] 55

Wall boundary layer thickness [mm]

35

Control Open-loop

Table 4.4 Properties of wind tunnel

Wind velocity is obtained from a pitot tube and this value is send to a processor that

can control and set the velocity. Using the pitot tube it is possible to compute the wind

speed measuring the wind kinetic pressure according to:

𝑣 = √2(𝑝𝑡𝑜𝑡 − 𝑝𝑠𝑡)

𝜌𝑎𝑖𝑟 (4.161)


94

(a) (b)

Figure 4.3 (a) wind tunnel with prototype, (b) instrumentation used in wind tunnel

4.2 Identification of the modal parameters

In this paragraph we are interested to compare experimental results with numerical

result.

Trough the motion-imposed tests we want to identify five modal parameters that we

need inside the numerical program. Natural frequency, modal mass and damping

ration can be obtained from experimental tests, as we will see in the second part of

this paragraph.

Instead using FRF it is possible to obtain the capacitance and the electromechanical

coupling value for the piezoelectric patch we are using.

These five parameters, that we estimate in this paragraph, will be used inside the

program to simulate the aero-electromechanical model in the third part of this

chapter.

In Table 4.5 are showed the main characteristics of prototypes used in our tests. In

Figure 3.36 and Figure 3.37 these prototypes are showed.


95

Prototype Configuration Buff body

length [cm]

Bluff body

density [𝑘𝑔

𝑚3]

Second domain length

[cm]

A40 Longitudinal 40 250 2



B40 Longitudinal 40 250 -



C5 Transversal 40 90 5

D5 Transversal 40 300 5

D9 Transversal 40 300 9

Table 4.5 Characteristics of the prototypes used

4.2.1 Transfer function

Transfer functions are used to estimate two parameters: the capacitance of PZT patch

and the electromechanical coupling.

To reach our purpose we use four FRF:

• Transfer function from base acceleration to the tip acceleration;

• Transfer function from base displacement to the tip displacement;

• Transfer function from voltage to the base acceleration;

• Transfer function from voltage to the tip displacement.

In Figure 4.4 is showed the scheme, as example we use only longitudinal model, where

all measurement acquisition points of the prototype are indicated. In particular we

indicate:

• Clamp position;

• PPA position;

• Tip position.

These points are used to measure displacements and accelerations with lasers and

accelerometers, to compute transfer functions.


96

Figure 4.4 Scheme of measurement acquisition points for longitudinal prototype

Transfer functions for motion-imposed tests are obtained from the model we

developed in the second chapter. The only difference is about the force acting on the

structure. In fact, here there is the shaker force instead aerodynamic forces.

{

𝑀∗�̈� + 𝑅�̇� + 𝐾𝑞 + 𝜒𝑉𝑝 = 𝐹𝑠ℎ𝑎𝑘𝑒𝑟

𝑉𝑝

𝑅+ 𝐶𝑝�̇�𝑝 − 𝜒�̇� = 0

(4.162)

Where 𝐹𝑠ℎ𝑎𝑘𝑒𝑟 is the input force. From this system we can see that aerodynamic force

is zero and we must consider only the force of the shaker.

We need to express this force using a coordinate system, and we can choose the

displacement of the shaker as the coordinate.

Taking the longitudinal prototype, we can show which is the effect of this force on the

body. The same procedure can be adopted for transversal configuration.

We have to rewrite the displacement for each domain, considering the shaker effect.

Because we are using the longitudinal model, there are to consider four domains.

𝑦1′(𝑥, 𝑡) = 𝑦1(𝑥, 𝑡) + 𝑧(𝑡)

𝑦2′(𝑥, 𝑡) = 𝑦2(𝑥, 𝑡) + 𝑧(𝑡)

𝑦3′(𝑥, 𝑡) = 𝑦3(𝑥, 𝑡) + 𝑧(𝑡)

𝑦4′(𝑥, 𝑡) = 𝑦4(𝑥, 𝑡) + 𝑧(𝑡)

(4.163)

Where z is the coordinate of the shaker displacement as it is possible to in Figure 4.5.


97

Figure 4.5 Prototype scheme and shaker force

Now It is possible to express also the shaker displacement with a phase and an

amplitude, because is a sinusoidal force:

𝑧(𝑡) = 𝑍0 cos(𝛺𝑡) (4.164)

It is possible to replace these equations inside the kinetic energy formula in this way

we can obtain a new kinetic part in modal coordinate:

𝐸𝑘 =

1

2∫ �̇�1


𝑉𝑠1

+ 21

2∫ �̇�1


𝑉𝑝1

+1

2∫ �̇�2


𝑉𝑠2

+1

2∫ �̇�3


𝑉𝑠3

+1

2∫ �̇�3


𝑉𝑏3

+1

2∫ �̇�4


𝑉𝑏4

(4.165)

And deriving this, we can obtain:

𝛿𝐸𝑐 = 𝛿�̇�𝑀∗�̇� + 𝛿�̇�𝑚∗�̇� (4.166)

From the last formula we can see the kinetic energy for shaker test, and the first part

is completely the same with the previous case of aerodynamic force.

The second part in associate with the shaker table and we have to express the value of

𝑚∗:

𝑚∗ = ∫ 𝑚1𝜙1𝑑𝑥1

𝐿1

0

+ ∫ 𝑚2𝜙2𝑑𝑥2

𝐿2

0

+ ∫ 𝑚3𝜙3𝑑𝑥3

𝐿3

0

+ ∫ 𝑚4𝜙4𝑑𝑥4

𝐿4

0

(4.167)


98

And at the end the final system is:

{

𝑀∗�̈� + 𝑅�̇� + 𝐾𝑞 + 𝜒𝑉𝑝 = −𝑚∗�̈�

𝑉𝑝

𝑅+ 𝐶𝑝�̇�𝑝 − 𝜒�̇� = 0

(4.168)

Now we can rewrite also the voltage of the piezo patch using the second formula of

the system. Before to obtain the final form, we need to pass through the state space

representation to Laplace form, and then:

𝑉𝑝 =𝜒𝑠

1𝑅

+ 𝐶𝑝𝑠𝑞 (4.169)

Writing the first equation in Laplace form, and substituting (4.168) to (4.167), it is

possible to obtain:

𝑀∗𝑞𝑠2 + 𝑅∗𝑞𝑠 + 𝐾∗𝑞 +

𝜒2𝑠

1𝑅

+ 𝐶𝑝𝑠𝑞 = −𝑚∗𝑠2𝑧 (4.170)

And at the end we can rewrite modal coordinate, damping and stiffness as:

• 𝑞 =𝑦

𝜙;

• 𝑅∗ = 2ℎ𝑀∗𝜔0;

• 𝐾∗ = 𝜔02𝑀∗.

Where the frequency is associate to the first mode of the structure and y is the

displacement of the prototype.

Of course, we have to specify also the 𝜙 value, because in this way It is possible to

select which point of the prototype we consider for displacement measurements.

Final form of transfer functions that we use in our experiment are expressed as:

• Transfer function from base acceleration to the tip acceleration:

�̈�𝑡𝑖𝑝

�̈�=

−𝑚∗ (1𝑅

+ 𝑐𝑝𝑠) 𝑠2

𝑀∗ [(1𝑅

+ 𝑐𝑝𝑠) (𝑠2 + 2ℎ𝜔0𝑠 + 𝜔02) +

𝜒2𝑠𝑀∗ ]

(4.171)


99

• Transfer function from base displacement to the tip displacement:

𝑦𝑡𝑖𝑝

𝑧=

−𝑚∗ (1𝑅

+ 𝑐𝑝𝑠) 𝑠2

𝑀∗ [(1𝑅

+ 𝑐𝑝𝑠) (𝑠2 + 2ℎ𝜔0𝑠 + 𝜔02) +

𝜒2𝑠𝑀∗ ]

(4.172)

• Transfer function from voltage to the base acceleration:

𝑉𝑝

�̈�=

−𝑚∗𝜒𝑠

𝑀∗ [(1𝑅

+ 𝑐𝑝𝑠) (𝑠2 + 2ℎ𝜔0𝑠 + 𝜔02) +

𝜒2𝑠𝑀∗ ]

(4.173)

• Transfer function from voltage to the tip displacement:

𝑉𝑝

𝑦𝑡𝑖𝑝=

𝜒𝑠

1𝑅

+ 𝑐𝑝𝑠 (4.174)

First two transfer functions are identical and for this reason we can use it to compare

the correctness of measurements.

In theory we should obtain, from experimental, two similar curves if both

measurements are correct.

Figure 4.6 Comparison between laser-accelerometer sensors for prototype A30


100

In Figure 4.6 it is possible to notice the same behaviour of two transfer function, using

an accelerometer and a laser in the same point of the bluff body.

It is clear that if we change the position of one of the two sensors, the results between

these will be a little bit different, because there is the effect of 𝜙(𝑥), that in our case is

1 because we consider the end of the tip body.

Another important consideration is about the transfer function between voltage and

tip displacement. In fact, if we see the equation we can notice that there is only a pole,

beyond that a zero in the origin, and this depends from the capacitance parameter.

We can write:

𝑠 = −

1

𝑐𝑝𝑅 (4.175)

In this formula R, that is the resistance load, is a constant value and we know it.

So, the pole can change his position only moving and set the value of the capacitance

of the piezoelectric patch. To obtain a correct value, we need to set it using different

values of resistance, in fact for each value we must obtain a good approximation

between numerical transfer function and an experimental one.

Transfer function between the base acceleration and the voltage of the piezo patch

can be used to set the correct value of the χ parameter, a value that we can’t find

using some tests.

(a) (b)

(c)


101

Figure 4.7 Motion-imposed tests: comparison between experimental and numerical results in term of FRF prototype A20 with R = 100 kΩ. (a) A Tip/A Constraint, (b) V PPA/A Constraint and (c) V PPA/L Tip

(a) (b)

(c)

Figure 4.8 Motion-imposed tests: comparison between experimental and numerical results in term of FRF prototype A20 with R = 10 kΩ. (a) A Tip/A Constraint, (b) V PPA/A Constraint and (c) V PPA/L Tip

(a) (b)


102

(c)

Figure 4.9 Motion-imposed tests: comparison between experimental and numerical results in term of FRF prototype A20 with R = 1 MΩ. (a) A Tip/A Constraint, (b) V PPA/A Constraint and (c) V PPA/L Tip

(a) (b)

(c)

Figure 4.10 Motion-imposed tests: comparison between experimental and numerical results in term of FRF prototype C5 with R = 100 kΩ. (a) A Tip/A Constraint, (b) V PPA/A Constraint and (c) V PPA/L Tip


103

(a) (b)

(c)

Figure 4.11 Motion-imposed tests: comparison between experimental and numerical results in term of FRF prototype D9 with R = 100 kΩ. (a) A Tip/A Constraint, (b) V PPA/A Constraint and (c) V PPA/L Tip

4.2.2 Estimation of modal parameters

In this section experimental tests for estimation of modal parameters are explained.

We will provide:

• Natural frequency;

• Modal mass;

• Modal shape.

To obtain these three parameters, we have to perform different experiments and in

this paragraph we will explain all procedures.

4.2.2.1 Natural frequency

At first it is important to remember that we work only with the first natural frequency

and for this reason we will provide only this value, without looking for more modes.


104

To do this we analyse the behaviour of the structure using a frequency sweep test. It is

appropriate to set a large range of frequency to study, because we can’t know the

correct value of the first mode. For a first estimate we can rely on numerical results.

In order to find the value of first natural frequency we should remember that natural

frequency is associate to the resonant phase and so:

• Maximum peak on the magnitude plot;

• Phase to 90°.

Prototype Analytical 𝝎𝟏[Hz] Experimental 𝝎𝟏[𝑯𝒛]

A40 4.55 3.49

A30 5.90 5.30

A20 8.50 8.70

B40 6.84 6.70

B30 8.90 7.47

B20 12.67 12.65

C5 19.30 21

D5 7.80 11.86

D9 4.95 6.50 Table 4.6 Comparison between numerical and experimental natural frequency

In Table 4.6 are showed the comparison between analytical and experimental results.

For longitudinal prototype good results are achieved for almost all prototypes. The

exceptions where we can find important differences are the prototypes A40 and B30

where the variation between analytical and experimental values are respectively

30.4% and 19.1%. These errors can be due to the approximation of the structure with

the beam theory.

For transversal prototypes we have three results with important variation between

expected and real values. Experimental results show a higher natural frequency value

and this can be explained with the rigidity introduced by the two beams that probably

analytical model can’t model it perfectly. The result is a less rigid structure in the

mathematical model, with a consequent lower value of first frequency.

Anyway also for transversal configuration the maximum difference between numerical

and real value is around 30%, as in the longitudinal configuration. We need to take in

to account this variation when we predict first frequency for a new prototype.

4.2.2.2 Modal mass

The effective modal mass provides a method for judging the significance of a vibration

mode.

We will study only the modal mass associated to the first natural frequency, because

this is the only mode we want to consider in our model.

In order to obtain this value, we can write the formula for the natural frequency:


105

𝜔 = √𝑘

𝑀∗ (4.176)

From the experimental we know the value of the frequency, but stiffness and modal

mass are unknow.

Now we can repeat the same experiment, with the same prototype, but increasing the

mass, adding to the end of the bluff body a concentrated mass. In this way we will

have a structure with the same stiffness, because the concentrated mass doesn’t

affect this value. We can repeat the experiment and it is possible to obtain the value

of the new natural frequency:

�̃� = √𝑘

𝑀∗ + 𝛥𝑚 (4.177)

Considering 𝑘 constant in these two formulas, we can rewrite:

𝑀∗ =

𝛥𝑚�̃�2

𝜔2 − �̃�2 (4.178)

Using this method, it is possible to obtain the experimental value of the modal mass.

For the analytical problem we can calculate this value from the modal shape, and we

are interested to study if the program can approximate well it.

In Figure 4.12 it is possible to see which type of mass we used in our experimental

tests. For each prototype we decided to use always the same concentrated mass in

order to have a repetitive research methodology.


106

(a) (b)

Figure 4.12 Example of concentrated mass for (a) prototype A20 and (b) prototype A40

Prototype Analytical 𝑴∗[Kg] Experimental 𝑴∗[Kg]

A40 0.0142 0.0156

A30 0.0152 0.0126

A20 0.0172 0.0092

B40 0.0141 0.0150

B30 0.0151 0.0125

B20 0.0171 0.0092

C5 0.0150 0.0201

D5 0.0371 0.0580

D9 0.0551 0.0721 Table 4.7 Comparison between numerical and experimental modal mass

In Table 4.7 are showed all the results for longitudinal and transversal prototypes.

From this comparison is clear to see that for both configurations, the estimation of the

real modal mass value is not correct. Only for prototypes A40 and B40 we obtained a

good approximation of the real value. For the rest of prototypes it is possible to see a

really high difference between the approximation and the real value, until 86.95% of

difference. This problem had already been observed in research by Marsetti [39], with

a difficult estimation of the modal mass.

At the end it is possible to conclude that for numerical simulation of the response in

time under the aerodynamic forces, it is better to take in to account this error and to

use the real value, if we are in possession of this data, for example from previous

experimental tests or exist prototypes.


107

4.2.2.3 Modal shape

The last modal parameter we need to obtain is the modal shape associated with the

first natural frequency.

For this test we don’t use sweep frequencies but a constant oscillation of the shaker at

the first natural frequency of the body. Under this forcing the structure will be

oscillate with a constant amplitude and shape.

To recreate the path associated to the modal shape, we need to collect more than two

measurement points displacement.

We repeat, at constant frequency, the test in different points of the structure, using a

laser to measure the relative displacement, and after we can reconstruct the shape of

the natural mode. To have a correctness of the results we need to set a constant

amplitude of the shaker, in this all the points will be referred to the same experiment.

Modal shape is used to calculate the electromechanical coupling, for this reason it is

really important that the numerical result can be similar to the real. If not we should

have also an error on the coupling estimation in numerical program with consequent

error also in the recovered power.

Figure 4.13 show an example of numerical modal shape, taking as example the

prototype A30. In the figure four region of longitudinal prototype are showed and it is

important to see which the deformation at the end of the first region is. In fact our

purpose is to increase the deformation at the end of the piezoelectric patch, in this

way we can increase the power output from the terminal.

Figure 4.13 Analytical modal shape prototype A30


108

In Figure 4.14 different analytical results for three prototypes are showed. In this

figure it is possible to observe the effect of the bluff body length on the PZT

deformation. It is legitimate to think that model B20 will have an higher coupling

factor and for this reason the possibility to recover more power.

Figure 4.14 Piezo patch deformation for models B

In Figure 4.15 all comparisons between analytical and experimental results are

provided.

(a) (b)


109

(c) (d)

(e) (f)

(g) (h)


110

(i)

Figure 4.15 Modal shape comparison between analytical and experimental results for (a) A20, (b) A30, (c) A40, (d) B20, (e) B30, (f) B40, (g) C5, (h) D9 and (i) D5

From results it is possible to confirm the good approximation of the numerical

program respect than the real behaviour of the system.

In Table 4.8 are provided PZT deformations for longitudinal prototypes obtained from

experimental test. As we expected in Figure 4.14 the bluff body length influences the

deformation at the end of the piezoelectric patch. This result is important to confirm

the validity of the numerical program for this value.

Piezo Deformation

A 40 0,007

A 30 0,009

A 20 0,014

B 40 0,011

B 30 0,019

B 20 0,022 Table 4.8 Experimental piezo patch deformation ratio longitudinal prototypes

4.2.3 Damping ratio

Damping ratio is not constant in our prototypes but it will increase with the oscillation

amplitude. There two things to consider to explain this effect. The first is about the

possible non-linearity of the mechanical system, that we can’t estimate it without

experimental test, and the second is the aerodynamic friction on the bluff body. In our

experiment we will consider both this effect only in one value.

In our program we won’t consider an increment of the damping ratio, but we will

model it as a constant value.

There are two different methods to obtain the damping ratio:


111

• Phase derivative: using the sweep excitation I can watch the behaviour of the

phase, and in this way, there is the possibility to compute the value of the

damping ratio:

ℎ = −

1

𝜔𝑛 |𝜕𝜙(𝑥)𝜕𝛺

|

(4.179)

• Decay: in this case we use an impulse excitation and from the free decay it is

possible to derive the damping ratio. In Figure 4.16 is showed an example of

free decay obtained during the tests.

Figure 4.16 Example of decay on prototype A30

𝛿 = ln

𝑈𝑛

𝑈𝑛+1 ℎ =

𝛿

2 ∗ 𝜋 (4.180)

To compute the damping value we used both methods at different amplitude

oscillations. In this way it was possible to interpolate all the points in order to obtain a

plot of the damping ratio in function of the displacement at the end of the PZT patch.

To have a unique method for all prototypes we decided to take, as constant value, the

damping ratio associated to a displacement of the piezoelectric patch of 0.5 mm.

In Figure 4.17 are showed some interpolations for different prototypes.


112

(a) (b)

(c) (d)

(e) (f)

Figure 4.17 Interpolation for damping ratio (a) A20, (b) B30, (c) A40, (d) B40, (e) C5 and (f) D9

Prototype Non-dimensional damping 𝒉

A40 1.5%

A30 1.2%

A20 1.7%

B40 2.4%

B30 2.1%

B20 3.1%


113

C5 1.1%

D5 2.4%

D9 2.2% Table 4.9 Damping ratio value for prototypes at 0.5 mm displacement of PPA

Looking at the Table 4.9 we see a dependence on the presence of the second region,

because we could expect an increment of mechanical stiffness when the second

region is reduced, and in fact this trend is confirm for all three configurations, in

models A and B.

4.2.4 Resistance effect

Resistance load is an important parameter for our electromechanical system. In fact

using this value it is possible to change the maximum recovered power. Our purpose is

to find which is the best resistance load solution in order to obtain the maximum

power from a piezoelectric patch. In our experiments we tested only 10 kΩ, 100 kΩ

and 1 MΩ and from Figure 4.18 we expect that the best solution is 100 kΩ.

Figure 4.18 Numerical simulation to show the effect of the resistance load on the power

Resistance Power [mW]

100 kΩ 3.66

10 kΩ 0.76

1 MΩ 1.43 Table 4.10 Power obtained at the same amplitude for model A20


114

Figure 4.19 Power response prototype A20

Table 4.10 and Figure 4.19 show experimental results obtained at a constant

amplitude for three different resistance load values. As we expected from the

numerical simulation, the best solution is 100 kΩ and from this point we will use it for

our experiment tests.

4.3 Validation of aero-electromechanical model

In this section we are going to analyse the response of the model under the

aerodynamic forces. Wind tunnel is used for this purpose, in this way we can

reproduce the aerodynamic forces on the bluff body.

In addition to the experimental results we will provide also numerical solutions

obtained from mathematical model developed in Chapter 2, where aerodynamic

model was proposed.

From numerical program we can obtain three values: galloping velocity, vortex

shedding velocity and speed ratio between these two velocities. To calculate these

values we need a correct estimation of modal parameters, for this reason it is possible

to use the precedent real values obtained during electromechanical validation tests. It

is impossible to compare analytical results with experimental results, because with the

interaction between two instabilities, as we studied in literature by Mannini [38], we

can’t distinguish the real galloping velocity.

Anyway numerical results can predict an evaluation about the interaction between

velocities through the speed ratio.

Numerical results are obtained from the instability of the matrix A.


115

For both configurations the state matrix A is expressed as:

[𝐴] =

[ −

𝑅∗ −12

𝜌𝑎𝑖𝑟𝑈𝐷𝑎1𝑘1

𝑀∗−

𝐾∗

𝑀∗−

𝜒

𝑀∗

1 0 0𝜒

𝐶𝑝0 −

1

𝐶𝑝𝑅]

(4.181)

To study the stability of the matrix we need to compute the eigenvalues of the system

and if there are one or more poles with a real value greater than zero, the system is

unstable. Of course the only value that can change in this matrix is the wind speed and

so when the system becomes unstable, the galloping speed is found.

4.3.1 Longitudinal configuration

In Table 4.11 are showed the analytical results for all prototypes we studied in wind

tunnel tests.

A40 A30 A20 B40 B30 B20

Galloping Velocity [m/s] 1.80 1.90 3.90 2.65 5.00 14.70

Vortex Shedding [m/s]

1.49 1.80 2.87 1.57 2.90 4.14

Ratio 𝑽𝒈

𝑽𝒔 1.21 1.06 1.36 1.69 1.70 3.35

Table 4.11 Analytical velocities and speed ratio for longitudinal prototypes

From previous considerations about the speed ratio, it is possible to predict the

behaviour of all prototypes, from numerical results showed in Table 4.11:

• Prototypes A40, A30, and A20 have an speed ratio really low, less than 1.5. For

this reason it is reasonable to think that these structures will have a perfect

coupling between vortex shedding and galloping velocity;

• Prototypes B30 and B40 have an speed ratio really similar and higher than 1.5.

In this case the possible behaviours is not easy to predict because we could be

in a limit case.

• The last is B20 model, in this case we have a high speed ratio and we expect a

decoupling between the two velocities. It will be interesting to analyse the

behaviour of this prototype, in order to compare our result, at high speed

ratio value, with results from literature.


116

In the next figures will show three plots per each prototype: tip displacement, piezo

patch displacement and power in function of wind speeds.

A comparison between numerical and experimental results will provide, where

numerical program is set with real modal parameters obtained in the previous

paragraph.

(a) (b)

(c)

Figure 4.20 Wind tunnel tests: comparison between analytical and experimental results for prototype A40 (a) tip displacement vs wind, (b) PPA displacement vs wind and (c) power vs wind

(a) (b)


117

(c)


(a) (b)

(c)



118

(a) (b)

(c)

Figure 4.23 Wind tunnel tests: comparison between analytical and experimental results for prototype B40 (a) tip displacement vs wind, (b) PPA displacement vs wind and (c) power vs wind

(a) (b)


119

(c)


(a) (b)

(c)


From experimental results it is possible to find different cases, as we expected from

the numerical predictions.

Prototypes A40, A30 and A20 are worked exactly as we expected. There is the perfect

coupling between galloping and vortex shedding. For this reason the prototype starts

to oscillate at vortex shedding velocity and it is impossible to find when galloping


120

instability is established. It is important to notice that using real modal parameters, in

the numerical program, it is possible to simulate in a good way the real behaviour of

these prototypes. This result can be used to validate the correctness of the program

used to simulate the prototypes behaviour. These behaviours are showed in Figure

4.20, Figure 4.21 and Figure 4.22.

Prototype B30 shows a particular behaviour, as it is possible to see in Figure 4.24. In

fact when the approach to the predicted galloping wind is slow, we can observe a

perfect coupling between galloping and vortex shedding. The behaviour changes if the

approach is fast, because the coupling is not established and there are not oscillations

of the prototype. As we expected this is a limit case and it is important to notice the

importance of the speed ratio on the longitudinal prototype.

In Figure 4.26 is it possible to see very well this effect. In this test we changed the

velocity of the approach to the galloping velocity and not in all cases we obtained the

same behaviour.

Figure 4.26 Different transient velocities for prototype B30

As for prototype B30 also prototype B40 was indicated as a limit case from numerical

results. Experimental results in Figure 4.23 confirmed this trend because after an

initial oscillation provide by vortex shedding, there is a decrement of amplitudes

before to restart to increase for galloping instability. This effect was showed also in

Mannini research [38] and it is perfectly confirmed in our work. With this second

prototype we can conclude that in limit cases of speed ratio value, there is only a

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

5.00E+00

6.00E+00

7.00E+00

8.00E+00

0.00 2.00 4.00 6.00 8.00

Po

wer

[m

W]

Wind Speed [m/s]

Rise

From 0 To 7.31 m/s

From 0 to 3.5 m/s

From 0 To 5.3 m/s

From 0 To 4.8 m/s

Galloping

Vortex Shedding


121

partial interaction and this can give some problems to predict which is the real

behaviour of the prototype.

At the end it is possible to see the results for prototype A20. Also in this case

experimental results confirmed our prediction based on the speed ratio. In fact for this

prototype there is a completely decoupling between galloping and vortex shedding

and from Figure 4.25 we can distinguish very well vortex shedding and galloping

instability. The problem for this prototype is about the unpredicted behaviour,

because when galloping is established, amplitude oscillations become immediately

divergent and the prototype could break after some cycles.

From experimental results it is possible to obtain important information for

longitudinal prototypes. In fact we obtained a confirmation about the importance of

the interaction between galloping and vortex shedding and it is suitable to design

prototypes with an speed ratio until 1.4.

This is fundamental because we want to guarantee a continuous work of the

prototype, without interruption, to design a device with all the characteristics we

explained at the start of this research.

Using a limit case, as B30 and B40 prototypes, can be a risk because we could not

predict exactly how it will works, with the possibility of the interruption during activity.

4.3.2 Transversal configuration

All the procedures we explained for longitudinal configuration are still valid for this

configuration. For this reason in Table 4.12 are showed all analytical results obtained

using real modal parameters inside numerical program.

C5 D5 D9

Galloping Velocity [m/s] 7.40 4.30 3.20

Vortex Shedding [m/s]

6.90 2.54 1.96

Ratio 𝑽𝒈

𝑽𝒔 1.06 1.69 1.63

Table 4.12 Analytical velocities and speed ratio for transversal prototypes

From analytical results we can see that only C5 prototype has a low speed ratio while

the other models are near 1.7, that was indicated as a limit value, with a partial

velocities interaction, in the longitudinal configuration. For this reason we can predict

a perfect coupling in C5 model, but for D5 and D9 models we cannot be sure about the

behaviour.


122

(a) (b)

(c)

Figure 4.27 Wind tunnel tests: comparison between analytical and experimental results for prototype C5 (a) tip displacement vs wind, (b) PPA displacement vs wind and (c) power vs wind

(a) (b)


123

(c)

Figure 4.28 Wind tunnel tests: comparison between analytical and experimental results for prototype C5 (a) tip displacement vs wind, (b) PPA displacement vs wind and (c) power vs wind

(a) (b)

(c)

Figure 4.29 Wind tunnel tests: comparison between analytical and experimental results for prototype D9 (a) tip displacement vs wind, (b) PPA displacement vs wind and (c) power vs wind

As we expected, the prototype C5 is in line with prediction and it is possible to observe

it in Figure 4.27. Also in this case it is impossible to distinguish the galloping velocity

because the oscillations begin with vortex shedding and it continue to increase in time.


124

For prototypes D5 and D9 we have an unexpected behaviour. In fact also if them has a

higher speed ratio value, the real behaviour presents a perfect coupling between

galloping and vortex shedding.

To understand which the reason of this difference is respect than longitudinal

prototype we need to see the natural frequencies of both prototypes.

In Table 4.13 are showed natural frequencies for prototype D9 in function of wind

speed and it is possible to see that the natural frequency decrease, with an effect on

galloping, when wind speed increase. The same effect there is for prototype D5. In this

way galloping speed moves near vortex shedding speed, with a redaction of the speed

ratio. At the end the real speed ratio will be lower respect our prediction and this is

the reason why these two prototypes have a perfect coupling.

The variation of natural frequency is due to a contribution of the adding mass from

wind and this parameter is in function of wind speed.

Wind Speed [m/s] Natural Frequency [Hz]

3.3 12.68

4.41 12.07

7.26 11.75 Table 4.13 Natural frequency variation due to wind contribution for prototype D9

As we said in third chapter, we decided to use two cantilever beams in order to reduce

the risk of the moment on the bluff body axis.

To study this behaviour it is possible to monitoring the displacement at the extreme

points of the bluff body. If bluff body is not affected by the moment, the two measures

will give the same value, because the displacement will be uniform for all structure.

In Figure 4.30 is showed the behaviour of prototype D9 and it is possible to conclude

that using two support beams reduce the risk of the moment and it may be preferable

if the bluff body has considerable dimensions, as in our case with 40 cm.


125

Figure 4.30 Moment study on prototype D9

4.3.3 Comparison between longitudinal and transversal model

At the end it is important to compare both configurations we tested in this research.

To provide a correct comparison we will consider only all prototypes with the same

bluff body volume and for this reason only model with a bluff body length of 40 cm.

We will focus on cut-in speed and on the power recovered for all prototypes. Also the

interaction between galloping instability and vortex shedding is an important effect to

observe for both configurations. All results are provided for a resistance of 100 kΩ. In

Table 4.14 all data are showed.

In order to obtain the efficiency we compare the power recovered by a piezoelectric

patch compared with the ideal power introduced by the wind:

𝜂 =

𝑃𝑒𝑙

𝑃𝑤𝑖𝑛𝑑 (4.182)

Where 𝑃𝑒𝑙 is obtained from wind tunnel experimental tests.

𝑃𝑤𝑖𝑛𝑑 is computed according to the definition given by Barrero-Gil et al. [25] as:

𝑃𝑤𝑖𝑛𝑑 =

1

2𝜌𝑎𝑖𝑟𝐷𝐿𝑏𝑈

3 (4.183)


126

Prototype Cut-in speed

[m/s] Maximum

power [mW]

Wind speed maximum

power [m/s] Efficiency

A40 1.22 1.69 6.14 0.07%

B40 1.56 0.63 4.75 0.06%

C5 5.21 2.87 8.62 0.05%

D9 1.87 3.06 6.17 0.13%

D5 3.54 7.07 7.91 0.19%

Table 4.14 Comparison between experimental results for prototypes with the same bluff body volume in both configurations

A first consideration is about the cut-in speed. Considering an equal bluff body volume

for all prototypes, for both models of longitudinal configurations we can observe a

lower cut-in speed value respect than transversal prototypes. Only D9 model can be

compared with longitudinal results, because C5 and D5 present higher values. This

concept is important in order to design a new prototype. If our interest is to design a

device with a really low cut-in speed, maybe it is better to use a longitudinal

configuration, as we observed in our tests. In the opposite case if we want to reach

high value of velocity, without interest to have a low value of cut-in speed, it is

possible to use a transversal prototype that it presents the possibility to increase the

cut-in velocity and to reach higher velocity of work, as we can see for model C5.

As we just concluded in the previous section, for all transversal prototype there is a perfect interaction between velocity instabilities and galloping is helped to establish by the vortex shedding. For longitudinal configuration only prototype A40 present a perfect interaction between instabilities and for this reason it is clear that with longitudinal prototypes we could have some problems due to three-dimensional effects. The problem is about the aerodynamic force. In fact in transversal prototype the force

is constant in every point of the bluff body length, as we consider inside the

mathematical model. Instead for longitudinal model we can have a different effect of

the aerodynamic force, respect to the distance from the clamp. This effect can explain

why with this prototype we could have problems to obtain a perfect interaction

between galloping and vortex shedding.

This behaviour is showed in Figure 4.31, where there is the power recovered, for only

a piezoelectric patch, in function of the wind speed for prototypes with similar cut-in

speed as A40, B40 and D9.


127

Figure 4.31 Comparison between prototypes A40, D9 and B40 in terms of power recovered in function of

wind speed

It is interesting to see also the efficiencies of all prototypes. In fact we can notice that

using transversal prototype it is possible to obtain higher values of efficiency, in

particular for prototypes D5 and D9. Prototype C5 shows an efficiency similar to the

longitudinal prototypes and this is because we used a very light bluff body. In fact

using a lower bluff body density, we obtain little deformation of the PZT patch respect

than prototype D5, as we can see in Figure 4.32.

Figure 4.32 Focused on PZT patch deformation from analytical modal shape for prototypes C5 and D5


128

The bluff body of D5 and D9 models are similar to the density for longitudinal

prototypes. For this reason we can conclude that for the same bluff body volume and

density, with transversal prototype we can obtain an higher efficiency.

CONCLUSION

129

CONCLUSION

In this work, a coupled non-linear distributed model for longitudinal and transversal

configurations of a GPEH has been developed. A sensitivity analysis was performed by

integrating the model equations with numerical methods. It was stated that the

objectives in the optimization phase of a GPEH were:

• increase power recovered;

• maximize power efficiency;

• low cut-in speed.

It was found that the main parameters which influence these targets are:

• length of the second domain;

• length of the bluff body;

• bluff body density.

The two experimental campaigns carried out on both longitudinal and transversal

configuration of GPEH have allowed to verify what it was found by the sensitivity

analysis. In particular, the following conclusions can be drawn:

• Increasing the length of the second domain there is a decrement of the

galloping velocity and, obviously, also of the speed ratio. This is because with a

reduction of the second region, there is a increment in the rigidity of the

structure and damping ratio increases;

• When there is a reduction of the bluff body length there is also a reduction of

the available area for the aerodynamic force, with a consequent effect on the

galloping velocity. In longitudinal configuration we also saw a variation of

damping ratio introduced by the bluff body length dimension;

• The last important parameter is bluff body density. Bluff body density is the

only parameter able to increase the speed ratio, that is the ratio between the

galloping onset speed and the vortex shedding velocity, leading to a

separation of the two instabilities.

It is possible to conclude this research providing a final comparison between two

prototypes configurations that it was our purpose for this work. Of course we can

compare only prototypes with the same bluff body volume, in our case all prototypes

with a length of 40 cm.

A first conclusion is about the interaction between galloping and vortex shedding

instabilities. In fact from experimental results we observed that for transversal

prototypes aerodynamic force is more efficient respect than longitudinal prototypes

and there aren’t three dimensional effects, that them are present in longitudinal

prototypes. For this reason we had always a perfect interaction between galloping and

vortex shedding.

CONCLUSION

130

This is an important effect because when there isn’t a perfect interaction, as in

prototype B40, the behaviour of the device is unpredictable and there isn’t continuity

of the work, as it is possible to see in Figure 5.0.1.

Figure 5.0.1 A non-perfect interaction between galloping and vortex shedding for prototype B40 in term of

power in function of the wind speed

An important comparison between two prototypes is about the efficiency. In this way

it is possible to study which is the value of power recovered respect than the ideal

power introduced by the wind.

To study this value we can take only prototypes with the same volume and bluff body

density, in this way it is possible to compare prototypes with the same bluff body.

Prototype Efficiency

A40 0.07%

B40 0.06%

D9 0.13%

D5 0.19%

Table 5.0.1 Efficiency for prototypes with the same bluff body volume and density

Form Table 5.0.1 we can see that transversal prototypes can give a higher value of

efficiency. This is a second advantage for transversal prototype because it means that

this configuration is able to convert more available energy in electrical energy.

Also in terms of maximum power recovered transversal prototypes is the best

solution, with a power peak of 7.07 mW, for D5 prototype, respect than 1.69 mW of

A40 prototype. In Table 5.0.2 are reported all values of power recovered for

prototypes we are investigating.

CONCLUSION

131

Prototype Maximum

power [mW]

A40 1.69

B40 0.63

D9 3.06

D5 7.07

Table 5.0.2 Power recovered for prototypes with the same bluff body volume and density

The only advantage of longitudinal prototype is about the cut-in speed. In fact, as we

can see in TTTT, both longitudinal models present a lower value respect transversal

prototype. The only comparable transversal model with longitudinal prototypes is the

D9, that it presents a second region of 9 cm, with a consequent to have a less compact

device. In conclusion for application where we need a low cut-in speed, longitudinal

configuration is preferable.

Prototype Cut-in speed

[m/s]

A40 1.22

B40 1.56

C5 5.21

D9 1.87

D5 3.54

Table 5.0.3 cut-in speed for prototypes with the same bluff body volume

A possible development for this research is on the maximum speed of work in both

configurations. This is the last important characteristic we should know in order to

understand which configurations can be suitable for real applications.

In fact another development is to create an active structure in order to control the

oscillation amplitude with a closed loop control, using a piezoelectric patch as an

active device to modify the damping ratio of the structure. In this case for each beam

we can us a PZT patch to recover power and the second patch to control the structure.

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132

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