Using topological modes to enhance the energy harvesting from elastic waves : Uso de modos topológicos para melhorar a coleta de energia de ondas elásticas

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Lucas Rafael Carneiro de Aguiar

USING TOPOLOGICAL MODES TO

ENHANCE THE ENERGY HARVESTING

FROM ELASTIC WAVES

USO DE MODOS TOPOLÓGICOS PARA

MELHORAR A COLETA DE ENERGIA DE

ONDAS ELÁSTICAS

CAMPINAS 2018

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USING TOPOLOGICAL MODES TO

ENHANCE THE ENERGY HARVESTING

FROM ELASTIC WAVES

USO DE MODOS TOPOLÓGICOS PARA

MELHORAR A COLETA DE ENERGIA DE

ONDAS ELÁSTICAS

Dissertation presented to the School of Mechanical Engineering of the University of Campinas in partial fulfillment of the requirements for the degree of Mas-ter in Mechanical Engineering, in the area of Solid Mechanics and Mechanical Design.

Dissertação de mestrado apresentada à Faculdade de Engenharia Mecânica da Universidade Estadual de Campinas como parte dos requisitos exigidos para obtenção do título de Mestre em Engenharia Mecânica, na Área de Mecânica dos Sólidos e Pro-jeto Mecânico.

Orientador: Prof. Dr. José Roberto de França Arruda

ESTE EXEMPLAR CORRESPONDE À VER-SÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELO ALUNO LUCAS RAFAEL CARNEIRO DE AGUIAR, E ORIENTADO PELO PROF. DR JOSÉ ROBERTO DE FRANÇA ARRUDA.

... ASSINATURA DO ORIENTADOR

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Ficha catalográfica

Universidade Estadual de Campinas Biblioteca da Área de Engenharia e Arquitetura

Luciana Pietrosanto Milla - CRB 8/8129

Aguiar, Lucas Rafael Carneiro de,

Ag93u AguUsing topological modes to enhance the energy harvesting from elastic

waves / Lucas Rafael Carneiro de Aguiar. – Campinas, SP : [s.n.], 2018.

AguOrientador: José Roberto de França Arruda.

AguDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade

de Engenharia Mecânica.

Agu1. Ondas elásticas. 2. Análise espectral - Métodos. I. Arruda, José Roberto

de França. II. Universidade Estadual de Campinas. Faculdade de Engenharia Mecânica. III. Título.

Informações para Biblioteca Digital

Título em outro idioma: Uso de modos topológicos para melhorar a coleta de energia de

ondas elásticas

Palavras-chave em inglês:

Elastic waves

Spectral analysis - Methods

Área de concentração: Mecânica dos Sólidos e Projeto Mecânico Titulação: Mestre em Engenharia Mecânica

Banca examinadora:

José Roberto de França Arruda [Orientador] José Maria Campos dos Santos

Carlos de Marqui Júnior

Data de defesa: 20-08-2018

Programa de Pós-Graduação: Engenharia Mecânica

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FACULDADE DE ENGENHARIA MECÂNICA

COMISSÃO DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA

DEPARTAMENTO DE MECÂNICA COMPUTACIONAL

DISSERTAÇÃO DE MESTRADO

USING TOPOLOGICAL MODES TO

ENHANCE THE ENERGY HARVESTING

FROM ELASTIC WAVES

USO DE MODOS TOPOLÓGICOS PARA

MELHORAR A COLETA DE ENERGIA DE

ONDAS ELÁSTICAS

Autor: Lucas Rafael Carneiro de Aguiar

Orientador: José Roberto de frança Arruda

A Banca Examinadora composta pelos membros abaixo aprovou esta dissertação:

Prof. Dr. José Roberto de França Arruda

DMC-Faculdade de Engenharia Mecânica - UNICAMP

Prof. Dr. José Maria Campos dos Santos

DMC-Faculdade de Engenharia Mecânica - UNICAMP

Prof. Dr. Carlos De Marqui Júnior

EESC - Escola de Engenharia de São Carlos - USP

A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.

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Primeiramente, gostaria de agradecer a Deus por me dar novamente a emoção de viver uma vida plena e desfrutar do real sabor da felicidade e de amor a vida.

Gostaria de agradecer a minha família por me dar todo o suporte que necessitei durante todo o mestrado.

Aos meus amigos de Brasília, Luiz Eduardo Carneiro e Paulo Marques por estarem me apoiando sempre.

Aos meus amigos que fiz em Campinas como o Rayston Werner, Max William, Luis Felipe Lima (Didi), Otávio Tovo, Fernando Simionato (Luciano Hulk), Vinicius Dias Lima (Whisky), Raimundo Lucena, Danilo Beli, Helder Dahia, José Ilmar (Inspetor), Hélio Cantanhede (PM), George Fernando, Breno Victor, Paulo Cassetari (Paulinho), Michael Serra (Billy), Edson Jansen Miranda e Fernando Ortolano. Gostaria de agradecer também a Gertrudes pelos cafés do final de semana e a Rose pelos cafés matinais na FEA.

Ao José Roberto de França Arruda, Mateus Rosa e Danilo Beli por integrarem o grupo de estudos sobre propagação de ondas em cristais fonônicos que tanto me ajudou

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A propagação de ondas elásticas dentro de uma estrutura está totalmente relacionada com suas propriedades mecânicas. Além disso, sabe-se que estruturas periódicas possuem características únicas de propagação de onda. Por exemplo, é possível criar bandas de frequências onde as ondas não podem se propagar dentro da estrutura (bandgaps). Esse conceito foi desenvolvido primeiramente na teoria eletrônica de bandas. Uma recente descoberta de estruturas periódicas ou quasi periódicas é a possibilidade de exploração de modos topológicos. Um desses modos é o modo de interface. Na interface entre duas estruturas unidimensionais periódicas (chamadas de Cristais Fonônicos) com uma certa relação de fase topológica, um modo isolado onde a vibração é concentrada na interface pode aparecer. Também, com um arranjo de cristais fonônicos com propriedades variando de maneira quase periódica, um modo de borda pode ser obtido, onde a vibração é concentrada em uma extremidade. Nesse trabalho, investiga-se estratégias para usar tais modos topológicos parar aumentar localmente a vibração visando aplicações de coleta de energia. Essa abordagem proposta consiste na inserção de um elemento piezoelétrico na zona onde as ondas são amplificadas com o intuito de coletar energia das ondas elásticas. A energia é extraida como potência elétrica estimada em uma carga resistiva conectada aos eletrodos do elemento. A análise computacional é executada usando o Método do Elemento Espectral (SEM). A potência elétrica coletada é comparada com a potência mecânica inserida na estrutura com o intuito de obter a eficiência total do sistema. Um experimento é realizado e comparado com as simulações teóricas.

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The elastic wave propagation inside a structure is totally related to its mechanical properties. Moreover, it is known that periodic structures have unique wave propagation features. It is possible, for instance, to create frequency bands where waves cannot propagate in the structure (bandgaps). This concept was first developed in electronic band theory. A more recently discovered property of periodic and quasi periodic structures is the possibility of finding topological modes. One such mode is the interface mode. In the interface between two one-dimensional periodic structures (called Phononic Crystals) with given topological phase relation, an isolated mode where the vibration is concentrated at the interface may appear. Also, with an arrangement of phononic crystals with quasi periodically varying properties, an edge mode can be obtained, where vibration is concentrated at one extremity. In this work we investigate strategies for using such topological modes to locally enhance vibration aiming at energy harvesting applications. The approach proposed here consists of inserting a piezoelectric element in the region where the waves are amplified in order to harvest the energy from the elastic waves. The energy is collected as electrical power estimated across an electrical load that is connected to the electrodes of the piezoelectric element. The computational analysis is performed using the Spectral Element Method (SEM). The electric power harvested is compared with the mechanical power inserted in the structure for obtaining the overall system efficiency. An experiment is performed and compared with theoretical simulation.

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1.1 (𝑎) Phononic Crystal of acoustic cylindrical waveguide with alternating cross-sectional areas described by Xiao et al. (2015) (𝑏) 2D Phononic Crystal

pro-posed by Laude et al. (2005). . . 19

1.2 Energy harvesting from vibration using a piezoelectric membrane (Ericka et al., 2005) . . . 20

2.1 Sign convention adopted (Lee, 2009) . . . 25

3.1 Periodic cell of the phononic crystals . . . 33

3.2 Topological transition plot . . . 33

3.3 Phononic Crystal 1: (𝑎) Dispersion relation of the cell (𝑏) 2D sketch. (𝑐) 3D sketch. . . 35

3.7 Comparison between the configurations PC1-PC1 and PC1-PC4 excited at the first end: Surface of the spatial distribution of the displacement in function of the frequency and X coordinate of (𝑎) PC1-PC1 configuration (𝑏)PC1-PC4 con-figuration; FRF of the displacement measured at the interface of (𝑐) PC1-PC1 configuration (𝑑)PC1-PC4 configuration; Spatial distribution of the displace-ment at the interface mode frequency of (𝑒) PC1 configuration (𝑓 ) PC1-PC4 configuration. . . 39

3.8 Comparison between the configurations PC2-PC2 and PC2-PC3 excited at the first end: Surface of the spatial distribution of the displacement in function of the frequency and X coordinate of (𝑎) PC2-PC2 configuration (𝑏)PC2-PC3 con-figuration; FRF of the displacement measured at the interface of (𝑐) PC2-PC2 configuration (𝑑)PC2-PC3 configuration; Spatial distribution of the displace-ment at the interface mode frequency of (𝑒) PC2 configuration (𝑓 ) PC2-PC3 configuration. . . 40

3.9 Polarization of the PZT . . . 41

3.10 Phononic Crystals with inclusion 1: (𝑎) Configuration PC2-PC3 (𝑏) Configura-tion PC1-PC4 . . . 41

3.11 Phononic Crystals with inclusion 2: (𝑎) Configuration PC2-PC3 (𝑏) Configura-tion PC1-PC4 . . . 42

3.12 Phononic Crystals with substitution: (𝑎) Configuration PC2-PC3 (𝑏) Configu-ration PC1-PC4 . . . 42

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power inserted (𝑏) Electrical power extracted . . . 43 3.14 Efficiency with 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 in PC1-PC4 configuration using

(𝑎) optimal resistance (𝑏) 𝑅 = 1Ω . . . 43

3.15 Efficiency with 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 in PC2-PC3 configuration using

(𝑎) optimal resistance (𝑏) 𝑅 = 1Ω . . . 44

3.16 (𝑎) Sketch of the PC1-PC4 free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes where the PZTs are in

short circuit condition (𝑅 = 1Ω) . . . 45

3.17 3D plot for excitation in all degrees of freedom of PC1-PC4 with inclusion 1 in (𝑎) 𝐶 natural mode with 𝑅 = 1Ω (𝑏) 𝐸 defect mode with 𝑅 = 1Ω (𝑐) 𝐷

interface mode with 𝑅 = 1Ω and (𝑑) 𝐷 interface mode with optimal resistance 46

3.18 (𝑎) Sketch of PC2-PC3 configuration in free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes where the

PZTs are in short circuit condition (𝑅 = 1Ω) . . . 47

3.19 3D plot for excitation in all degrees of freedom of PC2-PC3 inclusion 1 system in (𝑎) C natural mode with 𝑅 = 1Ω (𝑏) E defect mode with 𝑅 = 1Ω (𝑐) D

interface mode with 𝑅 = 1Ω and (𝑑) D interface mode with optimal resistance . 48

(𝑎) 𝑅 = 1Ω (𝑏) optimal resistance . . . 49

3.22 (𝑎) Sketch of the PC1-PC4 with inclusion 2 free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes with the

PZTs in short circuit condition (𝑅 = 1Ω) . . . 50

3.24 (𝑎) Sketch of the PC2-PC3 with inclusion 2 free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes where

the PZTs are in short circuit condition (𝑅 = 1Ω) . . . 52

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PZTs in short circuit condition (𝑅 = 1Ω) . . . 55 3.29 3D plot for excitation in all degrees of freedom of PC1-PC4 using substitution

in (𝑎) 𝐵 natural mode with 𝑅 = 1Ω (𝑏) 𝐹 second interface mode with 𝑅 = 1Ω (𝑐) 𝐶 first interface mode with 𝑅 = 1Ω and (𝑑) 𝐶 first interface mode with

optimal resistance . . . 56

3.30 (𝑎) Sketch of the PC2-PC3 in substitution free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes where

the PZTs are in short circuit condition (𝑅 = 1Ω) . . . 57

3.31 3D plot of excitation in all degrees of freedom of PC2-PC3 using substitution in (𝑎) 𝐶 natural mode with 𝑅 = 1Ω (𝑏) 𝐸 natural mode with 𝑅 = 1Ω (𝑐) 𝐷

interface mode with 𝑅 = 1Ω and (𝑑) 𝐷 interface mode with optimal resistance 58

4.1 Sketch of PC . . . 61

4.2 Topological transition plot for discrete symmetric cell . . . 61

4.3 Discrete approximation of PC1: (𝑎) Dispersion relation (𝑏) Dimensions of the

cell. (𝑐) 3D sketch. . . 63

4.7 Configuration PC1-PC4: (𝑎) Surface of spatial distribution of the displacement

(𝑏) FRF of the displacement. (𝑐) Spatial displacement at the interface mode

frequency. . . 67

4.8 Configuration PC2-PC3: (𝑎) Surface of spatial distribution of the displacement

(𝑏) FRF of the displacement. (𝑐) Spatial displacement at the interface mode

frequency. . . 68

4.9 Configurations with PZTs: (𝑎) Configuration PC1-PC4 (𝑏) Configuration

PC2-PC3 . . . 68

4.10 Efficiency with 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 in PC1-PC4 configuration

excited at the interface using (𝑎) optimal resistance (𝑏) 𝑅 = 1Ω . . . 69

4.11 Efficiency with 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 in PC2-PC3 configuration

excited at the first end using (𝑎) optimal resistance (𝑏) 𝑅 = 1Ω . . . 69

4.12 FRF of the PC1-PC4 configuration displacement at the interface exciting at the first end with the selected modes, where the PZTs are in short circuit condition

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face mode with 𝑅 = 1Ω in free-free condition (𝑐) 𝐷 first interface mode with optimal resistance in free-free condition and (𝑑) 𝐷 first interface mode with

𝑅 = 1Ω in Clamped-Clamped condition. . . 71

4.14 PC2-PC3 configuration displacement at the interface exciting the first end with

the selected modes where the PZTs are in short circuit condition (𝑅 = 1Ω) . . . 72

4.15 3D plot of excitation in all degrees of freedom of PC2-PC3 using substitution in (𝑎) 𝐵 natural mode with 𝑅 = 1Ω in free-free condition (𝑏) 𝐷 first inter-face mode with 𝑅 = 1Ω in free-free condition (𝑐) 𝐷 first interinter-face mode with optimal resistance in free-free condition and (𝑑) 𝐷 first interface mode with

𝑅 = 1Ω in Clamped-Clamped condition. . . 73

5.1 Discrete approximation of the left cell: (𝑎) Dispersion relation (𝑏) Dimensions

of the cell. (𝑐) 3D sketch. . . 76

5.2 Discrete approximation of the right cell: (𝑎) Dispersion relation (𝑏) Dimensions

of the cell. (𝑐) 3D sketch. . . 77

5.3 System analysed . . . 77

5.4 Experimental analysis: (𝑎) Accelerometer adopted (𝑏) Hammer used (𝑐)

Com-plete system without piezoelectric transducer with the interface highlighted . . 78

5.5 Response of the system shown in Figure (5.3) excited at the first end: (𝑎) and

(𝑏) are the numerical and experimental longitudinal displacements in dB as a function of the frequency and the 𝑋 coordinate, respectively, where the interface mode is highlighted.(𝑐) Comparison between the numerical and experimental FRF of the displacement at the interface (𝑑) Longitudinal displacement as a

function of the 𝑋 coordinate. . . 79

5.6 Response of the system shown in Figure (5.3) excited at the interface: (𝑎) and

(𝑏) are the numerical and experimental longitudinal displacements in dB as a function of the frequency and the 𝑋 coordinate, respectively, where the interface mode is highlighted.(𝑐) Comparison between the numerical and experimental FRF of the displacement at the interface (𝑑) Longitudinal displacement as a

function of the 𝑋 coordinate. . . 80

5.7 System analysed with a piezoelectric transducer acting as PZT . . . 81

5.8 Experimental setup: (𝑎) Piezoelectric transducer (𝑏) Aluminium adapter (𝑐)

Complete system with the piezoelectric transducer . . . 81

5.9 Effect of the inclusion of the aluminium adapter rod in the dynamic response:

(𝑎) Left cell scheme excited at the first end and measured at point 𝐾 (𝑏) Right cell scheme excited at the first end and measured at 𝐿 (𝑐) Comparison between

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experimental longitudinal displacement as a function of the frequency and the 𝑋 coordinate, where the interface mode is highlighted (𝑐) Comparison between the numerical and experimental FRF of the displacement measured at point 𝑀

(𝑑) Longitudinal displacement as a function of 𝑋 coordinate. . . 83

5.11 Voltage in dB with 1V reference as a function of coordinate of DOF excited and

frequency, adopting the open circuit case . . . 84

5.12 Response of the system with piezoelectric transducer connected to a 220Ω re-sistance shown in Figure (5.7) excited at all DOF individually; (𝑎) and (𝑏) are, respectively, the voltage and electrical power harvested in dB at the terminals of the resistance as a function of frequency and coordinate of the degree of free-dom excited (𝑐) Summation of the electrical power harvested exciting all DOF

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3.1 Geometry of the PCs . . . 34

3.2 Properties and geometry common to all the PCs . . . 34

3.3 Properties of the PZTs . . . 41

3.4 Modes chosen in PC1-PC4 configuration using inclusion 1 . . . 45

3.5 Modes of inclusion 1 in PC1-PC4 configuration, where the best extracted power values are highlighted . . . 46

3.7 Modes of inclusion 1 in PC2-PC3 configuration, where the best extracted power values are highlighted . . . 48

3.9 Modes of inclusion 2 in PC1PC4 configuration, where the highlighted modes are the highest extracted power values for each configuration . . . 51

3.11 Modes of inclusion 2 in PC2-PC3 configuration, where the highlighted modes are the highest extracted power values for each configuration . . . 53

3.12 Modes chosen in PC1-PC4 configuration using substitution . . . 55

3.13 Modes of substitution in PC1-PC4 configuration, where the best extracted power values for each configuration are highlighted . . . 56

3.15 Modes of substitution in PC2-PC3 configuration, where the highlighted modes are the highest extracted power values for each configuration . . . 58

4.1 Properties of the rod equivalent to a spring . . . 62

4.2 Properties of the rods equivalent to masses . . . 62

4.3 Properties of the masses of PC1 . . . 63

4.8 Modes of substitution in PC1PC4 configuration for low frequency interface mode, where the best harvested energy levels for each configuration are high-lighted . . . 71

4.10 Modes of substitution in PC2-PC3 configuration for low frequency interface mode, where the best harvested energy levels for each configuration are high-lighted . . . 73

5.1 Aluminium properties . . . 75

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1 INTRODUCTION 19

1.1 Motivation . . . 19

1.2 Literature review . . . 20

1.3 Objectives . . . 21

1.4 Outline . . . 21

2 SPECTRAL ELEMENT MODELS FOR ELASTIC AND PIEZOELECTRIC RODS 23 2.1 Spectral element of elastic rods . . . 23

2.1.1 Governing equation and constitutive relation . . . 23

2.1.2 Spectral stiffness matrix . . . 24

2.1.3 Transfer matrix . . . 26

2.2 Spectral element of piezoelectric rods . . . 27

2.2.1 Transfer matrix . . . 29

2.3 Topological modes . . . 32

3 ENERGY HARVESTING IN THE SECOND BANDGAP 33 3.1 Description of the problem . . . 33

3.1.1 Sketch of the PCs . . . 34 3.1.2 Configurations adopted . . . 38 3.1.3 Types of insertion of PZTs . . . 40 Inclusion 1 . . . 41 Inclusion 2 . . . 41 Substitution . . . 42 3.2 Inclusion 1 . . . 43 3.2.1 Resistive circuit . . . 43

3.2.2 Resistive circuit exciting all degrees of freedom . . . 44

3.3 Inclusion 2 . . . 49

3.3.1 Resistive circuit . . . 49

3.3.2 Resistive circuit exciting all degrees of freedom . . . 50

3.4 Substitution . . . 54

3.4.1 Resistive circuit . . . 54

3.4.2 Exciting all degrees of freedom . . . 54

3.5 Conclusion . . . 59

4 ENERGY HARVESTING IN THE FIRST BANDGAP 61 4.1 Description of the problem . . . 61

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4.3 Exciting all degrees of freedom . . . 70

5 EXPERIMENTAL ANALYSIS 75 5.1 System without piezoelectric transducer . . . 75

5.1.1 Description of the problem . . . 75

5.1.2 Results . . . 78

5.2 System with piezoelectric transducer . . . 80

5.2.1 Description of the problem . . . 80

5.2.2 Results . . . 82

6 CONCLUSION 87 6.1 Future work . . . 88

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1 INTRODUCTION

1.1 Motivation

Since the end of nineteenth century, periodic structures have fascinated the academic community due to its features. Phononic Crystals (PC) are systems in which the mechanical impedance varies periodically. This phenomenon allows to control the wave propagation along the structure (Zhang and Han, 2016). Using this characteristics, it is possible to create bands where waves do not propagate along the structure, called bandgap zones. These can be de-termined by computing the dispersion relation of the unitary periodic cell, obtained by Plane Wave Expansion (PWE), Spectral Finite Element (SFE) or Finite Element (FE) methods (Junyi et al., 2016). The PC are commonly used for sound insulation, wave guiding and acoustic cloaking (Junyi et al., 2016). The acoustic PCs shown in Figure (1.1) (𝑎) were implemented as acoustic lensing by Xiao et al. (2015) . The PCs may have the periodicity in one, two or three dimension. In Figure (1.1) (𝑏), it is possible to visualize the 2D PC performed by Laude et al. (2005).

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Figure 1.1: (𝑎) Phononic Crystal of acoustic cylindrical waveguide with alternating cross-sectional areas described by Xiao et al. (2015) (𝑏) 2D Phononic Crystal proposed by Laude

et al.(2005).

Energy harvesting, which is a process of collecting environment wasted energy and con-verting to electricity, is becoming largely used for microelectromechanical systems (MEMS) because the employment of batteries implies in large volume, environmental pollution and large maintenance requirements (Shen et al., 2008). Vibration sources such as human and ocean wave motion may supply mechanical energy which can be scavenged by adopting piezoelec-tric (Wang et al., 2012; Ericka et al., 2005; DuToit and Wardle, 2007; Minazara et al., 2006; Roundy et al., 2003), electromagnetic (Bouendeu et al., 2011; Wang and Chang, 2010; Ar-royo et al., 2013; Williams and Yates, 1996; Arnold, 2007; Glynne-Jones et al., 2004; Beeby et al., 2007; Mann and Sims, 2009), electrostatic (Roundy et al., 2002; Tashiro et al., 2000) and magnetostrictive (Wang and Yuan, 2008) energy harvesters.

Then, it is possible to make an association of the control of the wave propagation in periodic structures with the vibration energy harvesting in order improve the electrical power collected from the system.

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Figure 1.2: Energy harvesting from vibration using a piezoelectric membrane (Ericka et al., 2005)

1.2 Literature review

In 2004, Yang et al. performed an experimental and theoretical study of phonon focusing phenomena in a pass band above the complete band gap in a 3D PC. In this same year, Khe-lif et al. demonstrated experimentally the guiding and the bending of acoustic waves in highly confined waveguides obtained by removing rods from a periodic 2D lattice of steel cylinders immersed in water. Laude et al., in 2005, defined the analysis of surface-acoustic-wave in 2D piezoelectric PC using the Plane-Wave-Expansion (PWE) method. Vasseur et al. calculated the elastic band structures of 2D PC plates using supercell plane wave expansion (SC-PWE) in 2008. Also in 2008, Olsson III and El-Kady reviewed recent developments in the area of micro-phononic crystals including design techniques and characterization methods. Lin and Huang (2011) performed a theoretical study on the tunability of phononic band gaps in 2D PCs con-sisting of various anisotropic cylinders in an isotropic host. Wang et al. (2013) examined the ef-fects of geometric and material non-linearities introduced by deformation on the linear dynamic response of 2D PC. In 2015, Wang et al., investigated a new type of PCs with topologically non-trivial band gaps for longitudinal and transverse polarizations, resulting in protected one-way elastic edge waves. Xu and Tang (2017) explored a tunable acoustic prism featuring continuous beam steering for transverse waves at a single frequency using arrayed piezoelectric unit-cells with individually connected inductive shunt circuits in the prism. After the literature review of PCs, it is important to be familiar with the past works developed in scientific community in relation of energy harvesting.

In 2005, Ericka et al. investigate the capability of harvesting the electric energy from mechanical vibration in a dynamic environment through an unimorph piezoelectric membrane transducer shown in Figure (1.2). Erturk et al. (2008) analysed analytically and experimentally the power generation and shunt damping performance of the linear single crystal

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piezoelec-tric ceramic lead magnesium niobate-lead zirconate titanate (PMN-PZT). In 2009, Erturk et al. studied a piezomagnetoelastic device for substantial enhancement of piezoelectric power gen-eration in non-linear vibration energy harvesting. Joon Kim et al. (2010) presented thermal energy scavenging for improving wireless base station energy efficiency. In the same work, Joon Kim et al. also introduced mechanical energy scavenging for powering sensors in sensor networks, for machine-to-machine communication, and for smart grid applications. In 2012, Tang and Yang proposed a magnetic coupled piezoelectric energy harvester, in which the mag-netic interaction is introduced by a magmag-netic oscillator. Lv et al. (2013) investigated a vibration energy harvesting generator using a point-defect PC with piezoelectric material. In 2015, Sha-hab et al. explored the use of ultrasonic waves transmitted and received by piezoelectric devices in order to enable larger power transmission distances using acoustic waves. Tol et al. (2016) performed numerically and experimentally the enhancement of structure-borne elastic wave en-ergy harvesting by exploiting a Gradient-Index Phononic Crystal Lens (GRIN-PCL) structure. Also in 2016, Tol et al. analysed a detailed investigation of elastic wave mirror design, analy-sis and fabrication for enhanced elastic wave energy harvesting. Tol et al. (2017) investigated a Structurally Embedded Mirror design, analysis and experimental validation for enhanced elastic wave energy harvesting. Also in 2017, Tol et al. proposed 3D-printed Gradient-Index Phononic Crystals Lens (GRIN-PCL) for structure-borne focusing both numerically and experimentally. Given this research scenario, this work tries to combine the local vibration enhancement due to topological modes with an efficient energy harvesting using PZT shunt circuits.

1.3 Objectives

The main objective of this dissertation is investigating the use of topological modes to concentrate mechanical vibration in a piezoelectric ceramic element (PZT-5H) in order to har-vest electrical energy. For this purpose, analytical and numerical analyses are conducted. Mea-surements are performed by adopting a resistive circuit connected to the PZT. The values of resistance used are a resistance close to short circuit condition (𝑅 = 1Ω) and the optimal resis-tance for low electromechanical coupling, as defined by Carrara et al. (2013).

1.4 Outline

This dissertation is divided in 5 chapters. This chapter presents the introduction and liter-ature review about energy harvesting and topological modes.

The second chapter sketches the theoretical formulation of one dimensional wave propa-gation in elastic rods with the Spectral Element Method (SEM), obtaining the dynamic stiffness matrix and transfer matrix in the frequency domain. Later, the formulation of one dimensional wave propagation in piezoelectric rods is developed. A short explanation about topological modes of structures is also exhibited in this chapter.

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In the third chapter, two PCs with different topologic number are used in order to obtain the interface mode in the second bandgap. Three types of insertion of the PZT element are analysed, two inclusions and one substitution. A resistive circuit is placed at the terminals of the PZT and the efficiency (ratio between mechanical power inserted and the electrical power dissipated in resistor) for each type of insertion of PZT is investigated. The efficiency is also computed as a function of the frequency in purely resistive circuits with short circuit condition and with an optimal resistance. Next, the comparison between the power extracted in a resistive circuit exciting all degrees of freedom individually at the interface mode and natural modes is obtained. Finally, the discussion of the results achieved is performed.

The fourth chapter investigates the interface mode for the first bandgap. The type of in-sertion adopted is only substitution. The plot of the efficiency as a function of the frequency is performed for a resistive circuit with 𝑅 = 1Ω and for an optimal resistance in the first bandgap. The comparison between the power harvested in natural modes and in the interface mode in the first bandgap is performed with both loads. Then, the results obtained are discussed.

The fifth chapter describes the system used for the experiment to harvest power using a piezoelectric transducer. The dynamic response obtained experimentally and numerically are compared. The voltage obtained in the PZT transducer is evaluated in the frequency domain.

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2 SPECTRAL ELEMENT MODELS FOR ELASTIC AND

PIEZO-ELECTRIC RODS

Rods are important structural elements since they support axial forces and conduct longi-tudinal waves, so they can be seen as an waveguide.

2.1 Spectral element of elastic rods

2.1.1 Governing equation and constitutive relation

The elementary theory of rods (Lee, 2009) states that the one dimensional constitutive equation of the linear elastic rod may be written as

𝜎𝑥𝑥 = 𝐸

𝑑𝑢

𝑑𝑥 (2.1)

where 𝜎𝑥𝑥 is the longitudinal stress, 𝑢 = 𝑢(𝑥,𝑡) is the longitudinal displacement of the rod and

𝐸 is the Young’s modulus. Hamilton’s principle can be written as

𝛿[︁ ∫︁ 𝑡2 𝑡1 (𝑇 − 𝑈𝑠𝑡𝑟𝑎𝑖𝑛+ 𝑊 )𝑑𝑡 ]︁ = 0 (2.2)

where 𝑇 , 𝑈𝑠𝑡𝑟𝑎𝑖𝑛and 𝑊 are the kinetic energy, strain energy and the external work in the system,

respectively. The kinetic energy is defined as

𝑇 = 1 2 y Ω 𝜌 ˙𝑢𝑇 ˙𝑢𝑑Ω = 1 2 ∫︁ 𝐿 0 [︀𝜌𝐴 ˙𝑢2_]︀𝑑𝑥 (2.3)

while the strain energy is

𝑈𝑠𝑡𝑟𝑎𝑖𝑛 = 1 2 y Ω 𝜎𝑇𝑑𝑢 𝑑𝑡𝑑Ω = 1 2 ∫︁ 𝐿 0 𝐸𝐴(︁𝑑𝑢 𝑑𝑥 )︁2 𝑑𝑥 (2.4)

and the external work is

𝑊 =x

𝑆

𝑓𝑇𝑢𝑑𝑆 = (𝑁 𝑢)𝐿₀ (2.5)

where 𝜌 is the mass density of the material, 𝐴 is the cross-sectional area, 𝐿 is the length of the rod, Ω is the spacial region of the rod, 𝑆 signifies the boundary surface with specified surface forces 𝑓 of the system, 𝑁 is the axial force on the ends of the rod and the over-dot represents differentiation with respect to time. Substituting Equations (2.3), (2.4) and (2.5) into (2.2), computing the variation and applying the integration, one obtains the governing equation

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and the constitutive relation of the elastic rod:

𝑁 (𝑥,𝑡) = 𝐸𝐴𝑢′ (2.7)

where 𝑁 (𝑥,𝑡) is the internal axial force and the prime means the derivatives in relation to the spatial coordinate 𝑥.

2.1.2 Spectral stiffness matrix

Lee (2009) states that the solution of Equation (2.6) is assumed in spectral form:

𝑢(𝑥,𝑡) = 1 𝑁 𝑁 −1 ∑︁ 𝑛=0 𝑈𝑛(𝑥,𝜔𝑛)𝑒𝑖𝜔𝑛𝑡 (2.8)

where 𝑖 = √−1 is the imaginary unit. Making the substitution of Equation (2.8) into (2.6)

yields

𝐸𝐴𝑈′′− 𝜔2_{𝜌𝐴𝑈 = 0} _(2.9)

the general solution of Equation (2.9) is assumed to be

𝑈 (𝑥) = 𝑎𝑒−𝑖𝑘(𝜔)𝑥 (2.10)

substituting Equation (2.10) into (2.9), a dispersion relation is obtained

𝑘2− 𝑘2

𝐿 = 0 (2.11)

in which 𝑘𝐿is the wavenumber of longitudinal wavemode defined by:

𝑘𝐿= 𝜔

√︂ 𝜌

𝐸 (2.12)

the real roots of Equation (2.11) are:

𝑘𝐿 = 𝑘1 = −𝑘2 (2.13)

considering an element of rod of length 𝐿, the general solution of Equation (2.9) is defined by

𝑈 (𝑥) = 𝑎1𝑒−𝑖𝑘𝐿𝑥+ 𝑎2𝑒𝑖𝑘𝐿𝑥 = e(𝑥,𝜔)a (2.14)

where e(𝑥; 𝜔) = [𝑒−𝑖𝑘𝐿𝑥 _𝑒𝑖𝑘𝐿𝑥_{] and a = {𝑎}

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Figure 2.1: Sign convention adopted (Lee, 2009)

nodal displacement of the rod can be associated to the displacement field as

d = [︃ 𝑈1 𝑈2 ]︃ = [︃ 𝑈 (0) 𝑈 (𝐿) ]︃ (2.15) then d = [︃ 𝑒(0; 𝜔) 𝑒(𝐿; 𝜔) ]︃ a = HR(𝜔)a (2.16) where H𝑅(𝜔) = [︃ 1 1 𝑒−𝑖𝑘𝐿𝑥 _𝑒𝑖𝑘𝐿𝑥 ]︃ (2.17) by substituting Equation (2.14) into (2.15), one can represent the displacement field in terms of the nodal DOF as

𝑈 (𝑥) = NR(x; 𝜔)d (2.18)

where

NR(𝑥; 𝜔) = e(𝑥; 𝜔)H-1R(𝜔) =[︀𝑐𝑠𝑐(𝐾𝐿𝐿)𝑠𝑖𝑛[𝑘𝐿(𝐿 − 𝑥)] 𝑐𝑠𝑐(𝑘𝐿𝐿)𝑠𝑖𝑛(𝑘𝐿𝑥)

]︀

(2.19) According to Equation (2.7), the spectral components of the axial force are associated with 𝑈 (𝑥) by the following relation

𝑁 (𝑥) = 𝐸𝐴𝑈′(𝑥) (2.20)

the spectral nodal forces can be related to the forces defined by the strength of materials as

fc(𝜔) = {︃ 𝑁1 𝑁2 }︃ = {︃ −𝑁 (0) 𝑁 (𝐿) }︃ (2.21)

substituting Equations (2.18) and (2.20) into (2.21), one obtains

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where SR(𝜔) is the spectral element matrix for the rod element and it is given by SR(𝜔) = 𝐸𝐴 𝐿 [︃ 𝑆𝑅11 𝑆𝑅12 𝑆𝑅12 𝑆𝑅22 ]︃ (2.23) where 𝑆𝑅11 = 𝑆𝑅22 = (𝑘𝐿𝐿)𝑐𝑜𝑡(𝑘𝐿𝐿) 𝑆𝑅12 = −(𝑘𝐿𝐿)𝑐𝑠𝑐(𝑘𝐿𝐿) (2.24) 2.1.3 Transfer matrix

The dynamics of one dimensional structures in the frequency domain can also be repre-sented by a set of ordinary differential equations as

𝑑y

𝑑𝑥 = A(𝜔)y(𝑥) (0 ≤ 𝑥 ≤ 𝐿) (2.25)

where A(𝜔) is the frequency-dependent system matrix and y(𝑥) is the cross-sectional state-vector defined by y(𝑥) = {︃ 𝒟(𝑥) F(𝑥) }︃ (2.26) where 𝒟 is the displacement field vector and F(𝑥) is the internal forces vector. The general solution of Equation (2.25) is

y(𝑥) = 𝑒A(𝜔)𝑥y(0) (2.27)

The transfer matrix relates the quantities in input side (𝑥 = 0) to those in output side (𝑥 = 𝐿). It is defined as:

T = 𝑒A(𝜔)𝐿 (2.28)

for an element of rod of length 𝐿, the governing equation and force-displacement relation in frequency domain are

𝐸𝐴𝑈′′+ 𝜔2𝜌𝐴𝑈 = 0

𝑁 = 𝐸𝐴𝑈′ (2.29)

The Equation (2.29) can be transformed into 2 first-order ODE as

𝑁′ = 𝐸𝐴𝑈′′= −𝜔2𝜌𝐴𝑈

𝑈′ = 𝑁

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yR = {︃ 𝑈 𝑁 }︃ (2.31) and AR(𝜔) = [︃ 0 _𝐸𝐴1 −𝐸𝐴𝑘2 𝐿 0 ]︃ (2.32)

applying the Equation (2.32) in (2.28), it is possible to obtain the transfer matrix TR of an

element of rod TR(𝜔) = [︃ 𝑐𝑜𝑠(𝑘𝐿𝐿) (𝐸𝐴𝑘𝐿)−1𝑠𝑖𝑛(𝑘𝐿𝐿) −𝐸𝐴𝑘𝐿𝑠𝑖𝑛(𝑘𝐿𝐿) 𝑐𝑜𝑠(𝑘𝐿𝐿) ]︃ (2.33)

2.2 Spectral element of piezoelectric rods

According to Li and Guo (2016), the constitutive equations of a general linear piezoelec-tric material are

{︃ 𝜎𝑝 D }︃ = [︃ CE −eT e 𝛼S ]︃ {︃ 𝜀𝑝 E }︃ (2.34)

where 𝜎𝑝 is the the stress vector, D is the electric displacement vector, CEis the elastic

con-stant matrix measured at zero electric field, e is the piezoelectric concon-stant matrix and 𝛼Sis the

dielectric constant matrix for piezoelectric rod measured at zero strain. The strain vector 𝜀𝑝and

the electric field vector E are defined as

𝜀𝑝 =

𝑑𝑢𝑝

𝑑𝑥

E = −∇𝜙 = −d𝜙

dx (2.35)

where axial displacement 𝑢𝑝 and electric potential 𝜙 are

𝑢𝑝 = 𝑢𝑝(𝑥,𝑡)

𝜙 = 𝜙(𝑥,𝑡) (2.36)

respectively. The constitutive equations for a piezoelectric rod in 𝑥 direction are {︃ 𝜎𝑝 𝐷 }︃ = [︃ 𝐸𝑝 𝑒11 𝑒11 −𝛼 ]︃ {︃_𝑑𝑢 𝑝 𝑑𝑥 𝑑𝜙 𝑑𝑥 }︃ (2.37)

where 𝑒11, 𝛼, 𝐸𝑝, 𝜎𝑝 and 𝐷 mean the piezoelectric constant, dielectric constant, Young’s

mod-ulus, axial stress and electric displacement in x direction for piezoelectric rod, respectively. Considering the Hamilton’s principle for a linear piezoelectric continuum

𝛿[︁ ∫︁ 𝑡2 𝑡1 (𝑇𝑝− 𝐻 + 𝑊𝑝)𝑑𝑡 ]︁ = 0 (2.38)

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where 𝐻 is the electric enthalpy defined by 𝐻 = 1 2 y Ω (︀𝜎𝑇_𝜀 𝑝− DTE)︀dΩ = (2.39) 1 2 ∫︁ 𝐿𝑝 0 𝐸𝑝𝐴𝑝 (︁𝑑𝑢_𝑝 𝑑𝑥 )︁2 𝑑𝑥 + ∫︁ 𝐿𝑝 0 𝑒11𝐴𝑝 𝑑𝑢𝑝 𝑑𝑥 𝑑𝜙 𝑑𝑥𝑑𝑥 − 1 2 ∫︁ 𝐿𝑝 0 𝛼𝐴𝑝 (︁𝑑𝜙 𝑑𝑥 )︁2 𝑑𝑥 (2.40)

where Ω refers to all domain of the piezoelectric system, 𝐴𝑝 and 𝐿𝑝 are the cross sectional

area and length of the piezoelectric rod, respectively. The expression of the kinetic energy for

the piezoelectric rod 𝑇𝑝 is the same of the elastic rod (already defined in Equation (2.3)). The

external work consists of mechanical and electrical work

𝑊 =x

𝑆𝑝

(𝑓_𝑝𝑇𝑢𝑝− 𝑞𝜙)𝑑𝑆 = (𝑁𝑝𝑢𝑝− 𝑄𝜙) |

𝐿𝑝

0 (2.41)

where 𝑆𝑝 means the boundary surface with the surface forces 𝑓𝑝 of the piezoelectric system. 𝑞

is the electric charge density per area, 𝑄 and 𝑁𝑝 are electric charge and axial force on the end

of the piezoelectric rod, respectively. Substituting Equations (2.3), (2.40) and (2.41) into (2.38) and integrating by parts, one obtains

[︃ ∫︁ 𝐿𝑝 0 𝜌𝑝𝐴𝑝𝑢˙𝑝𝛿𝑢𝑝𝑑𝑥 ]︃𝑡2 𝑡1 + ∫︁ 𝑡2 𝑡1 [︃ ∫︁ 𝐿𝑝 0 [︁ 𝐸𝑝𝐴𝑝 𝑑2𝑢𝑝 𝑑𝑥2 + 𝑒11𝐴𝑝 𝑑2𝜙 𝑑𝑥2 − 𝜌𝑝𝐴𝑝𝑢¨𝑝 ]︁ 𝛿𝑢𝑝𝑑𝑥 ]︃ 𝑑𝑡+ ∫︁ 𝑡2 𝑡1 [︃ ∫︁ 𝐿𝑝 0 [︁ 𝑒11𝐴𝑝 𝑑2𝑢𝑝 𝑑𝑥2 − 𝛼𝐴𝑝 𝑑2𝜙 𝑑𝑥2 ]︁ 𝛿𝜙𝑑𝑥 ]︃ 𝑑𝑡 + ∫︁ 𝑡2 𝑡1 [︃ [︁ 𝑁𝑝− 𝐸𝑝𝐴𝑝 𝑑𝑢𝑝 𝑑𝑥 − 𝑒11𝐴𝑝 𝑑𝜙 𝑑𝑥 ]︁ 𝛿𝑢𝑝 ]︃𝐿𝑝 0 𝑑𝑡+ ∫︁ 𝑡2 𝑡1 [︃ [︁ − 𝑄 − 𝑒11𝐴𝑝 𝑑𝑢𝑝 𝑑𝑥 + 𝛼𝐴𝑝 𝑑𝜙 𝑑𝑥 ]︁ 𝛿𝜙 ]︃𝐿𝑝 0 𝑑𝑡 = 0 (2.42)

where 𝜌𝑝 means the mass density of the piezoelectric element. Notice that the variation of the

displacement functions vanishes at times 𝑡1 and 𝑡2. Then, considering that 𝛿𝑢𝑝 |𝑡=𝑡1= 0 and

𝛿𝑢𝑝 |𝑡=𝑡2= 0, the first part of Equation (2.42) will vanish. The second and third term mean the

variations of energy inside the piezoelectric rod due to the variations of 𝛿𝑢𝑝and 𝛿𝜙, respectively,

while the fourth and fifth part of Equation (2.42) represent the change of energy functional on

piezoelectric rod ends due to the variations of 𝛿𝑢𝑝and 𝛿𝜙. The second to fifth terms in Equation

(2.42) should disappear because both 𝛿𝑢𝑝 and 𝛿𝜙 are independent each other inside and at the

piezoelectric rod ends, yielding the governing equations

𝐸𝑝𝐴𝑝 𝑑2_𝑢 𝑝 𝑑𝑥2 + 𝑒11𝐴𝑝 𝑑2_𝜙 𝑑𝑥2 = 𝜌𝑝𝐴𝑝𝑢¨𝑝 𝑒11𝐴𝑝 𝑑2𝑢𝑝 𝑑𝑥2 − 𝛼𝐴𝑝 𝑑2𝜙 𝑑𝑥2 = 0 (2.43)

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and the constitutive relations of the piezoelectric rod 𝑁𝑝 = 𝐸𝑝𝐴𝑝 𝑑𝑢𝑝 𝑑𝑥 + 𝑒11𝐴𝑝 𝑑𝜙 𝑑𝑥 𝑄 = −𝑒11𝐴𝑝 𝑑𝑢𝑝 𝑑𝑥 + 𝛼𝐴𝑝 𝑑𝜙 𝑑𝑥 (2.44)

By applying Equation (2.44) into (2.43), one obtains the governing equations of the

piezo-electric rod where only the axial displacement 𝑢𝑝is involved.

(︁ 𝐸𝑝+ 𝑒2 11 𝛼 )︁ 𝐴𝑝 𝑑2_𝑢 𝑝 𝑑𝑥2 = 𝜌𝑝𝐴𝑝𝑢¨𝑝 (2.45) 2.2.1 Transfer matrix

Using the Fourier transform into Equations (2.45) and (2.44) leads to the frequency-domain formulation (︁ 𝐸𝑝+ 𝑒2 11 𝛼 )︁ 𝐴𝑝 𝑑2_𝑢_ˆ 𝑝 𝑑𝑥2 + 𝜔 2_𝜌 𝑝𝐴𝑝𝑢ˆ𝑝 (2.46) ˆ 𝑁𝑝 = 𝐸𝑝𝐴𝑝 𝑑 ˆ𝑢𝑝 𝑑𝑥 + 𝑒11𝐴𝑝 𝑑 ˆ𝜙 𝑑𝑥 (2.47) ˆ 𝑄 = −𝑒11𝐴𝑝 𝑑 ˆ𝑢𝑝 𝑑𝑥 + 𝛼𝐴𝑝 𝑑 ˆ𝜙 𝑑𝑥 (2.48)

where 𝜔 is the frequency and the caret (ˆ) over a physical variable means the corresponding

quantity in the frequency domain. Considering ˆ𝑄 = − ˆ𝐷𝑥𝐴𝑝as the scalar electric charge at the

initial end of the piezoelectric rod and ˆ𝐷𝑥 the uniform electric displacement, Equation (2.48)

leads to 𝑑 ˆ𝜙 𝑑𝑥 = 𝑒11 𝛼 𝑑 ˆ𝑢𝑝 𝑑𝑥 + ˆ 𝑄 𝛼𝐴𝑝 ˆ 𝑉 = ˆ𝜙(𝐿𝑝) − ˆ𝜙(0) = ∫︁ 𝐿𝑝 0 𝑑 ˆ𝜙 𝑑𝑥𝑑𝑥 = 𝑒11 𝛼 [︁ ˆ 𝑢𝑝(𝐿𝑝) − ˆ𝑢𝑝(0) ]︁ +𝑄𝐿ˆ 𝑝 𝛼𝐴𝑝 (2.49)

where ˆ𝑉 is the electric potential (voltage) between the two ends of the piezoelectric rod. Making

the substitution of 𝑑 ^_𝑑𝑥𝜙 of Equation (2.49) into (2.47) and including the solution to the spectral

axial displacement

ˆ

𝑢𝑝(𝑥) = 𝑎1𝑒𝑖𝑘𝑝𝑥+ 𝑑1𝑒−𝑖𝑘𝑝𝑥 (2.50)

one obtains the wave solution to the spectral axial force: ˆ 𝑁𝑝(𝑥) = 𝜉1𝑎1𝑒𝑖𝑘𝑝𝑥− 𝜉1𝑑1𝑒−𝑖𝑘𝑝𝑥+ 𝐵 [︁ ˆ 𝑢𝑝(𝐿𝑝) − ˆ𝑢𝑝(0) ]︁ (2.51)

where 𝑎1 and 𝑑1 are the corresponding wave amplitudes. 𝑘𝑝 and 𝜉1 are the wave number and

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boundary conditions defined by: 𝑘𝑝 = 𝜔 √︃ 𝜌𝑝 𝐸𝑝+ 𝑒2 11 𝛼 𝜉1 = 𝑖𝑘𝑝𝐴 (︁ 𝐸𝑝+ 𝑒2 11 𝛼 )︁ (2.52) The first two terms of Equations (2.50) and (2.51) represent waves propagating along the negative and positive 𝑥 axis that are called as the arriving and departing waves, respectively. The constant 𝐵 is the axial-force influence coefficient of the piezoelectric rod that depends on the electrical boundary condition and it will be defined later. Considering that between the terminals of the piezoelectric rod, it is placed a RLC series circuit. The impedance in frequency domain of each device is

ˆ

𝑍𝑅= 𝑅 𝑍ˆ𝐿 = 𝑖𝜔𝐿𝑖𝑛𝑑 𝑍ˆ𝐶 =

1

𝑖𝜔𝐶 (2.53)

where 𝑅, 𝐿𝑖𝑛𝑑 and 𝐶 are the resistance, inductance and capacitance of the load, respectively.

The equivalent impedance ˆ𝑍𝑒𝑞is defined as

ˆ

𝑍𝑒𝑞=

𝑖𝜔𝑅𝐶 − 𝜔2𝐿𝑖𝑛𝑑𝐶 + 1

𝑖𝜔𝐶 (2.54)

The electrical circuit theory states that ˆ

𝑉 = ˆ𝑍𝑒𝑞𝐼ˆ

ˆ

𝐼 = −𝑖𝜔 ˆ𝑄 (2.55)

where ˆ𝐼 is current along the impedance in frequency domain. Then, the voltage can be expressed

as ˆ 𝑉 =[︁𝑖𝜔𝑅𝐶 − 𝜔 2_𝐿 𝑖𝑛𝑑𝐶 + 1 𝑖𝜔𝐶 (︀ − 𝑖𝜔 ˆ𝑄 )︀]︁ (2.56) then ˆ 𝑄 = ˆ 𝑉 𝐶 𝜔2_𝐿 𝑖𝑛𝑑𝐶 − 𝑖𝜔𝑅𝐶 − 1 (2.57) substituting Equation (2.57) into (2.49) yields:

ˆ 𝑉 = 𝑉 𝐶𝐿ˆ 𝑝 𝛼𝐴𝑝 [︁ 𝜔2_𝐿 𝑖𝑛𝑑𝐶 − 𝑖𝜔𝑅𝐶 − 1 ]︁ + 𝑒11 𝛼 [︁ ˆ 𝑢𝑝(𝐿𝑝) − ˆ𝑢𝑝(0) ]︁ (2.58)

making some mathematical manipulations in Equation (2.59), the voltage in frequency domain is: ˆ 𝑉 = 𝑒11 𝛼 [︁ ˆ 𝑢𝑝(𝐿𝑝) − ˆ𝑢𝑝(0) ]︁ 1 − 𝐶𝐿𝑝 𝛼𝐴𝑝 [︁ 𝜔2_𝐿 𝑖𝑛𝑑𝐶−𝑖𝜔𝐶𝑅−1 ]︁ (2.59)

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By applying Equation (2.57) and first part of (2.49) into (2.47), one obtains: ˆ 𝑁𝑝 = [︁ 𝐸𝑝𝐴𝑝+ 𝑒2 11𝐴𝑝 𝛼 ]︁𝑑 ˆ𝑢_𝑝 𝑑𝑥 + 𝑒11𝑉 𝐶ˆ 𝛼[︁𝜔2_𝐿 𝑖𝑛𝑑− 𝑖𝜔𝐶𝑅 − 1 ]︁ (2.60)

substituting Equation (2.59) into (2.60), the axial force in frequency domain can be expressed as: ˆ 𝑁𝑝 = [︁ 𝐸𝑝𝐴𝑝+ 𝑒2 11𝐴𝑝 𝛼 ]︁𝑑 ˆ𝑢_𝑝 𝑑𝑥 + 𝑒2 11𝐶 [︁ ˆ 𝑢𝑝(𝐿𝑝) − ˆ𝑢𝑝(0) ]︁ 𝛼[︁𝜔2_𝐿 𝑖𝑛𝑑𝐶 − 𝑖𝜔𝐶𝑅 − 1 − 𝐶𝐿_𝛼𝐴𝑝_𝑝 ]︁ (2.61) considering 𝐵 = 𝑒 2 11𝐶 𝛼[︁𝜔2_𝐿 𝑖𝑛𝑑𝐶 − 𝑖𝜔𝐶𝑅 − 1 − 𝐶𝐿𝑝 𝛼𝐴𝑝 ]︁ (2.62)

it is possible to simplify Equation (2.61) to ˆ 𝑁𝑝 = [︁ 𝐸𝑝𝐴𝑝+ 𝑒2 11𝐴𝑝 𝛼 ]︁𝑑 ˆ𝑢_𝑝 𝑑𝑥 + 𝐵 [︁ ˆ 𝑢𝑝(𝐿𝑝) − ˆ𝑢𝑝(0) ]︁ (2.63) the transfer matrix for the piezoelectric rod is defined as

[︃ ˆ 𝑈p2 ˆ 𝑁p2 ]︃ = 1 ∆p [︃ 𝑡p11 𝑡p12 𝑡p21 𝑡p22 ]︃ [︃ ˆ 𝑈p1 ˆ 𝑁p1 ]︃ (2.64) where ∆p = −𝐵(𝑒−𝑖𝑘𝑝𝐿𝑝 − 𝑒𝑖𝑘𝑝𝐿𝑝) + 2𝜉1, 𝑡p11 = 𝑡p22 = (𝜉1 − 𝐵)𝑒−𝑖𝑘𝑝𝐿𝑝 + (𝜉1 + 𝐵)𝑒𝑖𝑘𝑝𝐿𝑝, 𝑡p12 = −(𝑒−𝑖𝑘𝑝𝐿𝑝−𝑒𝑖𝑘𝑝𝐿𝑝) and 𝑡p21 = −𝜉12(𝑒 −𝑖𝑘𝑝𝐿𝑝−𝑒𝑖𝑘𝑝𝐿𝑝_)+2𝜉 1𝐵(𝑒−𝑖𝑘𝑝𝐿𝑝+𝑒𝑖𝑘𝑝𝐿𝑝−2). Again,

it is possible to transform the transfer matrix into the dynamic stiffness matrix (Lee, 2009): [︃ ˆ 𝑁p1 ˆ 𝑁p2 ]︃ = Kˆp(𝜔) [︃ ˆ 𝑈p1 ˆ 𝑈p2 ]︃ where Kˆp(𝜔) = [︃ 𝑡−1_p12𝑡p11 (︀𝑡p12/∆p )︀−1 𝑡p21− 𝑡p22𝑡−1p21𝑡p11/∆p 𝑡p22𝑡−1p12 ]︃ (2.65) Finally, after obtaining the voltage between the terminals of PZT, the electric power dis-sipated by the resistor is performed by

𝑃 = 𝑅𝑒( ˆ𝑉 ˆ𝑉*𝑍_𝑅−1) (2.66) where 𝑍_𝑅−1 = 𝑅 𝑅2₊_{(︀𝑗𝜔𝐿} 𝑖𝑛𝑑− _𝑗𝜔𝐶1 )︀2 (2.67)

where 𝑅𝑒() means the real part of the argument. In case of purely resistive circuit, the optimal resistance is performed by (Carrara et al., 2013)

𝑅𝑜𝑝𝑡 =

1

𝜔𝐶𝑃 𝑍𝑇

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where 𝐶𝑃 𝑍𝑇 is the internal capacitance of PZT calculated by 𝐶𝑃 𝑍𝑇 = 𝛼𝐴_𝐿_{𝑃 𝑍𝑇}𝑃 𝑍𝑇. The mechanical

power inserted in the system is calculated by

𝑃𝑚𝑒𝑐= 1 2𝑅𝑒(︀𝑖𝜔𝐹 * 𝑈𝐹 )︀ (2.69) the term 𝑅𝑒(︀𝑖𝜔𝐹*_𝑈

𝐹)︀ is the real part of the product between velocity at the point of application

of the force and the complex conjugate of the force. The efficiency is obtained by

𝐸𝑓 𝑓 = 𝑃

𝑃𝑚𝑒𝑐

(2.70)

2.3 Topological modes

The geometric-phase in which describes the topological features of Bloch bands has an important duty in band theory of solids (Atala et al., 2013). That is defined by special invariants, expressed in terms of the Berry’s phase obtained by a particle during adiabatic motion through the band (Berry, 1984). The topological properties of 1D solids are performed by Zak phase (the Berry’s phases acquired by a particle moving across the Brillouin zone (Zak, 1989)). For

a certain Bloch wave 𝜓𝑘(𝑥) with a quasimomentum 𝑘, the Zak phase, 𝜙𝑍𝑎𝑘, can be expressed

through the cell-periodic Bloch function 𝑢𝑘(𝑥) = 𝑒−𝑖𝑘𝑥𝜓𝑘(𝑥):

𝜙𝑍𝑎𝑘= 𝑖

∫︁ 𝐺/2

−𝐺/2

⟨𝑢𝑘 | 𝜕𝑘 | 𝑢𝑘⟩𝑑𝑘 (2.71)

where 𝐺 = 2𝜋/𝑑 is the reciprocal lattice vector, 𝑑 is the lattice period and 𝜕𝑘 is the partial

derivative in relation to 𝑘. The geometric phase effects and the existence of topological transi-tions plots in acoustic system was demonstrated by Xiao et al. (2015) using acoustic systems. Xiao et al. (2015) also defined that PCs with different topologies exhibits an amplified mode of sound intensity at the interface state between both PCs. This rises inside the second bandgap of the system. Pal and Ruzzene (2017) explore elastic structures that imitate the Quantum Valley Hall Effect (QVHE) to obtain topologically protected edge modes. This effect employs valley states instead of spins, with the advantage of each lattice site needs to have only a single degree of freedom. Using discrete lattices, Pal and Ruzzene (2017) also show that a mass spring model can be used to obtain localized modes between two lattices with distinct topological indices. For one dimensional systems, it is possible to create localized modes at the interface of two periodic discrete structures whose parity of springs is flipped (Pal and Ruzzene, 2017).

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3 ENERGY HARVESTING IN THE SECOND BANDGAP

In this chapter, two Phononic Crystals (PC) are placed together with a commom interface in order to create a topological interface mode.

3.1 Description of the problem

In 2015, the phenomenon of interface mode in acoustic ducts was firstly studied by Xiao et al.. In the circumstance adopted by the author, the interface mode arises in the second bandgap and the same procedure is adopted. Two types of cross section (A and B) are used in each cell, according to the Figure (3.1).

Figure 3.1: Periodic cell of the phononic crystals

Figure 3.2: Topological transition plot

Figure (3.2) shows the topological transition plot in which each side has a distinct topo-logical number 𝐹 , characterized here as the bandwidth of the second bandgap of each PC. In order to create an interface mode (localized mode of vibration at the interface) between two PCs, both must be placed in opposite sides of the crossing of the topological number, according

to Figure (3.2). Using the parametric analysis of the bandgap as a function of ∆𝐿 = 𝐿𝑎−𝐿𝑏

2 and

considering that 𝐿𝑎+ 𝐿𝑏 = 85𝑚𝑚 (where 𝐿𝑎varies from 22.5𝑚𝑚 in PC1 to 62.5𝑚𝑚 in PC4

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(3.1)), the topological phase transition (Zak phase inversion) is noticed.

3.1.1 Sketch of the PCs

The Figures (3.3) (𝑏) and (𝑐) show the geometry of the PC1 according to the Tables (3.1) and (3.2). The Figure (3.3) (𝑎) exhibits the dispersion relation of the PC1. This plot shows the zones frequency that the imaginary part of the wavenumber is different of zero (called bandgaps). In theses zones, the elastic waves are evanescent and they do not propagate along the structure. In this plot, one may realize that the second bandgap is situated from 37𝑘𝐻𝑧 to 48𝑘𝐻𝑧, coinciding with the bandwidth of the red line of Figure (3.2).

Dimensions PC1 PC2 PC3 PC4

∆L (mm) -20 -5 5 20

𝐿𝑎(mm) 22.5 37.5 47.5 62.5

𝐿𝑏(mm) 62.5 47.5 37.5 22.5

Table 3.1: Geometry of the PCs

Dimensions PC

𝐿𝑐𝑒𝑙𝑙 (mm) Length of the cell 85

𝑅𝑎(mm) Radius of 𝑎 cross section 24

𝑅𝑏(mm) Radius of 𝑏 cross section 15

𝜌 (kg/m³) Density 7400

𝐸 (GPa) Young’s Modulus 96

𝜂 Loss factor 0.001

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(a)

(b) (c)

Figure 3.3: Phononic Crystal 1: (𝑎) Dispersion relation of the cell (𝑏) 2D sketch. (𝑐) 3D sketch.

The Figures (3.4) (𝑏) and (𝑐) present the geometry of PC2 in accordance with the infor-mations of the Tables (3.1) and (3.2). In Figure (3.4) (𝑎), the dispersion relation is shown, where the range of frequency of the second bandgap are from 40𝑘𝐻𝑧 to 45𝑘𝐻𝑧, matching with the magenta line of Figure (3.2).

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(a)

(b)

(c)

The geometry of PC3 is exhibited in Figures (3.5) (𝑏) and (𝑐), while (𝑎) shows the dis-persion relation of the structure. Comparing the Figure (3.5) (𝑎) with the Figure (3.4) (𝑎), it is possible to visualize that the dispersion relation plots are equal. This is due to the fact that both PCs are symmetric in relation to the crossing point in the Figure (3.2).

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(a)

(b) (c)

The Figures (3.6) (𝑏) and (𝑐) show the geometry of the PC4 according to the properties of the Tables (3.1) and (3.2). The Figure (3.6) (𝑎) exhibits the dispersion relation of the structure. One may notice that the dispersion relation of PC4 and PC1 are equal. This occur due to the fact that in Figure (3.2), the lines in which represent the PC1 and PC4 are symmetric in relation to the crossing point.

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(a)

(b) (c)

3.1.2 Configurations adopted

In Figure (3.7), two configurations is analysed. In PC1-PC1 configuration, both arrays have the same topological number, while in PC1-PC4, they have different topological number. This last condition implies in interface mode arising in the second bandgap as observed com-paring the Figures (3.7) (𝑐) and (𝑑). This mode is protected by the bandgap and its vibration is amplified by the mechanical energy concentration as seen in Figure (3.7) (𝑓 ).

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(a) (b) Frequency [Hz] _×104 0 2 4 6 8 10 12 dB [ref. m/N] -300 -250 -200 -150 -100 (c) Frequency [Hz] _×104 0 2 4 6 8 10 12 dB [ref. m/N] -300 -250 -200 -150 -100 (d) x coordinate [m] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 dB [ref. m/N] -280 -260 -240 -220 -200 (e) x coordinate [m] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 dB [ref. m/N] -320 -300 -280 -260 -240 -220 -200 -180 (f )

Figure 3.7: Comparison between the configurations PC1-PC1 and PC1-PC4 excited at the first end: Surface of the spatial distribution of the displacement in function of the frequency and X coordinate of (𝑎) PC1-PC1 configuration (𝑏)PC1-PC4 configuration; FRF of the displacement measured at the interface of (𝑐) PC1-PC1 configuration (𝑑)PC1-PC4 configuration; Spatial dis-tribution of the displacement at the interface mode frequency of (𝑒) PC1-PC1 configuration (𝑓 ) PC1-PC4 configuration.

The same situation is seen in Figure (3.8) when comparing the configurations PC2-PC2 and PC2-PC3.

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(a) (b) Frequency [Hz] _×104 0 2 4 6 8 10 12 dB [ref. m/N] -300 -250 -200 -150 -100 (c) Frequency [Hz] _×104 0 2 4 6 8 10 12 dB [ref. m/N] -300 -250 -200 -150 -100 (d) x coordinate [m] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 dB [ref. m/N] -280 -260 -240 -220 -200 (e) x coordinate [m] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 dB [ref. m/N] -230 -220 -210 -200 -190 -180 (f )

Figure 3.8: Comparison between the configurations PC2-PC2 and PC2-PC3 excited at the first end: Surface of the spatial distribution of the displacement in function of the frequency and X coordinate of (𝑎) PC2-PC2 configuration (𝑏)PC2-PC3 configuration; FRF of the displacement measured at the interface of (𝑐) PC2-PC2 configuration (𝑑)PC2-PC3 configuration; Spatial dis-tribution of the displacement at the interface mode frequency of (𝑒) PC2-PC2 configuration (𝑓 ) PC2-PC3 configuration.

Then, it is possible to conclude that the interface mode is attained in configurations PC2-PC3 and PC1-PC4. Now the challenge is placing the PZT (which is considered an inclusion) without destroying the interface mode in order to harvest energy. This situation is achieved by matching the elastic properties of the PZT and PCs. As the mechanical properties of piezo-electric rod are fixed, the material that has the closest mechanical properties of the PZT-5H is bronze. It can be verified in Tables (3.2) and (3.3) the proximity of both features.

3.1.3 Types of insertion of PZTs

Adopting the radius of 𝑅𝑃 𝑍𝑇 = 24𝑚𝑚, three types of insertion of PZTs are analysed

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Dimensions PZT-5H 𝜌 (𝑘𝑔/𝑚3₎ _Density ₇₅₀₀ 𝐸 (𝐺𝑃 𝑎) Young’s Modulus 117 𝜂 Damping ratio 0.001 𝛼 (𝑛𝐹/𝑚) Dielectric constant 13.02 𝑒11(𝐶/𝑚2) Piezoelectric constant 23.3

Table 3.3: Properties of the PZTs

thickness of the electrodes was neglected.

Figure 3.9: Polarization of the PZT

Inclusion 1

The first one, called inclusion 1, is the addition of 2 PZTs with 𝐿𝑃 𝑍𝑇 = 𝐿𝑐𝑒𝑙𝑙/2 between

the PCs, as sketched in Figure (3.10)

(a)

(b)

Figure 3.10: Phononic Crystals with inclusion 1: (𝑎) Configuration PC2-PC3 (𝑏) Configuration PC1-PC4

Inclusion 2

In second type of insertion, called inclusion 2, each PZT has the length of the adjacent rod of bronze PC, as seen in Figure (3.11).

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(a)

(b)

Figure 3.11: Phononic Crystals with inclusion 2: (𝑎) Configuration PC2-PC3 (𝑏) Configuration PC1-PC4

Substitution

In third insertion, the rod adjacent to interface is replaced by PZTs with the same dimen-sions, as shown in Figure (3.12).

(a)

(b)

Figure 3.12: Phononic Crystals with substitution: (𝑎) Configuration PC2-PC3 (𝑏) Configuration PC1-PC4

The purely resistive circuit is inserted in the terminals of each PZT with optimal resis-tance and with 𝑅 = 1Ω. The configurations PC1-PC4 (excited harmonically by an unitary force at the interface) and PC2-PC3 (excited harmonically by an unitary force at the first end) are adopted. The loss factors chosen are 𝜂 = 0, 𝜂 = 0.01 and 𝜂 = 0.001. It is also made a com-parison between the mechanical power inserted by the force and the electrical power extracted directly with the resistance of each circuit, in terms of the efficiency defined in Equation (2.70). Subsequently, all degrees of freedom of the entire system are excited individually with a cho-sen frequency and the electrical power obtained in resistive circuit with resistances 𝑅 = 1Ω

and 𝑅 = 𝑅𝑜𝑝𝑡 is summed. The power extracted from certain modes are compared in order to

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3.2 Inclusion 1

3.2.1 Resistive circuit

As pointed out before, after Equation (2.70), in case of a purely resistive circuit, an op-timal resistance is selected to obtain a maximum electric power efficiency with respect to me-chanical power injected. The case with 𝑅 = 1Ω is also adopted as a reference. Taking in con-sideration Equation (2.66), the electrical power obtained at the terminals of the piezoelectric rod, shown in Figure (3.13) (𝑏), depends on the excitation frequency, electrical properties of the PZT, mechanical properties of the complete phononic system and electrical parameters of the shunt circuit. The mechanical power inserted in the system using optimal resistance is sketched in Figure (3.13) (𝑎). Frequency [Hz] _×104 0 2 4 6 8 10 12 dB [ref. W/N] -300 -250 -200 -150 -100 -50 η=0 η=0.001 η=0.01 (a) Frequency [Hz] _×104 0 2 4 6 8 10 12 dB [ref. W/N] -300 -250 -200 -150 -100 -50 η=0 η=0.001 η=0.01 (b)

Figure 3.13: Comparison of energy for 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 using optimal resistance in PC1-PC4 configuration excited at the interface: (𝑎) Mechanical power inserted (𝑏) Electrical power extracted

The configuration PC1-PC4 excited at the interface is analysed with respect to its effi-ciency in Figure (3.14), where both resistances are performed .

Frequency [Hz] _×₁₀4 0 2 4 6 8 10 12 Efficiency dB [ref. W/W] -120 -100 -80 -60 -40 -20 0 20 η=0 η=0.001 η=0.01 (a) Frequency [Hz] _×₁₀4 0 2 4 6 8 10 12 Efficiency dB [ref. W/W] -200 -150 -100 -50 0 50 η=0 η=0.001 η=0.01 (b)

Figure 3.14: Efficiency with 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 in PC1-PC4 configuration using (𝑎) optimal resistance (𝑏) 𝑅 = 1Ω

The configuration PC2-PC3 excited at the first end is adopted in Figure (3.15) to evaluate the efficiency using optimal resistance and 𝑅 = 1Ω.

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Frequency [Hz] _×₁₀4 0 2 4 6 8 10 12 Efficiency dB [ref. W/W] -200 -150 -100 -50 0 50 η=0 η=0.001 η=0.01 (a) Frequency [Hz] _×₁₀4 0 2 4 6 8 10 12 Efficiency dB [ref. W/W] -250 -200 -150 -100 -50 0 50 η=0 η=0.001 η=0.01 (b)

Figure 3.15: Efficiency with 𝜂 = 0, 𝜂 = 0.001 and 𝜂 = 0.01 in PC2-PC3 configuration using (𝑎) optimal resistance (𝑏) 𝑅 = 1Ω

For the resistive circuit, the efficiency, defined by Equation (2.70), is equal to 1 along all frequencies in PC1-PC4 and PC2-PC3 sets when the loss factor is negligible, as shown in Figure (3.14). When increasing the loss factor, the efficiency decreases in all frequencies. For configuration PC1-PC4 excited at the interface, the efficiency is higher inside the bandgaps than in other ranges of frequency. Using the optimal resistance, the efficiency remains close to 1, while considering 𝑅 = 1Ω it decays considerably, as seen in Figure (3.14). In Figure (3.15), for PC2-PC3 set excited at the first end, the efficiency remains high in natural modes of vibration, mainly adopting optimal resistance. Considering 𝑅 = 1Ω, the efficiency reduces noticeably in all ranges of frequency, but the interface mode inside the bandgap prevails.

3.2.2 Resistive circuit exciting all degrees of freedom

Now, all degrees of freedom are excited individually and the power of the contribution of each excitation is summed and the total power achieved is compared regarding the resis-tance adopted (𝑅 = 1Ω and optimal resisresis-tance) and the boundary conditions used (free-free conditions and fixed-fixed conditions). In the 3D plot, the 𝑋 axis is the spatial coordinate of the system and 𝑌 is the degree of freedom excited. 𝑍 is the longitudinal displacement of the structure. Figure (3.16) shows the sketch of the configuration PC1-PC4 in free-free condition with inclusion 1 and the modes adopted for comparison.

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(a) Frequency [Hz] _×₁₀4 0 2 4 6 8 10 12 dB [ref. m/N] -350 -300 -250 -200 -150 -100 X: 3622 Y: -138.9 X: 1.78e+04 Y: -210.6 X: 3.635e+04 Y: -175.9 X: 4.531e+04 Y: -235.1 X: 6.517e+04 Y: -234.5 X: 8.737e+04 Y: -194.1 A B C D E F (b)

Figure 3.16: (𝑎) Sketch of the PC1-PC4 free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes where the PZTs are in short circuit condition (𝑅 = 1Ω)

The selected modes of the PC1-PC4 system with inclusion 1 are described in Table (3.4). Table 3.4: Modes chosen in PC1-PC4 configuration using inclusion 1

Modes Frequency (KHz) Free-Free Clamped-Clamped 𝐴 3.622 2.721 𝐵(defect mode) 17.8 17.8 𝐶 36.35 30.79 𝐷(interface mode) 45.31 45.31 𝐸(defect mode) 65.17 65.17 𝐹 87.37 90.02

The 3D plot of the displacement exciting all the degrees of freedom individually for modes 𝐶, 𝐸, 𝐷 with 𝑅 = 1Ω and 𝐷 with optimal resistance are shown in Figure (3.17).

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X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -250 -200 (a) X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -250 -200 (b) X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -250 -200 (c) X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -250 -200 (d)

Figure 3.17: 3D plot for excitation in all degrees of freedom of PC1-PC4 with inclusion 1 in (𝑎) 𝐶 natural mode with 𝑅 = 1Ω (𝑏) 𝐸 defect mode with 𝑅 = 1Ω (𝑐) 𝐷 interface mode with 𝑅 = 1Ω and (𝑑) 𝐷 interface mode with optimal resistance

The total electrical power harvested in chosen modes is given in Table (3.5).

Table 3.5: Modes of inclusion 1 in PC1-PC4 configuration, where the best extracted power values are highlighted

Inclusion 1 Free-free Clamped-Clamped Modes 𝑃 (𝑅 = 1Ω) 𝑃 (𝑅 = 𝑅𝑜𝑝𝑡) 𝑃 (𝑅 = 1Ω) 𝑃 (𝑅 = 𝑅𝑜𝑝𝑡) 𝐴 2.57 × 10−7𝑊 0.0045𝑊 1.12 × 10−7𝑊 0.0046𝑊 𝐵 (defect mode) 9.13 × 10−5𝑊 1.02 × 10−4𝑊 9.22 × 10−5𝑊 1.02 × 10−4𝑊 𝐶 9.12 × 10−6𝑊 5.54 × 10−4𝑊 9.67 × 10−6𝑊 4.68 × 10−4𝑊 𝐷 (interface mode) 7.75 × 10−5𝑊 8.30 × 10−5𝑊 7.75 × 10−5𝑊 8.30 × 10−5𝑊 𝐸 (defect mode) 4.60 × 10−6𝑊 4.66 × 10−4𝑊 4.60 × 10−6𝑊 4.66 × 10−4𝑊 𝐹 1.82 × 10−7𝑊 2.54 × 10−4𝑊 1.14 × 10−7𝑊 1.68 × 10−4𝑊

The Figure (3.18) shows the system PC2-PC3 in free-free condition with inclusion 1 and the respective modes studied.

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(a) Frequency [Hz] _×₁₀4 0 2 4 6 8 10 12 dB [ref. m/N] -300 -250 -200 -150 -100 X: 7062 Y: -152.4 X: 1.74e+04 Y: -233 X: 3.495e+04 Y: -178.7 X: 4.393e+04 Y: -199.3 X: 6.269e+04 Y: -229.5 X: 8.305e+04 Y: -223.5 A B C D E F (b)

Figure 3.18: (𝑎) Sketch of PC2-PC3 configuration in free-free system excited at the first end and (𝑏) the FRF measured at the interface with the selected modes where the PZTs are in short circuit condition (𝑅 = 1Ω)

For the system shown in Figure (3.18), the following modes were chosen, according to the Table (3.6).

Table 3.6: Modes chosen in PC2-PC3 configuration using inclusion 1

Modes Frequency (KHz) Free-Free Clamped-Clamped 𝐴 7.062 7.922 𝐵(defect mode) 17.4 17.4 𝐶 34.95 37.51 𝐷(interface mode) 43.93 43.93 𝐸(defect mode) 62.69 62.67 𝐹 (defect mode) 83.05 83.05

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X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -280 -260 -240 -220 -200 -180 -160 (a) X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -280 -260 -240 -220 -200 -180 -160 (b) X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -280 -260 -240 -220 -200 -180 -160 (c) X coordinate (m) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 DOF of excitation 10 20 30 40 50 60 -300 -280 -260 -240 -220 -200 -180 -160 (d)

Figure 3.19: 3D plot for excitation in all degrees of freedom of PC2-PC3 inclusion 1 system in (𝑎) C natural mode with 𝑅 = 1Ω (𝑏) E defect mode with 𝑅 = 1Ω (𝑐) D interface mode with 𝑅 = 1Ω and (𝑑) D interface mode with optimal resistance

The 3D plot of mode 𝐶, 𝐸 and 𝐷 with 𝑅 = 1Ω and 𝐷 with optimal resistance is described in Figure (3.19). The results of the total electrical power collected at the resistance for PC2-PC3 with inclusion 1 system are shown in Table (3.7).

Table 3.7: Modes of inclusion 1 in PC2-PC3 configuration, where the best extracted power values are highlighted

Inclusion 1 Free-free Clamped-Clamped Modes 𝑃 (𝑅 = 1Ω) 𝑃 (𝑅 = 𝑅𝑜𝑝𝑡) 𝑃 (𝑅 = 1Ω) 𝑃 (𝑅 = 𝑅𝑜𝑝𝑡) 𝐴 1.80 × 10−6𝑊 0.0089𝑊 1.79 × 10−6𝑊 0.0054𝑊 𝐵 (defect) 1.71 × 10−4𝑊 9.01 × 10−5𝑊 1.71 × 10−4𝑊 9.01 × 10−5𝑊 𝐶 1.12 × 10−5𝑊 4.33 × 10−4𝑊 8.34 × 10−6𝑊 5.05 × 10−4𝑊 𝐷 (interface) 1.72 × 10−5𝑊 1.73 × 10−4𝑊 1.62 × 10−5𝑊 1.74 × 10−4𝑊 𝐸 (defect) 4.63 × 10−5𝑊 1.09 × 10−4𝑊 3.78 × 10−5𝑊 1.06 × 10−4𝑊 𝐹 (defect) 5.96 × 10−6𝑊 3.45 × 10−4𝑊 5.95 × 10−6𝑊 3.44 × 10−4𝑊

In Figures (3.16) and (3.18), it is possible to see that big defect modes appear inside the bandgaps as a consequence of the elastic inclusion of PZT rods. As shown in Figure (3.17), it is possible to notice that both interface mode and defect modes concentrate mechanical energy in the interface region, independently of where the excitation is located, being worth for using as energy harvesting mode. This phenomenon can be confirmed comparing the results of Tables (3.5) and (3.7) for 𝑅 = 1Ω. The other advantage of adopting the interface and defect modes is that they remain in same frequency considering free-free or Clamped-Clamped conditions, i.e.,