Control and optimization on portal frame for current flow using piezoeletric on buoy

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DEPARTMENT OF ELECTRICAL ENGINEERING

WAGNER BARTH LENZ

CONTROL AND OPTIMIZATION ON PORTAL FRAME FOR

CURRENT FLOW USING PIEZOELECTRIC ON BUOY

DISSERTATION

PONTA GROSSA 2019

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CONTROL AND OPTIMIZATION ON PORTAL FRAME FOR

CURRENT FLOW USING PIEZOELECTRIC ON BUOY

Master’s dissertation presented as partial requirement for obtain a Master’s Degree in Electrical Engineering from the department of Electrical Engineering at Federal University of Technology of Parana - UTFPR

Advisor:Prof. Dr. Angelo Marcelo Tusset Co-advisor:Prof. Dr. José Manoel Balthazar

PONTA GROSSA 2019

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Ficha catalográfica elaborada pelo Departamento de Biblioteca

da Universidade Tecnológica Federal do Paraná, Campus Ponta Grossa n.06/20

Elson Heraldo Ribeiro Junior. CRB-9/1413. 07/02/2020. L575 Lenz, Wagner Barth

Control and optimization on portal frame for current flow using piezoelectric on buoy / Wagner Barth Lenz. - 2019.

70 f. : il. ; 30 cm.

Orientador: Prof. Dr. Angelo Marcelo Tusset Coorientador: Prof. Dr. José Manoel Balthazar

Dissertação (Mestrado em Engenharia Elétrica) - Programa de Pós-Graduação em Engenharia Elétrica. Universidade Tecnológica Federal do Paraná, Ponta Grossa, 2019.

1. Correntes elétricas. 2. Sensoriamento remoto. 3. Ondas sonoras. 4. Energia - Fontes alternativas. I. Tusset, Angelo Marcelo. II. Balthazar, José Manoel. III. Universidade Tecnológica Federal do Paraná. IV. Título.

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MODELO DE FOLHA DE APROVAÇÃO

(Este modelo deverá ser inserido no VERMELHO, DE ACORDO COM A CÓPIA

IMPRESSA E ASSINADA DESTA FOLHA, margens: Esquerda 3cm, Esquerda 2cm, Superior e Inferior 1,5cm).

FOLHA DE APROVAÇÃO Título de Dissertação Nº 49/2020

CONTROL AND OPTIMIZATION ON PORTAL FRAME FOR CURRENT FLOW USING PIEZOELECTRIC ON BUOY

por

Wagner Barth Lenz

Esta dissertação foi apresentada às 14:00 do dia 20 de janeiro de 2020 como requisito parcial para a obtenção do título de MESTRE EM ENGENHARIA ELÉTRICA, com área de concentração em Controle e Processamento de Energia, linha de pesquisa em Instrumentação e Controle do Programa de Pós-Graduação em Engenharia Elétrica. O candidato foi argüido pela Banca Examinadora composta pelos professores abaixo assinados. Após deliberação, a Banca Examinadora considerou o trabalho aprovado.

Prof. Dr. Rodrigo Tumonlin Rocha (KAUST)

Prof. Dr. Angelo Marcelo Tusset Orientador - (UTFPR)

Prof. Dr. Mauricio Aparecido Ribeiro (UTFPR)

Prof. Dr. (Nome do orientador) (UTFPR) -

Orientador

Prof. Dr. Angelo Marcelo Tusset (UTFPR)

Coordenador do PPGEE

A FOLHA DE APROVAÇÃO ASSINADA ENCONTRA-SE NO DEPARTAMENTO DE REGISTROS ACADÊMICOS DA UTFPR –CÂMPUS PONTA GROSSA

Paraná Campus de Ponta Grossa

Diretoria de Pesquisa e Pós-Graduação

PROGRAMA DE PÓS-GRADUAÇÃO EM

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First, I would like to thank my parents for supporting me during this tough time. To my advisor, Prof. Dr. Angelo Marcelo Tusset, for the friendship, patience and knowledge during years of work. It was a pleasure to work with you.

To my co-advisor, Prof. Dr. José Manoel Balthazar, for the friendship, and the advises.

To my girlfriend, Grasieli Oliveira for the patience during this project. To all the member of the thesis committee, for the correction and advises. To all my friends for the well spent time at the lab, and of course the contribution on this thesis.

To the Brazilian governments that through their development agencies could finance this work

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LENZ, Wagner Barth.CONTROL AND OPTIMIZATION ON PORTAL FRAME FOR CURRENT FLOW USING PIEZOELECTRIC ON BUOY. 2019. 70 p. Dissertation (Master’s Degree in Electrical Engineer) – Federal University of Technology – Parana. Ponta Grossa, 2019.

The growing demand for remote sensing for weather forecasts, telecommunication amplifiers, and monitoring long flights, has been generating demand for alternative power sources. Consequently, this leads to a micro self-power source on site, to avoid unnecessary batteries replacement and power line transmission. A common source of energy is wind or waves that are harvested using vibration transducer, thus actuating as generators and sensors. In this dissertation, a portal frame structure with piezoelectric material using waves as a source of vibration energy. First, an initial dynamic characterization was made for sine and beating excitation. Second, a parametric sensibility analysis was made and an optimization process was carried out using Particle Swarm Optimization (PSO). Third, to further improve the performance the State-Dependent Riccati Equation (SDRE) controller was used varying the external load. The results shows that the PSO did not choose the same parameter as present on the literature with an small gain (.5%) in performance. However, the SDRE controller shown an increase of 0.59𝐶 to 3𝐶 in a period of 200 𝑠.

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LENZ, Wagner Barth.Controle e Otimização de uma estrutura do tipo portal para geração de corrente elétrica embarcadas em boias. 2019. 70 f. Dissertação

(Mestrado em Engenharia Elétrica ) – Universidade Tecnológica Federal do Paraná. Ponta Grossa, 2019.

A crescente demanda por sensoriamento remoto para previsões meteorológicas, amplificadores de telecomunicações e monitoramento de voos longos tem gerado demanda por fontes alternativas de energia. Consequentemente, isso leva a uma micro fonte de auto-alimentação no local, para evitar a substituição desnecessária de baterias e a transmissão da linha de energia. Uma fonte comum de energia é o vento ou as ondas que são colhidas usando transdutores de vibração, agindo assim em geradores e sensores. Nesta dissertação, uma estrutura de estrutura de portal com material piezoelétrico usando ondas como fonte de energia vibratória. Primeiro, foi feita uma caracterização dinâmica inicial para excitação senoidal e pulsante. Segundo, uma análise de sensibilidade paramétrica foi realizada e um processo de otimização foi realizado usando o Particle Swarm Optimization (PSO). Terceiro, para melhorar ainda mais o desempenho do controlador da Equação de Riccati Dependente do Estado (SDRE), foi usada para variar a carga externa. Os resultados mostram que o PSO não escolheu o mesmo parâmetro presente na literatura com um pequeno ganho (0,5%) em desempenho. No entanto, o controlador SDRE mostrou um aumento de 0.59𝐶 para 3𝐶 em um periodo de 200 s.

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Figure 1 – Sample of Earthquake . . . 14 Figure 2 – Distributions permitted zones (A) ETOPS 60 min (B) ETOPS 180 min 15 Figure 3 – Map of fiber optic cables . . . 16 Figure 4 – Buoy example (A) Common buoy setup (B) Schematic of research

buoy . . . 16 Figure 5 – Examples of wave energy harvesting . . . 17 Figure 6 – Piezoelectric material (A) Under compression (Direct effect) (B)

Under external voltage (Reverse effect) . . . 24 Figure 7 – Piezoelectric actuator . . . 25 Figure 8 – Example of Portal Frame Structure (A) Truss representation (B)

Dynamic Model . . . 26 Figure 9 – Portal Frame Structure (A) Truss representation (B) Dynamic Model 27 Figure 10 – Spring comparison (A) Force (B) Work . . . 28 Figure 11 – Example of feedback control with contoller and actuator . . . 30

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Graph 1 – Optimization of PZT . . . 18

Graph 2 – Energy transfer (A) Distribution of energy (B) Wind transfer mechanism 22 Graph 3 – Wave forms (A) 1D-Sine wave (B) 2D-Sine wave . . . 23

Graph 4 – Beating Excitation (A) f1= 1 Hz df=0 Hz (B) (A) f1= 1 Hz df=.1 Hz (C) (A) f1= 1 Hz df=0.5 Hz (D) f1= 1 Hz df=1.5 Hz . . . 23

Graph 5 – PSO Behavior (A) First Interaction (B) Second Interaction (C) Few Interactions . . . 29

Graph 6 – Comparison between Linear and Quadratic Least Squares (A) Range of experimental data (B) Extrapolation Comparison . . . 35

Graph 7 – 0 − 1 Test 2D (A) For 𝑥1 (B) For 𝑥2 . . . 38

Graph 8 – Lyapunov for Beating (A) 2D varying df and f (B) 𝑑𝑓 = 4/ 𝐻𝑧 varying f 39 Graph 9 – Bifurcation for beating excitation . . . 39

Graph 10 – Test 0-1 for beating excitation for 𝑑𝑓 = 4 Λ𝐻𝑧 . . . 40

Graph 11 – Scalogram for beating excitaiton on 𝑑𝑓 = 4 Λ𝐻𝑧 (A) For 𝑥1 (B) For 𝑥2 . 40 Graph 12 – Bifurcation for initial parameter for sine excitation without PZT (A) 𝑋1 (B) 𝑋3 . . . 41

Graph 13 – Bifurcation for initial parameter for sine excitation with PZT (A) 𝑋1 (B) 𝑋3 . . . 41

Graph 14 – Test 0-1 (A) without PZT 𝑋1 (B) without PZT 𝑋2 (C) with PZT 𝑋1 (D) with PZT 𝑋2 . . . 42

Graph 15 – Lyapunov Expoent (A) Without (B) with 1 pzt . . . 43

Graph 16 – FFT -2d for sine before optimization (A) 𝑋1without pzt (B) 𝑋3 without pzt (C) 𝑋1 with pzt (D) 𝑋3 with pzt . . . 44

Graph 17 – Local Sensibility of 𝜔1 , best at 𝜔1 = 0.4592 . . . 45

Graph 18 – Local Sensibility of 𝑀 , best at 𝑀 = 3.413 . . . 45

Graph 19 – Local Sensibility of 𝐾𝑛𝑙, best at 𝐾𝑛𝑙 = 755.3893 . . . 46

Graph 20 – Local Sensibility of 𝐾𝑏, best at 𝐾𝑏 = 1.5730 . . . 46

Graph 21 – Local Sensibility of Ω1,best at Ω1 = 0.1 . . . 47

Graph 22 – Global Sensitivity . . . 47

Graph 23 – Temporal evolution PSO . . . 48

Graph 24 – Comparison between Swarm Size . . . 48

Graph 25 – 0-1 Test (A) For 𝑥1 (B) For 𝑥2 . . . 49

Graph 26 – Lyap for Beating (A) 2D varying df and f whole spectrum (B) 2D varying df and f zoom (C) 𝑑𝑓 = 4/; 𝐻𝑧 varying f . . . 50

Graph 27 – Bifurcation for beating excitation (A) 𝑥1 (B) 𝑥2 . . . 50

Graph 28 – 0-1 Test for beating excitation (A) 𝑥1 (B) 𝑥2 . . . 51

Graph 29 – Scalogram for beating excitation where 𝑑𝑓 = 4 (A) For 𝑥1 (B) For 𝑥2 . 51 Graph 30 – Bifurcation for sine excitation after optimization without PZT (A) 𝑋1 (B) 𝑋2 . . . 52

Graph 31 – Bifurcation for sine excitation after optimization with PZT (A) 𝑋1 (B) 𝑋2 52 Graph 32 – 0-1 Test for sine excitation after optimization (A) 𝑋1 without PZT (B) 𝑋3 without PZT (C) 𝑋1 with PZT (D) 𝑋3 with PZT . . . 53

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Graph 35 – After Optimization Lyapunov Expoent (A) without PZT (B) with PZT . 55 Graph 36 – signal control values for (A) Beating (B) Sine . . . 56 Graph 37 – 𝑅2 values for (A) Beating (B) Sine . . . 56

Graph 38 – Control efficiency for N on beating excitation (A) 𝑥1 (B) 𝑥2 . . . 57

Graph 39 – Control efficiency for N on beating excitation (A) Rms(𝑥3) (A) 𝐸200 . . 57

Graph 40 – Control efficiency for sine excitation (A) 𝑥1 (A) 𝑥2 . . . 58

Graph 41 – Control efficiency for sine excitation (A) Sum of currents, (A) N Sum of currents . . . 58 Graph 42 – SDRE efficiency for different types of excitaitons (A) 𝑥1 (B) 𝑥2 . . . . 59

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Table 1 – Parameters for portal frame . . . 34 Table 2 – Parameters for least squares method . . . 35 Table 3 – Swam Optimization Parameters . . . 49

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ABBREVIATIONS

ADBS Automatic dependent surveillance

PDI Proportional-integral–derivative Controller PSO Particle Swarm Opimization

SDRE State-Dependent Riccati Equation LQG Linear–quadratic–Gaussian control

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GREEK LETTERS

Ψ1 Coupling parameter [N_V]

Ψ2 Linear coupling parameter [N_V]

Ψ3 Nonlinear coupling parameter [N_V]

LATIN LETTERS

𝐴𝑚𝑝 Amplitude of wave [m]

𝑑𝑓 Differential of frequency of excitation [−s]

𝑘 Spring stiffens [_mN]

𝑏 Damping coefficient [Ns_m]

𝑠𝑤 Wave speed [m_s]

𝐶 Piezoelectric capacitance [F]

𝑓1 Frequency of excitation [−s]

𝑞 Charge on the piezoelectric [C]

𝑡 Time [s]

𝑥𝑤 Wave displacement [m]

𝑅 Electrical resistance [Ω]

𝐶0 Inertia of the swarm coefficient [ ]

𝐶1 Self knowledge coefficient [ ]

𝐶2 Group knowledge coefficient [ ]

𝑅𝑎1 and 𝑅𝑎2 random values [ ]

𝑃 1(𝑓 ) Normalized Power of Spectrum [ ]

𝐸200 Sum of charges over 200 𝑠 [C]

MATRICES AND VECTORS

A States matrix [ ]

A1 Linear Portion of States matrix [ ]

A2 Nonlinear Portion of States matrix [ ]

x State vector [ ]

B Input matrix [ ]

C Output matrix [ ]

D Feedthrough matrix [ ]

Q Cost of state matrix [ ]

R Cost of control matrix [ ]

K Cost of control matrix [ ]

e Error vector [ ]

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1 INTRODUCTION . . . . 14 1.1 GENERAL OBJECTIVE . . . 19 1.2 SPECIFIC OBJECTIVES . . . 19 1.3 JUSTIFICATION . . . 19 1.4 THESIS ORGANIZATION . . . 20 2 LITERATURE REVIEW . . . . 21 2.1 WAVE ENERGY . . . 21 2.1.1 Wave Model . . . 21 2.2 PIEZOELECTRIC MATERIALS . . . 24

2.3 PORTAL FRAME STRUCTURE . . . 26

2.4 MECHANICAL MODEL . . . 26

2.4.1 Non linear Stiffens of Springs . . . 27

2.5 PARTICLE SWARM OPTIMIZATION . . . 29

2.6 SDRE . . . 30

2.6.1 Controllability . . . 32

3 METHODOLOGY . . . . 33

3.0.1 Parameters of Portal Frame . . . 33

3.1 COUPLING PARAMETER OF PZT . . . 34

3.2 PARAMETRIC ANALYSES . . . 35

3.3 OPTIMIZATION PROCESS . . . 36

3.4 CONTROL ANALYSES . . . 36

3.4.1 SDRE Parameters . . . 37

4 RESULTS AND DISCUSSION . . . . 38

4.1 CURRENT MODELS . . . 38

4.1.1 Beating Excitation . . . 38

4.1.2 Sine Excitation . . . 41

4.2 PARAMETER SENSIBILITY . . . 44

4.3 OPTIMIZATION . . . 48

4.4 DYNAMICAL ANALYSIS OPTIMIZED MODEL . . . 49

4.4.1 Beating Excitation - 2D . . . 49 4.4.2 Beating Excitation - 1D . . . 50 4.4.3 Sine Excitation . . . 52 4.5 CONTROL METHOD . . . 54 4.6 CONTROL EFFICIENCY . . . 55 4.6.1 Beating Excitation . . . 56 4.6.2 Sine Excitation . . . 58

4.6.3 Beating and Sine . . . 59

5 CONCLUSION AND REMARKS . . . . 61

5.1 FUTURE RESEARCH . . . 61

BIBLIOGRAPHY . . . . 63

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1 INTRODUCTION

In remote locations, a vast of meteorologic and seismologic effects originate that directly affect the global population. For example of remote areas, the atomic disaster in Fukushima that was hit by a Tsunami and hurricanes that are generate by the heat evaporation on seas every summer in the Caribbeans, causing immense damage. To avoid future disaster, the demand for more deep ocean sensor in remote location increased. Thus, imposing inhospitable conditions to machines and sensors, subjecting to be off the electric grid, in sever weather conditions, and far for the conventional supply chain, for electric energy and maintenance. Fig. 1 shows the history of earthquakes.

Figure 1 – Sample of Earthquake

Source: National Oceanic and Atmospheric Administration (2018)

As shown in Fig. 1 it is the clear location of the geological faults, e.g. in the pacific and the Oceania. Thus, these remote areas could further improve the weather forecast, better models for oil spills and current on the sea (MORONI et al, 2016). Additionally, weather stations could collect wind information, and monitoring the temperature of the surface and undersea (VENKATESAN et al, 2016; BASTIEN et al, 2009).

Another use of buoy is to combine many censoring devices. For example, combining weather, earthquake, and flight radar. In a remote location, there is no conventional radar and the position depends on the self-reporting mechanism.

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Despite being not accurate as primary radars, it is acceptable for Extended operations (ETOPS).

ETOPS is one of the most important advances in aviation. This methodology allows twin engines to fly over remote areas in a distance greater than 60min of flight areas, based on the primary engines, where the reliability was low. After, the reliability of turbo engines started to increase in the 60’s some relaxation on the rule passed the maximum time to 95min. In 1985, the first approved flight with 180min rule ETOPS (MOORE, 1993; LIN; LIN, 2010; WANG, 2017). As shown in Fig. 2.

Figure 2 – Distributions permitted zones (A) ETOPS 60 min (B) ETOPS 180 min

(A) (B)

Source: Swartz (2019)

where shadows areas are of limit and bright areas are allowed for commercial flights. It is clear when contrasting Fig. 2 (A) and (B), the difference of areas allows to twin jets to operate in a more direct route. This new rule allowed a more efficient airplane to flight in routes that before were dominated by bigger and lesse efficient quadri-jets. Nevertheless, even in modern aviation, there are areas without radar coverage. At this stage, the solution is Automatic dependent surveillance (ADBS). It tries to mitigate the risks, by self-report positions procedures that count with satellites. Thus, avoiding incidents where airplanes vanish from the globe (ZHANG et al, 2018).

Other application of buoy is amplifying the signal, a considerable number of fiber optic cables are laid to connect the telecommunication system. In contrast with copper, fiber optics has a quarter of power requirements, with losses lower than only 0.2𝑑𝐵/𝑘𝑚and for the same amount of data transmit and without the electromagnetic interference and the inertial of signals over long cables (BOLETTI et al, 2013; AGRAWAL, 2010). Fig. 3 shows the interactivity of the Globe relying on submarine fiber optics.

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Figure 3 – Map of fiber optic cables

Source: PriMetrica (2019)

As shown in Fig. 3, there is a great amount of overlapping of areas that require a remote device to boost the signal. Over long distances, fiber optic needs a boost on signal every 50km or less to avoid degradation in the signal (AGRAWAL, 2010). The amplification of the signal requires an amplification unit that uses a battery, supercapacitor that recharges from some source of electric energy and needs to be changed every few years. An example of a generic buoy system is shown in Fig. 4.

Figure 4 – Buoy example (A) Common buoy setup (B) Schematic of research buoy

Source: Adapted from Eriksson (2007)

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stand vast types of weather. They could be installed on farms to leverage a good site energy source. In addition, in Fig. 4 (B), a generalized model for buoy apparatus, where the communication could be made via satellite link, 𝑏1 represent the dissipate element

that could be the linear hydraulic or linear motor, 𝑏2 represent the stiffness of the cable

and the stiffness of spring used to return the buoy mass, 𝑏3is the example of fiber optic

cable amplifier, 𝑏4 is the anchoring in the sea bed after the sand. This kind of censoring

could be even used to monitor tsunami.

Regarding power consumption, a few considerations must take place. Buoy system for weather observation requires low power (3.5 𝑚𝑊 ) and the power management can take full advantage of hibernates states, taking samples every 15 minutes and sending data once a day. For example, in a study of remote sonar buoy the active time only 10 days in a period of 450 days(AUSTIN; STOKEY; SHARP, ). Thus, PZT can be a self-powering sensor (MINAZARA et al, 2006). For more active, robust and military the power necessary grows to 100 𝑊 demanding more energy-dense methods (DEWAN et al, 2014).

Usually, wave energy harvesting is related to production on a macro scale with a big production site. Such site can have a significant output and many forms of transforming the wave energy into electric energy (JAPPE; MUSSA, 2009; BERNARD, 2012). Various forms of Buoys system are shown inf Fig. 5.

Figure 5 – Examples of wave energy harvesting

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where (A) is a surge device that works with tidal amplitude, (B) is heaving float that works with waves generated by wind. amplitude, (C) is heaving and pitching float that works by using the waves to create rotation between buoy, (D) is pitching device that works twisting the device, (E) is oscillating that works creating air pressure generate electricity. As shown before in Fig. 4, the selected model usually is (B). Because it allows the power extraction unit to at the bottom, and the use of a linear motor or linear hydraulic actuator. A review of Buoy generation could be seen in Eriksson (2007), Falcão (2010), Babarit (2017).

Another approach is the use of piezoelectric material. This material produces a differential of potential when subjected to mechanical stress. As a micro powering device, PZT materials can be embedded in nanodevices. On the comparison of two models on PZT and external load and power, delivery was investigated by Lu, Lee and Lim (2003). This comparison is summarized in Graph 1 (LU; LEE; LIM, 2003; PRIYA; INMAN, 2009). Graph 1 – Optimization of PZT 0 100 200 300 400 500 External Resitance [ k ] 0 0.2 0.4 0.6 0.8 Power [ mW ] (A) 0 200 400 600 800 1000 Frequency [ kHz ] 0 20 40 60 80 100 Power [ mW ] (B) PZt P1515 PZN-8% PT 0 50 100 150 200 250 External Resitance [ k ] 0 0.5 1 1.5 2 Power [ mW ] (C) 7000Hz 4000Hz 2939Hz

Source: Adapted from Lu, Lee and Lim (2003)

Another way to explore PZT is by tuning the frequency. Usually, the mechanical system has very low frequencies (below 100Hz), and PZT are designed to operate are higher frequencies. One way is to trade amplitude for frequencies or artificially increase the frequency of excitation by tuning forks (PRIYA; INMAN, 2009).

Another application for PZT is the use of actuators for flexible elements to avoid deflection. In these cases, the reverse effect is a desire effect and it could be used with the Genetic Algorithm (GA) to better locate and size the PZT. Thus, the State-Dependent Riccati Equation (SDRE) controller is used to drive the PZT to the require

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behavior (MOLTER et al, 2010).

On the same vein, PZT has been used for position control in different manners (SHEN; HOMAIFAR, 2001; XU; ONO; ESASHI, 2006; L.; LIU, 2012). In this application, the PZT element pump energy from and to the system to control the motion. Shen and Homaifar (2001) used a combination of fuzzy-PID with Linear–quadratic–Gaussian control (LQG) to control the motion of a plate, it concluded that both strategies are valid and robust to control the experimental test to better evaluate the performance. Thus, showing the complexity and difficulty to use PZT actuators.

1.1 GENERAL OBJECTIVE

The general objective of this thesis is the optimization of the parameters of the piezoelectric buoy sensor, using Particle Optimization Algorithm (PSO), and further increase the efficiency of the system with SDRE control.

1.2 SPECIFIC OBJECTIVES

• To propose and simulate the piezoelectric buoy.

• To investigate and optimize the parameters of the PZT generator.

• To propose and apply a SDRE control on the PZT generator.

• To analyze through numerical simulation the efficiency and efficacy of the system.

1.3 JUSTIFICATION

The continuous demand for more bandwidth is pushing the repeater technology to evolve and be more power-hungry. Thus sorting the lifespan of batteries and super-capacitors that are constantly doing deep cycles. The use of solar panels could be an alternative. Nevertheless, it has a clear discontinuous charging capability and it does not have a clear chain to be recyclable (XU et al, 2018).

The remote locations further constrains the power generation. Due to the fact that the buoy is in constant motion, it is a logical solution to harvest this energy. Due to power output the combination of several piezoelectric materials further enhances the

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reliability.

Piezoelectric material is reliable and it is not just used to collect energy. It is present on humidifiers, printers and high-performance diesel engines, proving the applicability in diverse fields and already consolidate technology.

1.4 THESIS ORGANIZATION

The Master Thesis was divided into five chapters, where Chapter 1 one has a brief introduction on energy remote locations, buoy types and piezoelectric.

Chapter 2 will present the buoy models, and mathematical models with piezoelectric models, in addition to the PSO optimization, and SDRE controller.

Chapter 3 will present the used methodology, the selected variables, the cost function for optimization, and the parameter for control.

On Chapter 4 the families of solutions are shown, and the discussions of the results. In Chapter 5 the of the conclusions and future researches.

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2 LITERATURE REVIEW

In this chapter, a brief review of wave energy, buoy elements and sensor, the piezoelectric material and the mechanical design will be explained.

2.1 WAVE ENERGY

Wave energy is renewable energy, with different mythological sources. The motion of the fluid is a constraint to the boundary and temporal conditions and it is extremely connected atmospheric conditions (PEDLOSKY, 2013; BOCCOTTI, 2000). One example is the El Nino effect, which has been extensible study and it has effects of the Global economy and weather (CANE, 1983). As much renewable energy, the heat of the sun is one of the main engines, but other factors such as wind and moon affect as well.

2.1.1 Wave Model

As most environmental behavior, the wave models it is not a t-distribution. For example, for wind, the wind speed distribution is Rayleigh distribution. This distribution shows that wind is biased toward lower speeds. In other words, the probability of lower wind occurring is greater than the higher wind speed. The distribution of wind directly impacts the ocean pattern because the wind is one of the main engines of wave moment (THURMAN et al, 1999). Graph 2 shows the distribution of frequency and amplitude and the relation of wind and water boundary layer (HASSELMANN et al, 1973).

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Graph 2 – Energy transfer (A) Distribution of energy (B) Wind transfer mechanism

Source: Adapted from Munk (1951), Thurman et al (1999), National Oceanic and Atmospheric Administration (2018)

As shown in Graph 2 there is a relation of the type of excitation and power transfer to an ocean wave. Also, most energy on waves is caused by the wind. On B, the element (𝐵1) represents the mechanism of transferring energy, where the

boundary layer of the loss energy and transfer energy to the boundary layer of the ocean, the same happens at the bottom, where there is no movement. On (𝐵2) there

is an additional factor of reducing area due to coastal approach. Thus there is turbulence and the returning current (Rip current). On (𝐵3) there are the earth

interactions that are usually drier than the air on the ocean.

Tradition energy harvest are design with sinusoidal excitation in mind, but in the real excitation has some significant white noise or random to the excitation (TVEDT; NGUYEN; HALVORSEN, 2010; NGUYEN et al, 2010). Thus, narrowing the results. Moreover, for a one-dimensional wave, two cases will be presented on Eq. 1.

𝑥𝑤 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝐴𝑚𝑝(𝑠𝑖𝑛(2𝜋𝑓1𝑡) + 𝑠𝑖𝑛(2𝜋𝑓1+ 𝑑𝑓𝑡)) Case 1 𝐴𝑚𝑝𝑠𝑖𝑛(2𝜋𝑓1𝑡) Case 2 (1)

where 𝐴𝑚𝑝 is the amplitude of the wave, 𝑓1 is the frequency that the wind excited the

water from the range from Graph 2, 𝑑𝑓 is the difference of frequencies, that is used to

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Graph 3 – Wave forms (A) 1D-Sine wave (B) 2D-Sine wave 8 10 12 14 16 t [s] -2 0 2 x w [m] (A) 8 10 12 14 16 t [s] -2 0 2 x w [m] (B) 8 10 12 14 16 t [s] -50 0 50 s w [m/s] (C) 8 10 12 14 16 t [s] -50 0 50 s w [m/s] (D)

Source: Adapted from Munk (1951), Thurman et al (1999), Talipova, Kharif and Giovanangeli (2015)

where (A) is the case of sine excitation and it represents a simplification of a direct excitation by the wind (Case 2), (B) in the case of beating, where two wind direction with very similar or different frequencies shakes the buoy (Case 1). The speed of the water excitation is 𝑠𝑤 and it is defined in Eq.

𝑠𝑤 = ⎧ ⎪ ⎨ ⎪ ⎩ 𝐴𝑚𝑝(2𝑓1𝜋𝑐𝑜𝑠(2𝜋𝑓1𝑡) + (𝑑𝑓 + 𝑓 1)2𝜋𝑐𝑜𝑠(2𝜋𝑡(𝑑𝑓 + 𝑓 1))) Case 1 𝐴𝑚𝑝2𝜋𝑓1𝑐𝑜𝑠(2𝜋𝑓1𝑡) Case 2 (2)

Graph shows 4 how the wave is distorced as the 𝑑𝑓 increases.

Graph 4 – Beating Excitation (A) f1= 1 Hz df=0 Hz (B) (A) f1= 1 Hz df=.1 Hz (C) (A) f1= 1 Hz df=0.5 Hz (D) f1= 1 Hz df=1.5 Hz 10 12 14 16 -2 0 2 (A) 4 6 8 10 12 14 16 -2 0 2 (B) 10 12 14 16 -2 0 2 (C) 10 12 14 16 -2 0 2 (D) Source:

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2.2 PIEZOELECTRIC MATERIALS

Piezoelectric (PZT) material was discovered by Curie brother in 1880 (CURIE; CURIE, 1880). This material has a unique characteristic that it polarizes due to mechanical compression and creates a trust under external differential of potential, an example is shown in Fig. 6.

Figure 6 – Piezoelectric material (A) Under compression (Direct effect) (B) Under external voltage (Reverse effect)

Source: Adapted from Ueberschlag (2001)

As shown in Fig. 6 (A), the compression on a piezoelectric material generates a Volt. The reverse can be Fig. 6 (B), where an external differential of potential generates a force to lift. This relationship electrical system and the mechanical systems are given by Eq. 3 (ILIUK et al, 2013).

⎧ ⎪ ⎨ ⎪ ⎩ 𝐹 = Ψ1 𝐶 𝑞 𝑅 ˙𝑞 − Ψ1 𝐶 ∆𝑋 + 𝑞 𝐶 = 0 (3)

where Ψ1 is the coupling parameter, 𝐶 is the piezoelectric capacitance, 𝑞 is the charge

on the piezoelectric, and 𝑅 is the resistance the circuit.

This ability is lost after pass a Currier temperature that changes rearrange the molecular structure. PZT only took off in 1980 in the revolution of Micro-Electro-Mechanical Systems (MEMS) that considerable shrink mechanical electrical systems, PZT is used as sensor and actuator. In electronics, it used to create oscillator clocks in modern electronic (CURIE; CURIE, 1880; K., ). Fig. 7 shows how PZT couple the mechanical and electrical systems.

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Figure 7 – Piezoelectric actuator

Source: Adapted from Y. et al (2018)

where (A) is the force that is applied to the system, in (B) it the representation in booth systems, (C) is the general current induced on the electrical system.

As express in Eq. 3, the first perception was that it was a linear relationship between the relative displacement and the Voltage across the PZT. However, further research indicates that is a nonlinear relation that is described on Eq. 15 (TRIPLETT; QUINN, 2009).

Ψ1 = Ψ2(1 + Ψ3|𝑋1|) (4)

Ψ2 is the linear term, Ψ3is the nonlinear term of coupling. This approximation was done

by Least Squares methods. The same methodology will be used in the Dissertation. Erturk and Inman (2011) has an extensive review of continuous PZT coupled with beams. For a single tuned mass, Nabavi, Farshidianfar and Afsharfard (2018) used PZT to harvest energy from the waves. In addition, PZT has already been used by N. et al (2019) to harvest energy from waves, where two tuned masses were used to extract energy from the first and second natural frequencies. Using a portal frame, to harvest energy it has been a common strategy because of the advantages of displacement and natural frequency.

Despite been famous for higher frequencies and low amplitude, PZT elements could be used at low frequencies (0.5 𝐻𝑧) to extract the hysteresis curves (XU; ONO; ESASHI, 2006). A new trend is the lower frequencies and higher displacements, up to 3x times the normal PZT (SHEN et al, 2007). These large actuators take advantage of large displacement due to levers, thus multiplying the displacement (XU; ONO; ESASHI, 2006). For natural excitation, e.g walking, low frequencies can not create voltage bigger than the forward voltage drop of the diodes. Thus, rectifying could be unnecessary and to increase the PZT voltage, to an usable level bridgeless converter

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is required. In addition, simulations and experimental data support the use of lower resistances to be near the Maximun Power Point Tracking (HATTI et al, 2011).

2.3 PORTAL FRAME STRUCTURE

Portal frame structure is primarily known as structures that support the weights of the structure, and external loads required by the project. It is made from a variety of product, since wood until steel and polymeric materials. An example is shown in Fig. 8.

Figure 8 – Example of Portal Frame Structure (A) Truss representation (B) Dynamic Model

Source: Self-Authorship

where: 𝑤1 is the weight of an external load, 𝐹𝑤 is the loading due to wind, 𝑚1,𝑚2,𝑚3

are the equivalent masses. Usually, when analyzing in representation Fig. 8 (A) is for structure sanity, so the truss can support the load that is designed for. On the other hand, Fig. 8 (B) is for dynamical analyze, to explore the dynamic of each mass.

2.4 MECHANICAL MODEL

In this thesis, the mechanical model will couple the mechanical structure with the piezoelectric, to harvest current from the structure. This methodology is wide used in portal frame. The Fig 9 shows the strcture:

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Figure 9 – Portal Frame Structure (A) Truss representation (B) Dynamic Model

where: 𝑚1 is the mass of the structure, 𝑚2 is the tuned mass,𝑘1 is the stiffness of the

structure, 𝑏1 is damping of the structure, 𝑘2 is the stiffness between the structure and

the tuned mass, 𝑏2 is damping of between the structure and the tuned mass. Thus,

applying the second Newton Law

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ¨ 𝑋1 = _𝑚1 1(− Ψ1 𝐶 𝑞 − 𝑘1∆1− 𝑏1∆˙1+ 𝑘2∆2+ 𝑏2∆˙2) ¨ 𝑋2 = _𝑚1 2(−𝑘2∆2− 𝑏2 ˙ ∆2) 𝑅 ˙𝑞 − Ψ1 𝐶∆𝑋1 + 𝑞 𝐶 = 0 (5) where, ∆1 = 𝑋1 − 𝑋𝑤, ∆2 = 𝑋2 − 𝑋1, ˙∆1 = ˙𝑋1 − ˙𝑋𝑤, ˙∆2 = ˙𝑋2 − ˙𝑋1, 𝐾1 = 3𝐸𝐼_ℎ3𝐶,

𝐾2 = 48𝐸𝐼_𝑙3 𝑏, 𝐼𝑐 = 𝐼𝑏. Nevertheless, a nonlinear stiffens was introduced in the 𝑘1

2.4.1 Non linear Stiffens of Springs

The linear spring is a common analogy to represents a region of actuation. It has been clearly described and cataloged the different applications and relationships with materials (SHIGLEY, 2011).

However, the stiffness increase due to nonlinear induced or not by fabrication, design or material. For example, springs usually have a constant pitch, but if the pitch is changed along with the spring the nonlinear effect changes the stiffens of the spring (TUSSET; BALTHAZAR; FELIX, 2013; ILIUK et al, 2013; YOUNIS, 2011). The Equation

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6 shows the linearization process.

∆𝑥𝑘1∆𝑥 = 𝑘𝑙1∆𝑥+ 𝑘𝑛𝑙1∆3𝑥 (6)

where: ∆𝑥 is the displacement, 𝑘1 is the linear model , 𝑘𝑙1 is the linear portion, 𝑘𝑛𝑙1 is

the nonlinear portion of the stiffness.

As shown on Eq. 6, the amount of force and energy storage in the system can quickly increase due to the exponential factor on the nonlinear stiffness. This relation is shown in Fig. 10.

Figure 10 – Spring comparison (A) Force (B) Work

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 x [ ] -60 -40 -20 0 20 40 60 Force [ ] (A) k_nl=0 k nl>0 k_nl<0 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 x [ ] 0 5 10 15 20 Energy [ ] (B) Source: Self-Authorship

As shown in Fig. 10 (A), in black the linear model, in blue the negative parameter (𝑘𝑛𝑙1) that softens the spring, in red the positive parameter (𝑘𝑛𝑙1) that

hardens the spring, and in Fig. 10 (B) the energy storage is vastly different as the displacement increase. This difference in energy lead systems to different comportment and orbits.

The Sommerfeld effect is associate to an unbalance mass that excites a flexible structures, such as a portal frame, until reaches a critical rotatio speed that matchs the natural frequency. At this point the rotation speeds remains contans and the displacement increase, until the "jump" occurs, where the rotation speed abruptly increase and the displacement reduces (EL-BADAWY, 2007). Thus, proving that for steady state phenomena sometimes it is inadequate to linearize the system (BALTHAZAR et al, 2003). Additionally, this can cause a energy sink on machinery that works close to resonance frequency, a small increase on power supply casue little effect on the rotation speed of the motor and, big displacements can cause

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damage to the structures. In addition, after the jump a increase the rotation speed and a sudden a big acceleration on the rotor shaft is specter after the jump.

2.5 PARTICLE SWARM OPTIMIZATION

Particle swarm optimization (PSO) is an optimization process, bases on mimicking the naturals phenomenons in nature, such as bee swarms and bird migrations. PSO is a stochastic and base-population method, that leverages the local and global perception of the best position. As usual for the optimization process, PSO suffered many contributions until the current format. It is attributed to Kennedy and Eberhart (1995) and Shi and Eberhart (1998). An example is shown in Graph 5.

Graph 5 – PSO Behavior (A) First Interaction (B) Second Interaction (C) Few Interactions

As shown in Graph 5 (A), the first interaction there is no speed, just a random distribution, this population will interact to generate the subsequent generations. Second, on Graph 5 (B) is possible to see a random movement of the particles, but they are converging to the global maximum from both sides. Third, on Graph 5 (C) there are a few particles closer to the maximum. But there are more particles closer to the local maximum. In addition, some random moment is presented. This behavior is desired, because involves the local and global confidence, and improves the search for the best position. The rule that explained the speed of each particle is shown in

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Eq. 7 (PARSOPOULOS; VRAHATIS, 2010). 𝑉_𝑛𝑖+1= 𝐶0𝑉𝑛𝑖+ 𝐶1𝑅𝑎1(𝑋𝑛𝑖 − 𝑋 𝑖 𝑔𝑏) + 𝐶2𝑅𝑎2(𝑋𝑛𝑖 − 𝑋 𝑖 𝑙𝑏) (7)

where: 𝐶0 is the inertia of the system, 𝑉𝑛 is the speed of the particle at the current

state, 𝐶1 is the self-knowledge, 𝑅𝑎1 and 𝑅𝑎2 are random values, 𝑋𝑙𝑏 is the direction for

the local best, 𝑋𝑝 is the current position, 𝐶2 is the group knowledge, 𝑋𝑝𝑏is the direction

for the group best, and 𝑖 is the interaction of the system. All the parameters in Eq. 7 could be considered a constant or a variable of interaction, a range of parameters can be found in Erik, Pedersen and Laboratories (2010). Thus, accelerating or retarding the optimization process. For each parameter to be optimized, the speed is calculated and it subject to lower and upper bound set by the designer.

2.6 SDRE

The State-dependent equation Riccati equation (SDRE) is a feedback control that optimizes the Cost-benefit function at the time infinity to calculate the K gains (NAIDU, 2002; ANDERSON; M., 2007). Fig. 11 shows a simple feedback control.

Figure 11 – Example of feedback control with contoller and actuator

where: K is the gain, u is the controller signal, 𝐹 is the force, y are the output, x are the states, e is the vector error

It can work as a regulator or as a tracker for a vector. It has as characteristics, multiple-input multi-output (MIMO), simple, optimal performance for the local region,

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and has a trade-off between variables and inputs. First, the system must be represented in state spaces as shown in Eq. 8.

⎧ ⎪ ⎨ ⎪ ⎩ ˙ x(𝑡) = A(𝑥,𝑡)x(𝑡) + B(𝑥,𝑡)u(𝑡) y(𝑡) = C(𝑥,𝑡)x(𝑡) + D(𝑥,𝑡)u(𝑡) (8)

where,A(𝑥,𝑡) is the state matrix, x is the state vector,B(𝑥,𝑡) is the input matrix, u(𝑡) is the input vector, C(𝑥,𝑡) is the output matrix, D(𝑥,𝑡) is the feedthrough matrix.

For most cases the matrix A(𝑥,𝑡) can be simplified as A, thus leading to Linear quadratic regulator (LQR) formulation. Nevertheless, the x must have all variables censored. If, x is recreated by some estimator such as Kalman filter (^x), as is common to improve cost-efficiency to reduce sensor deployment. There is no guarantee of robustness (DOYLE, 1978). Nevertheless, this technique is acceptable and is commonly known as LQG. However, when A(𝑥,𝑡) can not be simplified, the Eq. 8 can be rewrite as shown in Eq.9.

˙

x(𝑡) = A1(𝑥,𝑡)x(𝑡) + B(𝑥,𝑡)u(𝑡) + A2(𝑡) (9)

where: A1(𝑥,𝑡)have all the parameter that can be express in relation to the time and

states, and A2(𝑡) has the other variables that can represent. Consequently, the

optimization process can not be ideal, because characteristic and singularities are missing. Nevertheless, there are many ways to express A1(𝑥,𝑡), thus leading to

overcompensation and/or controllability issues and/or stability issues. The quadratic cost functions are defined on Eq. 10.

K = R−1(BTP + NT) (10)

and where P can be obtained on Eq. 11.

0 = ATP + PA − (PB + N)R−1(BTP + NT+ Q) (11)

by minimizing the J, on the Eq. 12.

J = ∫︁ ∞

0

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Thus, using the linear control law the signal control (u) can be determined by Eq. 13. ⎧ ⎪ ⎨ ⎪ ⎩ e = x − sp u = −Ke (13)

where, u is the signal control, and e is the error, spis the set-point vector, if the set-point

is set to zero, the controller is called regulator. This optimization can only be done if the system is controllable.

2.6.1 Controllability

Controllability is a property of the system at a state that can be move from two points. It expresses the level of authority that the actuators have upon the system, thus a good selection of A and B reflect the controllability. That means that it is possible to change the behavior of stable or unstable systems by applying a control signal. Thus, a system can be controlled at initial conditions but can pass to states that are not controllable. A system is considered controllable if the controllability matrix (Γ) is equal to the number of variables 𝑛.

Γ = [B AB A2B ... An−1B] (14)

where 𝑛 is the number of variables. The controllability of the system must be checked every step of integration because the SDRE controller is heavily dependable of A(𝑥,𝑡) and B(𝑥,𝑡). To overcome this issue, a few strategies could be adopted to overcome these issues, such as: using the last known K matrix, using the K at the desired orbit, or even just setting the signal control to zero.

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3 METHODOLOGY

In this chapter the methodology will be explored, the method used, the mathematical assumptions, the parameter selected.

First, a dynamical analysis was carried out with the current parameters present, for beating excitation to detected problematic regions that the response was big, or the chaotic motion need to be stabilized. Thus, 2D (𝑑𝑓 and 𝑓 variable ) and

1D(𝑑𝑓 = 4𝐻𝑧).

Second, a dynamical analysis with and without coupling the PZT. In other words, analyzing the bifurcation, Scalogram and chaotic states to create a baseline. This dynamical analysis was carried out to have a base line of the behaviour, thus identifying big amplitudes and chaotic behaviour that could be a problem if the system was implemented.

Third, a dimensionless process will carry out in order to analysis the linear sensitivity of each parameter. This process was carried out to possible eliminate variable that does not affect the current production.

Fourth, an optimization based on dimensionless dimensions were carried out with boundaries. Because of a dimension analyses to compare and contrast with initial parameters present in the literature, and physical consideration such as negative mass and negative coefficient of dissipation.

Fifth, and strategy using SDRE controller was used to vary the external load to further improve the current flow.

3.0.1 Parameters of Portal Frame

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Table 1 – Parameters for portal frame Parameters Value Unit

𝑔 9.81 [𝑚/𝑠2] 𝑀 42 [𝑘𝑔] 𝑚 0.5 [𝑘𝑔] 𝑐1 1.55 [𝑁 𝑠_𝑚] 𝑐2 3.14 [𝑁 𝑠_𝑚] 𝐸𝐼 128 [𝑚/𝑠2_] 𝐿 0.52 [𝑚] ℎ 40.36 [𝑚] 𝑅 101 _[Ω] 𝑘𝑙 8230.453 [𝑁 𝑚] 𝑘𝑛𝑙 8230.453 × 106 _[𝑁 𝑚] 𝑘2 43695.95 [𝑁 𝑠_𝑚] 𝑐 10−6 [𝐹 ] 𝐴 10−3 [m]

Source: Adapted from: Tusset,

Balthazar and Felix (2013), Rocha (2014), Triplett and Quinn (2009)

All the temporal analyses were made numerically, with a maximum step of 10−2. The variables will be noted as: [𝑋1 𝑋˙1 𝑋2 𝑋˙2 𝑞] = [𝑥1 ˙𝑥1 𝑥2 ˙𝑥2 𝑥3]

For the dynamical analyses, the following standers were used. 0-1 Test, 0 < 𝐾 < 0.2periodic, 0.2 < 𝐾 < 0.8 gray area, Limit cycle or weak chaos, 𝐾 > 0.8 chaos, for more information on A. FFT is normalized, thus 0 < |𝑃 1(𝑓 )| < 0.2 background noise, 0.2 < |𝑃 1(𝑓 )| < 0.8auxiliary frequency, |𝑃 1(𝑓 )| < 0.8 principal frequency, where 𝑃 1(𝑓 ) is the power spectrum of that frequency. Lyapunov exponent 0 > 𝜆 periodic, 0 < 𝜆 < 0.1 weak chaos, 0.1 < 𝜆 chaos. For, FFT, bifurcation and 0-1 the time for simulation was 200s with 100s of the transient time. For the Lyapunov, the time was 1000s with 500s, when test the converge time was below 100s.

3.1 COUPLING PARAMETER OF PZT

Following the methodology of Triplett and Quinn (2009), that proposed a least square method,

Ψ1 = Ψ2(1 + Ψ3|𝑥1|) == 𝑎 + 𝑏|𝑥1|(𝑎) (15)

Ψ1 = 𝑎 + 𝑏|𝑥1| + 𝑐|𝑥1|2 (𝑏)

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(b) and (c), and the results should yield similar performance. For a displacement of 10−1[m] .

Graph 6 – Comparison between Linear and Quadratic Least Squares (A) Range of experimental data (B) Extrapolation Comparison

0 1 2 3 4 5 x [m] 10-4 1.5 2 2.5 3 3.5 4 4.5 1 [N/V] 10-10 (A) -1 -0.5 0 0.5 1 x [m] 10-3 1 2 3 4 5 6 7 1 [N/V] 10-10 (B) Experimental Quadratic Linear

Source: Adapted from Triplett and Quinn (2009)

where: 𝑎 = −5, 90 × 10−04_{, 𝑏 = 7,69 × 10}−07_{, 𝑐 = 1,82 × 10}−10_{, for quadratic, and}

𝑎 = 4.76 × 10−07, 𝑏 = 2.056 × 10−10for linear. As shown on Fig. 6, the Quadratic prevail the Linear approximation. The performance parameter are shown on Table 2.

Table 2 – Parameters for least squares method

Performance parameter Quadratic Linear

Sum of squares due to error 2,10×10−20 _6,92×10−17

R-squared 1.000 0.9748

Degrees of freedom in the error 53 54

Degree-of-freedom adjusted coefficient of determination 1.000 0.9743 Root mean squared error 1,99×10−09 1,13×10−07

Both approximation were good, but the quadratic approximation will be used. Thus, the Ψ1 = 0at ≈15 mm.

3.2 PARAMETRIC ANALYSES

To better analyze the dynamical system, a parametric analysis will be carried out to investigate the performance of the power extracted at lower frequencies for each variable. Thus, the following parameters and ranges were stabilized as based on Tusset, Balthazar and Felix (2013).

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𝜔1 = 10𝑓 𝜔2 𝑀 = 𝑚2 𝑚1 𝐾𝑎 = 𝑘2 𝑘1 𝐾𝑏 = 𝑘2 𝑘1 (17) Ω1 = 𝑅

For the local analysis, the parameter 𝐸200 = ∑︀𝑡=200𝑠_𝑡=0 |𝑥3| [C] will be utilize to

indicate energy, and it will be normalized. Thus flat lines means a variable that does not affect the energy production. On the other hand, a variable that is more irregular it has more potential to improvement. In addition, a global comparison between all the variables will show the magnitude of each variable.

3.3 OPTIMIZATION PROCESS

The optimization process was carried out with 𝐸200with sine excitation for each

particle with the following optimization proposes.

minimize 𝐽 𝐽 (𝑓,𝐾𝑏,𝐾𝑛𝑙) = (100 × 𝐸200) −1 subject to 0.1 < 𝑓 < 30,0.1 < 𝑀 < 10, 0.1 < 𝐾𝑏 < 100,0.1 < 𝐾𝑛𝑙 < 107 (18)

Thus, 𝑥3 is maximizing, and the amount of current increase regardless of the

signal of the current. The constraints imposed base on parametric sensibility.

3.4 CONTROL ANALYSES

For the control analyses, the first step was to analyse the system without PZT. to check much force was required to have a maximum amplitude of 0.1 at the same frequency as excited.

Second, the amount of force that 1 PZT developed and the amount of PZT that will be required to restrain the 𝑚1 just varying the 𝑅 [load].

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Third, select the correct A matrix for a system. Thus, the following A was selected.

3.4.1 SDRE Parameters

For the SDRE control, the selection of A matrix was reduced to speed up the controlling process. Thus, the 𝑥2 is consider an external excitation.

A = ⎡ ⎢ ⎢ ⎢ ⎣ 0 1 0 𝑏1 𝑏2 𝑏3 𝑑1 0 𝑑3 ⎤ ⎥ ⎥ ⎥ ⎦ (19) where, 𝑏1 = − 𝑘𝑛1𝑥21 𝑚1 − 𝑘𝑙1 𝑚1 − 𝑘2𝑙 𝑚1,𝑏2 = − 𝑐1 𝑚1 − 𝑐2 𝑚1, 𝑏3 == − 𝑁 𝑎 𝐶𝑚1 − 𝑁 𝑐|𝑥1|2 𝐶𝑚1 − 𝑁 𝑏|𝑥1| 𝐶𝑚1 ,𝑑1 = 𝑏|𝑥1| 𝐶𝑅1 + 𝑎 𝐶𝑅1 + 𝑁 𝑐|𝑥1|2 𝐶𝑚1 ,𝑑3 = −(𝐶𝑅1) −1

The B matrix is consider as shown on Eq. 20.

B = ⎡ ⎢ ⎢ ⎢ ⎣ 0 0 𝑅1−1 ⎤ ⎥ ⎥ ⎥ ⎦ (20)

where, 𝑁 is the number of PZT elements,𝑅 = 𝑅1 + 𝑅2, and 𝑅1 = 10 and 𝑅2 can be

calculate by the following equations

𝑅2 = ⎧ ⎪ ⎨ ⎪ ⎩ 0, if|𝑥5| < 10−8 𝑢 𝑥5 (21)

Limited to the lower bound of 0.01, to simulate an instantaneous charge on a large array of battery. Thus, 𝑅1 works as a protective load limit the inrush current, and

the SDRE control can adjust the resistance as required.

The Matrix A, and B are not the full representation of the system. Because the numerical integration does not use the A, and B, it is possible to use the partial matrix to facilitate the rank. Q = 1010_I

3R = 100, where I, is the identity matrix. The sp was

chosen as [𝑎𝑚𝑝1𝑠𝑖𝑛(2𝜋𝑓 𝑡) 2𝜋𝑓 𝑎𝑚𝑝1𝑐𝑜𝑠(2𝜋𝑓 𝑡) 0], where 𝑎𝑚𝑝1= 10−3, Thus the controller

will try to reduce the amplitude, and this energy will be converted in current.

For the dynamical analyses, an 𝑁 number of PZT elements will be compared and contrast. Consequently, the ideal number of actuators, where there is no more trade-off of the amount of energy harvested.

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4 RESULTS AND DISCUSSION

In this chapter, the results and discussion will be presented. First, the current models and dynamical analyze.

4.1 CURRENT MODELS

Using the non-optimized model, and the excitation beating and sine.

4.1.1 Beating Excitation

In this section, the beating excitation will be analyzed before the optimization. Thus, for a general idea, the test 0-1 for 2 variables was done, as shown in Graph 7.

Graph 7 – 0 − 1 Test 2D (A) For 𝑥1(B) For 𝑥2

As Shown, on Graph, there is regions of chaos at high frequencies (10𝐻𝑧 to 15𝐻𝑧) and high 𝑑𝑓 (4𝐻𝑧 to 7𝐻𝑧) on both 𝑥1 and 𝑥3. To assure these chaotic regions the

Lyapunov exponent was calculated with a time of 1000s , with a convergence of 100s on Graph 8.

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Graph 8 – Lyapunov for Beating (A) 2D varying df and f (B) 𝑑𝑓 = 4/ 𝐻𝑧varying f (A) (B) 2 4 6 8 10 12 14 0 1 2 3 4 5 1 2 Source: Self-Authorship

As shown on Graph 8 (A), there are regions of chaos similar on Graph 8. Thus, evidencing the need for a controller to bring to a more stable orbit. In addition, on Graph 8 (A) a 𝑑𝑓 = 4 Λ𝐻𝑧 was stabilized to further investigate the dynamics in 1D and analyze

the maximum displacements. First, the bifurcation to investigate behavior at Graph 9.

Graph 9 – Bifurcation for beating excitation

As seen in Graph 9, there is clear region on lower frequencies of multiple periods and chaos due to beating excitation, and a chaotic behavior near to the end of the spectrum. The first step to determine the dynamics was the 0-1 test, as shown in Graph 10.

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Graph 10 – Test 0-1 for beating excitation for 𝑑𝑓 = 4 Λ𝐻𝑧 0.5 5 10 15 f [Hz] 0 0.5 0.8 1 K X 1 [ ] (A) 0.5 5 10 15 f [Hz] 0 0.5 0.8 1 K X 2 [ ] (B) Source: Self-Authorship

As expected, confronting with the Lyapunov similar results shows a periodic behavior at the beginning (0.5𝐻𝑧to 5𝐻𝑧), periodic and quasi-periodic to the middle (5𝐻𝑧to 10𝐻𝑧) and multiple periods toward the end (15𝐻𝑧to 10𝐻𝑧) of the selected spectrum. This could be explained by the FFT-2D as shown in Graph 11.

Graph 11 – Scalogram for beating excitaiton on 𝑑𝑓 = 4 Λ𝐻𝑧(A) For 𝑥1(B) For 𝑥2

(A) (B) 5 10 15 f [Hz] 10 20 30 40 50 |P1(f) [Hz] | 0 0.2 0.4 0.6 0.8 1 5 10 15 f [Hz] 10 20 30 40 50 |P1(f) [Hz] | 0 0.2 0.4 0.6 0.8 1 Source: Self-Authorship

As shown in Graph 11, between 1 − 0.8 𝐻𝑧 is defined as the main frequencies, between 0.8 𝐻𝑧 and 0.2 𝐻𝑧 are the secondary frequencies and bellow .2 are background noise. Thus, as expect, the two main frequencies of the beating are present and roughly space between 4𝐻𝑧. Comparing the FFT-2D with the Lyapunov it is possible to observe similar patterns. Every time the Lyapunov exponent increase a third frequency appears in the system. Thus, some other high frequency of the system

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was trying to prevail over the main excitation. Similar analyses were made by the sine excitation.

4.1.2 Sine Excitation

For the sine excitation, the influence of a single PZT was analyzed over the same frequencies is shown in Graph 12.

Graph 12 – Bifurcation for initial parameter for sine excitation without PZT (A) 𝑋1 (B)

𝑋3

As shown in Graph 12 the periodic behavior slowly transforms into Chaotic Behaviour and increases the magnitude of the displacement. Furthermore, after the jump, the movement appears to be periodic. Adding one PZT element, the bifurcation diagram changes to Graph 13.

Graph 13 – Bifurcation for initial parameter for sine excitation with PZT (A) 𝑋1(B) 𝑋3

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As Shown in Graph 13, the addition of a single PZT element apparently did not change drastically the trajectories. The most apparent change is the change for a double period to a single period. Nevertheless, it is possible to compare both 0-1 test, as is shown on Graph 14.

Graph 14 – Test 0-1 (A) without PZT 𝑋1(B) without PZT 𝑋2(C) with PZT 𝑋1(D) with PZT

𝑋2 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 1 [ ] (A) 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 2 [ ] (B) 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 1 [ ] (C) 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 2 [ ] (D) Source: Self-Authorship

As Shown in Graph 14, the addition of on PZT increase the overall results of 0-1 test. In addition, it is clearer than in the region between 20𝐻𝑧 to 25𝐻𝑧 the influence. To guarantee this behavior, the Lyapunov Exponent was calculated and it is shown on Graph 15.

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Graph 15 – Lyapunov Expoent (A) Without (B) with 1 pzt 5 10 15 20 25 30 -2 0 2 4 (A) 5 10 15 20 25 30 f [Hz] -2 0 2 4 (B) 1 2 Source: Self-Authorship

As Shown in Graph 15, despite of the difference between the 0-1 test anf the Lyapunov Exponent was very similar with a minor difference in spikes values. Similar to beating excitation the FFT was done. it is shown on Graph 16.

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Graph 16 – FFT -2d for sine before optimization (A) 𝑋1without pzt (B) 𝑋3without pzt (C) 𝑋1with pzt (D) 𝑋3with pzt (A) (B) 5 10 15 20 25 30 f [Hz] 10 20 30 40 50 |P1(f) [Hz] | 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 f [Hz] 10 20 30 40 50 |P1(f) [Hz] | 0 0.2 0.4 0.6 0.8 1 (C) (D) 5 10 15 20 25 30 f [Hz] 10 20 30 40 50 |P1(f) [Hz] | 0 0.2 0.4 0.6 0.8 1 5 10 15 20 25 30 f [Hz] 10 20 30 40 50 |P1(f) [Hz] | 0 0.2 0.4 0.6 0.8 1 Source: Self-Authorship

Similarly, spikes on Lyapunov exponent could be traced back to the FFT, especially between 20ℎ𝑧 and 25ℎ𝑧. Comparing the addition of PZT, small changes can be noted between Graph 16 with and without. The next analyze was the parametric sensibility.

4.2 PARAMETER SENSIBILITY

The parameter sensibility was analyzing the parameter 𝐸200 with 1 PZT

coupled. Thus, each analysis has his local maximum and stander deivation. The first analys was made with the parameter 𝑜𝑚𝑒𝑔𝑎1 shown on Graph 17.

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Graph 17 – Local Sensibility of 𝜔1, best at 𝜔1= 0.4592 0 0.5 1 1.5 2 2.5 3 1 [ ] 0.9975 0.998 0.9985 0.999 0.9995 1 Local Influence [ ] Source: Self-Authorship

This result shows that frequency does not greatly affect the current flow, with a standard deviation of 4.4348 × 10−06_{. Due to the low current flow and the exponential}

behavior of the PZT that decreases the voltage after a gain on amplitude as shown on Graph 6. The Graph 18 shows the sensibility to parameter 𝑀 .

Graph 18 – Local Sensibility of 𝑀 , best at 𝑀 = 3.413

1 2 3 4 5 6 7 8 9 10 M [ ] 0.99999975 0.9999998 0.99999985 0.9999999 0.99999995 1 Local Influence [ ] Source: Self-Authorship

This result shows that parameter 𝑀 does not greatly affect the production, with a standard deviation of 1.8269 × 10−10_{. The desire behavior for the parameter 𝑀 is to}

avoid resonant mass 𝑚2 to sink enough energy to bring the oscilation to the max of

production without sink much energy. The Graph 19 shows the sensibility to parameter 𝐾𝑛𝑙.

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Graph 19 – Local Sensibility of 𝐾𝑛𝑙, best at 𝐾𝑛𝑙= 755.3893

This result shows that parameter 𝐾𝑛𝑙 does not greatly affect the production,

with a standard deviation of 4.31 × 10−7 . The desire behavior for the parameter 𝐾𝑛𝑙

is to protect the PZT, and tunne the 𝑚1 with the excitation frequency. The Graph 20

shows the sensibility to parameter 𝐾𝑏.

Graph 20 – Local Sensibility of 𝐾𝑏, best at 𝐾𝑏= 1.5730

This result shows that parameter 𝐾𝑏 does not greatly affect the production,

with a standard deviation of 1.99 × 10−6. The desire behavior for the parameter 𝐾𝑏 is to

tunned the 𝑚2 to sink some energy to improve the current flow. The Graph 21 shows

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Graph 21 – Local Sensibility of Ω1,best at Ω1= 0.1

As Shown on Graph 21 the resistance picks at 0.1Ω, and presents similar results that (HATTI et al, 2011) which track the Maximum Power Point Tracking for PZT. Combine all the factors, to investigate the magnitude order among is shown on Graph 22.

Graph 22 – Global Sensitivity

As shown on Graph 22, there is a great difference on the scales between Ω1

and 𝜔1,𝑚2,𝐾𝑏,𝐾𝑛𝑙. The best performance was 𝜔1 with 𝐸200 = 0.00595𝐶. Thus, the

optimization process will use 𝜔1,𝑚2,𝐾𝑏,𝐾𝑛𝑙 to tune the 𝑚2 to gain energy, but a small

gain is expected due to the magnitude of the exponential values. The following variables 𝜔1, 𝐾𝑏 𝐾𝑛𝑙, with a lower bound of [.1,.1,.1,.1], and the upper bound

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4.3 OPTIMIZATION

The results of the optimization process for each interaction are shown in Graph 23.

Graph 23 – Temporal evolution PSO

As Shown in Graph 23 (A) the only that did not converge was the optimization with 7 particles. All the optimization saturate at 1.68. Thus, 𝐸200 = 5.95𝑚𝐶, an small

charge, but this type of device is for macro powering devices. Comparing the optimization for different Swarm Size is shown on Graph 24.

Graph 24 – Comparison between Swarm Size

As Shown in Graph 24, green is at least on time bigger than (Ω1), and blue

is the best performance, red is worst performance than 𝜔1. Thus, comparing different

sizes with the best single influence (Ω1) all the optimization improves the power output.

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Table 3 – Swam Optimization Parameters Swarm size 7 14 21 28 35 42 49 56 Interations 15 11 12 11 12 14 11 11 𝑓 15,30 18,14 13,58 30,00 18,21 9,38 6,56 9,38 m2 15,14 12,07 7,88 18,20 8,78 14,09 11,77 4,92 𝐾𝑏 13909 12488 23993 55986 3527 32136 20200 47075 𝐾𝑛𝑙(MN/m3₎ _39,86 _54,45 _62,23 _25,11 _19,16 _12,32 _3,36 _82,30 J 1,681 1,680 1,681 1,681 1,681 1,680 1,681 1,680 𝐸200(mC) 5,9500 5,9507 5,9505 5,9497 5,9506 5,9507 5,9505 5,9507 Source: Self-Authorship

As shown in Table 3, most of the parameters remain similar. Nevertheless, the best optimization parameter were with Swarm Size of 14. Thus, parameters will be used in the control.

4.4 DYNAMICAL ANALYSIS OPTIMIZED MODEL

After the optimization, the dynamical analyses and current flow was compared. Thus, it is possible to investigate if the chaotic regions diminish or expand over the same frequency spectrum. First, the Beating excitation -2D.

4.4.1 Beating Excitation - 2D

Before the optimization, the Beating - 2D had two major areas of chaos. The 0-1 test is shown on Graph 25.

Graph 25 – 0-1 Test (A) For 𝑥1(B) For 𝑥2

As shown in Graph 25, the chaotic region reduce, but this is misleading due to the jump. Calculating the Lyapunov exponent shown on Graph 26 shows the dynamical

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states.

Graph 26 – Lyap for Beating (A) 2D varying df and f whole spectrum (B) 2D varying df and f zoom

(C) 𝑑𝑓 = 4/; 𝐻𝑧varying f

The chaotic region vanishes, only small weak chaos regions remain. This is due to the mathematical model that relies on displacement. Thus, the optimization takes advantage of the jump phenomena introduced by the nonlinear spring. For the beating 1D variation the same analysis as before was done.

4.4.2 Beating Excitation - 1D

For the Beating Excitation - 1D, the first analyses is to investigate the general dynamics by the bifurcation, shown on Graph 27.

Graph 27 – Bifurcation for beating excitation (A) 𝑥1(B) 𝑥2

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shown in Graph 26, the behavior is periodic. It is expected to 0-1 test to have the same behavior. Thus, the test 0-1 is shown on Graph 28.

Graph 28 – 0-1 Test for beating excitation (A) 𝑥1(B) 𝑥2

0.5 5 10 15 f [Hz] 0 0.5 0.8 1 K X 1 [ ] (A) 0.5 5 10 15 f [Hz] 0 0.5 0.8 1 K X 2 [ ] (B) Source: Self-Authorship

As expected on Graph 28, most regions are classified in the periodic. Nevertheless, there is a false positive in the region between resonance. This could be explained by the FFT-2D presented at Graph 29.

Graph 29 – Scalogram for beating excitation where 𝑑𝑓 = 4 (A) For 𝑥1(B) For 𝑥2

(A) (B)

Graph 29 shows the switch on the mains frequency, and where the resonance effect occurs the secondary frequency vanish. Thus, this could be on the reasons in the overlapping of frequencies that the 0-1 Test did not identify correctly the behavior. The same anal yes was made for the Sine Excitation.

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For the Sine excitation, the dynamical analyses were made comparing the behavior with the addition of one PZT element. The first analyses were the bifurcation without PZT, shown on Graph 30.

Graph 30 – Bifurcation for sine excitation after optimization without PZT (A) 𝑋1(B) 𝑋2

0.5 5 10 15 20 25 30 f [Hz] -2 0 2 4 6 8 10 12 X 1 [ m ] 10-3 (A) 0.5 5 10 15 20 25 30 f [Hz] 0 0.005 0.01 0.015 0.02 0.025 X 2 [ m ] (B) Source: Self-Authorship

As shown on Graph 30 a single resonance frequency, with a little blur near to 0.5 𝐻𝑍for 𝑋2. Adding the PZT element to the Portal Frame, the bifurcation is presented

on Graph 31.

Graph 31 – Bifurcation for sine excitation after optimization with PZT (A) 𝑋1(B) 𝑋2

0.5 5 10 15 20 25 30 f [Hz] -2 0 2 4 6 8 10 12 X 1 [ m ] 10-3 (A) 0.5 5 10 15 20 25 30 f [Hz] -0.005 0 0.005 0.01 0.015 0.02 0.025 X 2 [ m ] (B) Source: Self-Authorship

As Shown on Graph 31, differently from Graph 30, there is no blur on the bifurcation on the 𝑋2. Thus it is possible to compare the simulations with the Test 0-1

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Graph 32 – 0-1 Test for sine excitation after optimization (A) 𝑋1 without PZT (B) 𝑋3

without PZT (C) 𝑋1with PZT (D) 𝑋3with PZT

0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 1 [ ] A1 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 2 [ ] A2 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 0.8 1 K X 1 [ ] A3 0.5 5 10 15 20 25 30 f [Hz] 0 0.5 K X 2 [ ] A4 Source: Self-Authorship

As expected, on Graph 32 both cases have a periodic behaviour. The only exception is the blur near to 0.5 𝐻𝑧 that is chaotic. Thus, the Lyapunov exponent was calculated and it shows on Graph 33.

Graph 33 – After Optimization Lyapunov Expoent (A) without PZT (B) with PZT

5 10 15 20 25 30 f [Hz] -1 -0.5 0 0.5 [ ] (A) 5 10 15 20 25 30 f [Hz] -1 -0.5 0 0.5 [ ] (B) Source: Self-Authorship

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Despite the Test 0-1 indicates chaotic behavior, the two higher Lyapunov exponent remains negative. Thus, all the chaotic behavior was removed due to optimization. Another comparison is the FFT-2D is with and without the PZT element shown on Graph 34.

Graph 34 – Scalogram for (A) 𝑋1without PZT (B) 𝑋3wihtout PZT (C) 𝑋1wiht PZT (D) 𝑋3wiht PZT

(A) (B)

(C) (D)

The Graph 34 shows a single dominant resonant frequency.

4.5 CONTROL METHOD

The control and the optimization are shown by the time series show on Graph 35.

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Graph 35 – After Optimization Lyapunov Expoent (A) without PZT (B) with PZT

As expected the optimization reduced the displacement of the 𝑚1to yield better

current flow due to the negative part of the nonlianarity of Ψ1. Consequently, the 𝑥3

increases. The SDRE tried to reduce the displacement of the 𝑥1, but the amount of

energy available depends on 𝑥1. Thus, increasing the 𝑥1 increase the current flow thus

the displacement of 𝑚2decrease. During actuation the amount of current flow increase

in spikes, because of the controller.

4.6 CONTROL EFFICIENCY

To evaluate the control efficiency, the signal Magnitude and behavior must be analyzed as well with the requirement of reducing the displacement and energy harvesting. The RMS, Max and Average values are shown in Graph 36.

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Graph 36 – signal control values for (A) Beating (B) Sine

As expected, due to saturation and Q, R values the Resistance value was big. Thus, showing that was a difficult task to reduce the displacement. For each excitation, a displacement and energy harvest was analyzed.

Graph 37 – 𝑅2values for (A) Beating (B) Sine

4.6.1 Beating Excitation

For the beating excitation a 𝑑𝑓 = 4 𝐻𝑧 and 𝑓 = 18.13 𝐻𝑧 was chosen. Thus,

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Graph 38 – Control efficiency for N on beating excitation (A) 𝑥1(B) 𝑥2

Graph 38 shows that there is a big difference between the RMS with SDRE control and without. Also, all the variables remain constant after 20 Units of PZT. Analyzing the amount of current, on Graph 39.

Graph 39 – Control efficiency for N on beating excitation (A) Rms(𝑥3) (A) 𝐸200

As expected, as shown in Graph 39 the number of PZT elements increases the amount of current for each element decrease. However, the collect amount of current saturates around 20 units. The same methodology was used for sine excitation.

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First, the RMS of SDRE and without control was compared for the same frequency as the beating excitation is shown on Graph 40.

Graph 40 – Control efficiency for sine excitation (A) 𝑥1(A) 𝑥2

As shown on Graph 40 and Graph 38 both excitation saturates around 20 units. Analyzing the amount of energy harvest is shown on Graph 41.

Graph 41 – Control efficiency for sine excitation (A) Sum of currents, (A) N Sum of currents

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As shown in Graph 41 the amount of energy harvest for each element reduces as the number increase. Nevertheless, the amount of energy increases and saturates around 44_{. Comparing both excitations it is possible to investigate a more}

general behavior.

4.6.3 Beating and Sine

As explained before, the SDRE prevails to the passive method. Now, investigate the similarity and contrast of both excitation for 𝑥1 on Graph 42.

Graph 42 – SDRE efficiency for different types of excitaitons (A) 𝑥1(B) 𝑥2

As shown in Graph 42 both excitation start to increase the displacement at 44_,

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Graph 43 – SDRE efficiency for 𝑥3(A) RMS for 𝑥3(B) 𝐸200for both excitations

For both excitation the current RMS plateaus at 34_{. Higher than this, the current}

spikes start to increase significantly, due to increase of displacement and the simulation no more represents a physical experiment.