UNIVERSIDADE ESTADUAL PAULISTA ‘‘JÚLIO DE MESQUITA FILHO’’ FACULDADE DE ENGENHARIA CAMPUS DE ILHA SOLTEIRA KAYC WAYHS LOPES THE EFFECT OF DAMAGE ON WAVE PROPAGATION IN PLATES WITH CIRCULAR PIEZOELECTRIC TRANSDUCERS ILHA SOLTEIRA 2023 KAYC WAYHS LOPES THE EFFECT OF DAMAGE ON WAVE PROPAGATION IN PLATES WITH CIRCULAR PIEZOELECTRIC TRANSDUCERS Thesis submitted to the Faculdade de Engenharia de Ilha Solteira - UNESP in partial fulfillment of the requirements for obtaining the Doctorate degree in Mechanical Engineering. Knowledge area: Solid Mechanics Advisor: Prof. Dr. Douglas Domingues Bueno Co-Advisor: Prof. Dr. Camila Gianini Gonsalez-Bueno ILHA SOLTEIRA 2023 Lopes The Effect of Damage on Wave Propagation in Plates with Circular Piezoelectric TransducersIlha Solteira2023 124 Sim Tese (doutorado)Engenharia MecânicaEngenharia Mecânica, Mecânica dos SólidosNão . . FICHA CATALOGRÁFICA Desenvolvido pelo Serviço Técnico de Biblioteca e Documentação Lopes, Kayc Wayhs. The effect of damage on wave propagation in plates with circular piezoelectric transducers / Kayc Wayhs Lopes. -- Ilha Solteira: [s.n.], 2023 124 f. : il. Tese (doutorado) - Universidade Estadual Paulista. Faculdade de Engenharia de Ilha Solteira. Área de conhecimento: Mecânica dos Sólidos, 2023 Orientador: Douglas Domingues Bueno Co-orientador: Camila Gianini Gonsalez Bueno Inclui bibliografia 1. Wave propagation. 2. Structural health monitoring. 3. Symmetric damage. 4. Circular piezoelectric transducers. 5. Optimal frequencies. L864e UNIVERSIDADE ESTADUAL PAULISTA Câmpus de Ilha Solteira THE EFFECT OF DAMAGE ON WAVE PROPAGATION IN PLATES WITH CIRCULAR PIEZOELECTRIC TRANSDUCERS TÍTULO DA TESE: CERTIFICADO DE APROVAÇÃO AUTOR: KAYC WAYHS LOPES ORIENTADOR: DOUGLAS DOMINGUES BUENO COORIENTADORA: CAMILA GIANINI GONSALEZ BUENO Aprovado como parte das exigências para obtenção do Título de Doutor em Engenharia Mecânica, área: Mecânica dos Sólidos pela Comissão Examinadora: Prof. Dr. DOUGLAS DOMINGUES BUENO (Participaçao Virtual) Departamento de Matematica / Faculdade de Engenharia de Ilha Solteira - UNESP Dr. DANILO BELI (Participaçao Virtual) Mechanical Engineering Department / Eindhoven University of Technology, Netherlands Prof. Dr. FABRICIO CESAR LOBATO DE ALMEIDA (Participaçao Virtual) Departamento de Engenharia Mecânica / Faculdade de Engenharia - UNESP Prof. Dr. TOBIAS SOUZA MORAIS (Participaçao Virtual) Departamento de Engenharia Aeronáutica / Universidade Federal de Uberlândia - UFU Prof. Dr. CARLOS DE MARQUI JUNIOR (Participaçao Virtual) Departamento de Engenharia Aeronáutica / Universidade de São Paulo - USP Ilha Solteira, 20 de setembro de 2023 Faculdade de Engenharia - Câmpus de Ilha Solteira - Avenida Brasil, 56, 15385000 www.ppgem.feis.unesp.brCNPJ: 48.031.918/0015-20. To my wonderful wife, Larissa. Acknowledgments I would like to thank God for everything He has done in my life. I would like to thank my wife and best friend, Dr. Larissa Drews Wayhs Lopes, for her support, friendship, and for always being by my side. I always look up to her and I’m very grateful for having her in my life. Many thanks to my parents, Gláucio Lopes and Sueli Lopes, for always encouraging me to keep studying. They did everything they could for me to have a good education. To my siblings, Carla, Kaio and Cauana, and my nephews, Iggor and Leonardo. I also would like to thank all the people from my family for all their support. I would like to give thanks to Prof. Dr. Douglas Bueno for more than five years of guidance and friendship. Prof Douglas always helped me when needed, and I could not do this thesis without his advice. He is always concerned about the well-being of his students. I’m glad I have him as a friend and advisor. Many thanks to Prof. Daniel Inman from the University of Michigan, who allowed me to work in his lab (AIMS laboratory) for one year. I had a great time there and I will remember it for the rest of my life. Prof Inman helped me a lot and I’m thankful for all the things he did for me and for having him as a mentor. I also would like to thank Prof. Dr. Camila Gonsalez-Bueno for her help, especially when I started in the SHM and wave propagation areas. Many thanks to my in-laws, Ricardo Wayhs, Márcia Wayhs, Gabriela and Bárbara, for all their support, especially during the pandemic time. I would like to thank my friends Dr. Renan Geronel and Dr. Frederico Ribeiro, with whom I had the pleasure to share the office during the first year of the Ph.D. Many thanks to my friend Jessé Paixão for all the coffees and talks we had in the first year of the Ph.D. Those talks were very helpful. I also thank my friends Eduardo Preto, Matheus Donatoni, Rubens Manrique, Matheus Lima and Luciano Maia, with whom I have been friends since we started undergraduate together. Many thanks to my friends from the University of Michigan for all the time we had together, especially my lab mates, Dr. Christina Harvey, Dr. Kevin Haughn, Dr. Piper Sigrest, Greta Colford and Elliot Kimmel. Thanks to all the employees from UNESP, PPGEM and the Department of Mechan- ical Engineering from FEIS/UNESP. I would like to thank the support of Carlos José Santana for his help in carrying out the experimental tests and José Carlos Melo for his help in processes related to FAPESP. I would like to give thanks to the São Paulo Research Foundation (FAPESP), grant numbers 2019/21149-9 and 2021/11493-4, and the Brazilian National Council for Scientific and Technological Development (CNPq), grant number 133397/2019-0, for providing financial support for my Ph.D. ‘‘Trust in the Lord with all your heart and lean not on your own understanding; in all your ways submit to Him, and He will make your paths straight.’’ Proverbs 3:5-6 Abstract Damage detection using Structural Health Monitoring (SHM) techniques is a challenge with increasing importance for the scientific community. SHM processes usually involve selecting actuators to excite the structure and sensors to measure outputs. The sensor outputs are post-processed to detect the damage. Usually, aspects such as the size and location of the actuators and sensors, and the choice of the excitation frequency are neglected in SHM campaigns, and they are very relevant to many damage detection algorithms. This thesis presents an approach to define the sensor’s size in terms of its position in the structure considering the scattering of longitudinal and flexural waves in damaged plate-like structures. Modeling is developed to compute each wave packet of reflected and transmitted waves separately, which allows one to describe the wave scattering in thin plates with symmetric damage. Numerical simulations are carried out and the results show that the sensor size can be adjusted to improve the damage detection process. Results from experimental tests are presented to demonstrate the approach considering circular actuators. A damage index R̄ is introduced and used to detect the damage. The modeling of circular piezoelectric transducers bonded to thin plates is also presented, and it demonstrates that there are optimal frequencies to create and measure these waves. In addition, new equations to compute the sensors’ output voltages in terms of the actuator input voltage applied are presented and demonstrated from experimental tests. The findings contribute to SHM systems based on longitudinal and flexural wave propagation to detect damage in plate-like structures. They contribute to the current state of the art in wave propagation SHM by investigating the effects of different excitation frequencies and the influence of the damage parameters and sensor sizing on the resulting waves. Keywords: wave propagation; structural health monitoring; symmetric damage; circular piezoelectric transducers; optimal frequencies. Resumo Detecção de danos via técnicas de Monitoramento da Integridade Estrutural (SHM, do inglês Structural Health Monitoring) é um desafio com crescente importância para a comunidade científica. Processos de SHM geralmente envolvem a seleção de atuadores para excitação da estrutura e sensores para medir as respostas. As respostas obtidas com os sensores são pós-processadas para detectar o dano. Normalmente, aspectos como o tamanho e localização dos atuadores e sensores, e a escolha da frequência de excitação são negligenciados nas campanhas de SHM, embora sejam muito relevantes para qualquer algoritmo de detecção de danos. Esta tese apresenta uma metodologia para definir o tamanho do sensor em termos de sua posição na estrutura, considerando a dispersão de ondas longitudinais e de flexão em estruturas do tipo placa. A modelagem é desenvolvida para se obter separadamente cada pacote de onda refletida e transmitida, o que permite descrever a dispersão da onda em placas finas com danos simétricos. Simulações numéricas são apresentadas e os resultados mostram que o tamanho do sensor pode ser ajustado para melhorar a detecção do dano. Resultados de testes experimentais são apresentados para demonstrar a abordagem considerando um atuador circular. Um índice de dano R̄ é proposto e usado para detectar o dano. A modelagem de transdutores piezoelétricos circulares para gerar e capturar ondas longitudinais e de flexão também é apresentada. Ela permite demonstrar que existem frequências ótimas para criar e medir essas ondas. Além disso, novas equações são apresentadas e demonstradas para calcular as tensões elétricas de saída medidas pelos sensores em termos da tensão de entrada aplicada ao atuador. Os resultados contribuem para a evolução de sistemas de SHM baseados na propagação de ondas longitudinais e flexurais para detectar danos em placas e, também para o estado da arte na propagação de ondas para SHM, abordando os efeitos de diferentes frequências de excitação e a influência dos parâmetros de dano e dimensão de sensor. Palavras-chave: propagação de ondas; monitoramento da integridade estrutural; dano simétrico; transdutores piezoelétricos circulares; frequências ótimas. List of Figures 1 Longitudinal waves propagating from a point o on the plate, where A(.) is the wave amplitude and the subscripts r, l, u and d indicate respectively right, left, up and down. The angle θ indicates the direction of propagation in relation to the x axis. Point o indicates the coordinate system origin. . . 36 2 Plate with cross-section change in the y direction. C represents an actuator responsible by the generation of a longitudinal wave that propagates in the x and y positive directions. Sl and Sr are the sensors responsible by the acquisition of reflected and transmitted waves. . . . . . . . . . . . . . . . . . 37 3 Thin plate with symmetrical damage of the width lx in the x axis and length Ly(d) (Ly(d) < Ly) in the y axis. Lx and Ly are the dimensions of the plate in the x and y directions, respectively. Point p represents the location of an actuator generating a longitudinal wave propagating from the point p to q. Circular sensors are considered at the points e and a, with radii rL and rR. Points a and e correspond to the first packets of transmitted and reflected waves, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4 Thin plate with symmetrical damage of the width lx in the x axis and length Ly(d) (Ly(d) < Ly) in the y axis. Point p represents the location of an actuator generating a longitudinal wave propagating from the point p to q. Circular sensors are considered at the points e and a, with radii rL and rR. Points a and e correspond to the first packets of transmitted and reflected waves, respectively. The points a, b and c capture the transmitted wave packets, while points e, f and g capture the reflected wave packets. . . . . 42 5 Thin plate with a circular actuator placed at the point p to create radial waves with amplitude Ai. A circular sensor at the point a captures the wave front for an angle range around the angle θ1. . . . . . . . . . . . . . . . 44 6 Thin plate with symmetric damage of length lx in the x axis. Points a and b indicate the first and second packets of amplitudes At∣θ1 1 and At∣θ1 2 , and points e and f indicate the first two reflected wave packets with amplitudes Ar∣θ1 1 and Ar∣θ1 2 , respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7 Plate with symmetric damage of length lx in the x direction. Point a indicates the sensor capturing the first wave packet (Āt∣θ1 1 ) for θ1 and the second wave packet (Āt∣θ2 2 ) for θ2. . . . . . . . . . . . . . . . . . . . . . . . . . 46 8 Magnitude of the transmitted tone burst with 5 cycles at the frequency 100 kHz. Continuous line is obtained for healthy structure and the dashed line for the unhealthy one. Symbols ◻ and ▽ are the maximum values obtained for each case, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9 Reflected wave amplitude ratios (∣Ar/Ai∣(n)) in terms of the damage depth (1 − h): first packet (n = 1, dashed line), second (n = 2, continuous line), third (n = 3, dotted line) and fourth packets (n = 4, dash-dotted line). . . . 49 10 Transmitted wave amplitude ratios (∣At/Ai∣(n)) in terms of the damage depth (1 − h): first packet (n = 1, dashed line), second (n = 2, continuous line), third (n = 3, dotted line) and fourth packets (n = 4, dash-dotted line). 50 11 Magnitude of the reflected wave amplitude ratio obtained for 2lx tan(θ) ≤ rL < 4lx tan(θ) (continuous line) and 4lx tan(θ) ≤ rL < 6lx tan(θ) (dotted line): (a) θ = 30○ and (b) θ = 60○. The dashed vertical lines indicate lx/λl(d) for the optimum frequencies f l opt. . . . . . . . . . . . . . . . . . . . . . . . . . 51 12 Magnitude of the transmitted wave amplitude ratio obtained for 2lx tan θ ≤ rR < 4lx tan θ (continuous line) and 4lx tan θ ≤ rR < 6lx tan θ (dotted line): (a) θ = 30○ and (b) θ = 60○. The dashed vertical lines indicate lx/λl(d) for the optimum frequencies f l opt. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 13 h̄d in terms of the damage depth (1−h). Continuous line corresponds to no noise condition. Symbols ◂, ● and ▸ respectively correspond to the results obtained by considering 3%, 5% and 10% of noise added to the signal energy. 54 14 (a) Schematic view and (b) photographs of the experimental setup (mea- surements in cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 15 Schematic representation of an equivalent section (material ‘‘c’’) from a damaged section composed by two different materials (‘‘a’’ and ‘‘b’’). . . . 56 16 Magnitude of the (a) reflected and (b) transmitted wave amplitude ratios obtained for 4lx tan(θ) ≤ r() < 6lx tan(θ), considering θ = 60o: Mass reduction by h = 0.5 (dashed-line) and mass increase by h = 1.5 (continuous line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 17 Damage Index R̄ computed using: the reference experimental results (symbol ◊); proposed approach [Equation 40] (symbol ◻); and theoretical prediction considering infinite wave packets arriving at the sensor, in which a) θ = 45o (symbol ▽) or θ = 42.4o (symbol ○), and b) θ = 29.92o (symbol ▽) or θ = 28.6o (symbol ○). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 18 Longitudinal waves in the time domain measured by sensor 1 for the healthy plate, considering as input a 5 cycles tone burst at the frequency 160 kHz. 60 19 Longitudinal waves in the time domain measured by sensor 1 for the damaged plate, considering as input a 5 cycles tone burst at the frequency 160 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 20 Magnitude in the frequency domain of the measured signals by sensor 2 for the healthy (continuous lines) and damaged structure (dash-dotted lines), considering as input a 5 cycles tone burst at the frequency 160 kHz. There are 10 signal for each condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 61 21 Sensor 1: boxplot of the damage index R̄ and threshold (dashed-line) R̄l = 0.99. Damage detected for all frequencies (below the threshold line). . . . 62 22 Sensor 2: boxplot of the damage index R̄ and threshold (dashed-line) R̄l = 0.99. Damage detected for all frequencies (below the threshold line). . . . 62 23 Infinitesimal element with area dxdy and thickness hp on the x-y plane with moments and forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 24 Flexural waves with amplitudes A4 and A3 that propagate in the quadrants I and II, respectively. A1 and A2 correspond to the wave amplitudes propagating in the y positive direction, attenuating in the left and right hand sides of the y direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 25 (a) Thin plate with reduction of symmetrical cross-section in the y direction and generated wave amplitudes considering incidence of flexural wave with amplitude Aq 4i at the point q. (b) Thin plate with increase of symmetrical cross-section in the y direction and generated wave amplitudes considering incidence of flexural wave (Ar 4i) and effect of an attenuating wave (Ār 2) in the x positive direction at the point r. . . . . . . . . . . . . . . . . . . . . . . 69 26 (a) Thin plate with symmetrical cross-section change reduction along the y direction if a flexural wave with amplitude As 3i is incident at the point s. At the same point, the effects of attenuating wave Ās 1 in the x negative direction and propagating waves along the y directions are considered. (b) Thin plate with cross-section change increasing along the y direction and the amplitudes generated due to flexural wave incidence with amplitude At 4i at the point t, in addition to presence of attenuating wave in the x positive direction (Āt 2) and propagating waves along the cross-section change. . . . 71 27 Angle ϕ in terms of α for different thickness ratio h of damages. h = 1 (continuous line), h = 0.75 (dashed line), h = 0.5 (dash-dotted line) and h = 0.25 (dotted line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 28 Zoomed view of the damage width and the corresponding wave amplitudes for each point q, r, s and t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 29 Thin plate with a circular actuator placed at the point p, which is respon- sible to create circular waves with amplitude Ai in all the directions of the x-y plane. At the point a is placed a circular sensor of radius equal to rR that captures the wave front with main angle α1 in relation to the x axis. 78 30 (a) Thin plate with symmetrical damage of length lx in the x axis. Points a and b are where the first and second packets of amplitudes At∣α1 1 and At∣α1 2 arrive, respectively. At the points e and f the first two reflected wave packets with amplitudes Ar∣α1 1 and Ar∣α1 2 respectively arrive. (b) Wave packets with amplitudes At∣α1 1 . At∣α2 2 and At∣α3 3 arrive at the point a. . . . . 79 31 Magnitude of the (a) transmitted (∣At/Ai∣) and (b) reflected (∣Ar/Ai∣) wave amplitude ratios obtained for different damage depth in relation to the ratio lx/λf(d), where: h = 1 (continuous line), h = 0.5 (dotted line), h = 0.7 (dash-dotted line) and h = 0.8 (dashed line). α = 0. . . . . . . . . . . . . . . . 80 32 Magnitude of the reflected wave amplitude ratio obtained for 2lx tan(ϕ) ≤ rL < 4lx tan(ϕ) (continuous line) and 4lx tan(ϕ) ≤ rL < 6lx tan(ϕ) (dotted line): (a) α = 30○ and (b) α = 15○. The symbol × indicates lx/λf(d) for the optimum frequencies f f opt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 33 Magnitude of the transmitted wave amplitude ratio obtained for 2lx tan(ϕ) ≤ rR < 4lx tan(ϕ) (continuous line) and 4lx tan(ϕ) ≤ rR < 6lx tan(ϕ) (dotted line): (a) α = 30○ and (b) α = 15○. The symbol × indicates lx/λf(d) for the optimum frequencies f f opt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 34 Ratio lx λf(d) obtained for the optimal frequencies of the (a) first and (b) second peaks of the resulting reflected wave amplitude ratio and (c) for the optimal frequencies of the first valley of the resulting transmitted wave amplitude ratio considering 4lx tan(ϕ) ≤ rL < 6lx tan(ϕ) as a function of α for different values of h, where h = 0.7 (▽), h = 0.8 (◻) and h = 0.9 (○). . . 84 35 (a) Schematic view and (b) photographs of the experimental setup (dimen- sions in cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 36 R̄ index obtained for: experimental data (box plot), transmitted resulting wave when the first three wave packets are summed considering α = 28o (dashed line) and for radial wave propagation with Equation (40) for α1 = 28o, α2 = 24.8o and α3 = 21.5o (continuous line). . . . . . . . . . . . . . . . . 87 37 R̄ index calculated for: healthy (blue boxes) and damaged (red boxes) plates. Dashed line represents the threshold R̄l = 0.95. . . . . . . . . . . . . 88 38 Flexural waves in the time domain collected for structure without symmet- rical damage using the sensor S considering as input a tone burst of 5 cycles and center frequency of 150 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . 89 39 Flexural waves in the time domain collected for structure with symmetrical damage using the sensor S considering as input a tone burst of 5 cycles and center frequency of 150 kHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 40 Magnitude in the frequency domain of the signals collected for healthy (continuous lines, from Fig. 18) and damaged (dash-dotted lines, from Figure39) plate using the sensor S2, considering as input a tone burst of 5 cycles and center frequency of 150 kHz. . . . . . . . . . . . . . . . . . . . . . 90 41 Thin plate with circular piezoelectric transducer generating a radial wave with amplitude Ar in all directions. . . . . . . . . . . . . . . . . . . . . . . . . 93 42 Frontal view of circular piezoelectric transducers bonded to the plate and working as actuator and sensor. The Vin is applied to the actuator, generating a wave with amplitude Ar that propagates to the sensor, which obtains the output voltage Vout. . . . . . . . . . . . . . . . . . . . . . . . . . . 95 43 Forces and moments applied to the plate due to the coupling between a circular piezoelectric transducer and the host structure when an input voltage Vin is applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 44 (a) Thin plate with one circular piezoelectric transducer generating a radial wave and another one capturing the waves. (b) Distances and parameters relation both actuator and sensor. . . . . . . . . . . . . . . . . . . . . . . . . . 97 45 Phase velocities obtained for longitudinal (continuous line, green) and flexural (dashed line, blue) and also for the S0 (dotted line, pink) and A0 (dash-dotted line) Lamb Wave modes. . . . . . . . . . . . . . . . . . . . 99 46 Flexural waves in time obtain obtained before (dash-dotted line, green) and after (continuous line, red) applying the factor Ic. . . . . . . . . . . . . . . . 99 47 Magnitudes of the output and input voltage ratios (∣Vout/Vin∣) for (a) longitudinal and (b) flexural waves in terms of the sensor radius rs and the wavelength λ() ratio (where l indicates the longitudinal wave and f the flexural one) for different values of actuator radius ra. ra/rs = 0.5 (continuous line, red), ra/rs = 1 (dash-dotted line, blue) and ra/rs = 2 (dashed line, green). Symbols 3, ◻ and ◯ indicate the maximum local magnitudes obtained at the peaks for ra/rs = 0.5, ra/rs = 1 and ra/rs = 2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 48 Maximum magnitudes of the output and input voltage ratios (∣Vout/Vin∣max) obtained at the first four peaks for (a) longitudinal and (b) flexural waves in terms of the ratios of the sensor rs and actuator ra radii. Continuous line (green) indicates the results for the first peak, dashed (red) line for the second peak, dash-dotted (blue) line for the third peak and dotted (black) line for the fourth one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 49 Magnitudes of the output and input voltage ratios (∣Vout/Vin∣) for (a) longitudinal and (b) flexural waves in terms of the sensor radius rs and the wavelength λ() ratio (where l indicates the lonfitudinal wave and f the flexural one) for different values of actuator thickness hs. hs/ha = 0.5 (continuous line, red), hs/ha = 1 (dash-dotted line, blue) and hs/ha = 2 (dashed line, green). Symbols 3, ◻ and ◯ indicate the maximum mag- nitudes obtained at the peaks for hs/ha = 0.5, hs/ha = 1 and hs/ha = 2, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 50 Maximum magnitudes of the output and input voltage ratios (∣Vout/Vin∣max) obtained at the 4 first peaks for (a) longitudinal and (b) flexural waves in terms of the sensor hs and actuator ha thicknesses ratio. Continuous line (green) indicates the results for the first peak, dashed (red) line for the second peak, dash-dotted (blue) line for the third peak and dotted (black) line for the fourth one. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 51 Magnitudes of the output and input voltage ratios (∣Vout/Vin∣) for (a) longitudinal and (b) flexural waves in terms of the sensor radius ra and the wavelength λ() ratio (where l indicates the longitudinal wave and f the flexural one) for different distances c. c/ra = 5 (continuous line, red), c/ra = 50 (dot-dashed line, green) and c/ra = 100 (dashed line, blue). . . . 106 52 (a) Schematic view and (b) photographs from the experimental setup (dimensions in cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 53 Magnitudes of the output and input voltage ratios (∣Vout/Vin∣) for longi- tudinal (dashed line) and flexural (continuous line) waves [obtained with Equation (103)] in terms of the sensor radius ra and the wavelength λ() ratio for each wave (where l indicates the longitudinal wave and f the flexural one). Symbols ◻ and 3 indicate the experimental results obtained for longitudinal and flexural waves when frequencies are equal to 130 kHz (ra/λl(p) = 0.24 and ra/λf(p) = 0.92), 160 kHz (ra/λl(p) = 0.29 and ra/λf(p) = 1.02), 180 kHz (ra/λl(p) = 0.33 and ra/λf(p) = 1.08) and 190 kHz (ra/λl(p) = 0.35 and ra/λf(p) = 1.11). . . . . . . . . . . . . . . . . . . . . . . . . 109 54 Wave amplitudes obtained experimentally (continous line, blue) and ana- lytically (dashed line, red) with Equation (103) for 130 kHz, 160 kHz, 180 kHz and 190 kHz. Rectangles (dotted line, green) and ellipses (dash-dotted line, yellow) highlight the longitudinal and flexural waves, respectively. R represents the ratios between experimental magnitudes of longuitudinal and flexural waves obtained from Figure 53. . . . . . . . . . . . . . . . . . . 110 List of Tables 1 Geometrical and physical properties of the plate and circular piezoelectric transducers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 List of Acronyms ASM - Average Scattering Matrix BEM - Boundary Element Method PZT - Piezoelectric transducer SEM - Spectral Element Method SHM - Structural Health Monitoring TFM - Total Focusing Method List of Symbols ā - Wave amplitude vector of longitudinal waves a - Wave amplitude vector of flexural waves A1 - Flexural wave amplitude propagating in the y positive direction and attenuating in the x negative direction A2 - Flexural wave amplitude propagating in the y positive direction and attenuating in the x positive direction A3 - Flexural wave amplitude propagating in the y positive direction and in the x negative direction A () 3 - Flexural wave amplitude A3 at the point () A4 - Flexural wave amplitude propagating in the x and y positive directions A () 4 - Flexural wave amplitude A4 at the point () A () r - Reflected wave amplitude at the point () Ar - Reflected wave amplitude A () r ∣(n) - Reflected wave amplitude of the n-th packet Ar∣ θ () (n) - Reflected wave amplitude of the n-th packet propagating with angle θ() A () t - Transmitted wave amplitude at the point () At - Transmitted wave amplitude A () t ∣(n) - Transmitted wave amplitude of the n-th packet At∣ θ () () - Transmitted wave amplitude of the n-th packet propagating with angle θ() At∣c - Amplitude of the resulting transmitted wave in case of circular wave propagation A () i - Incident wave amplitude at the point () Aru - Longitudinal wave amplitude propagating to the positive directions of x and y A () ru - Longitudinal wave amplitude Aru at the point () Arui - Incident longitudinal wave amplitude Aru A () rui - Longitudinal wave amplitude Arui at the point () Alu - Longitudinal wave amplitude propagating to the negative direction of x and positive direction of y A () lu - Longitudinal wave amplitude Alu at the point () Ald - Longitudinal wave amplitude propagating to the negative directions of x and y A () ld - Longitudinal wave amplitude Ald at the point () Arl - Longitudinal wave amplitude propagating to the positive direction of x and negative direction of y A () rl - Longitudinal wave amplitude Arl at the point () B() - Potential function amplitude c - Distance between centers of actuator and sensor cl - Phase velocity of the longitudinal wave cf - Phase velocity of the flexural wave Ct(.) - Coefficient defined by the ratio between amplitudes of the transmitted and incident longitudinal waves Cr(.) - Coefficient defined by the ratio between amplitudes of the reflected and incident longitudinal waves Dp - Flexural rigidity computed with properties of the undamaged region of the plate dx - Distance between two points in the x direction dy - Distance between two points in the y direction Eeq - Equivalent Young’s modulus Ea - Young’s modulus of the actuator Es - Young’s modulus of the sensor Ep - Young’s modulus of the undamaged region of the plate Ēp - Young’s modulus of the beam used to create the damaged section Ed - Young’s modulus of the damaged region of the plate Er - Energy of the reflected longitudinal wave Er∣(1) - Energy of the first reflected longitudinal wave packet Et - Energy of the transmitted longitudinal wave Et∣(1) - Energy of the first transmitted longitudinal wave packet f l opt - Optimal frequency for damage detection with longitudinal wave f f opt - Optimal frequency for damage detection with flexural wave f l o - Optimal frequency to create and measure longitudinal waves with circular piezoelectric transducers f f o - Optimal frequency to create and measure flexural waves with circular piezoelectric transducers Gp - Shear modulus of the undamaged region of the plate h - Ratio between hd and hp h̄ - Longitudinal wave state vector h - Flexural wave state vector H̄ - Longitudinal wave transformation matrix H - Flexural wave transformation matrix ( )p - ( ) computed with properties of the undamaged region of the plate ( )d - ( ) computed with properties of the damaged region of the plate ha - Actuator thickness hs - Sensor thickness hd - Damage thickness hp - Plate thickness kl - Longitudinal wavenumber i, j - Unitary vectors in the x and y directions, respectively i - Pure imaginary number k1 - Component of the longitudinal wavenumber in the x direction k2 - Component of the longitudinal wavenumber in the y direction kl(p) - Longitudinal wavenumber computed with properties of the undamaged region of the plate kl(d) - Longitudinal wavenumber computed with properties of the damaged region of the plate kf - Flexural wavenumber kf(p) - Flexural wavenumber computed with properties of the undamaged region of the plate kf(d) - Flexural wavenumber computed with properties of the damaged region of the plate Lx,Ly - Dimensions of the plate in the x and y directions, respectively lx - Damage length in the x direction Ly(d) - Damage length in the y direction Mx,My - Bending moments along the x and y directions, respectively Mxy,Myx - Twisting moments Nx,Ny - Line forces per unit of displacement in the x and y directions, respectively Qx,Qy - Shear forces rL - Circular sensor radius on the left-hand side of the damage rR - Circular sensor radius on the right-hand side of the damage ra - Circular piezoelectric transducer radius working as actuator rs - Circular piezoelectric transducer radius working as sensor R̄ - Damage index R̄l - Threshold value for damage detection t - Time T̄ - Longitudinal wave spatial transformation matrix T - Flexural wave spatial transformation matrix Vin - Input voltage applied to the actuator Vout - Output voltage measured by the sensor w - Transversal displacement u, v - Longitudinal displacements respectively in the x and y directions x, y, z - Coordinates of the reference system Greek Letters α, ϕ - Angles between direction of flexural wave propagation and the x axis ∇ - Gradient operator ∇2 - Laplacian φ - Scalar function representing the potential linear displacement ψψψ - Potential angular displacement ρeq - Equivalent density ρp - Plate material density of the undamaged region ρ̄p - Beam material density used to represent the damaged section ρd - Plate material density of the damaged region νp - Plate Poison’s coefficient of the undamaged region νd - Plate Poison’s coefficient of the damaged region νa - Actuator Poison’s coefficient νs - Sensor Poison’s coefficient λf(p) - Flexural wavelength computed with properties of the undamaged region of the plate λf(d) - Flexural wavelength computed with properties of the damaged region of the plate λl(p) - Longitudinal wavelength computed with properties of the undamaged region of the plate λl(d) - Longitudinal wavelength computed with properties of the damaged region of the plate θ - Angle between direction of longitudinal wave propagation and the x axis θ1 - Angle between direction of longitudinal wave propagation and the x axis for the first wave packet captured by the sensor in case of radial longitudinal wave θ2 - Angle between direction of longitudinal wave propagation and the x axis for the second wave packet captured by the sensor in case of radial longitudinal wave θx,θy - Slopes along the x and y directions, respectively ω - Angular frequency Contents 1 INTRODUCTION 25 1.1 MOTIVATION AND LITERATURE REVIEW . . . . . . . . . . . . . . . . . 25 1.2 OBJECTIVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.3 MAIN CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4 OUTLINE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2 SENSOR SIZE AND OPTIMUM FREQUENCIES TO DETECT DAMAGE WITH LONGITUDINAL WAVES 33 2.1 LONGITUDINAL WAVE PROPAGATION IN THIN PLATES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.2 CROSS-SECTION CHANGE ANALYSIS . . . . . . . . . . . . . . . . . . . . . 37 2.3 LONGITUDINAL WAVE SCATTERING IN THIN PLATES WITH SYM- METRICAL DAMAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.1 Wave Scattering due to Oblique Incident Wave Propagating in the x and y Positive Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.3.2 Resulting Wave due to Circular Longitudinal Wave Propagation . . . . . . . 44 2.3.3 Detecting the Thickness Reduction in a Damaged Section . . . . . . . . . . . 46 2.3.4 Index for Damage Detection in Plates using Transmitted Longitudinal Wave 47 2.4 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 EXPERIMENTAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.6 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3 FLEXURAL WAVES WITH OBLIQUE INCIDENCE 65 3.1 FLEXURAL WAVE PROPAGATION IN THIN PLATES . . . . . . . . . . . 65 3.2 TRANSMITTED AND REFLECTED WAVE COEFFICIENTS DUE TO CROSS-SECTION CHANGE . . . . . . . . . . . 68 3.3 FLEXURAL WAVE SCATTERING IN THIN PLATES WITH SYMMET- RICAL DAMAGE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.3.1 Wave Scattering due to Oblique Incident Wave Propagating in the x and y Positive Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.2 Resulting Wave due to Circular Flexural Wave Propagation . . . . . . . . . . 78 3.4 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.4.1 EXPERIMENTAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.5 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4 CIRCULAR PIEZOELECTRIC TRANSDUCERS FOR SHM 92 4.1 CIRCULAR PIEZOELECTRIC TRANSDUCER BONDED TO THE PLATE 93 4.1.1 Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 RESULTS AND DISCUSSION . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3 EXPERIMENTAL TESTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4 FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5 FINAL REMARKS 112 5.1 SUGGESTIONS FOR FUTURE WORK . . . . . . . . . . . . . . . . . . . . . 113 REFERENCES 114 25 1 INTRODUCTION This first chapter presents a brief motivation and the literature review, the objectives, the main contributions, and the outline of this thesis. 1.1 MOTIVATION AND LITERATURE REVIEW Engineers and researchers have developed numerous techniques for structural health monitoring (SHM) to detect incipient damage, mainly focused on contributing to reducing costs of maintenance and improving safety in a variety of structures and machines (BAP- TISTA et al., 2014; LOPES JR et al., 2000; FILHO; BAPTISTA; INMAN, 2011b; GONSALEZ et al., 2015; SILVA; GONSALEZ; LOPES JR., 2011). SHM systems are usually classified into the following levels: i) damage detection; ii) damage detection and location; iii) damage detection, location and quantification; iv) damage detection, location, quantification and prediction of useful life of structure (DOE- BLING; FARRAR; PRIME, 1998). Many approaches in the literature are focused on post-processing input and output signals obtained experimentally by using actuators and sensors placed on the structure (HENDRICK, 1988; POVICH; LIM, 1994; RHIM; LEE, 1995; STASZEWSKI, 2002; DASGUPTA et al., 2004; LEE; KIM, 2007; AI et al., 2018). Filho, Baptista and Inman (2011a) for example present a new methodology for damage detection based on the magnitude of the coherence function computed with the signals obtained by piezoelectric transducers coupled to the host structure. The authors carried out experimental tests on healthy and damaged aluminum plates, computing the coherence index for the measured signals and they concluded that the approach is efficient for damage detection. In SHM applications, computational tools can be also involved, as presented by references Wu, Ghaboussi and Garrett (1992), Spillman et al. (1993), Doebling et al. (1996), and others. SHM techniques based on post-processing input and output signals usually are demon- strated considering experimental data obtained from well-controlled tests conducted in 26 laboratories. In these last years, authors have considered more signals and included statistic-based procedures to infer a general efficiency of their methods based on their particular findings. However, SHM techniques have also been developed by investigating the wave propagation in the structure. Gonsalez-Bueno (2019) presents a strategy for monitoring beams with corrosion like-damage. The author considers longitudinal and flexural waves to demonstrate the wave interactions with symmetric and asymmetric damage. Optimum frequencies are defined to improve damage detectability. Piezoelectric transducers are employed to obtain experimental data to validate the accuracy of the proposed modeling. Modeling wave propagation in structures allows one to investigate different issues in engineering, as shown in references (SZEFI, 2003; NASCIMENTO, 2009; EBRAHIMIAN; TODOROVSKA, 2014; NANDA; KAPURIA, 2015; NUCERA et al., 2015; SANTOS, 2018), and others. Ahmida and Arruda (2002) obtain structural vibration modes from wave propagation using the Spectral Element Method (SEM). The authors introduce a parameter to measure the structural mode complexity. Mace (1984) investigates the vibration behavior of beams considering the effect of geometric discontinuity. The article shows the importance of near-field waves for computing the amplitude of motion accurately. Santos et al. (2021) present a strategy to design periodic rods for flexural wave propagation for vibration supressions in mechanical structures. The authors obtain mathematical relations to compute the transmitted wave in terms of the incident wave and involving physical and geometrical properties of the structure, contributing to the design of an efficient structure to exhibit desired stop band. However, relatively fewer efforts have investigated wave propagation focused on developing SHM techniques (LEE; STASZEWSKI, 2003; MAL et al., 2005; RYUE et al., 2011; GONSALEZ-BUENO, 2019). Luca et al. (2022) investigate the wave propagation phenomena in isotropic and anisotropic plates to understand the guided wave dispersion and highlight their differences. Their work contributes to an improvement in the design of SHM systems based on guided waves. In addition, the authors investigate the scattering due to a stiffener in aluminum and composite plates and show different effects between these two materials. Perfetto et al. (2023a) investigate guided waves on composite-type structures with curvature and compare the results with numerical simulations obtained for a flat plate. They noted more differences for the S0 lamb wave mode propagation at the low-frequency range of excitation. On the other hand, no differences are observed for the A0 lamb wave mode propagation. Mechanical structures commonly operate under effects of different environmental 27 conditions and load levels, which affect the wave propagation in the structures (PER- FETTO et al., 2022; PERFETTO et al., 2023b; LUCA et al., 2023). Then, Perfetto et al. (2022) investigated how guided waves (A0 and S0 lamb wave modes) propagate under the influence of temperature effects using the Finite Element Method for a composite-type plate. They also present results from experimental tests, which show that an increase in the temperature results in decreasing the group velocity of the S0 lamb wave mode due to the decrease in the mechanical properties of the composite. On the other hand, they show that the group velocity of the A0 lamb wave mode is less sensitive to changes in temperature. Their work provides an important contribution to the improvement of SHM systems based on guided waves considering the influence of temperature. The effects of geometric discontinuities on the coefficients of transmission and re- flection waves are investigated by Cho (2000), Schaal and Mal (2016) and Poddar and Giurgiutiu (2016) considering plate-like structures with cross-section changes. Thickness increasing and decreasing are employed to represent the discontinuity. On the other hand, different geometric shapes are considered in references Cho and Rose (2000), Lowe et al. (2002), Martin and Jata (2007), Benmeddour et al. (2008), Roberts (2008), Kim and Roh (2011), Ajith and Gopalakrishnan (2013), Glushkov et al. (2015), Pau and Achillopoulou (2017), Haider et al. (2018), Bhuiyan et al. (2018) and Kubrusly, Weid and Dixon (2019). Lowe and Diligent (2002) investigate the coefficient of reflection in a plate considering an asymmetrical rectangular crack considering the S0 Lamb wave. The authors show that resulting wave amplitudes exhibit periodic behavior, with maximum and minimum magnitude depending on the damage and excitation frequency. They highlight the importance of understanding this dynamic behavior for detecting the damage via non-destructive-based techniques considering the S0 lamb wave. Pain and Drinkwater (2013) highlight that composite-materials-based structures have fibers that can present waviness, which can affect their mechanical strength. The authors introduce two methodologies to detect the waviness: which involve a post-processing algorithm of images called TFM (Total Focusing Method) and the Average Scattering Matrix (ASM) method. Experimental tests are carried out considering different levels of waviness. According to the authors, the phase TFM imaging presents potential to obtain the image of the waviness, especially for larger levels. They indicate that ASM shows a good performance to investigate the scattering of the waves and conclude that additional experimental tests are required to investigate the limitations of both techniques. Considering the case of Lamb waves, Ng and Veidt (2011) studies the wave scattering due to incidence of an A0 Lamb Wave mode in the delamination of a composite plate 28 using a 3D model. They employ the Finite Element method for a low-frequency range to detect and characterize the damage using the wave scattered amplitudes. The approach predicts the wave scattering field, including its characteristics, such as amplitudes and directions in terms of the delamination size and wavelength. The wave interaction with stiffened plates has been investigated by Martin and Jata (2007), Roberts (2008), Ajith and Gopalakrishnan (2013), Haider et al. (2018) and Bhuiyan et al. (2018). Usually the studies consider incident wave perpendicular to the damage. However, if an oblique incidence is considered, He et al. (2020) and Santhanam and Demirli (2013) show that there are important variations in the transmitted and reflected wave amplitude and phase. Golato, Demirli and Santhanam (2014) consider a symmetric damage and the S0 and A0 Lamb wave modes with incidence of zero and ninety degrees. The authors verified the influence of this angle of incidence on the transmitted and reflected wave energy. They noted for the S0 Lamb wave mode that there are reflected shear waves for high incidence angles, and they are reduced for large damage. Piezoelectric transducers have been used to investigate the best monitoring setups, which includes defining shapes of actuators and sensors to improve the damage detection and decrease false diagnoses. Giurgiutiu (2003), for example, presents an isotropic plate with a piezoelectric transducer attached to its surface and acting as actuator. The author obtains the displacement and strain for different Lamb wave modes, and the waves are created due to external voltage applied to the actuator. The pin-force model is used to relate the host structure and the piezoelectric transducer. Raghavan and Cesnik (2004) demonstrate the displacement and strain for different Lamb wave modes considering circular piezoelectric actuators. They also introduce equations to compute the output voltage for a rectangular piezoelectric sensor. Raghavan and Cesnik (2005) present an approach to obtain the displacement and output voltage for arbitrary shapes of actuator and sensor, respectively. The transducers are bonded to an isotropic plate. The authors investigate two different shapes of actuators (rectangular and circular) and a rectangular sensor. Their results show that at certain frequencies some wave modes present higher amplitudes than other ones, and those frequencies depend on the actuator and sensor shapes and dimensions. Similar results are observed by Nienwenhui et al. (2005), Scalea, Matt and Bartoli (2007), Lin et al. (2012) and Lin and Giurgiutiu (2012). Damage detection processes based on waves using piezoelectric transducers usually involve defining the best kind of wave to detect each type of damage, and the most 29 convenient shape for the actuator. In addition, based on the actuator shape, it is important to establish the best excitation frequency to create the desirable wave. The shape for the sensor, and the relative position between the transducers are also relevant issues to be addressed. The Waveform Revealer 3.0 is a robust software developed by Shen and Giurgiutiu (2016) and Giurgiutiu (2022). The software allows one to investigate guided waves interactions with damages, and it provides different theoretical solutions for S0 and A0 lamb wave modes, including analytical waveforms for different shapes and kinds of piezoelectric transducers. Li, Khodaei and Aliabadi (2020) present a methodology called Boundary Element Method (BEM) and introduce an approach similar to the pin- force model to couple actuator and host structure to each other. The authors investigate damage effects on the waves using the Dual BEM. However, the effects of the piezoelectric transducers sizing on creating and collecting waves are neglected. Based on this context, the present thesis considers the modeling of longitudinal and flexural waves propagating in damaged plate-like structures to obtain S0 and A0 lamb wave modes for low-frequency range. The classical formulation is provided by Giurgiutiu (2014c), and different algebraic rearrangements are carried out to investigate the influence of the damage on the wave propagation. Circular piezoelectric transducers are considered and an analytical approach is introduced to demonstrate the sensor output voltage in terms of the input voltage applied to the actuator. A new damage detection index is proposed and theoretical and experimental results are presented to demonstrate the findings, and they contribute to the improvement of SHM processes to detect damage in plate-like structures. The work presented herein is currently situated at a relatively low Technology Readiness Level (TRL) (MANKINS, 1995). However, it holds substantial promise for advancing SHM systems and stands as a step towards addressing challenges and improving the reliability, accuracy, and efficiency of SHM technologies. 1.2 OBJECTIVES The main objective of this work is to investigate how longitudinal and flexural waves interact with symmetrical damages in plates with circular piezoelectric transducers. The specific objectives are summarized below: • Develop a dynamic model in the frequency domain to describe longitudinal and flex- ural wave propagation in a damaged plate-like structure with circular piezoelectric transducers; 30 • Determine an approach to compute optimal frequencies to detect damages in plate- like structures in terms of the damage physical and geometrical properties; • Obtain optimal frequencies to create and measure longitudinal and flexural waves using circular piezoelectric transducers, in terms of their physical and geometrical properties; • Demonstrate the effect of longitudinal and flexural waves considering oblique incidence into symmetrical damaged sections. 1.3 MAIN CONTRIBUTIONS The main contributions of this work to the literature are summarized below: • An analytical approach in the frequency domain to compute longitudinal and flexu- ral wave amplitudes in damaged plate-like structures including circular piezoelectric transducers; • Analytical equations to compute optimal frequencies to create and acquire longitu- dinal and flexural waves with circular piezoelectric transducers coupled to plates; • Optimal frequencies to detect damage using longitudinal and flexural waves in terms of the damage characteristics; • The number of wave packets collected by circular piezoelectric sensors depending on their dimensions and relative position. The main results of this work were published in the following journal articles: • Lopes, Gonsalez-Bueno and Bueno (2022): ‘‘On the Frequencies for Structural Health Monitoring in Plates with Symmetrical Damage: An Analytical Approach’’. Journal of Nondestructive Evaluation (2022). • Lopes et al. (2023b): ‘‘On the modeling of circular piezoelectric transducers for wave propagation-based structural health monitoring applications’’. Journal of Intelligent Material Systems and Structures (2023). • Lopes et al. (2023a): ‘‘Longitudinal wave scattering in thin plates with symmetric damage considering oblique incidence’’. Ultrasonics (2023). 31 and in the following conferences: • Lopes et al. (2021): ‘‘On the Frequencies for Structural Health Monitoring in Plates with Symmetrical Damage: An Analytical Approach’’. 26th International Congress of Mechanical Engineering (2021). • Lopes et al. (2022): ‘‘On the Frequencies for Structural Health Monitoring in Plates with Asymmetrical Damage: An Analytical Approach’’. Health Monitoring of Structural and Biological Systems XVI, SPIE (2022). • Lopes et al. (2023c): ‘‘On the Symmetric Damage Detection with Flexural Waves when using Circular Piezoelectric Transducers’’. Health Monitoring of Structural and Biological Systems XVII, SPIE (2023). In addition, the following awards and achievements were obtained in international conferences: • Best Student Paper Award - Bronze award, 32nd International Conference on Adaptive Structures and Technologies (ICAST 2022). • Health Monitoring of Structural and Biological Systems Best Student Paper Award - Third place, SPIE Smart Structures + Nondestructive Evaluation 2023. 1.4 OUTLINE This thesis is structured into the following chapters: • Chapter 1 - INTRODUCTION: it covers the motivation and literature review, objectives, main contributions, and the general content of the thesis. • Chapter 2 - SENSOR SIZE AND OPTIMUM FREQUENCIES TO DETECT DAMAGE WITH LONGITUDINAL WAVES: it presents the mathematical modeling of longitudinal wave propagation in thin plates with sym- metrical damage. The scattering of the longitudinal wave and equations to compute the wave amplitudes of the wave packets for oblique incidence are introduced. The effect of the number of wave packets collected by the sensor is written in terms of its dimensions. Optimal frequencies for damage detection are also introduced. Numerical simulations are presented to demonstrate the proposed approach, and https://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/noticias/v/id::10005/aluno-do-dem-recebe-premio-em-evento-internacional https://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/noticias/v/id::10005/aluno-do-dem-recebe-premio-em-evento-internacional https://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/noticias/v/id::10013/bolsista-do-dem-tem-artigo-premiado-nos-estados-unidos https://www.feis.unesp.br/#!/departamentos/engenharia-mecanica/noticias/v/id::10013/bolsista-do-dem-tem-artigo-premiado-nos-estados-unidos 32 experimental tests are carried out considering circular actuator and radial wave propagation. • Chapter 3 - FLEXURAL WAVES WITH OBLIQUE INCIDENCE: it provides a detailed modeling of flexural wave propagation in thin plates with symmetrical damages. • Chapter 4 - CIRCULAR PIEZOELECTRIC TRANSDUCERS FOR SHM: it shows the modeling of circular piezoelectric transducers coupled to thin plates used to create and measure longitudinal and flexural waves. New equations to compute the sensor output voltage in terms of the actuator input voltage are introduced. Optimal frequencies to create and measure the waves in terms of both PZTs and plate properties are also presented, and it is shown that their physical and geometrical parameters affect the measurements obtained by the sensors. • Chapter 5 - FINAL REMARKS: comprises the final conclusions, and sugges- tions for future work. 33 2 SENSOR SIZE AND OPTIMUM FREQUENCIES TO DETECT DAMAGE WITH LONGITUDINAL WAVES This chapter presents an approach to define the sensors size in terms of their position in the structure. The approach includes a strategy to determine the plate thickness reduction due to the damage presence by considering an oblique incidence of waves generated by the actuator. Modeling is developed to compute each wave packet of reflected and transmitted waves separately from each other, which allows one to describe the wave scattering in thin plates with symmetric damage. 2.1 LONGITUDINAL WAVE PROPAGATION IN THIN PLATES The longitudinal displacements are denoted u and v respectively in the x and y directions in a rectangular plate. The displacements are related by the following equations of motion (GIURGIUTIU, 2014a; LOPES; GONSALEZ-BUENO; BUENO, 2022) ∂2u ∂x2 + (1 − νp) 2 ∂2u ∂y2 + (1 + νp) 2 ∂2v ∂x∂y = (1 − ν2 p)ρp Ep ∂2u ∂t2 ∂2v ∂y2 + (1 − νp) 2 ∂2v ∂x2 + (1 + νp) 2 ∂2u ∂x∂y = (1 − ν2 p)ρp Ep ∂2v ∂t2 (1) where Ep is the Young’s modulus, ρp is the material density and νp is the Poison’s coefficient. The subscript p indicates the undamaged plate. The solution of both equations in Equation (1) is obtained by considering the vector d on the x-y plane which contains the plate, such that d = ui + vj, which allows one to rewrite the equation of motion as 34 follows (PARK et al., 2001; YEO et al., 2017) (1 − νp) 2 ∇2d + (1 + νp) 2 ∇∇ ● d = (1 − ν2 p)ρp Ep ∂2d ∂t2 (2) where i and j are unitary vectors in the x and y directions, respectively, ● indicates the scalar product, ∇ = ∂()i ∂x + ∂()j ∂y is the gradient operator and ∇2 = ∇ ●∇ is the Laplacian. The solution vector d can be conveniently given by considering potential functions φ(x, y, t) and ψψψ(x, y, t), such that (MIKLOWITZ, 1978) d(x, y, t) = ∇φ(x, y, t) +∇ ×ψ(x, y, t) (3) where φ(x, y, t) is a scalar function representing the potential linear displacement and ψψψ represents the potential angular displacement and normal to the plate plane, i.e., ∇●ψψψ = 0 (PARK et al., 2001). Substituting Equation (3) into (2), it is obtained: (1 − νp) 2 ∇2 [∇φ +∇ ×ψ] + (1 + νp) 2 ∇(∇ ● [∇φ +∇ ×ψ]) = (1 − ν2 p)ρp Ep ∂2 ∂t2 [∇φ +∇ ×ψ] (4) where ∇●∇φ = ∇2φ and ∇●(∇×ψψψ) = 0 (PARK et al., 2001), which allows one to rewrite (1 − νp) 2 ∇2 [∇φ +∇ ×ψ] + (1 + νp) 2 ∇(∇2φ) = (1 − ν2 p)ρp Ep ∂2 ∂t2 [∇φ +∇ ×ψ] (5) considering that ∇2(∇φ) = ∇(∇2φ) and ∇2(∇×ψψψ) = ∇× (∇2ψψψ). In this case, it is possible to obtain ∂2 ∂t2 (∇ × ψψψ) = ∇ × ( ∂2 ∂t2ψψψ) and ∂2 ∂t2 (∇φ) = ∇( ∂2 ∂t2φ), which allows one to write Equation (5) in the following form (1 − νp) 2 [∇(∇2φ) +∇ × (∇2ψ)] + (1 + νp) 2 ∇(∇2φ) = (1 − ν2 p)ρp Ep [∇(∂ 2φ ∂t2 ) +∇ × (∂ 2ψ ∂t2 )] . (6) From this last equation, it is possible to obtain ∇(∇2φ − (1 − ν2 p)ρp Ep ∂2φ ∂t2 ) +∇ × ((1 − νp) 2 ∇2ψ − (1 − ν2 p)ρp Ep ∂2ψ ∂t2 ) = 0. (7) Equation (7) allows one to write ∇2φ = 1 c2 l ∂2φ ∂t2 and ∇2ψψψ = 1 c2 s ∂2ψ ∂t2 , and they are respectively related to the longitudinal and the shear waves in the middle plane of the 35 plate. Their respective phase velocities are cl = √ Ep/[ρp(1 − ν2 p)] and cs = √ Gp/ρp, where Gp is the shear modulus. The shear waves can be neglected, which allows one to introduce the following equation to compute the potential linear displacement (PARK et al., 2001) φ(x, y,ω) = [B1ei(−k1x−k2y) +B2ei(k1x−k2y) +B3ei(k1x+k2y) +B4ei(−k1x+k2y)]e−iωt (8) where B1 to B4 are potential function amplitudes, k1 and k2 are components of the longitudinal wavenumber respectively in the x and y directions, and i2 = −1. They can be written such that k1 = kl cos θ and k2 = kl sin θ (GOLATO; DEMIRLI; SANTHANAM, 2014), where kl = ω/cl is the wavenumber and θ the angle between direction of wave propagation and the x axis. kl(p) ≡ kl for the plate undamaged region and kl(d) for the damaged one. Based on Equation (8) and considering that u = ∂φ/∂x e v = ∂φ/∂y (PARK et al., 2001), the longitudinal displacements are given by u(x, y,ω) = cos θ[Arue−− −Alue+− −Alde++ +Arde−+]e−iωt v(x, y,ω) = sin θ[Arue−− +Alue+− −Alde++ −Arde−+]e−iωt (9) where ei(±k1x±k2y) ≡ e±± is used to simplify the notation, i.e., for an instance e++ = ei(+k1x+k2y). The wave amplitudes are Aru = −B1ikl, Alu = −B2ikl, Ald = −B3ikl and Ard = −B4ikl, where subscripts r (right) and u (up) indicate positive directions of x and y axes, respectively. The subscripts l (left) and d (down) are related to the negative x and y directions, respectively, such as illustrated in Figure 1. The line forces per unit of displacement in the x and y directions are Nx and Ny, respectively, and they are given by (GIURGIUTIU, 2014a; RADWAŃSKA et al., 2017) Nx = Ephp 1 − ν2 p (∂u ∂x + νp ∂v ∂y ) Ny = Ephp 1 − ν2 p (νp ∂u ∂x + ∂v ∂y ) (10) where hp is the plate thickness. Substituting Equation (9) into Equation (10), it is possible to write Nx = Ephp 1 − ν2 p [AruLe−− +AluLe+− +AldLe++ +ArdLe−+] e−iωt Ny = Ephp 1 − ν2 p [AruL̄e−− +AluL̄e+− +AldL̄e++ +ArdL̄e−+] e−iωt (11) 36 Figure 1 – Longitudinal waves propagating from a point o on the plate, where A(.) is the wave amplitude and the subscripts r, l, u and d indicate respectively right, left, up and down. The angle θ indicates the direction of propagation in relation to the x axis. Point o indicates the coordinate system origin. o Aruy x Alu Ald Ard � � � � Source: elaborated by the author. where L = −i(k1 cos θ + νpk2 sin θ) and L̄ = −i(νpk1 cos θ + k2 sin θ). Based on Equations (9) and (11), the state vector h̄ = {u v Nx Ny}T is defined. Brennan (1994) noted that is convenient to introduce the transformation matrix H̄, the spatial transformation matrix T̄ and the vector of wave amplitudes ā to rewrite a state-vector notation such that h̄(x, y, θ, ω) = H̄(θ,ω)T̄(x, y,ω)ā(ω). (12) Note that the dependency of x, y, θ and ω indicated in Equation (12) is omitted in this chapter by simplicity. Although Brennan (1994) has investigated one dimensional structures, the equations for the plate considered herein can be similarly rearranged, and the following matrices and vector can be defined H̄ = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ cos θ sin θ EphpL (1 − ν2 p) EphpL̄ (1 − ν2 p) − cos θ sin θ EphpL (1 − ν2 p) EphpL̄ (1 − ν2 p) − cos θ − sin θ EphpL (1 − ν2 p) EphpL̄ (1 − ν2 p) cos θ − sin θ EphpL (1 − ν2 p) EphpL̄ (1 − ν2 p) ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (13) T̄ = diag ( ei(−k1x−k2y), ei(k1x−k2y), ei(k1x+k2y), ei(−k1x+k2y) ) (14) 37 ā = { Aru Alu Ald Ard } T (15) where diag( ) indicates a diagonal matrix. 2.2 CROSS-SECTION CHANGE ANALYSIS This section presents an investigation on the longitudinal wave propagation con- sidering a cross-section change, as shown in Figure 2. The idea is to determine the coefficients to express transmitted and reflected waves in relation to the incident wave. These coefficients are written in terms of the physical and geometric properties of the plate. Figure 2 – Plate with cross-section change in the y direction. C represents an actuator responsible by the generation of a longitudinal wave that propagates in the x and y positive directions. Sl and Sr are the sensors responsible by the acquisition of reflected and transmitted waves. h p h d C Sr dx1 q a Ap Aa dx3 p Aq Aq Aqru ru Sl b Ab dx2 rui ru lu lu d y2 y x d y1 d y3 Source: elaborated by the author. Considering an actuator at the point p shown in Figure 2. Assume that it can generate a longitudinal wave that propagates simultaneously in the x and y positive directions with an angle θ between its direction and the x axis. At the points a and b are considered two 38 sensors Sr and Sl to obtain the transmitted and reflected waves, respectively. Equation (9) is rewritten as shown below: u(x, y,ω) = cos θ[Arue−− −Alue+−]e−iωt v(x, y,ω) = sin θ[Arue−− +Alue+−]e−iωt. (16) Substituting Equation (16) in Equation (10), it is possible to obtain the line forces Nx and Ny as follows: Nx = Ephp 1 − ν2 p [−Aruikl(p)(cos2 θ + νp sin2 θ)e−− −Aluikl(p)(cos2 θ + νp sin2 θ)e+−] e−iωt Ny = Ephp 1 − ν2 p [−Aruikl(p)(νp cos2 θ + sin2 θ)e−− −Aluikl(p)(νp cos2 θ + sin2 θ)e+−] e−iωt. (17) The line force Nx is rewritten such that [Arue−− +Alue+−] e−iωt = −Nx (1 − ν2 p) Ephp 1 ik(cos2 θ + νp sin2 θ) (18) which allows one to write Ny and v in terms of Nx. Then, v(x, y,ω) = −Nx (1 − ν2 p) Ephp sin θ ik(cos2 θ + νp sin2 θ) (19) Ny(x, y,ω) = Nx (νp cos2 θ + sin2 θ) (cos2 θ + νp sin2 θ) . (20) Note that Ny and v are written in terms of Nx, which allows one to consider u and Nx to define the state vector h̄(x, y,ω) = {u Nx}T = H̄(ω)T̄(x, y,ω)ā(ω). In this case, H̄(ω) = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ cos θ −ikl(p)Ephp(cos2 θ + νp sin2 θ) 1 − ν2 p − cos θ −ikl(p)Ephp(cos2 θ + νp sin2 θ) 1 − ν2 p ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (21) T̄(x, y,ω) = diag (eikl(p)(−x cos θ−y sin θ), eikl(p)(x cos θ−y sin θ)) (22) ā(ω) = {Aru Alu}T . (23) 39 Considering the point q in Figure 2, it is possible to write Aru ≡ Aq rui and Alu ≡ Aq lu for section with thickness hp. On the other hand, Aru ≡ Aq ru for section with thickness hd. In this case h̄p = H̄pāq p, where āq p = {Aq rui Aq lu}T (at the point q). Similarly, h̄d = H̄dāq d for āq d = {A q ru 0}T . The subscripts indicate the corresponding section, i.e., d for section with thickness hd; and p for section with thickness hp. The superscripts indicate the evaluated point (see Figure 2). Considering h̄p = h̄d, it is obtained āp = Zād, where Z = [H̄p]−1H̄d. Then, the amplitudes Aq lu and Aq ru can be written in terms of the incident amplitude Aq rui, as follows: Aq ru Aq rui = 2 1 + H̄hP̂ for 0 ≤ θ ≤ 60o, (24) Aq lu Aq rui = H̄P̂ − 1 1 + H̄hP̂ for 0 ≤ θ ≤ 60o, (25) where H̄ = √ (Edpρdp)/ν̌ (Edp = Ed/Ep, ρdp = ρd/ρp), h = hd/hp, ν̌ = (ν̂d/ν̂p), ν̂p = (ν2 p − 1) and ν̂d = (ν2 d − 1). P̂ = (Pd/Pp). Pd = (νd sin2 θ + cos2 θ) and Pp = (νp sin2 θ + cos2 θ). In addition, if the sections of thicknesses hp and hd have different physical properties, the propagation angle of the transmitted wave is different from the angles of the incident and reflected waves (which are the same) due to differences in their velocities. For this case, it is necessary to take into account the Snell-Descartes’ law to compute the propagation angle of the transmitted wave. Let Ct(pd) be the coefficient defined by ratio between the amplitudes of the transmitted and incident longitudinal waves, and Cr(pd) for the reflected wave, it is possible to write the following equations Ct(pd) = 2 1 + h (26) Cr(pd) = h − 1 1 + h (27) for 0 ≤ θ ≤ 60o. If hp < hd, then, Ct(dp) = hCt(pd) = 2h 1 + h (28) Cr(dp) = −Cr(pd) = 1 − h 1 + h. (29) Equations (26) to (29) can be used to determine the relations between the wave amplitudes considering symmetrical cross-section change, and consequently, there is no conversion 40 of modes due to symmetry of the cross-section change (PAU; ACHILLOPOULOU, 2017; PODDAR; GIURGIUTIU, 2016; LOPES et al., 2022). 2.3 LONGITUDINAL WAVE SCATTERING IN THIN PLATES WITH SYMMETRICAL DAMAGE Local thickness reduction has been successfully used to represent structural damage, as presented in references Lowe et al. (2002), Lowe and Diligent (2002), Glushkov et al. (2015), Gonsalez-Bueno (2019) and others. This approach was previously introduced by Shen and Pierre (1990) to describe symmetric cracks and this type of damage representation has been often considered in the literature to investigate the interactions between damage and wave propagation (PAU; ACHILLOPOULOU; VESTRONI, 2016). In particular for the plate, it is possible to include a damage with width lx << Lx and length Ly(d) < Ly in the x and y directions, respectively, where Lx and Ly are the dimensions of the plate in the x and y directions. See a schematic illustration in Figure 3. Note that it is assumed lx << Ly(d) as discussed in the following sections. 2.3.1 Wave Scattering due to Oblique Incident Wave Propagating in the x and y Positive Directions Figure 3 shows a thin plate with thickness hp. The plate exhibits damage with physical properties equal to the undamaged section. However, it is characterized by thickness hd < hp. The point p represents the position of an actuator used for generating a longitudinal wave that propagates in the x and y positive directions with the angle θ in relation to the positive x axis, i.e., the wave is generated at the point p and propagates until the point q. Due to the damage presence, the incident wave is divided into transmitted (from point q to r) and reflected (from point q to e) waves. Health and damaged regions of the plate have the same physical properties, then, the velocities of the waves are the same in both regions and the propagation angle of the transmitted wave packet inside the damage is also θ. If the damage has different properties from the healthy region of the plate, it is necessary to take into account the Snell-Descartes’ law to compute the propagation angle of the transmitted wave packet (see Section 2.2). In addition, this approach is applied for low-frequency range. The point a indicates the position of a circular sensor with radius rR (subscript R indicates right-hand side of the plate). The first reflected wave packet is received at the point e, where encounters another circular sensor with radius rL (subscript L indicates left-hand side of the plate). 41 Figure 3 – Thin plate with symmetrical damage of the width lx in the x axis and length Ly(d) (Ly(d) < Ly) in the y axis. Lx and Ly are the dimensions of the plate in the x and y directions, respectively. Point p represents the location of an actuator generating a longitudinal wave propagating from the point p to q. Circular sensors are considered at the points e and a, with radii rL and rR. Points a and e correspond to the first packets of transmitted and reflected waves, respectively. h p lx hd L y( d ) C dx1 q dx3 p dx2 d y2 y x d y1 r e rL a d y3 rR Source: elaborated by the author. The wave transmitted from the point q is incident at point r. The wave generated at the point r propagates until the point s shown in Figure 4. From the point s, the second reflected wave packet propagates to the left-hand side of the plate until the point f . Similarly, the wave propagates from the point s to the right-hand side of the plate, until the point t. The second transmitted wave packet propagates from t until the point b. This process occurs continuously over time, and the third reflected wave packet arrives at the point g and the third transmitted wave packet arrives at the point c. Considering a longitudinal wave from point p to point q, it is possible to write that the first reflected wave packet incoming at the point e is equal to Ae r Ap i = Ae r Aq r Aq r Aq i Aq i Ap i , where A ( ) r and A ( ) i are the reflected and incident wave amplitudes at the point ( ). Aq r/Aq i is the ratio between the reflected and incident wave amplitudes due to the cross-sectional change at the point q. Then, it is possible to write Aq r/Aq i = Cr(pd), where Cr(pd) is given by Equation (27) (see Section 2.2 for details). The ratio of amplitudes Ae r/Aq r and Aq i /A p i are obtained using the spatial transformation matrix, as such Aq i /A p i = T p 1 and 42 Figure 4 – Thin plate with symmetrical damage of the width lx in the x axis and length Ly(d) (Ly(d) < Ly) in the y axis. Point p represents the location of an actuator generating a longitudinal wave propagating from the point p to q. Circular sensors are considered at the points e and a, with radii rL and rR. Points a and e correspond to the first packets of transmitted and reflected waves, respectively. The points a, b and c capture the transmitted wave packets, while points e, f and g capture the reflected wave packets. C q a p b s y x c e f g 2 l y 2 l y 2 ly 2 ly r t u w 2 l y 2 l y 2 ly 2 ly Source: elaborated by the author. Ae r/Ap i = T p 2 , where T p 1 = eikl(p)(−dx1 cos θ−dy1 sin θ) and T p 2 = eikl(p)(+(−dx2) cos θ−dy2 sin θ). kl(p) is computed for the undamaged section properties of the plate. In this case, Ae r Ap i = T p 1 T p 2Cr(pd), and considering dx1 = dx2 e dy1 = dy2, it is possible to obtain Ae r Ap i = (h − 1) (h + 1)e −2ikl(p)dx1 sec θ (30) where h = hd/hp and sec θ = (1/ cos θ). The amplitude ratio Aa t /Ap i at the point a can be written in terms of the generated wave by the actuator at the point p, such that Aa t Ap i = Aa t Ar t Ar t Ar i Ar i Aq t Aq t Aq i Aq i Ap i . In particular, Ar t Ar i = Ct(dp) (Equation 28) and Aq t Aq i = Ct(pd) because they correspond to the amplitudes resulting from the interaction between wave and cross- sectional change, respectively, at the points q and r. In addition, the amplitudes Aa t Ar t and Ar i Aq t can be obtained using the transformation matrix, such that Aa t /Ar t = T p 1 (if dx1 = dx3 and dy1 = dy3) and Ar i /A q t = T d 1 , in which T d 1 = eikl(d)(−lx cos θ−ly sin θ). kl(d) ≡ kl(p) because the damaged and undamaged sections both have equal physical properties. Also, note that ly = lx tan θ. Thus, it is possible to obtain Aa t Ap i = (T p 1 )2T d 1Ct(pd)Ct(dp), which allows one to write the following equation 43 Aa t Ap i = 4h (h + 1)2 e−2ikl(p)dx1 sec θe−ikl(d)lx sec θ. (31) The approach described above can be repeated to obtain the second and third wave packets. In this case, it is possible to write for the point f Af r Ap i = Af r As t As t As i As i Ar r Ar r Ar i Ar i Aq t Aq t Aq i Aq i Ap i , which describes the second reflected wave packet, where Ar r Ar i = Cr(dp), As t As i = Ct(dp) and As i Ar r = T d 2 , considering that T d 2 = eikl(d)(+(−lx) cos θ−ly sin θ). This ratio is written by Af r Ap i = T p 1 T p 2 T d 1 T d 2Ct(pd)Cr(dp)Ct(dp), and the following equation is obtained Af r Ap i = −4h (h − 1) (h + 1)3 e−2ikl(p)dx1 sec θe−2ikl(d)lx sec θ. (32) Similarly, the amplitude ratio at the point g is Ag r Ap i = T p 1 T p 2 (T d 1 T d 2 )2Ct(pd)(Cr(dp))3Ct(dp), which can be rewritten as follows Ag r Ap i = −4h (h − 1)3 (h + 1)5 e−2ikl(p)dx1 sec θe−4ikl(d)lx sec θ. (33) The second and third transmitted wave packets are obtained using similar approach. For the points b and c, it is possible to write Ab t Ap i = (T p 1 )2(T d 1 )2T d 2Ct(pd)(Cr(dp))2Ct(dp) and Ac t Ap i = (T p 1 )2(T d 1 )3(T d 2 )2Ct(pd)(Cr(dp))4Ct(dp), such that Ab t Ap i = 4h (h − 1)2 (h + 1)4 e−2ikl(p)dx1 sec θe−3ikl(d)lx sec θ (34) Ac t Ap i = 4h (h − 1)4 (h + 1)6 e−2ikl(p)dx1 sec θe−5ikl(d)lx sec θ. (35) Based on the modeling introduced in this present chapter, it is possible to compute the first reflected and transmitted wave packets (i.e., incoming respectively at the points e and a) using the following equations: Ar Ai ∣ (1) = (h − 1) (h + 1)e −2ikl(p)dx1 sec θ (36) At Ai ∣ (1) = 4h (h + 1)2 e−2idx1kl(p) sec θe−ikl(d)lx sec θ. (37) The other packets of reflected and transmitted waves can also be defined respectively by 44 the following equations Ar Ai ∣ (n) = −4h(h − 1)(2n−3) (h + 1)(2n−1) e−2idx1kl(p) sec θe−2(n−1)ilxkl(d) sec θ (38) At Ai ∣ (n) = 4h(h − 1)2(n−1) (h + 1)2n e−2idx1kl(p) sec θe−(2n−1)ikl(d)lx sec θ (39) where n = 2, 3, 4, .... 2.3.2 Resulting Wave due to Circular Longitudinal Wave Propagation According to Raghavan and Cesnik (2005), the use of circular actuators in plates generates propagation of radial waves with the same wave amplitude in all directions. Figure 5 presents a thin plate with a circular actuator placed at the point p which creates radial longitudinal waves with amplitude Ai, as presented by Shen and Giurgiutiu (2016). A circular sensor with radius rR is considered at the point a, and it captures the wave front created by the actuator for an angle range that depends on its size. The direction between the actuator and sensor centers is indicated by a solid line, with an angle θ1 in relation to the x axis. The corresponding angle (see two dotted lines) is used to compute the sensor output voltage (YEUM; SOHN; IHN, 2011). Figure 6 shows the plate with symmetric damage of length lx in the x direction. Ar∣θ1 () and At∣θ1 () represent respectively amplitudes for the reflected and transmitted waves, in which the subscript indicates the wave packet number. If the sensors radii are smaller than 2lx tan(θ1), these transducers capture just the first wave packet. Figure 5 – Thin plate with a circular actuator placed at the point p to create radial waves with amplitude Ai. A circular sensor at the point a captures the wave front for an angle range around the angle θ1. x y �� a Ai p C Source: elaborated by the author. 45 Figure 6 – Thin plate with symmetric damage of length lx in the x axis. Points a and b indicate the first and second packets of amplitudes At∣θ1 1 and At∣θ1 2 , and points e and f indicate the first two reflected wave packets with amplitudes Ar∣θ1 1 and Ar∣θ1 2 , respectively. f ��p a e b C 2 ly 2 l y At 1 At 2 �� �� Ar 1 �� x y lx Ar 2 �� Source: elaborated by the author. Considering that the circular actuator C generates radial waves in all directions, the first transmitted wave packet associated to the θ1 direction achieves the sensor at point a. The second wave packet associated to an angle θ2 also achieves the same sensor. Figure 7 illustrates the path of both wave packets, with amplitudes At∣θ1 1 and At∣θ2 2 , for θ2 < θ1. Due to their small amplitudes, other wave packets (such as the third, fourth, etc) can be neglected (see Figure 10 to clarify). Then, assuming sensor sizes such as r() < 2lx tan(θ1), the resulting wave ( At Ai ∣ c ) captured by the sensor at the point a can be accurately described by sum of the first and second packets, respectively associated to the angles θ1 and θ2. Note that the subscript c indicates the circular actuator. The wave amplitude for the first packet propagating in θ1 is computed by Equation (37). In addition, the second wave packet propagating in θ2 is computed by Equation (39) (for n = 2). Note that the angles θ1 and θ2 are determined from the relative positions between actuator and sensor. The resulting wave is accurately obtained by the following equation: At Ai ∣ c ≈ At Ai ∣ θ1 1 + At Ai ∣ θ2 2 . (40) 46 Figure 7 – Plate with symmetric damage of length lx in the x direction. Point a indicates the sensor capturing the first wave packet (Āt∣θ1 1 ) for θ1 and the second wave packet (Āt∣θ2 2 ) for θ2. �� �� p a C At 1 +At 1 �� At 2 x y lx b �� �� At 2 �� Source: elaborated by the author. 2.3.3 Detecting the Thickness Reduction in a Damaged Section According to Glushkov et al. (2015), the energy of a longitudinal wave is proportional to the square of its absolute value. Considering the ratio between reflected and incident waves, or transmitted and incident, it is possible to write the relative energy of the reflected wave by Er = ∣ Ar Ai ∣ 2 (41) and for the transmitted wave Et = ∣ At Ai ∣ 2 . (42) Considering a circular sensor with radius r( ) < 2ly, i.e., r( ) < 2lx tan θ, placed at the points a and e, both sensors measure the first wave packets separately from the other ones. Comparing Equations (41) and (42) with Equations (36) and (37), the relative energy related to the measured reflected and transmitted wave packets in these sensors corresponds to the following equations Er ≡ Er∣(1) = (1 − h)2 (h + 1)2 and Et ≡ Et∣(1) = 16h2 (h + 1)4 . (43) Based on these equations, the thickness of the plate in the damaged section can be 47 determined by using the following equation: hd = 1 − √ Er∣(1) 1 + √ Er∣(1) hp, (44) which provides accurate estimates of hd. Note that it is possible to write hd = hp (1 − √ 1 −Et∣(1)) 2 / √ Et∣(1). However, this last equation is mathematically limited to be applied for large damage depth (i.e., 1 − h > 20%), otherwise Et∣(1) > 1, and a complex number is obtained from the term with the square root, and this equation must be neglected. 2.3.4 Index for Damage Detection in Plates using Transmitted Longitu- dinal Wave The presence of a damage changes the transmitted wave amplitude at the frequency of excitation, which allows one to establish the damage detection index R̄ defined by the ratio between the maximum magnitudes obtained for an unknown structural condition (max(∣Au∣), Figure 8, symbol ▽) and for the healthy plate (max(∣Ak∣), Figure 8, symbol ◻). Then, the index R̄ = max(∣Au∣) max(∣Ak ∣) is proposed, which implies to the theoretical value R̄ = 1 if the unknown condition corresponds to the undamaged structure. In particular, a five cycles tone burst is employed as input signal, and the output amplitudes are obtained from the magnitude in the frequency domain at the frequency of excitation (see an illustrative example in Fig. 8). In a practical case, it is interesting to consider more than one signal for characterizing the healthy structure, and one of them is used as reference. The damage index R̄ can be computed considering the other signals, which allows one to establish a range [R̄l,1] for describing the healthy structure. Then, the minimum value of R̄ for the healthy condition is denoted by R̄l, such that a damage is detected for an unknown structural condition if a computed damage index is R̄ < R̄l. Note that R̄l establishes a threshold for the damage detection. 48 Figure 8 – Magnitude of the transmitted tone burst with 5 cycles at the frequency 100 kHz. Continuous line is obtained for healthy structure and the dashed line for the unhealthy one. Symbols ◻ and ▽ are the maximum values obtained for each case, respectively. 0 50 100 150 200 Frequency [kHz] 0 2 4 6 8 10 12 14 |A | Source: elaborated by the author. 2.4 RESULTS AND DISCUSSION This section presents numerical results to demonstrate the proposed approach to detect damage in an aluminum plate with Lx = Ly = 50 cm, Ep = 69 MPa, ρp = 2710 kg.m−3, hp = 1.5 mm, lx = 5 mm, dx1 = dx2 = dx3 = 10 cm and νp = 0.31. The longitudinal wave is generated from the point p, with oblique incidence in a damaged section. The physical properties of the damaged section are equal to the undamaged section, but with reduced thickness hd < hp. Reflected (∣Ar/Ai∣(n)) and transmitted (∣At/Ai∣(n)) wave amplitude ratios are con- stants over both frequency and incidence angle, and they depend on the ratio h = hd/hp. Figures 9 and 10 show the four first wave packets of these amplitudes in terms of the damage depth (1 − h). Based on Figure 9, it is noted that if (1 − h) < 30% the two first reflected wave packets present higher amplitudes in comparison with the other packets. These results indicate that if a circular sensor with radius rL < 2lx tan(θ) is placed on the plate at the point e, it can measure the first packet and the measurement contains important amount of energy once it is proportional to the square of the wave amplitude. 49 In addition, the sensor can measure the first wave separately from the other packets, which is interesting to damage detection, as discussed herein. Figure 9 – Reflected wave amplitude ratios (∣Ar/Ai∣(n)) in terms of the damage depth (1 − h): first packet (n = 1, dashed line), second (n = 2, continuous line), third (n = 3, dotted line) and fourth packets (n = 4, dash-dotted line). 0.0 0.2 0.4 0.6 0.8 1.0 1− h 0.0 0.2 0.4 0.6 0.8 1.0 |Ar Ai | Source: elaborated by the author. Figure 10 shows that the first transmitted wave packet is higher than the other packets. However, ∣At Ai ∣ is almost equal to 1 for low damage depth (i.e., small value of (1 − h)). In practice, this means that the transmitted wave amplitude for small (1 − h) is approximately equal to the amplitude of the wave generated by the actuator, which becomes difficult to detect the damage presence by comparing them. Figure 11 shows that two different sizes of circular sensors rL obtain amplitudes of reflected waves approximately equal to each other. Sensor sizes between the ranges 2lx tan(θ) ≤ rL < 4lx tan(θ) and 4lx tan(θ) ≤ rL < 6lx tan(θ) are compared. The incoming wave for the sensor size 2lx tan(θ) ≤ rL < 4lx tan(θ) corresponds to the first and second wave packets, whereas the three first wave packets are obtained if 4lx tan(θ) ≤ rL < 6lx tan(θ). However, there is a small contribution from the third and fourth wave packets, as shown in Figure 9. This characteristic is also verified for the transmitted wave, as shown in Figure 12. Based on these results, it is not necessary to consider a sensor with radius larger than 4lx tan(θ). Note that the maximum wave amplitude ratios occur at 50 Figure 10 – Transmitted wave amplitude ratios (∣At/Ai∣(n)) in terms of the damage depth (1 − h): first packet (n = 1, dashed line), second (n = 2, continuous line), third (n = 3, dotted line) and fourth packets (n = 4, dash-dotted line). 0.0 0.2 0.4 0.6 0.8 1.0 1− h 0.0 0.2 0.4 0.6 0.8 1.0 |At Ai | Source: elaborated by the author. specific optimum frequencies f l opt = (1+m) 4lx cos(θ) √ Ed/[ρd(1 − ν2 d)], where m = 0, 2, 4, ..., which also depends on the angle of incidence θ. The optimal frequencies for symmetric damage are slightly different from those obtained for asymmetric damage, for which wave mode conversion is observed. The difference between these cases increases according to the damage asymmetry, as presented by Lopes et al. (2022) and Gonsalez-Bueno (2019). Therefore, it is important to know these frequencies for the symmetric case, which can correspond with good accuracy to the optimal frequencies for the asymmetric case, mainly when considering small damage. Note that it is useful to consider this result to define the SHM process, which allows one to establish the excitation frequency of the input signal. 51 Figure 11 – Magnitude of the reflected wave amplitude ratio obtained for 2lx tan(θ) ≤ rL < 4lx tan(θ) (continuous line) and 4lx tan(θ) ≤ rL < 6lx tan(θ) (dotted line): (a) θ = 30○ and (b) θ = 60○. The dashed vertical lines indicate lx/λl(d) for the optimum frequencies f l opt. 0.0 0.2 0.4 0.6 0.8 1.0 lx λl(d) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 |Ar Ai | lx λl(d) → f l opt (a) θ = 30○ 0.0 0.2 0.4 0.6 0.8 1.0 lx λl(d) 0.00 0.05 0.10 0.15 0.20 0.25 0.30 |Ar Ai | lx λl(d) → f l opt lx λl(d) → f l opt (b) θ = 60○ Source: elaborated by the author. 52 Figure 12 – Magnitude of the transmitted wave amplitude ratio obtained for 2lx tan θ ≤ rR < 4lx tan θ (continuous line) and 4lx tan θ ≤ rR < 6lx tan θ (dotted line): (a) θ = 30○ and (b) θ = 60○. The dashed vertical lines indicate lx/λl(d) for the optimum frequencies f l opt. 0.0 0.2 0.4 0.6 0.8 1.0 lx λl(d) 0.96 0.97 0.98 0.99 1.00 |At Ai | lx λl(d) → f l opt (a) θ = 30○ 0.0 0.2 0.4 0.6 0.8 1.0 lx λl(d) 0.96 0.97 0.98 0.99 1.00 |At Ai | lx λl(d) → f l opt lx λl(d) → f l opt (b) θ = 60○ Source: elaborated by the author. The energy of the first reflected wave packet can be used to determine the plate thickness hd in the damaged section. The idea for practical applications is to obtain 53 the energy over time by measuring the wave amplitude. Note that the sensor size needs to be correctly defined, as presented previously herein. The damaged thickness hd is numerically obtained from Equation (44). The procedure is illustrated by computing the Inverse Fourier Transform (IFT) of the first reflected wave packet amplitude for different damage depths, which allows one to obtain the amplitudes of the first reflected wave packets in the time domain. Then, three different levels of noise are artificially added to the obtained signals. The three noisy signals are numerically obtained by considering 3%, 5% and 10% of the signal energy (by using the of signal-to-noise ratio (SNR) approach). A similar approach is used by Rébillat, Hajrya and Mechbal (2014) and Brogin, Faber and Bueno (2020). The energy for each noisy signal is computed, and the corresponding damaged thickness hdet d is determined by using Equation (44) for the different damage depths. Briefly, (i) the amplitude of the first reflected wave packet is obtained in the frequency domain; (ii) the IFT is applied to the result; (iii) the noise is added to the signal in the time domain and; (iv) the relative energy is computed, which allows one to obtain hdet d . Note that hdet d = hd if no noise is considered. However, the noise affects the damaged thickness estimate. Then, hdet d /hd = h̄d → 1 for low level of noise. Figure 13 shows h̄d for the three different levels of noise and considering different damage depth (1−h). Twenty signals were simulated for each boxplot shown in Fig. 13. The red line of the boxes indicate the median values obtained from the signals for each frequency, the lower and upper lines from the boxes show the first and third quartiles, respectively, and the vertical lines are called whiskers1. It is possible to obtain an accurate estimation of hd for low damage depth even if that higher level of noise is considered. The accuracy decreases with increasing of noise if the damage depth is (1 − h) > 50%. 1For more information about boxplot figures, see https://matplotlib.org/stable/api/_as_gen/ matplotlib.pyplot.boxplot.html. https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.boxplot.html https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.boxplot.html 54 Figure 13 – h̄d in terms of the damage depth (1 − h). Continuous line corresponds to no noise condition. Symbols ◂, ● and ▸ respectively correspond to the results obtained by considering 3%, 5% and 10% of noise added to the signal energy. 0.0 0.2 0.4 0.6 0.8 1.0 1− h 0.80 0.85 0.90 0.95 1.00 h̄ d 3% 5% 10% Analytical (no noise) Source: elaborated by the author. 2.5 EXPERIMENTAL TESTS The proposed approach is demonstrated through experimental tests. An aluminum plate with dimensions 1000×1000×1.5 mm is considered, for which Ep = 69 MPa, ρp = 2710 kg/m3 and νp = 0.33. Figure 14 presents a schematic view and photographies of the experimental setup, which considers 3 circular piezoelectric transducers coupled to the plate. They are disc bender with ceramic diameters equal to 19.4 mm and thickness 0.45 mm, such that one of them works as an actuator (C) and the other ones are employed as sensors (S1 and S2). The input signals (five cycles tone burst) are generated by using a LabVIEW-based application with a NI USB 6353 National Instruments hardware, and a power amplifier EL 1225 Mide QuicPack. An oscilloscope DSO7034B Keysight is employed to measure the signals from the actuator and sensors. 55 Figure 14 – (a) Schematic view and (b) photographs of the experimental setup (measure- ments in cm). C Oscilloscope Oscilloscope Amplifier Amplifier NI DAQ NI DAQ Laptop with Labview Laptop with Labview Plate Damaged Plate Upper beam Damage position Lower beam (a) C x y S1 S2 100 10 0 S2 25 .9 81 20 35 30 .3 C S1 0. 15 14.5 1.25 (b) Source: elaborated by the author. Considering the geometric center of the actuator at the origin of the coordinate system (x axis positive direction to the left-hand side and y direction up), the angles for the sensors S1 and S2 correspond to 45o and 29.92o in relation to the positive x axis, respectively. The symmetrical damage is created by two aluminum beams coupled to the faces of the plate (see their position in Fig. 14). Their geometric and physical properties are 1000 × 12.5 × 1.5 mm, Ēp = 70 MPa and ρ̄p = 2600 kg/m3. A double-sided adhesive tape model 430 of thickness 0.115 mm from Adelbras® was used to couple the beams to the plate. Note that the damaged section in this case is composed by two different materials, and because of this the Equivalent Cross Section Method is employed to compute resulting properties for the damaged section (HIBBELER, 2010). Figure 15 presents an illustrative representation of two different materials, ‘‘a’’ and ‘‘b’’, with cross sections Sm and Sn, Young’s modulus Em and En, and densities ρm and ρn. They have the same strain ε, but different stresses (σm,σn). ‘‘c’’ is the equivalent material, and let Feq be the equivalent force applied in the cross section of material ‘‘c’’, where Feq = Seqσeq. It corresponds to the sum of forces in both materials, i.e., Feq = Fm + Fn. In addition, considering the Hooke’s 56 Law for mechanical stresses (σ = Eε), and the equivalent area Seq = Sm +Sn, it is possible to obtain Eeq = Eo = EmSm +EnSn Sm + Sn . (45) The mass of the material ‘‘c’’ is given by the sum of masses ‘‘a’’ and ‘‘b’’ (mo =mm+mn), and its equivalent density is given by ρeq = ρo = ρmSm + ρnSn Sm + Sn . (46) Then, Equations (45) and (46) are used to determine the properties of the damaged section. Figure 15 – Schematic representation of an equivalent section (material ‘‘c’’) from a damaged section composed by two different materials (‘‘a’’ and ‘‘b’’). a b c Composed Material Equivalent Material Source: elaborated by the author. Note that the created damage corresponds to the local cross-section increasing, instead of reduction. This is a convenient strategy due to its relative simplicity to be obtained. Similar approach has been employed in the literature, as presented by Gonsalez et al. (2015) and Hou et al. (2020) for example. The main point of the equivalence between reduction or increase the local mass is that the optimal frequencies for damage detection are equals for both of them, although the wave amplitude ratios are different (compare Figure 16(a) and 16(b) for the same lx/λl(p)). Figure 16 shows the magnitudes of the reflected (Fig. 16a) and transmitted (Fig. 16b) wave amplitude ratios, respectively by dashed and continuous lines. The mass increasing is obtained by considering h = 1.5, whereas h = 0.5 is used for the mass reduction in the damaged section. Then, both strategies are equivalent to each other for the proposal in this chapter. In practice, the methodology presented herein can be used for both configurations, such that a mass addition is represented by h > 1, whereas a mass subtraction corresponds to h < 1, and the optimal frequencies are equal to each other for both ones, as presented in Figure 16. 57 Figure 16 – Magnitude of the (a) reflected and (b) transmitted wave amplitude ratios obtained for 4lx tan(θ) ≤ r() < 6lx tan(θ), considering θ = 60o: Mass reduction by h = 0.5 (dashed-line) and mass increase by h = 1.5 (continuous line). 0.0 0.1 0.2 0.3 0.4 0.5 lx λl(d) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 |Ar Ai | (a) Reflected wave amplitude ratio 0.0 0.1 0.2 0.3 0.4 0.5 lx λl(d) 0.80 0.85 0.90 0.95 1.00 |At Ai | (b) Transmitted wave amplitude ratio Source: elaborated by the author. 58 The resulting damaged section is composed by two different materials (from the plate and beams), and its equivalent physical properties are obtained by using the approach introduced previously, from which Ed = Eeq = 69.67 MPa and ρd = ρeq = 2636.67 kg/m3. The double-sided adhesive tapes are not taken into account to obtain this equivalent damaged section due to their small effect on these properties. The double-sided adhesive tape has Young’s modulus Eb = 10 MPa, material density ρb = 1000 kg.m−3 and thickness hb = 0.115 mm. Then, if they are considered with the plate and beams to define the equivalent damaged section, its equivalent properties are equal to Ẽeq = 69.66 MPa and ρ̃eq = 2636.58 kg.m−3. On the other hand, neglecting them, the equivalent properties of the damaged section are equal to Eeq = 69.67 MPa and ρ̃eq = 2636.67 kg.m−3, which allows one to show the small influence of the double-sided adhesive tape. In addition, note that a good agreement is observed when comparing experimental and theoretical results. The sensors radii are rS1 = rS2 = 9.7 mm and the damage length is lx = 12.5 mm, which allows one to verify that both sensors S1 and S2 capture just the first wave packet from their respective angle θ( )1 because rS1 < 2lx tan(θ(S1) 1 ) = 25 mm and rS2 < 2lx tan(θ(S2) 1 ) = 14.40 mm, where θ(S1) 1 = 45o and θ (S2) 1 = 29.92o (i.e., from the angles of the waves coming from the actuator directly to the respective sensors S1 and S2). The resulting wave (Eq. 40) is obtained by considering the second wave packet propagating in the direction of the angle θ( )2 , such that θ(S2) 2 < θ(S2) 1 and θ(S1) 2 < θ(S1) 1 and θ(S2) 2 = 28.6o and θ(S1) 2 = 42.4o. The angles were obtained from the drawing using a software CAD. In this case, healthy and damaged sections of the plate have different physical properties, then it is necessary to compute the angle of propagation inside the damage by using the Snell-Descartes’ law. In addition, Eqs. (25) and (24) have to be used to compute the wave amplitude ratios of the wave packets. Figure 17(a) shows the damage index to demonstrate that the experimental resulting wave in the sensor S1 is accurately predicted from Equation (40). The experimental results1 are presented through the symbol ◊ and the corresponding theoretical results are shown through the symbol ◻. On the other hand, the theoretical prediction fails if more wave packets are considered for the sensor S1, as shown through the symbols ▽ and ○, respectively corresponding to θ = 45o and θ = 42.4o. Similar conclusions are obtained from the sensor S2 for lx/λp ≥ 0.35, as shown in Fig. 17(b). 1These experimental data correspond to the reference signals, which are arbitrarily defined as the first measurement for each frequency. 59 Figure 17 – Damage Index R̄ computed using: the reference experimental results (symbol ◊); proposed approach [Equation 40] (symbol ◻); and theoretical prediction considering infinite wave packets arriving at the sensor, in which a) θ = 45o (symbol ▽) or θ = 42.4o (symbol ○), and b) θ = 29.92o (symbol ▽) or θ = 28.6o (symbol ○). 0.25 0.30 0.35 0.40 0.45 lx λl(p) 0.90 0.92 0.94 0.96 0.98 1.00 R̄ (a) Sensor S1 0.25 0.30 0.35 0.40 0.45 lx λl(p) 0.90 0.92 0.94 0.96 0.98 1.00 R̄ (b) Sensor S2 Source: elaborated by the author. 60 The experimental signals were measured by S1 for the healthy and damaged structures at the frequencies from 100 kHz up to 200 kHz, corresponding to the range 0.20 < lx/λl(p) < 0.50. The first measured signal for the healthy condition is used as reference to compute the damage index. Figures 18 and 19 show the longitudinal waves measured by using the sensor S1 respectively for both healthy and damaged conditions. Note that a 5 cycles tone burst at the frequency 160 kHz is used. Figure 20 presents the magnitude of these signals in the frequency domain, from which different magnitude can be noted when comparing the healthy and damaged conditions. Figure 21 presents the boxplot for these data, in which the outliers are omitted for easier viewing. A horizontal blue dashed line indicates the global threshold for all frequencies investigated, and then the results above it correspond to the healthy condition. On the other hand, the damage is detected for all signals obtained for the damaged structure - see the results below the threshold line. Similar results are found from the sensor S2, as shown in Figure 22. Figure 18 – Longitudinal waves in the time domain measured by sensor 1 for the healthy plate, considering as input a 5 cycles tone burst at the frequency 160 kHz. 70 80 90 100 110 120 Time [µs] −30 −20 −10 0 10 20 30 A m pl it u d e [m V ] Source: elaborated by the author. 61 Figure 19 – Longitudinal waves in the time domain measured by sensor 1 for the damaged plate, considering as input a 5 cycles tone burst at the frequency 160 kHz. 70 80 90 100 110 120 Time [µs] −30 −20 −10 0 10 20 30 A m pl it u d e [m V ] Source: elaborated by the author. Figure 20 – Magnitude in the frequency domain of the measured signals by sensor 2 for the healthy (continuous lines) and damaged structure (dash-dotted lines), considering as input a 5 cycles tone burst at the frequency 160 kHz. There are 10 signal for each condition. 50 100 150 200 250 Frequency [kH