In the past few years a new class of layered piezoceramic and piezoceramic-composite actuators, known as RAINBOW and GRAPHBOW, respectively, that are capable of achieving 100 times greater out-of-plane displacements than previously available has been developed. Prior to the development of RAINBOW and GRAPHBOW, large stacks of piezoelectric actuators, requiring complicated electronic drive circuits, were necessary to achieve the displacement now possible through the use of a single RAINBOW actuator. The major issues with both RAINBOW and GRAPHBOW are the prediction of their room-temperature shapes after processing, and their deformation response under application of electric field. In this research, a methodology for predicting the manufactured shapes of rectangular and disk-style RAINBOW and GRAPHBOW is developed. All of the predictive analyses developed are based on finding approximate displacement responses that minimize the total potential energy of the devices through the use of variational methods and the Rayleigh-Ritz technique. These analyses are based on classical layered plate theory and assumed the various layers exhibited linear elastic, temperature-independent behavior. Geometric nonlinearities are important and are included in the strain-displacement relations. Stability of the predicted shapes is determined by examining the second variation of the total potential energy. These models are easily modified to account for the deformations induced by actuation of the piezoceramic. The results indicate that for a given set of material properties, rectangular RAINBOW can have critical values of sidelength-to-thickness ratio (Lx/H or Ly/H) below which RAINBOW exhibits unique, or single-valued, spherical or domed shapes when cooled from the processing temperature to room temperature. For values of sidelength-to-thickness ratio greater than the critical value, RAINBOW exhibits multiple room-temperature shapes. Two of the shapes are stable and are, in general, near-cylindrical. The third shape is spherical and is unstable. Similarly, disk-style RAINBOW can have critical values of radius-to-thickness ratios (R/H) below which RAINBOW exhibits axisymmetric room-temperature shapes. For values of R/H greater than the critical value, disk-style RAINBOW exhibits two stable near-cylindrical shapes and one unstable axisymmetric shape. Moreover, it is found that for the set of material properties used in this study, the optimal reduced layer thickness would be at 55%, since the maximum change in curvature is achieved under the application of an electric field, while the relationship between the change in curvatures and the electric field is kept very close to being linear. In general, good agreement is found for comparisons between the predicted and manufactured shapes of RAINBOW. A multi-step thermoelastic analysis is developed to model the addition of the fiber-reinforced composite layer to RAINBOW to make GRAPHBOW. Results obtained for rectangular RAINBOW indicate that if the bifurcation temperature in the temperature-curvature relation is lower than the composite cure temperature, then a unique stable GRAPHBOW shape can be obtained. If the RAINBOW bifurcation temperature is above the composite cure temperature, multiple room-temperature GRAPHBOW shapes are obtained and saddle-node bifurcations can be encountered during the cooling to room temperature of [0°/RAINBOW], [RAINBOW/0o], and [0o2/RAINBOW]. Rectangular [RAINBOW/0o/90o] seems to be less likely to encounter saddle-node bifurcations. Furthermore, the unstable spherical RAINBOW configuration is converted to a stable near-cylindrical configuration. For the case considered of disk-style GRAPHBOW, three stable room-temperature shapes are obtained and the unstable axisymmetric RAINBOW configuration is also converted to a stable near-cylindrical configuration. For both rectangular and disk-style GRAPHBOW, the relationship between the major curvature and the electric field is shown to be very close to being linear. This characteristic would aid any dynamic analysis of RAINBOW or GRAPHBOW. / Ph. D.

This research discusses the mechanical characteristics of laminated piezoelectric actuators that are manufactured at an elevated temperature, to cure the adhesive bonding the layers together, or to cure the layers made of polymeric composite material, and then cooled to a service temperature. Mainly discussed are actuators that are composed of layers of passive materials and a layer of piezoelectric material. THUNDER (THin layer UNimorph ferroelectric DrivER and sensor) and LIPCA (LIghtweight Piezo-composite Curved Actuator) actuators, which consist of layers of metal, adhesive and piezoelectric material, and carbon-epoxy, glass-epoxy and piezoelectric material, respectively, are studied and investigated in detail to understand the thermal effects due to the elevated manufacturing temperature. Owing to the large out-of-plane deformations of the THUNDER actuators as a result of cooling to the service temperature, inclusion of geometric nonlinearities in the kinematic relations is taken into consideration for prediction of the thermally-induced deformations and residual stresses. The deformations and residual stresses are predicted by using a 23-term Rayleigh-Ritz approach and more rigorous, time-consuming, finite-element analyses performed with ABAQUS. The thermally-induced deformations of THUNDER actuators can result in multiple room-temperature manufactured shapes, whereas those of LIPCA actuators (LIPCA-C1 and LIPCA-C2) exhibit single room-temperature manufactured shape. Actuation responses of these actuators caused by a quasi-static electric field applied to the piezoelectric layer are also studied with the Rayleigh-Ritz approach. It is shown that geometrical nonlinearities play an important role in the actuation responses, and these nonlinearities can be controlled by the choice of actuator geometry and the materials in the passive layers. In addition, blocking forces representing load-carrying capability of THUNDER and LIPCA actuators are determined. Support conditions and again geometrical nonlinearities are vital factor in load-resisting performances. Amongst the actuators considered, the actuated deflection and blocking forces are compared. Finally, based on the outcome of this study, new criteria for designing a new type of laminated piezoelectric actuators with improvement of performance characteristics are proposed. / Ph. D.

Even with the advent of efficient, faster computers, researchers are often forced to scale down some of their problems, or pay a high price on huge computer resources for solving them. This has led to an intense effort directed towards developing efficient and rapid analysis procedures. This scenario is also true for many research-oriented finite element codes which have a limited element library or have a problem-size limit because of the use of a memory-resident equation solver. This study focuses on the development of a simple and accurate global/local method for calculating the static response of simply-supported, laminated composite panels, with cutouts and cracks. The approach primarily involves two steps. In the first step a global approach, the Rayleigh-Ritz method, is used to calculate the response of the structure. Displacement based Ritz functions for the plate without the cutout are augmented with a perturbation function. which is accurate for uniform thickness isotropic plates only, to account for the cutout. The Ritz solution does not accurately satisfy the natural boundary conditions at the cutout boundary. Therefore, a second step, local in nature is taken in which a small area in the vicinity of the hole and encompassing other points of singularities is discretized using a fine finite element mesh. The displacement boundary conditions for the local region are obtained from the global Ritz analysis. The chosen perturbation function is reliable only for circular cutout in uniform plates, therefore elliptical cutouts were suitably transformed to circular shapes using conformal mapping. Cracks are modeled as elliptical cutouts with high aspect ratios. The methodology is also extended to the analysis of composite plates. Composite structures have very high structural efficiency due to the high values of stiffness and fiber strength. But the design and analysis of composite structures is much more complicated than that of metallic structures because of the presence of highly directional properties which induces anisotropy in the mechanical characteristics and lead to a myriad of failure modes. Behavior of these structures in the presence of various discontinuities, particularly cutouts and cracks. could be even more complicated. Global/local analysis of composites with cracks and cutouts proved to be very useful and more efficient. The proposed approach resulted in considerable savings in both computational times and data storage without sacrificing the accuracy in the prediction of stresses. Results are compared with adaptive meshing technique, and it is seen that global/local analysis performed equally well. The method is very useful in design and optimization where a large number of analyses have to be performed. / Ph. D.

A pressurized arch-shell structural component made of flexible material is considered. The component is inflated with high internal pressure. The behavior of similar types of structures, such as a pair of leaning pressurized arches and pressurized arch-supported membrane shelters, has been investigated in the past. More recently, several types of pressurized structures have been incorporated as part of the framework for a variety of structural systems. Particularly, the U.S. Army has been investigating the use of large lightweight and transportable pressurized arch-shell structures to be used as maintenance shelters for vehicles, helicopters, and airplanes. The formulated equations using thin shell theory are applied to a pressurized arch-shell component. A numerical investigation based on the Rayleigh-Ritz method is utilized to determine the behavior of arch-shells under various types of loading. The types of loading include a uniformly distributed vertical load representing snow, a wind load, and a horizontal side load distributed along the arc length. Deflections, stress resultants, and moments at various locations are computed for two types of shapes: circular and non-circular arch-shells. / Master of Science

[pt] Algumas máquinas produzem solicitações dinâmicas que são transferidas às fundações por meio de movimentos vibratórios. É necessário analisar e estudar esses movimentos vibratórios durante o projeto da fundação para evitar danos aos equipamentos, à própria fundação e às estruturas vizinhas. Este tipo de analise também é muito importante para a segurança e saúde ocupacional de pessoas que venham a frequentar tais estruturas. Por esta razão as frequências naturais de vibração são importantes para o dimensionamento dinâmico de fundações, para se evitar efeitos de ressonância e amplificação de deslocamentos. Neste trabalho, estacas embutidas em fundações elásticas de Winkler foram estudadas. As vibrações longitudinais e transversais foram investigadas. As frequências naturais foram calculadas pelo método de Rayleigh-Ritz considerando estacas como elementos de barra embutidas em fundação elástica no caso da avaliação de vibrações longitudinais, e como vigas embutidas em base elástica no caso da avaliação das vibrações transversais. Na modelagem dos elementos de barra e viga utilizaram-se para aproximações dos deslocamentos funções convencionais enriquecidas com n funções adicionais. As constantes de mola de Winkler foram determinadas por provas de carga verticais e horizontais. Os resultados das vibrações longitudinais foram comparados com medições in situ com estacas reais. Os valores reais da vibração longitudinal das estacas foram obtidos pela aplicação da técnica Fast Fourier Transform nas acelerações resultantes de ensaios de carregamentos dinâmicos, que impõem uma vibração forçada no sistema estaca-solo. / [en] Some machines create dynamic efforts that are transferred to foundations by vibratory movements. Is necessary to analyze and study these movements during foundation design in order avoid damages in equipment, in foundation and in neighbor structures. This analysis is also important to safety and health of persons. For this reason natural vibration frequencies are important to dynamic design of foundations, to avoid resonance effects and displacements amplification. In this work, piles embedded in Winkler elastic foundations were studied. The longitudinal and transversal vibrations were investigated. The natural frequencies are calculated by Rayleigh-Ritz method modelling piles as bars embedded in elastic foundations in the case of longitudinal vibration evaluation, and beams embedded in elastic foundations in the case of transversal vibration evaluation. The bars and beams elements are modeled using conventional shape function enriched with n additional functions. The Winkler spring constants were evaluated by vertical and horizontal load tests. The longitudinal vibration results were compared with in situ measurements in real piles. The longitudinal vibration real values of piles were obtained by Fast Fourier Transform in acceleration data of dynamic load tests that impose a forced vibration in soil-pile system.

[pt] O presente trabalho mostra uma versão do método Rayleigh-Ritz com funções especializadas para a análise de placas circulares e anulares finas e espessas sujeitas a cargas fora do plano e em plano. As funções de aproximação para deslocamentos são polinómios em direção radial combinada com funções trigonométricas na direção circunferencial. Um recurso conveniente é o uso de funções nodais lineares, que permitem a fácil consideração de cargas nodais e condições de contorno (incluindo forças seguidoras), enriquecidos por polinômios de ordem superior, sem inclusão de nós adicionais. O modelo permite a variação da espessura e é aplicado em MAPLE18, possibilitando o cálculo de deslocamentos e tensões sob carregamento constante e de variação linear, as frequências de vibração, cargas de flambagem com alguns efeitos do nível de carga conservativa e não conservativa. Os exemplos mostram a eficácia desta abordagem na análise de tal estrutura e leva um novo enfoque a este problema clássico, que apresenta comparações interessantes e originais que descrevem o efeito de deformação de cisalhamento, no caso de vibrações o efeito das rotações inerciais e variação de espessura em placas circulares e anulares, incluindo deslocamentos, momentos e forças de cisalhamento, frequências de vibração, cargas de flambagem e uma análise de cargas seguidoras tangenciais não conservativas na estabilidade, utilizando o critério dinâmico é executada. / [en] The present work shows a version of the Rayleigh-Ritz method with specialized functions for the analysis of thin and thick circular and annular plates subjected to out-of-plane and in-plane loads. The approximation functions for displacements are polynomials in the radial direction combined with trigonometric functions in the circumferential direction. A convenient feature is the use of linear nodal functions, which allows for easy consideration of nodal loads and boundary conditions (including follower forces), enriched by higher order polynomials without inclusion of additional nodes. The model allows for thickness variation and was implemented in MAPLE18, enabling the calculation of displacements and stresses under constant and linearly varying load, frequencies of vibration, buckling loads with a few commands and the effect of the level of conservative and non-conservative on load the stability. The examples show the effectiveness of this approach in the analysis of such structures and bring new light to this classical problem, presenting interesting and novel comparisons illustrating the effect of shear deformation, in case of vibrations of the inertial rotations analysis and thickness variation in circular and annular plates, including displacements, moments and shear forces, vibration frequencies, buckling loads and a stability analysis of non-conservative tangential follower loads, using the dynamic criterion is performed.

[pt] Este estudo apresenta um método computacional para modelar campos eletromagnéticos em guias de onda retangulares curvados com seção transversal uniforme, usando o método variacional de Rayleigh-Ritz. Potenciais aplicações desta pesquisa em engenharia incluem o projeto de alimentadores para antenas, conversores de modais na faixa de micro-ondas, filtros, entre outros. Embora vários modelos tenham sido propostos para resolver este problema, as técnicas numéricas convencionais baseadas em elementos finitos, diferenças finitas e volumes finitos requerem altos custos computacionais. Para superar esses problemas, foi desenvolvida uma formulação variacional para resolver as equações de Maxwell em um sistema de coordenadas toroidal local, por meio de um novo funcional introduzido neste trabalho. O funcional foi adaptado para domínios uniformemente curvados com seção transversal arbitrária, e investigações analíticas foram conduzidas para confirmar suas características estacionárias. O formalismo Rayleigh-Ritz foi utilizado para converter o funcional em um problema equivalente de autovalores e autovetores, usando uma expansão em harmônicas retangulares de um guia de onda reto como funções de base para modelar um guia de onda retangular curvo. Um algoritmo numérico foi desenvolvido em Matlab para validar nosso modelo, e os resultados foram comparados com soluções perturbacionais e numéricas de referência, demonstrando alta precisão e menor custo computacional. / [en] This study presents a computational method for modeling electromagnetic fields in curved rectangular waveguides with uniform cross-section, using the variational Rayleigh-Ritz method. The potential applications of this research in engineering include the design of feeders for antennas, microwave mode converter devices, filters, among others. While various models have been proposed to solve this problem, conventional numerical techniques based on finite elements, finite differences, and finite volumes require high computational costs. To overcome these issues, a variational formulation for solving Maxwell s equations in a local toroidal coordinate system was developed via a novel functional introduced in this work. The functional was adapted to handle uniformly bend domains with arbitrary cross-section, and analytical investigations were conducted to confirm its stationary characteristics. The Rayleigh-Ritz formalism was employed to convert the functional into an equivalent problem of eigenvalues and eigenvectors using an expansion in terms of rectangular harmonics of a straight waveguide as basis functions for modeling a bend rectangular waveguide. A numerical algorithm was developed in Matlab to validate our model, and the results were compared against reference perturbational and numerical solutions, demonstrating high accuracy and lower computational costs.

This thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis.

Orientador: Jose Maria Campos dos Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-07-31T20:57:31Z (GMT). No. of bitstreams: 1 Ferraz_FabioGuilherme_M.pdf: 9315977 bytes, checksum: 2c619b06df862551ab474095f22421cc (MD5) Previous issue date: 2001 / Resumo: Este trabalho explora o uso de métodos de Síntese Modal dos Componentes (SMC), a Teoria da Perturbação por Mínimo Rank (MRPT) e o Algoritmo de Realização de vetores de Ritz (RRA) como uma ferramenta para a detecção de danos estruturais. A SMC consiste em modelar separadamente componentes individuais de uma estrutura e então acoplá-Ios num sistema único. Os métodos de SMC podem ser generalizados para se permitir o uso de outros vetores, tais como os vetores de Ritz em substituição aos modos normais. Os vetores de Ritz são obtidos a partir de relações de recorrência, e representam uma ótima base de vetores como alternativa aos modos naturais de vibrar dos componentes. O MRPT é um método de detecção de danos baseado em modelo, que usa o fato de que um dano discreto pode ser manifestado num modelo de elementos finitos (MEF) como uma pequena perturbação no rank das matrizes que compõe as propriedades da estrutura. O RRA converte as matrizes em tempo discreto identificadas pelo algoritmo ERA para o tempo contínuo, a partir das quais pode-se extrair os vetores de Ritz experimentais usados para se complementar a matriz de forças dinâmicas residuais na formulação do MRPT. Através do modelo sem danos sintetizado, e do modelo medido com danos, o MRPT com RRA é usado para se detectar a localização do dano e sua extensão, ou no mínimo o componente que contém o dano. Diferentes considerações de modelagem subestruturada são exploradas na detecção de danos usando exemplos numéricos. Em particular um MEF de vigas da estação espacial internacional é implementado e os efeitos de danos localizados em diferentes componentes são investigados / Abstract: This work explores the use of Component Mode Synthesis methods (CMS), Minimum Rank Perturbation Theory (MRPT), and Ritz Realization AIgorithm (RRA) as a structural damage detection tooI. The CMS consists in to modeling individual components of a structure separately and then to couple them to form an assembled system. CMS methods can be generalized to allow the use of other vectors, like Ritz vectors, rather than normal modes. The Ritz vectors are obtained from recurrence relations, and represent a suitable basis vectors as an alternative to component vibration modes. MRPT is a model-based damage detection method, which utilizes the fact that discrete damage is manifested in a structural finite element model (FEM) as a low rank perturbation to the structural property matrices. RRA converts the discrete time system matrices identified by Eigensystem Realization AIgorithm (ERA) to continuous time system matrices, from which one can extract the experimental Ritz vectors used to complement the matrix of dynamic residual forces in the MRPT formulation. With the coupled healthy model and the measured damaged model, the MRPT with RRA is used to detect the damage location and his extent, or at least the component that contains the damage. Different substructure modeling assumptions on damage detectability are explored using numerical examples. In particular a beam FEM of the international space station structure is implemented and the effects of different localized component damages are investigated. / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

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