Curved beams such as arches find ubiquitous applications in civil, mechanical and aerospace engineering, e.g., stiffened floors, fuselage, railway compartments, and wind turbine blades. The analysis of free vibrations of curved structures plays a critical role in their design to avoid transient loads with dominant frequencies close to their natural frequencies. One way to increase their areas of applications and possibly make them lighter without sacrificing strength is to make them of Functionally Graded Materials (FGMs) that are composites with continuously varying material properties in one or more directions. In this thesis, we study free vibrations of FGM circular beams by using a logarithmic shear deformation theory that incorporates through-the-thickness logarithmic variation of the circumferential displacement, and does not require a shear correction factor. The radial displacement of a point is assumed to depend only upon its angular position. Thus the beam theory can be regarded as a generalization of the Timoshenko beam theory. Equations governing transient deformations of the beam are derived by using Hamilton's principle. Assuming a time harmonic variation of the displacements, and by utilizing the generalized differential quadrature method (GDQM) the free vibration problem is reduced to solving an algebraic eigenvalue problem whose solution provides frequencies and the corresponding mode shapes. Results are presented for different spatial variations of the material properties, boundary conditions, and the aspect ratio. It is found that the radial and the circumferential gradation of material properties maintains their natural frequency within that of the homogeneous beam comprised of a constituent of the FGM beam. Furthermore, keeping every other variable fixed, the change in the beam opening angle results in very close frequencies of the first two modes of vibration, a phenomenon usually called mode transition. / Master of Science / Curved and straight beams of various cross-sections are one of the simplest and most fundamental structural elements that have been extensively studied because of their ubiquitous applications in civil, mechanical, biomedical and aerospace engineering. Many attempts have been made to enhance their material properties and designs for applications in harsh environments and reduce weight. One way of accomplishing this is to combine layerwise two or more distinct materials and take advantage of their directional properties. It results in a lightweight structure having overall specific strength superior to that of its constituents. Another possibility is to have volume fractions of two or more constituents gradually vary throughout the structure for enhancing its performance under anticipated applications. Functionally graded materials (FGMs) are a class of composites whose properties gradually vary along one or more space directions. In this thesis, we have numerically studied free vibrations of FGM circular beams to enhance their application domain and possibly use them for energy harvesting.

[pt] O estudo teve como objetivo abordar o ensino das Funções Exponenciais e Logarítmicas quanto ao aspecto teórico para o fortalecimento dos conceitos. Para alcançar este objetivo, este trabalho foi feito baseado em pesquisas bibliográficas em livros, artigos, dissertações entre outras fontes. Antes de abordarmos o assunto de exponenciais explanamos sobre o conceito de funções e citamos a importância das tabelas. Abordamos o assunto de exponenciais citando algumas de suas aplicações e algumas demonstrações. Mostramos também o processo de construção da tabela de logaritmos decimais. Para fortalecer o conhecimento sobre o assunto faremos uso de calculadoras e planilhas eletrônicas em algumas atividades propostas para mostrar propriedades das funções exponenciais e logarítmicas e a importante constante matemática e, que aparece naturalmente em fenômenos da Natureza. Esperamos contribuir de forma positiva para o interesse e aprimoramento de professores no assunto explanado e, principalmente, para a motivação dos alunos em estudar e compreender melhor a importância dos logaritmos. / [en] This study aims to approach the teaching of exponential and logarithmic functions regarding the theoretical aspects to the enhancement of concepts. In order to achieve this goal, this study was based on bibliographic research, articles, and dissertations, among other sources. Before we approached the issue on exponentials, we went over the concept of functions and highlited the importance of tables. We also approached the issue of exponentials citing from some of its applications and demonstrations. We brought up, as well, the process of setting up the decimal logarithmic table. In order to enhance the knowledge on the subject, we will make use of calculators and tables so as to reinforce the knowledge over the subject in some activities proposed to demonstrate the properties of the exponential and logarithmic functions as well as the important mathematical constant e which occurs naturally in natural phenomena. We hope to contribute positively to the bringing out of interest to teachers and likewise enhancement to their knowledge on the subject studied and, specially, motivation to students generating a better understanding over the importance of logarithms.

Submitted by JÉSSICA ANTONIO DELGADO null (jessica_dellgado@hotmail.com) on 2016-08-05T13:36:45Z No. of bitstreams: 1 DISSERTACAO_FINAL0308.pdf: 4117025 bytes, checksum: 0bc23567e553d5dba25e546bbeba3e55 (MD5) / Rejected by Ana Paula Grisoto (grisotoana@reitoria.unesp.br), reason: Solicitamos que realize uma nova submissão seguindo as orientações abaixo: A data de defesa informada na capa e folha de rosto do trabalho está diferente da data que consta na folha de aprovação. Corrija estas informações e realize uma nova submissão contendo o arquivo correto. Agradecemos a compreensão. on 2016-08-09T17:15:33Z (GMT) / Submitted by JÉSSICA ANTONIO DELGADO null (jessica_dellgado@hotmail.com) on 2016-08-09T20:16:39Z No. of bitstreams: 1 DISSERTACAO_FINAL0908.pdf: 4117292 bytes, checksum: 2fa555b044c52407e59e19d3d51f664d (MD5) / Approved for entry into archive by Ana Paula Grisoto (grisotoana@reitoria.unesp.br) on 2016-08-10T14:41:21Z (GMT) No. of bitstreams: 1 delgado_ja_me_bauru.pdf: 4117292 bytes, checksum: 2fa555b044c52407e59e19d3d51f664d (MD5) / Made available in DSpace on 2016-08-10T14:41:21Z (GMT). No. of bitstreams: 1 delgado_ja_me_bauru.pdf: 4117292 bytes, checksum: 2fa555b044c52407e59e19d3d51f664d (MD5) Previous issue date: 2016-06-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Este trabalho propõe uma abordagem que utiliza uma combinação de métodos baseados na função barreira logarítmica e barreira logarítmica modificada para a resolução do problema de Fluxo de Potência Ótimo Reativo. Na abordagem proposta as restrições de desigualdade são transformadas em igualdade introduzindo as variáveis de folga, as quais são tratadas pela função barreira logarítmica ou pela função barreira logarítmica modificada e as restrições de igualdade são tratadas pelos multiplicadores de Lagrange. Estes métodos são utilizados em duas etapas. Inicialmente aplica-se o método baseado na função barreira logarítmica até que um critério de transição seja satisfeita. Depois aplica-se o método baseado na função barreira logarítmica modificada até que a condição de convergência seja satisfeita. Testes numéricos utilizando os sistemas elétricos de 3 barras e IEEE 14, 30, 57 e 118 barras indicam que o método é eficiente na resolução do problema Fluxo de Potência Ótimo Reativo. Uma comparação dos resultados obtidos pela abordagem proposta com os dos métodos de barreira logarítmica e de barreira logarítmica modificada separadamente é realizada. / This work proposes an approach that uses a combination of methods based on loga-rithmic barrier function and modi˝ed logarithmic barrier function for the resolution of the reactive optimal power ˛ow problem. In this proposed approach the inequality cons-traints are transformed into equalities by adding the slack variables, which are handled by logarithmic barrier function or modified logarithmic barrier function, and the equality constraints are handled by means of Lagrange multipliers. These methods are used in two steps. Initially, the method based on the logarithmic barrier function is applied until the stop condition is satisfied. After the method based on modified logarithmic barrier function is used until the convergence condition is satisfied. Numerical tests carried out with the 3 bus, IEEE 14, 30, 57 and 118 bus electrical systems indicate that the proposed approach is efficient in the resolution of the reactive optimal power flow problem. The results obtained by the proposed approach were compared with the logarithmic barrier method and the logarithmic barrier method modified, separately.

本文旨在探討CHRISTENSEN.L.R.,D.W.JORGENSON,AND L.J.LAU（１９７１）（１９７ ３）提出二次對數（TRANSCENDENTAL LOGARITHMIC）函數形（簡稱TRANSLOG）二次對 數函數的實證應用，我們所選擇的實證對象為在現代經濟體系，扮演至為重要角色的 金融中介－－銀行。探討的方法：先選出擁有最好先驗理論特性的函數形，作為實證 函數形；其次，搜集資料，建立模型，選定適當估計方法，進行估計。第三，依據估 得結果，做銀行成本函數分析。本文的成本論基礎為BAMOUL（１９８２）競爭市場理 論，函數形的理論則部份應用對偶理論。 所以，本文首章旨在篩選模型的函數形；我們藉由FUSS,MCFADDEN,AND MUNDLAK （１ ９７８）及 LAU（１９８６）函數形標準，比較 CD,CES,……等九種函數形，從而得 出二次對數函數為本文模型最適實證函數形。 第二章則要說明實證資料，實證模型，估計方法及程式設計；這最後頸，本文擬將全 部估計過程製成附錄，以資採信或供有意做後續研究者參考，一改久遭疵議的黑箱實 證過程。 有了估計結果，經整理，第三章即可進行成本函數效率分析，但在分析之前要先交代 分析所依據的競爭市場理論。 最後的分析結果及建議，輯成末章，此即本文。

La simulation moléculaire est un outil indispensable pour comprendre le comportement de systèmes complexes pour lesquels les expériences s'avèrent coûteuses ou irréalisables à l'heure actuelle. Cette thèse est dédiée aux aspects méthodologiques de la simulation moléculaire et comprend deux volets. Le premier volet porte sur la recherche de chemins de réaction et de points col d'une surface d'énergie potentielle. Nous proposons, dans le chaptire 3, une amélioration d'une des méthodes de cette classe, appelée '"Activation Relaxation Technique"(ART). Nous donnons également une preuve de convergence pour un algorithme prototype. Le deuxieme volet porte sur le calcul d'énergie libre pour les transitions caractérisées par une coordonnée de réaction. Nous nous plaçons dans le cadre d'une méthode d'échantillonnage d'importance adaptative, appelée 'Adaptive Biasing Force' (ABF). Ce volet comprend en soi deux sous-parties. La première partie (chapitre 5) s'attache à montrer l'applicabilité à un système biomoléculaire, d'une nouvelle mise en oeuvre parallèle d'ABF, nommée 'multiple-walker ABF' (MW-ABF), consistant à utiliser plusieurs répliques. Cette mise en oeuvre s'est avérée utile pour surmonter des problèmes liés à un mauvais choix de coordonnée de réaction. Nous confirmons ensuite ces résultats numériques en étudiant la convergence théorique d'un algorithme d'ABF adapté. Le chapitre 6 comprend une étude de convergence en temps long utilisant les méthodes d'entropie relative et les inégalités de Sobolev logarithmiques / Molecular simulation is an essential tool in understanding complex chemical and biochemical processes as real-life experiments prove increasingly costly or infeasible in practice . This thesis is devoted to methodological aspects of molecular simulation, with a particular focus on computing transition paths and their associated free energy profiles. The first part is dedicated to computational methods for reaction path and transition state searches on a potential energy surface. In Chapter 3 we propose an improvement to a widely-used transition state search method, the Activation Relaxation Technique (ART). We also present a local convergence study of a prototypical algorithm. The second part is dedicated to free energy computations. We focus in particular on an adaptive importance sampling technique, the Adaptive Biasing Force (ABF) method. The first contribution to this field, presented in Chapter 5, consists in showing the applicability to a large molecular system of a new parallel implementation, named multiple-walker ABF (MW-ABF). Numerical experiments demonstrated the robustness of MW-ABF against artefacts arising due to poorly chosen or oversimplified reaction coordinates. These numerical findings inspired a new study of the longtime convergence of the ABF method, as presented in Chapter 6. By studying a slightly modified model, we back our numerical results by showing a faster theoretical rate of convergence of ABF than was previously shown

Orientador: Manoel Ferreira Borges Neto / Banca: Antônio Luís Venezuela / Banca: Sandra Regina Monteiro Masalshiene Roveda / Resumo: O objetivo deste trabalho ée estabelecer similaridades entre a análise complexa e os quatérnios. Nele é feito um estudo da regularidade de funções quaterniônicas e são estabelecidas as funções exponencial e logarítmica para os quatérnios sendo feito um estudo da hiperpe- riodicidade dessas funções. Outro resultado apresentado é a generalização quaterniônica da fórmula integral de Cauchy um dos principais teoremas da análise complexa. / Abstract: The objective of this work is to establish similarities between the complex analysis and the quaternions. In it is made a study of the regularity of quaternionic functions and are established the exponential and logarithmic functions for the quaternions being made a study of the hiperperiodicity of these functions. Another presented result is the quater- nionic generalization of the Cauchy's integral formula one of the main theorems of the complex analysis. / Mestre

Neste trabalho, calculam-se o calor específico e a entropia do Modelo de Anderson simétrico de uma impureza usando o Grupo de Renormalização Numérico (GRN). O método é baseado na discretização logarítmica da banda de condução do metal hospedeiro a qual a impureza está acoplada. Porém, esta discretização introduz oscilações nas propriedades termodinâmicas. Esta inconveniência, inerente ao método, é contornável para a suscetibilidade magnética, mas é crítica para o calor específico, restringindo o alcance do GRN. Para sobrepor essa dificuldade, é usado o novo procedimento denominado intercalado que foi desenvolvido para o cálculo da suscetibilidade magnética de modelos de duas impurezas. Para reduzir as matrizes e o tempo computacional, é usado, também, o operador carga axial, recentemente definido no contexto do Modelo de Kondo de duas impurezas, e que é conservado pelo Hamiltoniano de Anderson simétrico. As curvas obtidas são comparadas com resultados exatos obtidos por ansatz de Bethe e pelo Modelo de Nível Ressonante. / The specific heat and the entropy of the one-impurity symmetric Anderson Model are calculated using the Numerical Renormalization Group (NRG). The heart of the method is the logarithmic discretization of the metal conduction band where the impurity is coupled. However, this discretization, inherent in the method, introduces oscillations in the thermodynamical properties. For the susceptibility it is not so critical but for the specific heat the usual calculation is prohibitive. To overcome this difficulty, we use the new procedure called interleaved that was developed to calculate the susceptibility of two-impurity models. In order to reduce the matrices and computation time, use is made of the axial charge operator recently defined in the two-impurity Kondo Model context and that is conserved by the symmetric Anderson Hamiltonian. The curves obtained are compared with exacts results of Bethe ansatz and Resonant Level Model.

Submitted by Raniere Barreto (raniere.barros@ufvjm.edu.br) on 2018-04-11T19:02:13Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) lincoln_ferreira_nunes.pdf: 15147355 bytes, checksum: 2fa1f34defc7c2b96c3429455de8add1 (MD5) / Approved for entry into archive by Rodrigo Martins Cruz (rodrigo.cruz@ufvjm.edu.br) on 2018-04-20T14:19:37Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) lincoln_ferreira_nunes.pdf: 15147355 bytes, checksum: 2fa1f34defc7c2b96c3429455de8add1 (MD5) / Made available in DSpace on 2018-04-20T14:19:37Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) lincoln_ferreira_nunes.pdf: 15147355 bytes, checksum: 2fa1f34defc7c2b96c3429455de8add1 (MD5) Previous issue date: 2017 / Este trabalho relaciona a utiliza??o da Modelagem Matem?tica no processo de ensino-aprendizagem das fun??es exponenciais e logar?tmicas nos primeiros anos do Ensino Superior, com possibilidades de aplica??o no Ensino M?dio. Inicialmente, relacionou-se a import?ncia das fun??es supracitadas na descri??o de fen?menos das ci?ncias em geral. Em seguida, ap?s an?lise sobre os livros did?ticos, detectou-se defici?ncias em rela??o ? contextualiza??o do estudo das referidas fun??es nestes livros. Para propor sequ?ncias did?ticas que auxiliassem a suprir as lacunas desta contextualiza??o utilizou-se a Modelagem Matem?tica, com aux?lio do software GeoGebra. Para analisar a utiliza??o das sequ?ncias did?ticas propostas, foi escolhida a Engenharia Did?tica como metodologia de pesquisa. Finalmente, ap?s planejadas as sequ?ncias did?ticas, as interven??es did?ticas foram realizadas, tendo como p?blico-alvo um grupo de estudantes do curso de Bacharelado em Ci?ncia e Tecnologia do Campus Mucuri da Universidade Federal dos Vales do Jequitinhonha e Mucuri. Ao final das interven??es, question?rios e relatos produzidos foram analisados, em rela??o ao cumprimento do objetivo inicial da pesquisa. Constatou-se ao final das an?lises dos resultados obtidos, um progresso expressivo dos estudantes em rela??o ? compreens?o da utiliza??o das fun??es supracitadas nas ci?ncias. / Disserta??o (Mestrado Profissional) ? Programa de P?s-Gradua??o Matem?tica, Universidade Federal dos Vales do Jequitinhonha e Mucuri, 2017. / This work relates the use of Mathematical Modeling in the teaching-learning process of exponential and logarithmic functions in College Education, With possibilities of application in High School. The importance of the above-mentioned functions in phenomenas description in general was initially related. Then, after analyzing the textbooks, deficiencies were detected in relation to the contextualization of the study of these functions in these books. In order to propose didactic sequences that helps to fulfil the gaps of this contextualization, Mathematical Modelling with the help of software GeoGebra was used. In order to analyze the use of the proposed didactic sequences, Didactic Engineering was chosen as a research methodology. Finally, after the didactic sequences were planned, the didactic interventions were carried out targeting a group of students from the Bachelor?s Degree in Science and Technology of the Mucuri Campus of the Federal University of the Jequitinhonha and Mucuri Valleys. At the end of the interventions, the questionnaires and reports produced were analyzed in relation to the accomplishment of the initial objective of the research. At the end of the analysis of the reports and questionnaires, a very significant progress was made by the students in understanding the use of these functions in the sciences.

Neste trabalho, calculam-se o calor específico e a entropia do Modelo de Anderson simétrico de uma impureza usando o Grupo de Renormalização Numérico (GRN). O método é baseado na discretização logarítmica da banda de condução do metal hospedeiro a qual a impureza está acoplada. Porém, esta discretização introduz oscilações nas propriedades termodinâmicas. Esta inconveniência, inerente ao método, é contornável para a suscetibilidade magnética, mas é crítica para o calor específico, restringindo o alcance do GRN. Para sobrepor essa dificuldade, é usado o novo procedimento denominado intercalado que foi desenvolvido para o cálculo da suscetibilidade magnética de modelos de duas impurezas. Para reduzir as matrizes e o tempo computacional, é usado, também, o operador carga axial, recentemente definido no contexto do Modelo de Kondo de duas impurezas, e que é conservado pelo Hamiltoniano de Anderson simétrico. As curvas obtidas são comparadas com resultados exatos obtidos por ansatz de Bethe e pelo Modelo de Nível Ressonante. / The specific heat and the entropy of the one-impurity symmetric Anderson Model are calculated using the Numerical Renormalization Group (NRG). The heart of the method is the logarithmic discretization of the metal conduction band where the impurity is coupled. However, this discretization, inherent in the method, introduces oscillations in the thermodynamical properties. For the susceptibility it is not so critical but for the specific heat the usual calculation is prohibitive. To overcome this difficulty, we use the new procedure called interleaved that was developed to calculate the susceptibility of two-impurity models. In order to reduce the matrices and computation time, use is made of the axial charge operator recently defined in the two-impurity Kondo Model context and that is conserved by the symmetric Anderson Hamiltonian. The curves obtained are compared with exacts results of Bethe ansatz and Resonant Level Model.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2018-08-29T14:10:49Z No. of bitstreams: 2 Dissertação - Wagner Pereira dos Santos - 2018.pdf: 1504219 bytes, checksum: 43e4aba8347b7ddf4c5ae0fb5418b1c0 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2018-08-29T14:11:32Z (GMT) No. of bitstreams: 2 Dissertação - Wagner Pereira dos Santos - 2018.pdf: 1504219 bytes, checksum: 43e4aba8347b7ddf4c5ae0fb5418b1c0 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2018-08-29T14:11:32Z (GMT). No. of bitstreams: 2 Dissertação - Wagner Pereira dos Santos - 2018.pdf: 1504219 bytes, checksum: 43e4aba8347b7ddf4c5ae0fb5418b1c0 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2018-08-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This dissertation aims to analyze and propose a new adaptation of Summaê to the IFG - Câmpus Luziânia, with the aim of reducing the high rate of reprobation in Mathematics of the first year students of the Technical Courses Integrated to High School in Buildings, Computer Science and Chemistry. Summaê is a playful game of questions and answers and is inserted in one of the active methodologies idealized by Professor Ricardo Fragelli. The adaptation of the original proposal of the Summaê was for the content of exponential and logarithmic functions, where they present different situations and natural phenomena that are part of our daily life, such as: Half-life, rates of proportional variations, compound interest and intensity of a sound. It should be emphasized that the use of games can promote different teaching situations, presenting a mathematical language as well as formal mathematical concepts, as well as associating theory and practice in the subjects addressed. / Esta dissertação objetiva analisar e propor uma nova adaptação do Summaê para o IFG - Câmpus Luziânia, com o intuito de reduzir o alto índice de reprovação em Matemática dos alunos do primeiro ano dos Cursos Técnicos Integrados ao Ensino Médio em Edificações, Informática e Química. O Summaê é um jogo lúdico de perguntas e respostas e está inserido em uma das metodologias ativas idealizadas pelo professor Ricardo Fragelli. A adaptação da proposta original do Summaê foi para o conteúdo de funções exponencial e logarítmica, onde as mesmas apresentam diversas situações e fenômenos naturais que fazem parte do nosso dia a dia, como por exemplo: Meia vida, taxas de variações proporcionais, juros compostos e intensidade de um som. Convém salientar que, o uso de jogos pode promover diversas situações de ensino, apresentando uma linguagem matemática e também conceitos matemáticos formais, e ainda, associar teoria e prática nos assuntos abordados.