É realizado um estudo sobre o problema do segundo espectro de frequências no modelo de Timoshenko para uma viga bi-apoiada. A equação característica, associada às condições de contorno bi-apoiada, permite determinar dois autovalores que dependem de um inteiro n não negativo, sendo que o de menor módulo está associado ao valor predito pela teoria de Euler-Bernoulli, e o de maior módulo é usualmente referido como sendo do segundo espectro. Este último espectro foi salientado por Traill-Nash e Collar (1953) e desconsiderado por Timoshenko, que utilizou funções trigonométricas que satisfazem as condições de contorno para obter os modos de vibração. Neste trabalho, o modelo de Timoshenko é formulado matricialmente e permite que o estudo dos autovalores e autofunções seja realizado com o uso de uma base da equação modal matricial de segunda ordem completa, gerada por uma solução matricial fundamental. Devido a uma representação analítica desta solução fundamental, o estudo é realizado com o uso da solução de um problema de valor inicial escalar, cujo comportamento torna-se completamente oscilatório acima de um valor crítico. São obtidos resultados que permitem classificar os autovalores como simples e duplos, em ambos os espectros, e determinar seus correspondentes modos. Observa-se que autovalores que correspondem a dois inteiros diferentes e localizados em espectros distintos, porém relativamente próximos, estão associados a modos de vibração descritos por funções trigonométricas que apresentam número de nós bastante diferente. / A study about the problem of the second spectrum of frequencies that arises in the Timoshenko model for a bi-supported beam is accomplished . The characteristic equation associated with the bi-supported boundary conditions allows to determine two eigenvalues that depend of a non-negative integer n, being that the eigenvalue with smaller module is associated to the value predicted by theory of Euler-Bernoulli, and the one with greater module is usually referred as belonging to the second spectrum. This latter spectrum was emphasized by Traill-Nash and Collar (1953) but disregarded by Timoshenko who used trigonometric functions that satisfy the boundary conditions in order to obtain the vibration modes. In this work the model of Timoshenko is formulated in matrix terms and allows that the study of the eigenvalues and eigenfunctions be performed using a basis of a complete second order matrix modal equation, generated by a fundamental matrix solution. Due to an analytical representation of this fundamental solution, the study is done with the solution of a scalar initial value problem, whose behavior becomes completely oscillatory above a critical value. Results are obtained in such a way that allow to classify the eigenvalues as simple and double in both spectra and to determine their corresponding modes. It is observed that eigenvalues that correspond to different integers and localized on distinct spectra, but relatively close together, are associated with vibration modes described by trigonometric functions which have quite different number of nodes.

Este trabalho faz uma pesquisa das vibrações de uma viga elástica não linear de Timoshenko sobre a influência de força axial e com uso do método espectral de Galerkin. O modelo não-linear de Timoshenko é obtido através do principio estendido de Hamilton. A função de energia é derivada de maneira geral, incluindo o caso linear, e com identificação das condições de contorno de natureza conservativa. A determinação das autofunções do sistema linear é realizada através de um problema de autovalor descrito por uma equação diferencial linear de segunda ordem com coeficientes matriciais que dependem não - linearmente no autovalor. A ortogonalidade das autofunções é obtida para as condições de contorno clássicas. As correspondentes autofunções são obtidas na base de Euler e na base gerada pela função matricial de Green de valor inicial. O método de Galerkin foi formulado matricialmente e a existência e unicidade foi obtida para uma viga bi-apoiada com o uso da função da energia e dados regulares. / This work investigates the vibrations of a nonlinear elastic Timoshenko beam, subject to an axial force, using the spectral method of Galerkin. The nonlinear model of Timoshenko is obtained through extended Hamilton s principle. The energy function is derived in a general form, including the linear case, and with identi cation of the boundary conditions of conservative nature. The determination of the eigenfunctions of the linear system is done through an eigenvalue problem described by a second-order di erential equation with matrix coe cients that depend non-linearly upon the eigenvalue. The orthogonality of the eigenfunctions is obtained for classical boundary conditions. The corresponding eigenfunctions are determined in the Euler s base and with the base generated by the initial value matrix Green function. The method of Galerkin was formulated in matrix terms and the existence and uniqueness of solutions was obtained for the bi-supported beam with use of the function of the energy and smooth data.

O objetivo deste trabalho é a decomposição da resposta forçada de um sistema distribuído de quarta ordeM no tempo, representado pela equação de Timoshenko para vigas. O modelo em estudo considera os efeitos da inércia rotativa e do cisalhamento. E introduzida uma base dinâmica para a obtenção dos modos de vibração, considerando-se determinadas condições iniciais e de contorno. Desenvolvese uma metodologia parasa obtenção da resposta dinâmica. A resposta forçada e caracterizada como uma decomposição da resposta permanente e da resposta livre, que, por sua vez, é induzida pela resposta permanente. Realiza-se o cálculo da resposta forçada para entradas do tipo concentrado e dinâmico. Realizam-se simulações para os modos, para a resposta impulso e para vibrações forçadas em vigas com extremidades apoiada e livre, extremidades fixa e livre e extremidades livre deslizante sujeitas a cargas concentradas e com dinâmica. A análise da resposta freqüência é ilustrada com uma viga fixa-livre. / The goal of this work is the decomposition of the forced response of a fourth order time distributed system, described by Timoshenko model for beams.The model in study considers the effects of rotary inertia and shear deformation. A dynamic basis is introduced for obtaining the vibration modes and for considering initial and boundary conditions. It is developed a methodology for obtaining the dynamidal response. The forced response is characterized as a composition of the permanent and free responses. The free response is induced for the permanent response. The computation of the forced response is done for concentrated and dynamical inputs. Sirnulations were done for cornputing the modes, the impulse response and forced vibrations for supported-free, fixed-free and free-sliding beams under concentrated loads and loads mlth dynamics. Frequency response analysis is illustrated with a fixed-free beam.

En el presente trabajo, se estudia el problema de la estabilidad lineal para un sistema de Timoshenko. Este problema consiste en mostrar que el tipo de un semigrupo es igual a la cota superior del espectro asociado. Esta propiedad no se verifica en general para todo semigrupo, como es de conocimiento en las bibliografías especializadas. Esta es una propiedad que siempre es válida en espacios de dimensión finita. En dimensión infinita, el problema en general es un problema abierto. Esto es, se desconocen las propiedades que debe satisfacer un semigrupo para que la estabilidad lineal se verifique. En este trabajo se demuestra que esta propiedad es vàlida para el sistema de Timoshenko con disipación friccional, independientemente de las condiciones de frontera en las que el sistema esté subordinado. Este resultado, generaliza el resultado de Racke y Rivera. Palabras Clave: Semigrupos, Espacios de Sobolev, Problema de Cauchy, Estabilidad Polinomial, Estabilidad Lineal. / --- In this thesis we stude the linear stability of the Timoshenko system. This problem consist in to show that the type of the semigroup is equals to the upper bound of the spectrum of the infinitesimal generator. This property is not true in general as was showed by Pazy and in differents international papers. This property is always valid in finite dimensional spaces. In infinite dimensional spaces this problem is open. That is to say it is not known the necessary and sufficient condition that a semigroup must verify in order to get the linear staibility. In this thesis we will show that the linear stability holds to Timoshenko system with fricctional dissipation, no matter the boundary condition the system verifies. This result improve the result obtained by Racke and Rivera. Keywords: Semigroups, Sobolev Spaces, Cauchy Problem, Polinomial Stability, Linear Stability. / Tesis

No description available.

O principal objetivo deste trabalho é realizar a análise estrutural de edifícios de andares múltiplos que apresentam núcleos resistentes, considerando a deformação pelo esforço cortante nos mesmos, bem como nos pilares. Para atingir esse objetivo, será preciso que o comportamento à flexão dos elementos verticais de contraventamento passe a ser regido pela teoria de barras de Timoshenko e não mais pela de Euler-Bernoulli. Foram então desenvolvidos algoritmos que, utilizando o Método dos Elementos Finitos (MEF), permitem calcular os fatores de forma de quaisquer seções transversais abertas de paredes delgadas pertencentes a núcleos estruturais, bem como a distribuição da tensão de cisalhamento na seção transversa em função do esforço cortante atuante. As alterações acima descritas foram feitas em um programa de análise de edifícios denominado CEASO 01, de autoria de MATIAS JR (1997). Embora esse programa realize análise não-linear geométrica, a consideração da deformação por cortante só foi implementada na análise linear. Apresentam-se, ao final, exemplos numéricos que permitem avaliar a influência da deformação pelo esforço cortante sobre os deslocamentos e esforços de núcleos resistentes e pilares. / The main aim of this work is to perform structural analysis of multistory buildings with resistant cores, taking into account shear deformation in those elements, as well as in columns. To achieve this objective, the flexural behaviour of vertical elements must be governed by Timoshenko beam theory, rather than the Euler-Bernoulli theory. Procedures using the finite element method (FEM) were developped, which enable to evaluate shear correction factors of generic thin-walled open sections and shear stress distribution as a function of the shear resultant. Changes described above were made in a structural analysis program named CEASO 01, whose author is MATIAS JR (1997). Even though this program is able to perform nonlinear analysis, only in linear analysis the effect of shear deformation is taken into account. Numerical examples are provided, which enable to evaluate the influence of taking into account shear deformation on displacements and stress resultants of resistant cores and columns.

Made available in DSpace on 2015-05-15T11:46:12Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 804145 bytes, checksum: 80e03dda4533d0039f991cc668ca2c87 (MD5) Previous issue date: 2012-08-10 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We consider a dynamical one-dimensional nonlinear Von Kármán model for beams depending on the parameter " > 0 and we study their asymptotic behavior for t large, when " ! 0. Introducing appropriate damping mechanisms we show that the energy of solutions of the corresponding damped models decay exponential uniform with respect to the parameter ". In order for this to be true the damping mechanism has to have the appropriate scale with respect to ". In the limit as " ! 0 we obtain damped Berger- Timoshenko beam model for which the energy tends exponentially to zero. This is done both in the case of internal and boundary damping . / Consideramos a dinâmica unidimensional não linear do modelo de Von Kármán para vigas dependendo de um parâmetro " > 0, e estudamos o seu comportamento assintótico para t grande, quando " ! 0. Introduzindo mecanismos adequados de amortecimento, mostramos que a energia de soluções do correspondente modelo amortecido possui decaimento exponencial uniforme com respeito ao parâmetro ". Afim de que seja verdadeiro, o mecanismo de amortecimento tem que ter a escala apropriada em relação a ". No limite, quando " ! 0 obtemos o modelo de Berger-Timoshenko para viga amortecida, bem como quando a energia tende exponencialmente para zero. Isso é feito tanto no caso de amortecimento interno e na fronteira.

Ce travail de recherche est dédié à la simulation de la réponse transitoire des assemblages de poutres soumis à des chocs. De tels chargements entraînent la propagation d’ondes haute fréquence dans l’ensemble de la structure. L’énergie qu’elles transportent peut être dommageable pour son fonctionnement ou celui des équipements embarqués. Dans des études précédentes, il a été observé sur des structures expérimentales qu’un régime vibratoire diffusif tend à s’installer pour des temps longs. Le but de cette étude est donc de développer un modèle robuste de la réponse transitoire des assemblages de poutres soumis à des chocs permettant de simuler, entre autres, cet état diffusif. Les champs de déplacement étant très oscillants et la densité modale élevée, la simulation numérique de la réponse transitoire à des chocs peut difficilement être menée par une méthode d’éléments finis classique. Une approche utilisant un estimateur de la densité d’énergie de chaque mode de propagation a donc été mise en œuvre. Elle permet d’accéder à des informations locales sur les états vibratoires, et de contourner certaines limitations intrinsèques aux longueurs d’onde courtes. Après avoir comparé plusieurs modèles de réduction cinématique de poutre à un modèle de Lamb de propagation dans un guide d’ondes circulaire, la cinématique de Timoshenko a été retenue afin de modéliser le comportement mécanique haute fréquence des poutres. En utilisant ce modèle dans le cadre de l’approche énergétique évoquée plus haut, deux groupes de modes de propagation de la densité d’énergie vibratoire dans une poutre ont été isolés : des modes longitudinaux regroupant un mode de compression et des modes de flexion, et des modes transversaux regroupant des modes de cisaillement et un mode de torsion. Il peut être également montré que l’´evolution en temps des densités d’énergie associées obéit à des lois de transport. Pour des assemblages de poutres, les phénomènes de réflexion/transmission aux jonctions ont du être pris en compte. Les opérateurs permettant de les décrire en termes de flux d’´energie ont été obtenus grâce aux équations de continuité des déplacements et des efforts aux jonctions. Quelques caractéristiques typiques d’un régime haute fréquence ont été mises en évidence, tel que le découplage entre les modes de rotation et les modes de translation. En revanche, les champs de densité d’énergie sont quant à eux discontinus aux jonctions. Une méthode d’éléments finis discontinus a donc été développée afin de les simuler numériquement comme solutions d’´equations de transport. Si l’on souhaite atteindre le régime diffusif aux temps longs, le schéma numérique doit être peu dissipatif et peu dispersif. La discrétisation spatiale a été faite avec des fonctions d’approximation de type spectrales, et l’intégration temporelle avec des schémas de Runge-Kutta d’ordre élevé du type ”strong stability preserving”. Les simulations numériques ont donné des résultats concluants car elles permettent d’exhiber le régime de diffusion. Il a été remarqué qu’il existait en fait deux limites diffusives différentes : (i) la diffusion spatiale de l’´energie sur l’ensemble de la structure, et (ii) l’équirépartition des densités d’énergie entre les différents modes de propagation. Enfin, une technique de renversement temporel a été développée. Elle pourra être utile dans de futurs travaux sur le contrôle non destructif des assemblages complexes et de grandes tailles. / This research is dedicated to the simulation of the transient response of beam trusses under impulse loads. The latter lead to the propagation of high-frequency waves in such built up structures. In the aerospace industry, that phenomenon may penalize the functioning of the structures or the equipments attached to them on account of the vibrational energy carried by the waves. It is also observed experimentally that high-frequency wave propagation evolves into a diffusive vibrational state at late times. The goal of this study is then to develop a robust model of high-frequency wave propagation within three-dimensional beam trusses in order to be able to recover, for example, this diffusion regime. On account of the small wavelengths and the high modal density, the modelling of high-frequency wave propagation is hardly feasible by classical finite elements or other methods describing the displacement fields directly. Thus, an approach dealing with the evolution of an estimator of the energy density of each propagating mode in a Timoshenko beam has been used. It provides information on the local behavior of the structures while avoiding some limitations related to the small wavelengths of high-frequency waves. After a comparison between some reduced-order beam kinematics and the Lamb model of wave propagation in a circular waveguide, the Timoshenko kinematics has been selected for the mechanical modelling of the beams. It may be shown that the energy densities of the propagating modes in a Timoshenko beam obey transport equations. Two groups of energy modes have been isolated: the longitudinal group that gathers the compressional and the bending energetic modes, and the transverse group that gathers the shear and torsional energetic modes. The reflection/transmission phenomena taking place at the junctions between beams have also been investigated. For this purpose, the power flow reflection/transmission operators have been derived from the continuity of the displacements and efforts at the junctions. Some characteristic features of a high-frequency behavior at beam junctions have been highlighted such as the decoupling between the rotational and translational motions. It is also observed that the energy densities are discontinuous at the junctions on account of the power flow reflection/transmission phenomena. Thus a discontinuous finite element method has been implemented, in order to solve the transport equations they satisfy. The numerical scheme has to be weakly dissipative and dispersive in order to exhibit the aforementioned diffusive regime arising at late times. That is the reason why spectral-like approximation functions for spatial discretization, and strong-stability preserving Runge-Kutta schemes for time integration have been used. Numerical simulations give satisfactory results because they indeed highlight the outbreak of such a diffusion state. The latter is characterized by the following: (i) the spatial spread of the energy over the truss, and (ii) the equipartition of the energy between the different modes. The last part of the thesis has been devoted to the development of a time reversal processing, that could be useful for future works on structural health monitoring of complex, multi-bay trusses.

O principal objetivo deste trabalho é realizar a análise estrutural de edifícios de andares múltiplos que apresentam núcleos resistentes, considerando a deformação pelo esforço cortante nos mesmos, bem como nos pilares. Para atingir esse objetivo, será preciso que o comportamento à flexão dos elementos verticais de contraventamento passe a ser regido pela teoria de barras de Timoshenko e não mais pela de Euler-Bernoulli. Foram então desenvolvidos algoritmos que, utilizando o Método dos Elementos Finitos (MEF), permitem calcular os fatores de forma de quaisquer seções transversais abertas de paredes delgadas pertencentes a núcleos estruturais, bem como a distribuição da tensão de cisalhamento na seção transversa em função do esforço cortante atuante. As alterações acima descritas foram feitas em um programa de análise de edifícios denominado CEASO 01, de autoria de MATIAS JR (1997). Embora esse programa realize análise não-linear geométrica, a consideração da deformação por cortante só foi implementada na análise linear. Apresentam-se, ao final, exemplos numéricos que permitem avaliar a influência da deformação pelo esforço cortante sobre os deslocamentos e esforços de núcleos resistentes e pilares. / The main aim of this work is to perform structural analysis of multistory buildings with resistant cores, taking into account shear deformation in those elements, as well as in columns. To achieve this objective, the flexural behaviour of vertical elements must be governed by Timoshenko beam theory, rather than the Euler-Bernoulli theory. Procedures using the finite element method (FEM) were developped, which enable to evaluate shear correction factors of generic thin-walled open sections and shear stress distribution as a function of the shear resultant. Changes described above were made in a structural analysis program named CEASO 01, whose author is MATIAS JR (1997). Even though this program is able to perform nonlinear analysis, only in linear analysis the effect of shear deformation is taken into account. Numerical examples are provided, which enable to evaluate the influence of taking into account shear deformation on displacements and stress resultants of resistant cores and columns.

We discuss a systematic methodology that leads to the reconstruction of the material profile of either single, or assemblies of one-dimensional flexural components endowed with Timoshenko-theory assumptions. The probed structures are subjected to user-specified transient excitations: we use the complete waveforms, recorded directly in the time-domain at only a few measurement stations, to drive the profile reconstruction using a partial-differential-equation-constrained optimization approach. We discuss the solution of the ensuing state, adjoint, and control problems, and the alleviation of profile multiplicity by means of either Tikhonov or Total Variation regularization. We report on numerical experiments using synthetic data that show satisfactory reconstruction of a variety of profiles, including smoothly and sharply varying profiles, as well as profiles exhibiting localized discontinuities. The method is well suited for imaging structures for condition assessment purposes, and can handle either diffusive or localized damage without need for a reference undamaged state. / text