Nesta dissertação são abordados os modelos estruturais de Euler-Bernoulli e de Timoshenko com o uso da teoria de semigrupos de operadores fortemente contínuos. Um estudo do espectro do modelo de Timoshenko é realizado com o uso de uma base fundamental de valor inicial para a determinação das autofunções. Uma expansão assintótica é realizada para a equação característica dos autovalores no caso de condições de contorno clássicas. / The aim of this work is to study the structural models of the Euler- Bernoulli and Timoshenko beams. At first, they are analyzed by using the tools provided by the semigroup theory of strongly continuous operators. Next,the fun- damental basis of initial value is applied to determine the eigenfunctions. Finally, the characteristics equations of the Timoshenko beams with classical boundary values are asymptotic expanded.

Neste trabalho é proposta uma abordagem analítica para o cálculo modal de diversas configurações de vigas de Euler-Bernoulli com propriedades da seção transversal contínuas sujeitas a condições de contorno clássicas e não clássicas. A metodologia proposta está baseada em condições de normalização para as soluções da equação modal, originando uma fórmula modal associada com propriedades físicas e matemáticas da viga. Para o caso de vigas monossegmentadas não é requerido o uso do computador no cálculo dos modos. As formas modais dos diversos tipos de vigas consideradas podem ser expressas em termos das funções de Bessel, funções triangulares e funções hiperbólicas. Para fins de ilustração. é apresentado um caso de rigidez fiexural e massa linear do tipo polinomial de quarta ordem. / In this work, it is proposed an analytical approach for the modal determination of various configurations of Euler-Bernoulli bearns with continuous crosssection properties constrained with classical and non-classical boundary conditions. The proposed approach is based on normal conditions for the solutions of the modal equation, yielding a modal formula associated with mathematical and physical properties of the beam. For the mono-segmented case it is not required tbe use of computer for the modal determination. The modes of the various types of beam considered can be expressed in terms of Bessel, triangular and hyperbolic functions. For illustration, it is presented a case of fourth-order polynomial-type flexural stiffness and linear mass.

Nesta dissertação são abordados os modelos estruturais de Euler-Bernoulli e de Timoshenko com o uso da teoria de semigrupos de operadores fortemente contínuos. Um estudo do espectro do modelo de Timoshenko é realizado com o uso de uma base fundamental de valor inicial para a determinação das autofunções. Uma expansão assintótica é realizada para a equação característica dos autovalores no caso de condições de contorno clássicas. / The aim of this work is to study the structural models of the Euler- Bernoulli and Timoshenko beams. At first, they are analyzed by using the tools provided by the semigroup theory of strongly continuous operators. Next,the fun- damental basis of initial value is applied to determine the eigenfunctions. Finally, the characteristics equations of the Timoshenko beams with classical boundary values are asymptotic expanded.

Neste trabalho é proposta uma abordagem analítica para o cálculo modal de diversas configurações de vigas de Euler-Bernoulli com propriedades da seção transversal contínuas sujeitas a condições de contorno clássicas e não clássicas. A metodologia proposta está baseada em condições de normalização para as soluções da equação modal, originando uma fórmula modal associada com propriedades físicas e matemáticas da viga. Para o caso de vigas monossegmentadas não é requerido o uso do computador no cálculo dos modos. As formas modais dos diversos tipos de vigas consideradas podem ser expressas em termos das funções de Bessel, funções triangulares e funções hiperbólicas. Para fins de ilustração. é apresentado um caso de rigidez fiexural e massa linear do tipo polinomial de quarta ordem. / In this work, it is proposed an analytical approach for the modal determination of various configurations of Euler-Bernoulli bearns with continuous crosssection properties constrained with classical and non-classical boundary conditions. The proposed approach is based on normal conditions for the solutions of the modal equation, yielding a modal formula associated with mathematical and physical properties of the beam. For the mono-segmented case it is not required tbe use of computer for the modal determination. The modes of the various types of beam considered can be expressed in terms of Bessel, triangular and hyperbolic functions. For illustration, it is presented a case of fourth-order polynomial-type flexural stiffness and linear mass.

Nesta dissertação são abordados os modelos estruturais de Euler-Bernoulli e de Timoshenko com o uso da teoria de semigrupos de operadores fortemente contínuos. Um estudo do espectro do modelo de Timoshenko é realizado com o uso de uma base fundamental de valor inicial para a determinação das autofunções. Uma expansão assintótica é realizada para a equação característica dos autovalores no caso de condições de contorno clássicas. / The aim of this work is to study the structural models of the Euler- Bernoulli and Timoshenko beams. At first, they are analyzed by using the tools provided by the semigroup theory of strongly continuous operators. Next,the fun- damental basis of initial value is applied to determine the eigenfunctions. Finally, the characteristics equations of the Timoshenko beams with classical boundary values are asymptotic expanded.

Neste trabalho é proposta uma abordagem analítica para o cálculo modal de diversas configurações de vigas de Euler-Bernoulli com propriedades da seção transversal contínuas sujeitas a condições de contorno clássicas e não clássicas. A metodologia proposta está baseada em condições de normalização para as soluções da equação modal, originando uma fórmula modal associada com propriedades físicas e matemáticas da viga. Para o caso de vigas monossegmentadas não é requerido o uso do computador no cálculo dos modos. As formas modais dos diversos tipos de vigas consideradas podem ser expressas em termos das funções de Bessel, funções triangulares e funções hiperbólicas. Para fins de ilustração. é apresentado um caso de rigidez fiexural e massa linear do tipo polinomial de quarta ordem. / In this work, it is proposed an analytical approach for the modal determination of various configurations of Euler-Bernoulli bearns with continuous crosssection properties constrained with classical and non-classical boundary conditions. The proposed approach is based on normal conditions for the solutions of the modal equation, yielding a modal formula associated with mathematical and physical properties of the beam. For the mono-segmented case it is not required tbe use of computer for the modal determination. The modes of the various types of beam considered can be expressed in terms of Bessel, triangular and hyperbolic functions. For illustration, it is presented a case of fourth-order polynomial-type flexural stiffness and linear mass.

Neste trabalho, procuramos resolver o problema direto da equação da viga de Euler- Bernoulli bi-engastada com condições iniciais nulas. Estudamos o problema inverso da viga, que consiste em identificar a fonte de vibração, modelada como um elemento em L2, usando como dado a velocidade de um ponto arbitrário da viga, durante um intervalo de tempo arbitrariamente pequeno. A relevância deste trabalho na Engenharia encontra-se, por exemplo, na identificação de danos estruturais em vigas. / In this work, we try to solve the direct problem of the clamped-clamped Euler- Bernoulli beam equation, with zero initial conditions. We study the inverse problem of the beam, consisting in the identification of the source of vibration, shaped as an element in L2, using as data the speed from an arbitrary point of the beam, during a time interval arbitrarily small. The relevance of this work in Engineering, for example, is in the identification of structural damage in beams.

Neste trabalho, procuramos resolver o problema direto da equação da viga de Euler- Bernoulli bi-engastada com condições iniciais nulas. Estudamos o problema inverso da viga, que consiste em identificar a fonte de vibração, modelada como um elemento em L2, usando como dado a velocidade de um ponto arbitrário da viga, durante um intervalo de tempo arbitrariamente pequeno. A relevância deste trabalho na Engenharia encontra-se, por exemplo, na identificação de danos estruturais em vigas. / In this work, we try to solve the direct problem of the clamped-clamped Euler- Bernoulli beam equation, with zero initial conditions. We study the inverse problem of the beam, consisting in the identification of the source of vibration, shaped as an element in L2, using as data the speed from an arbitrary point of the beam, during a time interval arbitrarily small. The relevance of this work in Engineering, for example, is in the identification of structural damage in beams.

New measuring methods continue to be developed in the field of wind anemometry for various environments subject to low-speed and high-speed flows, turbulent-present flows, and ideal and non-ideal flows. As a result, anemometry has taken different avenues for these environments from the traditional cup model to sonar, hot-wire, and recent developments with sphere anemometers. Several measurement methods have modeled the air drag force as a quadratic function of the corresponding wind speed. Furthermore, by incorporating non-drag fluid forces in addition to the main drag force, a dynamic set of equations of motion for the deflection and strain of a spherical anemometer's beam can be derived. By utilizing the equations of motion to develop a direct relationship to a measurable parameter, such as strain, an approximation for wind speed based on a measurement is available. These ODE's for the strain model can then be used to relate directly the fluid speed (wind) to the strain along the beam’s length. The spherical anemometer introduced by the German researcher Holling presents the opportunity to incorporate the theoretical cantilevered Euler-Bernoulli beam with a spherical mass tip to develop a deflection and wind relationship driven by cross-area of the spherical mass and constriction of the shaft or the beam's bending properties. The application of Hamilton's principle and separation of variables to the Lagrangian Mechanics of an Euler-Bernoulli beam results in the equations of motion for the deflection of the beam as a second order partial differential equation (PDE). The boundary conditions of our beam's motion are influenced by the applied fluid forces of a relative drag force and the added mass and buoyancy of the sphere. Strain gauges will provide measurements in a practical but non-intrusive method and thus the concept of a measuring strain gauge is simulated. Young's Modulus creates a relationship between deflection and strain of an Euler-Bernoulli system and thus a strain and wind relation can be modeled as an ODE. This theoretical sphere anemometer's second order ODE allows for analysis of the linear and non-linear accuracies of the motion of this dynamic system at conventional high speed conditions.

Research is performed to assess the viability of applying the least squares model to one-dimensional heat transfer and Euler-Bernoulli Beam Theory problems. Least squares models were developed for both the full and mixed forms of the governing one-dimensional heat transfer equation along weak form Galerkin models. Both least squares and weak form Galerkin models were developed for the first order and second order versions of the Euler-Bernoulli beams. Several numerical examples were presented for the heat transfer and Euler- Bernoulli beam theory. The examples for heat transfer included: a differential equation having the same form as the governing equation, heat transfer in a fin, heat transfer in a bar and axisymmetric heat transfer in a long cylinder. These problems were solved using both least squares models, and the full form weak form Galerkin model. With all four examples the weak form Galerkin model and the full form least squares model produced accurate results for the primary variables. To obtain accurate results with the mixed form least squares model it is necessary to use at least a quadratic polynominal. The least squares models with the appropriate approximation functions yielde more accurate results for the secondary variables than the weak form Galerkin. The examples presented for the beam problem include: a cantilever beam with linearly varying distributed load along the beam and a point load at the end, a simply supported beam with a point load in the middle, and a beam fixed on both ends with a distributed load varying cubically. The first two examples were solved using the least squares model based on the second order equation and a weak form Galerkin model based on the full form of the equation. The third problem was solved with the least squares model based on the second order equation. Both the least squares model and the Galerkin model calculated accurate results for the primary variables, while the least squares model was more accurate on the secondary variables. In general, the least-squares finite element models yield more acurate results for gradients of the solution than the traditional weak form Galkerkin finite element models. Extension of the present assessment to multi-dimensional problems and nonlinear provelms is awaiting attention.