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.

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.

The field of inverse problems is an area in applied mathematics that is of great importance in several scientific and industrial applications. Since an inverse problem is typically founded on non-linear and ill-posed models it is a very difficult problem to solve. To find a regularized solution it is crucial to have a priori information about the solution. Therefore, general theories are not sufficient considering new applications. In this thesis we consider the inverse problem to determine the beam bending stiffness from measurements of the transverse dynamic displacement. Of special interest is to localize parts with reduced bending stiffness. Driven by requirements in the wood-industry it is not enough considering time-efficient algorithms, the models must also be adapted to manage extremely short calculation times. For the developing of efficient methods inverse problems based on the fourth order Euler-Bernoulli beam equation and the second order string equation are studied. Important results are the transformation of a nonlinear regularization problem to a linear one and a convex procedure for finding parts with reduced bending stiffness.

The conventional finite element models (FEM) of problems in structural mechanics are based on the principles of virtual work and the total potential energy. In these models, the secondary variables, such as the bending moment and shear force, are post-computed and do not yield good accuracy. In addition, in the case of the Timoshenko beam theory, the element with lower-order equal interpolation of the variables suffers from shear locking. In both Euler-Bernoulli and Timoshenko beam theories, the elements based on weak form Galerkin formulation also suffer from membrane locking when applied to geometrically nonlinear problems. In order to alleviate these types of locking, often reduced integration techniques are employed. However, this technique has other disadvantages, such as hour-glass modes or spurious rigid body modes. Hence, it is desirable to develop alternative finite element models that overcome the locking problems. Least-squares finite element models are considered to be better alternatives to the weak form Galerkin finite element models and, therefore, are in this study for investigation. The basic idea behind the least-squares finite element model is to compute the residuals due to the approximation of the variables of each equation being modeled, construct integral statement of the sum of the squares of the residuals (called least-squares functional), and minimize the integral with respect to the unknown parameters (i.e., nodal values) of the approximations. The least-squares formulation helps to retain the generalized displacements and forces (or stress resultants) as independent variables, and also allows the use of equal order interpolation functions for all variables. In this thesis comparison is made between the solution accuracy of finite element models of the Euler-Bernoulli and Timoshenko beam theories based on two different least-square models with the conventional weak form Galerkin finite element models. The developed models were applied to beam problems with different boundary conditions. The solutions obtained by the least-squares finite element models found to be very accurate for generalized displacements and forces when compared with the exact solutions, and they are more accurate in predicting the forces when compared to the conventional finite element models.

<p>The field of inverse problems is an area in applied mathematics that is of great importance in several scientific and industrial applications. Since an inverse problem is typically founded on non-linear and ill-posed models it is a very difficult problem to solve. To find a regularized solution it is crucial to have <em>a priori</em> information about the solution. Therefore, general theories are not sufficient considering new applications.</p><p>In this thesis we consider the inverse problem to determine the beam bending stiffness from measurements of the transverse dynamic displacement. Of special interest is to localize parts with reduced bending stiffness. Driven by requirements in the wood-industry it is not enough considering time-efficient algorithms, the models must also be adapted to manage extremely short calculation times.</p><p>For the developing of efficient methods inverse problems based on the fourth order Euler-Bernoulli beam equation and the second order string equation are studied. Important results are the transformation of a nonlinear regularization problem to a linear one and a convex procedure for finding parts with reduced bending stiffness.</p>

This thesis deals with developing a thermodynamically consistent model to simulate the electromechanical response of ionic polymer-metal composites based on Euler-Bernoulli beam theory. Constitutive assumptions are made for the Helmholtz free energy and the rate of dissipation. The governing equations involving small deformations are formulated using the conservation laws, the power theorem, and the maximum rate of dissipation hypothesis. The model is extended to solve large deformation cantilever beams involving pure bending which could be used in the characterization of the material parameters. A linear finite element solution along with a staggered time stepping algorithm is provided to numerically solve the governing equations of the small deformations problem under generalized electromechanical loading and boundary conditions. The results are in qualitative and quantitative agreement with the experiments performed on both Nafion and Flemion based Ionic Polymer-Metal Composite strips.

Analysis of vibration transmission and reflection in beam-like engineering structures requires better predictive models to optimise structural behaviour further. Numerous studies have used flexural and longitudinal structural wave motion to model the vibrational response of angled junctions in beam-like structures, to better understand the transmission and reflection properties. This study considers a model of a variable joint angle which joins two semi-infinite rectangular cross-section beams. In a novel approach, the model allows for the joint to expand in size as the angle between the two beams is increased. The material, geometric and dynamics properties were consistently being considered. Thus, making the model a good representation of a wide range of angles. Predicted results are compared to an existing model of a joint between two semi-infinite beams where the joint was modelled as a fixed inertia regardless of the angle between the beams, thus limiting its physical representation, especially at the extremes of angle (two beams lay next to each other at 180 degree joint). Results from experimentation were also compared to the modelling, which is in good agreement for the range of angles investigated. Optimum angles for minimum vibrational power transmission are identified in terms of the frequency of the incoming flexural or longitudinal wave. Extended analysis and effect of adding stiffness and damping (rubber material) at the joint are also reported.

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.

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