Dissertations / Theses on the topic 'Natural frequencies of free vibration'

Methods of determining natural frequencies of the D76D88, B76D88, A86E93, C86G90, C86L90 and C125L89 turbine wheel designs for various environmental conditions were investigated by application of Finite Element Analysis and beam theory. Modelling and simulation methods were developed ; the first method composed of 15 finite element simulations ; the second composed of 15 finite element simulations and a set of experimental frequency survey results; the third composed of 5 simulations , an incorporated mathematical model and a set of experimental frequency survey results. Each of these methods was designed to allow prediction of resonant frequency changes across a range of exhaust gas temperature and shaft rotational speed. For the new modelling and simulation methods, an analysis template and a plotting tool were developed using Microsoft Excel and MATLAB software. A graph showing a frequency-temperature-speed variations and a Campbell Diagram that incorporates material stiffening and softening effects across a range of rotational speeds was designed, and applied to the D76D88, B76D88, A86E93, C86G90, C86L90 and C125L89 turbine wheel designs. New design methodologies for turbine wheels were formulated and validated, showing a good agreement with a range of data points from frequency survey, strain-gauge telemetry and laser tip-timing test results. The results from the new design method were compared with existing single compensation factor methodology, and showed a great improvement in accuracy of prediction of modal vibration. A new nomenclature for the mode shapes of a turbocharger’s blade was proposed, designed and demonstrated to allow direct identification of associated mode shape. It is concluded that Finite Element Analysis combined with the frequency survey is capable of predicting changes in turbine natural frequencies and, when incorporated into the existing turbine design methodology, resulted in a major improvement in the accuracy of the predictions of vibration frequency.

Orientador: Leandro Palermo Jr<br>Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Civil, Arquitetura e Urbanismo<br>Made available in DSpace on 2018-08-09T15:27:34Z (GMT). No. of bitstreams: 1 Sakanaka_SandraHiromi_M.pdf: 2300706 bytes, checksum: bf530823f185b6dbee7f56dd83d01a67 (MD5) Previous issue date: 2006<br>Resumo: A análise de vibração livre e de instabilidade de placas finas e placas moderadamente espessas é apresentada através do método dos elementos de contorno (MEC) considerando o efeito da deformação pela força cortante e, particularmente para o cálculo de freqüências naturais, o efeito da inércia rotatória é também considerado. A formulação da solução fundamental é baseada na teoria de Mindlin (1951) mas resultados para a teoria de Kirchhoff (1850) também podem ser obtidos [Palermo Jr. (2000)]. O presente trabalho usa a técnica da iteração inversa através do coeficiente de Rayleigh para a determinação das menores freqüências naturais e cargas críticas de instabilidade das placas. A implementação numérica emprega elementos de contorno isoparamétricos lineares contínuos e descontínuos. Elementos constantes de domínio são usados. Os parâmetros nodais são posicionados nos extremos dos elementos e os pontos de carregamento dos elementos descontínuos são deslocados para o interior a uma distância igual a um quarto do comprimento do elemento. Expressões analíticas das integrais de contorno são desenvolvidas para os casos em que o elemento contém o ponto de carregamento e integração numérica de Gauss-Legendre é feita nos outros casos. As integrais de domínio foram transformadas em integrais de contorno para cada célula e foram tratadas como cargas de superfície atualizadas através de um processo iterativo. Os resultados obtidos foram comparados com valores encontrados na literatura para demonstrar a precisão do presente trabalho<br>Abstract: Free-vibration analysis and static buckling loads analysis of thin and thick plates considering the shear deformation effects using the Boundary Element Method (BEM) is presented. For the calculation of natural frequencies, the rotatory inertia is also counted. The formulation of the fundamental solution considers Mindlin¿s plates but results according to the classic theory can also be obtained [Palermo Jr. (2000)]. The present article makes use of the inverse iteration with Rayleigh coefficient to determine the smallest natural frequencies and the smallest static buckling loads of the plates. The numerical implementation employed continuous or discontinuous isoparametric linear boundary elements according to the characteristics of the problem to be solved. Constant domain elements are used. Nodal parameters have been placed at the ends of the elements and the source point of the discontinuous elements were positioned at a distance equal to one quarter of the element length. Analytical expressions have been employed in the integration on elements containing the source point and Gauss-Legendre numerical integration scheme otherwise. The domain integrals containing the inertia effects or nonlinear effect have been transformed into boundary integrals for each cell and were treated as surface loads updated in an iterative process. The obtained results were compared to those in literature to demonstrate the precision of this proposal<br>Mestrado<br>Estruturas<br>Mestre em Engenharia Civil

ABSTRACT Forced Vibration Testing and Analysis of Pre- and Post- Retrofit Buildings Erica Dawn Jacobsen The primary goal of the thesis was to detect the retrofit through vibration testing of both buildings. The secondary goal focused on correctly identifying the behavior of the building through FVT, comparing that behavior to computational model predictions, and determining the necessary level of detail to include in the computational modeling. Forced vibration testing (FVT) of two stiff-wall/flexible-diaphragm buildings yielded natural frequencies and mode shapes for the two buildings. The buildings were nearly identical with the exception that one had been retrofitted. Both buildings were comprised of concrete shearwalls and steel moment frames in the north/south direction and moment frames in the east/west direction. The retrofit strengthened the moment connections and added braces to the perimeter walls in the east/west direction. The natural frequencies were found through FVT by setting a 30-lb shaker on the roof of both buildings and sweeping through a range of frequencies in both the east/west and north/south directions. Accelerometers were placed on the building to detect the accelerations. The peaks on the Fast Fourier Transform (FFT) graphs indicated the frequencies at which the structure resonated. Mode shapes were tested for by placing the shaker in a position ideal for exciting the mode and setting the shaker to the natural frequency detected from the FFT graphs. The accelerometers were placed around the roof of the building to record the mode shape. After testing, computational models were created to determine if the models could accurately predict the frequencies and mode shapes of the buildings as well as the effect of the retrofit. A series of increasingly complex computational models, ranging from hand calculations to 3D models, were created to determine the level of detail necessary to predict the building behavior. Natural frequencies were the primary criteria used to determine whether the model accurately predicted the building behavior. The mid-diaphragm deflection and base shear from spectral analysis were the final criteria used to compare these select models. It was determined that in order to properly capture the modal behavior of the building, the sawtooth framing, major beams, and the lateral-force-resisting-system (LFRS) must be modeled. Though the mode shape of the building is dominated by the flexible diaphragm, the LFRS is necessary to model to accurately predict both the natural frequency of the building as well as the diaphragm deflection.

The study of rectangular cantilever plates with step discontinuities in properties is of interest in many areas of industry. Cantilever plates are assigned different properties to represent change in stiffness and mass creating therefore the concept of step discontinuities in properties. The step discontinuity in properties is best represented by dividing the cantilever plate into separate segments called spans, with each span having its own stiffness and/or mass distribution as one moves outwards from the clamped edge. This thesis presents the concept of dividing the cantilever plate into spans and provides accurate analytical solutions for free vibration frequencies and mode shapes. Chapter 1 introduces the reader to the theory of rectangular plates while chapter 2 concentrates on introducing the general solutions as applied to rectangular plates. Although there is no limit to the number of plate spans to be studied, three and four span cantilever plates were analysed in chapter 3. The computed eigenvalues are validated against previously published uniform rectangular cantilever plate free vibration results. They were found converging to the known values. Free vibration eigenvalues and mode shapes are then calculated for a variety of cantilever plates of different aspect ratios and with different span thicknesses creating a discontinuity in mass and flexural rigidity along the plate. The results are presented in chapter 4 and discussed in chapter 5.

Vibration is an inherent phenomenon in dynamic mechanical systems. The work undertaken in this thesis is to identify natural frequencies of a variable length blade. Therefore designers can ensure that natural frequencies will not be close to the frequency (or integer multiples) of the main excitation forces in order to avoid resonance. For a wind turbine blade, the frequency range between 0.5 Hz and 30 Hz is relevant. The turbine blade is approximated by a cantilever, therefore, it is fully constrained where attached to a turbine shaft/hub. Flap-wise, edge-wise and torsional natural frequencies are calculated. The MATLAB program “BEAMANALYSIS.m” has been developed for the finite element analysis of a one dimensional model of the beam. Similarly, a three dimensional model of the beam has been developed in a finite element program Unigraphics NX5. The results found using the MATLAB program are compared with those found with NX5. Satisfactory agreement between the results is found for frequencies up to almost 500 Hz. Additionally, the frequencies one might expect in an experiment are identified. Experimental modal analysis has been performed on a uniform and stepped beam made of mild steel to extract the first five flap-wise natural frequencies. The results found have been compared to numerical results and the exact solution of an Euler-Bernoulli beam. Concurrence is found for the frequency range of interest. Although, some discrepancies exist at higher frequencies (above 500 Hz), finite element analysis proves to be reliable for calculating natural frequencies. Finally, the fixed portion and moveable portion of the variable length blade are approximated respectively by a hollow and a solid beam which can be slid in and out. Ten different configurations of the variable length blade, representing ten different positions of the moveable portion are investigated. A MATLAB program named VARIBLADEANALYSIS.m was developed to predict natural frequencies. Similarly three dimensional models of the variable length blade have been developed in the finite element program Unigraphics NX5.<br>This work was supported by the Research office of CPUT.

Dynamic instrumentation and a series of ambient vibration tests were performed on a four storey strengthened R/C building within the scope of this study. Traffic load and wind load were accepted as natural dynamic loads and the vibrations were recorded by sensitive accelerometers.For that study, 12 uniaxial, 1 triaxial accelerometers and a 15 channel data logger system were used. Four sets of dynamic measurements were recorded over a period of 6 months. Recorded readings were analyzed using UPC, PC and CVA algorithms and Artemis software. The natural freqeuncies, mode shape of the tested building were determined. The experimental results were compared against each other. A 3D-FE model of the building was prepared and analytical results were also compared against experimental results.The calibration (updating) of the analytical model was carried out using the experimentally obtained mode shapes and freqeunices. The results of the study indicate that first few mode shapes and freqeuncies of the building can be obtained successfully within zero to 10 Hz range using ambient monitoring. Field calibrated FE models can effectively simulate the first translational and torsional modes of the building. Calibration studies indicate that the upper floor is more flexible than the nominal model and there are weaknesses between the shear wall and roof slab connections.

O Concreto Leve Estrutural com Pérolas de EPS (CLEPE), ou simplesmente Concreto com EPS, é uma alternativa na execução de lajes maciças. Como possui uma redução da ordem de 50% no peso próprio em relação ao Concreto Convencional (CC), suas características dinâmicas são diferentes. Neste trabalho descreve-se o comportamento dinâmico de lajes de CLEPE que são comparadas com as de CC por meio de ensaios dinâmicos e uma análise paramétrica. Nos ensaios dinâmicos focou-se a obtenção de três fatores imprescindíveis em uma análise dinâmica, que são as frequências naturais, os modos de vibração (deformadas modais) e o amortecimento estrutural, valor determinado experimentalmente que constitui dado de entrada em simulações numéricas. Os resultados experimentais indicam que o fator de amortecimento do CLEPE é ligeiramente maior que o do CC. Já os resultados da análise paramétrica revelam que a diminuição da rigidez é preponderante em relação à diminuição da massa do CLEPE, o que acarreta na diminuição das frequências naturais das lajes com este material. Mesmo com um amortecimento maior, as lajes de CLEPE apresentam maior sensibilidade às vibrações. Esta constatação, entretanto, não exclui o CLEPE como um material estrutural, apenas indica que, como no uso de qualquer material, as estruturas com CLEPE também devem ser verificadas com relação ao comportamento dinâmico.<br>The Structural Lightweight Concrete with EPS Beads (SLCEB), or simply EPS concrete, is an alternative in the execution of massive slabs. As it has a reduction in the order of 50% in self-weight in relation to the Ordinary Concrete (OC), its dynamic characteristics are different. In this work it is described the dynamic behavior of SLCEB slabs whose are compared with those of OC by means of dynamic tests and a parametric analysis. The obtainment of three essential factors in a dynamic analysis, which are natural frequencies, the ways of vibration (deformed modal), and the structural damping, experimentally determinate value which constitutes an input data in numerical simulations are focused in the dynamical tests. The experimental results indicate that the damping factor of SLCEB is a little bigger than the OC ones. Although, the results of the parametric analysis indicate that the decrease of stiffness is preponderant in relation to the decrease of the SLCEB mass, which result in some decrease of natural frequencies of slabs with this material. In despite of having a bigger damping, the SLCEB slabs expose a bigger sensibility to vibrations. However, this observation does not exclude the SLCEB as a structural material. It only means that as in use of any material, the SLCEB structures must be also verified in relation to the dynamic behavior.

A wide range of dynamic systems can be approximated as linear and time invariant, for which a wealth of tools are available to characterize or modify their dynamic characteristics. Experimental modal analysis (EMA) is a procedure whereby the natural frequencies, damping ratios and mode shapes which parameterize vibratory, linear, time invariant systems are derived from experimentally measured response data. EMA is commonly applied in a multitude of applications, for example, to generate experimental models of dynamic systems, validate finite element models and to characterize dissipation in vibratory systems. Recent EMA has also been used to characterize damage or defects in a variety of systems. The Algorithm of Mode Isolation (AMI), presented by Drexel and Ginsberg in 2001, employs a unique strategy for modal parameter estimation in which modes are sequentially identified and subtracted from a set of FRFs. Their natural frequencies, damping ratios and mode vectors are then refined through an iterative procedure. This contrasts conventional multi-degree-of-freedom (MDOF) identification algorithms, most of which attempt to identify all of the modes of a system simultaneously. This dissertation presents a hybrid multi-input-multi-output (MIMO) implementation of the algorithm of mode isolation that improves the performance of AMI for systems with very close or weakly excited modes. The algorithmic steps are amenable to semi-automatic identification, and many FRFs can be processed efficiently and without concern for ill-conditioning, even when many modes are identified. The performance of the algorithm is demonstrated on noise contaminated analytical response data from two systems having close modes, one of which has localized modes while the other has globally responsive modes. The results are compared with other popular algorithms. MIMO-AMI is also applied to experimentally obtained data from shaker excited tests of the Z24 highway bridge, demonstrating the algorithm's performance on a data set typical of many EMA applications. Considerations for determining the number of modes active in the frequency band of interest are addressed, and the results obtained are compared to those found by other groups of researchers.

A clássica teoria de Euler-Bernoulli para vibrações transversais de vigas elásticas é sabido não ser adequada para vibrações de altas freqüências, como é o caso de vibração de vigas curtas. Esta teoria assume que a deflexão deve-se somente ao momento fletor, uma vez que os efeitos da inércia de rotação e da força cortante são negligenciados. Lord Rayleigh complementou a teoria clássica demonstrando a contribuição da inércia de rotação e Timoshenko estendeu a teoria ao incluir os efeitos da força cortante. A equação resultante é conhecida como sendo a que caracteriza a chamada teoria de viga de Timoshenko. Usando-se a matriz de rigidez dinâmica, as freqüências naturais e a resposta dinâmica de estruturas de barras são determinadas e comparadas de acordo com resultados de quatro modelos de vibração. São estudados o problema de vibração flexional de vigas, pórticos e grelhas, bem como o problema de fundação elástica segundo o modelo de Winkler e também a versão mais avançada que é o modelo de Pasternak.<br>Classical Euler-Bernoulli theory for transverse vibrations of elastic beams is known to be inadequate to consider high frequency modes which occur for short beams, for example. This theory is derived under the assumption that the deflection is only due to bending. The effects of rotary inertia and shear deformation are ignored. Lord Rayleigh improved the classical theory by considering the effect of rotary inertia. Timoshenko extended the theory to include the effects of shear deformation. The resulting equation is known as Timoshenko beam theory. The natural frequencies and dynamic reponse of framed structures are determined by using the dynamic stiffness matrix and compered according to these theories. The flexional vibration problems of beams, plane frames and grids are analysed, as well problems of elastic foundation according the well known Winkler model and also the more general Pasternak model.

The subject of this diploma thesis is a dynamic analysis of phenomena occurring in the thin cylindrical region, ie in a gab, which flows liquid by the given parameters. Through the simplifications introduced, it comes to the problem of structural and forced vibration for both the horizontal pulsation in a tube and the area of a compensating piston. The described method uses the separation of variables for pressure function. The speed function is derived from the already known pressure field distribution using the Navier-Stokes equation.

L’objectif de ce travail est de développer des outils numériques utilisables dans la détermination de manière exacte des propriétés modales des structures sandwichs viscoélastiques composites au vue de la conception des structures sandwichs viscoélastiques légères mais à haut pouvoir amortissant. Pour cela nous avons tout d’abord développé un outil générique implémenté en Matlab pour la détermination des propriétés modales en vibration libre des plaques sandwichs viscoélastiques dont les faces sont en stratifié de plusieurs couches orientées dans diverses directions. L’intérêt de cet outil, basé sur une formulation éléments finis, réside dans sa capacité à prendre en compte l’anisotropie des couches composites, la non linéarité matérielle de la couche viscoélastique traduit par diverses lois viscoélastiques dépendant de la fréquence ainsi que diverses conditions aux limites. La résolution du problème aux valeurs propres non linéaires complexes se fait par le couplage entre la technique d’homotopie, la méthode asymptotique numérique et la différentiation automatique. Ensuite pour permettre une étude continue des effets d’un paramètre de modélisation sur les propriétés modales des sandwichs viscoélastiques, nous avons proposé une méthode générique de résolution de problème résiduel non linéaire aux valeurs propres complexes possédant en plus de la dépendance en fréquence introduite par la couche viscoélastique du coeur, la dépendance du paramètre de modélisation qui décrit un intervalle d’étude bien spécifique. Cette résolution est basée sur la méthode asymptotique numérique, la différentiation automatique, la technique d’homotopie et la continuation et prend en compte diverses lois viscoélastiques. Nous proposons après cela, deux formulations distinctes pour étudier les effets, sur les propriétés amortissantes, de deux paramètres de modélisation qui sont importants dans la conception de sandwichs viscoélastiques à haut pouvoir amortissement. Le premier est l’orientation des fibres des composites dans la référence du sandwich et le second est l’épaisseur des couches qui lorsqu’elles sont bien définies permettent d’obtenir non seulement des structures sandwichs à haut pouvoir amortissant mais très légères. Les équations fortement non linéaires aux valeurs propres complexes obtenues dans ces formulations sont résolues par la nouvelle méthode de résolution d’équation résiduelle développée. Des comparaisons avec des résultats discrets sont faites ainsi que les temps de calcul pour montrer non seulement l’utilité de ces deux formulations mais également celle de la méthode de résolution d’équations résiduelles non linéaires aux valeurs propres complexes à double dépendance<br>Modeling and design of composite viscoelastic structures with high damping powerThe aim of this thesis is to develop numerical tools to determine accurately damping properties of composite sandwich structures for the design of lightweight viscoelastic sandwichs structures with high damping power. In a first step, we developed a generic tool implemented in Matlab for determining damping properties in free vibration of viscoelastic sandwich plates with laminate faces composed of multilayers. The advantage of this tool, which is based on a finite element formulation, is its ability to take into account the anisotropy of composite layers, the material non-linearity of the viscoelastic core induiced by the frequency-dependent viscoelastic laws and various boundary conditions . The nonlinear complex eigenvalues problem is solved by coupling homotopy technic, asymptotic numerical method and automatic differentiation. Then for the continuous study of a modeling parameter on damping properties of viscoelastic sandwichs, we proposed a generic method to solve the nonlinear residual complex eigenvalues problem which has in addition to the frequency dependence introduced by the viscoelastic core, a modeling parameter dependence that describes a very specific study interval. This resolution is based on asymptotic numerical method, automatic differentiation, homotopy technique and continuation technic and takes into account various viscoelastic laws. We propose after that, two separate formulations to study effects on the damping properties according to two modeling parameters which are important in the design of high viscoelastic sandwichs with high damping power. The first is laminate fibers orientation in the sandwich reference and the second is layers thickness which when they are well defined allow to obtain not only sandwich structures with high damping power but also very light. The highly nonlinear complex eigenvalues problems obtained in these formulations are solved by the new method of resolution of eigenvalue residual problem with two nonlinearity developed before. Comparisons with discrete results and computation time are made to show the usefulness of these two formulations and of the new method of solving nonlinear complex eigenvalues residual problem of two dependances

碩士<br>大同大學<br>機械工程學系(所)<br>94<br>This thesis discusses the structure of the spatially beam as the difference conditions between curvature and torsion in the cross-section of the beam ask the change relation of the natural vibration. Every point which follows the centerline of the beam has three translation displacements and three rotational angles of the cross-section area. The Eigenvalue problem is solved the natural frequencies of varying pre-twisted angles of the rectangular cross-section area, slenderness ratio, E/G ratio, all kinds of end conditions. Using the method of Runge-Kutta integration gets the difference equations. Then, adjoint operator system operates them. Finally, the equations of the difference end conditions write the fortran program by controlling seven factors. The datum of the program draw the charts by using Tecplot. The datum discuss the relational change.