A Comparison of Higher-Order Difference Methods in the Solution of Beam-Vibration Problems

Several higher-order difference methods are investi gated and compared for the problem of finding the natural frequencies of the lateral vibration of a beam. All of the methods considered are applicable to either digital or analog computers, although particu lar reference is made to the analog computer. The methods considered in most detail use the same basic central difference approximation, the variations occurring in the method of representing boundary conditions. Three higher-order approaches to the problem of boundary conditions are pre sented. They are 1) the use of one-sided differences of fourth order, 2) the use of symmetry assumptions, and 3) the passive-circuit approach. Each method is shown to have its advantages, the final choice de pending upon the particular requirements of the problem. Results are presented in the form of curves of per centage mode-frequency error vs number of cells for the various approximation methods.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/68833/2/10.1177_003754976400300110.pd

Improved methods of vibration analysis of pretwisted, airfoil blades

Vibration analysis of pretwisted blades of asymmetric airfoil cross section is performed by using two mixed variational approaches. Numerical results obtained from these two methods are compared to those obtained from an improved finite difference method and also to those given by the ordinary finite difference method. The relative merits, convergence properties and accuracies of all four methods are studied and discussed. The effects of asymmetry and pretwist on natural frequencies and mode shapes are investigated. The improved finite difference method is shown to be far superior to the conventional finite difference method in several respects. Close lower bound solutions are provided by the improved finite difference method for untwisted blades with a relatively coarse mesh while the mixed methods have not indicated any specific bound

Correlating low energy impact damage with changes in modal parameters: diagnosis tools and FE validation

This paper presents a basic experimental technique and simplified FE based models for the detection, localization and quantification of impact damage in composite beams around the BVID level. Detection of damage is carried out by shift in modal parameters. Localization of damage is done by a topology optimization tool which showed that correct damage locations can be found rather efficiently for low-level damage. The novelty of this paper is that we develop an All In One (AIO) package dedicated to impact identification by modal analysis. The damaged zones in the FE models are updated by reducing the most sensitive material property in order to improve the experimental/numerical correlation of the frequency response functions. These approximate damage models(in term of equivalent rigidity) give us a simple degradation factor that can serve as a warning regarding structure safety

Smart FRP Composite Sandwich Bridge Decks in Cold Regions

INE/AUTC 12.0

An improved finite-difference analysis of uncoupled vibrations of tapered cantilever beams

An improved finite difference procedure for determining the natural frequencies and mode shapes of tapered cantilever beams undergoing uncoupled vibrations is presented. Boundary conditions are derived in the form of simple recursive relations involving the second order central differences. Results obtained by using the conventional first order central differences and the present second order central differences are compared, and it is observed that the present second order scheme is more efficient than the conventional approach. An important advantage offered by the present approach is that the results converge to exact values rapidly, and thus the extrapolation of the results is not necessary. Consequently, the basic handicap with the classical finite difference method of solution that requires the Richardson's extrapolation procedure is eliminated. Furthermore, for the cases considered herein, the present approach produces consistent lower bound solutions

Comparison of shearography to scanning laser vibrometry as methods for local stiffness identification of beams

Local stiffness of Euler–Bernoulli beams can be identified by dividing the bending moment of a deformed beam by the local curvature. Curvature and moment distributions can be derived from the modal shape of a beam vibrating at resonance. In this article, the modal shape of test beams is measured by both scanning laser vibrometry (SLV) and shearography. Shearography is an interferometric optical method that produces full-field displacement gradients of the inspected surface. Curvature can be obtained by two steps of derivation of the modal amplitude (in the case of SLV) or one step of derivation of the modal shape slope (in the case of shearography). Three specially prepared aluminium beams with a known stiffness distribution are used for the validation of both techniques. The uncertainty of the identified stiffness distributions with both techniques is compared and related to their signal-to-noise ratios. A strength and weakness overview at the end of the article reveals that the shearography is the technique that shows the most advantages

Damage identification in a concrete beam using curvature difference ratio

Previous studies utilising changes in mode shape or curvature to locate damage rely on the fact that the greatest change occurs around the defect. However, in concrete beams this fact is undermined due to the nature of the defect as distributed multi-site cracks. In addition, differences in mode shape and curvature as ways to locate the damage is unstable because of occurrence of modal nodes and inflection points. In this paper, one interesting solution to this problem is being tested by establishing a new non-dimensional expression designated the 'Curvature Difference Ratio (CDR)'. This parameter exploits the ratio of differences in curvature of a specific mode shape for a damaged stage and another reference stage. The expression CDR is reasonably used to locate the damage and estimate the dynamic bending stiffness in a successively loaded 6m concrete beam. Results obtained by the proposed technique are tested and validated with a case study results done by Ren and De Roeck [1] also by Maeck and De Roeck [2]. Another contribution of this work is that relating changes in vibration properties to the design bending moment at beam sections as defined in Eurocode 2 specifications [3]. Linking between a beam section condition and the change in vibration data will help to give a better comprehension on the beam condition than the applied load

Application of the Finite Element Method to Rotary Wing Aeroelasticity

A finite element method for the spatial discretization of the dynamic equations of equilibrium governing rotary-wing aeroelastic problems is presented. Formulation of the finite element equations is based on weighted Galerkin residuals. This Galerkin finite element method reduces algebraic manipulative labor significantly, when compared to the application of the global Galerkin method in similar problems. The coupled flap-lag aeroelastic stability boundaries of hingeless helicopter rotor blades in hover are calculated. The linearized dynamic equations are reduced to the standard eigenvalue problem from which the aeroelastic stability boundaries are obtained. The convergence properties of the Galerkin finite element method are studied numerically by refining the discretization process. Results indicate that four or five elements suffice to capture the dynamics of the blade with the same accuracy as the global Galerkin method

Modelo de vibrações de vigas de reforço utilizadas em plataformas ofshore

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico. Programa de Pós-Graduação em Engenharia MecânicaPlacas reforçadas por vigas são os principais componentes de estruturas de plataformas offshore, usadas na prospecção e produção de petróleo. As vibrações geradas pelas máquinas propagam-se através da estrutura gerando altos níveis de ruído nos alojamentos. Para determinar com precisão o fluxo de potência através de placas reforçadas por vigas, um modelo que inclua o efeito das ressonâncias próprias da alma e aba da viga e as ondas no plano deve ser considerado, sendo o efeito das ondas no plano importante principalmente nas altas freqüências. Este trabalho apresenta um modelo para determinar a resposta de placas reforçadas por vigas utilizando uma abordagem analítica. Os modelos incluem vigas L e T e uma placa reforçada por uma viga L submetida a um carregamento distribuído. Este modelo pode ser usado para a determinação dos fatores de acoplamento de estruturas a serem utilizadas em uma análise de Análise Estatística Energética (SEA). Este método é útil devido ao baixo tempo de processamento comparado aos outros métodos existentes e a precisão é consideravelmente boa especialmente nas altas freqüências. Os resultados obtidos são validados comparando-se com o Método de Elementos Finitos. Outros métodos para a determinação da mobilidade de estruturas tipo placa reforçada por vigas também são apresentados neste trabalho. Um destes métodos é o Método de Imagens, que envolve a distribuição de fontes e imagens de vibrações de forma que obedeça as condições de contorno da estrutura. Este método é aplicável principalmente para estruturas de grandes dimensões e alto amortecimento estrutural