Damage assessment of structures with uncertainty by using mode-shape curvatures and fuzzy logic

Uncertainties associated with the structural model and measured vibration data may lead to unreliable damage detection. In this paper, we show that geometric and measurement uncertainty cause considerable problem in damage assessment which can be alleviated by using a fuzzy logic-based approach for damage detection. Curvature damage factor (CDF) of a tapered cantilever beam are used as damage indicators. Monte Carlo simulation (MCS) is used to study the changes in the damage indicator due to uncertainty in the geometric properties of the beam. Variation in these CDF measures due to randomness in structural parameter, further contaminated with measurement noise, are used for developing and testing a fuzzy logic system (FLS). Results show that the method correctly identifies both single and multiple damages in the structure. For example, the FLS detects damage with an average accuracy of about 95 percent in a beam having geometric uncertainty of 1 percent COV and measurement noise of 10 percent in single damage scenario. For multiple damage case, the FLS identifies damages in the beam with an average accuracy of about 94 percent in the presence of above mentioned uncertainties. The paper brings together the disparate areas of probabilistic analysis and fuzzy logic to address uncertainty in structural damage detection.

A note on the inverse mode shape problem for bars, beams, and plates

Motivated by the idea of designing a structure for a desired mode shape, intended towards applications such as resonant sensors, actuators and vibration confinement, we present the inverse mode shape problem for bars, beams and plates in this work. The objective is to determine the cross-sectional profile of these structures, given a mode shape, boundary condition and the mass. The contribution of this article is twofold: (i) A numerical method to solve this problem when a valid mode shape is provided in the finite element framework for both linear and nonlinear versions of the problem. (ii) An analytical result to prove the uniqueness and existence of the solution in the case of bars. This article also highlights a very important question of the validity of a mode shape for any structure of given boundary conditions.

A mode-shape classification technique for robust speech rate estimation and syllable nuclei detection

Acoustic feature based speech (syllable) rate estimation and syllable nuclei detection are important problems in automatic speech recognition (ASR), computer assisted language learning (CALL) and fluency analysis. A typical solution for both the problems consists of two stages. The first stage involves computing a short-time feature contour such that most of the peaks of the contour correspond to the syllabic nuclei. In the second stage, the peaks corresponding to the syllable nuclei are detected. In this work, instead of the peak detection, we perform a mode-shape classification, which is formulated as a supervised binary classification problem - mode-shapes representing the syllabic nuclei as one class and remaining as the other. We use the temporal correlation and selected sub-band correlation (TCSSBC) feature contour and the mode-shapes in the TCSSBC feature contour are converted into a set of feature vectors using an interpolation technique. A support vector machine classifier is used for the classification. Experiments are performed separately using Switchboard, TIMIT and CTIMIT corpora in a five-fold cross validation setup. The average correlation coefficients for the syllable rate estimation turn out to be 0.6761, 0.6928 and 0.3604 for three corpora respectively, which outperform those obtained by the best of the existing peak detection techniques. Similarly, the average F-scores (syllable level) for the syllable nuclei detection are 0.8917, 0.8200 and 0.7637 for three corpora respectively. (C) 2016 Elsevier B.V. All rights reserved.

Experimental analysis of shape deformation of evaporating droplet using Legendre polynomials

Experiments involving heating of liquid droplets which are acoustically levitated, reveal specific modes of oscillations. For a given radiation flux, certain fluid droplets undergo distortion leading to catastrophic bag type breakup. The voltage of the acoustic levitator has been kept constant to operate at a nominal acoustic pressure intensity, throughout the experiments. Thus the droplet shape instabilities are primarily a consequence of droplet heating through vapor pressure, surface tension and viscosity. A novel approach is used by employing Legendre polynomials for the mode shape approximation to describe the thermally induced instabilities. The two dominant Legendre modes essentially reflect (a) the droplet size reduction due to evaporation, and (b) the deformation around the equilibrium shape. Dissipation and inter-coupling of modal energy lead to stable droplet shape while accumulation of the same ultimately results in droplet breakup. (C) 2013 Elsevier B.V. All rights reserved.

Tailoring the second mode of Euler-Bernoulli beams: an analytical approach

In this paper, we study the inverse mode shape problem for an Euler-Bernoulli beam, using an analytical approach. The mass and stiffness variations are determined for a beam, having various boundary conditions, which has a prescribed polynomial second mode shape with an internal node. It is found that physically feasible rectangular cross-section beams which satisfy the inverse problem exist for a variety of boundary conditions. The effect of the location of the internal node on the mass and stiffness variations and on the deflection of the beam is studied. The derived functions are used to verify the p-version finite element code, for the cantilever boundary condition. The paper also presents the bounds on the location of the internal node, for a valid mass and stiffness variation, for any given boundary condition. The derived property variations, corresponding to a given mode shape and boundary condition, also provides a simple closed-form solution for a class of non-uniform Euler-Bernoulli beams. These closed-form solutions can also be used to check optimization algorithms proposed for modal tailoring.

Rotating Beams and Nonrotating Beams With Shared Eigenpair

In this paper, we look for rotating beams whose eigenpair (frequency and mode-shape) is the same as that of uniform nonrotating beams for a particular mode. It is found that, for any given mode, there exist flexural stiffness functions (FSFs) for which the jth mode eigenpair of a rotating beam, with uniform mass distribution, is identical to that of a corresponding nonrotating uniform beam with the same length and mass distribution. By putting the derived FSF in the finite element analysis of a rotating cantilever beam, the frequencies and mode-shapes of a nonrotating cantilever beam are obtained. For the first mode, a physically feasible equivalent rotating beam exists, but for higher modes, the flexural stiffness has internal singularities. Strategies for addressing the singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test-functions for rotating beam codes and for targeted destiffening of rotating beams.

Aerodynamic characteristics of pretensioned elastic membrane rectangular sailwings

Presented in the paper are the details of a method for obtaining aerodynamic characteristics of pretensioned elastic membrane rectangular sailwings. This is a nonlinear problem governed by the membrane equation for the inflated sail and the lifting surface theory integral equation for aerodynamic loads on the sail. Assuming an admissible mode shape for the inflated elastic sail, an iterative procedure based on a doublet lattice method is employed to determine the inflated configuration as well as various aerodynamic characteristics. Application of the method is made to a typical nylon-cotton sailwing of AR = 6.0 and results are presented graphically to show the effect of various parameters. The results are found to tend to plane wing values when the pretensions are large in magnitude.

On determining the elastic modulus of a cylindrical sample subjected to flexural excitation in a resonant column apparatus

For resonant column tests conducted in the flexure mode of excitation, a new methodology has been proposed to find the elastic modulus and associated axial strain of a cylindrical sample. The proposed method is an improvement over the existing one, and it does not require the assumption of either the mode shape or zero bending moment condition at the top of the sample. A stepwise procedure is given to perform the necessary calculations. From a number of resonant column experiments on aluminum bars and dry sand samples, it has been observed that the present method as compared with the one available in literature provides approximately (i) 5.9%-7.3% higher values of the elastic modulus and (ii) 6.5%-7.3% higher values of the associated axial strains.

Decoupled three-dimensional finite element computation of thermoelastic damping using Zener's approximation

We consider three dimensional finite element computations of thermoelastic damping ratios of arbitrary bodies using Zener's approach. In our small-damping formulation, unlike existing fully coupled formulations, the calculation is split into three smaller parts. Of these, the first sub-calculation involves routine undamped modal analysis using ANSYS. The second sub-calculation takes the mode shape, and solves on the same mesh a periodic heat conduction problem. Finally, the damping coefficient is a volume integral, evaluated elementwise. In the only other decoupled three dimensional computation of thermoelastic damping reported in the literature, the heat conduction problem is solved much less efficiently, using a modal expansion. We provide numerical examples using some beam-like geometries, for which Zener's and similar formulas are valid. Among these we examine tapered beams, including the limiting case of a sharp tip. The latter's higher-mode damping ratios dramatically exceed those of a comparable uniform beam.

Stiff string approximations in Rayleigh-Ritz method for rotating beams

The governing differential equation of the rotating beam reduces to that of a stiff string when the centrifugal force is assumed as constant. The solution of the static homogeneous part of this equation is enhanced with a polynomial term and used in the Rayleighs method. Numerical experiments show better agreement with converged finite element solutions compared to polynomials. Using this as an estimate for the first mode shape, higher mode shape approximations are obtained using Gram-Schmidt orthogonalization. Estimates for the first five natural frequencies of uniform and tapered beams are obtained accurately using a very low order Rayleigh-Ritz approximation.

Rotating beams and non-rotating beams with shared eigenpair for pinned-free boundary condition

In this paper we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams.

Application of the random eigenvalue problem in forced response analysis of a linear stochastic structure

The random eigenvalue problem arises in frequency and mode shape determination for a linear system with uncertainties in structural properties. Among several methods of characterizing this random eigenvalue problem, one computationally fast method that gives good accuracy is a weak formulation using polynomial chaos expansion (PCE). In this method, the eigenvalues and eigenvectors are expanded in PCE, and the residual is minimized by a Galerkin projection. The goals of the current work are (i) to implement this PCE-characterized random eigenvalue problem in the dynamic response calculation under random loading and (ii) to explore the computational advantages and challenges. In the proposed method, the response quantities are also expressed in PCE followed by a Galerkin projection. A numerical comparison with a perturbation method and the Monte Carlo simulation shows that when the loading has a random amplitude but deterministic frequency content, the proposed method gives more accurate results than a first-order perturbation method and a comparable accuracy as the Monte Carlo simulation in a lower computational time. However, as the frequency content of the loading becomes random, or for general random process loadings, the method loses its accuracy and computational efficiency. Issues in implementation, limitations, and further challenges are also addressed.

Analytical test functions for free vibration analysis of rotating non-homogeneous Timoshenko beams

In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.

Finite element analysis of free vibration of the delaminated composite plate with variable kinematic multilayered plate elements

Composite laminates are prone to delamination. Implementation of delamination in the Carrera Unified Formulation frame work using nine noded quadrilateral MITC9 element is discussed in this article. MITC9 element is devoid of shear locking and membrane locking. Delaminated as well as healthy structure is analyzed for free mode vibration. The results from the present work are compared with the available experimental or/and research article or/and the three dimensional finite element simulations. The effect of different kinds and different percentages of area of delamination on the first three natural frequencies of the structure is discussed. The presence of open-mode delamination mode shape for large delaminations within the first three natural frequencies is discussed. Also, the switching of places between the second bending mode, with that of the first torsional mode frequency is discussed. Results obtained from different ordered theories are compared in the presence of delamination. Advantage of layerwise theories as compared to equivalent single layer theories for very large delaminations is stated. The effect of different kinds of delamination and their effect on the second bending and first torsional mode shape is discussed. (C) 2014 Elsevier Ltd. All rights reserved.

Modal tailoring and closed-form solutions for rotating non-uniform Euler-Bernoulli beams

In this paper, the free vibration of a rotating Euler-Bernoulli beam is studied using an inverse problem approach. We assume a polynomial mode shape function for a particular mode, which satisfies all the four boundary conditions of a rotating beam, along with the internal nodes. Using this assumed mode shape function, we determine the linear mass and fifth order stiffness variations of the beam which are typical of helicopter blades. Thus, it is found that an infinite number of such beams exist whose fourth order governing differential equation possess a closed form solution for certain polynomial variations of the mass and stiffness, for both cantilever and pinned-free boundary conditions corresponding to hingeless and articulated rotors, respectively. A detailed study is conducted for the first, second and third modes of a rotating cantilever beam and the first and second elastic modes of a rotating pinned-free beam, and on how to pre-select the internal nodes such that the closed-form solutions exist for these cases. The derived results can be used as benchmark solutions for the validation of rotating beam numerical methods and may also guide nodal tailoring. (C) 2014 Elsevier Ltd. All rights reserved.