Papers by John Mottershead
Mechanical Systems and Signal …, Jan 1, 2009
In structural finite element model (FEM) updating an important step is the comparison/correlation... more In structural finite element model (FEM) updating an important step is the comparison/correlation between the FEM and experimental data. Currently the most widely used method is the modal assurance criterion (MAC), which can be interpreted as the cosine of the angle between numerical (FE model) and measured eigenvectors. However, the eigenvectors only contain the displacement of discrete coordinates, so that the MAC index carries no explicit information on shape features. A new technique, based upon the well-developed philosophies of image processing (IP) and pattern recognition (PR) are considered in this paper. The Zernike moment descriptor (ZMD) is a region-based shape descriptor having outstanding properties in IP including rotational invariance, expression and computing efficiency, ease of reconstruction and robustness to noise. In this paper the ZMD is applied to the problem of mode-shape recognition for simple plate structures. Result shows that the ZMD has considerable advantages over the traditional MAC index when identifying the cyclically symmetric mode shapes that occur in axisymmetric structures at identical frequencies.
Keywords: Zernike moment; Mode-shape recognition; Model updating
Journal of Sound and Vibration, Jan 1, 2009
Currently the most widely used method for comparing mode shapes from finite elements and experime... more Currently the most widely used method for comparing mode shapes from finite elements and experimental measurements is the modal assurance criterion (MAC), which can be interpreted as the cosine of the angle between the numerical and measured eigenvectors. However, the eigenvectors only contain the displacement of discrete coordinates, so that the MAC index carries no explicit information on shape features. New techniques, based upon the well-developed philosophies of image processing (IP) and pattern recognition (PR) are considered in this paper. The Zernike moment descriptor (ZMD), Fourier descriptor (FD), and wavelet descriptor (WD) are the most popular shape descriptors due to their outstanding properties in IP and PR. These include (1) for the ZMD-rotational invariance, expression and computing efficiency, ease of reconstruction and robustness to noise; (2) for the FD—separation of the global shape and shape-details by low and high frequency components, respectively, invariance under geometric transformation; (3) for the WD—multi-scale representation and local feature detection. Once a shape descriptor has been adopted, the comparison of mode shapes is transformed to a comparison of multidimensional shape feature vectors. Deterministic and statistical methods are presented. The deterministic problem of measuring the degree of similarity between two mode shapes (possibly one from a vibration test and the other from a finite element model) may be carried out using Pearson's correlation. Similar shape feature vectors may be arranged in clusters separated by Euclidian distances in the feature space. In the statistical analysis we are typically concerned with the classification of a test mode shape according to clusters of shape feature vectors obtained from a randomised finite element model. The dimension of the statistical problem may often be reduced by principal component analysis. Then, in addition to the Euclidian distance, the Mahalanobis distance, defining the separation of the test point from the cluster in terms of its standard deviation, becomes an important measure. Bayesian decision theory may be applied to formally minimise the risk of misclassification of the test shape feature vector. In this paper the ZMD is applied to the problem of mode shape recognition for a circular plate. Results show that the ZMD has considerable advantages over the traditional MAC index when identifying the cyclically symmetric mode shapes that occur in axisymmetric structures at identical frequencies. Mode shape recognition of rectangular plates is carried out by the FD. Also, the WD is applied to the problem of recognising the mode shapes in the thin and thick regions of a plate with different thicknesses. It shows the benefit of using the WD to identify mode-shapes having both local and global components. The comparison and classification of mode shapes using IP and PR provides a ‘toolkit’ to complement the conventional MAC approach. The selection of a particular shape descriptor and classification method will depend upon the problem in hand and the experience of the analyst.
ICEM
It is good practice to validate analytical and numerical models used in stress analysis for engin... more It is good practice to validate analytical and numerical models used in stress analysis for engineering design by comparison with measurements obtained from real components either in-service or in the laboratory. In reality, this critical step is often neglected or reduced to placing a single strain gage at the predicted hot-spot of stress. Modern techniques of optical analysis allow full-field maps of displacement, strain and, or stress to be obtained from real components with relative ease and at modest cost. However, validations continued to be performed only at predicted and, or observed hot-spots and most of the wealth of data is ignored. It is proposed that image decomposition methods, commonly employed in techniques such as fingerprinting and iris recognition, can be employed to validate stress analysis models by comparing all of the key features in the data from the experiment and the model. Image decomposition techniques such as Zernike moments and Fourier transforms have been used to decompose full-field distributions for strain generated from optical techniques such as digital image correlation and thermoelastic stress analysis as well as from analytical and numerical models by treating the strain distributions as images. The result of the decomposition is 101 to 102 image descriptors instead of the 105 or 106 pixels in the original data. As a consequence, it is relatively easy to make a statistical comparison of the image descriptors from the experiment and from the analytical/numerical model and to provide a quantitative assessment of the stress analysis.
Journal of Sound and …, Jan 1, 2010
The comparison of structural responses, e.g. natural frequencies and mode shapes, between predict... more The comparison of structural responses, e.g. natural frequencies and mode shapes, between predictions and measurements is an important step in finite element (FE) model updating. Full-field measurement techniques such as high speed cameras with digital image correlation (DIC) algorithms provide detailed, global displacement data. It is necessary to compress huge amounts of full-field data before implementing the comparison procedures. Image processing and pattern recognition techniques offer effective ways of doing this. Image decomposition using integral transformation is one of the most common procedures. It is found that appropriate selection or construction of the transformation kernels usually generates succinct and effective shape feature terms. Thus, the discrepancies between the geometric mode shapes may be assessed by using distance measures between the shape feature vectors. In the present study, vibration mode shapes of a composite panel are measured by a DIC system and predicted by a FE model. Succinct and effective shape features of the full-field mode shapes were obtained by employing the Tchebichef moment descriptor. Mode shape discrepancies are clearly indicated by the resultant Tchebichef features. The FE model was then modified and updated. Results show that including only the shape features results in a better updated model than when natural frequencies only are used. The most improved model was obtained when both natural frequencies and shape features are included in the updating routine. © 2010 Elsevier Ltd. All rights reserved.
Journal of Physics: …, Jan 1, 2009
The most widely used method for comparing mode shapes from finite elements and experimental measu... more The most widely used method for comparing mode shapes from finite elements and experimental measurements is the Modal Assurance Criterion (MAC), which returns a single numerical value and carries no explicit information on shape features. New techniques, based on image processing (IP) and pattern recognition (PR) are described in this paper. The Zernike moment descriptor (ZMD), Fourier descriptor (FD), and wavelet descriptor (WD), presented in this article, are the most popular shape descriptors having properties that include efficiency of expression, robustness to noise, invariance to geometric transformation and rotation, separation of local and global shape features and computational efficiency. The comparison of mode shapes is readily achieved by assembling the shape features of each mode shape into multi-dimensional shape feature vectors (SFVs) and determining the distances separating them.
Recent advances in measurement techniques, including digital image correlation, automated photoel... more Recent advances in measurement techniques, including digital image correlation, automated photoelasticity, electronic speckle pattern interferometry and thermoelastic stress analysis, permit full-field maps of displacement or strain to be obtained easily. They provide large volumes of mostly redundant data, which should be condensed to the essential information to permit straightforward processes such as validations of computational models or damage assessments. A way to do this is by image processing, an important aspect of which is the definition of an orthogonal basis (orthogonal kernel functions). Generally, this is problem dependent and requires some skill from the analyst if the number of image features (the coefficients of the orthogonal basis) is to be restricted to a suitably small number. Advantage may be taken of patterns of symmetry, for example cyclically symmetric patterns are well-suited to treatment by Zernike polynomials and rectangular patterns are well-suited to treatment by Fourier series. The Zernike and Fourier kernels are continuous polynomials with orthogonality properties that require integration and must be discretised. The discrete Tchebichef polynomials are ideal for the treatment of full-field information at multiple discrete data points. In many cases the data field is localised around a particular feature, such as local strain around a hole in a tension-test specimen. In this case, the polynomial basis should similarly be localised by various forms of scaling – this requires the application of the Gram-Schmidt procedure to maintain orthogonality. The image features (sometimes called shape features) are meaningful and may be used to identify particular patterns in the data – e.g. for detecting cracks or other forms of damage. When assembled in a feature vector, the distance between feature vectors from measured and numerical results are useful for refining numerical models. In this paper the principles of image analysis, as applied to full-field displacement/strain data are explained and experimental examples are used to illustrate the practical usefulness of the method. The applications include (i) vibration mode shapes of laminated honeycomb structures and, (ii) strain in an aluminium plate with a central hole in tension.
The term damage can be generally defined as a change introduced into a system that affects its pe... more The term damage can be generally defined as a change introduced into a system that affects its performance. Its identification and characterization is a valid help in deciding amongst continuing the operation or performing a repair or replacement of the system. A valid support to this decision is based on the use of well-known measurement techniques from Non-Destructive Testing and Evaluation (NDT&E). A well-established correlation between damage and features extracted from the measured data makes these techniques capable of providing information about the extent of the damage. However the prediction of the remaining useful life of a system by comparing full-filed measurements techniques and FEM analysis results is the challenge of an increasing number of research studies. The need of a guide for enumerating the extent of the damage has been the thrust to perform this work. A common methodology developed for both numerical and experimental studies will be presented. It consists of three main parts: proper selection of the parameter capable of describing the damage in a quantitative manner; several approaches to obtain results from measurement techniques and FEM analysis; and damage assessment making use of a quantitative comparison of FEM results only, full-field experimental results only, or comparison of FEM to experimental results. A different approach in damage assessment will be also presented making use of Zernike moment descriptors from which the severity of the damage is inferred. An example to illustrate the methodology will be shown.
Finite element model updating of a Westland Lynx XZ649 helicopter tail is presented. Eigenvalue s... more Finite element model updating of a Westland Lynx XZ649 helicopter tail is presented. Eigenvalue sensitivities with respect to Young’s modulus and mass density are used. Large groups based on material input data were divided to form smaller subgroups so that those parts of the model responsible for errors in the predicted eigenvalues were located. A particular new development was the use of parameter clustering based on the similarity of different columns of the sensitivity matrix. Finally the finite element model was updated successfully with regard to the lower frequency tail-bending modes.
The achievement of high levels of confidence in finite element models involves their validation u... more The achievement of high levels of confidence in finite element models involves their validation using measured responses such as static strains or vibration mode shapes. A huge amount of data with a high level of information redundancy is usually obtained in both the detailed finite element prediction and the full-field measurements so that achieving a meaningful validation becomes a challenging problem. In order to extract useful shape features from such data, image processing and pattern recognition techniques may be used. One of the most commonly adopted shape feature extraction procedures is the Fourier transform in which the original data may be expressed as a set of coefficients (coordinates) of the decomposition kernels (bases) in the feature space. Localised effects can be detected by the wavelet transform. The acquired shape features are succinct and therefore simplify the model validation, based on the full-field data, allowing it to be achieved in a more effective and efficient way. In this paper, full-field finite element strain patterns of a plate with a centred circular hole are considered. A special set of orthonormal shape decomposition kernels based on the circular Zernike polynomials are constructed by the Gram-Schmidt orthonormalization process. It is found that the strain patterns can suitably be represented by only a very small number of shape features from the derived kernels.
This paper presents a methodology for the treatment of uncertainty in nonlinear, interference-fit... more This paper presents a methodology for the treatment of uncertainty in nonlinear, interference-fit, stress analysis problems arising from manufacturing tolerances. Image decomposition is applied to the uncertain stress field to produce a small number of shape descriptors that allow for variability in the location of high stress points when geometric parameters (dimensions) are changed within tolerance ranges. A meta-model, in this case based on the polynomial chaos expansion (PCE), is trained using a full finite element model to provide a mapping from input geometric parameters to output shape descriptors. Global sensitivity analysis using Sobol' indices provides a design tool that enables the influence of each input parameter on the observed variances of the outputs to be quantified. The methodology is illustrated by a simplified practical design problem in the manufacture of automotive wheels.
A full-field Digital Image Correlation (DIC) method with integrated Kriging regression is present... more A full-field Digital Image Correlation (DIC) method with integrated Kriging regression is presented in this paper. The displacement field is formulated as a best linear unbiased model that includes the correlations between all the locations in the Region of Interest (RoI). A global error factor is employed to extend conventional Kriging interpolation to quantify displacement errors of the control points. An updating strategy for the self-adaptive control grid is developed on basis of the Mean Squared Error (MSE) determined from the Kriging model. Kriging DIC is shown to outperform several other full-field DIC methods when using open-access experimental data. Numerical examples are used to demonstrate the robustness of Kriging DIC to different choices of initial control points and to speckle pattern variability. Finally Kriging DIC is tested on an experimental example.
For the analysis of vibrations and mode shape extraction in particular the use of optical full-fi... more For the analysis of vibrations and mode shape extraction in particular the use of optical full-field measurement techniques has grown during the last years. Beside techniques like Digital Speckle Pattern Interferometry, Moiré, Thermography or Photoelasticity the Digital Image Correlation techniques have already been successfully proven to be an accurate displacement analysis tool for a wide range of applications.
In 1972 AH Vincent, the then Chief Dynamicist at Westland Helicopters, discovered that when a str... more In 1972 AH Vincent, the then Chief Dynamicist at Westland Helicopters, discovered that when a structure excited at point p with a constant frequency is modified, for example by the addition of a spring between two points r and s, then the response at another point q traces a circle ...
Applied Mechanics and Materials, 2011
Recent advances in measurement techniques, including digital image correlation, automated photoel... more Recent advances in measurement techniques, including digital image correlation, automated photoelasticity, electronic speckle pattern interferometry and thermoelastic stress analysis, permit full-field maps of displacement or strain to be obtained easily. They provide large volumes of mostly redundant data, which should be condensed to the essential information to permit straightforward processes such as validations of computational models or damage assessments. A way to do this is by image processing, an important aspect of which is the definition of an orthogonal basis (orthogonal kernel functions). Generally, this is problem dependent and requires some skill from the analyst if the number of image features (the coefficients of the orthogonal basis) is to be restricted to a suitably small number. Advantage may be taken of patterns of symmetry, for example cyclically symmetric patterns are well-suited to treatment by Zernike polynomials and rectangular patterns are well-suited to t...
The Journal of Strain Analysis for Engineering Design, 2013
ABSTRACT Recent advances in measurement techniques such as digital image correlation, automated p... more ABSTRACT Recent advances in measurement techniques such as digital image correlation, automated photoelasticity, electronic speckle pattern interferometry and thermoelastic stress analysis allow full-field maps (images) of displacement or strain to be obtained easily. This generally results in the acquisition of large volumes of highly redundant data. Fortunately, image decomposition offers feasible techniques for data condensation while retaining essential information. This permits data processing such as the validation of computational models, modal testing or structural damage assessment efficiently and in a straightforward way. The selection, or construction, of decomposition bases (kernel) functions is essential to data reduction and has been shown to produce features, or attributes, of the full-field image that are effective in reproducing the measured information, succinct in condensation and robust to measurement noise. Among the most popular kernel functions are the orthogonal Fourier series, wavelets and Legendre polynomials, which are defined on continuous rectangular domains, and Zernike polynomials and Fourier–Mellin functions, which are defined on continuous circular domains. The discrete orthogonal polynomials include Tchebichef, Krawtchouk and Hahn functions that are directly applicable to digital images and avoid the approximate numerical integration that becomes necessary with the sampling of continuous kernel functions. In practice, full-field measurements of the engineering components are usually non-planar within irregular domains – neither rectangular nor circular, so that the classical kernel functions are not immediately applicable. To address this problem, a complete methodology is described, consisting of (1) surface parameterisation for the mapping of three-dimensional surfaces to two-dimensional planar domains, (2) Gram–Schmidt orthogonalisation for the construction of orthogonal kernel functions on arbitrary domains and (3) reconstruction of localised image features, such as regions of high strain gradient, by a windowing technique. Application of this methodology is demonstrated in a series of illustrative examples
Damage Assessment of Structures X, Pts 1 and 2, 2013
ABSTRACT Damage assessment of composite materials is crucial for health monitoring of engineering... more ABSTRACT Damage assessment of composite materials is crucial for health monitoring of engineering structures. It is particularly important to detect damage invisible to the human eye caused by low speed impact. Optical non-contact sensing techniques enable full-field measurements from structural responses. However, damage is generally associated with its local effect on deformation and strain patterns while full-field measured data is highly information redundant. It is possible to apply image processing techniques [1, 2] to extract succinct and efficient features, or attributes, from full-field data. Iterative reanalysis of large and detailed numerical models is generally very expensive and may not be feasible. Meta modeling is one of the practical ways to overcome the problem of high computational cost. In this paper, a case study of composite delamination assessment based on simulation is discussed. The damage mode in the composite plate is assumed to be a delamination of elliptical shape. Surface strain of a specimen under tensile loading is considered to be the measured structural output. The Krawtchouk moment descriptor is applied to extract a small number of features from the strain map. A meta model in the form of a Kriging predictor [3] is constructed to map the damage parameters to Krawtchouk shape features of the strain distribution. The delamination region is quantified using an inverse procedure based on the trained Meta model. Furthermore, a biomimicry algorithm, particle swarm optimisation, is applied to detect the location of the delamination.
Applied Mechanics and Materials, 2011
For the analysis of vibrations and mode shape extraction in particular the use of optical full-fi... more For the analysis of vibrations and mode shape extraction in particular the use of optical full-field measurement techniques has grown during the last years. Beside techniques like Digital Speckle Pattern Interferometry, Moiré, Thermography or Photoelasticity the Digital Image Correlation techniques have already been successfully proven to be an accurate displacement analysis tool for a wide range of applications.
Applied Mechanics and Materials, 2006
Page 1. Finite Element Model Updating of Large Structures by the Clustering of Parameter Sensitiv... more Page 1. Finite Element Model Updating of Large Structures by the Clustering of Parameter Sensitivities H. Shahverdi 1,a , C. Mares 2,b , W. Wang 1,c , CH Greaves 3,d and JE Mottershead 1,e 1Mechanical Engineering Division, University of Liverpool, Liverpool, L69 3GH, UK ...
International Journal of Solids and Structures, 2011
Finite element model updating is an inverse problem based on measured structural outputs, in this... more Finite element model updating is an inverse problem based on measured structural outputs, in this case maximum principal strain measured using digital image correlation. Full-field responses in the form of strain maps contain valuable information for model updating but within large volumes of highly-redundant data. In this paper, shape descriptors based on Zernike polynomials having the properties of orthogonality and rotational invariance are shown to be powerful decomposition kernels for defining the shape or map of the strain distribution. A square plate with a circular hole subject to a uniaxial tensile load is considered and effective shape features are constructed using a set of modified Zernike polynomials. The modification includes the application of a decaying weighting function to the Zernike polynomials so that high strain magnitudes around the hole are well-represented. The Gram-Schmidt process is then used to ensure orthogonality for the obtained decomposition kernels over the domain of the specimen, i.e. excluding the hole. Results show that only a very small number of Zernike moment descriptors are necessary and sufficient to represent the full-field data. The onset of yielding may be quantified using the descriptors. Furthermore, model updating of nonlinear elasto-plastic material properties is carried out using the Zernike moment descriptors derived from full-field strain measurements.
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Papers by John Mottershead
Keywords: Zernike moment; Mode-shape recognition; Model updating
Keywords: Zernike moment; Mode-shape recognition; Model updating