Magdalena Palacz

On approximate analytical solutions for vibrations in cracked plates

Recent NATO funded research on methods for detection and interpretation methodologies for damage detection in aircraft panel structures has motivated work on low-order nonlinear analytical modelling of vibrations in cracked isotropic plates, typically in the form of aluminium aircraft panels. The work applies fundamental aspects of fracture mechanics to define an elliptical crack, and the local stress field and loading conditions, arbitrarily located at some point in the plate, and then derives an analytical expression for this that can be incorporated into the PDE for an edge loaded plate with various possible boundary conditions. The plate PDE is converted into a nonlinear Duffing-type ODE in the time domain by means of a Galerkin procedure and then an arbitrarily small perturbation parameter is introduced into the equation in order to apply an appropriate solution method, in this case the method of multiple scales. This is used to solve the equation for the vibration in the cracked plate for the chosen boundary conditions, which, in turn, leads to an approximate analytical solution. The solution is discussed in terms of the perturbation approximations that have been applied and highlights the phenomenology inherent within the problem via the specific structures of the analytical solution.

Detection of Delamination in Multilayer Composite Beams

W pracy przedstawiono model belki kompozytowej z delaminacją. Omowiono czynniki propagacji fali sprezystej w belce i mozliwości wykorzystania zmian w propagującej fali wywolanych delaminacją do jej detekcji.

Flexural-Shear Wave Propagation in Cracked Composite Beam.

A spectral finite element model for analysis of flexural-shear coupled wave propagation in a multilayer composite beam with a transverse open and not propagating crack is presented. The concept of obtaining the exact spectral element dynamic stiffness matrix is discussed. Computation is performed in the frequency domain at FFT sampling points over a broad frequency band. Post processing of the response is made in the time domain, which is suitable for structural diagnostics and broad-band wave propagation problems. Implemented numerical examples illustrate the influence of crack on wave propagation in cantilever multilayer laminated composite beams. I N T R O D U C T I O N Composite materials play an increasing role in many engineering applications. High performance, strength, stiffness and low weight are the attractive factors which increase the use of these materials in aerospace, automobile, marine and rail industries. One of the major concerns associated with composites is their susceptibility to damage, which may occur during manufacture, service or maintenance. Among others, delamination, matrix and fibre cracking are the most common damages occurring in composite materials. Although such damages are barely visible, they can severely degrade the mechanical properties and the load carrying capability of the structure. Any growth of this damage can lead to fracture of the material. Most of the structural health monitoring methods can be classified into model-based and signal-based approaches. The former utilises structural physical parameters for damage detection and is rather related to modelling and identification problems; any change of structural physical parameters can indicate damage /I/. The signal-based approach uses different vibration, strain, acoustical and ultrasonic signals for damage detection and is related to signal processing. It is usually based on a relationship between a structure condition and a damage symptom or feature, where the problem is to find symptoms which are sensitive to damage and damage evolution. The paper is devoted to utilizing wave propagation as an efficient tool for damage detection in composite structures. Many numerical methods are applied for wave propagation modelling. The most efficient and convenient among them is spectral element method ( S E M ) 121. The S E M is based on exact solution to governing Partial Differential Equations (PDE) in the frequency domain /3/. This exact solution is used as interpolating function for spectral element formulation. The use of an exact solution in the element formulation ensures exact mass and stiffness distribution. A s a result, the element directly yields the exact dynamic

Spectral Plate Element for Crack Detection with the Use of Propagating Waves

The article presents a new spectral finite plate element with a crack. It is assumed that the crack having an arbitrary length, depth and location is parallel to one side of the plate. Elastic behaviour of the plate at the crack location is considered as a line spring with a varying stiffness along the crack length. Elaborated model is suitable for the analysis of wave propagation in platelike structures and the same for utilizing the differences in propagating waves for structural health monitoring.

Wavelet Analysis for Damage Identification in Composite Structures

The main aspect of the paper is to give an answer to the question of what specific kind of defect has actually occurred in a structure and how to distinguish between different kinds of discontinuities. For this purpose composite rods and beams with fatigue cracks, step changes in cross-sectional area and small changes in material properties have been investigated. The objective of the work has been to propose a signal processing methodology based on wavelet transformation for identification of specific discontinuity. The identification of a fatigue crack from other discontinuities has been demonstrated. It has been also found that the proposed methodology might be useful for precise indication of the size of the identified fatigue damage.

Transmission and Reflection Coefficients for Damage Identification in 1D Elements

According to the latest research results presented in the literature changes in propagating waves are one of the most promising parameters for damage identification algorithms. Numerous publications describe methods of damage identification based on the analysis of signals reflected from damage. They also include complicated signal processing techniques. Such methods work well for damage localisation, but it is rather difficult to use them in order to estimate the size of damage. It is natural that propagating wave reflects from any structural discontinuity. The bigger the disturbance the bigger part of a propagating wave reflects from it. The amount of energy reflected and transmitted through any discontinuity can expressed as reflection and transmission coefficients. In the literature different application for these coefficients may be found – the most often cited application is connected with localising changes in the geometry of structures. Changes in the coefficients due to cross section variations in rods and beams or due to existence of stiffeners in plates are well documented. However there are no application of using the reflection and transmission coefficients for damage size identification. For this reason the analysis presented in this paper has been carried out. The article presents a method of damage identification in 1D elements based on the wave propagation phenomenon and changes in reflection and transmission coefficients. The changes in transmission and reflection coefficients for waves propagating in isotropic rods with different types of damage have been analysed. The rods have been modelled with the elementary, two and three mode theories or rods. For numerical modelling the Spectral Finite Element Method has been used. Several examples are given in the paper.

On the Problem for Damage Detection of Vibrating Cracked Plates

This work was motivated by the recent NATO funded research on preventing disasters from collapse and improving the safety of aircraft structures. It considers the problem for vibrationbased damage detection in aircraft panels modelled as isotropic plates. The explored method does not use any assumptions of model or linearity, it is simply based on pure signal analysis of the vibration response of plates. FE modelling is used to model the plate’s dynamic response in its intact and in its damaged state. The signals obtained are analysed using multivariate analysis applied in the measured frequency domain. This reduces the data dimensionality and is expected to have a clustering effect. At this stage the measured data is transformed into features – new variables- which have smaller dimension than the initial ones and make the categories more distinguishable. Then a very simple pattern recognition (PR) method is applied to discriminate between the two categories of data -data coming from the undamaged plate and data coming from the damaged plate. This is the second stage when the obtained features are used for the actual recognition between the defined categories. The paper suggests the use of the Karhunen-Loeve transform in order to extract features from the measured frequency response functions of the plate. When the data dimensionality is brought down to two the response of the plate can be visualised. The clustering effect on the features coming from undamaged plate and those from the damaged is obvious.

Damage Identification in 1D Structures

The paper summarizes experimental and numerical analysis directed towards identification of damages in one-dimensional constructional elements. The presented method of structural damage identification makes use of high frequency signals, which are sensitive to small discontinuities in structures. The experimental set is presented. Comparison of numerical and experimental results is discussed. Promising conclusions are derived.

Experimental and Numerical Investigation of Wave Propagation in Composite Beam with an Additional Mass

The article is to show results of numerical and experimental examination of changes in wave propagation in a composite rod with additional mass. For numerical modelling the spectral element method is used. For experimental verification the IFFM PAS laboratory equipment was used. As actuators and sensors PZT elements were utilised. The results obtained via numerical and experimental simulations are compared and discussed.