Application of continuous wavelet transform in vibration based damage detection method for beams and plates
Introduction
The application of wavelet transforms to a wide variety of problems is so plentiful that they have emerged as the most promising techniques in the past decade. Wavelets help to analyse variations of values at financial markets. The biologists use them for cell membrane recognition. The Federal Bureau of Investigation (FBI) considers wavelet application for storage of 30 million sets of criminal fingerprints [1]. The computer scientists exploit them in image processing like edge recognition, image searching, animation control, image compression and even Internet traffic description. The engineers use wavelet transforms for time phenomena study in transient processes. Recently, wavelets have been tested for structural health monitoring and damage detection. The ability to monitor a structure and detect damage at earliest possible stage becomes an important issue throughout the aerospace, mechanical and civil engineering communities.
The literature on wavelet transforms in the one-dimensional case is very extensive. Applicability of various wavelets in detection of cracks in beams has been studied by Douka et al. [2], Quek et al. [3] as well as Gentile and Messina [4]. Frame structures have been analysed by Ovanesova and Suarez [5]. For a practical application of the wavelet damage detection techniques research on experimental data is the most important. Hong et al. [6] and Douka et al. [2] showed that the effectiveness of wavelets for damage localization is limited by the measurements precision and the sampling distances. They used the dynamic mode shapes extracted from the acceleration measurements. One accelerometer was kept as a reference input, while the second one was moved along the beam. They preformed the measurements in 39 points of the beam. For wavelet analysis the signal was oversampled to 390 points by a cubic spline interpolation. Rucka and Wilde [7] used the optic displacement measurement technique that allowed the high precision measurements of the beam static displacements in 81 points. Although current works show that only relatively large cracks can be detected, the search for the structural damage by wavelets is a promising and developing field of research.
The two-dimensional damage detection problems were addressed by Wang and Deng [8]. They analysed a steel plate with an elliptical hole and subjected to a uniform tensile loading. The static displacement field was determined by the analytical formula and was considered as input for the wavelet transform. The location of the crack tip was found by a variation of the Haar wavelet coefficients. Douka et al. [9] studied vibrations of a rectangular plate with a crack running parallel to one side of the plate. The one-dimensional wavelet transform was successfully applied to the analytically determined mode shapes along their vertical lines at different locations. Cracks of a relative depth varied from 10% up to 50% have been considered. The proposed intensity factor allowed estimation of the damage size. The works based on numerically computed plate mode shapes were presented by Chang and Chen [10] and Rucka and Wilde [11]. The wavelet transforms of the two-dimensional plate problems [8], [9], [10], [11] were addressed by the one-dimensional wavelet analysis since the signal lines at the different locations have been treated separately. The two-dimensional discrete wavelet transform for detection of cracks in plates based on numerical data was presented by Loutridis et al. [12].
The experimental researches on plate damage detection have been presented by Wilde and Rucka [13]. The experimental mode shapes of the cantilever plate have been determined by the acceleration measurement in one point and impact excitation in 66 points. The relative depth of the introduced rectangular defect was about 19%. The location of the damage was determined by the Gaussian wavelet with four vanishing moments. However, the problem was approached by the one-dimensional wavelet formulation.
In this paper a method for estimating damage localization in a beam and plate is presented. The damage localization is based on the experimentally determined mode shapes of a cantilever beam and a plate with four fixed supports. For the plate problem the two-dimensional formulation of the wavelet transform is derived.
Section snippets
One-dimensional wavelet transform
A wavelet is an oscillatory, real or complex-valued function of zero average and finite length. Function ψ(x) is called a mother wavelet and denotes the Hilbert space of measurable, square-integrable one-dimensional functions. In this paper, apart from general definition, only the real wavelets and the space domain will be considered. The function ψ(x) localized in both space and frequency domains is used to create a family of wavelets ψu,s(x) formulated as
Experimental set-up
A cantilever beam and a plate with four fixed supports are considered. The beam (Fig. 4) of length L=480 mm, width B=60 mm and height H=20 mm is made of polymethyl methacrylate (PMMA), sold by the tradenames Plexiglas. The experimentally determined material properties are: Young's modulus E=3420 MPa, Poisson ratio ν=0.32 and mass density ρ=1187 kg/m3. The beam contains an open crack of length Lr=2 mm and height a=7 mm at a distance L1=120 mm from the clamped end. The depth of the crack is 35% of beam
Numerical mode shapes of beam and plate
The mode shapes for the notched beam and plate were computed by the commercial FEM program SOFiSTiK. The beam mode shape was calculated using a solid six-sided element of length 2 mm. The mode shapes of the plate were computed using square plane element of the size 40×40 mm. The first calculated frequency for the beam is f1=23.62 Hz while the plate first frequency is f1=65.21 Hz. The computed frequencies of both structures are very similar to the experimentally obtained frequencies. A comparison
Border distortion problem
The CWT is defined as integration of the product of a wavelet and a signal of infinite length. Since mode shape of the beam f (x) as well as the mode shape of the plate f (x, y) are signals of finite length a border distortion problem appears. The wavelet coefficients achieve an extremely high value at the ends of a signal and those values do not indicate damage. Therefore, the border of the signal should be treated independently from the rest of the signal. The influence of boundary effects can
Conclusions
The presented work is devoted to the wavelet-based damage detection techniques in beam and plate structures. The wavelet transforms are applied to fundamental mode shapes of the beam and plate. The mode shapes are determined experimentally and numerically. The one-dimensional wavelet analysis has been extended for application in two-dimensional structures. The formulation of the wavelet transform for two-dimensional plate problems is presented.
The study on wavelet analysis applied in damage
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