Crack identification in beams using wavelet analysis

Abstract

In this paper a simple method for crack identification in beam structures based on wavelet analysis is presented. The fundamental vibration mode of a cracked cantilever beam is analyzed using continuous wavelet transform and both the location and size of the crack are estimated. The position of the crack is located by the sudden change in the spatial variation of the transformed response. To estimate the size of the crack, an intensity factor is defined which relates the size of the crack to the coefficients of the wavelet transform. An intensity factor law is established which allows accurate prediction of crack size. The viability of the proposed method is investigated both analytically and experimentally in case of a cantilever beam containing a transverse surface crack. In the light of the results obtained, the advantages and limitations of the proposed method as well as suggestions for future work are presented and discussed.

Introduction

Cracks present a serious threat to the performance of structures since most of the structural failures are due to material fatigue. For this reason, methods allowing early detection and localization of cracks have been the subject of intensive investigation the last two decades. As a result, a variety of analytical, numerical and experimental investigations now exist. A review of the state of the art of vibration based methods for testing cracked structures has been published by Dimarogonas (1996).

A crack in a structure induces a local flexibility which affects the dynamic behaviour of the whole structure to a considerable degree. It results in reduction of natural frequencies and changes in mode shapes of vibration. An analysis of these changes makes it possible to identify cracks. In the pioneering work of Dimarogonas (1976) and Paipetis and Dimarogonas (1986) the crack was modelled as a local flexibility and the equivalent stiffness was computed using fracture mechanics methods. In that vein, Chondros and Dimarogonas (1980) developed methods to identify cracks in various structures relating the crack depth to the change in natural frequencies for known crack position. Adams and Cawley (1979) developed an experimental technique to estimate the location and depth of a crack from changes in natural frequencies. Gudmunson (1982) used a perturbation method to predict changes in natural frequencies of structures resulting from cracks, notches and other geometrical changes. Further work on crack identification via natural frequency changes was done by Anifantis et al. (1985). Using a similar approach Masoud et al. (1998) investigated the vibrational characteristics of a prestressed fixed–fixed beam with a symmetric crack and the coupling effect between crack depth and axial load. Narkis (1993) developed a closed form solution for the problem of a cracked beam, which he applied to study the inverse problem of localization of cracks on the basis of natural frequency measurements.

The main reason for the popularity of natural frequencies as damage indicators is that natural frequencies are rather easy to determine with a high degree of accuracy. A sensor placed on a structure and connected to a frequency analyzer gives estimates of several natural frequencies. Problems exist, however, when the size of the damage is small. The presence of measurement errors results in a degradation of the ability to predict the size of the crack accurately. The existing methods give a proper estimation of moderate cracks (about 20% of the height of the beam).

To overcome the aforementioned difficulties related to natural frequencies, many research studies have been focused on utilizing changes in mode shapes (Stubbs and Kim, 1966; Kim and Stubbs, 2002). The idea of using mode shapes as crack identification tool is the fact that the presence of a crack causes changes in the modal characteristics. Rizos et al. (1990) suggested a method for using measured amplitudes of two points of a cantilever beam vibrating at one of its natural modes to identify crack location and depth. Recently, an interesting comparison between a frequency––based and mode shape––based method for damage identification in beam like structures has been published by Kim et al. (2003). The advantage of using mode shapes is that changes in mode shapes are much more sensitive compared to changes in natural frequencies. Using mode shapes, however, has some drawbacks. The presence of damage may not significantly influence mode shapes of the lower modes usually measured. Furthermore, environmental noise and choice of sensors used can considerably affect the accuracy of the damage detection procedure. To overcome these difficulties, modal testing using scanning laser vibrometers have been developed (Stanbridge and Ewing, 1999). The laser vibrometer, used as a vibration transducer, has the advantage of being non-contacting and measures at a controlled position with high accuracy.

In the last few years, wavelet analysis has become a promising damage detection tool due to the fact that it is very accurate to detect localized abnormalities in a mode shape caused by the presence of a crack. It has useful localization characteristics and does not require the numerical differentiation of the measured data (Newland, 1994a, Newland, 1994b). Wavelet transform can be implemented as fast as the Fourier transform and its main advantage is the fact that the local features in a signal can be identified with a desired resolution.

Deng and Wang (1998) applied the discrete wavelet transform to locate a crack along the length of a beam. Wang and Deng (1999) extended the analysis to a plate with a through-thickness crack. In the last study, the Haar wavelet were used with success. However, a method for estimating the crack extend has not been proposed. Haar wavelet were also used in the study of Quek et al. (2001). The authors were able to accurately detect relatively small cracks under both simply-supported and fixed-ended conditions. Here again, the estimation of the size of the crack is not discussed. Hong et al. (2002) used the Lipschitz exponent for the detection of singularities in beam modal data. The Mexican hat wavelet was used throughout the study and the crack size has been related to different values of the exponent. The correlation of the crack extent with the Lipschitz exponent is sensitive to both sampling distance and noise resulting in limited accuracy of the prediction.

In the present work, a method for crack identification in beam structures based on wavelet analysis is presented. The fundamental vibration mode of a cracked beam is wavelet transformed and both the location and size of the crack are estimated. For this purpose, a “symmetrical 4” wavelet having two vanishing moments is utilized. The position of the crack is located by the variation of the spatial signal at the site of the crack due to the high resolution property of the wavelet transform. To estimate the size of the crack, an intensity factor is defined which relates the size of the crack to the coefficients of the wavelet transform. An intensity factor law is established which allows accurate prediction of the crack size. The feasibility of the proposed method is investigated both analytically and experimentally in case of a cantilever beam containing a transverse surface crack. The influence of noise on the estimation of crack size has been also investigated. It is shown that noise added to mode shape increases the estimated crack size. In view of the results obtained, the limitations and the advantages of the proposed method as well as suggestions for future work are presented and discussed.