11.6 Damage Detection

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Properties of the wavelet transform are also quite appealing for damage-detection purposes. Early

investigations in this field [93,94] used wavelet analysis to detect local faults in machineries. Specifically,

visual inspection of the modulus and phase of the wavelet transform was used to localize the fault [93].

Further, it was shown that transient vibrations due to developing damage are disclosed by the local

maxima of the mean-square wavelet map [94]. These investigations gave a qualitative approach to

damage detection as no estimate of the damage amplitude was provided. Additional studies have

confirmed the correlation between local maxima of the wavelet transform and damage in beams and

plates [95 – 97], and a first attempt to estimate the damage amplitude was made by Okafor and Dutta

[98]. Specifically, Daubechies wavelets were used to wavelet transform the mode shapes of a damaged

cantilever beam, and a regression analysis by a least-square method was conducted to correlate the peaks

of the wavelet coefficients with the corresponding damage amplitude.

A consistent mathematical framework for wavelet analysis of damaged beams is due to Hong et al.

[99]. The focal concept is that defects in structures, even if small, may affect significantly the vibration

mode shapes, depending on the location and the kind of damage. Such variations may not be apparent in

the measured data but become detectable as singularities if wavelet analysis is used due to its high

resolution properties. Specifically, Hong et al. have shown that the singularity of the vibration modes can

be described in terms of Lipschitz regularity, a concept also encountered in the theory of differential

equations, widely used in image processing where object contours correspond to irregularities in the

intensity [100,101]. In mathematical terms, a function f ðxÞ is Lipschitz a $ 0 at x ¼ x0 if there exists

K . 0, and a polynomial of order m (m is the largest integer satisfying m # a), pmðxÞ, such that and a

polynomial of order m; pmðxÞ; such that

f ðxÞ ¼ pmðxÞ þ 1ðxÞ ð11:77Þ 􀀈 􀀈

1ðxÞ

􀀈 􀀈

# K

􀀈 􀀈

x 2 x0

􀀈 􀀈

a

ð11:78Þ

The wavelet transform of Lipschitz a functions enjoys some properties. Mallat and Hwang [100] have

shown that for a wavelet basis with a number of vanishing moments a # n; a local Lipschitz singularity

at x0 corresponds to maxima lines of the wavelet transform modulus. That is, local maxima with

asymptotic decay across scales. Near the cone of influence x ¼ x0; such moduli satisfy the equation

􀀈 􀀈

Wf ða; xÞ

􀀈 􀀈

# Aaaþ1=2; A . 0 ð11:79Þ

from which the Lipschitz exponent is computed as

log2

􀀈 􀀈

Wf ða; xÞ

􀀈 􀀈

# log2A þ a þ

1

2

􀀏 􀀐

log2a ð11:80Þ

By plotting the wavelet coefficients on a logarithmic scale, A and a may be computed by setting the

equality sign in Equation 11.80 and minimizing the error in the least-square sense. Hong et al. have

Wavelets — Concepts and Applications 11-17

© 2005 by Taylor & Francis Group, LLC

applied Equation 11.79 to the first mode shape of a damaged cantilever beam via a Mexican Hat wavelet

transform. The first mode shape is preferable since it is the most accurately determined by modal testing;

it features the lowest curvature; and sets off the singularity better. A correlation between damage size and

the magnitude of the Lipschitz exponent has been found from a number of beams with different damage

parameters.

Some of the ideas presented by Hong et al. may also be found in the work by Douka et al., who

have pursued crack identification in beams and plates using Daubechies wavelets [102,103]. The first

mode vibration response has been considered and the singularity induced by local defects has been

characterized in terms of Equation 11.79. The Lipschitz exponent has been used to describe the kind

of singularity, and the parameter A has been taken as the factor relating the depth of the crack to the

amplitude of the wavelet transform. Specifically, a second-order polynomial law has been found for

the intensity factor as a function of the crack depth. The work by Douka et al. has pointed out

the importance of the number of vanishing moments M of the chosen wavelet basis. It is intuitive

that the capability of setting off singularities in a regular function increases with M: However,

wavelet functions with high M exhibit a long support and lack space resolution. A compromise,

then, must be achieved, depending on the application in hand. Further insight into some

mathematical details of both the methods developed by Hong et al. and Douka et al. may be

found in Haase and Widjajakusuma [50]. Specifically, a fast algorithm to determine the maxima

lines of the wavelet transform has been devised. Also, the performance of various wavelet bases, such as

the Gaussian family of wavelets, has been assessed versus Daubechies wavelets used by Douka et al.

Another approach for damage-detection problems has been proposed by Yam et al. [104]. Clearly,

detection of small and incipient damage cannot be pursued by computing modal parameters that change

only if the amount of damage is significant. Thus, a method has been devised based on the energy

variation of the vibration response due to the occurrence of damage. The method is implemented in two

steps. The first involves the construction of damage feature proxy vectors using the energy at various

scales of the wavelet transformed vibration response. Then, classification and identification of the

structural damage status is pursued by using artificial neural networks (ANNs), which offer

significant advantages compared with genetic algorithms (GAs) developed by Moslem and Nafaspour

for damage-identification purposes [105]. Genetic-algorithm-based damage detection, in fact,

requires repeatedly searching among numerous damage parameters to find the optimal solution of the

objective function.

Yet another approach for applications of wavelet analysis to damage detection has been discussed by

Paget et al. [106], who have developed a procedure to detect impact damage in composite plates. It is based

on Lamb waves generated and received by embedded piezoceramic transducers. The Lamb waves can be

quite effective since they can propagate over long distances in the composite material and can interfere with

damage. To characterize the damage, the Lamb waves are wavelet transformed using an original wavelet

basis, devised from the recurrent waveforms of the Lamb waves. The changes in the Lamb waves

interacting due to the occurrence of damage are captured by the amplitude change of the wavelet

coefficients. From this effect, an estimate of the impact energy and the damage level is obtained based on

experimental results.

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