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Crackdetectioninbeamsusingkurtosis

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DOI:10.1016/j.compstruc.2004.11.010

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Computers and Structures 83 (2005) 909–919

www.elsevier.com/locate/compstruc

Crack detection in beams using kurtosis

Leontios J. Hadjileontiadis a,*, Evanthia Douka b, Athanasios Trochidis c

a Department of Electrical and Computer Engineering, Division of Telecommunications, Aristotle University of Thessaloniki,

Thessaloniki GR-54124, Greeceb Department of Engineering Sciences, Division of Mechanics, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greecec Department of Engineering Sciences, Division of Physics, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greece

Received 19 April 2004; accepted 8 November 2004

Available online 12 January 2005

Abstract

A new technique for crack detection in beam structures based on kurtosis is presented. The fundamental vibration

mode of the cracked cantilever beam is analyzed and both the location and size of the crack are estimated. The location

of the crack is determined by the abrupt changes in the spatial variation of the analyzed response, while the size of the

crack is related to the estimate of kurtosis. The proposed prediction method was validated by experiments on cracked

Plexiglas beams. The influence of added noise on the estimation accuracy of the method has been investigated. Com-

pared to existing methods for crack detection, the proposed kurtosis-based prediction scheme is attractive due to low

computational complexity and inherent robustness against noise.

� 2004 Elsevier Ltd. All rights reserved.

Keywords: Cracked beam; Crack location; Crack depth; Vibration mode; Kurtosis-based crack detector; Noise robustness

1. Introduction

The presence of a crack in a structural member re-

duces the stiffness and increases the damping of the

structure. As a consequence, there is a decrease in natu-

ral frequencies and modification of the modes of vibra-

tion. Many researchers have used the above

characteristics to detect and locate cracks and a plethora

of vibration-based methods for crack detection has been

developed [1].

Several approaches have been used to model the

problem of a cracked beam. Natural frequencies of

0045-7949/$ - see front matter � 2004 Elsevier Ltd. All rights reserv

doi:10.1016/j.compstruc.2004.11.010

* Corresponding author. Tel.: +302310 996340; fax: +302310

996312.

E-mail address: leontios@auth.gr (L.J. Hadjileontiadis).

cracked beams can be obtained by numerical analysis

using the finite element method [2–4]. Alternatively, sim-

plified procedures are available to evaluate the influence

of crack location and size on the natural frequencies.

Among these methods are those proposed by Christides

and Barr [5], who developed a continuous vibration the-

ory for a cracked Euler–Bernoulli beam and by Shen

and Pierre [6,7] using either a Rayleigh–Ritz method

or the Garleking method. Fernandez-Saez et al. [8] pro-

posed a method to calculate the fundamental frequency

of cracked beams, providing a closed-form solution. In

other cases, the presence of the crack and the corre-

sponding reduction of the stiffness have been modeled

by linear springs, whose stiffness may be related to the

size of the crack using fracture mechanics theory [9].

This model has been successfully applied to simply sup-

ported [10], cantilever [11], and fixed–fixed [12] cracked

ed.

910 L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919

beams. Natural frequencies have been the most appeal-

ing damage indicator because they can be easily mea-

sured and are less contaminated by experimental noise.

The main disadvantage of using natural frequency

changes for crack detection is the fact that significant

cracks may cause small changes in natural frequencies,

which may go undetected due to measurements errors.

In an effort to overcome these difficulties, research

has been focused on using changes in mode shapes

[13,14]. Mode shapes are more sensitive to local damage

compared to changes in natural frequencies. Further-

more, mode shapes inherently posses the geometry of

the system and damage may be potentially determined

by directly processing the geometrical changes of the

structure. However, using mode shapes also has some

drawbacks. The presence of a crack may not signifi-

cantly influence lower modes that are usually measured

from vibration tests. Extracted mode shapes are usually

affected by experimental noise and the duration of mea-

surements increases considerably if a detailed mode

shape has to be estimated.

Recently, the application of wavelet transform meth-

ods has emerged as a promising crack detection tool.

The advantage of the wavelet transform is its multireso-

lution property, which allows efficient identification of

local features of the signal [15]. The wavelet transform

has been successfully applied for crack localization in

beam like structures [16,17]. Hong et al. [18] used the

Lipschitz exponent to estimate the size of the crack. Re-

cently, Douka et al. [19] proposed a method for estimat-

ing both the location and size of the crack by defining an

intensity factor which relates the size of the crack to the

coefficients of the wavelet transform. It seems, however,

that the key issue to the efficient practical application of

the wavelet analysis to damage detection is the availabil-

ity of free of noise signals with high spatial resolution.

Therefore, sensors and measurement techniques able to

pick up perturbations caused by the presence of a crack

are needed. In that vein, laser scanning vibrometers are

used for non-contacting accurate measurement of mode

shapes [20].

In this work, an alternative method for crack detec-

tion using kurtosis analysis is proposed. Among other

statistical parameters, kurtosis has been proposed to

measure and analyze the vibration signal in the time do-

main and has been proved the most effective for the

detection of defects in rolling element bearings [21]. A

kurtosis value greater than the one measured for an

undamaged bearing can be considered as an indication

of incipient damage. In case of multiple bearing defects,

it has been observed that kurtosis value increases with

increasing number of bearing defects [22]. Kurtosis has

also been used to monitor the condition of gearboxes.

When one or more teeth are defected kurtosis value in-

creases, thus indicating the amount of localized damage

[23].

In this paper, the fundamental vibration mode of a

cracked cantilever beam is analyzed and both the loca-

tion and size of the crack are estimated. The location

of the crack is detected by a sudden change in the spatial

variation of the analyzed response, while the size of the

crack is related to the kurtosis measure. The proposed

technique forms a Kurtosis-based crack detector, which

takes into account the non-Gaussianity of the vibration

signal in order to efficiently detect both the location and

the size of the crack. Although it is based on higher-

order statistics, the method is straightforward, simple

in implementation and has low computational complex-

ity. The proposed prediction scheme was validated by

experimental investigations on a cracked cantilever

beam. The influence of added noise on the estimation

accuracy of the method has been also investigated. It

turns out that the proposed technique is more robust

against noise or measurement errors compared to other

techniques such as wavelet analysis, for example. In view

of the results obtained, the limitations and advantages of

the proposed kurtosis-based technique as well as sugges-

tions for future work are presented and discussed.

2. Mathematical background

Let {X(k)} be a real random zero-mean process that

is fourth-order stationary. Its second and fourth-order

moments, i.e., RX2 ðs1Þ and RX

4 ðs1; s2; s3Þ, respectively,

are defined as [24]

RX2 ðs1Þ � EfX ðkÞX ðk þ s1Þg; ð1Þ

RX4 ðs1; s2; s3Þ � E X ðkÞX ðk þ s1ÞX ðk þ s2ÞX ðk þ s3Þf g;

ð2Þ

where E{Æ} denotes the expectation operator. The

fourth-order cumulants sequence CX4 ðs1; s2; s3Þ of the

random process {X(k)} is defined as [24]

CX4 ðs1; s2; s3Þ ¼ RX

4 ðs1; s2; s3Þ � 3ðRX2 ðs1ÞÞ

2: ð3Þ

The fourth-order cumulants for zero lags, i.e.,

s1 = s2 = s3 = 0, is the kurtosis parameter, i.e., cX4 ¼CX

4 ð0; 0; 0Þ. For an N-sample sequence X(k), such as real

observations, the estimate of the kurtosis c4 is calculatedas [25]

c4 ¼

PNk¼1

½X ðkÞ � mX �4

ðN � 1Þr4X

� 3; ð4Þ

where mX and rX are the estimates of the mean and stan-

dard deviation of the N-sample sequence X(k). It is note-

worthy that c4 is equal to zero for Gaussian or

symmetrically distributed random variables [24].

The kurtosis is a measure of the heaviness of the tails

in the distribution of the X(k) sequence [24]; hence,

L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919 911

outliers or abrupt changes in the X(k) sequence have

high values and accordingly appear in the tails of the dis-

tribution. Consequently, the non-Gaussianity of the sig-

nal is powered, making heavier the tails of the

distribution and destroying its symmetry, resulting in

high values of the kurtosis parameter. In this way, the

kurtosis could be used to establish an effective statistical

test in identifying abrupt changes in signals, such as

those produced in the vibration signals from cracked

beams due to the existence of a crack.

3. The kurtosis crack detector (KCD)

The KCD is based on the property of kurtosis to

identify deviations from Gaussianity in band-limited

random process [24]. To this vein, the kurtosis could

be used as a measure of the non-Gaussianity of the sig-

nal in the vibration domain. This non-Gaussianity could

vary with different structural conditions, i.e., reduction

of the stiffness due to the occurrence of a crack, and,

thus, changes in the mode shapes of vibration. From this

perspective, the crack is seen as a factor that shifts the

vibration signal towards non-Gaussian behaviour. Con-

sequently, by means of kurtosis the deviations from

Gaussianity are linked with the changes in the vibration

signal, providing a fast computational tool that tracks

the non-Gaussianity, i.e., the existence of a crack.

In the examined case, the considered signal, X(k),

k = 1, . . . , N, consists of spatial data, i.e., displacement

measured via an accelerometer, from a single vibration

mode. In our case, the fundamental vibration mode

was selected, as the one that due to its shape has resulted

in the least edge effects in the kurtosis-based analysis.

Using (4), an estimate of the kurtosis c4 can be derived.

However, although this single value provides an overall

index of the non-Gaussianity of the examined sequence

X(k), with respect to the deviation of c4 from the zero va-

lue (Gaussian case), it does not give any particular infor-

mation regarding the crack characteristics (e.g. location,

depth). This was also justified by the work of Rivola [26],

who, although has used kurtosis as a measure of system

linearity, he found quite small differences between the

kurtosis estimates derived from vibration signals corre-

sponding to an uncracked beam and a cracked one with

a severe crack of 60% normalized depth. However, simi-

larly to our results, Rivola [26] has found, in both cases,

kurtosis estimates significantly deviating from the zero

value. This clearly shows a deviation from the Gaussia-

nity ofX(k). Considering that the primary aim of the pro-

posed analysis is the construction of an efficient crack

detector, the latter observation could place the effort on

tracking the changes seen in the degree of non-Gaussia-

nity due to the existence of a crack by calculating the kur-

tosis at a more local level (e.g. in a windowed signal).

This approach would allow the X(k) data, which corre-

spond to the crack location, to significantly influence

the c4 estimate, and give rise to its value at a local level,

hence, revealing the location of the crack.

Based on the above observation and focusing on

maximising the detectability of the proposed approach,

the KCD scheme is constructed as follows. Initially, a

sliding window of M = int(0.4/dx) sample length is em-

ployed, where int( Æ ) indicates the integer part of the

argument and dx denotes the distance resolution in the

acquisition of the vibration signal. The constant 0.4 is

empirically set and justified by the efficient performance

of the algorithm. Actually, we have noticed that whenM

was greater than 15 samples, the estimated kurtosis time

series was smoothed and the correct crack peak could

not easily be revealed. On the other hand, when M

was smaller than 15 samples, a sharp peak at the true

location of the crack was generated, clearly enhancing

the KCD detectability. In our experiments we have used

a value of dx = 0.1 cm, and, hence, of M = 4 samples.

Next, this M-sample window is right shifted along the

N-sample section of the vibration signal (M � N), with

a 99% of overlap, i.e., one data-point from the beginning

of the windowed sequence leaves and a new one is added

at its end. This procedure is selected to obtain point-to-

point values of the estimated kurtosis. Hence, the exact

point of the location of the crack is accurately captured.

Over each vibration signal segment obtained from the

sliding window, the c4 parameter is computed using (4)

and its estimated value, c4, is assigned to the midpoint

of the sliding window. Finally, the output vector k4 of

the KCD scheme is constructed as

k4 ¼j c4 � �c4 j; ð5Þ

where c4 is the vector with the estimated c4 values derivedat each position of the sliding M-sample window across

the N-sample vibration signal, �c4 is the sample mean va-

lue of c4, and j Æ j denotes the absolute value. From a clo-

ser look, it is easy to realise that the subtraction of the

sample mean value �c4 is almost equivalent to the subtrac-

tion of the c4 estimate derived when (4) is applied to the

whole sequence X(k). In this way, the values of the out-

put vector k4 outside the area of the crack location are

almost zeroed, obviously enhancing the visual inspection

of the existence of a crack in the KCD output.

Clearly, the window length affects the dynamic range

of the estimated c4 with respect to the true c4. However,

in problems where the detection is the primary aim, the

window length that provides with increased detectability

and repeatable results is preferable. In this case, it appears

that the true value of the c4 is not as important as the

changes in c4 associated with the existence of the crack.

In our experiments we have used a value of dx =0.1 cm,

and, hence, ofM = 4 samples. To elaborate on this selec-

tion, different values of dx, i.e., dx = 0.05, 0.01 and 0.005,

were tested, resulting in window length values of M = 8,

912 L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919

40 and 80 samples, respectively. It must be noted that in

practice, the difficulty to perform vibration mode mea-

surements is highly increased as the distance resolution

dx is also increased, taking values less than 0.1 cm. How-

ever, the aforementioned values of dx were adopted to

produce the vibration mode signal derived from a theo-

retical model (see subsequent section) to test the perfor-

mance of the KCD scheme. As it was expected, the c4was more accurately estimated, yet the detectability and

the sensitivity of the KCD to crack location and depth,

respectively, was severely reduced as the M value was in-

creased (especially when M was equal to 40 and 80 sam-

ples). Consequently, the selection of dx = 0.1 cm and

M = 4 samples can be seen as a trade-off between the

accurate estimation of the true value of the c4 and the effi-

cient performance of the KCD, in terms of high detect-

ability and simplicity in practical implementation.

The computational complexity of the KCD for an N-

sample recorded signal consists of (N � M + 1)(M +

1)(M + 6) + 2 additions and (N � M + 1)(5M + 4) + 1

multiplications. For the maximum values (worst case)

of N = 301 and M = 5 (99% overlap) used, the total

number of additions and multiplications needed is

ADDt = 19604 and MULTt = 8614, respectively. From

these values it is clear that the KCD scheme has negligi-

ble computational load, hence, it can easily be imple-

mented within a real-time context using either an

ordinary PC or dedicated hardware.

4. Kurtosis analysis of a cracked beam

4.1. Vibration model of a cracked beam

A cantilever beam of length L, of uniform rectangu-

lar cross-section w · w with a crack located at Lc is con-

sidered as shown in Fig. 1(a). The crack is assumed to be

open and have uniform depth a.Due to the localized crack effect, the beam can be

simulated by two segments connected by a massless

spring (Fig. 1(b)). For general loading, a local flexibility

matrix relates displacements and forces. In the present

analysis, since only bending vibrations of thin beams

are considered, the rotational spring constant is assumed

to be dominant in the local flexibility matrix [9].

The bending constant KT in the vicinity of the

cracked section is given by [9]

(a)

Fig. 1. (a) Cantilever beam under study;

KT ¼ 1

cwith c ¼ 5:346

wEI

Jaw

� �; ð6Þ

where a is the depth of the crack, E is the modulus of

elasticity of the beam, I is the moment of inertia of the

beam cross-section and J(a/w) is the dimensional local

compliance function, given by

Jða=wÞ ¼ 1:8624ða=wÞ2 � 3:95ða=wÞ3

þ 16:37ða=wÞ4 � 37:226ða=wÞ5

þ 76:81ða=wÞ6 � 126:9ða=wÞ7

þ 172ða=wÞ8 � 43:97ða=wÞ9

þ 66:56ða=wÞ10: ð7Þ

The displacement in each part of the beam is

part 1: u1ðxÞ ¼c1 coshKBxþ c2 sinhKBx

þ c3 cosKBxþ c4 sinKBx;

part 2: u1ðxÞ ¼c5 coshKBxþ c6 sinhKBx

þ c7 cosKBxþ c8 sinKBx

ð8Þ

with K4B ¼ x2qA‘4

EI , where A is the cross-sectional area, x is

the vibration angular velocity, q is the material density

and ci, i = 1, 2, . . . , 8, are constants to be determined

from the boundary conditions. The boundary conditions

at both ends are

at x ¼ 0: u1ð0Þ ¼ 0; u01ð0Þ ¼ 0;

at x ¼ L: M2ðLÞ ¼ 0; F 2ðLÞ ¼ 0;ð9Þ

where the prime denotes derivative with respect to x.

For each connection between the two segments, condi-

tions can be introduced which impose continuity of dis-

placement, bending moment and shear. Moreover, an

additional condition imposes equilibrium between trans-

mitted bending moment and rotation of the spring repre-

senting the crack. Consequently, the boundary conditions

at the crack positions can be expressed as follows:

u1ðLcÞ ¼ u2ðLcÞ; M1ðLcÞ ¼ M2ðLcÞ;F 1ðLcÞ ¼ F 2ðLcÞ;

� EIo2

ox2u1ðLcÞ ¼ KT

o

oxu1ðLcÞ �

o

oxu2ðLcÞ

� �:

ð10Þ

The resulting characteristic equation for the above-

described system can be solved numerically and both

the natural frequencies and mode shapes of the beam

can be obtained.

(b)

(b) cracked cantilever beam model.

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5x 10

-3

Distance from clamped end (cm)

Am

plitu

de

Fig. 2. Simulated fundamental vibration mode of a cracked

cantilever beam (crack location x = 4 cm; normalized crack

depth 20%).

L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919 913

For numerical simulations a Plexiglas beam of total

length 30 cm and rectangular cross-section 2 · 2 cm2 is

considered. A crack of relative depth 20% is introduced

at x = 4 cm from clamped end. Based on the theoretical

model presented above, the fundamental vibration mode

of the beam was calculated. The results are shown in

Fig. 2. Response data follow a sampling distance of

1mm resulting in a sequence of 301 point available. It

can be seen that the displacement data reveal no local

features that directly indicate the existence of the crack.

4.2. Determination of crack location

To determine the location of the crack, the scheme

presented in Section 3 was employed on simulated re-

sponse data and the estimate of kurtosis was evaluated.

Considering the Plexiglas beam examined previously,

the fundamental vibration mode was calculated for fixed

crack location at x = 4 cm from the fixed end and three

different crack depths, namely 20%, 30% and 40%. The

estimated kurtosis versus distance along the beam is pre-

sented in Fig. 3. It can be seen that in all cases the esti-

mate of kurtosis exhibits a peak value at x = 4 cm where

the crack is located. It can be also observed that the peak

value increases with increasing crack depth indicating

that kurtosis is related to crack depth.

4.3. Estimation of crack depth

To estimate the size of the crack, the dependence of

the kurtosis estimate on both crack location and depth

was systematically investigated. For that purpose, the

vibration modes of the beam were calculated for relative

crack depths varying from 5% up to 50% in steps of 5%,

while the crack location was varied from 2 cm to 10 cm

from the clamped end. The results are presented in Fig.

4. It can be seen that kurtosis increases with increasing

crack depth. For a given crack depth, the rate of increase

depends on crack location. The increment of the increase

is higher for cracks close to the clamped end and de-

creases gradually as the crack location is shifted towards

the free end. Consequently, cracks in the vicinity of the

clamped end could be more easily and accurately deter-

mined. However, statistical analysis (non-parametric

Wilcoxon signed ranks test) of these results using SPSS

12.0 (SPSS Inc.) shows statistically significant difference

among the k4 values estimated for different crack loca-

tions i.e., 2:2:10 cm, and among those estimated for dif-

ferent normalized crack depths, i.e., 5:5:50%, since it

results in maximum probabilities of error equal to

ploc = 0.006 < 0.05 and pdepth = 0.043 < 0.05, respec-

tively. This justifies further the overall efficiency of the

KCD to discriminate among cracks in either different

location and/or with different normalized depth.

5. Noise stress test results

In order to test the noise robustness of the KCD

scheme, a noise stress test has been set up. In this test,

the original and the noise signals are a priori known,

thus true signal-to-noise ratios (SNRs) could be mea-

sured. In particular, zero-mean Gaussian noise of differ-

ent levels was added to all vibration signals, resulting in

contaminated signals with SNRs ranging from 100 to 30

dB (step of �0.1 dB). Since the location of the crack is a

priori known in the original vibration signal, the addi-

tive Gaussian noise was constructed in such a way that

ensured the aforementioned SNR values at the exact

location of the crack in each original vibration signal.

This means that the SNR could vary across the contam-

inated vibration signal, since the noise and the original

signal are uncorrelated and the latter follows an expo-

nential profile, but in the true location of the crack, its

desired level, namely localized SNR (LSNR), was se-

cured. In this way, the efficiency of the KCD scheme

was actually tested against noise. The different ampli-

tude noise was produced by

nLSNRðkÞ ¼ ALSNR � nðkÞ; k ¼ 1; . . . ;N ; ð11Þ

where n(k) is an N-sample zero-mean unit variance

Gaussian noise and ALSNR is a multiplicative factor that

varies the noise amplitude according to the desired

LSNR level, given by

ALSNR ¼ xðklocÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi10LSNR=10

p� nðklocÞ

;

LSNR ¼ 100 : �0:1 : 30 dB;

ð12Þ

where kloc denotes the kth sample that provides the a

priori known location of the crack in the N-sample

0 4 5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

1.2Normalized crack depth 20%

0 4 5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

1.2

Est

imat

ed K

urt

osis

k4 Normalized crack depth 30%

0 4 5 10 15 20 25 30

0.2

0.4

0.6

0.8

1

1.2

Distance from clamped end (cm)

Normalized crack depth 40%

(a)

(b)

(c)

Fig. 3. Estimated kurtosis as a function of distance along the cracked beam (crack location x = 4 cm). Normalized crack depth: (a)

20%, (b) 30%, (c) 40%.

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

Normalized crack depth (%)

Est

imat

ed K

urto

sis

k 4

2 cm 4 cm 6 cm 8 cm

10 cm

Fig. 4. Estimated kurtosis versus normalized crack dept

(5:5:50%) for different crack locations: (s) 2 cm, (h) 4 cm,

(�) 6 cm, (,) 8 cm, (n) 10 cm.

914 L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919

original vibration signal x(k), and LSNR denotes the

desired LSNR level. This procedure was repeated for

different normalized crack depths, a, ranging from 5%

to 50% with a step of 5%, and different crack locations,

d, ranging from 2 to 10 cm, with a step of 2 cm. The con-

taminated vibration signals were then produced by

xLSNRd;a ðkÞ ¼ xd;aðkÞ þ nLSNR;d;aðkÞ; k ¼ 1; . . . ;N ;

LSNR ¼ 100 : �0:1 : 30 dB;

a ¼ 5 : 5 : 50%; d ¼ 2 : 2 : 10 cm;

ð13Þ

and were subjected to the kurtosis analysis. In particu-

lar, the KCD was applied to the xLSNRd;a signals and the

k4 vector was estimated for each LSNR level and for

the selected range of crack depth a and location d.

The important issue examined with this noise stress

test was the ability of the KCD to efficiently detect both

the true location and depth of the crack, despite the

noise presence in the vibration signal. From the estima-

tion of the location of the maximum peak in the k4

curves and from visual inspection of the test results it

was confirmed that there were no false positive detec-

tions regarding the estimation of the crack location in

all records analyzed during the noise stress test. Never-

L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919 915

theless, missed detections were observed for high-level

noise (LSNR values below 40 dB). Regarding the noise

robustness of the KCD to differentiate among the vari-

ous crack depths in the noise presence, the analysis

showed an excellent performance in the range of

80–100 dB, a stable one in the range of 60–80 dB, a

moderately deteriorated one in the range of 50–60 dB,

a deteriorated one in the range of 40–50 dB, and a

severely deteriorated one in the range of 30–40 dB. This

is clearly shown from an example of the noise stress test

results for the case of artificially contaminated vibration

signals obtained from a cracked beam with the crack lo-

cated at 4 cm, i.e., xLSNR4;a , with LSNR = 100:�0.1:30 dB

and a = 5:5:50%; this example is illustrated in Fig. 5. In

all subfigures of Fig. 5, i.e., Fig. 5(a)–(e), the k4(kloc) val-

ues, corresponding to the estimated kurtosis at the klocth

sample (4 cm), derived from the original vibration sig-

nals x4,a are denoted with a dashed line, while the mean

and the standard deviation of the k4(kloc) values ob-

tained from the contaminated signals xLSNR4;a are denoted

with a solid line and a grey area, respectively. In partic-

ular, Fig. 5(a) depicts the results from the kurtosis anal-

ysis for low contaminated vibration signals, i.e., for

x100:�0:1:904;5:5:50% . From this figure, it is clear that the effect of

the noise with LSNR values within 90–100 dB in the per-

formance of the KCD is negligible, since the estimated

k4(kloc) values from the x100:�0:1:904;5:5:50% are almost identical

to the ones derived from the x4,5:5:50% signals. Fig. 5(b)

shows the results from the kurtosis analysis for more

contaminated vibration signals than in the previous

case, i.e., for x70:�0:1:604;5:5:50% . This figure shows that the higher

noise level increases the standard deviation and de-

creases the values of the overall kurtosis curve, mainly

in the range of the normalized depth of 15–40%; the

k4(kloc) values for a normalized depth of 5%, however,

are increased. Fig. 5(c) shows the results from the kurto-

sis analysis for contaminated vibration signals with

LSNR in the range of 60:�0.1:50 dB, i.e., for x60:�0:1:504;5:5:50% .

It is noteworthy that the noise effect shown in the previ-

ous noise level is inverted, resulting in overestimation of

the k4(kloc) values across the overall kurtosis curve, but

mainly in the range of the normalized depth of 5–40%.

This overestimation of the k4(kloc) values is augmented

with the increase of the noise level, as it is clearly shown

in Fig. 5(d), where the results from the kurtosis analysis

for contaminated vibration signals with LSNR in the

range of 40:�0.1:30 dB, i.e., for x40:�0:1:304;5:5:50% , are shown.

From this figure it is clear the severe alteration of the

kurtosis curve from its original structure (dashed line)

in the normalized depth range of 5–40%, resulting in

an almost horizontal line with high standard deviation

when the LSNR level lies between 30 and 40 dB. This

degrades the ability of the KCD to differentiate among

cracks with varying depth; hence, when the kurtosis

curve is compared with its original form (dashed line),

degradation of the KCD performance is concluded.

One possible reason for the deterioration of the per-

formance of the KCD in the latter case is the fact that

the additive noise might split the one peak seen in the

kurtosis curve derived from the original signal to many

small ones, resulting in the overestimated k4(kloc) values

seen in the noisy case (see Fig. 5(d)). To validate this

assumption, a wider (double) sliding window was em-

ployed (M = 8 samples) to analyze the vibration signals

under the same noisy conditions, i.e., x40:�0:1:304;5:5:50% . Fig. 5(e)

shows the results of this analysis, where the k4(kloc) val-

ues from the original vibration signal when using a slid-

ing window ofM = 8 samples are depicted with a dashed

line; the mean and the standard deviation of the k4(kloc)

values from the noisy vibration signal are noted with a

solid line and a grey area, respectively. For comparison

reasons, the k4(kloc) values from the original vibration

signal when employing a sliding window of M = 4 sam-

ples are also depicted with a dashed-diamond line. As it

can be seen from the comparison between the dashed

and the dashed-diamond lines, the increase in the win-

dow length results in smaller k4(kloc) values (mostly in

the depth range of 15–50%), when the original vibration

signal is analyzed. This fact confirms the initial selection

of the window with small length, i.e., M = 4 samples, for

the kurtosis analysis in the noise-free case. Nevertheless,

in the severe contamination case, the use of wider win-

dow may provide better results. This is due to the

smoothing that is noted in the kurtosis values when a

wider window is employed, eliminating the effect of the

kurtosis overestimation. This is clear from Fig. 5(e),

where the kurtosis curve from the noisy data has smaller

standard deviation compared to the one in Fig. 5(d),

resembles much better the kurtosis curve from the origi-

nal vibration signal (dashed line), and retains a gradient

that could differentiate cracks with different normalized

depths, mainly within the range of 15–50%. Conse-

quently, when noisy vibration data are analyzed with

the KCD, wider sliding window could contribute to

the elimination of the noise effect in the crack depth

identification procedures. In this way, the KCD could

be used in noisy cases with lower (<30 dB) LSNR values,

reducing, accordingly, the required accuracy in the mea-

surement of the mode shapes and increasing, simulta-

neously, its practicality.

It is noteworthy that there is no significant difference

in the performance of the KCD for the rest of the exam-

ined locations of the crack, i.e., 2:2:6 cm and 10 cm, and

in all cases, the morphology of the curves depicted in

Fig. 4 is retained (for an adequate LSNR level, i.e.,

LSNR above 40 dB). This implies that the proposed

method could be successfully applied in a variety of

cracks, with respect to their depth and/or location.

These results also indicate that the KCD scheme is ro-

bust enough to the presence of noise, so it can efficiently

be used in the noisy environment usually met in practical

problems.

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

Normalized crack depth (%)

5 10 15 20 25 30 35 40 45 50

Est

imat

ed K

urt

osis

k4

Localized SNR: 50-60 dB

(c)

Normalized crack depth (%)

Est

imat

ed K

urto

sis

k 4

Localized SNR:30-40 dB

M=4 samples (noise-freecase)

M=8 samples(noise-free case)

M=8 samples (noisy case)

(e)

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

Normalized crack depth (%)

Est

imat

ed K

urto

sis

k 4

Localized SNR:30-40 dB

(d)

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

Normalized crack depth (%)

Est

imat

ed K

urt

osis

k4

Localized SNR: 90-100 dB

(a)

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

Normalized crack depth (%)

Est

imat

ed K

urt

osis

k4

Localized SNR: 60-70 dB

(b)

Fig. 5. Estimated kurtosis versus normalized crack depth for different localized SNR values. Crack location x = 4 cm. (a) Localized

SNR: 90:0.1:100 dB; sliding window size M = 4 samples. (b) Localized SNR: 60:0.1:70 dB; sliding window size M = 4 samples. (c)

Localized SNR: 50:0.1:60 dB; sliding window sizeM = 4 samples. (d) Localized SNR: 30:0.1:40 dB; sliding window sizeM = 4 samples.

(e) Localized SNR: 30:0.1:40 dB; sliding window size M = 8 samples; the dashed-diamond line denotes the noise-free case for M = 4

samples. In all subfigures, the dashed line denotes the estimated kurtosis for the noise-free case, whereas the solid line and the grey area

denote the mean value and the standard deviation of the estimated kurtosis for each localized SNR range, respectively.

916 L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919

L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919 917

6. Experimental results

To validate the analytical results of the kurtosis-

based crack detector, an experiment on a Plexiglas beam

has been performed. A 30-cm long Plexiglas beam of

rectangular cross-section 2 · 2 cm2 was clamped at the

vibrating table. A crack of relative crack depth 30%

was introduced at x = 6 cm from the clamped end. An

electromagnetic vibrator by Link and two B&K acceler-

ometers were used for the experiment. Harmonic excita-

tion was utilized via a 2110 B&K analyzer and the

fundamental mode of vibration was investigated.

The vibration amplitude was measured with a sam-

pling distance of 7.5 mm, which was the effective diame-

ter of the accelerometer used, so that a total number of 39

measuring points were obtained. Mode shape was mea-

sured by using two calibrated accelerometers mounted

on the beam. One accelerometer was kept at the clamped

end as the reference input, while the second one was

0 5 6 10 10

20

40

60

80

100

Distance from

Am

plitu

de

0 5 6 10 10

0.05

0.1

0.15

0.2

Est

imat

ed K

urto

sis

k 4

Distance from

0 5 6 10 1-5

0

5

10x 10

-4

Distance from

Est

imat

ed (

RM

S)"

(a)

(b)

(c)

Fig. 6. (a) Measured fundamental vibration mode (s: actual measure

(crack location x = 6 cm; normalized crack depth 30%). (b) Estimate

Second derivative of the estimated RMS signal when using a sliding

moved along the beam to measure the mode amplitude.

For that purpose, a miniature accelerometer weighting

2.5 g was used. Its mass is small compared to the mass

per unit length of the test beam. Therefore, the presence

of the accelerometer had no significant effect on the mea-

sured response. A plot of the measured fundamental

mode of the beam is shown in Fig. 6(a) with circles (s).

Because of the spared sampling, the kurtosis analysis if

implemented directly would detect many points of sam-

ple data as singularities. Therefore, to smooth the transi-

tion from one point to another an over-sampling

procedure was necessary. For that purpose, a cubic

spline interpolation was used resulted in a total number

of 301 points available (solid line in Fig. 6(a)).

Fig. 6(b) presents the estimate of kurtosis calculated

by analyzing the measured vibration mode shown in

Fig. 6(a). It can be seen that there is a main clear peak

at x = 6 cm and smaller in different positions. The

obtained results are not as smooth as in case of the

5 20 25 30clamped end (cm)

5 20 25 30clamped end (cm)

interpolated dataactual data

5 20 25 30clamped end (cm)

ments, -: curve after interpolation) of a cracked cantilever beam

d kurtosis when using a sliding window of M = 5 samples. (c)

window of M = 5 samples.

15 20 25 27.51 30 35 400

0.1

0.216

0.3

0.4

0.5

0.6

0.7

Normalized crack depth (%)

Est

imat

ed K

urto

sis

k 4

Fig. 7. Mapping of the kurtosis value derived from experimen-

tal data [see Fig. 6(b), x = 6 cm] on the relevant theoretical

kurtosis curve, resulting in an estimated normalized crack depth

of 27.5% (true value 30%).

918 L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919

simulated data. Obviously, measurement errors and

noise corrupted the response data. For comparison rea-

sons, the measured vibration mode shown in Fig. 6(a)

was also analyzed with a standard technique, such as

root mean square (RMS) value estimation [27], which

is a measure of the power content in the vibration signa-

ture included each time in the sliding window; the sec-

ond derivative of the estimated RMS signal is depicted

in Fig. 6(c). By comparing the analysis results shown

in Fig. 6(b) and (c), it is clear that KCD outperforms

RMS, since although both approaches result in unde-

sired peaks in the area outside the true location of the

crack (x = 6), the KCD, unlike the RMS, produces

undesired peaks with significantly smaller amplitude

compared to the one of the main peak located at x = 6

cm. This is due to the inherent property of the kurtosis,

as a fourth-order statistic, to be equal to zero for any

Gaussian distributed process (such as additive Gaussian

noise) [24]; on the contrary, the RMS scheme is prone to

abrupt signal amplitude changes due to the noise pres-

ence, thus, it exhibits higher false peaks (see Fig. 6(c)).

After locating the position of the crack, the estimate

of kurtosis was calculated at the estimated crack loca-

tion. The resulting value, equal to 0.216 is placed next

on the kurtosis estimate versus crack depth curve pro-

duced from noise-free analytical results. This leads to a

crack depth of 27.51%, as it is shown in Fig. 7. The pre-

dicted crack depth is in good agreement with the true va-

lue, which was equal to 30%.

7. Conclusions

A new method for crack identification in beam struc-

tures based on kurtosis analysis is presented. A cracked

cantilever beam having a transverse surface crack was

investigated both analytically and experimentally. The

fundamental vibration mode of the beam was analyzed

and both the location and size of the crack were esti-

mated. The location of the crack was determined by

the abrupt change in the spatial variation of the ana-

lyzed response, while the size of the crack was related

to the kurtosis estimate.

Before applying themethod tomeasured data, the sen-

sitivity to noise was investigated. A noise test performed

on simulated data proved the ability of the method to

accurately identify cracks for localized SNR values down

to 40 dB. The proposed KCD can be used in noisy cases

with lower LSNR values as well, by adopting a wider slid-

ing window, resulting, however, in reduced identification

accuracy.

The numerical results were confirmed by the applica-

tion of the method to experimental mode shapes of a

cracked cantilever beam. Using the noisy experimental

data, the location and size of a crack were detected with

reasonable accuracy. For comparison reasons, the data

were analyzed using RMS value estimation as well.

The results show that the proposed KCD is superior

when compared to the RMS technique, in terms of

detection accuracy of crack characteristics.

In conclusion, the presented results provide a foun-

dation of using kurtosis as an efficient crack detection

tool. Compared to existing methods for crack detection,

it is attractive due to its low computational complexity

and robustness against noise. Further work is needed,

however, to enhance the reliability and accuracy of the

proposed method. A key issue is the high spatial resolu-

tion and accuracy of the measured data. In that direc-

tion, existing modern techniques, like laser scanning

vibrometers, allowing for non-contacting accurate mea-

surements can be employed.

The results of the present work refer to a cantilever

beambut they canbe easily extended to includemore com-

plex structures and boundary conditions.Work is already

under way to explore the application of the proposed

kurtosis-based detector to more complicated structures.

These include multicracked beams and cracked plates.

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