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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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