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Crack detection in beams using kurtosis
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, Greece b Department of Engineering Sciences, Division of Mechanics, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greece c 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. Compared 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 reduces the stiffness and increases the damping of the structure. As a consequence, there is a decrease in natural frequencies and modification of the modes of vibration. 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
* Corresponding author. Tel.: +302310 996340; fax: +302310 996312. E-mail address: [email protected] (L.J. Hadjileontiadis).
cracked beams can be obtained by numerical analysis using the finite element method [2–4]. Alternatively, simplified 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 theory for a cracked Euler–Bernoulli beam and by Shen and Pierre [6,7] using either a Rayleigh–Ritz method or the Garleking method. Fernandez-Sa´ez et al. [8] proposed 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 corresponding 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 supported [10], cantilever [11], and fixed–fixed [12] cracked
0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.11.010
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beams. Natural frequencies have been the most appealing damage indicator because they can be easily measured 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. Furthermore, 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 significantly influence lower modes that are usually measured from vibration tests. Extracted mode shapes are usually affected by experimental noise and the duration of measurements increases considerably if a detailed mode shape has to be estimated. Recently, the application of wavelet transform methods has emerged as a promising crack detection tool. The advantage of the wavelet transform is its multiresolution 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. Recently, Douka et al. [19] proposed a method for estimating 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 availability 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 detection using kurtosis analysis is proposed. Among other statistical parameters, kurtosis has been proposed to measure and analyze the vibration signal in the time domain 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 increases, 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 location 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 higherorder statistics, the method is straightforward, simple in implementation and has low computational complexity. 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 suggestions 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 ðs1 Þ and RX4 ðs1 ; s2 ; s3 Þ, respectively, are defined as [24] RX2 ðs1 Þ EfX ðkÞX ðk þ s1 Þg;
ð1Þ
RX4 ðs1 ; s2 ; s3 Þ EfX ðkÞX ðk þ s1 ÞX ðk þ s2 ÞX ðk þ s3 Þg; ð2Þ where E{Æ} denotes the expectation operator. The fourth-order cumulants sequence C X4 ðs1 ; s2 ; s3 Þ of the random process {X(k)} is defined as [24] C X4 ðs1 ; s2 ; s3 Þ ¼ RX4 ðs1 ; s2 ; s3 Þ 3ðRX2 ðs1 ÞÞ2 :
ð3Þ
The fourth-order cumulants for zero lags, i.e., s1 = s2 = s3 = 0, is the kurtosis parameter, i.e., cX4 ¼ C X4 ð0; 0; 0Þ. For an N-sample sequence X(k), such as real observations, the estimate of the kurtosis ^c4 is calculated as [25] N P
^c4 ¼
^ X 4 ½X ðkÞ m
k¼1
ðN 1Þ^ r4X
3;
ð4Þ
^X are the estimates of the mean and stan^ X and r where m dard deviation of the N-sample sequence X(k). It is noteworthy 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,
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outliers or abrupt changes in the X(k) sequence have high values and accordingly appear in the tails of the distribution. Consequently, the non-Gaussianity of the signal 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 signal 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. Consequently, 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 value (Gaussian case), it does not give any particular information 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 corresponding to an uncracked beam and a cracked one with a severe crack of 60% normalized depth. However, similarly 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 Gaussianity of X(k). Considering that the primary aim of the proposed 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-Gaussianity due to the existence of a crack by calculating the kurtosis at a more local level (e.g. in a windowed signal). This approach would allow the X(k) data, which corre-
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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 employed, 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 when M 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-topoint 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 ^4 of of the sliding window. Finally, the output vector k the KCD scheme is constructed as ^4 ¼j ^c4 c4 j; k
ð5Þ
where ^c4 is the vector with the estimated ^c4 values derived at each position of the sliding M-sample window across the N-sample vibration signal, c4 is the sample mean value of ^c4 , and j Æ j denotes the absolute value. From a closer look, it is easy to realise that the subtraction of the sample mean value c4 is almost equivalent to the subtraction of the ^c4 estimate derived when (4) is applied to the whole sequence X(k). In this way, the values of the out^4 outside the area of the crack location are put vector k 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, of M = 4 samples. To elaborate on this selection, different values of dx, i.e., dx = 0.05, 0.01 and 0.005, were tested, resulting in window length values of M = 8,
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40 and 80 samples, respectively. It must be noted that in practice, the difficulty to perform vibration mode measurements is highly increased as the distance resolution dx is also increased, taking values less than 0.1 cm. However, the aforementioned values of dx were adopted to produce the vibration mode signal derived from a theoretical model (see subsequent section) to test the performance of the KCD scheme. As it was expected, the ^c4 was 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 increased (especially when M was equal to 40 and 80 samples). 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 efficient performance of the KCD, in terms of high detectability and simplicity in practical implementation. The computational complexity of the KCD for an Nsample 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 negligible computational load, hence, it can easily be implemented 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 rectangular cross-section w · w with a crack located at Lc is considered 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)
KT ¼
1 c
with c ¼ 5:346
w a J ; EI w
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 J ða=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: u1 ðxÞ ¼c1 cosh K B x þ c2 sinh K B x þ c3 cos K B x þ c4 sin K B x; part 2: u1 ðxÞ ¼c5 cosh K B x þ c6 sinh K B x þ c7 cos K B x þ c8 sin K B x 2
ð8Þ
4
qA‘ with K 4B ¼ x 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: at x ¼ L:
u1 ð0Þ ¼ 0; u01 ð0Þ ¼ 0; M 2 ðLÞ ¼ 0; F 2 ðLÞ ¼ 0;
ð9Þ
where the prime denotes derivative with respect to x. For each connection between the two segments, conditions can be introduced which impose continuity of displacement, bending moment and shear. Moreover, an additional condition imposes equilibrium between transmitted bending moment and rotation of the spring representing the crack. Consequently, the boundary conditions at the crack positions can be expressed as follows: u1 ðLc Þ ¼ u2 ðLc Þ; M 1 ðLc Þ ¼ M 2 ðLc Þ; F 1 ðLc Þ ¼ F 2 ðLc Þ; o2 o o u1 ðLc Þ u2 ðLc Þ : EI 2 u1 ðLc Þ ¼ K T ox ox ox
ð10Þ
The resulting characteristic equation for the abovedescribed system can be solved numerically and both the natural frequencies and mode shapes of the beam can be obtained.
(b)
Fig. 1. (a) Cantilever beam under study; (b) cracked cantilever beam model.
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2.5
x 10
Amplitude
2
1.5
1
0.5
0 0
5
10
15
20
25
30
Distance from clamped end (cm)
Fig. 2. Simulated fundamental vibration mode of a cracked cantilever beam (crack location x = 4 cm; normalized crack depth 20%).
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 response 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 presented in Fig. 3. It can be seen that in all cases the estimate 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
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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 decreases 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 determined. However, statistical analysis (non-parametric Wilcoxon signed ranks test) of these results using SPSS 12.0 (SPSS Inc.) shows statistically significant difference ^ 4 values estimated for different crack locaamong the k tions i.e., 2:2:10 cm, and among those estimated for different 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, respectively. 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 measured. In particular, zero-mean Gaussian noise of different 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 additive 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 contaminated vibration signal, since the noise and the original signal are uncorrelated and the latter follows an exponential profile, but in the true location of the crack, its desired level, namely localized SNR (LSNR), was secured. In this way, the efficiency of the KCD scheme was actually tested against noise. The different amplitude noise was produced by nLSNR ðkÞ 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 xðk loc Þ ; ALSNR ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LSNR=10 nðk loc Þ 10
ð12Þ
LSNR ¼ 100 : 0:1 : 30 dB; where kloc denotes the kth sample that provides the a priori known location of the crack in the N-sample
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(a)
Normalized crack depth 20%
1 0.8 0.6 0.4 0.2
Estimated Kurtosis k 4
0
1.2
4 5
10
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(b)
20
25
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4 5
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(c)
20
25
30
Normalized crack depth 40%
1 0.8 0.6 0.4 0.2 0
4 5
10 15 20 Distance from clamped end (cm)
25
30
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%.
3.5 3
Estimated Kurtosis k4
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 contaminated vibration signals were then produced by
2 cm 4 cm 6 cm 8 cm 10 cm
2.5
xLSNR ðkÞ ¼ xd;a ðkÞ þ nLSNR;d;a ðkÞ; d;a
2
k ¼ 1; . . . ; N ;
LSNR ¼ 100 : 0:1 : 30 dB; a ¼ 5 : 5 : 50%; d ¼ 2 : 2 : 10 cm;
1.5 1 0.5 0 5
10
15
20
25
30
35
40
45
50
Normalized crack depth (%)
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.
original vibration signal x(k), and LSNR denotes the desired LSNR level. This procedure was repeated for
ð13Þ
and were subjected to the kurtosis analysis. In particusignals and the lar, the KCD was applied to the xLSNR d;a ^ 4 vector was estimated for each LSNR level and for k 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^4 tion of the location of the maximum peak in the k curves and from visual inspection of the test results it was confirmed that there were no false positive detections regarding the estimation of the crack location in all records analyzed during the noise stress test. Never-
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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 various 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 located at 4 cm, i.e., xLSNR , with LSNR = 100:0.1:30 dB 4;a 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) values, corresponding to the estimated kurtosis at the klocth sample (4 cm), derived from the original vibration signals x4,a are denoted with a dashed line, while the mean and the standard deviation of the k4(kloc) values obtained from the contaminated signals xLSNR are denoted 4;a with a solid line and a grey area, respectively. In particular, Fig. 5(a) depicts the results from the kurtosis analysis for low contaminated vibration signals, i.e., for x100:0:1:90 4;5:5:50% . From this figure, it is clear that the effect of the noise with LSNR values within 90–100 dB in the performance of the KCD is negligible, since the estimated k4(kloc) values from the x100:0:1:90 are almost identical 4;5:5:50% 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 x70:0:1:60 4;5:5:50% . This figure shows that the higher noise level increases the standard deviation and decreases 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 kurtosis analysis for contaminated vibration signals with LSNR in the range of 60:0.1:50 dB, i.e., for x60:0:1:50 4;5:5:50% . It is noteworthy that the noise effect shown in the previous 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 x40:0:1:30 4;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.
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One possible reason for the deterioration of the performance 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 employed (M = 8 samples) to analyze the vibration signals under the same noisy conditions, i.e., x40:0:1:30 4;5:5:50% . Fig. 5(e) shows the results of this analysis, where the k4(kloc) values from the original vibration signal when using a sliding window of M = 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 samples 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 window 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 window 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 original vibration signal (dashed line), and retains a gradient that could differentiate cracks with different normalized depths, mainly within the range of 15–50%. Consequently, 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 measurement of the mode shapes and increasing, simultaneously, its practicality. It is noteworthy that there is no significant difference in the performance of the KCD for the rest of the examined 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 robust enough to the presence of noise, so it can efficiently be used in the noisy environment usually met in practical problems.
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15
20
25
30
35
40
45
0 5
50
10
15
Normalized crack depth (%)
Estimated Kurtosis k4
2.5
20
25
30
35
40
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50
Normalized crack depth (%)
(e)
Localized SNR:30-40 dB
2
M=4 samples (noise-freecase)
1.5
1 M=8 samples (noisy case)
0.5 M=8 samples(noise-free case)
0
5
10
15
20
25
30
35
40
45
50
Normalized crack depth (%)
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 size M = 4 samples. (d) Localized SNR: 30:0.1:40 dB; sliding window size M = 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.
L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919
6. Experimental results To validate the analytical results of the kurtosisbased 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 accelerometers were used for the experiment. Harmonic excitation was utilized via a 2110 B&K analyzer and the fundamental mode of vibration was investigated. The vibration amplitude was measured with a sampling distance of 7.5 mm, which was the effective diameter of the accelerometer used, so that a total number of 39 measuring points were obtained. Mode shape was measured 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
100
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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 measured 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 sample data as singularities. Therefore, to smooth the transition 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
(a)
Amplitude
80 60 40 interpolated data actual data
20 0 0
5 6
10
15
20
25
30
25
30
25
30
Estimated Kurtosis k4
Distance from clamped end (cm) (b)
0.2 0.15 0.1 0.05 0 0
5 6
10
15
20
Distance from clamped end (cm) -4
Estimated (RMS)"
10
x 10
(c) 5
0
-5 0
5 6
10
15
20
Distance from clamped end (cm) Fig. 6. (a) Measured fundamental vibration mode (s: actual measurements, -: curve after interpolation) of a cracked cantilever beam (crack location x = 6 cm; normalized crack depth 30%). (b) Estimated kurtosis when using a sliding window of M = 5 samples. (c) Second derivative of the estimated RMS signal when using a sliding window of M = 5 samples.
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Estimated Kurtosis k4
0.6 0.5 0.4 0.3 0.216
0.1 0 15
20
25 27.51 30
35
40
Normalized crack depth (%)
Fig. 7. Mapping of the kurtosis value derived from experimental 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%).
simulated data. Obviously, measurement errors and noise corrupted the response data. For comparison reasons, 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 signature included each time in the sliding window; the second 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 undesired 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 presence, 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 location. The resulting value, equal to 0.216 is placed next on the kurtosis estimate versus crack depth curve produced from noise-free analytical results. This leads to a crack depth of 27.51%, as it is shown in Fig. 7. The predicted crack depth is in good agreement with the true value, which was equal to 30%.
7. Conclusions A new method for crack identification in beam structures 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 estimated. The location of the crack was determined by the abrupt change in the spatial variation of the analyzed response, while the size of the crack was related to the kurtosis estimate. Before applying the method to measured data, the sensitivity 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 sliding window, resulting, however, in reduced identification accuracy. The numerical results were confirmed by the application 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 foundation 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 resolution and accuracy of the measured data. In that direction, existing modern techniques, like laser scanning vibrometers, allowing for non-contacting accurate measurements can be employed. The results of the present work refer to a cantilever beam but they can be easily extended to include more complex 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.
References [1] Doebling SW, Farrar CR, Prime MB. A summary review of vibration-based damage identification method. Shock Vib Dig 1998;30(2):91–105. [2] Friswell MI, Mottershead JE. Finite element model updating in structural dynamics. Dordrecht: Kluwer Academic Publishers; 1995. [3] Morasi A. Crack-induced changes in eigenfrequencies of beam structures. J Eng Mech 1993;119:1768–803. [4] Sinha JK, Friswell MI, Edwards S. Simplified models for the location of cracks in beam structures using measured vibration data. J Sound Vib 2002;251:13–38.
L.J. Hadjileontiadis et al. / Computers and Structures 83 (2005) 909–919 [5] Christides S, Barr ADS. One dimensional theory of cracked Bernoulli–Euler beams. Int J Mech Sci 1984;26:639–48. [6] Shen MHH, Pierre C. Natural modes of Bernoulli–Euler beams with symmetric cracks. J Sound Vib 1990;138(1): 115–34. [7] Shen MHH, Pierre C. Free vibrations of beams with singleedge crack. J Sound Vib 1994;170(2):237–59. [8] Fernandez-Sa´ez J, Rubio L, Navarro C. Approximate calculation of the fundamental frequency for bending vibrations of cracked beams. J Sound Vib 1999;225: 345–52. [9] Paipetis SA, Dimarogonas AD. Analytical methods in rotor dynamics. London: Elsevier Applied Science; 1986. [10] Narkis Y. Identification of crack location in vibrating simple supported beams. J Sound Vib 1993;172:549–58. [11] Bamnios Y, Douka E, Trochidis A. Crack identification in beam structures using mechanical impedance. J Sound Vib 2002;256:287–97. [12] Masoud A, Jarrad MA, Al-Maamory M. Effect of crack depth on natural frequency of prestressed fixed–fixed beam. J Sound Vib 1998;214:201–12. [13] Rizos P, Aspragathos N, Dimarogonas AD. Identification of crack location and magnitude in a cantilever beam from vibration modes. J Sound Vib 1990;138(3):381–8. [14] Kim T, Stubbs N. Improved damage identification. J Sound Vib 2002;252:223–38. [15] Newland DE. Wavelet analysis of vibration. J Vib Acoust 1994;116:409–16. [16] Wang Q, Deng X. Damage detection with spatial wavelets. Int J Solids Struct 1999;36:3443–68. [17] Quek S, Wang Q, Zhang L, Ang K. Sensitivity analysis of crack detection in beams by the wavelet technique. J Mech Sci 2001;43:2899–910.
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[18] Hong JC, Kim YY, Lee C, Lee YW. Damage detection using the Lipschitz exponent estimated by the wavelet transform. Int J Solids Struct 2002;39:1803–16. [19] Douka E, Loutridis S, Trochidis A. Crack identification in beams using wavelet analysis. Int J Solids Struct 2003;40:3557–69. [20] Stanbridge AB, Ewins DJ. Modal testing using a scanning laser vibrometer. Mech Syst Signal Process 1999;13: 255–70. [21] Tandon N, Choudhury A. A review of vibration and acoustic measurement methods for the detection of defects in rolling element bearings. Tribol Int 1999;32:469–80. [22] Prabhakar S, Mohanty AR, Sekhar AS. Application of discrete wavelet transform for detection of ball bearing race faults. Tribol Int 2002;35:793–800. [23] James Li C, Limmer JD. Model-based condition index for tracking gear wear and fatigue damage. Wear 2000;241: 26–32. [24] Nikias CL, Petropulu AP. Higher-order spectra analysis: A nonlinear signal processing framework. Englewood Cliffs, NJ: PTR, Prentice-Hall; 1993. [25] Bickel PJ, Doksum KA. Mathematical statistics. San Francisco, CA: Holden-Day; 1977. [26] Rivola A. Comparison between second and higher order spectral analysis in detecting structural damages. Available from. [27] Lebold M, McClintic K, Campbell R, Byington C, Maynard K. Review of vibration analysis methods for gearbox diagnostics and prognostics. In: Proceedings of the 54th Meeting of the Society for Machinery Failure Prevention Technology, Virginia Beach, VA, May 1–4, 2000. p. 623–34.
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, Greece b Department of Engineering Sciences, Division of Mechanics, Aristotle University of Thessaloniki, Thessaloniki GR-54124, Greece c 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. Compared 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 reduces the stiffness and increases the damping of the structure. As a consequence, there is a decrease in natural frequencies and modification of the modes of vibration. 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
* Corresponding author. Tel.: +302310 996340; fax: +302310 996312. E-mail address: [email protected] (L.J. Hadjileontiadis).
cracked beams can be obtained by numerical analysis using the finite element method [2–4]. Alternatively, simplified 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 theory for a cracked Euler–Bernoulli beam and by Shen and Pierre [6,7] using either a Rayleigh–Ritz method or the Garleking method. Fernandez-Sa´ez et al. [8] proposed 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 corresponding 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 supported [10], cantilever [11], and fixed–fixed [12] cracked
0045-7949/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2004.11.010
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beams. Natural frequencies have been the most appealing damage indicator because they can be easily measured 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. Furthermore, 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 significantly influence lower modes that are usually measured from vibration tests. Extracted mode shapes are usually affected by experimental noise and the duration of measurements increases considerably if a detailed mode shape has to be estimated. Recently, the application of wavelet transform methods has emerged as a promising crack detection tool. The advantage of the wavelet transform is its multiresolution 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. Recently, Douka et al. [19] proposed a method for estimating 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 availability 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 detection using kurtosis analysis is proposed. Among other statistical parameters, kurtosis has been proposed to measure and analyze the vibration signal in the time domain 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 increases, 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 location 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 higherorder statistics, the method is straightforward, simple in implementation and has low computational complexity. 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 suggestions 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 ðs1 Þ and RX4 ðs1 ; s2 ; s3 Þ, respectively, are defined as [24] RX2 ðs1 Þ EfX ðkÞX ðk þ s1 Þg;
ð1Þ
RX4 ðs1 ; s2 ; s3 Þ EfX ðkÞX ðk þ s1 ÞX ðk þ s2 ÞX ðk þ s3 Þg; ð2Þ where E{Æ} denotes the expectation operator. The fourth-order cumulants sequence C X4 ðs1 ; s2 ; s3 Þ of the random process {X(k)} is defined as [24] C X4 ðs1 ; s2 ; s3 Þ ¼ RX4 ðs1 ; s2 ; s3 Þ 3ðRX2 ðs1 ÞÞ2 :
ð3Þ
The fourth-order cumulants for zero lags, i.e., s1 = s2 = s3 = 0, is the kurtosis parameter, i.e., cX4 ¼ C X4 ð0; 0; 0Þ. For an N-sample sequence X(k), such as real observations, the estimate of the kurtosis ^c4 is calculated as [25] N P
^c4 ¼
^ X 4 ½X ðkÞ m
k¼1
ðN 1Þ^ r4X
3;
ð4Þ
^X are the estimates of the mean and stan^ X and r where m dard deviation of the N-sample sequence X(k). It is noteworthy 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,
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outliers or abrupt changes in the X(k) sequence have high values and accordingly appear in the tails of the distribution. Consequently, the non-Gaussianity of the signal 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 signal 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. Consequently, 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 value (Gaussian case), it does not give any particular information 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 corresponding to an uncracked beam and a cracked one with a severe crack of 60% normalized depth. However, similarly 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 Gaussianity of X(k). Considering that the primary aim of the proposed 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-Gaussianity due to the existence of a crack by calculating the kurtosis at a more local level (e.g. in a windowed signal). This approach would allow the X(k) data, which corre-
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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 employed, 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 when M 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-topoint 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 ^4 of of the sliding window. Finally, the output vector k the KCD scheme is constructed as ^4 ¼j ^c4 c4 j; k
ð5Þ
where ^c4 is the vector with the estimated ^c4 values derived at each position of the sliding M-sample window across the N-sample vibration signal, c4 is the sample mean value of ^c4 , and j Æ j denotes the absolute value. From a closer look, it is easy to realise that the subtraction of the sample mean value c4 is almost equivalent to the subtraction of the ^c4 estimate derived when (4) is applied to the whole sequence X(k). In this way, the values of the out^4 outside the area of the crack location are put vector k 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, of M = 4 samples. To elaborate on this selection, different values of dx, i.e., dx = 0.05, 0.01 and 0.005, were tested, resulting in window length values of M = 8,
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40 and 80 samples, respectively. It must be noted that in practice, the difficulty to perform vibration mode measurements is highly increased as the distance resolution dx is also increased, taking values less than 0.1 cm. However, the aforementioned values of dx were adopted to produce the vibration mode signal derived from a theoretical model (see subsequent section) to test the performance of the KCD scheme. As it was expected, the ^c4 was 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 increased (especially when M was equal to 40 and 80 samples). 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 efficient performance of the KCD, in terms of high detectability and simplicity in practical implementation. The computational complexity of the KCD for an Nsample 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 negligible computational load, hence, it can easily be implemented 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 rectangular cross-section w · w with a crack located at Lc is considered 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)
KT ¼
1 c
with c ¼ 5:346
w a J ; EI w
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 J ða=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: u1 ðxÞ ¼c1 cosh K B x þ c2 sinh K B x þ c3 cos K B x þ c4 sin K B x; part 2: u1 ðxÞ ¼c5 cosh K B x þ c6 sinh K B x þ c7 cos K B x þ c8 sin K B x 2
ð8Þ
4
qA‘ with K 4B ¼ x 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: at x ¼ L:
u1 ð0Þ ¼ 0; u01 ð0Þ ¼ 0; M 2 ðLÞ ¼ 0; F 2 ðLÞ ¼ 0;
ð9Þ
where the prime denotes derivative with respect to x. For each connection between the two segments, conditions can be introduced which impose continuity of displacement, bending moment and shear. Moreover, an additional condition imposes equilibrium between transmitted bending moment and rotation of the spring representing the crack. Consequently, the boundary conditions at the crack positions can be expressed as follows: u1 ðLc Þ ¼ u2 ðLc Þ; M 1 ðLc Þ ¼ M 2 ðLc Þ; F 1 ðLc Þ ¼ F 2 ðLc Þ; o2 o o u1 ðLc Þ u2 ðLc Þ : EI 2 u1 ðLc Þ ¼ K T ox ox ox
ð10Þ
The resulting characteristic equation for the abovedescribed system can be solved numerically and both the natural frequencies and mode shapes of the beam can be obtained.
(b)
Fig. 1. (a) Cantilever beam under study; (b) cracked cantilever beam model.
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2.5
x 10
Amplitude
2
1.5
1
0.5
0 0
5
10
15
20
25
30
Distance from clamped end (cm)
Fig. 2. Simulated fundamental vibration mode of a cracked cantilever beam (crack location x = 4 cm; normalized crack depth 20%).
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 response 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 presented in Fig. 3. It can be seen that in all cases the estimate 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
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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 decreases 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 determined. However, statistical analysis (non-parametric Wilcoxon signed ranks test) of these results using SPSS 12.0 (SPSS Inc.) shows statistically significant difference ^ 4 values estimated for different crack locaamong the k tions i.e., 2:2:10 cm, and among those estimated for different 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, respectively. 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 measured. In particular, zero-mean Gaussian noise of different 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 additive 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 contaminated vibration signal, since the noise and the original signal are uncorrelated and the latter follows an exponential profile, but in the true location of the crack, its desired level, namely localized SNR (LSNR), was secured. In this way, the efficiency of the KCD scheme was actually tested against noise. The different amplitude noise was produced by nLSNR ðkÞ 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 xðk loc Þ ; ALSNR ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LSNR=10 nðk loc Þ 10
ð12Þ
LSNR ¼ 100 : 0:1 : 30 dB; where kloc denotes the kth sample that provides the a priori known location of the crack in the N-sample
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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%.
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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 contaminated vibration signals were then produced by
2 cm 4 cm 6 cm 8 cm 10 cm
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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.
original vibration signal x(k), and LSNR denotes the desired LSNR level. This procedure was repeated for
ð13Þ
and were subjected to the kurtosis analysis. In particusignals and the lar, the KCD was applied to the xLSNR d;a ^ 4 vector was estimated for each LSNR level and for k 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^4 tion of the location of the maximum peak in the k curves and from visual inspection of the test results it was confirmed that there were no false positive detections 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
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 various 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 located at 4 cm, i.e., xLSNR , with LSNR = 100:0.1:30 dB 4;a 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) values, corresponding to the estimated kurtosis at the klocth sample (4 cm), derived from the original vibration signals x4,a are denoted with a dashed line, while the mean and the standard deviation of the k4(kloc) values obtained from the contaminated signals xLSNR are denoted 4;a with a solid line and a grey area, respectively. In particular, Fig. 5(a) depicts the results from the kurtosis analysis for low contaminated vibration signals, i.e., for x100:0:1:90 4;5:5:50% . From this figure, it is clear that the effect of the noise with LSNR values within 90–100 dB in the performance of the KCD is negligible, since the estimated k4(kloc) values from the x100:0:1:90 are almost identical 4;5:5:50% 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 x70:0:1:60 4;5:5:50% . This figure shows that the higher noise level increases the standard deviation and decreases 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 kurtosis analysis for contaminated vibration signals with LSNR in the range of 60:0.1:50 dB, i.e., for x60:0:1:50 4;5:5:50% . It is noteworthy that the noise effect shown in the previous 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 x40:0:1:30 4;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.
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One possible reason for the deterioration of the performance 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 employed (M = 8 samples) to analyze the vibration signals under the same noisy conditions, i.e., x40:0:1:30 4;5:5:50% . Fig. 5(e) shows the results of this analysis, where the k4(kloc) values from the original vibration signal when using a sliding window of M = 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 samples 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 window 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 window 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 original vibration signal (dashed line), and retains a gradient that could differentiate cracks with different normalized depths, mainly within the range of 15–50%. Consequently, 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 measurement of the mode shapes and increasing, simultaneously, its practicality. It is noteworthy that there is no significant difference in the performance of the KCD for the rest of the examined 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 robust enough to the presence of noise, so it can efficiently be used in the noisy environment usually met in practical problems.
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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 size M = 4 samples. (d) Localized SNR: 30:0.1:40 dB; sliding window size M = 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.
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6. Experimental results To validate the analytical results of the kurtosisbased 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 accelerometers were used for the experiment. Harmonic excitation was utilized via a 2110 B&K analyzer and the fundamental mode of vibration was investigated. The vibration amplitude was measured with a sampling distance of 7.5 mm, which was the effective diameter of the accelerometer used, so that a total number of 39 measuring points were obtained. Mode shape was measured 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
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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 measured 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 sample data as singularities. Therefore, to smooth the transition 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
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Fig. 7. Mapping of the kurtosis value derived from experimental 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%).
simulated data. Obviously, measurement errors and noise corrupted the response data. For comparison reasons, 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 signature included each time in the sliding window; the second 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 undesired 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 presence, 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 location. The resulting value, equal to 0.216 is placed next on the kurtosis estimate versus crack depth curve produced from noise-free analytical results. This leads to a crack depth of 27.51%, as it is shown in Fig. 7. The predicted crack depth is in good agreement with the true value, which was equal to 30%.
7. Conclusions A new method for crack identification in beam structures 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 estimated. The location of the crack was determined by the abrupt change in the spatial variation of the analyzed response, while the size of the crack was related to the kurtosis estimate. Before applying the method to measured data, the sensitivity 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 sliding window, resulting, however, in reduced identification accuracy. The numerical results were confirmed by the application 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 foundation 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 resolution and accuracy of the measured data. In that direction, existing modern techniques, like laser scanning vibrometers, allowing for non-contacting accurate measurements can be employed. The results of the present work refer to a cantilever beam but they can be easily extended to include more complex 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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