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Improvement of damage detection methods based on experimental modal parameters
Mechanical Systems and Signal Processing 25 (2011) 2169–2190
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Improvement of damage detection methods based on experimental modal parameters Maciej Radzien´ski a, Marek Krawczuk a,b, Magdalena Palacz a,b,n a b
´ sk University of Technology, Faculty of Electrical and Control Engineering, Narutowicza 11/12, 80-233 Gdan ´ sk, Poland Gdan ´ sk, Poland Institute of Fluid Flow Machinery PAS, Fiszera 14, 80-231 Gdan
a r t i c l e i n f o
abstract
Article history: Received 11 June 2010 Received in revised form 16 November 2010 Accepted 11 January 2011 Available online 2 February 2011
The objective of this paper is to introduce a new method for structural damage detection based on experimentally obtained modal parameters. The new method is suitable for detection of fatigue damage occurring in an aluminium cantilever beam. The damage has been practically realised as saw cuts of different sizes and at different locations. The first step of analysis included an attempt of damage identification with the most often used damage indicators based on measured modal parameters. For that purpose special signal processing technique has been proposed improving the effectiveness of indicators tested. However the results obtained have not been satisfactory. That was the motivation for defining new damage indicators (frequency change based damage indicator, Hybrid Damage Detection method), utilising the change of natural frequencies and any mode shape (measured or modelled) as the measurement of frequencies is much less time consuming in comparison to total mode shape measurement. It has been shown that the proposed technique is suitable for damage localisation in beam-like structures. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Modal analysis Damage detection Signal processing
1. Introduction According to the definition, the structural health monitoring is a scientific procedure comprising of several nondestructive processes including identification of operational and environmental loads acting on the component, recognition of the mechanical damage caused by that loading, and observation of damage growth as the component operates. Finally, the health monitoring deals with assessing the future performance of the component as damage develops. The health monitoring techniques should be non-destructive and ideally implemented online in an automated manner with embedded hardware/software, as a system operates [1]. The modern health monitoring techniques utilise such parameters as the electromechanical and high frequency impedance changes, the strain, stress or temperature fields, and the vibration changes. So as to measure these responses different transducers have been used. The most frequently used are eddy current/electromagnetic sensors, fibre optic based sensors, piezoelectric and piezoceramic sensors, thermal imaging sensors, conductive polymers, MEMS (Micro Electro-Mechanical Systems), and optical lasers. An integral part of damage identification algorithm is adequate signal processing and the feature extraction techniques. The signal processing methods are applied to raw data that is acquired from the sensors in order to produce significant information for diagnostics. One of the main challenges faced in signal processing is the variability of symptoms caused by operational conditions and the resulting changes in the quantities used for diagnostics. Such variations may cause false positive or negative indications of damage, which should be excluded according to the end user’s health monitoring
n Corresponding author at: Gdan´sk University of Technology, Faculty of Electrical and Control Engineering, Narutowicza 11/12, 80-233 Gdan´sk, Poland. Tel.: + 48 58 347 1521; fax: + 48 58 347 2136. E-mail address: [email protected] (M. Palacz).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.01.007
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specifications. For example, while selecting the signal processing technique and the feature extraction method in the health monitoring algorithms, it is essential to consider carefully the extent to which these methods amplify or suppress sources of variability [1]. In addition to detecting the presence of damage, the damage identification algorithms should also ideally provide an indication of the damage location and magnitude. The task of damage localisation usually requires a model of the component and the data from multiple sensors. Similarly, the task of damage quantification almost always requires a physical-based model of the component. The damage is also more quantifiable if active sensing is applied, where the sensor signals can be compared continuously to a measured signal [1]. Structural damage is defined as a permanent change in the mechanical state of a structural material or component that could potentially affect their performance [1]. Common sources of damage in material or structural components include micro-structural defects (dislocations, voids, inclusions), corrosion (loss of material), residual stress, cracking (fatigue, matrix, ply), fastening fault (weld crack, bold preload, broken rivet), adhesive fault (de-bonding, delamination, separation), and instability (thermo-mechanical buckling) [1]. The challenge is to interpret the changes of the response parameters due to the type of damage and to correlate them with the corresponding measured parameters. Specification of the relationship between the damage and the characteristic parameters provides the foundation of the identification and assessment algorithm required for the SHM system [2,3]. In the last two decades the non-destructive examination (NDE) techniques, and the structural health monitoring (SHM) techniques as well, have received a considerable amount of interest. Among them vibration analysis for damage detection has been the most popular, due to its implementation simplicity [1,2]. Development of the Scanning Laser Doppler Vibrometers (SLDV) brings about that those techniques have been even more attractive because of non-contact precise vibration measurements with high resolution in a considerably small period of time. Vibration-based damage detection methods make use of dynamic responses of a structure, like natural frequencies and mode shapes, and frequency response functions as well. Occurrence of the damage leads to changes in dynamics of structure. These changes may be used for damage localisation and assessment of its magnitude. Recently, many studies have been performed proposing different vibration-based damage detection methods. In the majority of cases those methods have been tested only numerically. In addition, it is hard to evaluate applicability of a specific damage detection method with no comparison to the results obtained from the same set of data for other techniques. Thus, the purpose of the present study is to determine benefits and drawbacks of the most popular and promising damage detection methods by mining the direct comparison between them. We would also like to show that the digital signal processing based on measured data may enhance the results of damage localisation. It has been shown that the introduced signal processing has improved the sensitivity of analysed damage indicators, however the results obtained have not been satisfactory. For that reason new damage indicators have been introduced. They are based on experimental data only and contain the proposed signal processing. One of the advantages of the approach proposed is the easiness of measuring the parameter needed—natural frequency change. There is also a curvature of the mode shape needed, but it may be simulated as well. The measurement of natural frequency only makes the whole procedure much less time consuming in comparison to classical modal analysis. Another positive aspect of the indicators proposed is that they may be used for improving the efficiency of already existing methods based on mode shapes changes. 2. Experimental set for data gathering It is commonly known that the presence of damage influences vibration parameters of the examined component. Different structural responses may be used for the damage detection and estimation. One of vibration parameters very sensitive to damages, are the mode shapes. Any mode shape is a unique characteristic of a mechanical structure and represents spatial distribution of the amplitude corresponding to each resonance. A local damage causes the change in the derivatives of the mode shapes at the position of the damage. In past times it was difficult to measure mode shape characteristics with proper precision. Recently, available technological devices, like SLDV, enable exact measurement of mode shapes with the accuracy needed, in relatively short time. Section 4 presents the analysis of several mode shape based damage detection methods, being the most popular according to [2]. Mode shapes have been obtained experimentally. All experiments have been done for a steel cantilever beam (Fig. 1) of dimensions: L H W= 1 0.02 0.01 m3. Various damage scenarios have been analysed. As the damage a saw cutting located at L1 = 10%, 30%, 50%, 70%, or 90% of the beam length L (starting from the fixed end) with cut depth Hc equal to 10% and 20% of the beam height has been tested. The width of the crack has been kept constant and equal to Lc =0.001 m. In addition, the cut depth Hc has been changed by a step change from 5% to 50% with 5% step, for the cut located at L1 = 10% of the beam length. Fig. 2 shows the experimental set-up used for gathering modal parameters from the examined element. All data has been measured using a Polytec PSV-400 SLDV. For system excitation an electromechanical shaker (GWV B100) has been used. So as to obtain the modal parameters there has been performed a measurement of Frequency Response Functions (FRFs) via Fast Fourier Transformation and the periodic chirp excitation. It has provided determination of natural frequencies by finding local maxima in averaged FRF graph. Those frequencies have been then successively used as a single-frequency excitation to examine ODS (operational deflection shapes) instead of mode shapes measured via FRF due to better precision obtained. Every point has been measured with 10 registered periods of sinusoidal excitation and mean value has been calculated. Such operation leads to better signal to noise ratio in comparison to traditional modal analysis. The measurements have been performed along the beam length at 125 equally spaced points. The noise level of the measured mode shapes has been controlled by utilising frequency bandwidth Bw and the signal level. The noise level is
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excitation rod crack Hc
L1
Lc
H
W L
Fig. 1. A scheme of the specimen and excitation methodology.
shaker
scanning head
SLDV System
laser beam measured specimen
computer DC field generator
Fig. 2. A scheme of the experimental set-up.
pffiffiffiffiffiffi proportional to Bw but the time of acquisition is inversely proportional to Bw, so it should be considered how to find a compromise between the acquisition time and the precision [4]. The signal received by the SLDV can be strengthened by coating the measured surface with a light back-scattering layer, such as bright acrylic paint used in this research. In order to obtain the best scanning parameters the scanning head should be positioned in a way providing maximal visibility. For the PSV-400 used, the maximum visibility is at the distance of 0.099 m+ n 0.204 m, where n = 1, 2, 3, which has been 1731 m with n= 8 in the presented setup. 3. Basic signal processing technique The measured mode shapes have been recorded including all necessary conditions in order to measure the minimal noise level. To obtain the mode shape curvature second order central difference has been used. In order to overcome the possible inconveniency of obtaining wrong damage identification results with methods based on modal parameters, a simple signal processing (SP) technique has been implemented. The first step of the proposed technique has been a linear signal extrapolation (for two points of both ends of the measured signal vector). This operation is introduced so as to overcome inconveniences of the boundary reflections. If needed more precise mathematical background the reader is kindly asked to follow [5]. Then an automatic denoising ‘‘wden’’ MATLABs function has been used in order to get a de-noised version of a 1D input signal. This function is based on wavelet decomposition and it is
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realized in three steps. First the signal is decomposed on detail and approximation coefficients at level Ld. Next the detail coefficients from levels 1 to Ld is thresholded. Finally signal is reconstructed based on approximation coefficient at level Ld and modified detail coefficients at level 1 to Ld. The parameters used in this ‘‘wden’’ function have been as follows: the decomposition level Ld = 2 for detail coefficients, the wavelet mother function ‘‘sym8’’ and the threshold selection rule— heuristic without rescaling. Several different denoising methods have been tested, like moving average, or median filter, but the best in this particular application has been the wavelet decomposition based function. The next step of the analysis has been the elimination of additional points from linear extrapolation being the first step of analysis of all vectors. The final step, before implementation of measured vectors into damage detection algorithms, has been triple cubic spline interpolation of all mode shapes in order to increase the number of points (except the wavelet transformation procedure). 4. Analysis results of damage detection with methods based on natural vibration 4.1. Mode shape curvature (MSC) and damage index (DI) When a damage is introduced in a structure, the bending stiffness at the location of the damage is reduced while at the same time the magnitude of the mode curvatures increase. The absolute differences between the mode curvature of the intact and damaged structures are the highest in the region of the damage and negligibly small outside this region [2]. As the changes of the mode shape curvatures are localised in the region of the damage, the changes may be used effectively to identify damage location in structures. According to literatures [2,3,6–8], the mode shape curvature (MSC) criterion may be defined as the difference in absolute curvatures of the healthy and damaged structures, for each mode, and may be represented as MSCi ¼
Nm X 00 d 00 fi,j fi,j
ð1Þ
j¼1 00
d
where f00 and f stands for mode shape second derivative of undamaged and damaged state, respectively, Nm is the number of measured mode shape curvatures vectors, and index i represents the measured point number and j is the mode shape number. At a certain damage location, the value of MSC is significantly higher than the ones at other locations. Based on the curvature difference values of measured data of damaged and healthy structures, and the MSCs, the location of damage in the structure can be identified. The most popular approach is the utilisation of only a few lower order curvature modes to identify the damage location. MSC has been successfully used so far for identification of various damages in the isotropic beams [8,3], laminated composite beams [2,6], or the wooden board [7]. Mode shape curvatures are also used for determination of the damage index (DI). This parameter utilises mode shape curvatures of a structure as the main variable in the damage localisation algorithm. The algorithm is based on the relative differences in modal strain energy, before and after the damage. The DI parameter for more than one mode is defined as 2 Nm 00 00 P P d 2 d 2 PN 00 DIi ¼ ðfi,j Þ þ N i ¼ 1 ðfi,j Þ i ¼ 1 ðfi,j Þ j¼1
P PN 00 d N 2 00 Þ2 þ 00 2 ðfi,j i ¼ 1 ðfi,j Þ i ¼ 1 ðfi,j Þ
ð2Þ
where Nm signifies the number of measured mode shapes and N means the number of measurement points. The damage index has been successfully applied to damage localisation in the composite [2,6], concrete [9], and isotropic [8,10] beams, mostly with analytical data. The work presented in [10] shows additionally an example of damage identification based on experimental data obtained for a real I-40 bridge Rio Grande in Albuquerque, NM, USA. The graphs from Figs. 3 and 4 present differences between the results of damage identification with MSC and DI obtained for pure signals of measured mode shapes (parts (a)) and for signals with proposed processing technique (parts (b)). In case shown the calculations have been based on measurements of a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end. It may be noticed that implementation of the simple signal processing technique for the two first damage indicators has improved the sensitivity of both methods. 4.2. Coordinate modal assurance criterion (COMAC) The Coordinate Modal Assurance Criterion (COMAC) identifies the co-ordinates at which two sets of the mode shape do not agree. The COMAC factor at a point i between two sets of the same mode shape, in two states, is defined by 2 P d Nm j ¼ 1 fi,j fi,j COMACi ¼ ð3Þ PNm PNm d 2 j ¼ 1 fi,j j ¼ 1 fi,j
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Fig. 3. MSC—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
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Fig. 4. DI—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
where Nm is the number mode shapes, fi,j and fdi,j denote the values of jth mode shape at a point i for undamaged and damaged states, respectively. The COMAC has been developed from the original MAC concept in such a way that the correlation is related to degrees of freedom of the structure rather than to the mode numbers. The resultant COMAC values have been shown to be of considerable help in showing where the damage or inhomogeniety within the structure occur.
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From the literature it emerges that the MAC and COMAC are highly dependent on the geometry of a structure and the location of a damage. It is also proven that these criteria are not sensitive enough to detect damages at their early stages of growth. The MAC and COMAC have been successfully used for numerical and experimental damage detection in the reinforced concrete beam [11,12], the wooden board [7] and the isotropic beam [3]. Fig. 5 presents the results of damage identification with COMAC obtained for pure signals of measured mode shapes (part (a)) and for signals with proposed processing technique (part (b)). As in previous case the calculations have been done for mode shape recorded for a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end. It may be noticed that for this particular damage indicator even the implementation of the proposed signal processing technique does not provide reliable information about the damage. 4.3. Strain energy damage index (SEDI) Strain energy damage index allows to evaluate the health state of the structure as the index expresses the degradation of the flexural rigidity in a certain zone of the structure. The index accounts for the change in strain energy of the structure, when it deforms in accordance with its particular mode shape. This method requires the baseline element state mode shapes, but only a few modes are required [13]. The localisation of a damage is based on the decrease in modal strain energy of two structural degrees of freedom, as defined by the form of the mode shapes. Another word if there is a damage in a structure, changes in flexural rigidity for specific area may be used to localise that damage. Assuming that a linear elastic beam structure may be divided into a set of i-elements, the damage index SEDIi for the ith element centred around ith degree of freedom can be written as [7] PNm d
mij j ¼ 1 mij j¼1
SEDIi ¼ PN m
ð4Þ
where mdij and mij are the values of the experimentally determined fractional strain energy for mode j between the endpoints of element i. More details on this method can be found in [13]. The successful damage localisation with SEDI has been presented for the isotropic beams and plates in [13], the RC beams by [12], the wooden board in [7], the composite laminated beams and plates of [14] and for an aluminium plate with the use of experimental data in [15]. The example presented in the graphs from Fig. 6 comes from measurements done for a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end. The graphs show differences between the results of damage identification with SEDI obtained for pure signals of measured mode shapes (part (a)) and for signals with proposed
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Fig. 5. COMAC—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
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Fig. 6. SE—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
processing technique (part (b)). It may be noticed that implementation of the simple signal processing technique for the two first damage indicators has improved the sensitivity of the method. 4.4. Modified Laplacian operator (MLO) The modified Laplacian operator may be also called Gapped Smoothing Method (GSM). The localised changes of element stiffness, due to the presence of a damage, result in a mode shape that has a localised change in slope [16]. Experimental mode shape data are discrete in space and therefore the change in slope can be estimated using a finite difference approximation. As a beam can be analysed as a one-dimensional structure, the one-dimensional Laplacian ‘i of the discrete mode shape fi can be introduced: ‘i ¼ ðfi þ 1 þ fi1 Þ2fi
ð5Þ
As the small damage influences the Laplacian insignificantly there have been attempts to improve the methodology. One of them, considered as the most effective, has been to fit a cubic polynomial to the Laplacian and calculate a difference function between the cubic and Laplacian ones [16]. The cubic function has been determined for every point of the Laplacian. Then the difference function between the cubic and Laplacian ones has been calculated, using only two points on either side of considered element. The difference function dI has been obtained in the form di ¼ a0 þ a1 xi þ a2 x2i þ a3 x3i ‘i ð6Þ with coefficients a0, determined using four Laplacian elements (‘i2 ,‘i1 ,‘i þ 1 ,‘i þ 2 ) [16]. Qiao et al. [17] describe an extended to 2D Gapped Smoothing Method (GSM) proposed in [18]. The GSM is based on the idea that a mode shape for an undamaged structure is smooth. If the measured mode shape from a damaged structure could be used for approximation of the ideal one then the difference might be calculated. The damage parameter based on this method has been simply defined as the square of the difference between the measured data and the smoothed fitting values, with the maximum value of GSM indicating the location of damage. The most important advantage of this approach is that the information from the undamaged structure is not necessarily required [17]. Both methods have been successfully used for identification of damages in the uniform isotropic [16] and composite [17,19] beams, isotropic [20] and composite [17] plates. Fig. 7 shows the difference of damage identification obtained for MLO parameter. As in previous examples the mode shapes were measured for a cantilever beam with saw cut of size 10%, located at the distance of 0.3 m from the fixed end. The data presented in part (a) of the figure has been obtained for MLO without proposed signal processing technique. On the contrary part (b) shows MLO after implementation the proposed signal processing. Although it is visible that there
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Fig. 7. MLO—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
is an improvement of the quality of damage prediction this indicator does not provide reliable information due to not very clean information occurring at the ends.
4.5. Generalised fractal dimension (GFD) The concept of fractal dimension (FD) and its relevant mathematical model were originally introduced by Mandelbrot [21]. The approach presented in this paper adopts Katz’s estimation of the FD, since it exhibits high-noise insusceptibility [22]. According to Katz [23], the FD of a curve defined by a sequence of points is estimated by FD ¼
log10 ðnÞ log10 d=L þlog10 ðnÞ
ð7Þ
where d= maxdist(l,i) is the diameter estimated as the distance between the first point of the sequence and the point of the sequence of points i that provides the farthest distance [23], L is the total length of the curve or the sum of distances between successive points. The number of steps in the curve is n = La¯, where a¯ is the average step or average distance between successive points. The latter forms a general unit or ‘‘yardstick’’ that eliminates the dependence of the FD estimates provided by [23] on the measurement units used. As it is clearly explained in [24], estimation of FD in a windowed vibration mode signal gives rise to its value at a local level, enhancing, at the same time, its potential for crack detection. The method shown by Katz [23] may give false peaks in higher mode shapes in the points of the maximum and minimum values of the mode shape first derivative. To overcome this problem Wang and Qiao [19,22] have proposed the application of a scale factor s in the FD algorithm. They have called their improved method generalised fractal dimension (GFD) and have defined it as follows: logðnÞ logðnÞ þ logðds ðxi ,M=Ls ðxi ,MÞÞÞ M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðyðxi þ j Þyðxi þ j1 ÞÞ2 þ s2 ðxi þ j xi þ j Þ2 Ls ðxi ,MÞ ¼
GFDM ðxÞ ¼
j¼1
ds ðxi ,MÞ ¼ max
1rjrM
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðyðxi þ j Þyðxi þ j1 ÞÞ2 þ s2 ðxi þ j xi þ j Þ2
ð8Þ
where n =1/a, with a standing for the average distance between successive points, while xi and yi are the coordinate values of the analysed curve. The term M represents the sliding window length.
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The fractal dimension approach has been used for damage location in the isotropic [19,22] and anisotropic [25] beams, as well as the isotropic [26] and anisotropic [17] plates. An illustration of improvement the GFD damage indicator is shown in Fig. 8. It may be noticed that implementation of the signal processing proposed has slightly improved the reliability of the indicator. Part (a) shows the indicator calculated without implementation of the proposed signal processing, part (b)—the results obtained for data calculated with the proposed signal processing (b). In this case it may be concluded
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that it is possible to badly interpret the data as additional picks have appeared at the ends. This example has also been based on the mode shapes measured for a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end.
Fig. 10. Comparison of mode shape (——), square of mode shape 2nd derivative (....) and of a natural frequency change (- - -) as a function of damage location (a) first, (b) third, (c) sixth, (d) ninth.
1
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Fig. 11. Comparison of mode shape, square of mode shape 2nd derivative and of a natural frequency change as a function of damage size and location (a) first, (b) third, (c) sixth, and (d) ninth.
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4.6. Wavelet transformation (WT) A wavelet is a function used to decompose a signal f(x) into the series components cu,s ðxÞderived from a mother wavelet
cðxÞby the scaling and translating, as given by 1 xu s s
cu,s ðxÞ ¼ pffiffi c
ð9Þ
Considering the mode shape j as a spatial one-dimensional signal, the continuous wavelet transform (CWT) can be obtained as [10,11] 1 W jðu,sÞ ¼ pffiffi s
Z
þ1
jðxÞ 1
xu s
dx
ð10Þ
The wavelet decomposition has the ability to find singularities in considered functions. Therefore, an abrupt change or peak in wavelet coefficient function can be used as an indicator of the damage location. Before numerical processing the measured mode shapes with wavelet decomposition an extrapolation and reduction of additional points after analysis has been done—no wden de-noising. After extrapolation 10-times cubic spline has been done as suggested in [27]. The next step has been the calculation of decomposition coefficients. The parameters used in the analysis have been as follows: the wavelet mother function ‘gauss4’ with scales of the range 0–70.
Fig. 12. Graphs of damage occurrence probability function for the first six mode shapes.
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The results of damage identification with WT is shown in Fig. 9. The graphs present differences between the results of damage identification with WT obtained for pure signal (part (a)) and for extrapolated signal (part (b)). All calculations have been based on measurements of a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end (Fig. 10).
4.7. Analysis results From the results of analysis shown at graphs in Figs. 3–9 it can be seen that almost every method tested has generally pinpointed the damage correctly, except of COMAC. It is worth mentioning that increasing the number of measurement points does not automatically improve the effectiveness of the analysed method. It is caused by the fact that in such case the signal to noise ratio is bigger in second derivative of mode shapes. This effect has been noticed and overcome by Sazonov and Klinkhachorn [28]. From the figures shown it is also visible that, although the proposed signal processing technique is simple, it improves in a significant manner the reliability of the analysed damage indicators. However, when based on experimental data only, it is not possible to provide universal and effective damage prediction. After all calculations it has appeared that only GFD and WT damage indicators have given a reliable identification result. It has been proven that those methods are efficient even in the case of the measurement noise present in the signal. The MLO is also able to correctly localise crack position but in less distinctive manner. The above stated might be used as a hint how to choose the detection method, but it would
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Fig. 13. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.3 x/L).
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be better to have more versatile procedure. The last statement has been the motivation of introducing another new damage indicator and totally new damage detection procedure. 5. Potential strain energy change based damage indicator 5.1. Theoretical background It is commonly known that for a Timoshenko beam the strain energy is given by [13] 2 2 Z Z 1 l @y 1 l @w y dx U¼ EI dx þ kGA 2 0 2 0 @x @x
ð11Þ
where EI is the bending stiffness, GA is shear stiffness, y is the cross sectional transverse rotation, w is cross sectional transverse displacement, k is so called shear-correction factor and l is the beam length. The fact that damage occurring in a structure causes some changes in strain energy, and as a consequence in structural mode shapes, is normally used for damage prediction by numerous damage detection methods. In more detail there is a clear relation between the strain change and the second derivative of every mode shape. One more interesting relation has been noticed by the authors. The variation of a function of natural frequency change, depending on a damage location along the beam length, is similar to the change of second derivative of mode shape. The
Fig. 14. Graphs of damage occurrence probability function for the first six mode shapes.
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graphs show results of simulation done for a FEM model of an analysed cantilever beam (Matlabs, beam element based on Timoshenko theory, 120 elements with total 4801 of freedom). The natural frequency change is defined as
Doj ðxÞ ¼ oj odj ðxÞ
ð12Þ
where oj is the natural frequency obtained for reference (undamaged) model and odj ðxÞ signifies the frequency simulated for an element with damage at location x. As the damage a change in stiffness has been assumed. If one takes into consideration the damage size as well 3d pictures may be obtained. Graphs from Fig. 11 present the natural frequency change as a function of damage location and size. The above presented data suggests that natural frequency change only might be used for damage detection; however this paper introduces also a method enhancing the efficiency of already existing ones. 5.2. Frequency change based damage indicator The proposed damage detection method has been realised in the following steps:
Calculation of the natural frequency change: Doj ¼ oj odj
ð13Þ
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Fig. 15. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.9 x/L).
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Normalising of every calculated frequency change to [0,1] range: doj ¼
Doj minj ðDoj Þ maxj ðDoj Þminj ðDoj Þ
ð14Þ
Validation of the following condition: if maxj Doj min Doj o 2e, with e being the threshold, then it is assumed that there is no damage or the existing damage is too small to be identified.
Linear extrapolation of measured signal with the number of points has been equal to the half of a filtering window. Signal smoothing with moving average: fi,j ¼
X
f~ ik,j Fk
ð15Þ
k
~ is the measured mode shape matrix, f is the smoothed mode shape and F is the averaging filter mask with where f i,j i,j k-elements equal to 1/k. Optional interpolation with cubic spline polynomials—for those mode shapes which have not enough measurement points. Mode shape extrapolation (linear or polynomial) with one point at both vector ends for obtaining the same number of points in analysed vectors and its derivatives (from Eq. (15)).
Fig. 16. Graphs of damage occurrence probability function for the first six mode shapes.
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Calculation of the curvatures (second derivatives) of the mode shapes with central difference method: 00 fi,j ¼
fi þ 1,j 2fi,j þ fi1,j
ð16Þ
h2
where f00 is the mode shape curvature, h is the distance between measurement points i of the Jth mode shape f.
Normalising of every calculated curvature to [0,1] range: 00 Fi,j ¼
00 00 fi,j minj ðfi,j Þ 00 00 Þ maxj ðfi,j Þminj ðfi,j
ð17Þ
Calculation of a function describing the change of strain along the beam length: si,j ¼ ðFi,j00 Þ2
ð18Þ
Determination of the function (P) defining the probability of damage location: Pi,j ¼ 19si,j doj 9
ð19Þ
Determination of the sum of aforementioned function (SP): SPi ¼
Nm 1 X P Nm j ¼ 1 i,j
ð20Þ
with Nm meaning the number of measured mode shapes. 2
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Fig. 17. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.3 x/L).
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Determination of the Frequency change based Damage Indicator (FDI) as the difference of SP and its mean value: FDIi ¼ SPi mðSPi Þ
ð21Þ
5.3. Hybrid Damage Detection method The signal processing technique described above may be utilised as a component of new Hybrid Damage Detection method (HDD) utilising natural frequencies and mode shapes for fatigue damage detection. The novelty proposed here is utilising the FDI method for improvement of damage identification method based on wavelet transformation, but it could be implemented into any damage indicator function. The algorithm consists of:
determination of the damage occurrence probability function P as in Eq. (19). normalisation of P function to get values on the range [01]: NPi,j ¼
Pi,j mini ðPi,j Þ maxi ðPi,j Þmini ðPi,j Þ
Fig. 18. Graphs of damage occurrence probability function for the first six mode shapes.
ð22Þ
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determination of the relation of NP function and do coefficient for every mode shape: Vi,j ¼
NPi,j
ð23Þ
d oj
where Vi,j is the jth mode shape ith point weight matrix. The reason for this operation is that bigger frequency change (in comparison to other changes) signifies bigger changes in mode shapes. If the bigger frequency or mode shape could be identified—the better damage prediction should be obtained. It is why there is a weight matrix needed for distinguishing the most useful mode shapes. Finally HDD is obtained by HDDi ¼
Nm X
Vi,j DIi,j
ð24Þ
j¼1
where Vi,j is the jth mode shape, ith point weight matrix, DIi,j is damage index calculated separately for jth mode shape. For calculating HDD from wavelets the following equation has been used (for every scale, respectively):
HDDi,s ¼
Nm X
Vi,j WTi,j,s
ð25Þ
j¼1
where WT is the wavelet transformation at i point for j mode shape and s scale.
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Fig. 19. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.3 x/L).
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5.4. Results Several sets of simulated data have been used to investigate the correctness of proposed approach in identification of damage in cantilever beam structure as well as to study the effect of changing the number of mode shapes taken for the analysis. The first two examples show results of numerical data analysis, latter two—results of analysis done with experimentally measured data. Experimental data has been used to show the efficiency of the method presented in Section 5.3 above (the last two examples given). The first example shows the comparison of the variation of the second derivative of a mode shape and the natural frequency change (Eq. (14)). Fig. 12 presents damage occurrence probability function for the first six mode shapes. The data has been obtained for a model of an aluminium cantilever beam with the crack simulated as 20% stiffness reduction and localised at the distance of 0.3 from the free end. The curves represent first six mode shapes, s given by Eq. (17) and P given by Eq. (19). Fig. 13 represents the variability of FDI criterion calculated for three, six and fifteen mode shapes respectively. It may be concluded that with the bigger number of mode shapes and frequencies is taken into account the better damage prediction is. The following Figs. 14 and 15 show similar results of numerical investigation but for a beam with damage located at the relative distance of 0.9 from the fixed end and for 20% reduction of the element stiffness. As in previous example one can conclude that growing number of mode shapes taken into analysis improves the sensitivity of the method. Following two more examples of utilisation of FDI criterion have been done for experimental data. Mode shapes and frequencies have been measured for a cantilever beam with a damage realised as a saw cut at the relative distance of 0.3 from the fixed end. Fig. 16 reveals the change of mode shape, s and r functions for the first six measured mode shapes, whereas Fig. 17 presents the damage identification with the proposed FDI criterion. To calculate the FDI three, six and fifteen measured mode shapes have been taken into account. As in previous cases it is clearly visible that greater number of mode shapes improves the sensitivity of results obtained. Figs. 18 and 19 show the experimental results of damage identification in an aluminium cantilever beam with a saw cut located at the distance of 0.9 from the fixed end. The data presented has analogous meaning as in Figs. 16 and 17. Also for this case it may be seen that increasing the number of mode shapes taken into calculations (regarding signal processing of measured mode shapes) improves the sensitivity of the method.
Wavelet Damage Index - only last scale 8
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Fig. 20. Graphs of wavelet damage index without HDD (a) and for wavelet damage index with HDD (b) (damage size—10% of the beam height located at 0.3 x/L).
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Wavelet Damage Index - only last scale 8
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Fig. 21. Graphs of wavelet damage index without HDD (a) and for wavelet damage index with HDD (b) (damage size—10% of the beam height located at 0.9 x/L).
The last three presented figures give a comparison of utilisation WT with and without HDD approach. The criteria presented here have been obtained while analysing 15 experimentally obtained mode shapes. Figs. 20 and 21 present the damage identification for a beam with a damage of depth of 20% located at the distance 0.3 and 0.9 from the fixed end. Both figures give an impression of the influence of proposed HDD technique on the effectiveness of the WT method. It may be noticed that including the proposed HDD signal analysis technique it is possible to improve the sensitivity of WT method. Similar conclusion may be driven on the bases of graphs shown in Fig. 22, which presents the results obtained for a beam with the saw cut of 10% of the beam height located at the relative distance of 0.3 from the fixed end. Also in this case it is clearly shown that utilising the HDD technique improves the damage location prediction.
6. Conclusions In this paper a new method for structural damage detection based on experimentally obtained modal parameters has been proposed. The new method has been tested while analysing different fatigue damages occurring in a beam like structure. The first step of analysis has been devoted to an attempt of damage identification with the most often used damage indicators based on measured modal parameters. Simple signal processing technique has been proposed. The proposed digital signal processing has enhanced the effectiveness of most tested damage detection methods. The best sensitivity improvement has been received for the damage index, but the most effective, noise independent and versatile has been the wavelet transform. Although the sensitivity of methods tested has been improved there has been proposed another damage detection indicator utilising mode shapes and natural frequencies. It has been shown that with the proposed technique, it is possible to localise a damage with experimentally obtained characteristics. Although the baseline information regarding natural frequencies is necessary it is relatively easy to provide this parameter with numerical simulation. Another conclusion may be derived according to the measurement itself. Special surface preparation and proper positioning of scanning head are very important for providing high quality of measured signals. If the resolution of measurement points is high it is worth to check how much descends the laser light. The most important conclusion is that proposed damage detection algorithms work well with experimental data and baseline information needed is the natural frequency set.
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Stress Based Damage Localization Index
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Wavelet Damage Index - only last scale 8
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Fig. 22. Graphs of FDI (a), for wavelet damage index without HDD (b) and for wavelet damage index with HDD (c) (damage size—10% of the beam height located at 0.3 x/L).
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/jnlabr/ymssp
Improvement of damage detection methods based on experimental modal parameters Maciej Radzien´ski a, Marek Krawczuk a,b, Magdalena Palacz a,b,n a b
´ sk University of Technology, Faculty of Electrical and Control Engineering, Narutowicza 11/12, 80-233 Gdan ´ sk, Poland Gdan ´ sk, Poland Institute of Fluid Flow Machinery PAS, Fiszera 14, 80-231 Gdan
a r t i c l e i n f o
abstract
Article history: Received 11 June 2010 Received in revised form 16 November 2010 Accepted 11 January 2011 Available online 2 February 2011
The objective of this paper is to introduce a new method for structural damage detection based on experimentally obtained modal parameters. The new method is suitable for detection of fatigue damage occurring in an aluminium cantilever beam. The damage has been practically realised as saw cuts of different sizes and at different locations. The first step of analysis included an attempt of damage identification with the most often used damage indicators based on measured modal parameters. For that purpose special signal processing technique has been proposed improving the effectiveness of indicators tested. However the results obtained have not been satisfactory. That was the motivation for defining new damage indicators (frequency change based damage indicator, Hybrid Damage Detection method), utilising the change of natural frequencies and any mode shape (measured or modelled) as the measurement of frequencies is much less time consuming in comparison to total mode shape measurement. It has been shown that the proposed technique is suitable for damage localisation in beam-like structures. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Modal analysis Damage detection Signal processing
1. Introduction According to the definition, the structural health monitoring is a scientific procedure comprising of several nondestructive processes including identification of operational and environmental loads acting on the component, recognition of the mechanical damage caused by that loading, and observation of damage growth as the component operates. Finally, the health monitoring deals with assessing the future performance of the component as damage develops. The health monitoring techniques should be non-destructive and ideally implemented online in an automated manner with embedded hardware/software, as a system operates [1]. The modern health monitoring techniques utilise such parameters as the electromechanical and high frequency impedance changes, the strain, stress or temperature fields, and the vibration changes. So as to measure these responses different transducers have been used. The most frequently used are eddy current/electromagnetic sensors, fibre optic based sensors, piezoelectric and piezoceramic sensors, thermal imaging sensors, conductive polymers, MEMS (Micro Electro-Mechanical Systems), and optical lasers. An integral part of damage identification algorithm is adequate signal processing and the feature extraction techniques. The signal processing methods are applied to raw data that is acquired from the sensors in order to produce significant information for diagnostics. One of the main challenges faced in signal processing is the variability of symptoms caused by operational conditions and the resulting changes in the quantities used for diagnostics. Such variations may cause false positive or negative indications of damage, which should be excluded according to the end user’s health monitoring
n Corresponding author at: Gdan´sk University of Technology, Faculty of Electrical and Control Engineering, Narutowicza 11/12, 80-233 Gdan´sk, Poland. Tel.: + 48 58 347 1521; fax: + 48 58 347 2136. E-mail address: [email protected] (M. Palacz).
0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.01.007
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specifications. For example, while selecting the signal processing technique and the feature extraction method in the health monitoring algorithms, it is essential to consider carefully the extent to which these methods amplify or suppress sources of variability [1]. In addition to detecting the presence of damage, the damage identification algorithms should also ideally provide an indication of the damage location and magnitude. The task of damage localisation usually requires a model of the component and the data from multiple sensors. Similarly, the task of damage quantification almost always requires a physical-based model of the component. The damage is also more quantifiable if active sensing is applied, where the sensor signals can be compared continuously to a measured signal [1]. Structural damage is defined as a permanent change in the mechanical state of a structural material or component that could potentially affect their performance [1]. Common sources of damage in material or structural components include micro-structural defects (dislocations, voids, inclusions), corrosion (loss of material), residual stress, cracking (fatigue, matrix, ply), fastening fault (weld crack, bold preload, broken rivet), adhesive fault (de-bonding, delamination, separation), and instability (thermo-mechanical buckling) [1]. The challenge is to interpret the changes of the response parameters due to the type of damage and to correlate them with the corresponding measured parameters. Specification of the relationship between the damage and the characteristic parameters provides the foundation of the identification and assessment algorithm required for the SHM system [2,3]. In the last two decades the non-destructive examination (NDE) techniques, and the structural health monitoring (SHM) techniques as well, have received a considerable amount of interest. Among them vibration analysis for damage detection has been the most popular, due to its implementation simplicity [1,2]. Development of the Scanning Laser Doppler Vibrometers (SLDV) brings about that those techniques have been even more attractive because of non-contact precise vibration measurements with high resolution in a considerably small period of time. Vibration-based damage detection methods make use of dynamic responses of a structure, like natural frequencies and mode shapes, and frequency response functions as well. Occurrence of the damage leads to changes in dynamics of structure. These changes may be used for damage localisation and assessment of its magnitude. Recently, many studies have been performed proposing different vibration-based damage detection methods. In the majority of cases those methods have been tested only numerically. In addition, it is hard to evaluate applicability of a specific damage detection method with no comparison to the results obtained from the same set of data for other techniques. Thus, the purpose of the present study is to determine benefits and drawbacks of the most popular and promising damage detection methods by mining the direct comparison between them. We would also like to show that the digital signal processing based on measured data may enhance the results of damage localisation. It has been shown that the introduced signal processing has improved the sensitivity of analysed damage indicators, however the results obtained have not been satisfactory. For that reason new damage indicators have been introduced. They are based on experimental data only and contain the proposed signal processing. One of the advantages of the approach proposed is the easiness of measuring the parameter needed—natural frequency change. There is also a curvature of the mode shape needed, but it may be simulated as well. The measurement of natural frequency only makes the whole procedure much less time consuming in comparison to classical modal analysis. Another positive aspect of the indicators proposed is that they may be used for improving the efficiency of already existing methods based on mode shapes changes. 2. Experimental set for data gathering It is commonly known that the presence of damage influences vibration parameters of the examined component. Different structural responses may be used for the damage detection and estimation. One of vibration parameters very sensitive to damages, are the mode shapes. Any mode shape is a unique characteristic of a mechanical structure and represents spatial distribution of the amplitude corresponding to each resonance. A local damage causes the change in the derivatives of the mode shapes at the position of the damage. In past times it was difficult to measure mode shape characteristics with proper precision. Recently, available technological devices, like SLDV, enable exact measurement of mode shapes with the accuracy needed, in relatively short time. Section 4 presents the analysis of several mode shape based damage detection methods, being the most popular according to [2]. Mode shapes have been obtained experimentally. All experiments have been done for a steel cantilever beam (Fig. 1) of dimensions: L H W= 1 0.02 0.01 m3. Various damage scenarios have been analysed. As the damage a saw cutting located at L1 = 10%, 30%, 50%, 70%, or 90% of the beam length L (starting from the fixed end) with cut depth Hc equal to 10% and 20% of the beam height has been tested. The width of the crack has been kept constant and equal to Lc =0.001 m. In addition, the cut depth Hc has been changed by a step change from 5% to 50% with 5% step, for the cut located at L1 = 10% of the beam length. Fig. 2 shows the experimental set-up used for gathering modal parameters from the examined element. All data has been measured using a Polytec PSV-400 SLDV. For system excitation an electromechanical shaker (GWV B100) has been used. So as to obtain the modal parameters there has been performed a measurement of Frequency Response Functions (FRFs) via Fast Fourier Transformation and the periodic chirp excitation. It has provided determination of natural frequencies by finding local maxima in averaged FRF graph. Those frequencies have been then successively used as a single-frequency excitation to examine ODS (operational deflection shapes) instead of mode shapes measured via FRF due to better precision obtained. Every point has been measured with 10 registered periods of sinusoidal excitation and mean value has been calculated. Such operation leads to better signal to noise ratio in comparison to traditional modal analysis. The measurements have been performed along the beam length at 125 equally spaced points. The noise level of the measured mode shapes has been controlled by utilising frequency bandwidth Bw and the signal level. The noise level is
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excitation rod crack Hc
L1
Lc
H
W L
Fig. 1. A scheme of the specimen and excitation methodology.
shaker
scanning head
SLDV System
laser beam measured specimen
computer DC field generator
Fig. 2. A scheme of the experimental set-up.
pffiffiffiffiffiffi proportional to Bw but the time of acquisition is inversely proportional to Bw, so it should be considered how to find a compromise between the acquisition time and the precision [4]. The signal received by the SLDV can be strengthened by coating the measured surface with a light back-scattering layer, such as bright acrylic paint used in this research. In order to obtain the best scanning parameters the scanning head should be positioned in a way providing maximal visibility. For the PSV-400 used, the maximum visibility is at the distance of 0.099 m+ n 0.204 m, where n = 1, 2, 3, which has been 1731 m with n= 8 in the presented setup. 3. Basic signal processing technique The measured mode shapes have been recorded including all necessary conditions in order to measure the minimal noise level. To obtain the mode shape curvature second order central difference has been used. In order to overcome the possible inconveniency of obtaining wrong damage identification results with methods based on modal parameters, a simple signal processing (SP) technique has been implemented. The first step of the proposed technique has been a linear signal extrapolation (for two points of both ends of the measured signal vector). This operation is introduced so as to overcome inconveniences of the boundary reflections. If needed more precise mathematical background the reader is kindly asked to follow [5]. Then an automatic denoising ‘‘wden’’ MATLABs function has been used in order to get a de-noised version of a 1D input signal. This function is based on wavelet decomposition and it is
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realized in three steps. First the signal is decomposed on detail and approximation coefficients at level Ld. Next the detail coefficients from levels 1 to Ld is thresholded. Finally signal is reconstructed based on approximation coefficient at level Ld and modified detail coefficients at level 1 to Ld. The parameters used in this ‘‘wden’’ function have been as follows: the decomposition level Ld = 2 for detail coefficients, the wavelet mother function ‘‘sym8’’ and the threshold selection rule— heuristic without rescaling. Several different denoising methods have been tested, like moving average, or median filter, but the best in this particular application has been the wavelet decomposition based function. The next step of the analysis has been the elimination of additional points from linear extrapolation being the first step of analysis of all vectors. The final step, before implementation of measured vectors into damage detection algorithms, has been triple cubic spline interpolation of all mode shapes in order to increase the number of points (except the wavelet transformation procedure). 4. Analysis results of damage detection with methods based on natural vibration 4.1. Mode shape curvature (MSC) and damage index (DI) When a damage is introduced in a structure, the bending stiffness at the location of the damage is reduced while at the same time the magnitude of the mode curvatures increase. The absolute differences between the mode curvature of the intact and damaged structures are the highest in the region of the damage and negligibly small outside this region [2]. As the changes of the mode shape curvatures are localised in the region of the damage, the changes may be used effectively to identify damage location in structures. According to literatures [2,3,6–8], the mode shape curvature (MSC) criterion may be defined as the difference in absolute curvatures of the healthy and damaged structures, for each mode, and may be represented as MSCi ¼
Nm X 00 d 00 fi,j fi,j
ð1Þ
j¼1 00
d
where f00 and f stands for mode shape second derivative of undamaged and damaged state, respectively, Nm is the number of measured mode shape curvatures vectors, and index i represents the measured point number and j is the mode shape number. At a certain damage location, the value of MSC is significantly higher than the ones at other locations. Based on the curvature difference values of measured data of damaged and healthy structures, and the MSCs, the location of damage in the structure can be identified. The most popular approach is the utilisation of only a few lower order curvature modes to identify the damage location. MSC has been successfully used so far for identification of various damages in the isotropic beams [8,3], laminated composite beams [2,6], or the wooden board [7]. Mode shape curvatures are also used for determination of the damage index (DI). This parameter utilises mode shape curvatures of a structure as the main variable in the damage localisation algorithm. The algorithm is based on the relative differences in modal strain energy, before and after the damage. The DI parameter for more than one mode is defined as 2 Nm 00 00 P P d 2 d 2 PN 00 DIi ¼ ðfi,j Þ þ N i ¼ 1 ðfi,j Þ i ¼ 1 ðfi,j Þ j¼1
P PN 00 d N 2 00 Þ2 þ 00 2 ðfi,j i ¼ 1 ðfi,j Þ i ¼ 1 ðfi,j Þ
ð2Þ
where Nm signifies the number of measured mode shapes and N means the number of measurement points. The damage index has been successfully applied to damage localisation in the composite [2,6], concrete [9], and isotropic [8,10] beams, mostly with analytical data. The work presented in [10] shows additionally an example of damage identification based on experimental data obtained for a real I-40 bridge Rio Grande in Albuquerque, NM, USA. The graphs from Figs. 3 and 4 present differences between the results of damage identification with MSC and DI obtained for pure signals of measured mode shapes (parts (a)) and for signals with proposed processing technique (parts (b)). In case shown the calculations have been based on measurements of a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end. It may be noticed that implementation of the simple signal processing technique for the two first damage indicators has improved the sensitivity of both methods. 4.2. Coordinate modal assurance criterion (COMAC) The Coordinate Modal Assurance Criterion (COMAC) identifies the co-ordinates at which two sets of the mode shape do not agree. The COMAC factor at a point i between two sets of the same mode shape, in two states, is defined by 2 P d Nm j ¼ 1 fi,j fi,j COMACi ¼ ð3Þ PNm PNm d 2 j ¼ 1 fi,j j ¼ 1 fi,j
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Fig. 3. MSC—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
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Fig. 4. DI—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
where Nm is the number mode shapes, fi,j and fdi,j denote the values of jth mode shape at a point i for undamaged and damaged states, respectively. The COMAC has been developed from the original MAC concept in such a way that the correlation is related to degrees of freedom of the structure rather than to the mode numbers. The resultant COMAC values have been shown to be of considerable help in showing where the damage or inhomogeniety within the structure occur.
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From the literature it emerges that the MAC and COMAC are highly dependent on the geometry of a structure and the location of a damage. It is also proven that these criteria are not sensitive enough to detect damages at their early stages of growth. The MAC and COMAC have been successfully used for numerical and experimental damage detection in the reinforced concrete beam [11,12], the wooden board [7] and the isotropic beam [3]. Fig. 5 presents the results of damage identification with COMAC obtained for pure signals of measured mode shapes (part (a)) and for signals with proposed processing technique (part (b)). As in previous case the calculations have been done for mode shape recorded for a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end. It may be noticed that for this particular damage indicator even the implementation of the proposed signal processing technique does not provide reliable information about the damage. 4.3. Strain energy damage index (SEDI) Strain energy damage index allows to evaluate the health state of the structure as the index expresses the degradation of the flexural rigidity in a certain zone of the structure. The index accounts for the change in strain energy of the structure, when it deforms in accordance with its particular mode shape. This method requires the baseline element state mode shapes, but only a few modes are required [13]. The localisation of a damage is based on the decrease in modal strain energy of two structural degrees of freedom, as defined by the form of the mode shapes. Another word if there is a damage in a structure, changes in flexural rigidity for specific area may be used to localise that damage. Assuming that a linear elastic beam structure may be divided into a set of i-elements, the damage index SEDIi for the ith element centred around ith degree of freedom can be written as [7] PNm d
mij j ¼ 1 mij j¼1
SEDIi ¼ PN m
ð4Þ
where mdij and mij are the values of the experimentally determined fractional strain energy for mode j between the endpoints of element i. More details on this method can be found in [13]. The successful damage localisation with SEDI has been presented for the isotropic beams and plates in [13], the RC beams by [12], the wooden board in [7], the composite laminated beams and plates of [14] and for an aluminium plate with the use of experimental data in [15]. The example presented in the graphs from Fig. 6 comes from measurements done for a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end. The graphs show differences between the results of damage identification with SEDI obtained for pure signals of measured mode shapes (part (a)) and for signals with proposed
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Fig. 5. COMAC—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
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Fig. 6. SE—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
processing technique (part (b)). It may be noticed that implementation of the simple signal processing technique for the two first damage indicators has improved the sensitivity of the method. 4.4. Modified Laplacian operator (MLO) The modified Laplacian operator may be also called Gapped Smoothing Method (GSM). The localised changes of element stiffness, due to the presence of a damage, result in a mode shape that has a localised change in slope [16]. Experimental mode shape data are discrete in space and therefore the change in slope can be estimated using a finite difference approximation. As a beam can be analysed as a one-dimensional structure, the one-dimensional Laplacian ‘i of the discrete mode shape fi can be introduced: ‘i ¼ ðfi þ 1 þ fi1 Þ2fi
ð5Þ
As the small damage influences the Laplacian insignificantly there have been attempts to improve the methodology. One of them, considered as the most effective, has been to fit a cubic polynomial to the Laplacian and calculate a difference function between the cubic and Laplacian ones [16]. The cubic function has been determined for every point of the Laplacian. Then the difference function between the cubic and Laplacian ones has been calculated, using only two points on either side of considered element. The difference function dI has been obtained in the form di ¼ a0 þ a1 xi þ a2 x2i þ a3 x3i ‘i ð6Þ with coefficients a0, determined using four Laplacian elements (‘i2 ,‘i1 ,‘i þ 1 ,‘i þ 2 ) [16]. Qiao et al. [17] describe an extended to 2D Gapped Smoothing Method (GSM) proposed in [18]. The GSM is based on the idea that a mode shape for an undamaged structure is smooth. If the measured mode shape from a damaged structure could be used for approximation of the ideal one then the difference might be calculated. The damage parameter based on this method has been simply defined as the square of the difference between the measured data and the smoothed fitting values, with the maximum value of GSM indicating the location of damage. The most important advantage of this approach is that the information from the undamaged structure is not necessarily required [17]. Both methods have been successfully used for identification of damages in the uniform isotropic [16] and composite [17,19] beams, isotropic [20] and composite [17] plates. Fig. 7 shows the difference of damage identification obtained for MLO parameter. As in previous examples the mode shapes were measured for a cantilever beam with saw cut of size 10%, located at the distance of 0.3 m from the fixed end. The data presented in part (a) of the figure has been obtained for MLO without proposed signal processing technique. On the contrary part (b) shows MLO after implementation the proposed signal processing. Although it is visible that there
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Fig. 7. MLO—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
is an improvement of the quality of damage prediction this indicator does not provide reliable information due to not very clean information occurring at the ends.
4.5. Generalised fractal dimension (GFD) The concept of fractal dimension (FD) and its relevant mathematical model were originally introduced by Mandelbrot [21]. The approach presented in this paper adopts Katz’s estimation of the FD, since it exhibits high-noise insusceptibility [22]. According to Katz [23], the FD of a curve defined by a sequence of points is estimated by FD ¼
log10 ðnÞ log10 d=L þlog10 ðnÞ
ð7Þ
where d= maxdist(l,i) is the diameter estimated as the distance between the first point of the sequence and the point of the sequence of points i that provides the farthest distance [23], L is the total length of the curve or the sum of distances between successive points. The number of steps in the curve is n = La¯, where a¯ is the average step or average distance between successive points. The latter forms a general unit or ‘‘yardstick’’ that eliminates the dependence of the FD estimates provided by [23] on the measurement units used. As it is clearly explained in [24], estimation of FD in a windowed vibration mode signal gives rise to its value at a local level, enhancing, at the same time, its potential for crack detection. The method shown by Katz [23] may give false peaks in higher mode shapes in the points of the maximum and minimum values of the mode shape first derivative. To overcome this problem Wang and Qiao [19,22] have proposed the application of a scale factor s in the FD algorithm. They have called their improved method generalised fractal dimension (GFD) and have defined it as follows: logðnÞ logðnÞ þ logðds ðxi ,M=Ls ðxi ,MÞÞÞ M qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X ðyðxi þ j Þyðxi þ j1 ÞÞ2 þ s2 ðxi þ j xi þ j Þ2 Ls ðxi ,MÞ ¼
GFDM ðxÞ ¼
j¼1
ds ðxi ,MÞ ¼ max
1rjrM
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðyðxi þ j Þyðxi þ j1 ÞÞ2 þ s2 ðxi þ j xi þ j Þ2
ð8Þ
where n =1/a, with a standing for the average distance between successive points, while xi and yi are the coordinate values of the analysed curve. The term M represents the sliding window length.
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The fractal dimension approach has been used for damage location in the isotropic [19,22] and anisotropic [25] beams, as well as the isotropic [26] and anisotropic [17] plates. An illustration of improvement the GFD damage indicator is shown in Fig. 8. It may be noticed that implementation of the signal processing proposed has slightly improved the reliability of the indicator. Part (a) shows the indicator calculated without implementation of the proposed signal processing, part (b)—the results obtained for data calculated with the proposed signal processing (b). In this case it may be concluded
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Fig. 8. GFD—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
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Fig. 9. WT—without (a) and with (b) proposed signal processing (damage size—10% of the beam height located at 0.3 x/L).
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that it is possible to badly interpret the data as additional picks have appeared at the ends. This example has also been based on the mode shapes measured for a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end.
Fig. 10. Comparison of mode shape (——), square of mode shape 2nd derivative (....) and of a natural frequency change (- - -) as a function of damage location (a) first, (b) third, (c) sixth, (d) ninth.
1
1
0.5
0.5
0
0
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0 0
10
20
30
40
50
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40 30 20 Stiffness loss [%]
50
0 0
10
0 0
10
20
30
40
Stiffness loss [%]
0 1 0.5
Damage location
50
40 30 20 Stiffness loss [%]
50
Fig. 11. Comparison of mode shape, square of mode shape 2nd derivative and of a natural frequency change as a function of damage size and location (a) first, (b) third, (c) sixth, and (d) ninth.
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4.6. Wavelet transformation (WT) A wavelet is a function used to decompose a signal f(x) into the series components cu,s ðxÞderived from a mother wavelet
cðxÞby the scaling and translating, as given by 1 xu s s
cu,s ðxÞ ¼ pffiffi c
ð9Þ
Considering the mode shape j as a spatial one-dimensional signal, the continuous wavelet transform (CWT) can be obtained as [10,11] 1 W jðu,sÞ ¼ pffiffi s
Z
þ1
jðxÞ 1
xu s
dx
ð10Þ
The wavelet decomposition has the ability to find singularities in considered functions. Therefore, an abrupt change or peak in wavelet coefficient function can be used as an indicator of the damage location. Before numerical processing the measured mode shapes with wavelet decomposition an extrapolation and reduction of additional points after analysis has been done—no wden de-noising. After extrapolation 10-times cubic spline has been done as suggested in [27]. The next step has been the calculation of decomposition coefficients. The parameters used in the analysis have been as follows: the wavelet mother function ‘gauss4’ with scales of the range 0–70.
Fig. 12. Graphs of damage occurrence probability function for the first six mode shapes.
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The results of damage identification with WT is shown in Fig. 9. The graphs present differences between the results of damage identification with WT obtained for pure signal (part (a)) and for extrapolated signal (part (b)). All calculations have been based on measurements of a cantilever beam with a crack of size 10%, located at the distance of 0.3 m from the fixed end (Fig. 10).
4.7. Analysis results From the results of analysis shown at graphs in Figs. 3–9 it can be seen that almost every method tested has generally pinpointed the damage correctly, except of COMAC. It is worth mentioning that increasing the number of measurement points does not automatically improve the effectiveness of the analysed method. It is caused by the fact that in such case the signal to noise ratio is bigger in second derivative of mode shapes. This effect has been noticed and overcome by Sazonov and Klinkhachorn [28]. From the figures shown it is also visible that, although the proposed signal processing technique is simple, it improves in a significant manner the reliability of the analysed damage indicators. However, when based on experimental data only, it is not possible to provide universal and effective damage prediction. After all calculations it has appeared that only GFD and WT damage indicators have given a reliable identification result. It has been proven that those methods are efficient even in the case of the measurement noise present in the signal. The MLO is also able to correctly localise crack position but in less distinctive manner. The above stated might be used as a hint how to choose the detection method, but it would
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Fig. 13. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.3 x/L).
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be better to have more versatile procedure. The last statement has been the motivation of introducing another new damage indicator and totally new damage detection procedure. 5. Potential strain energy change based damage indicator 5.1. Theoretical background It is commonly known that for a Timoshenko beam the strain energy is given by [13] 2 2 Z Z 1 l @y 1 l @w y dx U¼ EI dx þ kGA 2 0 2 0 @x @x
ð11Þ
where EI is the bending stiffness, GA is shear stiffness, y is the cross sectional transverse rotation, w is cross sectional transverse displacement, k is so called shear-correction factor and l is the beam length. The fact that damage occurring in a structure causes some changes in strain energy, and as a consequence in structural mode shapes, is normally used for damage prediction by numerous damage detection methods. In more detail there is a clear relation between the strain change and the second derivative of every mode shape. One more interesting relation has been noticed by the authors. The variation of a function of natural frequency change, depending on a damage location along the beam length, is similar to the change of second derivative of mode shape. The
Fig. 14. Graphs of damage occurrence probability function for the first six mode shapes.
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graphs show results of simulation done for a FEM model of an analysed cantilever beam (Matlabs, beam element based on Timoshenko theory, 120 elements with total 4801 of freedom). The natural frequency change is defined as
Doj ðxÞ ¼ oj odj ðxÞ
ð12Þ
where oj is the natural frequency obtained for reference (undamaged) model and odj ðxÞ signifies the frequency simulated for an element with damage at location x. As the damage a change in stiffness has been assumed. If one takes into consideration the damage size as well 3d pictures may be obtained. Graphs from Fig. 11 present the natural frequency change as a function of damage location and size. The above presented data suggests that natural frequency change only might be used for damage detection; however this paper introduces also a method enhancing the efficiency of already existing ones. 5.2. Frequency change based damage indicator The proposed damage detection method has been realised in the following steps:
Calculation of the natural frequency change: Doj ¼ oj odj
ð13Þ
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Fig. 15. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.9 x/L).
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Normalising of every calculated frequency change to [0,1] range: doj ¼
Doj minj ðDoj Þ maxj ðDoj Þminj ðDoj Þ
ð14Þ
Validation of the following condition: if maxj Doj min Doj o 2e, with e being the threshold, then it is assumed that there is no damage or the existing damage is too small to be identified.
Linear extrapolation of measured signal with the number of points has been equal to the half of a filtering window. Signal smoothing with moving average: fi,j ¼
X
f~ ik,j Fk
ð15Þ
k
~ is the measured mode shape matrix, f is the smoothed mode shape and F is the averaging filter mask with where f i,j i,j k-elements equal to 1/k. Optional interpolation with cubic spline polynomials—for those mode shapes which have not enough measurement points. Mode shape extrapolation (linear or polynomial) with one point at both vector ends for obtaining the same number of points in analysed vectors and its derivatives (from Eq. (15)).
Fig. 16. Graphs of damage occurrence probability function for the first six mode shapes.
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Calculation of the curvatures (second derivatives) of the mode shapes with central difference method: 00 fi,j ¼
fi þ 1,j 2fi,j þ fi1,j
ð16Þ
h2
where f00 is the mode shape curvature, h is the distance between measurement points i of the Jth mode shape f.
Normalising of every calculated curvature to [0,1] range: 00 Fi,j ¼
00 00 fi,j minj ðfi,j Þ 00 00 Þ maxj ðfi,j Þminj ðfi,j
ð17Þ
Calculation of a function describing the change of strain along the beam length: si,j ¼ ðFi,j00 Þ2
ð18Þ
Determination of the function (P) defining the probability of damage location: Pi,j ¼ 19si,j doj 9
ð19Þ
Determination of the sum of aforementioned function (SP): SPi ¼
Nm 1 X P Nm j ¼ 1 i,j
ð20Þ
with Nm meaning the number of measured mode shapes. 2
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Fig. 17. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.3 x/L).
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Determination of the Frequency change based Damage Indicator (FDI) as the difference of SP and its mean value: FDIi ¼ SPi mðSPi Þ
ð21Þ
5.3. Hybrid Damage Detection method The signal processing technique described above may be utilised as a component of new Hybrid Damage Detection method (HDD) utilising natural frequencies and mode shapes for fatigue damage detection. The novelty proposed here is utilising the FDI method for improvement of damage identification method based on wavelet transformation, but it could be implemented into any damage indicator function. The algorithm consists of:
determination of the damage occurrence probability function P as in Eq. (19). normalisation of P function to get values on the range [01]: NPi,j ¼
Pi,j mini ðPi,j Þ maxi ðPi,j Þmini ðPi,j Þ
Fig. 18. Graphs of damage occurrence probability function for the first six mode shapes.
ð22Þ
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determination of the relation of NP function and do coefficient for every mode shape: Vi,j ¼
NPi,j
ð23Þ
d oj
where Vi,j is the jth mode shape ith point weight matrix. The reason for this operation is that bigger frequency change (in comparison to other changes) signifies bigger changes in mode shapes. If the bigger frequency or mode shape could be identified—the better damage prediction should be obtained. It is why there is a weight matrix needed for distinguishing the most useful mode shapes. Finally HDD is obtained by HDDi ¼
Nm X
Vi,j DIi,j
ð24Þ
j¼1
where Vi,j is the jth mode shape, ith point weight matrix, DIi,j is damage index calculated separately for jth mode shape. For calculating HDD from wavelets the following equation has been used (for every scale, respectively):
HDDi,s ¼
Nm X
Vi,j WTi,j,s
ð25Þ
j¼1
where WT is the wavelet transformation at i point for j mode shape and s scale.
2
zscore
1 0 -1 -2
0
0.1
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0.5 x/L
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0
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0.5 x/L
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1
0
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0.3
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0.5 x/L
0.6
0.7
0.8
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1
4
zscore
2
0
-2
6
zscore
4 2 0 -2 -4
Fig. 19. Graphs of FDI for three six and fifteen mode shapes (damage size—10% of the beam height located at 0.3 x/L).
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5.4. Results Several sets of simulated data have been used to investigate the correctness of proposed approach in identification of damage in cantilever beam structure as well as to study the effect of changing the number of mode shapes taken for the analysis. The first two examples show results of numerical data analysis, latter two—results of analysis done with experimentally measured data. Experimental data has been used to show the efficiency of the method presented in Section 5.3 above (the last two examples given). The first example shows the comparison of the variation of the second derivative of a mode shape and the natural frequency change (Eq. (14)). Fig. 12 presents damage occurrence probability function for the first six mode shapes. The data has been obtained for a model of an aluminium cantilever beam with the crack simulated as 20% stiffness reduction and localised at the distance of 0.3 from the free end. The curves represent first six mode shapes, s given by Eq. (17) and P given by Eq. (19). Fig. 13 represents the variability of FDI criterion calculated for three, six and fifteen mode shapes respectively. It may be concluded that with the bigger number of mode shapes and frequencies is taken into account the better damage prediction is. The following Figs. 14 and 15 show similar results of numerical investigation but for a beam with damage located at the relative distance of 0.9 from the fixed end and for 20% reduction of the element stiffness. As in previous example one can conclude that growing number of mode shapes taken into analysis improves the sensitivity of the method. Following two more examples of utilisation of FDI criterion have been done for experimental data. Mode shapes and frequencies have been measured for a cantilever beam with a damage realised as a saw cut at the relative distance of 0.3 from the fixed end. Fig. 16 reveals the change of mode shape, s and r functions for the first six measured mode shapes, whereas Fig. 17 presents the damage identification with the proposed FDI criterion. To calculate the FDI three, six and fifteen measured mode shapes have been taken into account. As in previous cases it is clearly visible that greater number of mode shapes improves the sensitivity of results obtained. Figs. 18 and 19 show the experimental results of damage identification in an aluminium cantilever beam with a saw cut located at the distance of 0.9 from the fixed end. The data presented has analogous meaning as in Figs. 16 and 17. Also for this case it may be seen that increasing the number of mode shapes taken into calculations (regarding signal processing of measured mode shapes) improves the sensitivity of the method.
Wavelet Damage Index - only last scale 8
z-score
6 4 2 0 -2 0
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0.5 x/L
0.6
0.7
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1
Wavelet Damage Index - only last scale 8
z-score
6 4 2 0 -2 0
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0.5 x/L
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0.7
Fig. 20. Graphs of wavelet damage index without HDD (a) and for wavelet damage index with HDD (b) (damage size—10% of the beam height located at 0.3 x/L).
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Wavelet Damage Index - only last scale 8
z-score
6 4 2 0 -2 0
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0.5 x/L
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Wavelet Damage Index - only last scale 10
z-score
8 6 4 2 0 -2 0
0.1
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0.5 x/L
0.6
0.7
0.8
Fig. 21. Graphs of wavelet damage index without HDD (a) and for wavelet damage index with HDD (b) (damage size—10% of the beam height located at 0.9 x/L).
The last three presented figures give a comparison of utilisation WT with and without HDD approach. The criteria presented here have been obtained while analysing 15 experimentally obtained mode shapes. Figs. 20 and 21 present the damage identification for a beam with a damage of depth of 20% located at the distance 0.3 and 0.9 from the fixed end. Both figures give an impression of the influence of proposed HDD technique on the effectiveness of the WT method. It may be noticed that including the proposed HDD signal analysis technique it is possible to improve the sensitivity of WT method. Similar conclusion may be driven on the bases of graphs shown in Fig. 22, which presents the results obtained for a beam with the saw cut of 10% of the beam height located at the relative distance of 0.3 from the fixed end. Also in this case it is clearly shown that utilising the HDD technique improves the damage location prediction.
6. Conclusions In this paper a new method for structural damage detection based on experimentally obtained modal parameters has been proposed. The new method has been tested while analysing different fatigue damages occurring in a beam like structure. The first step of analysis has been devoted to an attempt of damage identification with the most often used damage indicators based on measured modal parameters. Simple signal processing technique has been proposed. The proposed digital signal processing has enhanced the effectiveness of most tested damage detection methods. The best sensitivity improvement has been received for the damage index, but the most effective, noise independent and versatile has been the wavelet transform. Although the sensitivity of methods tested has been improved there has been proposed another damage detection indicator utilising mode shapes and natural frequencies. It has been shown that with the proposed technique, it is possible to localise a damage with experimentally obtained characteristics. Although the baseline information regarding natural frequencies is necessary it is relatively easy to provide this parameter with numerical simulation. Another conclusion may be derived according to the measurement itself. Special surface preparation and proper positioning of scanning head are very important for providing high quality of measured signals. If the resolution of measurement points is high it is worth to check how much descends the laser light. The most important conclusion is that proposed damage detection algorithms work well with experimental data and baseline information needed is the natural frequency set.
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Stress Based Damage Localization Index
3 2 zscore
1 0 -1 -2 -3 0
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Wavelet Damage Index - only last scale 6 5
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4 3 2 1 0 -1 -2 0
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Wavelet Damage Index - only last scale 8
z-score
6 4 2 0 -2 0
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Fig. 22. Graphs of FDI (a), for wavelet damage index without HDD (b) and for wavelet damage index with HDD (c) (damage size—10% of the beam height located at 0.3 x/L).
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