Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 3, pp. 579-604, Warsaw 2010 NEURO-WAVELET DAMAGE DETECTION TECHNIQUE IN BEAM, PLATE AND SHELL STRUCTURES WITH EXPERIMENTAL VALIDATION Magdalena Rucka Krzysztof Wilde Gdansk University of Technology, Faculty of Civil and Environmental Engineering, Gdańsk, Poland e-mail: mrucka@pg.gda.pl; wild@pg.gda.pl The paper presents a new neuro-wavelet damage detection technique for structural health monitoring. The proposed method combines the abi- lity of the continuous wavelet transform to detect abnormalities in the structure dynamic parameters with the artificial neural network possi- bility of learning, remembering and recognition. The effectiveness of the method is verified on experimental mode shapes of a beam, plate and shell structures. The results of the study show that the neural network trained on the data from a simple structure can effectively improve the search of the location of the same type of damage in complex structures. Key words: damage detection, continuous wavelet transform, artificial neural networks 1. Introduction All structures raised by humans wear out and undergo self-destruction in the course of time. Fatigue, corrosion, dynamic phenomena, overloading and envi- ronmental conditions can cause their degradation. In order to improve the reliability and safety of structures, various damage detection techniques and health monitoring systems have been intensively studied over the last few de- cades. A simple, quick and nondestructive structural diagnostic system that could facilitate traditional diagnostic procedures is of great importance for solving many problems with maintenance of engineering structures. A relatively recent area of research in damage detection and localization is based on the continuous wavelet transform (CWT). This technique can be performed onmode shapes or static deflections of structure elements. An im- portant feature of thewavelet transform is the ability to characterize the local 580 M. Rucka, K. Wilde irregularity introduced by a defect and to react to subtle changes of the struc- ture response. Most of the reported research, e.g., Douka et al. (2003), Hong et al. (2002) is limited to crack identification bywavelets in beams.Only a few studies (Wang and Deng, 1996; Douka et al., 2004; Chang and Chen, 2004) are devoted to damage detection in plate structures based on numerically de- termined data. The experimental researches on plate damage detection were presented by Rucka and Wilde (2005). The experimental mode shapes of the cantilever plate were determined by the acceleration measurement in one po- int and impact excitation in 66 points. The relative depth of the introduced rectangular defect was 19%. The location of the damage was determined by the one-dimensional Gaussian wavelet with 4 vanishingmoments. The formu- lation of the two-dimensional continuous wavelet transform for plate damage detection was presented byRucka andWilde (2006a). Damage localization by the Reverse Biorthogonal wavelet on the experimental mode shapes of a plate with four fixed supportswas proposed. In the presented research, the location of thepeak in the response indicated the damage location in the structure.Ho- wever, in the case of the experimental data contaminated by noise, the precise identification of the wavelet peaks becomes possible only for serve damage. In Fig.1a the wavelet transform of the beam static deflection line (cf. Rucka and Wilde, 2006b) for the crack of the relative depth of 50% is shown. In this case, the localization of the damage is easy and precise. Although, the decision of damage location in the case of the crack depth of 35% (Fig.1b) cannot be drawnwithout doubts. Fig. 1. Experimental wavelet transformmodulus of the beam deflection lines: (a) visible defect position, (b) hidden defect position Damage detection schemes can be enhanced by the use of the artificial neural networks (ANN). In earlier studies, damage recognition and localization based on the ANN were applied to different structures like ships (Zubaydi et al., 2002) or helicopters (Cabell et al., 1998), crates of beverages (Zacharias et al., 2004), joints in steel structures (Yun et al., 2001), bridges (Barai and Neuro-wavelet damage detection technique in beam... 581 Pandey, 1997; Yeung and Smith, 2005). From among many types of artificial neural networks the backpropagation neural network is the most commonly used to the analysis of civil engineering problems (Kuźniar andWaszczyszyn, 2002; Hoła and Schabowicz, 2005). In the previous studies, as inputs of ANN were used: the ultrasonic signals (Liu et al., 2002; Okafor and Dutta, 2001), broad-band spectral signals (Garg et al., 2004), natural frequencies (Sahin and Shenoi, 2003; Waszczyszyn and Ziemiański, 2001; Zapico et al., 2003) or vibration responses (Barai and Pandey, 1997; Kao and Hung, 2003; Yam et al., 2003). Sanz et al. (2007) as well as Yam et al. (2003) applied wavelet transforms of vibration signals as inputs to ANN. In this study, thewavelet transforms of static deflection lines aswell as the wavelet transforms of vibrationmode shapes are used to the search of damage localization. The artificial neural network is trained and tested on a simple cantilever beam with a rectangular notch on static deflection lines. Then the method is validated on the experimentally determined first threemode shapes of the beam, plate and shell with damage of similar type. The possibility of enhancing the very simple peak-pickingmethodby an artificial neural network is studied. 2. Damage detection strategy of the neuro-wavelet system The practical application of the neuro-wavelet (NW) system for a real engine- ering structure is practically impossible since the training of the NW system requires data of the structure with a large number of damage combinations. On existing structures in use, there is very limited space for extensive static or dynamic tests, not tomention the possibility of imposing damage in a large number of locations. Therefore, the notion of derivation of the NW system on very simple tests is hereafter presented. It is proposed that the data from the static experiments on a simple crac- ked cantilever beam facilitated by the numerical data obtained from updated FEMmodels is sufficient to define theNWsystem formonitoring the complex structures. It is expected that the NW system can detect damage regardless of the structure geometry and the type of the measurements. It is assumed that the defect geometry is the same in all considered structures. Thewavelet transformmoduli computed fromthemeasurement signals are used as inputs to the neural network system. The advantage of the wavelet transform over the Fourier transforms or measurement records is the ability to extract the selected information from the various types of signals. Thus, 582 M. Rucka, K. Wilde it is possible to combine, for example, the data from static experiments with the data from dynamic tests. The disadvantage of wavelets is the problem with selection of the most appropriate waveform and its number of vanishing moments. The choice of the best wavelet function usually requires trial and error simulations. TheproposedNWsystem is intended to have the ability of recognizing two levels of damage detection precision, namely, the presence of damage and its position. The search of damage presence and prediction of damage position is considered independently. If the NW system gives the information that there is no damage in the structure, the results on damage location are neglected. Based on the Authors’ experience, various experimental data transformed by wavelets contains information with different intensity. For example, the mode shapes extracted fromthe ambient vibrations (Wilde et al., 2006)might notbe sufficiently precise to indicate the damage location, but are good for drawing the conclusion on damage presence. The cantilever beamof plexiglass with the rectangular cross-section is con- sidered as the simple testing structure. The data for the NW system training are obtained form the static tests conducted on the beamwith a defect. In this paper, a rectangular notchobtainedbyacutbyahighprecision saw is conside- red as the defect. The static deflection lines for different damage configuration were determined by a photogrametric method (cf. Rucka and Wilde, 2006b). The NW system training data also consist of the numerical static deflection lines of the beam. Although the experiments on the cantilever beam are very simple, it is still complicated to test all possible defects positions and defect depths. Therefore, the results from the experimentally verified FEMmodel of the cracked beamare employed. In the trainingdata the ”noise free” aswell as the deflection lines with added noise are used. Next theNW system is verified on the experimental mode shapes of the beam, plate and shell structures. The neural part of the NW system is a simple two hidden layer artificial neural network (Fig.2). The procedure of damage prediction in the complex structure consists of the following stages: 1. Find the mode shapes of the monitored structures. 2. Calculate the wavelet transformmoduli of the mode shapes. 3. Feed the CWTmoduli to the NW system. 4. Evaluate the output regarding defect presence (0 means there is no de- fect, 1 means defect exists). 5. Evaluate output regardingdefect localization (0means there is nodefect, xi denotes the defect position). Neuro-wavelet damage detection technique in beam... 583 Fig. 2. Outline of the neuro-wavelet damage detection system 3. Continuous wavelet transform in damage detection For a given one-dimensional signal f(x), the continuous wavelet transform (CWT) can be obtained by integration of the product of the signal function with thecomplexconjugate ψ∗(x) of thewavelet functions.Consideringthede- flection line ormode shape of the structures as a one-dimensional signal f(x), the continuous wavelet transform can be defined as (e.g. Mallat, 1998) Wf(u,s)= 1√ s +∞ ∫ −∞ f(x)ψ∗ (x−u s ) dx (3.1) Wf(u,s) is called the wavelet coefficient for the wavelet ψu,s(x) and it me- asures the variation of the signal in the vicinity of the position u whose size is proportional to the scale s. In detection of signal singularities, the vanishing moments play an important role. A wavelet has n vanishing moments if it carries out the following condition +∞ ∫ −∞ xkψ(x) dx =0 k =0,1,2, . . . ,n−1 (3.2) Awavelet with n vanishingmoments can be rewritten as the n-th order deri- vative of a smoothing function θ(x). The resulting wavelet transform can be expressed as a multiscale differential operator 584 M. Rucka, K. Wilde Wf(u,s)= sn dn dun (f(x)∗θs(x))(u) θs(x)= 1√ s θ (−x s ) (3.3) where f(x)∗θs(x) denotes convolution of functions and can be interpreted as an average of f(x) over a domain proportional to the scale s. Thus wavelet transform is the n-th derivative of the signal f(x) smoothed by the function θs(x) at the scale s. The singularities are detected by finding the abscissa where the maxima of the wavelet transform modulus |Wf(u,s)| converge at fine scales (Mallat, 1998). The selection of anappropriate typeofwavelet and thechoice of its number of vanishing moments is essential for effective use of the wavelet analysis. The use of wavelets that create the maximum number of wavelet coefficients that are close to zero is proposed. For the first mode shape of a cantilever or simply supported beam, wavelets with 4 vanishing moments should be used. For structural responses that are similar toapolynomial ofhigher order than4, the use of wavelets with a higher number of vanishing moments is necessary. In this paper, the Gaussian wavelets gaus4 and gaus6 having four and six vanishing moments, respectively, have been chosen as the best candidates to damagedetectionwith theone-dimensional continuouswavelet transform.The advantage of the Gaussian wavelets was discussed by Gentile and Messina (2003), Mallat (1998) as well as Rucka andWilde (2006b, 2007). In the case of the plate mode shape, i.e. two-dimensional spatial signal f(x,y), the two-dimensional wavelet transformof the function f(x,y) is given by (Rucka andWilde, 2006a) W if(u,v,s)= 1 s ∞ ∫ −∞ ∞ ∫ −∞ f(x,y)ψi (x−u s , y−v s ) dx dy = (3.4) = 1 s f ∗ψi (−u s , −v s ) = f ∗ψis(u,v) i =1,2 where the horizontal wavelet ψ1(x,y) and the vertical one ψ2(x,y) are con- structed with separable products of the scaling φ and wavelet function ψ ψ1(x,y)= φ(x)ψ(y) ψ2(x,y)= ψ(x)φ(y) (3.5) However, in this approach, the number of vanishing moments is the same in both directions. In fact, the two-dimensional wavelet transform is not favoura- ble for themore complicated platemode shapes, which can be interpolated by polynomials of different order in both directions. Therefore, instead of (3.4) we use one-dimensional integrals Neuro-wavelet damage detection technique in beam... 585 W1f(u,v,s)= 1√ s ∞ ∫ −∞ f(x,y)ψ∗ (y−v s ) dy (3.6) W2f(u,v,s)= 1√ s ∞ ∫ −∞ f(x,y)ψ∗ (x−u s ) dx Integration in (3.6)1 is performed for each column of the signal f(x,y) and integration in (3.6)2 is performed for each rowof the signal f(x,y).Coefficients W1f(u,v,s) and W2f(u,v,s) are called horizontal and vertical, respectively, and they are used to formulate the modulus of the wavelet transform at the scale s Mf(u,v,s)= √ |W1f(u,v,s)|2+ |W2f(u,v,s)|2 (3.7) Note that formula (3.6) indicates independent treatment of the horizontal and vertical coefficients, and therefore, the choice of the number of vanishingmo- ments has to satisfy Eq. (3.2) independently for each direction. In this paper, threemode shapes of the plate fixed at four edges are considered. To analysis of the first mode shape, gaus4 wavelet having four vanishing moments was applied in both directions. Analysis of the second and thirdmode shapes was conducted using gaus4wavelet along the plate width, and gaus6wavelet along the plate length. The additional advantage of applying the two-dimensional wavelet transform in terms of one dimensional integrals is the possibility of using Gaussian wavelets, which do not have scaling functions φ(x). 4. Artificial neural network 4.1. Architecture and backpropagation algorithm The used neural network (Fig.3) consists of three layers – two hidden layers and one output layer. This model includes an input vector z, weight Fig. 3. Architecture of the applied neural network 586 M. Rucka, K. Wilde vector w, bias b and output vector o. A hyperbolic tangent sigmoid function was selected for the hidden layers and a linear transfer function was selected for the output layer. The input of the neural network is a 401-elements vector of wavelet coefficients. The first hidden layer has 100 neurons whereas the second hidden layer has 20 neurons. These numbers were chosen by trial and error simulations and the previous experiences. The outputs indicate both, the presence and position of the structural defect. The backpropagation algorithm is explained in details in many sources, e.g. Bow (2002), Hu andHwang (2002). The net function for the hidden layer neurons and net function for the output layer neurons are given in the form neth(p)m = R ∑ r=1 whmrz (p) r + b h m net o(p) k = M ∑ m=1 wokmi (p) m + b o k (4.1) where R and M denote the number of neurons in the input andhidden layers, respectively. The superscript p refers to the p-th input pattern and b is the bias. The input from the m-th hidden layer neuron i (p) m to the output layer neuron is given as a sigmoid function, whereas the output for the output layer neurons o (p) k is given as an identity function. The error minimized by the training algorithm is defined as E(p) = 1 2 K ∑ k=1 (d (p) k −o(p) k )2 (4.2) where K denotes the number of neurons in the output layer and d (p) k is the known target. The global error can be expressed as a sum of errors of all patterns E = P ∑ p=1 E(p) (4.3) The weight update equations for both output and hidden layers are given in the form wokm(t+1)= w o km(t)−η ∂E(p) ∂wo km whmr(t+1)= w h mr(t)−η ∂E(p) ∂whmr (4.4) where t is the iteration step and η denotes the learning rate assumed as the same on all neurons in all layers. In this study, the search for the optimal weights is conducted by the scaled conjugate gradient algorithm developed by Moller (1993). The optimization is performed along the conjugate direction, which provides generally faster convergence than the direction of the gradient Neuro-wavelet damage detection technique in beam... 587 steepest descent. The step size is modified at each iteration. This algorithm avoids the time consuming line-search per learning iteration. To overcome the overfitting problem, the regularization was used. Overfit- tingmeans that the error on the training data has a very small value but the error is large on the new simulated data. The performance function, chosen as the mean sum of squares of the network errors mse = 2E P (4.5) is modified by adding a term that consists of the mean of the sum of squares of the network weights and biases (Demuth and Beale, 2003) msereg = γmse =(1−γ)msw (4.6) where γ is the performance ratio andmsw can be written as msw = 1 n n ∑ j=1 w2j (4.7) It is difficult to determine the optimumperformance parameter since too large parameter γ provides overfitting, whereas too small parameter provides no fit in the training data. In the performed tests, the ratio γ was set to 0.5. Additionally, the correlation coefficient R was used. The correlation co- efficient R is a normalized measure of the strength of the linear relationship between the target and predicted value R = ∑P i=1(oi−o)(di−d) √ ∑ P i=1(oi−o)2 √ ∑ P i=1(di−d)2 (4.8) where oi and di are the output and target values, and o and d denote mean values. The perfectly correlated data result in a correlation coefficient equal to 1, whereas no correlation results in R equal to 0. 4.2. Training of NW system TheNWsystemhasbeen trained on the experimental, numerical andnoise corrupted numerical data. The training was conduced for the cantilever beam (seeFig.6a) of length L =400mm,height H =20mmandwidth B =60mm (Rucka and Wilde, 2006b). The beam had one notch of length Lr = 2mm and depth a equal to 7, 10 and 13mm. In the numerical simulations, the defect location L1, described by the distance from the clamped end, changed 588 M. Rucka, K. Wilde from 80mm to 320mmwith step of 4mm, giving 61 different defect positions. The numerical deflection lines were computed by SOFiSTiK and the noise added to the data hadGaussiandistributionwith standarddeviation changing from 0.02% to 0.05% of themaximumamplitude of the deflection line. In the experimental research, thenotchpositionwas L1 =101mm.Theexperimental static deflection lines were obtained in 81 points by the fast photogrammetric technique based on digital photo and image processing (Rucka and Wilde, 2006b). The experimental program is described in Table 1. Beam case 4 from Table 1 was used to the training process. Table 1.Beam static deflection lines: experimental program Case No. Dimensions of beam Size of defect length L heigh H width B length Lr heigh a [mm] [mm] [mm] [mm] [mm] 1 400 20 60 0 0 2 2 7 3 2 10 4 2 13 Thewavelet transformmodulus of the numerical and experimental deflec- tion lines was calculated using gaus4 wavelet for scales 10, 20, 30 and 40. The total number of input patterns was 748. The ANN was trained on the wave- let transformmoduli, which clearly indicated damage location, as well as the wavelet transformmoduli that did not explicitly point the damage position. To speed up the ANN learning process, the search of the weights and biases has been divided into five stages. The weights and biases computed in one stage have been used as the initial values for the following stage. In Stage 1, the initial ANN parameters have been randomly selected and the ANNtraining has been conducted only on the numerical data.Then, to obtain ANN insensitive to noise, the experimental data have been fed (Stage 2) and in Stage 3 all the input patterns have been used. The calculated weights and biases have been slightly corrected on the experimental data (Stage 4) and the final tuning has been done on the noisy numerical wavelet coefficients. The trainingwas stopped either when the error msereg achieved 10−3 or for a maximumof 700 epochs.Theproposed stages of theANNselection process are the result of the compromise between the search for theminimum calculation time and the best ANN performance (highest correlation coefficient R). The NW system performance for the best network, with the best recogni- tion ability, is presented in Fig.4. On the horizontal axis the actual values Neuro-wavelet damage detection technique in beam... 589 are introduced, whereas on the vertical axis they describe the neural network predictions. It can be seen that all points lie very near the line called the best linear fit. The correlation coefficient is 0.99989 and is very close to 1, which indicates very good compatibility between the outputs and targets. Fig. 4. Results of the trainingmode; (a) crack presence, (b) crack localization 4.3. Testing of NW system The wavelet transformmoduli, computed for the experimentally determi- ned static deflection profiles (cases 1, 2 and 3 from Table 1), were tested in the trainedNWsystem.Additionally, the numerical patternswere tested. The set of numerical data (noisy and noise-free) was prepared for different defect positions. The distance L1 from the clamped end to the notch was changed with sampling distance 4mm, from 82mm to 318mm giving 60 different de- fect locations. The set of numerical data consisted of 488 patterns. Defect localizations in the numerical data of the testingmodewere different from the defect localizations in the numerical data of the training mode. However, the predicted values are in good agreementwith the actual values. The correlation coefficient reaches 0.99986. The results of the testingmodeare illustrated inFig.5.Thedetailed values for the experimental data are contained in Table 2. It can be noted that the wavelet transform is not able to recognize whether the defect exists or not, in cases 1 and 2. The ANN prediction for the defect presence was −0.0018 for the beam without notch, what means no defect. For beam case 2, the ANN value for the defect existencewas 0.9925, what is in agreementwith the target value equal to 1. For beam case 3, where thewavelet transform recognized the notch presence, the ANN prediction also confirms this result with the value 590 M. Rucka, K. Wilde 0.9976. The maximum error between the actual and predicted values of the defect presence is 0.65%. Fig. 5. Results of the testing mode for the beam deflection lines; (a) crack presence, (b) crack localization Table 2.Actual, recognized bywavelets and neurally predicted defect identi- fication Input patterns Patterns descrip- tion Defect presence Defect centre position [mm] Recog- Predic- Recog- Predic- Actual nized by ted by Actual nized ny ted by CWT ANN CWT ANN case 1 gaus4 s=30 0 – −0.0018 0 – −0.1449 case 2 1 – 0.9935 101 – 124.6851 case 3 1 1 0.9976 101 101 109.7300 TheNWsystemenabled topredictdefect localization for thebeamwithout defect as well as for the beamswith defect. The differences are from 8.64% to 23.15%. It can be concluded that the neural network can predict the defect location even if the wavelet transformmoduli are not legible (case 1 and 2). 5. Validation of neuro-wavelet system on experimental examples 5.1. Experimental investigations Experimental tests have been performed on beam, plate and shell structu- res (Fig.6). The cantilever beam of length L, width B and height Hmm is Neuro-wavelet damage detection technique in beam... 591 madeof polymethylmethacrylate (PMMA).Thebeamcontains an opennotch of length Lrmm and depth amm at a distance L1 from the clamped end. Thedepth of the defect is 35%of the beamheight. The steel plate of length L, width B and height H has a fixed support on each side. The plate contains a rectangular defect of length Lr, width Br anddepth a. The distance from the defect left-down corner to the plate left-down corner in horizontal and vertical directions are L1 and B1, respectively. The area of the flaw amounts to 2.4% of the plate area and the depth of the flaw is 25% of the plate height. The cylindrical shell of diameters D1 = 293mm and D2 = 300mm is made of steel. The cylindrical shell is welded to a steel plate. The surface area of the cylinder is given in Fig.6c. The length of the surface is denoted by L, width by B and height by H. The shell contains a rectangular defect of length Lr, height Hr and depth a. The defect is situated from the inside of the cylinder. The distance from the defect left-down corner to the shell left-down corner in horizontal and vertical direction are L1 and H1, respectively. The area of the flaw amounts to about 0.2% of surface area and the depth of the flaw is about 30% of the shell thickness. Descriptions of structure geometry, defect sizes and locations as well as experimentally determined material properties are given in Table 3. Table 3.Beam, plate and shell mode shapes: experimental program Type Dimensions of Size of defect Material Location of of structure [mm] [mm] properties defect [mm] struc- L B H Lr Br a E ν ρ L1 B1 ture [GPa] [–] [kg/m3] beam 480 60 20 2 20 7 3.42 0.32 1187 120 – plate 560 480 2 80 80 0.5 192 0.25 7430 200 200 shell 930 180 3.4 5 60 1 190 0.25 7850 285 60 The experimental setup of the beamandplatewas presented byRucka and Wilde (2006a). In this paper, the experiments, including the shell structure, are conducedwithmore advanced signal processing techniques and a noise ro- bust method for the estimation of frequency response functions (FRFs). The structures were subjected to a dynamic pulse load applied by themodal ham- mer PCB086C03 at selected points. In the case of the beam, 48measurement points were distributed along the length of the beam axis whereas in the ca- se of the plate, 143 measurement points were distributed on its surface. The measurement points on the shell were spread out along the central ring and its number amounted to 62. To record the response of the structure one B&K 592 M. Rucka, K. Wilde Fig. 6. Experimental setup: (a) beam; (b) plate; (c) shell accelerometer was used. It was kept in one position throughout the measure- ments. The acceleration and force data were collected by the data acquisition system Pulse type 3650C. Each acceleration and force measurement was re- peated five times and the data were averaged in the frequency domain. The H2(ω) estimator was applied to minimize the noise problem in the input si- gnal from modal hammer. The imaginary part of the FRF estimator H2(ω) for the beam, plate and shell, is presented in Fig.7. The obtained FRFs al- lowed precise identification of the first, second and third mode shapes for all the considered structures. Themode shapes for the beam, plate and shell were also computed by the commercial FEMprogramSOFiSTiK.Thebeammode shapeswere computed using a solid six-sided element of length 2mm. A square plane element of the size 40×40mmwas used to calculate platemode shapes. Themode shapes of the shell were computed using a plane element of SOFiSTiK of size 5×5mm. Neuro-wavelet damage detection technique in beam... 593 Fig. 7. Imaginary part of FRF estimator H2(ω): (a) beam, 1© – f1exp =23.375Hz, f1num =23.01Hz; 2© – f2exp =157.5Hz, f2num =149.91Hz; 3© – f3exp =433.00Hz, 32num =410.89Hz; (b) plate, 1© – f1exp =64.875Hz, f1num =65.10Hz; 2© – f2exp =114.875Hz, f2num =120.00Hz; 3© – f3exp =195.25Hz, 32num =207.09Hz; (c) shell, 1© – f1exp =295.4Hz, f1num =337.3Hz; 2© – f2exp =575.8Hz, f2num =614.6Hz; 3© – f3exp =877.2Hz, 32num =928.9Hz Numerical frequencies are compared with the experimental ones in the plots of FRFs (Fig.7). The difference between themeasured frequencies relative to calculated frequencies ranges from 0.35% to 12.42%. The experimental and numerical mode shapes are compared in Figs. 8, 9 and 10 for the beam, pla- te and shell, respectively. Most visible differences between the experimental and numerical modes come form the modelling of the boundary condition. For example, the experimental modes of the beam show some rotation on the support while the numerical model assumes an ideal cantilever beam. Never- theless, the MAC values range from 0.7411 to 0.9998 indicating a very good agreement. 5.2. Wavelet analysis of structural mode shapes The continuous wavelet transforms are applied to structural mode shapes. For analysis of beammode shapes, one-dimensional CWTwas used. The wa- velet analysis of the firstmode shape is performed using gaus4 wavelet having 594 M. Rucka, K. Wilde Fig. 8. Experimental and numerical beammode shapes: (a) first, (b) second, (c) third Fig. 9. Experimental and numerical platemode shapes: (a) first, (b) second, (c) third Neuro-wavelet damage detection technique in beam... 595 Fig. 10. Experimental and numerical shell mode shapes: (a) first, (b) second, (c) third 4 vanishingmoments, whereas the second and thirdmode shapes are analyzed using gaus6 wavelet having 6 vanishingmoments. The wavelet transform modulus computed for the numerical and experi- mental data are shown in Fig.11. In the numerical simulations, themaximum value of the modulus grows with the increase of the scale and clearly points to the defect position at 121mm from the clamped end. However, the experi- mental results have additional maxima lines resulting from the measurement noise. Nevertheless, it is possible to locate the damage position, since the do- minant maximum line increases monotonically and for larger scales achieves the largest values. Theposition of the defect determined bywavelet analysis is 132mm in the case of the firstmode shape and 129mm in the case of the se- condmode. Thewavelet analysis of the thirdmode shapemakes it impossible to detect damage presence and its localization. The plate analysis was conducted using two-dimensional wavelet analysis expressed in terms of one-dimensional integrals. The presence of the defect is detected bya suddenchange in a spatial variation of the transformed response. The wavelet transform moduli for the numerical and experimental data are given in Fig.12. The results are computed for scale s = 40. Analysis of the first mode shape was conducted using gaus4 wavelet in both directions. Both peaks in the wavelet transform modulus for the numerical and experimental 596 M. Rucka, K. Wilde Fig. 11.Wavelet transformmodulus of the beam numerical and experimental mode shapes: (a) first, (b) second, (c) third Neuro-wavelet damage detection technique in beam... 597 data are capable to indicate location of the defect. The real defect position is x = 240mm and y = 240mm and it is in agreement with the recognition of the defect position equal to x = 240mm and y = 243mm. Analysis of the second and third mode shapes was performed using gaus4 wavelet along the plate width, and gaus6 wavelet along the plate length. The numerical results enable one to identify defect position, however in the case of the experimental data the noise masks the defect position. In the last example, the wavelet analysis is conducted on three mode sha- pes of the cylindrical shell. Gaussian wavelet gaus6 is considered. The one- dimensional CWTof the numerical and experimentalmode shapes (Fig.13) is performed for scales s =1-10. In the case of the numerical data, themaximum modulus growswith the increase of the scale and clearly points to the damage position at 288mm. TheCWT results based on the experimental thirdmode shape (Fig.13c) have additionalmaxima lines resulting from themeasurement noise. Nevertheless, the constant slope of the maximum line clearly indicates the damage location at 298mm. Wavelet transforms of the first and second experimental mode shapes do not allow for detection of the defect position (Figs. 13a and 13b). 5.3. Predictions of NW system TheNWdamage identification ability has been tested on the experimental beam, plate and shell mode shapes described in Section 5.1. In the case of the beam and shell mode shapes, selected lines of the wavelet transformmodulus were used as inputs. The set of data for the beam consisted of 3 patterns. For the beam first mode shape, the wavelet transformmodulus computed for scale s = 60 was used, whereas for the second and third modes, the wavelet transformmodulus computed for the scale s =40 was applied. For the shell, the set of input data consisted of 3 patterns, calculated for the scale s = 10. In the case of plate, the two-dimensional wavelet transform moduli were converted into a one-dimensional input signal since lines at different locations can be treated separately. In this simulation, only the lines passing through the centre of the defect have been considered. The set of the experimental input data consists of 6 patterns, i.e. 3 patterns along the length of the plate and 3 patterns along thewidth of the plate. The results are plotted in Fig.14. It is visible that all patterns are in agreement with the actual values with the correlation coefficient equal to 0.95247. The detailed values for all examples are contained in Table 4. The defect presence identification by the CWT was possible only for 5 of 12 input patterns. TheNW system prediction was completely successfully for 598 M. Rucka, K. Wilde Fig. 12.Wavelet transformmodulus of the plate numerical and experimental mode shapes: (a) first, (b) second, (c) third Neuro-wavelet damage detection technique in beam... 599 Fig. 13.Wavelet transformmodulus of the shell numerical and experimental mode shapes: (a) first, (b) second, (c) third 600 M. Rucka, K. Wilde Fig. 14. Results of the NW system for the experimental beam, plate and shell mode shapes; (a) crack presence, (b) crack localization Table 4.Actual, recognized bywavelets and neurally predicted defect identi- fication Struc- ture Input patterns Patterns description Defect Defect centre presence position [mm] A ct u a l Recog. Predic- A ct u a l Recog. Predic- by ted by by ted by CWT ANN CWT ANN mode 1 gaus4, s =60 1 1 0.9892 121 132 143.9611 beam mode 2 gaus6, s =40 1 1 0.9931 121 129 130.5971 mode 3 gaus6, s =40 1 – 0.7371 121 – 118.4275 plate (along x) mode 1 gaus4, s =40 1 1 0.9983 240 240 243.0938 mode 2 gaus6, s =40 1 – 1.0043 240 – 311.3410 mode 3 gaus6, s =40 1 – 0.9994 240 – 246.5563 plate (along y) mode 1 gaus4, s =40 1 1 0.9998 240 243 244.9450 mode 2 gaus4, s =40 1 – 0.9993 240 – 243.3374 mode 3 gaus4, s =40 1 – 0.9127 240 – 247.6269 mode 1 gaus6, s =10 1 – 0.9978 287.5 – 325.7603 shell mode 2 gaus6, s =10 1 – 0.9987 287.5 – 303.4104 mode 3 gaus6, s =10 1 1 0.9981 287.5 298 280.7378 10 patterns (with difference ranging from 0.02% to 1.08%) and quite success- fully for 2 patterns (with difference 8.73% and 26.27%). TheNWsystem could identify the damage presence in all the considered cases with the average error of 3.17%. Neuro-wavelet damage detection technique in beam... 601 The identification of defect position by the CWT was also possible only for 5 of 12 input patterns. The errors of theCWTbased peak-pickingmethod reach from 0% to 9.09%. The NW was able to predict defect localization for all cases, including cases, in which it was impossible to locate the defect by the CWT. The NW prediction was successful for 9 patterns (with difference ranging from 1.29% to 7.93%) and satisfactory for 3 patterns (with differences 13.31%, 18.98% and 29.73%). TheNWsystem could indicate damage location in all the considered cases with the average error of 7.55%. 6. Conclusions In this paper, the neuro-wavelet damage detection method for defect identifi- cation is presented. The technique is studied on experimental mode shapes of beam, plate and shell structures.The search for the defect location, conducted by picking the largest wavelet transform modulus, is enhanced by the artifi- cial neural network. A simple, backpropagation neural network was trained on static deflection lines of the cantilever beam. Although, the network was taught on the static data, it could effectively localize the damage of the same type in more complex structures, like plate or shell. The results of the wavelet damage detection and analysis of the neural network simulations lead to the following conclusions: • The wavelet damage detection method might be suitable for detecting relatively small damage (with damage area being 0.2% of the total shell area). In more complex structures, the sensitivity of higher modes to small defectsmight be high (the shell highest consideredmode appeared the best for damage detection). • The NW system can predict damage location with relatively good pre- cision even for the cases with no visible peaks in the wavelet transforms. • TheNWsystem for a complex engineering object can be trained ondata obtained on a simple structure. In such a case, it is easy to conduct an experiment and numerical simulations to collect theANN training data. • The reliability of theNWsystemprediction can be improved by applica- tion of amore advancedANNarchitecture and the use of a large number of training patterns corresponding to the cases encountered in the real structures. • Theartificial neural network solutions belong to the class of the so-called soft computing and theANNcan predict defects onlywith a certain pro- bability. Hence, the application of the ANN provides sometimes worse 602 M. Rucka, K. Wilde identification of defect positions than the identification of defect locali- zations byCWT.Nevertheless, the ANNpredictions are relatively good in the cases when it is impossible to locate the defect by analysis of the wavelet transforms of mode shapes. • The system supporting the decisionmaking process on damage location should not rely solely on the neural network. 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ZubaydiA.,HaddaraM.R., SwamidasA.S.J., 2002,Damage identification in a ship’s structure using neural networks,Ocean Engineering, 29, 1187-1200 Wykrywanie uszkodzeń w konstrukcjach belkowych, płytowych i powłokowych przy użyciu systemu neuro-wavelet Streszczenie Niniejsza praca poświęcona jest technice diagnostyki konstrukcji bazującej na transformacie falkowej oraz sztucznych sieciach neuronowych (tzw. system neuro- wavelet). Zastosowanie analizy falkowej pozwala na lokalizację uszkodzeń wymaga- jącą minimalnej ilości danych wejściowych. W tym celu niezbędna jest tylko odpo- wiedź konstrukcji pomierzona w wielu punktach. Poprawę efektywności lokalizacji zniszczeń uzyskano poprzez użycie sztucznej sieci neuronowej. Nauczona sieć neuro- nowa poprawnie rozpoznajemiejsce położenia uszkodzeń, nawet w przypadkach, gdy określenie położenia uszkodzenia nie byłomożliwe bezpośrednio z obliczonychwspół- czynników falkowych. Zaproponowanametoda została sprawdzona eksperymentalnie na przykładach konstrukcji belkowych, płytowych i powłokowych. Manuscript received October 4, 2009; accepted for print December 2, 2009