The Use of ANN for Cracks Predictions in Curvilinear Beams Based on their Natural Frequencies and...

8/3/2019 The Use of ANN for Cracks Predictions in Curvilinear Beams Based on their Natural Frequencies and Frequency Res

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The Use of ANN for Cracks Predictions inCurvilinear Beams Based on their Natural

Frequencies and Frequency Response

FunctionsR. A. Saeed, L. E. George

Abstract This paper presents the results of using different arrangements of feed forward neural network (ANN) for crack

prediction in curvilinear beams. The prediction process is based on the feeding the vibration characteristic as input data. The

adopted vibration characteristics include: Natural Frequencies (n) and Frequency Response Functions (FRFs). The introduced

ANNs were used to estimate the size of the crack and its location. The finite element method (FEM) has been applied to

compute the vibration characteristics for intact and damaged beams. To avoid the establishment of largeANNarrangements the

data reduction techniques were utilized to reduce the input set of computed natural frequencies (n) and Frequency Response

functions (FRFs). The analysis results revealed that the reduced arrangements ofANNcan give a satisfactory prediction results;

this reflects encouraging sign about the use ofANNfor damage identification.

Index Terms Crack identification, Artificial Neural Network, Data reduction techniques, Vibration analysis.

1 INTRODUCTION

The existence of damage introduces a local flexibility in astructure [1] which leads to the reduction of the stiffnessand stability, and could also affect its damping properties[2]. Consequently, the vibration characteristics of thestructure would also change. These changes can be meas-ured accurately and then used in a non-destructive me-thod to evaluate the damage state [3]. Significant efforts

have been spend for developing different fault detectionand diagnosis approaches for damage identification instructures based on vibration characteristics [4, 5] includ-ing natural frequencies (n) and Frequency ResponseFunctions (FRFs). Measured amplitudes of FRFs are oftenused for damage identification with satisfactory accuracy[6]. For instance, Huynh et al. [7] used measured FRFsdata for damage detection in truss structure and a platestructure. Apart from amplitudes of FRFs, some authorsuse the changes in natural frequencies to estimate the sizeand locations of the cracks in different structures [1, 5, 8,9, 10]. Analytical and experimental investigations for thecrack identification in a cantilever beam based on the

change in natural frequencies have been conducted by [4].Several authors presented analytical and numerical mod-els to investigate the influence of cracks on vibration cha-racteristics of the structure [9, 10]. A detailed literaturereview of the various methods for damage detection indifferent structures by using natural frequencies as a di-agnostic parameter has been reported by [11]. Several

authors employed both natural frequencies and FRFs si-multaneously to identify faults in structures [2, 5, 12, 13].All these studies indicate that there is a link between thechange in the natural frequencies and amplitude of FRFswith the size and location of the crack. Therefore, vibra-tion-based methods have been intended for the identifica-tion of damage in structures.

Different Artificial Intelligent (AI) techniques are ap-plied for fault detection, such as fuzzy logic, artificialneural network (ANN) and genetic algorithms (GA) [14].It is reported thatANNapproach can be applied for dam-age detection [14] by training the neural network to rec-ognise given patterns and to classify other untrained pat-terns [15]. TheANNtechnique based on vibration charac-teristics have been used for damage identification in dif-ferent structures, such as: beam structure [12], truss andframe structures [16], rectangular plate [14], truss struc-ture [15] and composite frames [17]. They showed that thevibration characteristics change when structure compo-nents deteriorate or defects start to develop.

The large number of input nodes is the main obstacleof training of ANN technique with input data sets. Selec-tion of all spectral lines of the FRFs and natural frequen-cies is excessive for training ANN models directly. Toavoid large models, a subset of the FRFs data sets can beused by considering fewer measurement points [18]. An-gelakis et al. [19] reduced the neural network input to 27selected frequencies from the whole frequency range.

Jnior and Turra [20] subdivided the range of frequency,which contained 512 points, into five intervals and se-lected the maximum amplitude from each interval. Zanget al. [18] proposed the PCA approach for reduction of thecomputed FRFs data sets.

R. A. Saeed, University of Sulaimani, Sulaimani, Iraq.

L. E. George, University of Baghdad, Baghdad, Iraq

JOURNAL OF COMPUTING, VOLUME 3, ISSUE 12, DECEMBER 2011, ISSN 2151-9617

https://sites.google.com/site/journalofcomputing

WW.JOURNALOFCOMPUTING.ORG 113


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High numbers of input data may make the ANNper-formance unstable. Therefore, in order to reduce dimen-sionality in the input layer, different techniques havebeen proposed. The calculated amplitudes of FRFs andnatural frequencies, as shown in figure 4 and 6, are ar-ranged into a matrix with 541 rows representing naturalfrequencies and amplitudes of FRFs for the damaged (540cases) and intact (1 case) beams. The number of spectrallines and natural frequencies are taken 500 and 20, respec-tively, which represent the number of columns. The re-duced FRFs and natural frequencies data sets are ob-tained by gradually reduction the number of natural fre-quencies and spectral lines.

These techniques allowed us to significantly reduce theamount of required FRFs and natural frequencies datasets. These approaches are applied to FRFs matrix(541x500) and n matrix (541x20) to achieve an acceptabledata reduction level. The new data sets of natural fre-quencies and amplitudes of FRFs are used as input vec-tors to the FFANN classifier models as an alternative tousing the original data sets. Such truncation of data sets

has very little effect on the accuracy of the classifier re-sults but considerably reduces the number of inputs intothe models.

In this study, in order to investigate the required num-ber of neurons in the input layer of the NN, two differenttechniques have been proposed, and we have used MAT-LAB to implement and simulate each technique.

2.2.1 The First Proposed Technique

In the first proposed technique, to determine the requirednumber of frequencies/neurons in the input layer thetraining, testing and validation data sets are loaded. Inthe first step, for training NN the number of frequen-

cies/neurons in the input layer is reduced sequentially, 1input frequency/neuron at each attempt. For the first at-tempt, 1st frequency/neuron is eliminated from the inputdata sets then ANN is trained and tested with the newdata sets. In the second attempt, 1st frequency/neuron isreturned back to the input data sets and the 2nd frequen-cy/neuron is eliminated from the input data sets then

ANN is trained and tested. In the third attempt, 2nd fre-quency/neuron is retained back to input data sets and the3rd frequency/neuron is eliminated from the input datasets and then ANN is trained and tested. This process iscontinuing until the last frequency/neuron is reduced. Inthe next step, the summation of correlation coefficientsbetween predicted and existence data for each attempthave been determined. In the final step, the maximumcorrelation coefficients are identified which are corres-ponding to one of the attempts, and then the frequen-cy/neuron that is eliminated in this attempt is removedfrom the input data sets permanently. The explained pro-cedure is continuing until reaching the satisfactory reduc-tion number of frequencies/neurons in the input layer oftheANN.

2.2.2 Second Proposed Technique

In the second proposed technique, to determine the re-quired number of frequencies/neurons in the input layer

the training, testing and validation data sets are loaded.The applied method for selecting the nodes to be deletedis based on the values of best weights matrix betweeninput and first hidden.

In this technique and after trainingANN, the varianceof each column of weight in the best weight matrix is de-termined which is representing the variance of weightsbetween each input frequency/neuron with all neuron inthe first hidden layer. The input frequency/neuron withthe low variance weights in the best weight matrix hasless effect on the model; therefore this frequency/neuronis removed from the training, testing, and validation datasets permanently. In the next attempt, the ANN will betraining again with the new input data sets after remov-ing one of the input frequency/neuron from the inputdata sets. Then the variance of each column of weight inthe best weight matrix is determined. Next, the input fre-quency/neuron with the minimum variance in the weightmatrix is eliminated from the training, testing, and valida-tion data sets permanently. The explained procedurescontinue until the satisfactory reduction number of fre-

quencies/neurons in the input layer of the NN.

2.3 Crack Identification

Based on the fact that the measured response containsinformation on the crack parameters (length and posi-tion), the adopted FFANN classifiers have been imple-mented using MATLAB Code. There is no feasible algo-rithmic solution for the crack identification in the com-plex structure [23]. In this case, anANNcan be consideredas the crack detector/classifier tool.Each data set of natu-ral frequencies and FRFs was used separately for trainingof an appropriate single ANNto identify the crack para-meters, and then the two data sets were aggregated to-

gether and used for training a multiple ANN to improvethe crack identification.

2.3.1 Implementation of ANN Using Natural Frequencyand FRFs

The implementation of a neural network model includesthe training and testing phases. In the training phase, twomodels (Model 1 and 2) of the feed-forward neural net-works were trained, separately, using BP algorithm.These two ANNs are trained by feeding predeterminednatural frequencies and FRFs data sets. These data setsreflect the response of all possible cracks sizes and loca-tions, as well as the response of the beam without cracks.A sufficient number of combinations of input and output

pairs were used for training purpose to make the networkwell trained network and capable do make good pre-dictions. Afterwards, the trainedANNmodel is tested ondata sets that have not been used in the training process.

For crack identification, changes in natural frequenciesor FRFs alone may not be sufficient, because the change ina particular frequency could be attributed to differentcrack lengths and locations. Also, cracks of the samelengths but at different locations may cause similarchanges in frequencies, which are too close to differen-tiate. It is expected that the overall accuracy can be im-proved by combining the two FFANN networks. The





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combination of multiple neural networks by using datafusion techniques can improve neural network generali-sation capability, as reported by [24]. The architecture ofthe proposed combined multipleANNmodel (Model-3) ispresented in figure 3.

2.2.2 Multiple Artificial Neural Networks

The combination of several neural networks has beentermed stacked neural network (SNN) or multiple neuralnetworks (MNN) [25, 26]. All individual neural networksare combined instead of training several networks andchoosing the best network, which improves the overallprediction error that could be made smaller than the in-dividual errors for each individual NN. As shown in fig-ure 3, the model consists of two individual neural net-

works where natural frequencies are used as input datafor the first neural network and the amplitudes of FRFsare used as input data for the second neural network.

3 RESULT AND DISCUSSION

3.1 Beam Natural Frequencies

The natural frequencies of the intact curvilinear beamcorresponding to the first twenty modes are listed in table1. The natural frequencies were computed for 30 differentcrack lengths, from 2mm to 60mm with a step of 2mm, at18 different locations, from (90 to 124) with a step of 2.

The variations of the first twenty natural frequenciesfor 541 simulations of the left half of the curvilinear beamare shown in figure 4.

For all considered simulations, the natural frequenciesundergo a noticeable change at each crack location, espe-cially for long cracks. However, the frequencies dependon the crack length change in different patterns for differ-ent crack locations depending upon the mode shapes ofthe beam.

3.2 Forced Vibration Analyses of the Beam

Forced vibration analyses of a curvilinear beam with var-ious cracks have been performed by using FEMto obtainamplitudes of FRFs. Figure 5 shows the amplitudes ofFRFs versus the excitation frequencies depending oncrack locations and sizes. The first 500 frequencies werecalculated and 500 spectral lines were selected for thefrequency range 1-500 Hz. These procedures have beenrepeated for the prescribed 541 different cases.

As evident from figure 5 both the crack location andthe crack length affect the amplitudes of FRFs, but thechange in amplitudes is different for every crack location

for the same crack size. Furthermore, the amplitude ofFRFs increases as the crack grows in size. Figure 6 showsa three-dimensional plot of the changes, in decibels, forthe curvilinear beam with a crack depending on its loca-tion and lengths. As seen, the changes of the measuredamplitude of FRFs of the beams are very sensitive to thecrack location and size.

Changes in amplitudes of the FRFs are obtained fromthe forced vibration analysis providing necessary infor-mation for the crack identification in the beams by corre-

Fig. 5. FRFs amplitudes of a beam for 20 cracks at 14 locations

Fig. 4. The changes in natural frequencies (for twenty modes) of thebeam depending on crack length and crack location

TABLE1

NATURAL FREQUENCIES OF THE BEAM IN HZ

Fig. 3. Architecture of multipleANN

Fig. 6. FRFs amplitudes in dB for the beam and for 20 cracks at 14locations





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lating dynamic response data of a structure with the crackcharacteristics.

3.3 Classifier Results

Natural frequencies and FRFs data sets have been used totrain three different models of ANN in order to identifythe crack parameters (length and location). First, the datasets of natural frequencies and FRFs have been used sepa-

rately for training two different ANN models (Model 1and 2). Then, both the natural frequency and FRFs datasets have been used together to train multiple ANN(Model-3).

3.3.1 Learning Algorithm

Natural frequencies and FRFs data sets are used as inputfor training the adopted FFANNclassifier; the crack pa-rameters (crack length and location) represent the output.The complete data sets (541 patterns) are divided intothree subsets. The first subset, 80% (433 patterns) of thecomplete data sets was selected, randomly, to be used fortraining, while the remaining 10% (54 patterns) have been

used for validation and the remaining 10% for testing.The effectiveness and the accuracy of the trained modelare examined using the testing data sets, while the valida-tion data set is used to optimise the performance of theclassifier model by stopping the training process whenthe validation error starts to increase as the classifier be-comes over fitted [27, 28].

During the learning process, a number of differentmulti-layer FFANNarrangements have been investigatedto determine the best possible number of hidden layersand the neurons number in each hidden layer. The ANNhas been tested for many combinations of training rates:(0.005, 0.01, 0.05, 0.1, and 0.3) and momentum: (0.6, 0.7,

0.8 and 0.9). For the neural network model using naturalfrequencies (Model-1), it was found that the lowest valueofMSE could be further reduced by decreasing the learn-ing rate value to 0.005, the momentum rate was set 0.6.For the neural network model using FRFs (Model-2), themomentum and the learning rate were fixed at 0.6 and0.001, respectively. Hyperbolic tangent function is used asthe activation function for hidden layer neurons. Thesigmoid function is used as the activation function foroutput layer neurons. After that, instead of training twonetworks and choosing the best network, Model-1 andModel-2 are combined to produce Model-3. Data sets ofFRFs and natural frequencies are assembled together fortraining Model-3 in order to improve the overall estima-

tion error. The neural networks were able to learn therelationship between natural frequencies and amplitudesof FRFs with the corresponding damage state. The train-ing process is terminated when there is no error im-provement for a limited number of cycles, or when themean square error (MSE) for training samples falls belowa specified limit, typically below 0.001 [29, 30] , or whenit reaches the maximum number of iterations. In thisstudy the maximum number of iterations is set 10000epochs.

3.3.3 Implementation of ANN Using Natural Frequencyand FRFs

It was revealed from experiments that the neural networkmay not be able to represent the system adequately if thenumber of hidden layers and hidden neurons in the neur-al network is too small. On the other hand, it becomesover-trained if the network is too big [15]. In the follow-

ing subsections the number of the hidden layer and num-ber of neuron in the hidden layer(s) and also the requirednumber of neurons in the input layer for training Model1&2 are investigated.

The performance of Model-1 with one hidden layer hasbeen examined with a different number of neurons. Thenumber of hidden neurons varied starting from 1 neuronto twice number of input frequencies (i.e., 20). The naturalfrequencies data sets (each consist of 20 natural frequen-cies) have been used as input for training ANNmodels;the crack parameters (crack length and location) representthe output. After training FFANN, the correlation coeffi-cient and MSE values for the prediction results of thecrack length and location have been determined, and the

results are shown in the figure 7 and 8.

Fig. 7. The correlation coefficient for the prediction results of thecrack length and location using natural frequencies data sets (20natural frequencies)

Fig. 8. The MSEfor the prediction results of the crack length andlocation using natural frequencies data sets (20 natural frequen-cies)





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As evident from the obtained results, the correlationcoefficients for the estimation of crack length and locationof all NNs are around 0.97 and 0.78, respectively. The in-vestigations showed that one hidden layer NN with 9hidden neurons at the hidden layer had produced thebest predictions of the crack parameters, where the corre-lation coefficients are 0.99 and 0.88 for the prediction ofthe crack length and location, respectively. Additionally,the corresponding MSE to estimate the crack parametersare 8.01 and 23.25, respectively.

In this section, the performance of Model-1 with twohidden layers has been examined for different number ofneurons (from 1 neuron to 40) in each hidden layer, toselect the required number of neurons in each of the hid-

den layers. Natural frequencies data sets (20 natural fre-quencies) have been used as input for training FFANNmodels; the crack parameters (crack length and location)represent the output. After training FFANN, the correla-tion coefficient andMSE values for prediction of the cracklength and location for different numbers of the neuronsin each hidden layer have been computed and the resultsare presented in the figure 9 and 10.

As demonstrated in the above figures, the correlationcoefficients obtained from testing all trained NNs aremore than 0.95 and 0.75 for the prediction of crack lengthand location, respectively. The achieved results showedthat the use of two hidden layers NN(with 7 neurons inthe first hidden layer and 9 neurons in the second hiddenlayer) are suitable to get best predictions of the crack pa-rameters; where the correlation coefficients are 0.99 and0.90 for the prediction of the crack length and location,respectively. Moreover, the corresponding MSE to esti-mate the crack length and location are 6.67 and 18.83, re-spectively.

Based on the investigation on the number of hidden

layers and number of neurons in each hidden layer, it canbe concluded that one hidden layer NNwith 9 neurons issufficient to estimate the crack length and predict the lo-cation of the crack. Therefore, one hidden layer NNwith 9hidden neurons has been used in the next section. Theperformance behaviour of one hidden layer NN with 9neurons in the hidden layer for the prediction of the cracklength and location is illustrated in figure 11.

3.3.4 Testing Number of Natural Frequencies forTraining ANN

In this section, the required number of frequen-cies/neurons in the input layer of Model-1 is investigated.In order to reduce the number of frequencies/neurons inthe input layer of the NNand also to obtain a satisfactory

number of frequencies/neurons, data reduction tech-niques have been applied. The details of these two tech-niques are explained in the previous sections. The imple-mentation of these techniques have been utilized forModel-1, the correlation coefficients and MSE values ob-tained from testing a trained NNcorresponding to elimi-nate natural frequency number are presented in table 2(for first technique) and table 3 (for second technique).

From table 2 it can be seen that all NNs demonstratedrelatively good accuracy for estimation of the crack lengthand location by implementation first data reduction tech-nique. As illustrated in the table, the model gives the

Fig. 9. The correlation coefficient to estimate the: (A) crack lengthand (B) crack location using natural frequencies data sets (20natural frequencies)

Fig. 11. Prediction of the: (A) crack length and (B) crack locationusing natural frequencies data sets (20 natural frequencies)

Fig. 10. The MSEof the estimation results for: (A) the crack lengthand (B) crack location using natural frequencies data sets (20natural frequencies)





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highest correlation coefficients and lowest value of MSEwhen the frequency/neuron number (12) is eliminatedfrom the training, testing and validating input data sets.Next, by removing frequency/neuron number (15) fromthe input data sets, the trained NNgives the best results.This procedure is continuing until the number of inputs isreduced to 3 inputs frequencies/neurons. The investiga-tion shows that reducing the number of frequen-cies/neurons in the input layer to 7 neurons (2, 3, 4, 5, 8,10, and 16) makes the NNto predict the crack parametersmore efficiently. However, it is evident from table 2 thatthe last 3 natural frequencies remaining from the inputdata sets (4, 8, and 16) are sufficient to identify the crackin the beam. Furthermore, reducing the number of inputsto three frequencies/neurons in the input layer of the

ANNgives satisfactory resultss.

While for the second dimensional reduction mechan-

ism, after training the NNand computing the variance ofeach column of weights in the best weight matrix, theresults show that the frequency/neuron number (11) hasa minimum variance, as shown in table 3, this indicatesthat this frequency/neuron has less effect than other neu-rons. Therefore, eliminating this frequency/neuron in thetraining, testing and validation input data has little effectson the results. In the next step, the NNis retrained withnew training, testing, and validation input data sets (with19 neurons). After training the NNwith new data sets, thevariance of each column of weight in the best weight ma-trix were calculated, the results show that the neu-ron/frequency number (14) has a minimum variance.Therefore this frequency/neuron is eliminated from thetraining, testing, and validating input data sets. This pro-cedure continued till the number of input is reduced to 3input frequencies/neurons. The investigations showedthat reducing the number of inputs to 3 frequen-cies/neurons (3, 8, and 16) in the input layer of the NNgives satisfactory results, and the NN can predict thecrack parameters efficiently. These three frequen-cies/neurons (3, 8, and 16) are sufficient to reflect the in-formatics part of the natural frequency data set that rele-vant to crack parameters. Thus these 3 natural frequencieshave been used further on. Consequently, the originalnatural frequencies array of 541x20 is reduced to 541x3.

By comparing the obtained results in table 2 and 3, onecan conclude that the reducing number of input frequen-cies/neurons based on the variance of each column ofweights in the best weight matrix give satisfactory resultscompared with the reducing number of input frequen-cies/neurons based on the best correlation coefficients. Itis evident from table 2 and 3 that computation of the 3natural frequencies is enough to identify the crack in thebeam, which is in agreement with Owolabi et al. (2003)and Yang at el. (2001). The determined correlation coeffi-cients and MSE of the prediction results of the crack pa-rameters are listed in table 2 and 3; which indicate thereliability of the results obtained by these techniques.

As, a next test step, the obtained 3 natural frequenciesdata sets are used as input data sets for examining theperformance of the FFANNwith one hidden layer (Mod-el-1) the number of neurons in the hidden layer was va-

ried (from 1 to 40 neurons), to determine the best numberof neurons in the hidden layer. The learning rate andmomentum are fixed at 0.005 and 0.9, respectively. Thecorrelation coefficient and MSE values for prediction re-sults of the crack length and location for testing NNusingreduced natural frequencies data sets with different num-bers of the neurons in the hidden layer have been deter-mined and the results are presented in figure 12 and 13.

Fig. 12. The correlation coefficient for the prediction results of thecrack length and location using natural frequencies data sets (3 natu-ral frequencies)

TABLE2THE CORRELATION COEFFICIENTS AND MSEVALUES OBTAINEDFROM TESTING A TRAINED NNCORRESPONDING TO ELIMINAT-

ED THE NATURAL FREQUENCYS NUMBER

TABLE3THE CORRELATION COEFFICIENTS AND MSEVALUES OBTAINED

FROM TESTING TRAINED NNCORRESPONDING TO THE SUB-SEQUENTIAL ELIMINATION OF NATURAL FREQUENCIES





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As evident from the obtained results, the correlationcoefficients for all cases are more than 0.96 and 0.87 forthe estimation of the crack length and location, respec-tively. The investigations revealed that one hidden layer

NN with 6 hidden neurons in the hidden layer give thebest results, where the correlation coefficients are 0.98 and0.94 for the prediction of the crack length and location,respectively. Furthermore, the corresponding MSE to es-timate the crack length and location are 10.59 and 12.96,respectively.

The predicted results of the crack length and locationby using one hidden layer NNwith 6 neurons in the hid-den layer are shown in figure 14 when the number of theneurons in the input layer is fixed to 3 frequen-cies/neurons. The tests showed that the lowest value of

MSE could be further reduced by changing the learningrate value to 0.005. The momentum rate was set 0.9. Fig-

ure 14 shows the accuracy of the crack length and locationprediction during the testing process. The testing resultsreveal that Model-1 can predict the existence of a crack inthe curvilinear beam effectively.

The test results show that NN can reliably recognizethe extent of the damage (see figure 14), it shows a betterclassification performance compared to the NN(see figure11). Furthermore, the comparison between correlationcoefficients (0.99 and 0.88, 0.98 and 0.94) andMSE values(8.01 and 23.25, 10.59 and 12.96) of the prediction resultsrelated to of the crack length and location before and afterreducing input datarespectively. This indicates the relia-bility of data reduction technique.

3.3.5 Testing of ANN Using Amplitudes of FRFs

In this section, in order to select the required number ofneurons in the hidden layer for the ANN using ampli-tudes of FRFs (Model-2), one hidden layer NNwith dif-ferent number of neurons in its hidden layer (from 1 neu-ron to 40) has been examined. FRFs data sets (500 spectrallines) are used as input for training ANN models; the

crack parameters (crack length and location) represent theoutput. The correlation coefficient and MSE values forpredicting the crack length and location for differentnumbers of the neurons in the hidden layer have beendetermined and the results are shown in figure 15 and 16.

As illustrated in the above figures, the calculated corre-lation coefficients are around 0.94 and 0.84 for the predic-tion of crack length and location, respectively. The inves-tigations showed that the use of one hidden layer NNconsist of 8 neurons in the hidden layers give the bestresults, where the correlation coefficients are 0.97 and 0.92for the prediction of the crack length and location, respec-tively, and the corresponding MSE to estimate the cracklength and location are 20.50 and 15.67, respectively.

Fig. 14. Scatter plot of prediction results of the: (A) crack length and(B) crack location, using natural frequencies data sets (3 naturalfrequencies)

Fig. 16. The MSEfor the prediction results of the crack length and

location using FRFs data sets (500 spectral lines)

Fig. 13. MSE for the prediction of the crack length and locationusing natural frequencies data sets (3 natural frequencies)

Fig. 15. The correlation coefficient for the prediction results of thecrack length and location using FRFs data sets (500 spectral lines)





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In this section, the performance of FFANN Model-2with two hidden layers has been examined with differentnumber of neurons (from 1 neuron to 40) in each hiddenlayer. After testing all trained NNs, the correlation coeffi-cient andMSE values for predicting the crack length andlocation for different numbers of the neurons in each hid-den layer have been computed and the results are pre-sented in figure 17 and 18.

As demonstrated in above figures, the correlation coef-ficients obtained from testing all trained NNs are morethan 0.92 and 0.79 for the prediction of crack length andlocation, respectively. However, the achieved resultsshowed that two hidden layers NNwith 8 neurons in thefirst hidden layer and 10 neurons in the second hiddenlayer are sufficient for learning, and they give the bestpredictions for the crack parameters, where the correla-tion coefficients are 0.97 and 0.89 for the prediction of thecrack length and location, respectively. The correspond-ing MSE values for estimation of the crack length andlocation are 19.22 and 21.28, respectively.

Based on the conducted tests results to estimate theeffect of number of neurons in each hidden layer, it can beconcluded that one hidden layer ANNwith 8 neurons issufficient to estimate the crack length and predict the lo-cation of the crack. Therefore, one hidden layer NNwith 8hidden neurons has been used in the next section. Theperformance of one hidden layer NN with 8 neurons inthe hidden layer for the prediction of the crack length andlocation is illustrated in figure 19.

The size of the FRFs data sets obtained from numericalanalysis is represented by 500 spectral lines in each mea-surement. The use of all spectral lines of the FRFs datasets causes long training time (i.e., high computationcomplexity). In order to avoid large complexity and over-fitting, the data reduction technique was used to reducethe size of input data (i.e., FRF spectra). This techniqueallows to significantly reducing the amount of FRFdata.The data reduction technique is applied on the FRFma-trix (541x500) to achieve an acceptable data reductionlevel. The new data sets of FRFs amplitudes are used asinput vectors to the FFANN classifiers as an alternativefor using the original FRFs data sets.

3.3.6 Testing Number of Spectral Lines of the FRFsData for Training ANN

In this section, the required number of spectral lines ofthe FRFs/neurons in the input layer of Model-2 has beeninvestigated. In order to reduce the number of spectrallines of the FRFs/neurons in the input layer to a satisfac-tory number of neurons, the data reduction mechanismhas been implemented. The first data reduction techniqueis a time consuming task, which attempts to improve theperformance by speeding up the training stage, thesecond data reduction technique have been used, and theresults are shown in figure 20.

Fig. 17. The calculated correlation coefficient values to estimatethe: (A) crack length and (B) crack location for different ANNar-rangements using FRFs data sets (500 spectral lines)

Fig. 20. The results due to reducing the spectral lines of the FRFsusing the second data reduction technique

Fig. 19. The prediction results of the crack length (A) and crack loca-tion (B) using FRFs data sets (500 spectral lines)

Fig. 18. The MSE for the estimation results of: (A) crack lengthand (B) crack location for different ANNarrangements using FRFsdata sets (500 spectral lines)





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The figure at the right side presents a close view forthe last 10 attempts (from 471 to 480) of reducing data ofthe FRFs data sets. Based on the data reduction technique,different FFANN arrangements have been trained withdifferent input data sets, where the number of input datasets gradually decreases 1 input spectral lines/neuron ateach attempt. The calculated correlation coefficients and

MSE for the prediction results of FFANNcorrespondingto each attempt of spectral line reduction are presented infigure 21 and 22 respectively.

The above figures demonstrate that most of the NNsshow, relatively, good accuracy for estimation of thecrack length and location. However, the accuracy of thecrack identification decreased with increase number ofreduced spectral lines, mainly after 300 spectral lines, asillustrated in figure 21 and 22. The 10 attempts (from 471to 480) for reducing input data size is shown in figure 23.The investigations show that the last remaining 20 spec-tral lines of the FRFs data sets are sufficient to identify thecrack in the curvilinear beam. These remaining spectrallines of the FRFs data, as shown in figure 23 are presentedin table 4. These 20 spectral lines capture the largest

amount of variations in the original data and they havebeen used further on. Therefore, the original FRFs dataarray of 541x500 is reduced to 541x20.

The new data sets of FRFs are used as input vectors tothe classifier models as an alternative to using the originalFRFs data sets. Such truncation of data sets has very littleeffect on the accuracy of the classifier results but consi-derably reduces the input data.

Figure 24 shows the locations of the remaining 20 spec-tral lines of the FRFs data sets with amplitude of FRFs

that have been obtained from the vibration analyses (541simulations) of a curvilinear beam with various crackslength and location. The figure shows the amplitudes ofFRFs versus the excitation frequencies depending oncrack locations and sizes. As indicated in the figure, theselocations are in the regions where the graph is highly un-iformly changed with change of the crack length and loca-tion. Therefore, these spectral lines of the FRFs data arethe most important features to identify the cracks.

In order to determine the best number of neurons inthe hidden layer of Model-2, one hidden layer NNwithdifferent number of neurons in the hidden layer (from 1neuron to 40) has been examined. The number of neurons

TABLE4THE LAST 20 REMAINING SPECTRAL LINES OF THE FRFSDATA

Fig. 23. The results of 10 attempts (from 471 to 480) for reducingsize of FRFs data setsFig. 21. The correlation coefficients for the estimation results of the

crack length and location corresponding to each reduced spectralline of the FRFs data set

Fig. 22. The MSEfor the estimation results of the crack length andlocation corresponding to each reduced spectral line of the FRFsdata set

Fig. 24. The locations of the remaining 20 spectral lines together withall FRFs data sets





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in the input layer was fixed to 20 spectral lines of theFRFs/neurons, and also the learning rate and momentumvalues are set 0.005 and 0.9 respectively. The correlationcoefficient andMSE values for the prediction of the cracklength and location for testing a trained NNwith differentnumbers of the neurons in the hidden layer have beendetermined and the results are presented in figure 25 and26.

As evident from the obtained results, the correlationcoefficient values for all cases are more than 0.95 and 0.83for the estimation of the crack length and location respec-tively. The results show that one hidden layer with 5 neu-rons is sufficient to estimate the crack length and predict

the location of the crack, where the correlation coefficientsandMSE are (0.98 & 0.90) and (11.27 & 17.52) for the iden-tification of crack length and location respectively.

Figure 27 shows the prediction of the crack length andlocation results by using one hidden layer NNwith 5 neu-rons in the hidden layer. The number of neurons in theinput layer is fixed to 20 spectral lines of theFRFs/neurons in the input layer. The test showed that theminimum of MSE could be further reduced by changingthe learning rate value to 0.01. The momentum was set0.9. The testing results reveal that Model-2 can predict theexistence of a crack in the curvilinear beam.

After successful training of the classifier models (i.e.,Model 1 & 2), a new data sets (54 data sets) of reducednatural frequencies (consist of 3 natural frequencies + 20amplitudes of FRFs data sets) corresponding to bothhealthy and damaged beam conditions were selected andfed simultaneously into the classifier models. The resultsobtained from the tested classifier models (Model 1 & 2)for the prediction of the crack length and location are pre-sented in figure 14 and 27. These figures demonstrate thecomparison between the predicted crack length and loca-tion using the classifier models versus the existence cracklength and location. As evident from the figures, the pre-dicted results are in good agreement with the actual crackparameters.

The high value of the correlation coefficients and lowvalues of MSE for the prediction results of the crack pa-rameters indicate the high performance of the adopted

ANN models. The sufficient number of neurons in thehidden layer is 6 and 5 for Model-1 and Model-2, respec-tively. The correlation coefficient values andMSE for NNusing natural frequencies are (0.98 & 0.92) and (11.13 &16.04) and for NN using FRFs are (0.976 & 0.926) and

(18.08 & 14.69) for the identification of crack length andlocation respectively. Based on obtained results one canconclude that by knowing the natural frequencies andamplitudes of FRFs the trained classifier models are ableto adequately predict the crack length and location in thecurvilinear beam. The presented results also support theidea that large cracks can be detected more accuratelythan small cracks, specifically the cracks smaller than10mm. This can be attributed to the fact that the naturalfrequencies and amplitudes of FRFs are fairly not enoughto reflect the reasons beside this kind of cracks. The com-parison between the predicted results, as shown in figure14 and 27, indicates that Model-1 is more accurate than

Model-2 for the identification of crack length and loca-tion.Since, a limited number of samples are used for train-

ing of the neural network, soANNwas not so sensitive ascompared with some other methods, such as ultrasound,X ray, dye penetrants, magnetic particle and acousticemission. On the other hand, the implementations ofthese methods on damage detection are limited [7].Therefore, ANN can be used as an initial assessment,which can be followed by one of the other methods todetermine the crack characteristics more precisely. Toprovide improved model for the crack identification, themultipleANN(MNN) model has been proposed.

Fig. 26. The MSEfor the prediction results of the crack length andlocation using FRFs data sets (20 spectral lines)

Fig. 27. The prediction results of the crack length (A) and cracklocation (B) using FRFs data sets (20 spectral lines)

Fig. 25. The correlation coefficient values for the prediction resultsof the crack length and location using FRFs data sets (20 spectral





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3.3.7 Implementation of MNN Using Natural Frequencyand FRFs

In this section, instead of training two networks andchoosing the best network, Model-1 and Model-2 arecombined to produce Model-3. A combined data sets(consist of FRFs, 20 spectral lines and natural frequencies,3 natural frequencies) are assembled together for training

Model-3 to improve the overall estimation error thatcould be smaller than the individual errors for each indi-vidual NN(Model 1 and 2). The multiple ANNwas ableto learn the relationship between the amplitudes of FRFsand natural frequencies with the corresponding damagestate. The convergence has been achieved after 10000epochs. The number of neurons in the hidden layer is 6and 5 for 1stANNand 2ndANN, respectively. The learningrate and momentum are assumed to be 0.005 and 0.9, re-spectively, for 1stANNand 2ndANN, where the values ofthe learning rate and momentum are set 0.05 and 0.6 forthe 3rdANN.

After successful training of the multipleANN(Model-3), new data sets are fed to the trained FFANN(Model-3).

Natural frequencies (3 parameters) and amplitude ofFRFs (20 parameters) data were fed simultaneously into1stANNand 2ndANN, respectively. The results obtainedfrom Model-3 are presented in figure 28. The figureshows the scatter plots between the predicted results ofModel-3. As displayed in the figure, it can be seen that thecrack estimation by the trained Model-3 show goodagreement with the existence of the crack length and loca-tion.

The obtained results, as shown in figure 28, reveal thatModel-3 demonstrates highly improved performance andthe prediction accuracy is more accurate than Model-1and Model-2. The correlation coefficient between the pre-

diction values and actual values of the crack length andlocation are 0.99 and 0.98. Additionally, the correspond-ingMSE to estimate the crack length and location are 4.13and 4.62, respectively.

4 CONCLUSIONS

This paper aimed at developing of efficient damage iden-tification technique for the identification of the crack pa-rameters (length and location) in curvilinear beams basedon changes in vibration characteristics, using data givenfrom numerical methods.The following conclusions are drawn:

1. The application of FEM to the curvilinear beamshowed that the changes in natural frequenciesand amplitude of FRFs are sensitive to the cracksize and the crack location.

2. FFANNwhich have single and multiple ANNar-rangements have been successfully employed forcrack identification in the curvilinear using natu-ral frequencies data sets or FRFs data sets

3. The data reduction techniques have effectivelyreduced the input data (number of natural fre-quencies and spectral lines of the FRFs). The re-sults revealed that the pruning mechanism whichis based on the data reduction mechanism canled to satisfactory results

4. Results obtained from the neural network modelshow that a neural network (NN) with propertraining is a good mean to estimate the lengthand location of the crack in the curvilinear beam.

5. The high values of the correlation coefficientsand low values of MSE for predicting the crackparameters have shown the accuracy of the ANN

models using natural frequencies (Model 1) andamplitudes of FRFs (Model 2). Based on obtainedresults it can be concluded that, by knowing thenatural frequencies and amplitudes of FRFs thetrained classifier, models can adequately predictthe crack length and location in the curvilinearbeam.

6. The comparison between the FFANN predictedresults indicates that the FFANNModel-1 is moreaccurate than Model-2 for the identification ofcrack length and location in the curvilinear beam.

7. The results show that the multiple ANN(Model-3) can improve the precision and can provide

better results compared to the single ANN. Thecorrelation coefficient andMSE between the pre-diction values and actual ones for the cracklength and location in the curvilinear beam are(0.99 and 0.98) and (4.13 and 4.62) respectively,which verify the validity of the present methodfor the crack identification.

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