Computationally efficient delamination detection in composite beams using Haar wavelets

Mechanical Systems and Signal Processing 25 (2011) 2257–2270

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Computationally efficient delamination detection in composite beams using Haar wavelets H. Hein , L. Feklistova Institute of Computer Science, University of Tartu, Liivi 2, 50409 Tartu, Estonia

a r t i c l e i n f o

abstract

Article history: Received 21 July 2010 Received in revised form 23 November 2010 Accepted 2 February 2011 Available online 19 February 2011

The paper presents an integrated vibration-based method for delaminations detection in homogeneous and composite beams. The method is based on Haar wavelets and artificial neural networks (ANNs). Firstly, scaled modal responses of the structure are expanded into Haar series by Chen–Hsiao method (CHM), and a delamination feature index is constructed. The database of 68 datasets built on Haar wavelet and frequencybased approaches was utilized by different ANNs to establish the mapping relationship between the delamination status and the delamination feature index or frequencies. The results are compared to each other. The simulations show the proposed complex method with delamination index detects the location of delaminations and identifies the delamination extent with high precision (4 90%); the approach requires less computations and processing time than the frequency-based approach. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Composite beams Delamination Haar wavelets Neural networks

1. Introduction Fiber-reinforced multi-layer composite materials are increasingly being used in civil infrastructure, vehicles, aerospace and light industry. Damage detection in composite structures and machinery is an important issue in terms of safety and functionality. One of the commonly encountered types of damage in laminated composites is delamination. Delaminations are caused by production stresses or service-induced strains, such as impact of foreign objects, exposure to unusual level of excitation or oscillating load over an extended period of time, etc. [1,2]. The early detection and the continuous monitoring of the delamination for the growth and location are the most important issues in the automatic delamination inspection of in-service composite structures [3]. The vibration-based structural damage detection is a relatively new research topic. The approach has several classifications. According to the structural model, the vibration-based structural damage detection approach can be divided into model-based and signal-based methods [4,5]. The model-based methods reveal the damage locations and severities through the comparison of data obtained during the experiments and with the aid of mathematical model of the structure since structural damages cause changes in the dynamic characteristics. This approach was proposed and experimentally tested by several authors [6–9]; the review and classification can be found in [10]. Contrary to the modelbased methods, the signal-based methods do not use the structural model and detect damage by comparing the structural responses before and after the damage. In [11], the frequency response functions and in [12] the shear horizontal waves, derived from mode conversion of the fundamental Lamb wave, were successfully used to extract the damage detection index.

 Corresponding author.

E-mail address: [email protected] (H. Hein). 0888-3270/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ymssp.2011.02.003

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According to the second classification of the vibration-based structural damage detection approach, it can be divided into traditional and modern-type methods [13]. The traditional methods use only dynamic characteristics of structure, e.g. natural frequencies, mode shapes, modal damping, modal strain energy, etc. The location and severity of damage can be determined by the differences between the structural dynamic characteristics of the damaged and intact structures. The advantage of the method is that insignificant changes in physical properties of a structure due to damage result in detectable variations in modal parameters (natural frequencies, mode shapes and modal damping) [14]. The main insufficiency of the method is how to extract the most important features from the vibration response with the purpose of damage detection. Furthermore, the method depends on experimental data analysis, and therefore it is not convenient for online damage detection. A comprehensive review on the vibration-based damage detection methods was presented by Zou et al. [15]. The modern-type damage detection methods are based on real-time measured structural response signals. This approach overcomes the drawbacks of the common non-destructive testing techniques, such as acoustic emission, scanning, X-ray, etc. [4]. The methodology can include neural networks, genetic algorithms, wavelet analysis, etc. The back-propagation ANN was trained to predict the delamination size and location from the natural frequencies of the beam in [1,2]. The wavelet transform has been applied in structural damage detection by many authors [3,16–20]. The advantage of the wavelet-based methods is that they do not require the analysis of the complete structure. The methods are independent from the time-frequency analysis. The wavelet transform decomposes a signal into a set of basis functions. The product of the transform is wavelet coefficients for different scales. Due to time-frequency localization the wavelet transform has ability to reveal some hidden parts of data that other signal analysis techniques fail to detect [13]. In the recent year articles, non-sufficient interest has been paid to the Haar wavelet functions, which are mathematically the simplest wavelets. Chen and Hsiao [21,22] demonstrated that these wavelets can be successfully approximated the derivatives of functions for solving differential equations. The approach has been developed further by Lepik [23]. In the present work an attempt to apply the Haar wavelets for delamination detection based on beam modal responses is made. The fundamental mode shapes are chosen since they are most accurately determinated by a standard modal testing method [15]. The feasibility of using frequency changes for damage detection is limited since significant damage may cause very small changes in natural frequencies [4,24]. To overcome these difficulties, the present work is focused on the changes in mode shapes as they are much more sensitive to local delaminations in comparison with the changes in natural frequencies [24]. The sensitivity of fundamental mode shapes to damage in cantilever beams with cracks is studied in [25]. The application of the proposed method in real structures includes the measurement of the modal responses, the construction of the delamination feature index and the implementation of the measurements and model into the delamination detection process. The paper is organized into six sections. Section 2 outlines the Haar wavelet method CHM. Section 3 presents the dynamic response of vibrating composite beams with multiple delaminations. In Section 4, ANN modeling for delamination detection is explained. In Section 5, several numerical experiments with different beam models, boundary conditions and delamination cases are described. The predictions of delamination parameters made during the computer simulations are compared with the results obtained by the frequency-based delamination detection. Different ANN learning methods are examined and compared. Section 6 summarizes the main finding and conclusions.

2. Chen–Hsiao Haar wavelet method (CHM) The Haar wavelets belong to a special class of discrete orthonormal wavelets. The orthonormal wavelets generated from the same mother wavelet form a basis whose elements are orthonormal to each other and are normalized to unit length. This property allows each wavelet coefficient to be computed independently of other wavelets. The Haar wavelet family is a group of square waves:   8 k 2kþ 1 > > , , 1 for x 2 > > m 2m > <   2kþ 1 k þ1 ð1Þ hi ðxÞ ¼ > 1 for x 2 , , > > 2m m > > : 0 elsewhere: Integer m ¼ 2j , j ¼ 0,1, . . . ,J indicates the level of the wavelet; k ¼ 0,1, . . . ,m1 is the translation parameter. Integer J determines the maximal level of resolution. The index i in (1) is calculated according to the formula i= m+ k+1; the minimal value for i is 2, in this case m= 1, k= 0; the maximal value of i is i ¼ 2M ¼ 2J þ 1 . The value i= 1 corresponds to the scaling function for which h1 ðxÞ  1. Any function y(x) which is square integrable in the interval [0,1] can be expanded into a Haar series with an infinite number of terms [21]: yðxÞ ¼

1 X i¼1

ci hi ðxÞ,

i ¼ 2j þ k, j Z0, 0 rk r 2j , x 2 ½0,1,

ð2Þ

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where the Haar coefficients Z 1 ci ¼ 2j yðxÞhi ðxÞ dx

ð3Þ

are determined minimizing the integral square error: #2 Z 1" m X yðxÞ ci hi ðxÞ dx, m ¼ 2j , j 2 N: e¼

ð4Þ

0

0

i¼1

Let the collocation points be defined as follows: xl ¼ ðl0:5Þ=ð2MÞ (l ¼ 1,2, . . . ,2M). According to [22], a coefficient matrix Hði,lÞ ¼ ðhi ðxl ÞÞ with dimension 2M  2M is introduced. If y(x) is a piece-wise constant or it can be approximated as a piece-wise constant, the sum in (2) can be terminated after 2M terms, that is yðxÞ ¼

2M X

ci hi ðxÞ:

ð5Þ

i¼1

The discrete form of this equation is yðxl Þ ¼

2M X

ci hi ðxl Þ ¼ cTð2MÞ hð2MÞ :

ð6Þ

i¼1

The coefficient vector cð2MÞ can be calculated from the following equation [22,23]: cTð2MÞ ¼ yð2MÞ H1 ð2M2MÞ ,

ð7Þ

where        1 3 4M1 y ...y : yð2MÞ ¼ y 4M 4M 4M

ð8Þ

Eq. (7) is called the forward transform which transforms the function y2M into the coefficient vector cTð2MÞ , and Eq. (8) is called the inverse transform. Since H2M2M and H1 2M2M contain many zeros, the Haar transform is much faster than the Fourier transform. It should be noted that calculations for Hð2M2MÞ and H1 ð2M2MÞ must be carried out only once. 0 Assume that the i-th scaled mode shapes of the delaminated and intact beams are WD i and Wi , respectively. The nonlþ1 dimensional delamination feature index vector of level l (the vector has 2 ¼ 2M components) is introduced as follows: l l , . . . ,Vi,2M Þ¼ Vli ¼ ðVi,1

0 1 ðWD ið2MÞ Wið2MÞ ÞHð2M2MÞ 0 1 JðWD ið2MÞ Wið2MÞ ÞHð2M2MÞ J

,

ð9Þ

0 where JJ denotes the Euclidean norm and WD ið2MÞ , Wið2MÞ are the vectors corresponding to the i-th mode shape of the 0 delaminated (WD ) and intact (W ) beams, respectively. The components of the vectors are calculated in the collocation i i points with the aid of cubic splines.

3. Dynamic response of vibrating composite beams with multiple delaminations The basic idea of the present research is to establish directly an input–output relationship between the modal responses and the delamination locations/sizes using back-propagation neural networks. In order to employ the approach, it is necessary to obtain the vibration response data of composite beams with different delaminations. For this purpose, a number of authors [1,3,6,20,26] adopted an appropriate finite element model. In the present work, a structural dynamic model of delaminated composite beam is established and the modal response data with various delaminations are obtained through numerical simulations. Let us consider the free vibrations of a composite laminated beam with n nonoverlapping delaminations. The geometry of the beam is shown in Fig. 1. The delaminated beam is considered as a combination of 3n +1 beam sections, connected at the delamination boundaries. Each beam section is treated as a classical Euler–Bernoulli beam model with Li bhi [9,10,27]. In the present research, the Euler–Bernoulli beam theory with a constrained mode, rigid connector and bending-extension coupling [10,27] is considered. The governing equations for the intact beam sections are Di

@4 wi @2 w þ ri Ai 2 i ¼ 0, @x4 @t

i ¼ 1, . . . ,3n þ 1,

ð10Þ

where wi ðx,tÞ is the vertical displacement of the i-th beam section; Di is the bending stiffness; ri is the density of material; Ai is the cross-sectional area; x is the axial coordinate and t is the time. Using the classical laminate theory [28], the bending stiffness Di is given by Di ¼ DðiÞ 11 

2 ðBðiÞ 11 Þ

AðiÞ 11

,

ð11Þ

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Fig. 1. The geometry of the beam.

Fig. 2. The i-th beam laminate.

where DðiÞ 11 ¼

ni bX ðQ k Þ ðz3 z3 Þ, 3 k ¼ 1 11 k k k1

ð12Þ

BðiÞ 11 ¼

ni bX ðQ k Þ ðz2 z2 Þ, 2 k ¼ 1 11 k k k1

ð13Þ

AðiÞ 11 ¼ b

ni X

k Þ ðz z ðQ11 k1 Þ, k k

ð14Þ

k¼1 k ¼ Q k cos4 j þ Q k sin4 j þ2ðQ k þ2Q k Þsin2 jcos2 j, Q11 11 22 11 66

Q11 ¼

E11 , 1n12 n21

Q22 ¼

E22 , 1n12 n21

Q66 ¼ G12 ,

n21 ¼

ð15Þ

n12 E22 E11

,

ð16Þ

ðiÞ ðiÞ where DðiÞ 11 is the bending stiffness, B11 is the coupling stiffness, A11 is the extensional stiffness of the lamina, b is the width, ni is the number of plies, n12 and n21 are the longitudinal and transverse Poisson’s ratios, respectively, E11 and E22 are the longitudinal and transverse Young’s moduli, respectively, j is the angle of k-th lamina orientation and zk and zk1 are the locations of the k-th lamina with respect to the midplane of i-th beam section (Fig. 2). According to the constrained model,

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the beam sections in the delaminated zone are forced to vibrate together, and the governing equations are ðDj1 þ Dj2 Þ

@4 wj @2 wj þðrj1 Aj1 þ rj2 Aj2 Þ 2 ¼ 0, 4 @x @t

j1 ¼ 2,5, . . . ,3n1; j2 ¼ 3,6, . . . ,3n:

ð17Þ

For free vibrations the solution can be sought in the form: wi ðx,tÞ ¼ Wi ðxÞsinðotÞ,

ð18Þ

where o is the natural frequency and Wi(x) is the mode shape of the i-th beam section. Substituting (18) into (10), taking into account xi ¼ x=Li and eliminating the trivial solution sinðotÞ ¼ 0, one can obtain the solutions of (10) in the following form: Wi ðxÞ ¼ Ci1 sinki xi þ Ci2 coski xi þ Ci3 sinhki xi þ Ci4 coshki xi ,

ð19Þ

where k4i ¼

o2 ri Ai L4i

ð20Þ

Di

and Ci1 , . . . ,Ci4 are the arbitrary integrating constants. The solution of (17) can be obtained in a similar way. The solution for the beam as a whole is obtained in terms of solutions of all the component beams by enforcing the appropriate boundary and continuity conditions. Eq. (10) is solved in the intact regions and (17) in the delaminated regions. The boundary conditions that can be applied at the supports x =0, x =L are the following. If the beam is clamped at x= 0, then W1 =0 and W 01 ¼ 0; if simply supported, then W1 =0 and W 00 1 ¼ 0; if free, then W 00 1 ¼ 0 and W 000 1 ¼ 0; if guided, then W 01 ¼ 0 and W 000 1 ¼ 0. The analogous boundary conditions can be established at x = L. The continuity conditions for deflection, slope and shear force at x =a1 are W1 ¼ W2 ,

W 01 ¼ W 02 ,

000 D1 W 000 1 ¼ ðD2 þ D3 ÞW 2 :

ð21Þ

The continuity condition for bending moments at x= a1 can be presented as [10]: M1 ¼ M2 þ M3 12 P2 ðh1 h2 Þ þ 12P3 ðh1 h3 Þ, Mi ¼ Di W 00 i ,

i ¼ 1,2,3:

ð22Þ

The axial forces P2 and P3 are established from the compatibility between the stretching/shortening of the delaminated layers and the axial equilibrium [8], which result P3 L2 EA1

2 L2  PEA ¼ ðW 01 ða1 ÞW 04 ða2 ÞÞ h21 , 2

P2 þ P3 ¼ 0,

ð23Þ

where a1 denotes the coordinate of cross-section between (s1) and (s2)–(s3), whereas a2 is the coordinate between (s2)–(s3) and (s4) beam sections. Similarly, the continuity conditions are derived at x ¼ a2 , . . . ,x ¼ am ; m ¼ 2, . . . ,2n. In the case of n delaminations, the boundary conditions and continuity conditions provide 8n +4 homogeneous equations for determination of integration constants. 4. Modeling of the ANN for delamination identification The back-propagation neural network (BPNN) was used in the present study because of its simple approach and precise generalization capability. The neural network was created using MATLAB version 6 and on computer Intel Core 2 Duo Processor, 2.4 GHz, 3.46 GB RAM. In order to identify the delamination status using BPNN, a database of sample data consisting of delamination feature indexes and corresponding delamination status (locations and lengths) was obtained using the Haar wavelet method. Three delamination feature index vectors for the clamped–clamped beam with one nonsymmetric delamination are in Fig. 3. The results proved that the element values of the index vectors vary markedly with the delamination status. In order to get a stable BPNN, a parametric study on the number of hidden layers and neurons in each hidden layer was performed. The log-sigmoid transfer function was used in this study. The mean-square network error was calculated as follows: MSE ¼

P X N 1X ðyðpÞ tiðpÞ Þ2 , Pp¼1i¼1 i

ð24Þ

where P is the number of patterns in the training or testing sets, N is the number of output neurons, yðpÞ and tiðpÞ denote the i actual output and the computed target value in the pattern, respectively. After numerous tests the network architecture was selected by the following rule: the number of hidden layers is equal to the number of output neurons; the number of neurons in all hidden layers overall is equal to the number of patterns. The MSE of the test set versus the number of hidden neurons in the case of three output neurons (two equal delaminations at the arbitrary position in one layer) is presented in Fig. 4. It can be seen that the network with three hidden layers gives the best result. For every network, the coefficient of

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1.00 0.80 0.60 0.40 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00

L1/L = 0.1; h2/h = 0.1 L1/L = 0.3; h2/h = 0.4 L1/L = 0.7; h2/h = 0.2

Fig. 3. Delamination feature index vector components for different delamination lengths.

Fig. 4. MSE with different numbers of hidden neurons.

multiple determination 2

R ¼ 1

PP

p¼1 PP p¼1

PN

ðpÞ ðpÞ 2 i ¼ 1 ðyi ti Þ ðpÞ 2 i ¼ 1 ðyi ym Þ

PN

and the variance account for (VAF)   varðytÞ  100 VAF ¼ 1 varðyÞ

ð25Þ

ð26Þ

have been calculated [29]. In (25) ym is the mean of output values y and var in (26) denotes the variance. The coefficient of multiple determination R2 shows the closeness of fit, whereas VAF shows the overall performance of the network. Ideally, R2 is equal to 1 and VAF is equal to 100%.

5. Numerical simulations To demonstrate the capability of the proposed technique, five different beam models with various delamination schemes were examined. Four sets of boundary conditions (clamped–clamped (CC), clamped–simply supported (CS), clamped–free (CF), simply supported–simply supported (SS)) were considered. For each case three sets of patterns with varying locations and lengths of delaminations were calculated. Two sets correspond to the wavelet level one and two, respectively, whereas the third set contained six natural frequencies with the same delamination status. The test sets contained ten patterns for the network efficiency assessment. Five patterns were known to the ANN and five patterns were unknown. In the first three ANN models (A, B, C), it is assumed that the beam is homogeneous, whereas the other two models (D, E) are about a T300/934 graphite/epoxy beam with a ½00 =900 2s stacking sequence. The dimensions of the 8-ply beam are 127  12.7  1.016 mm3. The material properties for the lamina are E11 =134 GPa, E22 = 10.3 GPa, G12 = 5 GPa, n12 ¼ 0:33 and r ¼ 1:48  103 kg=m3 . This beam with CF boundary conditions was considered as a benchmark in [10–13,28,30]. The first frequencies calculated for the composite beam with CF boundary conditions (cantilever) are shown in Table 1. It was found that a good agreement was obtained between the frequencies calculated by the present model and the experimental [30], analytical [12,27] and FEM results [30].

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Table 1 Primary frequencies for cantilever composite beam E. Delamination length (mm)

Present (Hz)

Shu and Della [27]

Shen and Grady [30]

Luo and Hanagud [9]

0.0 25.4 50.8 76.2 101.6

82.02 79.93 74.36 65.07 54.75

81.88 80.47 75.36 66.14 55.67

82.04 80.13 75.29 66.94 57.24

81.86 81.84 76.81 67.64 56.95

Table 2 Structure of evaluated networks. Network

Number of neurons in hidden layer 1

Number of neurons in hidden layer 2

Number of neurons in hidden layer 3

Number of output neurons

A B C D E

92 56 35 118 49

– 56 35 – 49

– – 35 – –

1 2 3 1 2

Table 3 Delamination localization errors for six first mode shapes. Location

Error (%) Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

0.15 0.30 0.48 0.52 0.56 0.59 0.64 0.68 0.75

0.67 1.50 0.27 0.21 0.09 0.47 0.23 0.06 0.11

4.27 0.60 0.69 0.58 1.07 0.53 0.91 0.60 1.33

0.47 0.63 1.02 0.04 0.87 0.58 0.02 0.31 0.27

3.33 2.00 0.33 0.60 0.20 0.42 0.70 0.43 0.55

4.60 0.27 1.96 0.12 2.79 2.32 1.91 0.04 0.61

4.60 0.57 0.29 0.92 0.59 0.31 4.53 0.16 20.83

Average

0.35

0.77

0.45

0.76

1.17

3.39

Beams A and B had symmetric midplane and non-symmetric delamination in arbitrary depth, respectively; beam C had two delaminations with the same length in the arbitrary depth; composite beam D had symmetric midplane delamination, and composite beam E had non-symmetric delamination. The network structures are presented in Table 2. 5.1. Sensitivity analysis Before applying the Haar wavelet approach to different physical models, the sensitivity analysis was performed. Numerous tests showed that the Haar wavelet approach is not sensitive to the boundary conditions, whereas the frequency-based method is sensitive (see next section). Nevertheless, both approaches are sensitive to the nature of the data: if the test patterns where known to the ANN, the predictions where more accurate; if the test data have not been used in training, the accuracy of computations decreased. Next, a study on the mode shapes was required in order to find the most informative and trainable mode for delamination detection. The case was examined using a delaminated cantilever beam and a ANN with 61 neurons on the hidden layer. The same amount of patters was used to train the ANN. The system was tested by ten patterns that had not been shown priori to the network. The midplane delamination localization errors (in percents) of cantilever beam for the first six modes are presented in Table 3. The comparisons confirmed that the most appropriate mode for the delamination detection is the first one. This is in accordance with the results of [25]. Therefore, the first mode shape was utilized in all next numerical examples. Finally, a study on noise was conducted in order to assure that the Haar wavelet approach is capable of making accurate calculations despite various distorters. To obtain the pseudo-experimental data, the calculated mode shapes were corrupted with additive noise: ~ ðxÞ ¼ WðxÞ þ gxðxÞ, W

ð27Þ

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where xðxÞ is the Gaussian white noise with zero mean and standard deviation one. The quantity g stands for the noise intensity. The output signal was filtered with the moving average filter with different number of points and then expanded into 16-element vector with Haar coefficients. Numerous tests were carried out using the homogenous beam with clamped–free ends. The length of midplane delamination was taken 0.1 and the noise intensity g ¼ 0:15. The ANN with 142 training patterns was repeated 20 times using the testing data obtained from the training data. As the testing patterns had previously been loaded to the system, the error of predictions was smaller than in the cases where the patterns were not shown to the network during the training. The average results of the delamination localization are shown in Table 4. 5.2. Homogenous beam with a symmetric midplane delamination Assume that the symmetric delamination occurred in the midplane of the beam. An ANN with 92 neurons placed in a hidden layer was used to study the case. The ANN was trained by six different algorithms (resilient, Fletcher–Reeves, Polak–Ribiere, Powell–Beale, Levenberg–Marquardt and Bayesian) using 92 training patterns. The accuracy of delamination length prediction is shown in Tables 5–8. Proceeding from the results, the following conclusions were made. Firstly, the boundary conditions of the beam did not have significant influence on the ANN ability to make predictions of the parameter. Secondly, in most tests the Haar wavelet approach with eight coefficients and ANN trained by Bayesian method made the most accurate predictions for the delaminations smaller than 0.01 units. A visual comparison of the Haar wavelet approach and six natural frequencies is shown in Fig. 5. The abscissa exposes the number of the test pattern, the axis of ordinates shows the length of the delamination. The circles indicate the target values, the crosses show the values computed by ANN. Thirdly, in the case of clamped beam, mere four Haar coefficients were enough to make accurate predictions. This proves the hypotheses that the signal can be decomposed by Haar wavelets into a certain number of components; the increased number of components results in the decrease of accuracy [23]. Fourthly, the accuracy of predictions depended on the training algorithm. The comparison of the results in the tables shows that different training Table 4 Average error after 20 tests on delamination localization using different filters with 3 and 10 points. Location

Error Point 3

Point 10

0.15 0.20 0.32 0.36 0.48 0.53 0.60 0.62 0.70 0.75

0.04 0.36 0.24 0.05 0.04 0.03 0.05 0.01 0.02 0.01

0.33 0.36 0.03 0.03 0.49 0.01 0.25 0.26 0.01 0.14

Average

0.08

0.19

Table 5 Accuracy of delamination length predictions of beam with clamped ends. Method Resilient Fr H4 Fletcher–Reeves Fr H4 Polak–Ribiere Fr H4 Powell–Beale Fr H4 Levenberg–Marquardt Fr H4 Bayesian Fr H4

MSE

VAF

R2

Time (s)

0.0020 0.0109

0.9975 0.8699

0.9969 0.8523

9.0470 6.0460

0.0007 0.0001

0.9934 0.9990

0.9904 0.9988

1.5780 5.6720

0.0038 0.0002

0.9503 0.9978

0.9479 0.9977

1.5780 6.6710

0.0070 0.0002

0.9905 0.9972

0.9903 0.9970

1.7030 4.2190

0.0000 0.0033

0.9995 0.9573

0.9995 0.9553

6.9840 0.9220

0.0000 0.0051

0.9999 0.9376

0.9999 0.9307

2.9e+ 2 7.8e+ 1

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Table 6 Accuracy of delamination length predictions of beam with clamped–simply supported ends. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Levenberg–Marquardt Fr H8 Bayesian Fr H8

MSE

VAF

R2

Time (s)

0.0002 0.0010

0.9968 0.9862

0.9968 0.9862

1.3e + 1 7.1710

0.0009 0.0023

0.9886 0.9707

0.9878 0.9687

2.0310 5.7340

0.0069 0.0013

0.9110 0.9399

0.9058 0.9305

1.4220 2.9690

0.0013 0.0051

0.9850 0.9955

0.9818 0.9946

4.0780 3.5780

0.0001 0.0001

0.9991 0.9986

0.9989 0.9985

7.9370 2.6090

0.0000 0.0000

0.9998 0.9995

0.9997 0.9995

1.4e + 2 2.7e + 2

Table 7 Accuracy of delamination length predictions of beam with clamped–free ends. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Levenberg–Marquardt Fr H8 Bayesian Fr H8

MSE

VAF

R2

Time (s)

0.0005 0.0003

0.9939 0.9968

0.9935 0.9966

2.5e + 1 2.9540

0.0007 0.0008

0.9928 0.9898

0.9906 0.9890

1.7030 4.7340

0.0009 0.0007

0.9883 0.9899

0.9883 0.9899

2.2030 8.4530

0.0013 0.0032

0.9830 0.9604

0.9824 0.9571

1.7340 4.0310

0.0000 0.0002

0.9994 0.9976

0.9993 0.9969

7.2960 3.3280

0.0000 0.0001

0.9999 0.9990

0.9999 0.9990

9.3e + 2 3.3e + 3

Table 8 Accuracy of delamination length predictions of beam with simply supported ends. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Levenberg–Marquardt Fr H8 Bayesian Fr H8

MSE

VAF

R2

Time (s)

0.0004 0.0008

0.9959 0.9910

0.9948 0.9896

3.4530 2.2e + 1

0.0010 0.0008

0.9878 0.9919

0.9863 0.9893

1.5470 0.0064

0.0021 0.0011

0.9754 0.9857

0.9711 0.9857

1.7190 9.9530

0.0015 0.0012

0.9801 0.9865

0.9801 0.9838

1.6250 5.7500

0.0002 0.0006

0.9971 0.9925

0.9971 0.9921

6.3750 1.9e3

0.0000 0.2970

0.9995 0.9884

0.9995 0.9857

2.2e + 2 9.9530

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TEST: o - given, x - test

0.9

0.9

0.8

0.8

0.7 0.6 0.5 0.4 0.3

0.7 0.6 0.5 0.4 0.3

0.2

0.2

0.1

0.1

0

1

2

3

4 5 6 7 Test Pattern number

TEST : o - given, x - test

1

Delamination length

Delamination length

1

8

9

10

0

1

2

3

4 5 6 7 Test pattern number

8

9

10

Fig. 5. Delamination length prediction using Fletcher–Reeves training method: (a) input, six natural frequencies and (b) input, eight Haar coefficients.

Table 9 Accuracy of non-symmetric delamination position predictions of cantilever beam. Method

MSE

Resilient Fr 0.001 H8 0.002 Fletcher–Reeves Fr 0.000 H8 0.001 Polak–Ribiere Fr 0.000 H8 0.004 Powell–Beale Fr 0.000 H8 0.013 Levenberg–Marquardt Fr 0.000 H8 0.807 Bayesian Fr 0.001 H8 0.000

VAF (L1/L)

VAF (h2/h)

VAF (mean)

R2 (L1/L)

R2 (h2/h)

R2 (mean)

Time (s)

0.869 0.848

0.997 0.874

0.933 0.861

0.857 0.848

0.997 0.869

0.927 0.859

4.7e+ 1 8.1e+ 1

0.981 0.928

0.999 0.905

0.990 0.917

0.982 0.925

0.999 0.905

0.991 0.915

1.3e+ 2 1.7e+ 2

0.981 0.770

0.999 0.785

0.990 0.777

0.981 0.672

0.999 0.785

0.990 0.672

1.3e+ 2 0.7852

0.966 0.011

0.999 0.299

0.983 0.155

0.966 0.103

0.999 0.272

0.983 0.084

8.0e+ 1 1.2e+ 2

0.982 NA

0.999 NA

0.991 NA

0.966 NA

1.000 NA

0.983 NA

1.8e+ 2 1.6e+ 5

0.878 0.999

1.000 0.990

0.939 0.995

0.889 0.999

1.000 0.989

0.944 0.994

2.7e+ 4 8.6e+ 4

algorithms work better using either frequencies (resilient, Fletcher–Reeves, Levenberg–Marquardt) or Haar coefficients (Polak–Ribiere, Powell–Beale). Finally, in terms of the computing time needed for data generation and ANN training, the overall process was considerably shorter in the case of Haar coefficients. During that time, the ANN passed less than 5000 epochs and reached the performance goal of training equal to 0.0001.

5.3. Homogenous beam with a non-symmetric delamination The following tests were dedicated to the non-symmetric model of the beam with a delamination of 0.2 units. The ANN predicted two parameters: the location of delamination along the length and height of the beam. The ANN contained two hidden layers with 56 neurons on each. The ANN was trained by 112 patterns. The results for the cantilever beam are shown in Table 9. The accuracy of predictions of each coordinate is calculated separately; the mean error is computed as well. According to the results, the height is easier to be predicted than the length. In terms of the approaches, the frequency-based calculations are faster and more accurate than Haar wavelet-based ones. The graphic comparison of the approaches and training methods is given in Fig. 6. The abscissa exposes the location of the delamination along the length of the beam; the axis of ordinates shows the location of the delamination along the height of the beam. Comparing Levenberg–Marquardt and Bayesian methods, it has to be highlighted that Levenberg–Marquardt method is very precise in

TEST : o - given, x - test

0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Delamination location along length of the beam

TEST : o - given, x - test 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 Delamination location along length of the beam

Delamination location along height of the beam

0.55

Delamination location along height of the beam

Delamination location along height of the beam

Delamination location along height of the beam

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TEST : o - given, x - test 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Delamination location along length of the beam TEST : o - given, x - test 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 Delamination location along length of the beam

Fig. 6. Delamination location prediction: (a) input, six natural frequencies; method, resilience; (b) input, eight Haar coefficients; method, resilience; (c) input, six natural frequencies; method, Bayesian; and (d) input, eight Haar coefficients; method, Bayesian.

making predictions if the patterns are known to the network. However, the method is inferior to Bayesian method in prediction of the patterns which are unknown to the ANN.

5.4. Homogenous beam with two delaminations at arbitrary height of the beam This ANN predicted the location of two delaminations occurred within the same layer but at arbitrary height of the homogenous beam. The network calculated three parameters: the location of the first delamination along the length of the beam L1/L, the distance between two delaminations L4/L and the height at which delamination developed h2/h. The ANN consisted of three hidden layers with 35 neurons on each. The ANN was trained by 140 patterns and tested by eight patterns. The patterns were composed analogically to the previous examples. The tests were conducted for the cantilever (Table 10) and simply supported beam (Table 11). In the case of cantilever beam, all approaches and methods worked effectively. Three desired parameters were predicted with more than 97% accuracy. However, in the case of resilience, Polak–Ribiere, Powell–Beale methods, the predictions were provided slightly better by the Haar wavelet approach; conversely, the calculations were a few seconds longer. In the case of simply supported beam, the boundary conditions of the beam influenced the accuracy of the predictions. The frequency approach was capable to predict precisely only two parameters, whereas the Haar wavelet approach managed to compute all three parameters with at least 90.00% accuracy. The later method took significantly less time to accomplish the task. These are the main advantages of the Haar wavelets approach. One of the best training methods for making predictions of three parameters is Bayesian. The accuracy was 100.00%.

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Table 10 Accuracy of predictions of two delaminations placed in the same layer at arbitrary height of the cantilever. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Bayesian Fr H8

MSE

VAF (h2 =h)

VAF (L1 =L)

VAF (L4 =L)

VAF (mean)

Time (s)

0.0000 0.0000

0.9999 0.9999

0.9995 1.0000

0.9999 0.9996

0.9994 0.9998

1.1e+ 2 1.2e+ 2

0.0001 0.0001

0.9998 0.9991

0.9980 0.9973

0.9853 0.9769

0.9944 0.9911

1.5e+ 2 1.8e+ 2

0.0002 0.0000

0.9985 1.0000

0.9852 0.9998

0.9726 0.9998

0.9854 0.9999

1.3e+ 2 2.0e+ 2

0.0000 0.0000

0.9995 0.9994

0.9963 0.9999

0.9951 0.9999

0.9970 0.9998

1.2e+ 2 1.2e+ 2

0.0000 0.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

1.0000 1.0000

7.2e+ 3 7.3e+ 3

Table 11 Accuracy of predictions of two delaminations placed in the same layer at arbitrary height of the beam with simply supported ends. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Bayesian Fr H8

MSE

VAF (h2/h)

VAF (L1/L)

VAF (L4/L)

VAF (mean)

Time (s)

0.0041 0.0003

0.9996 0.9571

0.3177 0.9989

0.9930 0.9962

0.7701 0.9841

1.0e+ 2 1.1e+ 2

0.0046 0.0005

0.9911 0.9177

0.3230 0.9996

0.9007 0.9980

0.7396 0.9718

1.9e+ 2 1.6e+ 2

0.0045 0.0005

0.9990 0.9196

0.2820 0.9997

0.9833 0.9649

0.7188 0.9533

1.9e+ 2 1.5e+ 2

0.0049 0.0012

0.9998 0.8034

0.1488 0.9997

0.9903 0.9981

0.7130 0.9337

1.7e+ 2 1.8e+ 2

0.0100 0.0000

0.9965 1.0000

0.1842 1.0000

0.2229 1.0000

0.4678 1.0000

1.7e+ 5 1.2e+ 4

5.5. Composite beam with one delamination in midplane The ANN computed the length of the delamination occurred in the midplane of the composite beam. The ANN consisted of one hidden layer with 140 neurons in it. The ANN was trained by 140 patterns and tested by ten patterns. The tests were conducted for the cantilever (Table 12) and the beam with clamped ends (Table 13). The advantage of the Haar wavelet approach is that in the case of a composite beam, it worked with any training method; whereas the frequency approach worked only with Levenberg–Marquardt and Bayesian training algorithm. Both methods are accurate but time-consuming. The Haar wavelet approach made calculations significantly faster than the frequency approach.

5.6. Composite beam with non-symmetric delamination at the arbitrary height of the beam The following ANN computed the length of the delamination and its location along the fixed height (h2/h =0.125) of the composite symmetric beam. The ANN consisted of two hidden layer with 49 neurons on each. The ANN was trained by 98 patterns and tested by ten patterns. The results for the cantilever beam are presented in Table 14. As in the case of homogenous non-symmetric model, the Haar wavelet approach did not show trustworthy results. Consequently, the frequency approach is a better approach if two parameters are to be predicted. Furthermore, as in the case of the composite symmetric beam, the boundary conditions of the beam influenced the accuracy of the predictions using frequency-based approach. A beam with clamped ends was easier to be modeled than the one with simply supported ends. The frequency approach was capable of making predictions for two parameters of the beam with clamped ends, but was not able to provide reasonable predictions for the beam with simply supported ends. The Haar approach did not show accurate predictions of two parameters.

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Table 12 Accuracy of delamination length predictions of the composite beam with clamped ends. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Levenberg–Marquardt Fr H8 Bayesian Fr H8

MSE

VAF

R2

Time (s)

0.0005 0.0002

NA 0.6658

NA 0.6125

1.8750 6.8440

0.0179 0.0001

NA 0.7594

NA 0.7084

1.5470 4.6720

0.0464 0.0003

NA 0.5210

NA 0.4799

1.5470 8.4370

0.0500 0.0003

NA 0.4553

NA 0.4519

1.4380 3.7970

0.0000 0.0001

0.9612 0.9300

0.9568 0.8336

1.7e + 1 1.8430

0.0000 0.0000

0.9991 0.9970

0.9984 0.9965

1.5e + 3 1.8e + 2

Table 13 Accuracy of delamination length predictions of the composite beam with clamped ends. Method Resilient Fr H8 Fletcher–Reeves Fr H8 Polak–Ribiere Fr H8 Powell–Beale Fr H8 Levenberg–Marquardt Fr H8 Bayesian Fr H8

MSE

VAF

R2

Time (s)

0.0000 0.0002

0.9355 0.6706

0.9221 0.5166

6.7340 8.0780

0.1029 0.0005

NA NA

NA NA

1.1400 5.3590

0.0423 0.0002

NA 0.5852

NA 0.5028

1.2030 1.1e + 1

0.0435 0.0009

NA NA

NA NA

1.3120 5.4220

0.0000 0.0001

0.9790 0.9380

0.9197 0.8316

1.7e + 1 2.4690

0.0000 0.0000

0.9993 0.9991

0.9993 0.9982

1.5e + 3 1.3e + 2

Table 14 Accuracy of predictions of the delamination length and position of composite cantilever beam. Method

MSE

Resilient Fr 0.000 H8 0.000 Fletcher–Reeves Fr 0.000 H8 0.000 Polak–Ribiere Fr 0.000 H8 0.000 Powell–Beale Fr 0.000 H8 0.000 Levenberg–Marquardt Fr 0.000 H8 0.000 Bayesian Fr 0.000 H8 0.000

VAF (L1/L)

VAF (h2/h)

VAF (mean)

R2 (L1/L)

R2 (h2/h)

R2 (mean)

Time (s)

0.989 0.595

0.993 0.546

0.991 0.571

0.989 0.565

0.993 0.281

0.991 0.422

2.2e + 1 1.1e + 1

0.968 0.646

0.991 0.761

0.979 0.704

0.968 0.637

0.990 0.567

0.979 0.602

1.3e + 2 2.6e + 1

0.991 0.496

0.996 0.593

0.994 0.545

0.991 0.303

0.996 0.512

0.994 0.408

5.7e + 1 1.6e + 1

0.987 0.631

0.995 0.809

0.991 0.720

0.982 0.630

0.995 0.791

0.988 0.711

4.8e + 1 1.2e + 1

0.983 0.435

0.992 0.260

0.987 0.347

0.977 0.003

0.990 0.144

0.984 0.070

4.4e + 3 1.1e + 2

0.986 0.888

0.990 0.500

0.988 0.694

0.982 0.8422

0.989 0.144

0.986 0.493

1.3e + 4 1.1e + 4

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6. Conclusions A new method of delamination identification in homogeneous and composite beams is presented. Neural networks are applied to establish the mapping relationship between frequencies, Haar series expansion of fundamental mode shapes of vibrating beam and delamination status. If only one parameter of a homogenous or composite beam is to be predicted, Bayesian training method along with frequency-based or Haar wavelets-based training patters can be used. If two parameters are to be predicted, the frequency approach makes the most reliable predictions. However, for the case of three parameters, the Haar wavelets approach with eight coefficients is more efficient. Secondly, in terms of time, the Haar wavelet approach is less time-consuming. Thirdly, in the case of homogenous models of the beam and Haar wavelet approach, the boundary conditions of the beam do not influence the accuracy of the predictions. Fourthly, the Levenberg–Marquardt training method makes accurate predictions if the test patterns are known to the network; whereas, the Bayesian training method provides accurate predictions in both cases, whether the test patterns are known to the network or not.

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