Applying deep learning and wavelet transform for predicting the vibration behavior in variable thickness skew composite plates with intermediate elastic support
Wael A. Altabey^{1}
^{1}International Institute for Urban Systems Engineering (IIUSE), Southeast Universi, Nanjing, 210096, Jiangsu, China
^{1}Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, 21544, Egypt
Journal of Vibroengineering, Vol. 23, Issue 4, 2021, p. 770783.
https://doi.org/10.21595/jve.2020.21480
Received 18 May 2020; received in revised form 20 December 2020; accepted 30 December 2020; published 6 February 2021
In this paper, the vibration behavior features are extracted from the combination between Wavelet Transform (WT), and Finite Strip Transition Matrix (FSTM) of skew composite plates (SCPs), with variable thickness, and intermediate elastic support. Although, the results of this technique and based on the previous work done by the authors, that show the method can reflect the vibration behavior of the composite plates. Due to the method's difficulty in terms of, a lot of calculations with a large number of iterations these results may not be good choices for quick and accurate vibration behavior extracting. Thus, the new deep neural network (NN) is designed to learn and test these results carrying out by extracting vibration behavior features that reflect the important and essential information about the mode shapes in SCP. The results give high indications about the proposed technique of deep learning is a promising method, particularly when the type structures are complicated and the ambient environment is variable.
 Extract the vibration behavior features of skew composite plates, with variable thickness, and intermediate elastic support from the combination between wavelet transform, and finite strip transition matrix.
 Design the new deep neural network to learn and test these results carrying out by extracting vibration behavior features that reflect the important and essential information about the mode shapes in skew composite plate.
 The results give high indications about the proposed technique of deep learning is a promising method, particularly when the type structures are complicated and the ambient environment is variable.
Keywords: free vibration, deep learning, wavelet transform (WT), variable thickness plates, skew composite plates (SCPs), BFRP.
1. Introduction
The importance of to use of composite materials in many fields of technology, such as aerospace industries, marine engineering, and civil engineering is due to special features, e.g. high strength/weight ratio and corrosion resistance property, particularly under the effects of the harsh environment. Although the structures are made of these types of materials have some drawback are subject to matrix cracks, fiber breakage, and delamination. These invisible faults can lead to catastrophic structural failures [14].
Other major modes of failure of fiberreinforced polymer (FRP) have a temperature, bending, tensile, stress, impact failure, and failure of the installation, etc. These types of failures are complicated and are not easy to assets, mostly when subjected to associate effects of multiple factors [57].
The user of an active system of structural health monitoring (SHM) to observe the safety and potential damage detection in composite plates are essential and most seriously. The function of SHM is consists of three main subfunctions, including system identification, features extraction for algorithms for detection and prediction, and reliability and risk evaluation [819].
The vibration behavior of composite structures is the traditional method for the intelligent detecting the defects in the composite. The mode shapes of the structure are one of the essential tools in Structural health monitoring (SHM) in the last decades, where, extracted the vibration response of composite structure for each damage type and position and analysis based on the variance in vibration parameters. In recent years, different techniques to extract the natural frequencies of composite plates have become a field of great interest in the scientific society [2034].
To find the mode shapes for different boundary conditions with IES, numerical methods or experimental methods must be used. Some researchers have been interested in the vibration of multispan plates using different approaches. In previous works, Altabey [35, 36] used the FSTM as one of the common use of semianalytical approaches to extract vibration response of basalt FRP laminated variable thickness rectangular plates with IES, and he tried to improve the results accuracy and by the way, decrease the calculations efforts due to a large number of iterations by combined his method with artificial neural networks (ANNs) and response surface (RS) methods.
In the present research, the new deep NN is designed to predict the vibration behavior of SCPs with variable thickness and IES with a different elastic restraint coefficient (${K}_{T}$) and four cases of boundary conditions (BCs) of plate edges, namely SSSS, CCCC, SSFF, and CCFF. The plate is a rectangular SCP with variable thickness function $h\left(y\right)$, the locations of the IES is at midline of the presented plate, and the plate was manufactured from basalt fiber reinforced polymer (BFRP) by using five symmetrically layers with the stacking angle [45°/–45°/45°/–45°/45°] as shown in Fig. 1. First, review the illustrated results of the utilized method by the combination of these WT and FSTM methods (WTFSTM) to convergence the studies by checking the agreement with the results available in the literature. Second, the trained deep NN is used to predict the outcome of the extracted vibration behavior of SCPs from WTFSTM at certain values of elastic restraint coefficients (${K}_{T}$) for IES, and then it is subsequently used to predict the vibration behavior for different levels of elastic restraint coefficients (${K}_{T}$) for IES. The results are predicted from the deep NN model are in very good agreement with the WTFSTM results. Hence, the results give high indications about the proposed technique of deep learning is a promising method.
Fig. 1. The geometry of rectangular SCP with variable thickness and IES
2. Model overview
The composite plate material has corresponding elastic and shear modulus values are shown in Table 1.
Table 1. Model property
${E}_{11}$ (GPa)  ${E}_{22}$ (GPa)  ${\upsilon}_{12}$  ${\upsilon}_{21}$  ${G}_{12}$_{}(GPa)  ${G}_{21}$ (GPa)  rho kg/m^{3} 
96.74  22.55  0.3  0.6  10.64  8.73  2700 
The normalized partial differential equation of vibration behavior for the plates system illustrated in Fig.1 under the assumption of the classical deformation theory in terms of the plate deflection ${w}_{o}\left(x,y,t\right)$ using the nonDimensional variables $\xi $ and $\eta $ related to the skew coordinate system $\left(u,\nu ,\varphi \right)$ defined by $u=xsec\left(\varphi \right)$, $\nu =yx\mathrm{t}\mathrm{a}\mathrm{n}\left(\varphi \right)$, and $\xi =\frac{u}{a}$, $\eta =\frac{\nu}{b}$, and after some derivation, the governing equation can be written as follows:
$+2{\beta}^{2}{\psi}_{2}\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\varphi \right)\frac{\partial {h}^{3}\left(\eta \right)}{\partial \eta}{W}_{\xi \xi \eta}+2{\beta}^{2}{\psi}_{2}{h}^{3}\left(\eta \right)\left(3\mathrm{s}\mathrm{i}{\mathrm{n}}^{2}\varphi +\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\varphi \right){W}_{\xi \xi \eta \eta}$
$+2{\beta}^{4}\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\varphi \right)\frac{\partial {h}^{3}\left(\eta \right)}{\partial \eta}{W}_{\eta \eta \eta}2{\beta}^{3}{\psi}_{4}\nu \left(\mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\varphi \right)\frac{{\partial}^{2}{h}^{3}\left(\eta \right)}{\partial {\eta}^{2}}{W}_{\xi \eta}$
$4{\beta}^{3}{\psi}_{4}\left(\mathrm{s}\mathrm{i}\mathrm{n}\varphi \right){h}^{3}\left(\eta \right){W}_{\xi \eta \eta \eta}+{\beta}^{4}\left(\nu \mathrm{t}\mathrm{a}{\mathrm{n}}^{2}\varphi +1\right)\mathrm{c}\mathrm{o}{\mathrm{s}}^{4}\varphi \frac{{\partial}^{2}{h}^{3}\left(\eta \right)}{\partial {\eta}^{2}}{W}_{\eta \eta}$
$+{\beta}^{4}{h}^{3}\left(\eta \right){W}_{\eta \eta \eta \eta}4{\beta}^{3}{\psi}_{4}\left(\mathrm{s}\mathrm{i}\mathrm{n}\varphi \mathrm{c}\mathrm{o}{\mathrm{s}}^{2}\varphi \right)\frac{\partial {h}^{3}\left(\eta \right)}{\partial \eta}{W}_{\xi \eta \eta}={\mathrm{\Omega}}^{2}h\left(\eta \right){h}_{o}^{2}\left(\mathrm{c}\mathrm{o}{\mathrm{s}}^{4}\varphi \right){W}_{tt},$
where ${\psi}_{1}=\frac{{D}_{11}}{{D}_{22}}$, ${\psi}_{2}=\frac{({D}_{12}+2{D}_{66})}{{D}_{22}}$, ${\psi}_{3}=\frac{{D}_{16}}{{D}_{22}}$, ${\psi}_{4}=\frac{{D}_{26}}{{D}_{22}}$.
Since the treatment of IES conditions are the main objective of this paper we presented it in more detail. The line of the IES$\mathrm{}y=b/2$, the displacement must vanish and the normal moment must be continuous, i.e.
where: ${\psi}_{1}=\frac{{D}_{11}}{{D}_{22}}$, ${\psi}_{2}=\frac{({D}_{12}+2{D}_{66})}{{D}_{22}}$, ${\psi}_{3}=\frac{{D}_{16}}{{D}_{22}}$, ${\psi}_{4}=\frac{{D}_{26}}{{D}_{22}}$, ${K}_{T}=\frac{{T}_{b/2}{b}^{3}}{{D}_{22}}$, ${\psi}_{5}=\frac{({D}_{12}+4{D}_{66})}{{D}_{22}}$.
3. Determination of vibration behavior using WT and FSTM
In This section, the mode shapes of the SCP will be extracted using a new method by combined between the WT and FSTM methods with an adjusting frequency parameter, in order to improve the estimated accuracy of extracting by optimized the WT entropy for adjusting frequency parameter.
3.1. Continuous wavelet transform (CWT)
Continuous wavelet transform (CWT) is a convolution process of the data sequence with a set of continuous scaled and translated versions of the mother wavelet (MW) $\psi \left(t\right)$. The translating process is a smoothing effect over the length of the data sequence to localize the wavelet in time domain $x\left(t\right)$, whereas the scaling process is compressing or stretching of analyzed wavelet which indicates various resolutions. The stretched wavelet is used to capture the slow changes; while the compressed wavelet is used to capture abrupt changes in the signal. The tradeoff of enhancing resolution is between increased computational cost and memory by computing wavelet components and multiplying each component by the correctly dilated and translated wavelet, resulting in the constituent wavelet of the analyzed signal [3745].
The $\psi \left(t\right)$ is stretched or squeezed through varying its dilation parameter s and moved through its translation parameter $\tau $ (i.e. along the localized time index $\tau $):
Let $x\left(t\right)$ be the system shape function response of FSTM, where $t$ denotes time. CWT of a function $x\left(t\right)\in {L}^{2}\left(R\right)$, where ${L}^{2}\left(R\right)$ is the set of squareintegrable functions is denoted as ${W}_{s,\tau}$ and defined as:
where the wavelet scale s and the period $\tau $ are used to adjust the frequency and time location. ${W}_{s,\tau}$shows how closely ${\psi}_{s,\tau}\left(t\right)$ correlated with $x\left(t\right)$. By inverse CWT, the signal $x\left(t\right)$ can be regenerated as:
For a plate striped in the $\xi $direction by divided into $N$ discrete longitudinal strips spanning between supports as shown in Fig. 1, the freeresponse equation for one striped beam system may be assumed in the form:
The WT of Eq. (7) is:
The logarithm of Eq. (8) gives:
By using the straight line of the slope of the logarithm of WT modulus, we can be obtained the natural frequency of the system and it is given by:
The plot of $\frac{d}{d\tau}Arg\left({W}_{{s}_{0},\tau}\right)$ is constant in the time domain and is equal to the natural frequency $\omega $. The nondimensional frequency parameter (NDFP) $\left(\mathrm{\Omega}\right)$ are addressed in the form:
4. Deep neural networks (NNs)
Recently, deep learning, which is a network with multiple hidden layers of neurons, has also been applied in solving and identifying the ordinary and partial differential equations [46, 47].
Deep neural networks (NNs) are one of the artificial intelligence (AI) algorithms used for solving advanced nonlinear problems [48]. The networks are consist of computational nodes that connected together to create one individual network, each node is processing a calculation on input and sends the result to output connections, and maybe a node output is an input to one other node or more.
In this section, we use the outcome of the results in Section 3 of vibration behavior of SCP extracted by WTFSTM at certain values of elastic restraint coefficients (${K}_{T}$) to obtain the training data and to predict the vibration behavior for different levels of elastic restraint coefficients (${K}_{T}$) not included in the results.
The proposed deep NN architecture connection is presented in Fig. 2. The steps of the NDFP $\left(\mathrm{\Omega}\right)$ prediction can be described through the following steps in Table 2.
Fig. 2. The architecture of the proposed deep NN for the NDFP ($\mathrm{\Omega}$) prediction
Table 2. The steps of deep NN training to predict NDFP ($\mathrm{\Omega}$)
NN steps  Step remark 
Data collecting  Extract the training data from the WTFSTM of the SCP at certain values of elastic restraint coefficients (${K}_{T}$) 
Training model  Divide the extracting data into three groups of data, the first one will be used for training in MSNN for mode shapes ${\lambda}_{i}$ prediction of SCP, and the second group of a dataset will be used for training in WNN for predicting deflection ${w}_{o}\left(x,t\right)$, this network without hidden layers 
Testing model  The last part of the data will use to test the trained model in the training model. If the model is welltrained, the predicted results by the WNN and MSNN will be convergence to the real value. The training performance of suggested Deep NN is presented in Fig. 3 
Prediction response  The response MSNN will be used to predict the ${\lambda}_{i}$ under random deflection ${w}_{o}\left(x,t\right)$. WNN will be used to predict the ${w}_{o}\left(x,t\right)$ at any location coordinate $x$ along with the SCP including the IES location presented in Section 2 
It is important for the NN designer to check his proposed deep NN performance is suitable or not from the formula of mean square error (MSE):
Therefore, only one global minimum for performance index based on the features of the input vectors, but the minimum local minimum of a function at finite input values, and it cannot be omitted when attaching deep NN. Therefore, we can judge on accuracy a local minimum, if it has a low closer range to global minimum and low MSE. Anyway, the designer must be selected as a suitable method to solve this problem in order to descent the local minimum with momentum. Momentum allows a network to respond not only to the local gradient but also to recent trends in the error surface. Without momentum, a network may get stuck in a shallow local minimum. Fig. 3 shows the performance curves of training with three groups for learning data.
Fig. 3. Training performance of proposed NN
5. Results and discussion
In this section, after reviewing the results available in the literature, the approach of WTFSTM are used to extract the vibration behavior of SCP with variable thickness are presented in Section 2 at certain values of elastic restraint coefficients (${K}_{T}$) for IES, on the other hand, to provide the active training data to proposed deep NN, in order to extract the influence of the IES on the natural frequencies with different elastic restraint coefficients (${K}_{T}$) of such plates.
5.1. Convergence study and accuracy
The importance for review of presented work results with the results available in the literature in order to validate the accuracy and reliability of the proposed technique. In this subsection, the WTFSTM technique has been applied on a CCCC variable thickness SP with $\beta =\text{0.5,}$$\mathrm{\Delta}=$ (0, 0.2, 0.4, 0.5) and $\varphi =$ (30°, 45°, 60°), and then the convergence between the results in Fig. 4 with the results from FSTM [35] will be done.
As shown in the Fig. 4, after convergence, we can see clearly generally, that the results of the presented method WTFSTM in excellent agreement with the other accurate methods in references [2326, 35]. On the other hand, we can see the effects of plate Skew angles ($\varphi $), tapered ratio ($\mathrm{\Delta}$) and aspect ratio ($\beta )$ on the NDFP ($\mathrm{\Omega}$) it has been increased with increasing of the $\varphi $, $\beta $, and $\mathrm{\Omega}$, in all methods WTFSTM and the methods in the literature.
5.2. Proposed method (WTFSTM) results
In the present study, the numerical computations using the WTFSTM approach is applied to extract vibration behavior. Due to the method difficulty in terms of, a lot of calculations with a large number of iterations these results may not be good choices for quickly and accurate vibration behavior extracting, the new deep NN is designed to learn and test these results carrying out by extracting vibration behavior features that reflect the important and essential information about the mode shapes in SCP. The proposed method target achieved using only two different ${K}_{T}$ in computations of the NDFP $\left(\mathrm{\Omega}\right)$, the first one is located at ${K}_{T}=$ 50 the second is${K}_{T}=$ 750, respectively. The first six frequencies are presented in Table 3, the NDFP $\left(\Omega \right)$has been computed with different values of skew angle ($\varphi $) at aspect ratio ($\beta =$ 0.5), and tapered ratio ($\mathrm{\Delta}=$ 0.5) to study the behavior of natural frequencies under a different skew angle for different four BCs namely SSSS, CCCC, SSFF, and CCFF.
Fig. 4. Comparison of the first four natural frequencies of CCCC skew plates $\beta =$ 0.5
Figs. (56) represent the comparison between the WTFSTM data and the deep NN predicted data NDFP $\left(\Omega \right)$ for ${K}_{T}=$ 50 and ${K}_{T}=$750 respectively of four different BCs are SSSS, CCCC, SSFF, and CCFF. The results of the proposed deep NN show much satisfactory prediction quality for this case study.
Fig. 5. Comparison between the WTFSTM data and deep NN predicted data for ${K}_{T}=$ 50
a)$\varphi =$ 0°
b)$\varphi =$ 30°
c)$\varphi =$ 45°
d)$\varphi =$ 60°
Fig. 6. Comparison between the WTFSTM data and deep NN predicted data for ${K}_{T}=$ 750
a)$\varphi =$ 0°
b)$\varphi =$ 30°
c)$\varphi =$ 45°
d)$\varphi =$ 60°
Table 3. The first six frequencies of SCP, $\beta =$ 0.5, $\mathrm{\Delta}=$ 0.5
BCs  $\varphi $  ${K}_{T}$  ${\mathrm{\Omega}}_{1}$  ${\mathrm{\Omega}}_{2}$  ${\mathrm{\Omega}}_{3}$  ${\mathrm{\Omega}}_{4}$  ${\mathrm{\Omega}}_{5}$  ${\mathrm{\Omega}}_{6}$ 
SSSS  0°  50  22.157  36.232  53.594  78.249  105.594  138.710 
750  55.078  69.159  86.525  111.179  138.523  171.633  
30°  50  39.507  49.713  69.226  98.139  135.487  179.658  
750  72.428  82.640  99.1570  131.069  166.416  212.581  
45°  50  57.936  69.256  87.408  119.321  156.736  201.733  
750  90.857  102.183  120.339  152.251  194.665  234.656  
60°  50  112.564  127.407  148.221  177.583  204.928  238.044  
750  145.485  160.334  181.152  210.513  237.857  270.967  
CCCC  0°  50  28.446  46.511  68.806  100.457  135.565  178.079 
750  61.361  79.435  101.737  133.372  168.490  211.102  
30°  50  45.796  59.992  85.438  120.347  166.458  219.027  
750  78.711  92.916  114.369  153.262  196.383  252.050  
45°  50  64.225  79.535  102.620  141.529  191.707  241.102  
750  97.140  112.459  135.551  174.444  224.632  274.125  
60°  50  118.853  137.686  163.433  199.791  234.899  277.413  
750  151.768  170.610  196.364  232.706  267.824  310.436  
SSFF  0°  50  12.286  20.083  29.716  43.375  58.538  76.887 
750  45.206  53.013  62.639  76.305  91.465  109.813  
30°  50  31.636  36.564  48.348  66.265  90.431  117.835  
750  62.556  66.494  75.271  96.195  119.358  150.761  
45°  50  48.065  53.107  63.530  84.447  110.680  139.910  
750  80.985  86.037  96.453  117.377  143.607  172.836  
60°  50  102.693  111.258  124.343  140.709  157.872  176.221  
750  135.613  144.188  157.266  173.639  190.799  209.147  
CCFF  0°  50  19.603  32.054  47.423  69.228  93.425  122.724 
750  52.529  64.981  80.346  102.154  126.355  155.641  
30°  50  36.953  47.535  64.055  91.118  126.318  163.672  
750  69.879  78.462  92.978  122.044  154.248  196.589  
45°  50  55.382  65.078  81.237  110.300  149.567  185.747  
750  88.308  98.005  114.160  143.226  182.497  218.664  
60°  50  110.010  123.229  142.050  168.562  192.759  222.058  
750  142.936  156.156  174.973  201.488  225.689  254.975 
5.3. Deep NN performance
The performances of suggested deep NN are presented in Table 4 and Fig. 3, the MSE and accuracy of predicted data are calculated from Eq. 12 for NDFP $\left(\mathrm{\Omega}\right)$. From Table 4 and Fig. 3 the value of MSE and accuracy of training data are 7.2 E5 and 99.7 % respectively and validating data are 6.2 E5 and 99.8 % respectively. From NN performance shows in Table 4 and Fig. 3, the proposed deep NN gave a good prediction for vibration behavior data in the presented SCP.
Table 4. Mean square error (MSE) and accuracy values
Data  MSE  Accuracy 
Training  7.2 E5  99.7 % 
Validating  6.2 E5  99.8 % 
5.4. Deep NN predicting results
In this subsection, the main target of design the deep NN of predicting the vibration behavior data of SCP under different elastic restraint coefficients (${K}_{T}$) is achieved, chosen 7 different ${K}_{T}$ for different four BCs (SSSS, CCCC, SSFF, and CCFF). The deep NN predicted results of the first six frequencies of SCP with $\beta =$ 0.5 and $\mathrm{\Delta}=$ 0.5 are shows in Fig. 7.
Moreover, the influence of the IES on the vibration behavior of the SCPs with variable thickness is shown in Fig. 7. As shown in the Fig. 7 for all values of skew angle ($\varphi $) and all types of BCs, the first six frequencies are increasing with increasing of the value of elastic restraint coefficient (${K}_{T}$). whereas the frequencies rapidly increase with for small values of elastic restraint coefficient (${K}_{T}$), and the influence of IES becomes negligible at high values. On the other hand for all values of skew angle ($\varphi $), the first six frequencies for fully clamped (CCCC) plate are the highest frequencies, and the semisimply supported (SSFF) plate is the lowest one, while, the other two boundaries (SSSS and CCFF) were rested between them. also, we can see the effects of plate skew angles ($\varphi $) on the NDFP ($\mathrm{\Omega}$) it has been increased with increasing of the. skew angles.
Fig. 7. The deep NN predicted results of NDFP $\left(\mathrm{\Omega}\right)$
a) SSSS
b) CCCC
c) SSFF
d) CCFF
6. Conclusions
By a combination of the WT and FSTM method (WTFSTM) was used to extract the vibration behavior of SCP with variable thickness, and IES, the plate is made from BFRP laminated. First, To investigate from accuracy and reliability of the proposed technique, the convergence between the proposed study results with the results available in the literature has been checked, thus validating the accuracy and reliability of the proposed technique. Then, due to the proposed method's difficulty in terms of, a lot of calculations with a large number of iterations, these results may not be good choices for quick and accurate vibration behavior extracting. Thus, the new deep neural network (NN) is designed to learn and test these results carrying out by extracting vibration behavior features that reflect the important and essential information about the mode shapes in SCP. The influence of $\beta $, $\mathrm{\Delta}$, $\varphi $, and ${K}_{T}$ on the predicted NDFP $\left(\mathrm{\Omega}\right)$ of the plate, has been studied, with four different support conditions (SSSS, CCCC, SSFF, and CCFF).
Based on the WTFSTM and the deep NN predicted results, we conclude that the deep NN predicted results of NDFP ($\mathrm{\Omega}$) are in very good agreement with the proposed method results WTFSTM with an accuracy of training and validating data are 99.7 % and 99.8 % respectively.
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 Zhao Y., Noori M., Altabey W. A. Reaching law based sliding mode control for a frame structure under seismic load. Journal of Earthquake Engineering and Engineering Vibration, Vol. 20, Issue 1, 2021, p. 2537, https://doi.org/10.1007/s1180302120031. [Publisher]
 Narjabadifam P., Hoseinpour R., Noori M., Altabey W. Practical seismic resilience evaluation and crisis management planning through GISbased vulnerability assessment of buildings. Journal of Earthquake Engineering and Engineering Vibration, Vol. 20, Issue 2, 2021, p. 2537, https://doi.org/10.1007/s1180302120031. [Publisher]
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 Altabey W. A. Prediction of natural frequency of basalt fiber reinforced polymer (FRP) laminated variable thickness plates with intermediate elastic support using artificial neural networks (ANNs) method. Journal of Vibroengineering, Vol. 19, Issue 5, 2017, p. 36683678. [Publisher]
 Noori M., Haifegn W., Altabey W. A., Ahmad I. H. S. A modified wavelet energy rate based damage identification method for steel bridges. Scientia Iranica, International Journal of Science and Technology, Transactions on Mechanical Engineering (B), Vol. 25, Issue 6, 2018, p. 32103230. [Search CrossRef]
 Zhao Y., Noori M., Altabey W. A., Seyed B. B. Mode shape based damage identification for a reinforced concrete beam using wavelet coefficient differences and multiresolution analysis. Journal of Structural Control and Health Monitoring, Vol. 25, Issue 3, 2017, p. e2041. [Publisher]
 Zhao Y., Noori M., Altabey W. A. Damage detection for a beam under transient excitation via three different algorithms. Journal of Structural Engineering and Mechanics, Vol. 63, Issue 6, 2017, p. 803817. [Search CrossRef]
 Zhao Y., Noori M., Altabey W. A., Awad T. A Comparison of three different methods for the identification of hysterically degrading structures using BWBN model. Journal of Frontiers in Built Environment, 2019, Vol. 4, p. 80. [Publisher]
 Silik A., Noori M., Altabey W. A., Ghiasi R. Comparative analysis of wavelet transform for timefrequency analysis and transient localization in structural health monitoring. Journal of Structural Durability and Health Monitoring, 2020, (in press). [Publisher]
 Silik A., Noori M., Altabey W. A. Ghiasi R. Choosing optimum levels of wavelet multiresolution analysis for timevarying signals in structural health monitoring. Journal of Structural Control and Health Monitoring, 2020, (in press). [Publisher]
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 Miranda F. J. Wavelet analysis of lightning return stroke. Journal of Atmospheric and SolarTerrestrial Physics, Vol. 70, Issues 1112, 2008, p. 14011407. [Publisher]
 Kumar R., Ismail M., Zhao W., Noori M., Yadav A. R., Chen S., Singh V., Altabey W. A., Silik A. I. H., Kumar G., Kumar J., Balodi A. Damage detection of wind turbine system based on signal processing approach: a critical review. Journal of Clean Technologies and Environmental Policy, 2021, https://doi.org/10.1007/s1009802002003w. [Publisher]
 Wang T., Altabey W. A., Noori M., Ghiasi R. A deep learning based approach for response prediction of beamlike structures. Structural Durability and Health Monitoring, Vol. 14, Issue 4, 2020, p. 315338. [Publisher]
 Kost A., Altabey W. A., Noori M., Awad T. Applying neural networks for tire pressure monitoring systems. Structural Durability and Health Monitoring, Vol. 13, Issue 3, 2019, p. 247266. [Publisher]
 Zhao Y., Noori M., Altabey W. A., Ghiasi R., Wu Z. Deep learningbased damage, load and support identification for a composite pipeline by extracting modal macro strains from dynamic excitations. Journal of Applied Sciences, 2018, Vol. 8, Issue 12, p. 2564. [Publisher]
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Lecture Notes in Civil Engineering
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Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science
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Composite Structures
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Engineering Applications of Artificial Intelligence
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Sensors
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2022

Applied Sciences
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Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications
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Structural Health Monitoring
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Applied Sciences
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Materials
Wael A. Altabey, Mohammad Noori, Zhishen Wu, Mohamed A. AlMoghazy, Sallam A. Kouritem

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Composites Part C: Open Access
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2022

Wael A. Altabey, Sallam A. Kouritem, Mohammed I. Abouheaf, Nabil Nahas 
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Wael A. Altabey, Sallam A. Kouritem, Mohammed I. Abouheaf, Nabil Nahas 
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Energy Conversion and Management
Sallam A. Kouritem, Wael A. Altabey

2022

Weixing Hong, Ju Wang, Dangui Guo, Xiaodong Wang, Gongzuo Shi, Yanliang Xu, Chuanyue Zhang, Mohammad Noori, Wael A. Altabey 
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Applied Sciences
Wael A. Altabey, Mohammad Noori, Tianyu Wang, Ramin Ghiasi, SinChi Kuok, Zhishen Wu

2021

Earthquake Engineering and Engineering Vibration
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2021

Applied Sciences
Wael A. Altabey, Mohammad Noori, Zele Li, Ying Zhao, Seyed Bahram Beheshti Aval, Ehsan Noroozinejad Farsangi, Ramin Ghiasi, Ahmed Silik

2021

Structural Durability & Health Monitoring
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2021
