Prediction of natural frequency of basalt fiber reinforced polymer (FRP) laminated variable thickness plates with intermediate elastic support using artificial neural networks (ANNs) method
Wael A. Altabey^{1}
^{1}International Institute for Urban Systems Engineering, Southeast University, Nanjing, 210096, China
^{1}Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, 21544, Egypt
Journal of Vibroengineering, Vol. 19, Issue 5, 2017, p. 36683678.
https://doi.org/10.21595/jve.2017.18209
Received 26 January 2017; received in revised form 2 May 2017; accepted 3 May 2017; published 15 August 2017
JVE Conferences
The paper is focused on the application of artificial neural networks (ANNs) in predicting the natural frequency of basalt fiber reinforced polymer (FRP) laminated, variable thickness plates. The author has found that the finite strip transition matrix (FSTM) approach is very effective to study the changes of plate natural frequencies due to intermediate elastic support (IES), but the method difficulty in terms of, a lot of calculations with large number of iterations is the main drawback of the method. For training and testing of the ANN model, a number of FSTM results for different classical boundary conditions (CBCs) with different values of elastic restraint coefficients (${K}_{T}$) for IES have been carried out to training and testing an ANN model. The ANN model has been developed using multilayer perceptron (MLP) Feedforward neural networks (FFNN). The adequacy of the developed model is verified by the regression coefficient (${R}^{2}$) and Mean Square error (MSE) It was found that the R2 and MSE values are 0.986 and 0.0134 for train and 0.9966 and 0.0122 for test data respectively. The results showed that, the training algorithm of FFNN was sufficient enough in predicting the natural frequency in basalt FRP laminated, variable thickness plates with IES. To judge the ability and efficiency of the developed ANN model, MSE has been used. The results predicted by ANN are in very good agreement with the FSTM results. Consequently, the ANN is show to be effective in predicting the natural frequency of laminated composite plates.
Keywords: artificial neural networks (ANNs), free vibration, finite strip transition matrix, variable thickness plate, basalt FRP.
1. Introduction
Continuous laminated plates and laminated composite plates with intermediate stiffeners are one of very common in composite structures that used in many engineering fields such as aerospace, civil and marine industries.
Natural frequencies of laminated composite structures are the classical tools for the intelligent diagnosis of composite defects (e.g. fiber breakage, matrix cracks, debonding and delamination). Vibration response has become an important tool in Structural health monitoring (SHM) in the last decades, the natural frequencies of composite structure are extracted and analyzed under each damage scenario as damage detection and localization technique based on the changes in vibration parameters. In the recent years, several approaches to find the natural frequencies and the mode shapes of laminated composite plates became a field that has attracted a lot of interest in the scientific community.
In order to find the natural frequencies and the mode shapes for different boundary conditions with IES, a numerical approach or an approximate method must be used. Several researchers are attracted to vibration of plates with IES. FSTM is one of a semianalytical methods are welcomed in the many literatures as an alternative to the exact solution. In our previous paper by Altabey [1], a semianalytical method, the FSTM approach has been used to investigate the free vibration of basalt FRP laminated variable thickness rectangular plates with IES. Since all of the coefficients in the FSTM method can be obtained, a lot of calculations with large number of iterations must be performed to obtain a frequency parameters, i.e. the time of solutions will be increased, and the large amount of data are required to study the changes of plate natural frequencies due to IES. This is the main drawback of the method identified so far.
In the present study, the laminated composite plate shown in Fig. 1 has been modeled and analyzed using FSTM method. The laminated composite plate was manufactured using five symmetrically, angleply, laminates with the fiber orientations [45º/45º/45º/45º/45º] of basalt fiber and a polymer resin matrix. A predictive model for natural frequency in terms of fiber orientations is then developed using artificial neural networks. The developed model is tested with the FSTM data which were never used for developing the model. The FSTM results show that R2 and MSE are 0.986 and 0.0134 for train and 0.9966 and 0.0122 for test data respectively. Hence ANN model predicted results are in very good agreement with the FSTM results. Consequently, ANN are shown to be very effective in predicting the natural frequency of laminated composite plates under Four different CBCs are used in the analysis with different KT for IES.
Fig. 1. The geometrical model of Basalt FRP laminated variable thickness rectangular plate with IES
2. Governing Equations
The normalized partial differential equation governing the vibration of symmetrically, angleply laminated, variable thickness, rectangular plates under the assumption of the classical deformation theory in terms of the plate deflection ${w}_{o}(x,y,t)$ using the nonDimensional variables $\xi $ and $\eta $ is given by [1, 2]:
where $\beta =a/b$ is the aspect ratio, also:
${W}_{\xi \xi \xi \xi}=\frac{{\partial}^{4}{w}_{o}}{\partial {\xi}^{4}},{W}_{\eta \eta \eta \eta}=\frac{{\partial}^{4}{w}_{o}}{\partial {\eta}^{4}},{W}_{\xi \xi \eta \eta}=\frac{{\partial}^{4}{w}_{o}}{\partial {\xi}^{2}\partial {\eta}^{2}},$
${W}_{\xi \xi \xi \eta}=\frac{{\partial}^{4}{w}_{o}}{\partial {\xi}^{3}\partial \eta},{W}_{\eta \eta \eta \xi}=\frac{{\partial}^{4}{w}_{o}}{\partial {\eta}^{3}\partial \xi},{W}_{\eta \eta}=\frac{{\partial}^{2}{w}_{o}}{\partial {t}^{2}},{m}_{o}=\rho {h}_{o}$
${D}_{ij}$ is the flexural rigidities matrix of the plate.
Since the treatment of IES conditions are the main objective of this paper we presented it in more details. At IES, $y=b/2$, the displacement must vanish and the moment must be continuous, i.e. [1]:
3. Finite strip transition matrix (FSTM) with artificial neural network (ANN) method
3.1. Finite strip transition matrix (FSTM) method
The method is made when such a shape function is not conveniently obtained in case of discussing the plate problems by series. The plate may be divided into N discrete longitudinal strips spanning between supports as shown in Fig. 2. By basic displacement interpolation functions may then be used to represent displacement field within and between individual strips.
For a plate striped in the $\xi $direction as shown in Fig. 2, the shape function $W(\xi ,\eta ,t)$ may be assumed in the form:
where: ${Y}_{i}\left(\eta \right)$ is unknown function to be determined and ${X}_{i}\left(\xi \right)$ is chosen a priori, the basic function in $\xi $direction.
Fig. 2. Finite strip simulation on plate
3.2. Artificial neural network (ANN) modeling
Artificial neural network (ANN) is an attractive inductive approach for modeling nonlinear and complex systems without explicit physical representation and thus provides an alternative approach for modeling hydrologic systems. Artificial neural network was first developed in the 1940s. Generally speaking, ANNs are information processing systems. In recent decades, considerable interest has been raised over their practical applications. Training of artificial neural network enables the system to capture the complex and nonlinear relationships that are not easily analyzed by using conventional methods such as linear and multiple regression methods and the network is built directly from experimental or numerical data by its selforganizing capabilities. Based on the different applications, various types of neural network with various algorithms have been employed to solve the different problems. In this work will be using the preferred ANN structures to predict the frequency parameter of presented numerical conditions.
3.2.1. ANN configuration
Since the ANN configuration has a great influence on the predictive quality, various arrangements have been considered in previous work. It is necessary to define a simple code to describe the ANN configuration, as follows:
where ${N}_{in}$ and ${N}_{out}$ are the element numbers of input and output parameters, respectively, and $e$ is the number of hidden layers. ${N}_{h1}$ and ${N}_{h2}$ are numbers of neurons in each hidden layer, respectively. For example, $\left\{7{\left[21\right]}_{1}1\right\}$ means a one hidden layer ANN with seven input and one output parameters, with the hidden layer containing 21 elements (neurons); $\left\{9{\left[\begin{array}{ll}15& 10\end{array}\right]}_{2}1\right\}$ denotes a nine input and one output ANN, with 15 and 10 neurons, respectively, in two hidden layers.
The powerful function of an ANN is due to the neurons within the hidden layers, as well as to the related interconnections. Networks are also sensitive to the number of neurons in their hidden layers. It is believed that an ANN can represent any reasonable relationship between input and output if the hidden layers have enough neurons. However, for the practical case, more hidden neurons bring more interconnections, which require, in turn, larger training datasets for learning the relationships. It is therefore always necessary to optimize the number of neurons of the ANN hidden layers, as demonstrated by Demuth and Beale [3].
3.2.2. Performance evaluation measures
It is very useful from the designer point of view to have a neural system aids to decide whether his suggested design is suitable or not by Compute the MSE from equation:
where: ${\left(\mathrm{\Omega}\right)}_{nn}$ is the predicted frequency parameter from ANN, $\mathrm{\Omega}$ is the target or computed values from FSTM method of frequency parameter, and $n$ is the number of FSTM computed data values.
Thus, the performance index will either have one global minimum, depending on the characteristics of the input vectors. Local minimum is the minimum of a function over a limited range of input values. Local minimum is an unavoidable when the ANN is fitted. So, a local minimum may be good or bad depending on how close the local minimum is to the global minimum and how low an MSE is required. In any case, the method applied to solve this problem and 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.
The estimation performances of frequency parameter $\left(\mathrm{\Omega}\right)$ is evaluated by the lack of fit with the regression coefficient of the multiple determination ${R}^{2}$; ${R}^{2}$ is defined as:
The value of ${R}^{2}$ is equal to or lower than 1.0. A higher value of ${R}^{2}$ implies a better fit. When the ANN shows a very good fit, ${R}^{2}$ approaches 1.0. A good fit of the ANN means that the ANN gives good estimations for the dielectric properties change used for the regression. Lower ${R}^{2}$ values means poorer estimations and the error band of the estimated result is wider.
3.3. Feedforward neural networks (FFNN)
In this work we will use the suggested feed forward neural network, FFNN, to predict natural frequency of basalt FRP composite plate, because it has minimum mean square error (MSE), in composite structure applications [4, 5]. FFNN in general consist of a layer of input neurons, a layer of output neurons and one or more layers of hidden neurons [6]. Neurons in each layer are interconnected fully to previous and next layer neurons with each interconnection have associated connection strength or weight. The activation function used in the hidden and output layers neurons is nonlinear, where as for the input layer no activation function is used since no computation is involved in that layer. Information flows from one layer to the other layer in a feedforward manner. Various functions are used to model the neuron activity such as liner transfer function ($purelin\left(n\right)$), TanSigmoid transfer function ($\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{g}\left(n\right)$) or Radial Basis (Gaussian) transfer functions ($radbas\left(n\right)$).
The training process is terminated either when the (MSE) between the observed data and the ANN outcomes for all elements in the training set has reached a prespecified threshold or after the completion of a prespecified number of learning epochs.
4. Results and discussion
In this work, The FSTM with ANNs techniques are used to predict the free vibration of the laminated composite plate shown in Fig. 1. The plate was manufactured using five symmetrically, angleply, laminates with the fiber orientations [$\theta $/$\theta $/$\theta $/$\theta $/$\theta $] is [45°/–45°/45°/–45°/45°] of basalt fiber and a polymer resin matrix. Physical and mechanical properties of the basalt FRP laminate composite plate are shown in Table 1.
Table 1. Physical and mechanical properties of the basalt FRP
${E}_{1}$

${E}_{2}={E}_{3}$

${G}_{1}={G}_{3}$

${G}_{2}$

${\upsilon}_{1}={\upsilon}_{3}$

${\upsilon}_{2}$

$\rho $

96.74 GPa

22.55 GPa

10.64 GPa

8.73GPa

0.3

0.6

2700 kg/m^{3}

The nondimensional frequency parameter $\mathrm{\Omega}$ are addressed in form [1]:
The plate has linear deformation in thickness $h\left(y\right)$ woth nondimensional form [1]:
where: $\mathrm{\Delta}$ is the tapered ratio of plate given by $\mathrm{\Delta}={(h}_{b}{h}_{o})/{h}_{o}$, (${h}_{o}$) is the thickness of the plate at $\eta =$0 and (${h}_{b}$) is the thickness of the plate at $\eta =$1.
4.1. Convergence study and accuracy
In this subsection, the author has carried out for convergening the proposed method, first six frequencies are calculated and compared with available results in literatures [1]. He compared the computational results of FSTM method from previous work [1] with values available from literatures [7, 8] for Eglass/ epoxy, square ($\beta =$1.0), uniform thickness ($\mathrm{\Delta}=$0) plates with elastic foundation support, the mechanical properties of the plate material are ${\upsilon}_{1}=$${\upsilon}_{2}=$ 0.23, $D=E{h}^{3}/\left[12\left(1{\upsilon}^{2}\right)\right]$, and ${D}_{66}=\left(1\upsilon \right)D/2$. Two different CBCs (SSSS and CCCC) are used in calculations, KT is taken equal to 500, 1390.2 for SSSS and CCCC respectively. In this study are addressed in form $\mathrm{\Omega}={\left(\rho h{\omega}^{2}{a}^{4}/D\right)}^{1/2}$. He found the results by FSTM method are in a very close agreement with results in References [7, 8] and the convergence speed can be achieved with terms of series solution ($N=$ 3 to 7).
4.2. Finite strip transition matrix (FSTM) results
In the present study, the numerical computations using FSTM approach is applied with an ANNs, which are combined to decrease detection effort to discern for different KT by minimizing the number of FSTM computations in order to keep save the time of calculations to a minimum. The method has successfully computations of the frequency parameter using only two different ${K}_{T}$, the first one is located at ${K}_{T}=$ 50 the second is ${K}_{T}=$ 750, respectively, as shown in Table 2.
Table 2 presents the first six frequencies of the laminated composite plate shown in Fig. 1. The plate has the parameters of aspect ratio ($\beta $) and thickness tapered ratio ($\Delta $) are 0.5. The Four different type of CBCs (SSSS, CCCC, SSFF and CCFF) and two different values of KT of IES are used in the calculations to study the changes of natural frequencies due to IES. The locations of the IES is at midline of the presented plate.
Table 2. The first six frequencies of laminated, plate shown in Fig. 1 for two different values of ${K}_{T}$, $(\mathrm{\Delta}=\text{0.5})$, $(\beta =\text{0.5})$
${K}_{T}$

${\mathrm{\Omega}}_{1}$

${\mathrm{\Omega}}_{2}$

${\mathrm{\Omega}}_{3}$

${\mathrm{\Omega}}_{4}$

${\mathrm{\Omega}}_{5}$

${\mathrm{\Omega}}_{6}$


SSSS

50

22.1450

36.2210

53.5870

78.2360

105.5870

138.6970

750

55.0692

69.1452

86.5112

111.1602

138.5112

171.6212


CCCC

50

28.4310

46.5025

68.7979

100.4437

135.5584

178.0668

750

61.3551

79.4267

101.7221

133.3678

168.4826

210.9910


SSFF

50

12.2750

20.0773

29.7033

43.3663

58.5270

76.8799

750

45.1992

53.0015

62.6275

76.2905

91.4511

109.8041


CCFF

50

19.5928

32.0466

47.4112

69.2194

93.4182

122.7123

750

52.5170

64.9707

80.3353

102.1436

126.3424

155.6365

4.3. A feedforward neural network (FFNN) design for predicting frequency parameters
A FFNN configuration in this case was $\left\{2{\left[\begin{array}{ll}5& 5\end{array}\right]}_{2}1\right\}$, with tansigmoid neurons for the first layer while the second layer has pure linear ones. FFNN is trained by measuring values of ${K}_{T}$, $\theta $ to predict $\mathrm{\Omega}$ for four different CBCs are SSSS, CCCC, SSFF and CCFF. Determination of the hidden layer, in addition to the number of nodes in the input and output layers, for providing the best training results, was the initial phase of the training procedure. The target for MSE to be reached at the end of the simulations was 0.001. Since the second step was largely a trialanderror process, and involved FFNNs with the number of hidden layer neurons more than five, it did not show any sizeable improvement in prediction accuracy. Thus, the number of neurons for the single hidden layer was selected as five neurons. Selection of the number of hidden layer neurons, with respect to the MSE term is shown in Fig. 3. In the first FFNN structure is applied for training the results data of FSTM. Fig. 4 shows the training performance of suggested FFNN.
Fig. 5 represents the comparison between the FSTM data and the feedforward neural network FFNN predicted data ($\mathrm{\Omega}$) for ${K}_{T}=$50 of four different CBCs are SSSS, CCCC, SSFF and CCFF. The results of this NN show much satisfactory predication quality for this case study. Fig. 6 shows the comparison between the FSTM data and the feed forward neural network FFNN expected (tested) data for ${K}_{T}=$750 of four different CBCs are SSSS, CCCC, SSFF and CCFF. From Fig. 6, it noted that the expected data from the suggested FFNN are applicable with the FSTM data.
Fig. 3. Plot of MSE terms corresponding to the number of hidden layer neurons, used for selecting the optimum number of hidden layer neurons
Fig. 4. Training performance of suggested FFNN$\mathrm{}$
Fig. 5. Comparison between the FSTM data and FFNN predicted data for $\mathrm{\Omega}$
Fig. 6. Comparison between the FSTM data and FFNN expected data for $\mathrm{\Omega}$
Fig. 7. The performances of the present FFNN to predict data of $\mathrm{\Omega}$
Fig. 8. The performances of the present FFNN to expect data of $\mathrm{\Omega}$
Table 3. Mean square error (MSE) and regression coefficient (${R}^{2}$) values
Data

MSE

${R}^{2}$

Predicted

0.0134

0.986

Expected

0.0122

0.9966

Table 3 and Figs. 7 and 8 represent the performances of the present FFNN by calculating the values of MSE and ${R}^{2}$ (see Eqs. (5, 6)) between FSTM data for both FFNN predicted and expected data for $\mathrm{\Omega}$. As shown in Fig. 7 the value of MSE and ${R}^{2}$ between the FSTM and FFNN predicted data are 0.0134 and 0.986 respectively. From Fig. 8 the value of MSE and ${R}^{2}$ between the FSTM and FFNN expected data are 0.0122 and 0.9966 respectively.
4.4. The use of present FFNN for predicting nonFSTM data of nondimensional frequency parameter
The main goal of the artificial neural network design is predicting non FSTM data. In this section we will use the suggested FFNN to predict some non FSTM data not included in FSTM evaluation. It is selected to use seven different ${K}_{T}$, for four types of CBCs (SSSS, CCCC, SSFF and CCFF). The previous parameters ${K}_{T}$, $\theta $ are the input vectors for artificial neural network, while the output is the signal vector is $\mathrm{\Omega}$.
Fig. 9. The FFNN predicted nonFSTM results of nondimensional frequency parameter $\mathrm{\Omega}$
a) SSSS
b) CCCC
c) SSFF
d) CCFF
Table 4. The predicted nonFSTM results of the first six frequencies by FFNN
${K}_{T}$

${\mathrm{\Omega}}_{1}$

${\mathrm{\Omega}}_{2}$

${\mathrm{\Omega}}_{3}$

${\mathrm{\Omega}}_{4}$

${\mathrm{\Omega}}_{5}$

${\mathrm{\Omega}}_{6}$


SSSS

150

Ref [1]

34.6580

48.7340

66.1000

90.7490

118.1000

151.2100

FFNN

34.8971

48.1343

66.6159

90.5891

118.8782

148.7286


400

Ref [1]

45.8453

59.9213

77.2873

101.9363

129.2873

162.3973


FFNN

45.8453

60.1268

77.2873

100.2624

129.2873

162.3973


1500

Ref [1]

62.4709

76.5469

93.9129

118.5619

145.9129

179.0229


FFNN

65.5428

76.7674

93.799

117.6289

147.1225

178.4941


2500

Ref [1]

67.5270

81.6030

98.9690

123.6180

150.9690

184.0790


FFNN

67.6715

80.0752

98.248

122.7383

152.2729

183.4709


5000

Ref [1]

70.7832

84.8592

102.2252

126.8742

154.2252

187.3352


FFNN

71.5277

83.8535

101.5647

125.5413

154.9449

186.8396


10000

Ref [1]

72.7065

86.7825

104.1485

128.7975

156.1485

189.2585


FFNN

72.8107

86.2942

105.2392

128.2564

156.1388

189.2703


1E+06

Ref [1]

73.1195

87.1955

104.5615

129.2105

156.5615

189.6715


FFNN

72.6947

87.0396

105.4209

128.5854

156.8348

189.6596


CCCC

150

Ref [1]

40.9440

59.0155

81.3109

112.9567

148.0714

190.5798

FFNN

41.0549

57.0992

80.3422

111.2997

148.2179

186.4952


400

Ref [1]

52.1313

70.2028

92.4983

124.1440

159.2587

201.7672


FFNN

55.4754

70.442

92.6468

123.0928

160.6131

201.1848


1500

Ref [1]

68.7569

86.8284

109.1239

140.7696

175.8843

218.3928


FFNN

69.7659

85.3865

109.1148

141.4497

179.6854

218.0006


2500

Ref [1]

73.8130

91.8845

114.1800

145.8257

180.9404

223.4489


FFNN

73.4808

92.2119

113.7449

143.3314

180.9021

222.5692


5000

Ref [1]

77.0692

95.1407

117.4361

149.0819

184.1966

226.7050


FFNN

77.8273

94.092

117.0825

147.6263

184.9599

226.1968


10000

Ref [1]

78.9925

97.0640

119.3594

151.0052

186.1199

228.6283


FFNN

79.3118

96.3864

119.7441

150.1071

187.0576

228.5796


1E+06

Ref [1]

79.4055

97.4770

119.7724

151.4182

186.5329

229.0413


FFNN

79.1422

97.2499

120.7791

150.5119

186.7872

229.1948


SSFF

150

Ref [1]

24.7880

32.5903

42.2163

55.8793

71.0400

89.3929

FFNN

24.4094

32.9047

41.7941

54.6268

71.2814

89.0171


400

Ref [1]

35.9753

43.7777

53.4036

67.0666

82.2273

100.5802


FFNN

35.9713

42.5469

52.8278

67.4484

83.9108

98.563


1500

Ref [1]

52.6009

60.4033

70.0293

83.6922

98.8529

117.2058


FFNN

52.904

60.2689

70.3874

83.7121

100.3213

119.6639


2500

Ref [1]

57.6570

65.4594

75.0853

88.7483

103.9090

122.2619


FFNN

57.9028

64.964

75.361

89.2589

105.6161

122.2164


5000

Ref [1]

60.9132

68.7155

78.3415

92.0045

107.1652

125.5181


FFNN

61.3157

69.5938

79.1805

91.7539

107.5848

125.4066


10000

Ref [1]

62.8365

70.6388

80.2648

93.9278

109.0885

127.4414


FFNN

63.0778

70.2994

80.2805

93.3565

109.3353

127.2809


1E+06

Ref [1]

63.2495

71.0518

80.6778

94.3408

109.5015

127.8544


FFNN

63.3054

70.5596

80.69

93.9744

110.0274

127.6158


CCFF

150

Ref [1]

32.1058

44.5596

59.9242

81.7324

105.9312

135.2253

FFNN

32.1827

43.2269

59.686

81.9174

108.2676

135.0946


400

Ref [1]

43.2931

55.7469

71.1115

92.9197

117.1185

146.4127


FFNN

45.562

57.8813

73.8431

93.8403

117.8631

145.3086


1500

Ref [1]

59.9187

72.3725

87.7371

109.5453

133.7442

163.0383


FFNN

59.6275

72.4773

88.4423

108.7844

134.0482

163.8665


2500

Ref [1]

64.9748

77.4286

92.7932

114.6014

138.8002

168.0944


FFNN

65.0645

76.7108

92.8809

114.0148

139.5554

167.6554


5000

Ref [1]

68.2310

80.6848

96.0494

117.8576

142.0564

171.3505


FFNN

68.3337

79.7057

96.0889

117.8545

143.8554

171.2413


10000

Ref [1]

70.1543

82.6081

97.9727

119.7809

143.9797

173.2738


FFNN

70.9017

82.7021

98.6556

119.3321

144.5419

173.0477


1E+06

Ref [1]

70.5673

83.0211

98.3857

120.1939

144.3927

173.6868


FFNN

70.6424

81.6968

98.1541

120.3744

146.7246

173.5624

Fig. 10. The performances of the present FFNN to predicted nonFSTM results of nondimensional frequency parameter $\mathrm{\Omega}$
a) SSSS
b) CCCC
c) SSFF
d) CCFF
Fig. 9 shows the FFNN predicted results of the first six frequencies of the laminated composite plate are presented in this study. The plate has $\beta =$0.5 and $\mathrm{\Delta}=$0.5 with four different types of CBCs.
The predicted nonFSTM results of the first six frequencies by FFNN and a comparison with the previous work results in literature of FSTM method by Altabey [1] are presented in Table 4. As a result, a FFNN gave good prediction for nonFSTM data even for extrapolations $\mathrm{\Omega}$ in presented laminated composite plate.
From Table 4 and Fig. 9, we can see, the first six frequencies increase with the increasing of the value of ${K}_{T}$ and we are observed that the gap between values of frequencies in the small ${K}_{T}$ and next values of ${K}_{T}$ are higher than the gap between values of frequencies in the large ${K}_{T}$, and the frequencies at high values of ${K}_{T}$ are almost constant, for all conditions (SSSS, CCCC, SSFF and CCFF). On the other hand, the fully clamped (CCCC) and semisimply supported (SSFF) condition have the higher and lower values of frequencies respectively and the other two conditions (SSSS) and (CCFF) are lie between them with an intermediate value. This conclusion was found in other work [1].
${R}^{2}$ of nonFSTM results are 0.9995, 0.9994, 0.9992 and 0.9993 for SSSS, CCCC, SSFF and CCFF plate respectively. All of the predicting results are plotted on the diagonal line (Fig. 10) to observe the performances of the present FFNN to predict nonFSTM of $\mathrm{\Omega}$. The MSE is defined of the predicted non FSTM. The MSE is of nonFSTM results are 0.006714, 0.0084, 0.007428 and 0.0086 for SSSS, CCCC, SSFF and CCFF plate respectively.
5. Conclusions
The natural frequency values were found by analyses which were done of basalt FRP laminated variable thickness rectangular plates with IES by FSTM method and ANNs algorithm, which are combined to decrease computational effort to discern natural frequency in basalt FRP laminated composite plates, in order to save the time of the FSTM computational data to a minimum with high accuracy and easy. Based on the FSTM results and the results predicted by artificial neural networks, the following conclusions are drawn for laminated composite material plates.
1) The FFNN model has been developed by considering the ${K}_{T}$ and ply angle ($\theta $) as the input for predicting $\mathrm{\Omega}$. The developed FFNN model could predict $\mathrm{\Omega}$ with the ${R}^{2}$ and MSE are 0.986 and 0.0134 for training data set and 0.9966 and 0.0122 for test data respectively.
2) The FFNN model could predict nonFSTM of $\mathrm{\Omega}$ with ${R}^{2}$ of nonFSTM results are 0.9995, 0.9994, 0.9992 and 0.9993 for SSSS, CCCC, SSFF and CCFF plate respectively. The MSE is defined of the predicted nonFSTM. The MSE is of nonFSTM results are 0.006714, 0.0084, 0.007428 and 0.0086 for SSSS, CCCC, SSFF and CCFF plate respectively.
3) The ANN predicted results are in very good agreement with the FSTM results.
References
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