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. Altabey1

1International Institute for Urban Systems Engineering, Southeast University, Nanjing, 210096, China

1Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, 21544, Egypt

Journal of Vibroengineering, Vol. 19, Issue 5, 2017, p. 3668-3678. 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

Copyright © 2017 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

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 (KT) for IES have been carried out to training and testing an ANN model. The ANN model has been developed using multilayer perceptron (MLP) Feed-forward neural networks (FFNN). The adequacy of the developed model is verified by the regression coefficient (R2) 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, de-bonding 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 semi-analytical methods are welcomed in the many literatures as an alternative to the exact solution. In our previous paper by Altabey [1], a semi-analytical 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, angle-ply, 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

 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, angle-ply laminated, variable thickness, rectangular plates under the assumption of the classical deformation theory in terms of the plate deflection wo(x,y,t) using the non-Dimensional variables ξ and η is given by [1, 2]:

(1)
ψ 1 1 a 4 W ξ ξ ξ ξ + 2 ψ 2 h 3 ( η ) 1 a 2 b h 3 ( η ) η W ξ ξ η + 2 ψ 2 1 a 2 b 2 W ξ ξ η η + ψ 3 1 a 3 b W ξ ξ ξ η             + 4 ψ 4 1 a b 3 W ξ η η η + 1 a b 4 ψ 4 h 3 ( η ) 2 h 3 ( η ) η 2 W ξ η + 8 ψ 4 h 3 ( η ) 1 a b 2 h 3 ( η ) η W ξ η η             + 1 b 2 1 h 3 ( η ) 2 h 3 ( η ) η 2 W η η + 1 b 4 W η η η η + 2 h 3 ( η ) 1 b 3 h 3 ( η ) η W η η η = - m o D 22 h o 2 h 2 η W t t ,

where β=a/b is the aspect ratio, also:

ξ = x a ,         η = y b ,         ψ 1 = D 11 D 22 ,         ψ 2 = ( D 12 + 2 D 66 ) D 22 ,         ψ 3 = D 16 D 22 ,         ψ 4 = D 26 D 22 ,
W ξ ξ ξ ξ = 4 w o ξ 4 ,         W η η η η = 4 w o η 4 ,         W ξ ξ η η = 4 w o ξ 2 η 2 ,
W ξ ξ ξ η = 4 w o ξ 3 η ,         W η η η ξ = 4 w o η 3 ξ ,         W η η = 2 w o t 2 ,       m o = ρ h o

D i j 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]:

(2)
1 b w 0 η η = 1 - / 2 = 1 b w 0 η η = 1 + / 2

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 ξ-direction as shown in Fig. 2, the shape function W(ξ,η,t) may be assumed in the form:

(4)
W ξ , η , t = i = 0 N X i ξ Y i η e i ω t ,

where: Yi(η) is unknown function to be determined and Xi(ξ) is chosen a priori, the basic function in ξ-direction.

Fig. 2. Finite strip simulation on plate

 Finite strip simulation on plate

3.2. Artificial neural network (ANN) modeling

Artificial neural network (ANN) is an attractive inductive approach for modeling non-linear 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 non-linear 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 self-organizing 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:

(5)
N i n N h 1 N h 2 e N o u t ,

where Nin and Nout are the element numbers of input and output parameters, respectively, and e is the number of hidden layers. Nh1 and Nh2 are numbers of neurons in each hidden layer, respectively. For example, 72111 means a one hidden layer ANN with seven input and one output parameters, with the hidden layer containing 21 elements (neurons); 9151021 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:

(6)
M S E = Ω n n - Ω 2 n ,

where: Ωnn is the predicted frequency parameter from ANN, Ω 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 Ω is evaluated by the lack of fit with the regression coefficient of the multiple determination R2; R2 is defined as:

(7)
R 2 = 1 - n Ω n n - Ω 2 n Ω 2 .

The value of R2 is equal to or lower than 1.0. A higher value of R2 implies a better fit. When the ANN shows a very good fit, R2 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 R2 values means poorer estimations and the error band of the estimated result is wider.

3.3. Feed-forward 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 feed-forward manner. Various functions are used to model the neuron activity such as liner transfer function (purelin(n)), Tan-Sigmoid transfer function (tansig(n)) or Radial Basis (Gaussian) transfer functions (radbas(n)).

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 pre-specified threshold or after the completion of a pre-specified 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, angle-ply, laminates with the fiber orientations [θ/-θ/θ/-θ/θ] 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
υ 1 = υ 3
υ 2
ρ
96.74 GPa
22.55 GPa
10.64 GPa
8.73GPa
0.3
0.6
2700 kg/m3

The non-dimensional frequency parameter Ω are addressed in form [1]:

(8)
Ω = m o h ( η ) ω 2 a 4 h o D 22 .

The plate has linear deformation in thickness hy woth non-dimensional form [1]:

(9)
h η = 1 + Δ η ,

where: Δ is the tapered ratio of plate given by Δ=(hb-ho)/ho, (ho) is the thickness of the plate at η= 0 and (hb) is the thickness of the plate at η= 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 E-glass/ epoxy, square (β= 1.0), uniform thickness (Δ= 0) plates with elastic foundation support, the mechanical properties of the plate material are υ1=υ2= 0.23, D=Eh3/121-υ2, and D66=1-υD/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 Ω=ρhω2a4/D1/2. 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 KT, the first one is located at KT= 50 the second is KT= 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 (β) and thickness tapered ratio (Δ) 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 mid-line of the presented plate.

Table 2. The first six frequencies of laminated, plate shown in Fig. 1 for two different values of KT, (Δ=0.5), (β=0.5)

K T
Ω 1
Ω 2
Ω 3
Ω 4
Ω 5
Ω 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 feed-forward neural network (FFNN) design for predicting frequency parameters

A FFNN configuration in this case was 25521, with tan-sigmoid neurons for the first layer while the second layer has pure linear ones. FFNN is trained by measuring values of KT, θ to predict Ω 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 trial-and-error 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 feed-forward neural network FFNN predicted data (Ω) for KT= 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 KT= 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

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 

 Training performance of suggested FFNN 

Fig. 5. Comparison between the FSTM data and FFNN predicted data for Ω

 Comparison between the FSTM data and FFNN predicted data for Ω

Fig. 6. Comparison between the FSTM data and FFNN expected data for Ω

 Comparison between the FSTM data and FFNN expected data for Ω

Fig. 7. The performances of the present FFNN to predict data of Ω

 The performances of the present FFNN  to predict data of Ω

Fig. 8. The performances of the present FFNN to expect data of Ω

 The performances of the present FFNN  to expect data of Ω

Table 3. Mean square error (MSE) and regression coefficient (R2) 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 R2 (see Eqs. (5, 6)) between FSTM data for both FFNN predicted and expected data for Ω. As shown in Fig. 7 the value of MSE and R2 between the FSTM and FFNN predicted data are 0.0134 and 0.986 respectively. From Fig. 8 the value of MSE and R2 between the FSTM and FFNN expected data are 0.0122 and 0.9966 respectively.

4.4. The use of present FFNN for predicting non-FSTM data of non-dimensional frequency parameter Ω

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 KT, for four types of CBCs (SSSS, CCCC, SSFF and CCFF). The previous parameters KT, θ are the input vectors for artificial neural network, while the output is the signal vector is Ω.

Fig. 9. The FFNN predicted non-FSTM results of non-dimensional frequency parameter Ω

 The FFNN predicted non-FSTM results of non-dimensional frequency parameter Ω

a) SSSS

 The FFNN predicted non-FSTM results of non-dimensional frequency parameter Ω

b) CCCC

 The FFNN predicted non-FSTM results of non-dimensional frequency parameter Ω

c) SSFF

 The FFNN predicted non-FSTM results of non-dimensional frequency parameter Ω

d) CCFF

Table 4. The predicted non-FSTM results of the first six frequencies by FFNN

K T
Ω 1
Ω 2
Ω 3
Ω 4
Ω 5
Ω 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 non-FSTM results of non-dimensional frequency parameter Ω

 The performances of the present FFNN to predicted non-FSTM results  of non-dimensional frequency parameter Ω

a) SSSS

 The performances of the present FFNN to predicted non-FSTM results  of non-dimensional frequency parameter Ω

b) CCCC

 The performances of the present FFNN to predicted non-FSTM results  of non-dimensional frequency parameter Ω

c) SSFF

 The performances of the present FFNN to predicted non-FSTM results  of non-dimensional frequency parameter Ω

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 β= 0.5 and Δ= 0.5 with four different types of CBCs.

The predicted non-FSTM 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 non-FSTM data even for extrapolations Ω 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 KT and we are observed that the gap between values of frequencies in the small KT and next values of KT are higher than the gap between values of frequencies in the large KT, and the frequencies at high values of KT are almost constant, for all conditions (SSSS, CCCC, SSFF and CCFF). On the other hand, the fully clamped (CCCC) and semi-simply 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 non-FSTM 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 non-FSTM of Ω. The MSE is defined of the predicted non- FSTM. The MSE is of non-FSTM 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 KT and ply angle (θ) as the input for predicting Ω. The developed FFNN model could predict Ω with the R2 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 non-FSTM of Ω with R2 of non-FSTM 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 non-FSTM. The MSE is of non-FSTM 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

  1. Altabey W. A. Free vibration of basalt FRP laminated variable thickness plates with intermediate elastic support using finite strip transition matrix (FSTM) method. Journal of Vibroengineering, Vol. 19, Issue 4, 2017, (in Press). [Publisher]
  2. Chakraverty S. Vibration of Plates. CRC Press, Taylor and Francis Group, 2009. [Search CrossRef]
  3. Demuth H., Beale M. Neural Network Toolbox User’s Guide for use with MATLAB Version 4.0. The Math Works, Inc., 2000. [Search CrossRef]
  4. Al-Tabey W. A. The Fatigue Behavior of Woven-Roving Glass Fiber Reinforced Epoxy under Combined Bending Moment and Hydrostatic Pressure. Ph.D. Theses, Alexandria University, Egypt, 2015. [Search CrossRef]
  5. Altabey W. A. Fatigue life prediction for carbon fiber/epoxy laminate composites under spectrum loading using two different neural network architectures. International Journal of Sustainable Materials and Structural Systems, 2017, (in Press). [Search CrossRef]
  6. Skapura D. Building Neural Networks. ACM Press, Addison-Wesley Publishing Company, New York, 1996. [Search CrossRef]
  7. Zhou D., Cheung Y. K., Lo S. H., Au F. T. K. Three dimensional vibration analysis of rectangular thick plates on Pasternak foundation. International Journal of Numerical Methods in Engineering, Vol. 59, 2004, p. 1313-1334. [Search CrossRef]
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