High performance estimations of natural frequency of basalt FRP laminated plates with intermediate elastic support using response surfaces method

Wael A. Altabey1

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

1Nanjing Zhixing Information Technology Co., Ltd., Andemen Street, Nanjing, Jiangsu, China

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

Journal of Vibroengineering, Vol. 20, Issue 2, 2018, p. 1099-1107. https://doi.org/10.21595/jve.2017.18456
Received 10 April 2017; received in revised form 12 August 2017; accepted 20 August 2017; published 31 March 2018

Copyright © 2018 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.

Studying the changes of the natural frequencies due to intermediate elastic support of laminated composites is usually need a lot of computational processes or difficult to estimate. The present study employs a new high performance method for natural frequency estimating in basalt fiber reinforced polymer (FRP) laminated, variable thickness plates with intermediate elastic support based on the finite strip transition matrix (FSTM) with response surfaces (RS) method. Author has found that the FSTM method is very effective. However, a large error of estimation remains for estimation of natural frequency due to the large number of an iteration implemented in FSTM algorithm to estimate the natural frequency. In the present study, a new data processing procedure is proposed to improve performance of estimations of natural frequency. The estimation responses for four of classical boundary conditions at the plate ends with different elastic restraint coefficients (KT) are computed to obtain the first six frequency parameters (Ω). As a result, the method reveals excellent performance of estimations of natural frequencies.

Keywords: response surfaces (RS) method, free vibration, finite strip transition matrix, variable thickness plate, basalt FRP.

1. Introduction

The classical and non-classical plates (i.e. Continuous and with intermediate stiffeners plates) are most common in a wide range of engineering fields such as aerospace, structures, vehicles and marine industries. In general, the approximate [1-10] or numerical [11-14] method must be used to compute the natural frequencies and find mode shapes for different boundary conditions for plate ends.

The author has investigated applicability of the FSTM method through a semi-analytical; natural frequency was estimated for basalt FRP laminated, variable thickness plates [15] and with intermediate elastic support. However, a large error of estimation remains due to a lot of computational processes must be performed for estimation of natural frequency. This is the main drawback of the FSTM method. Author also has already applied a predictive model for natural frequency in terms of fiber orientations is then developed using artificial neural networks (ANNs) to improve the estimations performance of FSTM by decreasing the effort to obtain frequency parameters in order to treatment the main drawback of the FSTM method [16]. The developed model via ANNs is trained and tested with a number of the FSTM results then used this predictive model for predicting non- FSTM data not included in FSTM evaluation. An ANNs model is shown to be very effective for predicting the natural frequency of basalt FRP laminated, variable thickness plates with intermediate elastic support.

In light of those difficulties, the present study uses RS method of various elastic restraint coefficients (KT) of basalt FRP laminated, variable thickness plates with intermediate elastic support (Fig. 1) to investigate effect of intermediate elastic support on the natural frequencies; furthermore, an improvement to obtain higher estimation performance is proposed for the FSTM method before making response surfaces. This new method is applied to estimate natural frequencies of plate type under four different ends of the plate; performance of the new estimation method is evaluated analytically herein.

Fig. 1. The geometrical model of Basalt FRP laminated variable thickness rectangular plate with intermediate elastic support

 The geometrical model of Basalt FRP laminated variable thickness rectangular plate  with intermediate elastic support

2. Governing equations

The formula that governing the dynamics of presented plate model in this study using the classical deformation theory in terms of the plate deflection δo(x,y,τ) after normalized to non-Dimensional variables ξ and η can be written as: [15, 17]:

(1)
ϕ 1 1 a 4 δ ξ ξ ξ ξ + 2 ϕ 2 t 3 ( η ) 1 a 2 b t 3 ( η ) η δ ξ ξ η + 2 ϕ 2 1 a 2 b 2 δ ξ ξ η η + ϕ 3 1 a 3 b δ ξ ξ ξ η + 4 ϕ 4 1 a b 3 δ ξ η η η
            + 1 a b 4 ϕ 4 t 3 ( η ) 2 t 3 ( η ) η 2 δ ξ η + 8 ϕ 4 t 3 ( η ) 1 a b 2 t 3 ( η ) η δ ξ η η + 1 b 2 1 t 3 ( η ) 2 t 3 ( η ) η 2 δ η η + 1 b 4 δ η η η η
            + 2 t 3 ( η ) 1 b 3 t 3 ( η ) η δ η η η = - m o D 22 t o 2 t 2 η δ τ τ ,

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

ξ = x a ,             η = y b ,             ϕ 1 = D 11 D 22 ,             ϕ 2 = ( D 12 + 2 D 66 ) D 22 ,             ϕ 3 = D 16 D 22 ,             ϕ 4 = D 26 D 22 ,
δ ξ ξ ξ ξ = 4 δ o ξ 4 ,             δ η η η η = 4 δ o η 4 ,             δ ξ ξ η η = 4 δ o ξ 2 η 2 ,             δ ξ ξ ξ η = 4 δ o ξ 3 η ,
δ η η η ξ = 4 δ o η 3 ξ ,             δ τ τ = 2 δ o τ 2 ,             m o = ρ t o ,

ρ is the material density, Dij is the flexural rigidities matrix are present by:

(2)
D i j = 1 3 t 3 ( y ) t o 3 k = 1 n ( q ¯ i j ) k ( t o k 3 - t o k - 1 3 ) ,           i , j = 1,2 , 3 , ,

where tok as shown in Fig. 1 is measuring between the middle-plane approbating to to to the layer bottom toth. And qijk¯ are the plane stress transformed reduced stiffness coefficients, they can be computed in terms of the longitudinal and transverse Young’s moduli E11, E22, plane shear modulus of elasticity G12 and Poisson coefficients υ12 and υ21 as:

(3)
q i j = q 11 q 12 q 13 q 12 q 22 q 23 q 13 q 23 q 66 = E 11 1 - υ 12 υ 21 υ 21 E 11 1 - υ 12 υ 21 0 υ 21 E 11 1 - υ 12 υ 21 E 22 1 - υ 21 υ 12 0 0 0 G 12 .

2.1. Boundary conditions

In this study, author applied new method to estimate natural frequencies of plate type under four different classical boundary conditions at the ends of the plate are SSSS, CCCC, SSFF and CCFF (see Fig. 2).

Since the treatment of the intermediate elastic line support conditions are the goal of the present study we can see in Fig. 1, the intermediate elastic line support, y=b/2 , the solution of the governing must be continuous at this line, i.e. [2]:

(4)
1 b δ o η η = 1 - / 2 = 1 b δ o η η = 1 + / 2 .

Fig. 2. Schematic representation of different plate ends

 Schematic representation of different plate ends

3. Finite strip transition matrix (FSTM) with response surfaces (RS) method

3.1. Finite strip transition matrix (FSTM) method

In this approach, the plate is divided into N space of strips between the ends of the plate as shown in Fig. 3. Each plate strips are considered as beam strip, and we can represent the displacement between individual strips by solve the shape function δ(ξ,η,τ), this shape function can be assumed after normalized to non-Dimensional variables ξ and η depending on the unknown function Yi(η) to be determined and basic function Xi(ξ) in ξ-direction, in the form:

(5)
δ ξ , η , τ = i = 0 N X i ξ Y i η e i ω τ .

3.2. Response surface (RS) method

The RS is more dependent tool for quality engineering areas [18, 19]. The methodology of the RS have two benefits, the approximately of the inverse problems can be solved without take in your account the engineering modeling, and the approximately results of RS can be estimated using powerful statistical tools. In this estimation work, the methodology of the RS was applied to predict the Ω from Eigen function. For widly applications of the RS, the functions for the approximately of the inverse problems are polynomials, in order to simplify. For the cases of quadratic polynomials, the RS is addressed as follow:

(6)
y = β 0 + j = 1 k β j x j + j = 1 k β j j x j 2 + i = 1 k - 1 j = i + 1 k β i j x i x j ,

where: k is the number of strips. In this case, there are N discrete longitudinal strips spanning (beam strips), y is the estimation results of the Ω by RS method and the coefficients β are obtained with the least square errors method [18, 19]. The estimations of the Ω by RS function in Eq. (6) is expressed as follow:

(7)
Ω = β 0 + j = 1 N β j S j + j = 1 N β j j S j 2 + i = 1 N - 1 j = i + 1 N β i j S i S j ,

where: Si, Sj are a standard eigenvalue can be obtained from the shape function W(ξ,η,t) by FSTM, i= 1, 2, 3,…, N, j= 1, 2, 3,…, M.

Fig. 3. FSTM model on plate

 FSTM model on plate

In the case that varies Ω, the total number of parameters is n, the RS can be computed as follows using matrix expression:

(8)
Y = X β + e ,

where:

Y = y 1 y 2 y n ,           X = 1 x 11 x 12 x 1 k 1 x 21 x 22 x 2 k 1 x n 1 x n 2 x n k ,           β = β 0 β 1 β k ,           e = e 1 e 2 e n ,

where: e is an error vector. The unbiased estimator b of the coefficient vector β is obtained using the well-known least square error method as follows:

(9)
b = X T X - 1 + X T Y .

The variance-covariance matrix of the b is obtained as follows:

(10)
C o v b i , b j = σ 2 X T X - 1 ,

where: the σ is the error of Y. The estimated value of σ is obtained as follows:

(11)
σ 2 = S S E n - k - 1 .

S S E is a square sum of errors, and expressed as follows:

(12)
S S E = Y T Y - b T X T Y .

In order to judge the goodness of the approximation of the RS, the adjusted coefficient of multiple determination Radj2 is used:

(13)
R a d j 2 = 1 - S S E n - k - 1 / S y y n - 1 ,

where: Syy is the total sum of squares:

(14)
S y y = Y T Y - i = 1 n y i 2 / n .

The value of Radj2 is equal to or lower than 1.0. A higher value of Radj2 implies a better fit. When the RS shows a very good fit, Radj2 approaches 1.0. A good fit of the RS means that the RS gives good estimations for the FSTM technique used for the regression. Lower Radj2 values means poorer estimations and the error band of the estimated result is wider.

4. Results and discussion

In this part, the present method is used to estimate the natural frequency of the plate was presented in Fig. 1. The intermediate elastic line support at one direction and we can be expressed as follow, KT=Tb/2b3/D22, T is translational stiffness. The plate model was designed of five symmetrically, angle-ply, laminates with the fiber angles θ/-θ/θ/-θ/θ is [45°/–45°/45º/–45°/45°] and has linear variable thickness, δy as form δ(η)=1+Δη, Δ=δb-δo/δo is the tapered ratio (see Fig. 1). The physical and mechanical properties of the basalt FRP laminate composite plate are shown in Table 1. The frequency parameter Ω is evaluated in non-dimensional form, expressed as: Ω=mot(η)ω2a4/toD221/2.

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
8.73 GPa
0.3
0.6
2700 kg/m3

4.1. The presented study verification

It is very important from the author to verify for the proposed method to decide whether his suggested method is suitable or not by compute the first six frequencies and compared with available results in literatures.

Table 2 shows a comparison between the first six frequency parameters from presented study and the corresponding results from literatures [5, 20-22]. literatures are used isotropic, square (β= 1.0), uniform thickness (Δ= 0) plates with a mid-line support in each direction, the physical and mechanical properties of the plate material are υ1=υ3=0.3,D11=D22=D=Et3/121-υ2,D66=1-υD/2. As results we can see a very close agreement is observed with a high value of adjusted coefficient Radj2.

Table 2. Comparison between the first six frequency parameters Ω=ρtω2a4/D1/2

R a d j 2
Ω 1
Ω 2
Ω 3
Ω 4
Ω 5
Ω 6
SSSS
78.957
94.590
94.590
108.240
197.392
197.392
Ref. [5]
1.0
78.957
94.590
94.590
108.240
197.392
197.392
Ref [20]
0.98
78.96
94.68
94.72
108.44
197.40
198.96
Ref [21]
0.99
78.958
94.826
94.826
108.41
197.50
197.50
Ref [22]
1.0
78.957
94.585
94.585
108.22
197.39
197.33
CCCC
108.229
127.417
127.417
144.109
242.818
242.778
Ref [5]
1.0
108.299
127.417
127.417
144.109
242.818
243.778

4.2. The proposed technique results

Fig. 4 presents the amount of changes of the natural frequencies due to intermediate elastic support of laminated, the plate has the aspect ratio β= 0.5 and tapered ratio Δ= 0.5. As shown in the figure, for each frequency of six frequencies, we can observe the increases in the frequency with the increasing of the value of elastic restraint coefficient (KT)

Fig. 5 presents changes of the vibration behavior due to variation in the types of the plate ends (SSSS, CCCC, SSFF and CCFF). As shown in the figure, we can see the variation of the frequencies values, where, the plate with ends of fully clamped (CCCC) and semi-simply supported (SSFF) have been recorded the higher and lower values of the frequencies respectively, and the other types of the plate ends (SSSS and CCFF) are located in intermediate.

Fig. 6 represents the estimation results of the first six frequencies by RS. The Radj2 of estimation results are 0.999, 0.996, 0.987 and 0.964 for SSSS, CCCC, SSFF and CCFF plate respectively.

Fig. 4. The relation between non-dimensional frequencies parameter (Ω) and elastic restraint coefficient (KT)

 The relation between non-dimensional frequencies parameter (Ω) and elastic  restraint coefficient (KT)

Fig. 5. The effect of the variation boundary conditions on non-dimensional frequencies parameter (Ω)

The effect of the variation  boundary conditions on non-dimensional  frequencies parameter (Ω)

All of the estimation results are plotted on the diagonal line (Fig. 6) to observe the performances of the present RS method to estimate non-dimensional frequencies (Ω). The error band is defined as the maximum error of the estimated Ω. The error band from the diagonal line is less than 0.01 for all types of boundary conditions. The results of the estimated Ω is presented in Table 3. As a result, the RS gave good estimations for FSTM to calculate Ω in composite plate.

Table 3. The first six frequencies of Basalt FRP rectangular plate with intermediate elastic line support

K T
Ω 1
Ω 2
Ω 3
Ω 4
Ω 5
Ω 6
R a d j 2
SSSS
50
22.2
36.2
53.6
78.2
105.6
138.7
0.999
150
34.7
48.7
66
90.8
118
151.3
0.999
400
45.9
59.9
77.3
102
129
162.4
0.999
750
55.1
69.2
86.6
111.2
138.5
171.6
1.0
1500
62.5
76.5
94
118.6
146
179
1.0
2500
67.5
81.6
99
124
151
184
1.0
5000
70.8
84.9
103
127
154
187
1.0
10000
72.7
86.8
104
129
156.2
189
1.0
1E+06
73.2
87.2
104.6
129
156.6
190
1.0
CCCC
50
28.5
46.5
68.9
100.5
135.6
178
0.988
150
41
59
81.4
113
148
190.6
0.992
400
52
70
92.5
124
160
201
0.995
750
61.4
79.4
101.7
133.4
168.5
212
0.998
1500
69
87
110
141
176
218.5
0.999
2500
74
92
114
146
181
223.6
0.999
5000
77
95
117.6
149
184.3
227
1.0
10000
79
97
119.4
151
186.2
229
1.0
1E+06
79.5
97.5
120
151.5
187
229.4
1.0
SSFF
50
12.3
20.1
29.7
43.4
58.5
77
0.967
150
25
32.6
43
56
71
89.5
0.973
400
36
43.8
53.5
67
82.5
100.7
0.979
750
45.2
53.1
62.6
76.3
91.4
109.8
0.987
1500
53
60.5
70
83.8
99
117
0.991
2500
58
65.7
75
89
104
122
0.994
5000
61
69
78.6
92
107.6
125.6
0.997
10000
63
71
80
94
109
127.7
0.999
1E+06
63.5
71.4
81
94.8
109.5
128
1.0
CCFF
50
19.6
32
47.4
69.2
93.4
122.7
0.899
150
32
44.8
60
82
106
135.4
0.926
400
43.4
55.7
71
93
117
146.8
0.94
750
52.5
65
80.4
102.2
126.4
155.7
0.958
1500
60
72.7
88
110
133.8
163
0.972
2500
65
77.9
93
114.8
139
168
0.989
5000
68.6
81
96
118
142
171.7
0.999
10000
70
83
98
120
144
173.5
1.0
1E+06
70.6
83.4
98.7
121
144.6
174
1.0

Fig. 6. The estimation results of non-dimensional frequency parameter Ω

 The estimation results of non-dimensional frequency parameter Ω

a) SSSS

 The estimation results of non-dimensional frequency parameter Ω

b) CCCC

 The estimation results of non-dimensional frequency parameter Ω

c) SSFF

 The estimation results of non-dimensional frequency parameter Ω

d) CCFF

5. Conclusions

In the present study, the FSTM with RS approach is applied. The method has successfully estimation of the natural frequency (Ω) in one step of computing using the FSTM output of standard eigenvalue, in order to eliminate the iteration that implemented in FSTM algorithm to compute the natural frequency and decrease the error of estimation. Based on the FSTM results and the results estimated by RS, the following conclusions are drawn for basalt FRP laminated variable thickness rectangular plates with intermediate elastic support.

1) The RS data are provided excellent estimations of the natural frequency of basalt FRP plates for each case of boundary conditions and different KT.

2) The increases in the frequency with the increasing of the value of elastic restraint coefficient (KT) is observed.

3) On other hand, the variation of the frequencies values, where, the plate with ends of fully clamped (CCCC) and semi-simply supported (SSFF) have been recorded the higher and lower values of the frequencies respectively, and the other types of the plate ends (SSSS and CCFF) are located in intermediate.

4) The RS technique could estimate Ω with high adjusted coefficient Radj2 for KT= 50 are 0.999, 0.988, 0.967 and 0.899 for SSSS, CCCC, SSFF and CCFF plate respectively, for KT= 1E+06 are 1.0 for each boundary conditions.

5) It is observed that the adjusted coefficient Radj2 increases with increasing values of KT of intermediate elastic support.

6) The boundary condition SSSS has the highest performance of estimate Ω, the boundary condition CCFF has the lowest performance of estimate Ω, while the other boundary conditions are laid in between, with descending order of CCCC and SSFF respectively.

7) As a result, the method reveals excellent performance of estimations of natural frequencies.

8) Finally, author have revealed that the response surfaces are better than artificial neural networks for this inverse problem.

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