Dynamic stiffness method for free vibration analysis of thin functionally graded rectangular plates

Manish Chauhan1 , Vinayak Ranjan2 , Prabhakar Sathujoda3

1, 2, 3Bennett University, Greater Noida, India

1Corresponding author

Vibroengineering PROCEDIA, Vol. 29, 2019, p. 76-81. https://doi.org/10.21595/vp.2019.21111
Received 22 October 2019; accepted 29 October 2019; published 28 November 2019

Copyright © 2019 Manish Chauhan, et al. 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.

In this present work, the dynamic stiffness method (DSM) is used to analyze the free vibration of a thin functionally graded rectangular plate. Classical plate theory (CPT) is used to develop the dynamic stiffness matrix of a functionally graded material (FGM) plate. For free vibration analysis, the natural frequencies of the functionally graded material plate are estimated by using DSM with Wittrick-Williams algorithm for different aspect ratios and different boundary conditions. The present research compared the DSM natural frequencies results with those available in the published literature.

Keywords: dynamic stiffness method, free vibration, functionally graded material, CPT.

1. Introduction

The concept of functionally graded materials was first time introduced by Yamanoushi et.al [1] in 1980 during the advancement of thermal resistance material for aerospace engineering applications. Functionally graded materials are known as a new class of composite materials, which is a mixture of ceramics and metal constituents. The ceramic constituents give high-temperature resistance, whereas metal constituents enhance the mechanical performance and decrease the failure possibility of the structure. Leissa [2] used the Ritz method to analyze free vibration behaviour of the rectangular isotropic plate under applied twenty-one possible boundary conditions. Bercin [3] analyze free vibration and mode shape of the orthotropic plate by using finite element method. Bercin and Langley [4] continued to this work to develop the dynamic stiffness matrix for vibration analysis of plate structures. Boscolo and Banerjee [5] used DSM for analysis of free transverse vibration of the rectangular isotropic plate by using classical plate theory and first-order shear deformation theory. Chauhan et al. [6] used classical plate theory to analyze the free vibration of isotropic plate for different boundaries by using DSM Shen and Yang [7] applied CPT to investigate free vibration behavior of initially stressed elastically founded functionally graded material (FGM) plates under impetuous lateral loading. Baferani et al. [8] used Navier and Levy type solution for the free vibration analysis of functionally graded plate under different boundary conditions by using CPT. Kumar et al. [9] used CPT to formulate the DSM with Wittrick-Williams algorithm to extarct the eigen value of the FGM plates.

In this paper, we have analyzed the free vibration behavior of functionally graded material plates by using dynamic stiffness method with Wittrick-Williams algorithm to extract the natural frequencies under different boundary conditions.

2. Governing differential equation of the functionally graded material plate

Fig. 1. shows a rectangular functionally graded plate of length a, width b and thickness h, where material properties vary along with the thickness as a power-law distribution [9] as given by Eq. (1):

(1)
V c z = z h + 1 2 k ,           V m z = 1 - V c z ,           - 0.5 h z 0.5 h ,

where Vc and Vm denotes the volume fractions of ceramics and metal constituents, k represent the power-law index that takes a positive real number in Eq. (1).

Fig. 1. Material geometry and coordinates system of the functionally graded plate

 Material geometry and coordinates  system of the functionally graded plate

Fig. 2. Boundary conditions for displacements and forces for a plate element

 Boundary conditions for displacements and forces for a plate element

The displacement components of thin rectangular functionally graded plate uox,y,z,vo(x,y,z) and wo(x,y,z) by using classical plate theory are given by Eq. (2):

(2)
u o x , y , z = u ' x , y - z - z 0 w ' x ,       v o ( x , y , z ) = v ' ( x , y ) - ( z - z 0 ) w ' y ,
w o x , y , z = w ' x , y ,

where u'(x,y), v'(x,y) and w'(x,y) are the mid-plate (i.e, z=0) displacement components.

Fig. 1. shows that the material properties are nonhomogeneous in the transverse direction, due to this the middle surface of the geometry has in-plane displacement, which cannot be neglected. Therefore, the middle surface of FGM plate geometry does not concur with the neutral surface. In this condition, the neutral surface must be changed to zn=z-z0, where z0 is the distance between mid-surface to the neutral surface of the plate as shown in Fig. 1.

Hamilton’s principle is used to drive the fourth-order differential equation for transverse deflection of a thin rectangular functionally graded plate under free vibration condition and is given by Eq. (3):

(3)
D e f f 4 w ' x 4 + 2 4 w ' x 2 y 2 + 4 w ' y 4 + ρ h 4 w ' t 4 = 0 .

The boundary conditions for Levy-type solution in Fig. 2., are given as:

(4)
V x : - D e f f 3 w ' x 3 + 2 - υ 3 w ' x y 2 δ w ' ,           M x x : - D e f f 2 w ' x 2 + υ 2 w ' y 2 δ y ,

where Deff=Eh3/12(1-υ2) is the effective bending stiffness, h plate thickness, E Young’s Modulus of Elasticity, υ Poisson’s ratio of the given material, Vx, Mxx, and y are the shear force, bending moment and rotation of the bending plate.

3. Formulation of dynamic stiffness

A levy type solution of Eq. (3) which satisfies the boundary condition of Eq. (4) can be expressed in the following form [8]:

(5)
w ' x , y , t = m = 1 W m x e i ω t sin m y ,         m = m π L ,           m = 1,2 , , ,

where ω is unknow natural frequency. By putting Eq. (5) into Eq. (3) we get Eq. (6):

(6)
d 4 W m d x 4 - 2 m 2 d 2 W m d x 2 + m 4 - ρ h ω 2 D e f f W m = 0 ,         m = 1,2 , , .

The two possible solutions of the ordinary differential Eq. (6) are obtained, depending on the nature of all roots. Here we show only one possible solution:

Case 1: m2ωI0Deff all roots are real (1m,-1m,2m,-2m):

(7)
1 m = m 2 + ω I 0 D e f f ,         2 m = m 2 - ω I 0 D e f f .

The solution is:

(8)
W m x = A m cosh ( 1 m x ) + B m sinh ( 1 m x ) + C m cosh ( 2 m x ) + D m sinh ( 2 m x ) .

The displacement w' in Eq. (8) and Eq. (5), shear force Vx, rotation y and the bending moment Mxx can be expressed in the following form using Eq. (4) as shown below:

(9)
ϕ y m x , y = ϕ y m x sin ( m y ) ,
(10)
V x m x , y = V x m x sin ( m y ) ,
(11)
M x x m x , y = M x x m x sin ( m y ) .

The displacements boundary conditions for the plate are:

(12)
x = 0 ,           W m = W 1 ,             ϕ y m = ϕ y 1 ,
x = b ,           W m = W 2 ,           ϕ y m = ϕ y 2 ,

similarly, the forces boundary conditions are:

(13)
x = 0 ,           V x m = - V 1 ,             M x x m = - M 1 ,
x = b ,           V x m = - V 2 ,             M x x m = M 2 .

The displacement boundary conditions are applied, i.e., putting Eq. (12) into Eqs. (8) and (9), the following matrix relationship is obtained:

(14)
W 1 ϕ y 1 W 2 ϕ y 1 = 1 0 1 0 0 - 1 m 0 - 2 m C h 1 S h 1 C 2 S 2 - 1 m S h 1 - 1 m C h 1 - 1 m S h 2 - 1 m C h 1 A m B m C m D m ,
(15)
δ = A C ,

where Ch1=cosh(imb), Sh1=sinh(imb), Ci=cos(imb), Si=sin(imb), (i= 1, 2).

The force boundary conditions are applied, i.e., putting Eq. (13) into Eqs. (10) and (11), the following matrix relationship is obtained:

(16)
V 1 M 1 V 2 M 2 = 0 R 1 0 R 2 L 1 0 L 1 0 - R 1 S h 1 - R 1 C h 1 - R 1 S 2 - R 1 C 2 - L 1 C h 1 - L 1 S h 1 - L 2 C h 1 - L 2 S 2 A m B m C m D m ,
(17)
F = R C ,

where Ri=Deff(im3-2im(2-ν)), Li=Deff(im2-2ν) with i= 1, 2. Using Eqs. (15) and (17), the dynamic stiffness matrix K for functionally graded (FG) plate can be formulated by eliminating the constant vector C to get Eq. (18):

(18)
F = K δ ,

where:

(19)
K = R A - 1 .

By using Eq. (19), the generalized dynamic stiffness matrix (K) as given by Eq. (20):

(20)
K = s v v   s v m f v v f v m s m m - f v m f m m S y m s v v   - s v m s m m ,

where six variable terms svv, svm, smm, fvv, fvm, fmm can be expressed in the following form [9].

Table 1. Non-dimensional natural frequencies (ϖ=ωa2 ρch/Dc ) for Functionally graded square plates with S-S-S-S and S-F-S-F boundary conditions using DSM method

S-S-S-S
m n
k =   0
k =   0.2
k =   0.5
k =   1
k =   2
k =   5
k =   10
1 1
19.7392
18.3137
16.7142
15.0610
13.6930
12.9831
12.5724
1 2
49.3480
45.7843
41.7855
37.6525
34.2326
32.4578
31.4311
2 1
49.3480
45.7843
41.7855
37.6525
34.2326
32.4578
31.4311
2 2
78.9568
73.2550
66.8568
60.2440
54.7722
51.9324
50.2898
1 3
98.6960
91.5687
83.5710
75.3050
68.4652
64.9156
62.8623
3 1
98.6960
91.5687
83.5710
75.3050
68.4652
64.9156
62.8623
2 3
128.3048
119.0393
108.6423
97.8965
89.0048
84.3902
81.7210
3 2
128.3048
119.0393
108.6423
97.8965
89.0048
84.3902
81.7210
4 1
167.7832
155.6668
142.0708
128.0186
116.3909
110.3565
106.8660
S-F-S-F
1 1
9.6313
8.9358
8.1553
7.34874
6.6812
6.33487
6.13450
2 1
16.1347
14.9696
13.6621
12.3108
11.1926
10.6123
10.2767
1 3
36.7256
34.0735
31.0975
28.0216
25.4765
24.1556
23.3916
2 1
38.9449
36.1325
32.9767
29.7149
27.0160
25.6153
24.8051
2 2
46.7381
43.3629
39.5756
35.6611
32.4221
30.7412
29.7688
2 3
70.7401
65.6316
59.8993
53.9746
49.0722
46.5280
45.0564
1 4
75.2833
69.8468
63.7463
57.4412
52.2239
49.5163
47.9501
3 1
87.9866
81.6327
74.5029
67.1338
61.0361
57.8717
56.0412
3 2
96.0405
89.1049
81.3224
73.2788
66.6231
63.1689
61.1709

4. Numerical results

The dynamic stiffness matrix is used to obtain natural frequencies of the functionally graded plate by applying the Wittrick-Williams algorithm [5]. The above procedure is used to formulate DSM and this procedure has been implemented in MATLAB program to compute the natural frequencies of the FGM plate for different boundary conditions with different power-law index (k) values as shown in Tables 1-3, where ρc and Dc are denotes the density, bending stiffness of the ceramic material. The letter m denotes the number of half-sine wave in x direction, whereas n represents the nth lowest frequency of a given value of m.

Table 2. Comparison of Non-dimensional natural frequencies (ϖ=ωa2 ρch/Dc ) with results reported in the available published literature of the functionally graded plate

S-S-S-S
S-C-S-C
Mode
a b
Source
k =   0
k = 0.5
k =   1
k =   2
k =   0
k =   0.5
k =   1
k =   2
1
1
DSM
19.7392
16.7142
15.0610
13.6930
28.9508
24.5141
22.0894
20.0831
Ref [11]
19.7398
16.7141
15.0609
13.6930
28.9468
24.5122
22.0840
20.0809
Ref [10]
19.7381
16.7127
15.0595
13.6917
28.9485
24.5122
22.0874
20.0809
Ref [8]
19.7281
16.6879
15.0357
13.6808
28.9478
24.4867
22.0743
20.0586
Ref [2]
19.7392
28.9508
0.5
DSM
12.3370
10.4463
9.4131
8.5581
13.6857
11.5884
10.4422
9.4937
Ref [8]
12.3259
10.4424
9.3849
8.5257
13.6808
11.5659
10.4093
9.484
Ref [2]
12.3370
13.6858
2
1
DSM
49.3480
41.7855
37.6525
34.2326
54.7430
46.3537
41.7689
37.9751
Ref [11]
49.3487
41.7852
37.6530
34.2334
54.7395
46.3525
41.7667
37.9740
Ref [10]
49.3486
41.7868
37.6446
34.2250
54.7328
46.3424
41.7600
37.9656
Ref [8]
49.3468
41.7894
37.6387
34.2020
54.7232
46.3297
41.7364
37.9362
Ref [2]
49.3480
54.7431
0.5
DSM
19.7392
16.7142
15.061
13.6930
23.6463
20.0225
18.0421
16.4034
Ref [8]
19.7281
16.7142
15.0610
13.6931
23.6463
19.9925
18.0098
16.3905
Ref [2]
19.7392
23.6463
3
1
DSM
78.9568
66.8568
60.2440
54.7721
94.5852
80.0902
72.1685
65.6136
Ref [11]
78.9559
66.8569
60.2428
54.7714
94.5854
80.0902
72.1687
65.6134
Ref [10]
78.9307
66.8351
60.2243
54.7546
94.5552
80.0633
72.1435
65.5882
Ref [8]
78.9125
66.8173
60.2088
54.6721
94.5430
80.0360
72.1382
65.5621
Ref [2]
78.9568
94.5853
0.5
DSM
32.0762
27.1605
24.4741
22.2512
38.6939
32.7641
29.5234
26.8419
Ref [8]
32.0541
27.1303
24.4536
22.2396
38.6932
32.7480
29.5096
26.8329
Ref [2]
32.0762
38.6939

Table 3. Comparison of Non-dimensional natural frequencies (ϖ=ωa2 ρch/Dc  of square FGM plate with published results in Chakraverty and Pradhan [12]

S-S-S-S
k =   0
k   =   0.5
k   =   1.0
m n
DSM
Ref [12]
%Err
DSM
Ref [12]
%Err
DSM
Ref [12]
%Err
1 1
19.7392
19.739
0.00
16.7142
17.337
3.726
15.0610
16.424
9.049
1 2
49.3480
49.349
0.00
41.7855
43.344
3.729
37.6525
41.061
9.052
2 1
49.3480
49.349
0.001
41.7855
43.344
3.729
37.6525
41.061
9.052
2 2
78.9568
79.401
0.450
66.8568
69.738
4.309
60.2440
66.065
9.662
1 3
98.6960
100.17
1.493
83.5710
87.983
5.279
75.3050
83.349
10.681
3 1
98.6960
100.19
1.513
83.5710
87.995
5.293
75.3050
83.360
10.681
S-F-S-F
1 1
9.6313
9.632
0.007
8.1553
8.460
3.736
7.34874
8.014
9.052
2 1
16.1347
16.135
0.001
13.6621
14.172
3.732
12.3108
13.425
9.050
1 3
36.7256
37.181
1.24
31.0975
32.656
5.011
28.0216
30.936
10.400
2 1
38.9449
38.972
0.069
32.9767
34.229
3.797
29.7149
32.427
9.127
2 2
46.7381
47.281
1.161
39.5756
41.527
4.930
35.6611
39.340
10.316
2 3
70.7401
72.053
1.855
59.8993
63.285
5.652
53.9746
59.952
11.074

From Table 1, we observed that with increase in k value, the natural frequencies decrease. This is because as the k value increase, the metal constituent in the FGM plate and the stiffness of the plate is reduced.

When we compared the natural frequency results of the FGM plates with those available in the published literature, we found that the reported natural frequencies values at k=0 in Tables 2-3 are nearly same with those available in the literature [2, 11, 12]. While increasing the k value from 0.5 to 1.0, the maximum error increases 5 % to 11 % as given by Chakravarty and Pradhan [12] in Table 3. The possible reasons for these reported results are discussed below.

Chakravarty and Pradhan [12] have considered mid-plane surface geometry instead of the neutral surface for solving the effective bending stiffness (Deff), which increases the percentage error. Due to this reason, we have observed that error is smaller for k= 0 and higher for k= 1.

5. Conclusions

The impetus of the present work is to formulate the dynamic stiffness matrix to estimate the natural frequencies of a thin rectangular functionally graded plate, where two different sides of the plate are simply supported. Classical plate theory is used to develop the dynamic stiffness matrix of a functionally graded material plate whereas the transcendental nature of dynamic stiffness matrix is solved by using Wittrick-Williams algorithm and this formulation has been employed into MATLAB to extract natural frequency of the FGM plate with the desired accuracy. The natural frequencies calculated by DSM are compared with those available in literature.

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