Dynamic stiffness method for free vibration analysis of thin functionally graded rectangular plates
Manish Chauhan^{1} , Vinayak Ranjan^{2} , Prabhakar Sathujoda^{3}
^{1, 2, 3}Bennett University, Greater Noida, India
^{1}Corresponding author
Vibroengineering PROCEDIA, Vol. 29, 2019, p. 7681.
https://doi.org/10.21595/vp.2019.21111
Received 22 October 2019; accepted 29 October 2019; published 28 November 2019
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 WittrickWilliams 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 hightemperature 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 twentyone 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 firstorder 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 WittrickWilliams 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 WittrickWilliams 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 powerlaw distribution [9] as given by Eq. (1):
where ${V}_{c}$ and ${V}_{m}$ denotes the volume fractions of ceramics and metal constituents, $k$ represent the powerlaw index that takes a positive real number in Eq. (1).
Fig. 1. Material geometry and coordinates system of the functionally graded plate
Fig. 2. Boundary conditions for displacements and forces for a plate element
The displacement components of thin rectangular functionally graded plate ${u}_{o}\left(x,y,z\right)\text{,}$${v}_{o}(x,y,z)$ and ${w}_{o}(x,y,z)$ by using classical plate theory are given by Eq. (2):
${w}_{o}\left(x,y,z\right)={w}^{\text{'}}\left(x,y\right),$
where ${u}^{\text{'}}(x,y)$, ${v}^{\text{'}}(x,y)$ and ${w}^{\text{'}}(x,y)$ are the midplate (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 inplane 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 ${z}_{n}=z{z}_{0}$, where ${z}_{0}$ is the distance between midsurface to the neutral surface of the plate as shown in Fig. 1.
Hamilton’s principle is used to drive the fourthorder differential equation for transverse deflection of a thin rectangular functionally graded plate under free vibration condition and is given by Eq. (3):
The boundary conditions for Levytype solution in Fig. 2., are given as:
where ${D}_{eff}=E{h}^{3}/12(1{\upsilon}^{2})$ is the effective bending stiffness, $h$ plate thickness, $E$ Young’s Modulus of Elasticity, $\upsilon $ Poisson’s ratio of the given material, ${V}_{x}$, ${M}_{xx}$, and ${\varnothing}_{y}$ 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]:
where $\omega $ is unknow natural frequency. By putting Eq. (5) into Eq. (3) we get Eq. (6):
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: ${\propto}_{m}^{2}\ge \omega \sqrt{\frac{{I}_{0}}{{D}_{eff}}}\Rightarrow $ all roots are real (${\propto}_{1m},{\propto}_{1m},{\propto}_{2m},{\propto}_{2m}$):
The solution is:
The displacement ${w}^{\text{'}}$ in Eq. (8) and Eq. (5), shear force ${V}_{x}$, rotation ${\varnothing}_{y}$ and the bending moment ${M}_{xx}$ can be expressed in the following form using Eq. (4) as shown below:
The displacements boundary conditions for the plate are:
$x=b,{W}_{m}={W}_{2},{\varphi}_{ym}={\varphi}_{y2},$
similarly, the forces boundary conditions are:
$x=b,{V}_{xm}={V}_{2},{M}_{xxm}={M}_{2}.$
The displacement boundary conditions are applied, i.e., putting Eq. (12) into Eqs. (8) and (9), the following matrix relationship is obtained:
where ${C}_{h1}=\mathrm{cosh}{(\propto}_{im}b)$, ${S}_{h1}=\mathrm{sinh}{(\propto}_{im}b)$, ${C}_{i}=\mathrm{cos}{(\propto}_{im}b)$, ${S}_{i}=\mathrm{sin}{(\propto}_{im}b)$, ($i=$ 1, 2).
The force boundary conditions are applied, i.e., putting Eq. (13) into Eqs. (10) and (11), the following matrix relationship is obtained:
where ${R}_{i}={D}_{eff}({{\propto}_{im}}^{3}{\propto}^{2}{\propto}_{im}(2\nu \left)\right)$, ${L}_{i}={D}_{eff}({{\propto}_{im}}^{2}{\propto}^{2}\nu )$ 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):
where:
By using Eq. (19), the generalized dynamic stiffness matrix ($K$) as given by Eq. (20):
where six variable terms ${s}_{vv}$, ${s}_{vm}$, ${s}_{mm}$, ${f}_{vv}$, ${f}_{vm}$, ${f}_{mm}$ can be expressed in the following form [9].
Table 1. Nondimensional natural frequencies ($\varpi =\omega {a}^{2}\sqrt{{\rho}_{c}h/{D}_{c}})$ for Functionally graded square plates with SSSS and SFSF boundary conditions using DSM method
SSSS


$mn$

$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

SFSF


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 WittrickWilliams 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 powerlaw index ($k$) values as shown in Tables 13, where ${\rho}_{c}$ and ${D}_{c}$ are denotes the density, bending stiffness of the ceramic material. The letter m denotes the number of halfsine wave in $x$ direction, whereas $n$ represents the $n$th lowest frequency of a given value of $m$.
Table 2. Comparison of Nondimensional natural frequencies ($\varpi =\omega {a}^{2}\sqrt{{\rho}_{c}h/{D}_{c}}$) with results reported in the available published literature of the functionally graded plate
SSSS

SCSC


Mode

$\frac{a}{b}$

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 Nondimensional natural frequencies ($\varpi =\omega {a}^{2}\sqrt{{\rho}_{c}h/{D}_{c}}$ of square FGM plate with published results in Chakraverty and Pradhan [12]
SSSS

$k=$0

$k=$0.5

$k=$1.0


$mn$

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

SFSF


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 23 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 midplane surface geometry instead of the neutral surface for solving the effective bending stiffness (${D}_{eff})$, 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 WittrickWilliams 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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