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 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.
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. 1Material geometry and coordinates system of the functionally graded plate
Fig. 2Boundary 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 1Nondimensional natural frequencies (ϖ=ωa2 ρch/Dc ) 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 2Comparison of Nondimensional natural frequencies (ϖ=ωa2 ρch/Dc ) 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 3Comparison of Nondimensional natural frequencies (ϖ=ωa2 ρch/Dc 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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