Document Type : Research Article
Authors
Department of Mechanical Engineering, University College of Engineering(A), JNTUK, Kakinada, 533003, Andhra Pradesh, India
Abstract
Keywords
Main Subjects
Research Article
Nathi Venkatalakshmi , Kalidindi^{ }Krishnabhaskar ^{*} , Koppanati Meerasaheb
Department of Mechanical Engineering, University College of Engineering(A), JNTUK, Kakinada, 533003, Andhra Pradesh, India
ARTICLE INFO 

ABSTRACT 
Article history: Received: 20230820 Revised: 20240530 Accepted: 20240705 

In this paper, a coupled displacement field (CDF) method was proposed to examine the free vibration behavior of a functionally graded (FG) rectangular plate with simply supported (SSSS) and clamped (CCCC) boundary conditions. The composition of the functionally graded rectangular plate is ceramic on the top and metal on the bottom. According to the powerlaw exponent form, the rectangular plate material properties vary continuously in the thickness direction. The trial functions signifying the displacement constituents of the crosssections of the plate are stated in simple algebraic polynomial forms. The lateral displacement field is derived in terms of the total rotations with the help of coupling equations. By utilizing the energy formulation, the undetermined coefficients are obtained. The frequency parameters with various aspect ratios, thickness ratios, and powerlaw for all edges are simply supported and clamped boundary conditions are derived. To validate the numerical results, a comparison of frequency parameters is done with other literature. 



Keywords: Aspect ratio; Firstorder shear deformation theory; Frequency parameters; Power law; Thickness ratio. 


© 2025 The Author(s). Mechanics of Advanced Composite Structures published by Semnan University Press. This is an openaccess article under the CCBY 4.0 license. (https://creativecommons.org/licenses/by/4.0/) 
Several engineering disciplines like automobile, aerospace, mechanical, and nuclear fields use complex structures made of structural members like plates and beams. Plates can be thick or thin, depending on the purpose. When these plates are subjected to internal or external force, they may vibrate with large amplitudes. The design of a structural member using a rectangular plate must consider the free vibration behavior under various environmental conditions. A functionally graded plate composition can be a metal, ceramic, or polymer. The properties of these materials continuously vary in the direction of thickness from one surface to another. The FG plate behavior will be analyzed under different boundaries to reduce vibrations. The fundamental frequency parameters of the plate are to be analyzed to prevent any damage caused by vibrations.
The firstorder shear deformation theory (FSDT) is based on the displacement field, which uses shear correction factors to set the differences between the actual transverse shear stress distribution and those evaluated by using the FSDT kinematic relations. To find the frequencies of the FG rectangular plates, FSDT was used to analyze and derive the equations of motion [1]. Significant results on the behavior of the FG plate are found in the path of material gradient stiffness [2]. The vibration frequencies of the FG plate based on amplitude and volume fraction have significant effects [3]. The governing equations of the plates are derived analytically by using FSDT under consideration of transverse shear stresses and rotational inertial effects [4, 5]. By implementing Hamilton’s rule, fundamental governing equations are derived [6, 7]. The interpolation functions of higher order are utilized to separate spatial derivatives [8].
The RayleighRitz (RR) method and the CDF method were used for solving the Eigenvalue problem [9]. The RR method is used to develop admissible functions for the analysis of vibrations in thick plates with similar elastic edge constraints [10, 11]. The RR method is used to find frequencies based on Mindlin's theory [12]. The Mindilin theory is used for vibration analysis on plates that are rectangular and thick [13]. The characteristic functions are studied for isotropic rectangular thick plates [14]. The observation is done on governing equilibrium equations of forces and forcedisplacement relations that are reduced to three partial differential equations of motion with total deflection [15]. An elasticity solution of FG simply supported 3D plate is obtained based on transverse loading [16]. By eliminating the integration constants from the projections of the general boundary conditions, the stiffness matrix has been derived [17]. An investigation is done on the nonlinear forced vibrations of thin FG circular plates under classical clampedclamped boundary conditions [18]. The governing equations for the boundary conditions are derived by differential rules [19]. Based on the strain linear elasticity theory, 3D vibration solutions are derived for FG rectangular plates under various boundary conditions [20]. Young's modulus varies throughout the direction of thickness, where Poisson's ratio is assumed to be constant [21]. Based on relative displacement and rotational degrees of freedom, the mass and stiffness matrix are derived [22].
To meet the outcome of the corresponding Kirchhoff frequencies, plates with various thickness ratios have been considered [23]. The vibration attributes of FG plates are verified based on power law, aspect, and thickness ratios [24]. Based on the numerical method, the mixed boundary conditions of a plate for differential equations are obtained [2527]. Eigenfrequencies are obtained for a broad range of thicknesses and aspect proportions [28]. The ordinary differential equation is resolved from the Eigen differential equation [29]. The analysis is done on a functionally graded cantilever beam to perceive the behavior of deformation and variations in stress [30]. Without changing the shape parameters Meshfree method is used to analyze the vibration response of rectangular plates [31]. The effects of variations in the Poisson's ratio are studied [32].
In the CDF method, the fields for lateral displacement and total rotations are coupled through the static equilibrium equation [33]. The CDF method uses only one undetermined coefficient. In the CDF method, a singleterm admissible function is used in the principle of conservation of total energy. The admissible trial function was assumed, where the lateral displacement function is attained by using coupling equations [34, 35]. The axial, bending, and shear displacements of a thick clampedclamped functionally graded material under a uniform load are developed [36]. Due to the utilization of coupling equations, the transverse displacement distribution comprises the identical undetermined coefficient as existing in the rotation direction. Material properties vary continuously through thickness according to a power law distribution in terms of the volume fraction of the constituents [37, 38]. The RR method uses two undetermined coefficients, which are reduced to one determined coefficient in the CDF method, which significantly minimizes the complexity of vibrations. The effects of the powerlaw, aspect ratio, thicknesslength ratio, and various boundary conditions on the vibration characteristics of the FG rectangular plate are examined [3941]. Free vibration analysis of rectangular plates under various boundary conditions is done [42]. The results of a plate on the natural frequencies under clamped and simply supported conditions are observed [43].
The objective of the present work is to study the free vibration analysis of an FG plate subjected to simply supported and clamped boundary conditions using the CDF method. To satisfy the essential boundary conditions the trail functions that denote the displacement fields are expressed in simple algebraic polynomial forms. The results obtained under simply supported and clamped boundary conditions are compared with the frequencies obtained in 8, 23, 24, [2629], 32 and [4143] are found to be in good agreement.
FG plate length (a), breadth (b), and thickness (h) are displayed in Fig. 1.
Fig. 1. Geometry of a functionally graded rectangular plate.
The FG plate used is a combination of ceramic on the upper and metal on the lower, where the mechanical attributes differ continuously in axis z. Since the thickness property varies, the upper surface and lower surface are treated as ceramic and metal respectively. It is observed that the properties of the FG plate become pure ceramic at k = 0 and metallic at a very high equivalent of k.
The powerlaw function is written as
(1) 
where and are the attributes of ceramic and metal, h is thickness and k is the powerlaw exponent of the FG plate. Accordingly, E and M vary continuously along the z direction as shown below.
(2) 
The displacements , and are given by
(3) 
(4) 
(5) 
where are unknown functions that are to be resolved. indicates the displacements of the midplane and t denotes the time. and denotes rotations of the transverse normal about the y and x axis.
Axial strain and shear strain are
(6) 
(7) 
here, , and indicate normal strains whereas , and indicate shear strains. Strain and kinetic energies represented by and are
(8) 
(9) 
Using the above equations, The undetermined coefficients are derived by
By considering and , estimate the transverse displacement denoted by w along the x and y directions.
(10) 
(11) 
where

(12) 
(13) 
Transverse lateral displacement w is obtained by applying Eqs. (12) and (13) in Eqs. (10) and (11). After integration and evaluation of the constant, we get
(14) 
here, is the undetermined coefficient and , and are the admissible functions.
(15) 
here, , where indicates the number of polynomials. The boundaries are controlled by the exponent’s r, s, t, and u of the function which can be 0, 1, or 2. Here, 0 indicates free (F), 1 indicates simply supported (S) and 2 indicates clamped (C). Using Pascal’s triangle, parameters are given in Table 1.
Table 1. Ten parameters of [24]
i 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

1 
x 
y 
x^{2} 
xy 
y^{2} 
x^{3} 
x^{2}y 
xy^{3} 
y^{3} 
Using Eqs. (12), (13), and (14) in Eqs. (8) and (9) we get
(16) 

(17) 

(18) 
Reducing the lagranzian concerning c_{i }
_{ } (19) 
where
The governing equation is given by
(20) 
and indicate stiffness and inertia matrices and represent unknown coefficients in the column vector. where

The frequency parameters obtained by Eq. (21) are discussed in the next chapter.
The behavior of vibrations in an FG rectangular plate using CDF with respect to thickness ratio (h/a) is obtained. The FG plate Nondimensional frequency parameters may be expressed as
(21) 
The properties of the materials used in the FG plate differ, i.e., for aluminum = 70 GPa, = 2700 kg/m^{3 }and = 0.3 and for alumina = 380 GPa, = 3800 kg/m^{3 }and = 0.3 respectively.
Table 2. Frequency parameters for all edges of the SSSS FG plate with k = 0 and h/a = 0.001 using CDF.
a/b 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
0.2 
10.264 
10.264^{③} 
11.466 
11.449^{③} 
13.504 
13.495^{③} 
16.423 
16.433^{③} 
28.319 
28.514^{③} 
0.5 
12.337 
12.337^{③} 
19.74 
19.739^{ ③} 
32.423 
32.421^{③} 
41.945 
41.947^{③} 
49.654 
49.659^{③} 
12.337^{⑦} 
19.739^{⑦} 
32.076^{⑦} 
41.945^{⑦} 
49.348^{⑦} 

1 
19.739 
19.739^{①} 
49.34 
49.348^{①} 
49.345 
49.348^{①} 
79.386 
78.956^{①} 
100.17 
98.696^{①} 
19.739^{③} 
49.348^{③} 
49.348^{③} 
79.401^{③} 
100.17^{③} 

19.739^{⑤} 
49.349^{⑤} 
49.349^{⑤} 
78.9633^{⑤} 
98.719^{⑤} 

19.739^{⑥} 
49.347^{⑥} 
49.3475^{⑥} 
78.955^{⑥} 
98.694^{⑥} 

19.739^{⑧} 
49.348^{⑧} 
49.348^{⑧} 
78.956^{⑧} 
98.696^{⑧} 

19.739^{⑨} 
49.348^{⑨} 
49.348^{⑨} 
78.957^{⑨} 
99.304^{⑨} 

19.74^{⑩} 
49.35^{⑩} 
 
79.03^{⑩} 
99.25^{⑩} 

19.74^{⑪} 
49.35^{⑪} 
49.35^{⑪} 
78.96^{⑪} 
 

1.5 
32.076 
32.078^{②} 
61.684 
61.688^{②} 
98.698 
98.697^{②} 
111.48 
111.03^{②} 
129.06 
128.31^{②} 
32.076^{⑧} 
61.685^{⑧} 
98.696^{⑧} 
111.03^{⑧} 
128.30^{⑧} 

32.08^{⑩} 
61.71^{⑩} 
98.76^{⑩} 
111.57^{⑩} 
 

2 
49.353 
49.348^{③} 
78.942 
78.958^{③} 
129.69 
129.68^{③} 
167.77 
167.79^{③} 
198.63 
198.63^{③} 
49.348^{④} 
78.957^{④} 
128.30^{④} 
167.78^{④} 
197.39^{④} 

49.348^{⑦} 
78.956^{⑦} 
128.30^{⑦} 
167.78^{⑦} 
197.39^{⑦} 

2.5 
71.558 
71.556^{⑧} 
101.161 
101.16^{⑧} 
151.83 
150.51^{⑧} 
220.74 
219.59^{⑧} 
256.62 
256.60^{⑧} 
71.555^{⑨} 
101.16^{⑨} 
150.99^{⑨} 
222.91^{⑨} 
256.61^{⑨} 

71.55^{⑩} 
101.19^{⑩} 
150.95^{⑩} 
219.71^{⑩} 
 
^{①, ②, ③, ④, ⑤, ⑥, ⑦, ⑧, ⑨,⑩,⑪}^{ }parameters are captured from RR, Ref. Papers [8, 23, 24, 26, 27, 28, 29, 32, 41, 42, 43].
Table 3. Frequency parameters for all edges of the CCCC FG plate with k = 0 and h/a = 0.001 using CDF.
a/b 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
0.2 
22.633 
22.633^{③} 
23.443 
23.440^{③} 
24.625 
24.877^{③} 
26.752 
27.039^{③} 
62.046 
30.816^{③} 
0.5 
24.585 
24.579^{③} 
31.831 
31.829^{③} 
44.954 
44.819^{③} 
64.021 
63.598^{③} 
64.868 
63.986^{③} 
1 
35.996 
35.997^{②} 
73.385 
73.432^{②} 
73.42 
73.432^{②} 
108.28 
108.38^{②} 
132 
131.65^{②} 
35.989^{③} 
73.399^{③} 
73.399^{③} 
108.27^{③} 
131.89^{③} 

35.992^{⑧} 
73.413^{⑧} 
73.413^{⑧} 
108.27^{⑧} 
131.64^{⑧} 

35.985^{⑨} 
73.395^{⑨} 
73.395^{⑨} 
108.22^{⑨} 
131.78^{⑨} 

35.99^{⑩} 
73.41^{⑩} 
 
108.26^{⑩} 
131.66^{⑩} 

37.22^{⑪} 
76.24^{⑪} 
76.24^{⑪} 
113.4^{⑪} 
 

1.5 
60.813 
60.782^{②} 
93.72 
93.901^{②} 
148.8 
148.85^{②} 
149.86 
149.76^{②} 
179.63 
179.86^{②} 
60.762^{③} 
98.841^{③} 
148.78^{③} 
149.68^{③} 
179.57^{③} 

60.772^{⑧} 
93.860^{⑧} 
148.82^{⑧} 
149.74^{⑧} 
179.66^{⑧} 

60.762^{⑨} 
93.835^{⑨} 
148.78^{⑨} 
149.85^{⑨} 
179.57^{⑨} 

60.77^{⑩} 
93.87^{⑩} 
148.83^{⑩} 
149.88^{⑩} 
 

2 
98.28 
98.318^{③} 
127.4 
127.32^{③} 
179.32 
179.28^{③} 
255.19 
254.39^{③} 
255.78 
255.95^{③} 
2.5 
147.73 
147.8^{⑧} 
174.04 
173.85^{⑧} 
221.62 
221.54^{⑧} 
291.36 
291.89^{⑧} 
394.29 
384.71^{⑧} 
147.78^{⑩} 
173.84^{⑩} 
221.52^{⑩} 
291.87^{⑩} 
 
^{ ②, ③, ⑧, ⑨, ⑩, ⑪}^{ }parameters are captured from RR, Ref. Papers [23, 24, 32, 41, 42, 43].
Table 4. Frequency parameters for all edges of the SSSS FG plate with k = 1 and h/a = 0.001 using CDF.
a/b 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
0.2 
8.54 
8.5405^{③} 
9.5408 
9.5260^{③} 
11.236 
11.228^{③} 
13.675 
13.673^{③} 
23.563 
23.725^{③} 
0.5 
10.265 
10.265^{③} 
16.425 
16.424^{③} 
26.976 
26.976^{③} 
35.039 
34.902^{③} 
41.315 
41.319^{③} 
1 
16.423 
16.424^{③} 
41.065 
41.061^{③} 
41.066 
41.061^{③} 
66.061 
66.065^{③} 
83.353 
83.349^{③} 
1.5 
26.688 
 
51.3238 
 
82.124 
 
93.572 
 
149.085 
 
2 
41.061 
41.060^{③} 
65.692 
65.697^{③} 
107.93 
107.9^{③} 
139.61 
139.61^{③} 
165.249 
165.27^{③} 
41.059^{④} 
65.697^{④} 
107.93^{④} 
139.61^{④} 
 
165.27^{④} 

2.5 
59.54 
 
84.177 
 
126.37 
 
126.37 
 
213.53 
 
^{, ③, ④,}^{ }parameters are captured from RR, Ref. Papers [24, 26].
Table 5. Frequency parameters for all edges of the CCCC FG plate with k = 1 and h/a = 0.001 using CDF.
a/b 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
0.2 
18.829 
18.832③ 
19.506 
19.503③ 
20.489 
20.699③ 
22.258 
22.498③ 
51.626 
25.642③ 
0.5 
20.452 
20.451③ 
26.493 
26.484③ 
37.391 
37.292③ 
53.263 
52.916③ 
58.976 
53.239③ 
1 
29.959 
29.945③ 
61.049 
61.072③ 
61.049 
61.072③ 
90.084 
90.082③ 
100.82 
100.75 
1.5 
50.512 
 
78.11 
 
123.79 
 
124.77 
 
149.45 
 
2 
81.857 
81.805③ 
106.01 
105.93③ 
149.17 
149.17③ 
212.12 
211.67③ 
213.24 
212.96③ 
2.5 
123.17 
 
144.16 
 
184.16 
 
244.28 
 
327.99 
 
^{③ }parameters are captured from RR, Ref. Paper [24].
Table 6. Frequency parameters for all edges of the SSSS FG plate with k = 2 and h/a = 0.001 using CDF.
a/b 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
0.2 
8.1635 
8.1639③ 
9.164 
9.106③ 
10.733 
10.733③ 
17.111 
13.069③ 
22.736 
22.679③ 
0.5 
9.1823 
9.8125③ 
15.701 
15.7③ 
25.788 
25.787③ 
33.496 
33.363③ 
39.493 
39.497③ 
1 
15.701 
15.699③ 
39.247 
39.25③ 
39.25 
39.251③ 
63.149 
63.153③ 
79.675 
79.674③ 
1.5 
25.511 
 
49.071 
 
78.502 
 
89.463 
 
102.64 
 
2 
39.25 
39.249③ 
62.808 
62.8③ 
103.16 
103.15③ 
133.45 
133.45③ 
157.98 
157.99③ 
2.5 
56.918 
 
80.429 
 
120.77 
 
179.67 
 
204.09 
 
^{③ }parameters are captured from RR, Ref. Paper [24].
Table 7. Frequency parameters for all edges of the CCCC FG plate with k = 2 and h/a = 0.001 using CDF.
a/b 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
CDF 
Ref. value 
0.2 
18.001 
18.002^{③} 
18.642 
18.643^{③} 
19.71 
19.786^{③} 
21.231 
21.506^{③} 
49.315 
24.512^{③} 
0.5 
19.556 
19.549^{③} 
25.326 
25.316^{③} 
35.764 
35.648^{③} 
50.933 
50.583^{③} 
56.506 
50.893^{③} 
1 
28.635 
28.624^{ ③} 
58.373 
58.379^{③} 
58.402 
58.379^{③} 
86.108 
86.111^{③} 
104.96 
104.91^{③} 
1.5 
48.343 
 
74.67 
 
118.43 
 
119.21 
 
142.87 
 
2 
78.161 
78.199^{③} 
101.25 
101.26^{③} 
142.6 
142.59^{③} 
202.81 
202.33^{③} 
203.44 
203.57^{③} 
2.5 
117.44 
 
138.4 
 
176.31 
 
233.23 
 
313.84 
 
^{③ }parameters are captured from RR, Ref. Paper [24].
Table 8. Frequency parameters for all edges of the SSSS FG plate with k = 0 for using CDF.
h/a 
Aspect ratio(a/b) 

0.2 
0.4 
1/2 
2/3 
1 
1.5 
2 
2.5 
3 
5 

0.001 
10.263 
11.448 
12.336 
14.266 
19.738 
32.075 
49.353 
71.558 
98.705 
256.61 
0.01 
10.206 
11.433 
12.32 
14.247 
19.731 
32.067 
49.336 
71.539 
98.676 
256.56 
 
11.446^{⑥} 
12.33^{⑥} 
14.252^{⑥} 
19.732^{⑥} 
32.057^{⑥} 
49.304^{⑥} 
71.463^{⑥} 
 
 

0.02 
9.8526 
11.374 
11.689 
14.221 
19.709 
32.041 
49.305 
71.493 
98.615 
256.41 
0.03 
9.2671 
11.210 
11.688 
14.163 
19.638 
31.99 
49.242 
71.412 
98.508 
256.15 
0.04 
8.594 
10.903 
11.688 
14.047 
19.593 
31.791 
48.705 
71.277 
98.317 
255.72 
0.05 
7.7076 
10.518 
11.671 
13.841 
19.476 
31.792 
48..690 
70.847 
97.992 
255.12 
^{⑥ }parameters are captured from RR, Ref. Paper [28].
Table 9. Frequency parameters for all edges of the CCCC FG plate with k = 0 for using CDF.
h/a 
Aspect ratio(a/b) 

0.2 
0.4 
1/2 
2/3. 
1 
1.5 
2 
2.5 
3 
5 

0.001 
22.633 
23.648 
24.585 
27.006 
35.996 
60.813 
98.28 
147.73 
208.84 
568.16 
0.01 
22.181 
23.521 
24.496 
26.969 
35.956 
60.751 
98.307 
147.75 
208.78 
565.96 
0.02 
20.890 
23.155 
24.241 
26.804 
35.843 
60.670 
98.258 
147.72 
208.76 
566.12 
0.03 
19.091 
22.555 
23.813 
26.504 
35.663 
60.545 
98.182 
147.61 
208.73 
566.55 
0.04 
17.160 
21.743 
23.237 
26.107 
35.388 
60.279 
97.884 
146.18 
208.59 
566.92 
0.05 
15.337 
20.809 
22.535 
25.609 
35.025 
60.065 
97.706 
145.45 
207.98 
566.90 
Table 10. Frequency parameters for all edges of the SSSS FG plate with k = 1 using CDF.
h/a 
Aspect ratio(a/b) 

0.2 
0.4 
1/2 
2/3 
1 
1.5 
2 
2.5 
3 
5 

0.001 
8.54 
9.5259 
10.264 
11.862 
16.423 
26.688 
41.061 
59.545 
82.113 
213.47 
0.01 
8.4918 
9.513 
10.256 
11.854 
16.417 
26.681 
41.050 
59.524 
82.103 
213.47 
0.02 
8.1979 
9.4640 
9.7259 
11.832 
16.399 
26.660 
41.025 
59.485 
82.052 
213.34 
0.03 
7.7107 
9.3278 
10.142 
11.784 
16.340 
26.623 
40.971 
59.418 
81.964 
213.13 
0.04 
7.1506 
9.0723 
9.9741 
11.688 
16.302 
26.452 
36.364 
59.306 
81.805 
212.77 
0.05 
6.1131 
8.7518 
9.7114 
11.516 
16.205 
26.452 
16.205 
58.948 
81.534 
212.27 
Table 11. Frequency parameters for all edges of the CCCC FG plate with k = 1 using CDF.
h/a 
Aspect ratio(a/b) 

0.2 
0.4 
1/2 
2/3. 
1 
1.5 
2 
2.5 
3 
5 

0.001 
18.829 
19.664 
20.452 
22.478 
29.959 
60.511 
81.856 
123.16 
173.92 
471.93 
0.01 
18.455 
19.571 
20.382 
22.431 
29.917 
50.548 
81.795 
122.95 
173.72 
470.89 
0.02 
16.816 
19.266 
20.169 
22.302 
29.823 
50.480 
81.756 
122.91 
173.70 
471.84 
0.03 
15.885 
18.767 
19.814 
22.053 
29.674 
50.377 
81.692 
122.82 
173.67 
471.80 
0.04 
14.278 
18.091 
19.334 
21.723 
29.445 
50.155 
81.444 
123.29 
173.55 
471.70 
0.05 
12.761 
17.314 
18.750 
21.308 
29.142 
49.977 
81.297 
123.52 
173.05 
471.69 
Table 12. Frequency parameters for all edges of the SSSS FG plate with k = 2 using CDF.
h/a 
Aspect ratio(a/b) 

0.2 
0.4 
1/2 
2/3 
1 
1.5 
2 
2.5 
3 
5 

0.001 
8.1635 
9.1058 
9.1823 
11.339 
15.700 
25.511 
39.25 
56.918 
78.494 
204.09 
0.01 
8.1098 
9.0924 
9.8025 
11.331 
15.693 
25.507 
39.24 
56.898 
78.481 
204.05 
0.02 
7.7901 
9.037 
9.7650 
11.309 
15.674 
23.507 
32.24 
50.898 
78.481 
203.05 
0.03 
7.2882 
8.8843 
9.6739 
11.254 
11.254 
15.621 
56.788 
56.788 
78.333 
203.71 
0.04 
7.1507 
9.0723 
9.9741 
11.688 
16.302 
26.452 
36.364 
59.306 
81.805 
212.77 
0.05 
5.8457 
8.2744 
9.2028 
10.947 
15.448 
25.256 
38.915 
56.470 
77.918 
202.57 
Table 13. Frequency parameters for all edges of the CCCC FG plate with k = 2 using CDF.
h/a 
Aspect ratio(a/b) 

0.2 
0.4 
1/2 
2/3. 
1 
1.5 
2 
2.5 
3 
5 

0.001 
18.001 
18.809 
19.555 
21.487 
28.635 
48.343 
78.260 
117.44 
165.80 
458.66 
0.01 
17.600 
18.684 
19.475 
21.434 
28.595 
48.312 
78.194 
117.37 
166.08 
450.63 
0.02 
16.468 
18.369 
19.251 
21.287 
28.521 
48.228 
78.139 
117.36 
166.07 
450.36 
0.03 
14.928 
17.451 
18.875 
21.036 
28.341 
48.109 
77.781 
117.34 
166.06 
450.02 
0.04 
13.311 
17.138 
18.367 
20.695 
28.117 
47.910 
77.614 
117.32 
166.04 
450.01 
0.05 
11.817 
16.332 
17.748 
20.248 
27.767 
20.431 
77.521 
117.30 
165.03 
450.00 


(a) 
(b) 

Fig. 2. Effect of aspect ratio on frequency parameters (The first five frequencies) of functionally graded simplysupported plate with k = 0.2 and h/a with (a) 0.01 (b) 0.02


(a) 
(b) 

Fig. 3. Effect of aspect ratio on frequency parameters (The first five frequencies) of the functionally graded clamped plate with k = 0.2 and h/a with (a) 0.01 (b) 0.02
The vibration behavior of functionally graded plates was evaluated with different thickness ratios (h/a), aspect ratios (a/b), and power law index (k) subjected to different boundary conditions. The fundamental frequencies for all edges are simply supported and clamped with a thickness ratio of h/a = 0.001, different aspect ratios and power law index are presented in Tables 27. The results obtained in the present method are compared with the RR method [24] and it was observed that they are accurate with a maximum variation of 0.05%, which shows the efficacy of the proposed method.
The fundamental frequencies for all edges simply supported and clamped with different thickness ratios, aspect ratios, and power law indexes are presented in Tables 813. It is observed that the fundamental frequency parameters decrease with an increase in the plate thickness ratio and frequencies increase with an increase in the aspect ratio. Fundamental frequencies are decreasing with an increase in powerlaw for a fixed aspect ratio, irrespective of boundary conditions.
The effect of aspect ratios (a/b) on frequency parameters (The first five frequencies) of a simply supported functionally graded plate and a clamped functionally graded plate is plotted in Figs. 2 and 3, respectively with k = 0.2 and different h/a. It is observed that the frequency parameters increase with the increase in aspect ratio.
The vibration characteristics are investigated for an FG rectangular plate subjected to all edges SSSS and CCCC boundary conditions using the CDF method. The energy formulations in the CDF method contain half the number of undetermined coefficients when compared with the RR method. To inspect the vibration characteristics of the FG rectangular plate, various aspect ratios, thickness ratios, and powerlaw indexes are utilized. It is observed that the frequency parameters are decreasing with increasing k and increasing with increasing aspect ratios. The numerical results acquired from the present work are validated with other literature and are found to be similar.
Other shear deformation plate theories can be easily handled in the above analysis to compare the results obtained from FSDT. Further, the CDF method can be extended to study the free vibration behavior of isotropic shells, cylindrical panels, laminate composite plates, and nonlinear dynamic responses of the structures.
Nomenclature
a 
Dimension of the plate in x direction 
b 
Dimension of the plate in y direction 
h 
Thickness of the plate 
k 
Material variation profile 
E_{m} 
Metal Young’s modulus 
E_{c } 
Ceramic Young’s modulus 
G 
Shear modulus at functionally graded material 
M_{c } 
Density of Ceramic 
M_{m} 
Density of Metal 

yaxis rotation 

xaxis rotation 
w 
Transverse displacement 
a/b 
Aspect ratio 
h/a 
Thickness ratio 
k 
Shear correction factor (= 5/6) 
U 
Strain energy 
T 
Kinetic energy 

Poisson’s ratio 
The authors thank the officials of our university for their encouragement in producing this paper and assure you that the authors do not have any affiliations with other organizations or any financial competing interests in the content discussed in this paper.
This research did not receive any specific grant from funding agencies in the public, commercial, or notforprofit sectors.
The author declares that there is no conflict of interest regarding the publication of this article.
[1] HosseiniHashemi, S., Taher, H.R.D., Akhavan, H. and Omidi, M., 2010. Free vibration of functionally graded rectangular plates using firstorder shear deformation plate theory. Applied Mathematical Modelling, 34(5), pp.12761291.
[2] Amirpour, M., Das, R. and Flores, E.I.S., 2017. Bending analysis of thin functionally graded plate under inplane stiffness variations. Applied Mathematical Modelling, 44, pp.481496.
[3] Amini, M.H., Soleimani, M., Altafi, A. and Rastgoo, A., 2013. Effects of geometric nonlinearity on free and forced vibration analysis of moderately thick annular functionally graded plate. Mechanics of Advanced Materials and Structures, 20(9), pp.709720.
[4] Yousefzadeh, S., Jafari, A. and Mohammadzadeh, A., 2019. Hydroelastic vibration analysis of functionally graded circular plate in contact with bounded fluid by Ritz method. Journal of Science and Technology of Composites, 5(4), pp.529538.
[5] Yousefzadeh, S., Akbari, A. and Najafi, M., 2019. Dynamic response of FG rectangular plate in contact with stationary fluid under moving load. Journal of Science and Technology of Composites, 6(2), pp.213224.
[6] Akavci, S.S. and Tanrikulu, A.H., 2015. Static and free vibration analysis of functionally graded plates based on a new quasi3D and 2D shear deformation theories. Composites Part B: Engineering, 83, pp.203215.
[7] Rahmani, B. and Naghmehsanj, M.R., 2015. Robust vibration control of a functionally graded beam with a variable crosssection. Journal of Science and Technology of Composites, 2(2), pp.1729.
[8] Eftekhari, S.A. and Jafari, A.A., 2012. Highaccuracy mixed finite elementRitz formulation for free vibration analysis of plates with general boundary conditions. Applied Mathematics and Computation, 219(3), pp.13121344.
[9] Leissa, A.W., 2005. The historical bases of the Rayleigh and Ritz methods. Journal of Sound and Vibration, 287(45), pp.961978.
[10] Cheung, Y.K. and Zhou, D., 2000. Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions. Computers & Structures, 78(6), pp.757768.
[11] Kumar, Y., 2022. Effect of Elastically Restrained Edges on Free Transverse Vibration of Functionally Graded Porous Rectangular Plate. Mechanics of Advanced Composite Structures, 9(2), pp.335348.
[12] Dawe, D.J. and Roufaeil, O.L., 1980. RayleighRitz vibration analysis of Mindlin plates. Journal of Sound and Vibration, 69(3), pp.345359.
[13] Sadrnejad, S.A., Daryan, A.S. and Ziaei, M., 2009. Vibration equations of thick rectangular plates using mindlin plate theory. Journal of Computer Science, 5(11), p.838.
[14] Lee, J.M. and Kim, K.C., 1995. Vibration analysis of rectangular Isotropic thick plates using Mindlin plate characteristic functions. Journal of Sound and Vibration, 187(5), pp.865867.
[15] Senjanović, I., Tomić, M., Vladimir, N. and Cho, D.S., 2013. Analytical solution for free vibrations of a moderately thick rectangular plate. Mathematical Problems in Engineering, p.207460.
[16] Kashtalyan, M., 2004. Threedimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of MechanicsA/Solids, 23(5), pp.853864.
[17] Kolarevic, N., NefovskaDanilovic, M. and Petronijevic, M., 2015. Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies. Journal of Sound and Vibration, 359, pp.84106.
[18] Ghaheri, A. and Nosier, A., 2015. Nonlinear forced vibrations of thin circular functionally graded plates. Journal of Science and Technology of Composites, 1(2), pp.110.
[19] Torabi, K., Afshari, H. and HeidariRarani, M., 2014. Free vibration analysis of a rotating nonuniform blade with multiple open cracks using DQEM. Universal Journal of mechanical Engineering, 2(3), pp.101111.
[20] Uymaz, B. and Aydogdu, M., 2007. Threedimensional vibration analyses of functionally graded plates under various boundary conditions. Journal of Reinforced Plastics and Composites, 26(18), pp.18471863.
[21] Chi, S.H. and Chung, Y.L., 2006. Mechanical behavior of functionally graded material plates under transverse load—Part I: Analysis. International Journal of Solids and Structures, 43(13), pp.36573674.
[22] Ma, Y.Q. and Ang, K.K., 2006. Free vibration of Mindlin plates based on the relative displacement plate element. Finite elements in analysis and design, 42(11), pp.10211028.
[23] Verma, Y. and Datta, N., 2018. Comprehensive study of free vibration of rectangular Mindlin’s plates with rotationally constrained edges using dynamic Timoshenko trial functions. Engineering Transactions, 66(2), pp.129160.
[24] Chakraverty, S. and Pradhan, K.K., 2014. Free vibration of functionally graded thin rectangular plates resting on Winkler elastic foundation with general boundary conditions using Rayleigh–Ritz method. International Journal of Applied Mechanics, 6(04), p.1450043.
[25] Sakiyama, T. and Matsuda, H., 1987. Free vibration of rectangular Mindlin plate with mixed boundary conditions. Journal of Sound Vibration, 113(1), pp.208214.
[26] Kumar, S., Ranjan, V. and Jana, P., 2018. Free vibration analysis of thin functionally graded rectangular plates using the dynamic stiffness method. Composite Structures, 197, pp.3953.
[27] Pratap Singh, P., Azam, M.S. and Ranjan, V., 2019. Vibration analysis of a thin functionally graded plate having an out of plane material inhomogeneity resting on Winkler–Pasternak foundation under different combinations of boundary conditions. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 233(8), pp.26362662.
[28] Hashemi, S.H. and Arsanjani, M., 2005. Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. International Journal of Solids and Structures, 42(34), pp.819853.
[29] Bert, C.W. and Malik, M., 1996. Free vibration analysis of tapered rectangular plates by differential quadrature method: a semianalytical approach. Journal of Sound and Vibration, 190(1), pp.4163.
[30] Biswas, A., Mahapatra, D., Mondal, S.C. and Sarkar, S., 2024. Higher Order Approximations for Bending of FG Beams Using BSpline Collocation Technique. Mechanics of Advanced Composite Structures, 11(1), pp.159176.
[31] Srivastava, M.C. and Singh, J., 2023. Assessment of RBFs Based Meshfree Method for the Vibration Response of FGM Rectangular Plate Using HSDT Model. Mechanics Of Advanced Composite Structures, 10(1), pp.137150.
[32] Leissa, A.W., 1973. The free vibration of rectangular plates. Journal of sound and vibration, 31(3), pp.257293.
[33] Rao, G.V., Saheb, K.M. and Janardhan, G.R., 2006. Concept of coupled displacement field for large amplitude free vibrations of shear flexible beams. Journal of and acoustics, 128(2), pp.251255.
[34] KrishnaBhaskar, K. and MeeraSaheb, K., 2017. Effect of aspect ratio on large amplitude free vibrations of simply supported and clamped rectangular Mindlin plates using coupled displacement field method. Journal of Mechanical Science and Technology, 31(5), pp.20932103.
[35] Reddy, G. and Kumar, N.V., 2023. Free vibration analysis of 2d functionally graded porous beams using novel higherorder theory. Mechanics Of Advanced Composite Structures, 10(1), pp.6984.
[36] Razouki, A., Boutahar, L. and El Bikri, K., 2020. A New Method of Resolution of the Bending of Thick FGM Beams Based on Refined Higher Order Shear Deformation Theory. Universal Journal of Mechanical Engineering. 8(2), pp.105113.
[37] Khorshidi, K., Fallah, A. and Siahpush, A., 2017. Free vibrations analaysis of functionally graded composite rectangular nanoplate based on nonlocal exponential shear deformation theory in thermal environment. Journal of Science and Technology of Composites, 4(1), pp.109120.
[38] Talha, M. and Singh, B., 2010. Static response and free vibration analysis of FGM plates using higher order shear deformation theory. Applied mathematical modelling, 34(12), pp.39914011.
[39] Baferani, A.H., Saidi, A.R. and Jomehzadeh, E., 2011. An exact solution for free vibration of thin functionally graded rectangular plates. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 225(3), pp.526536.
[40] Chauhan, M., Ranjan, V. and Sathujoda, P., 2019. Dynamic stiffness method for free vibration analysis of thin functionally graded rectangular plates. Vibroengineering Procedia, 29, pp.7681.
[41] Bhat, R.B., 1985. Natural frequencies of rectangular plates using characteristic orthogonal polynomials in RayleighRitz method. Journal of sound and vibration, 102(4), pp.493499.
[42] Liew, K.M., Lam, K.Y. and Chow, S.T., 1990. Free vibration analysis of rectangular plates using orthogonal plate function. Computers & Structures, 34(1), pp.7985.
[43] Boay, C.G., 1993. Free vibration of rectangular isotropic plates with and without a concentrated mass. Computers & structures, 48(3), pp.529533.