Research Article

Free Vibration Analysis of Simply Supported and Clamped Functionally Graded Rectangular Plate Using Coupled Displacement Field Method

Nathi Venkatalakshmi , Kalidindi Krishnabhaskar ^* , Koppanati Meerasaheb

Department of  Mechanical Engineering, University College of Engineering(A), JNTUK, Kakinada, 533003, Andhra Pradesh, India

1.     Introduction

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 first-order 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 Rayleigh-Ritz (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 force-displacement relations that are reduced to three partial differential equations of motion with total deflection [15]. An elasticity solution of FG simply supported 3-D 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 clamped-clamped boundary conditions [18]. The governing equations for the boundary conditions are derived by differential rules [19]. Based on the strain linear elasticity theory, 3-D 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 [25-27]. 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 single-term 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 clamped-clamped 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 power-law, aspect ratio, thickness-length ratio, and various boundary conditions on the vibration characteristics of the FG rectangular plate are examined [39-41]. 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, [26-29], 32 and [41-43] are found to be in good agreement.

2.     Functionally Graded Plate

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 power-law function is written as

where  and  are the attributes of ceramic and metal, h is thickness and k is the power-law exponent of the FG plate. Accordingly, E and M vary continuously along the z direction as shown below.

3.      First-Order Shear Deformation Theory

The displacements  ,  and are given by

where  are unknown functions that are to be resolved. indicates the displacements of the mid-plane  and t denotes the time. and denotes rotations of the transverse normal about the y and x axis.

Axial strain and shear strain are

here, ,  and  indicate normal strains whereas ,  and  indicate shear strains. Strain and kinetic energies represented by  and  are

Using the above equations, The undetermined coefficients are derived by  

4.      Coupled Displacement Field Method

By considering  and , estimate the transverse displacement denoted by w along the x and y directions.

where

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

here, is the undetermined coefficient and , and are the admissible functions.

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]

Using Eqs. (12), (13), and (14) in Eqs. (8) and (9) we get

Reducing the lagranzian concerning c_i  

where

The governing equation is given by

 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.

5.     Results and Analysis

The behavior of vibrations in an FG rectangular plate using CDF with respect to thickness ratio (h/a) is obtained. The FG plate Non-dimensional frequency parameters may be expressed as

The properties of the materials used in the FG plate differ, i.e., for aluminum  = 70 GPa,  = 2700 kg/m³ and  = 0.3 and for alumina  = 380 GPa,  = 3800 kg/m³ 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.

^{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.

 ^{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.

^{, 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.

^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.

^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.

^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.

^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.

Table 10. Frequency parameters for all edges of the SSSS FG plate with k = 1 using CDF.

Table 11. Frequency parameters for all edges of the CCCC FG plate with k = 1 using CDF.

Table 12. Frequency parameters for all edges of the SSSS FG plate with k = 2 using CDF.

Table 13. Frequency parameters for all edges of the CCCC FG plate with k = 2 using CDF.

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

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 2-7. 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 8-13. 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 power-law 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.

6.     Conclusions

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 power-law 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 non-linear dynamic responses of the structures.

Nomenclature

Acknowledgments

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.

Funding Statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this article.

References

[1]   Hosseini-Hashemi, S., Taher, H.R.D., Akhavan, H. and Omidi, M., 2010. Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory. Applied Mathematical Modelling, 34(5), pp.1276-1291.

[2]   Amirpour, M., Das, R. and Flores, E.I.S., 2017. Bending analysis of thin functionally graded plate under in-plane stiffness variations. Applied Mathematical Modelling, 44, pp.481-496.

[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.709-720.

[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.529-538.

[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.213-224.

[6]   Akavci, S.S. and Tanrikulu, A.H., 2015. Static and free vibration analysis of functionally graded plates based on a new quasi-3D and 2D shear deformation theories. Composites Part B: Engineering, 83, pp.203-215.

[7]   Rahmani, B. and Naghmehsanj, M.R., 2015. Robust vibration control of a functionally graded beam with a variable cross-section. Journal of Science and Technology of Composites, 2(2), pp.17-29.

[8]   Eftekhari, S.A. and Jafari, A.A., 2012. High-accuracy mixed finite element-Ritz formulation for free vibration analysis of plates with general boundary conditions. Applied Mathematics and Computation, 219(3), pp.1312-1344.

[9]   Leissa, A.W., 2005. The historical bases of the Rayleigh and Ritz methods. Journal of Sound and Vibration, 287(4-5), pp.961-978.

[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.757-768.

[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.335-348.

[12] Dawe, D.J. and Roufaeil, O.L., 1980. Rayleigh-Ritz vibration analysis of Mindlin plates. Journal of Sound and Vibration, 69(3), pp.345-359.

[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.865-867.

[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. Three-dimensional elasticity solution for bending of functionally graded rectangular plates. European Journal of Mechanics-A/Solids, 23(5), pp.853-864.

[17] Kolarevic, N., Nefovska-Danilovic, M. and Petronijevic, M., 2015. Dynamic stiffness elements for free vibration analysis of rectangular Mindlin plate assemblies. Journal of Sound and Vibration, 359, pp.84-106.

[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.1-10.

[19] Torabi, K., Afshari, H. and Heidari-Rarani, M., 2014. Free vibration analysis of a rotating non-uniform blade with multiple open cracks using DQEM. Universal Journal of mechanical Engineering, 2(3), pp.101-111.

[20] Uymaz, B. and Aydogdu, M., 2007. Three-dimensional vibration analyses of functionally graded plates under various boundary conditions. Journal of Reinforced Plastics and Composites, 26(18), pp.1847-1863.

[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.3657-3674.

[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.1021-1028.

[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.129-160.

[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.208-214.

[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.39-53.

[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.2636-2662.

[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(3-4), pp.819-853.

[29] Bert, C.W. and Malik, M., 1996. Free vibration analysis of tapered rectangular plates by differential quadrature method: a semi-analytical approach. Journal of Sound and Vibration, 190(1), pp.41-63.

[30] Biswas, A., Mahapatra, D., Mondal, S.C. and Sarkar, S., 2024. Higher Order Approximations for Bending of FG Beams Using B-Spline Collocation Technique. Mechanics of Advanced Composite Structures, 11(1), pp.159-176.

[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.137-150.

[32] Leissa, A.W., 1973. The free vibration of rectangular plates. Journal of sound and vibration, 31(3), pp.257-293.

[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.251-255.

[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.2093-2103.

[35] Reddy, G. and Kumar, N.V., 2023. Free vibration analysis of 2d functionally graded porous beams using novel higher-order theory. Mechanics Of Advanced Composite Structures, 10(1), pp.69-84.

[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.105-113.

[37] Khorshidi, K., Fallah, A. and Siahpush, A., 2017. Free vibrations analaysis of functionally graded composite rectangular na-noplate based on nonlocal exponential shear deformation theory in thermal environment. Journal of Science and Technology of Composites, 4(1), pp.109-120.

[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.3991-4011.

[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.526-536.

[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.76-81.

[41] Bhat, R.B., 1985. Natural frequencies of rectangular plates using characteristic orthogonal polynomials in Rayleigh-Ritz method. Journal of sound and vibration, 102(4), pp.493-499.

[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.79-85.

[43] Boay, C.G., 1993. Free vibration of rectangular isotropic plates with and without a concentrated mass. Computers & structures, 48(3), pp.529-533.