Nonlinear vibration analysis of functionally graded plate in contact with fluid: Analytical study

Document Type : Research Paper


1 Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran

2 K.N Toosi University


In this paper, the nonlinear vibrations analysis of functionally graded (FG) rectangular plate in contact with fluid are investigated analytically using first order shear deformation theory (FSDT). The pressure exerted on the free surface of the plate by the fluid is calculated using the velocity potential function and the Bernoulli equation. With the aid of von Karman nonlinearity strain-displacement relations and Galerkin procedure the partial differential equations of motion are developed. The nonlinear equation of motion is then solved by modified Lindstedt-Poincare method. The effects of some system parameters such as vibration amplitude, fluid density, fluid depth ratio, volume fraction index and aspect ratio on the nonlinear natural frequency of the plate are discussed in detail.


Main Subjects

[1] Zhang, D.G., and Zhou, Y.H., “A Theoretical Analysis of FGM Thin Plates Based on Physical Neutral Surface”, Comp. Mater. Sci. Vol. 44, pp. 716–720, (2008).
[2] Abrate, S., “Functionally Graded Plates Behave Like Homogeneous Plates”, Compos. Part B-Eng. Vol. 39, pp. 151–158, (2008).
[3] Nguyen, T.K., Sab, K., and Bonnet, G., “First-order Shear Deformation Plate or Functionally Graded Materials”, Compos. Struct. Vol. 83, pp. 25–36, (2008).
[4] Timoshenko, S.P., “On the Transverse Vibrations of Bars of Uniform Cross-section”, Philos. Mag. Vol. 43, pp. 125–131, (1922).
[5] Ferreira, A.J.M., Batra, R.C., Roque, C.M.C., Qian, L.F., and Jorge, R.M.N., “Natural Frequencies of Functionally Graded Plates by a Meshless Method”, Compos. Struct. Vol. 75, pp. 593–600, (2006).
[6] Efraim, E., and Eisenberger, M., “Exact Vibration Analysis of Variable Thickness Thick Annular Isotropic and FGM Plates”, J. Sound Vib. Vol. 299, pp. 720–738, (2007).
[7] Zhao, X., Lee, Y.Y., and Liew, K.M., “Free Vibration Analysis of Functionally Graded Plates using the Element-free Kp-Ritz Method”, J. Sound Vib. Vol. 319, pp. 918–939, (2009).
[8] Abrate, S., “Free Vibration, Buckling, and Static Deflections of Functionally Graded Plates”, Compos. Sci. Technol.  Vol. 66, pp. 2383–2394, (2006).
[9] Rani, R., and Lal, R., “Free Vibrations of Composite Sandwich Plates by Chebyshev Collocation Technique”, Composites Part B: Engineering, Vol. 165, pp. 442–455, (2019).
[10] Hosseini-Hashemi, S., Taher, H.R., Akhavan, H., Omidi, M., “Free Vibration of Functionally Graded Rectangular Plates using First-order Shear Deformation Plate Theory”, Applied Mathematical Modelling, Vol. 34, pp. 1276–1291, (2010).
[11] Zhao, X., Lee, Y.Y., and Liew, K.M., “Free Vibration Analysis of Functionally Graded Plates using the Element-free Kp-Ritz Method”, Journal of Sound and Vibration, Vol. 319, pp. 918–939, (2009).
[12] Yang, J., and Shen, H.S., “Vibration Characteristics and Transient Response of Shear-deformable Functionally Graded Plates in Thermal Environments”, Journal of Sound and Vibration, Vol. 255, pp. 579–602, (2002).
[13] Gupta, A., Talha, M., and Singh, B.N., “Vibration Characteristics of Functionally Graded Material Plate with Various Boundary Constraints using Higher Order Shear Deformation Theory”, Composites Part B: Engineering, Vol. 94, pp. 64–74, (2016).
[14] Wang, Y.Q., and Zu, J.W., “Large-amplitude Vibration of Sigmoid Functionally Graded Thin Plates with Porosities”, Thin-Walled Structures, Vol. 119, pp. 911–924, (2017).
[15] Yazdi, A.A., “Homotopy Perturbation Method for Nonlinear Vibration Analysis of Functionally Graded Plate”, Journal of Vibration and Acoustics, Vol. 135, pp. 12–21, (2013).
[16] Woo, J., Meguid, S.A., and Ong, L.S., “Nonlinear Free Vibration Behavior of Functionally Graded Plates”, Journal of Sound and Vibration, Vol. 289, pp. 595–611, (2006).
[17] Malekzadeh, P., and Monajjemzadeh, S.M., “Nonlinear Response of Functionally Graded Plates under Moving Load”, Thin-Walled Structures, Vol. 96, pp. 120–129, (2015).
[18] Duc, N.D., and Cong, P.H., “Nonlinear Vibration of Thick FGM Plates on Elastic Foundation Subjected to Thermal and Mechanical Loads using the First-order Shear Deformation Plate Theory”, Cogent Engineering, Vol. 2, pp. 1045222, (2015).
[19] Fung, C.P., and Chen, C.S., “Imperfection Sensitivity in the Nonlinear Vibration of Functionally Graded Plates”, European Journal of Mechanics-A/Solids, Vol. 25, pp. 425–461, (2006).
[20] Fakhari, V., Ohadi, A., and Yousefian, P., “Nonlinear Free and Forced Vibration Behaviour of Functionally Graded Plate with Piezoelectric Layers in Thermal Environment”, Composite Structures, Vol. 93, pp. 2310–2321, (2011).
[21] Hao, Y.X., Zhang, W., and Yang, J., “Nonlinear Oscillation of a Cantilever FGM Rectangular Plate Based on Third-order Plate Theory and Asymptotic Perturbation Method”, Composites Part B: Engineering, Vol. 42, pp. 402–415, (2011).
[22] Zhang, W., Hao, Y., Guo, X., and Chen, L., “Complicated Nonlinear Responses of a Simply Supported FGM Rectangular Plate under Combined Parametric and External Excitations”, Meccanica, Vol. 47, pp. 985–1014, (2012).
[23] Dinh Duc, N., Tuan, N.D., Tran, P., and Quan, T.Q., “Nonlinear Dynamic Response and Vibration of Imperfect Shear Deformable Functionally Graded Plates Subjected to Blast and Thermal Loads”, Mechanics of Advanced Materials and Structures, Vol. 24, pp. 318-347, (2017).
[24] Talebitooti, R., Johari, V., and Zarastvand, M., "Wave Transmission Across Laminated Composite Plate in the Subsonic Flow Investigating Two-variable Refined Plate Theory", Latin American Journal of Solids and Structures, Vol. 15, No. 5, (2018).
[25] Talebitooti, R., Zarastvand, M., and Rouhani, A.H., "Investigating Hyperbolic Shear Deformation Theory on vibroacoustic Behavior of the Infinite Functionally Graded Thick Plate", Latin American Journal of Solids and Structures, Vol. 16, No. 1, (2019).
[26] Talebitooti, R., Zarastvand, M., and Darvishgohari, H., "Multi-objective Optimization Approach on Diffuse Sound Transmission through Poroelastic Composite Sandwich Structure", Journal of Sandwich Structures & Materials, Jun 12:1099636219854748, (2019).
[27] Motaharifar, F., Ghassabi, M., and Talebitooti, R., "Vibroacoustic Behavior of a Plate Surrounded by a Cavity Containing an Inclined Part–through Surface Crack with Arbitrary Position", Journal of Vibration and Control, Jun 18:1077546319853666, (2019).
[28] Uğurlu, B., A., Kutlu, A., Ergin, and Omurtag, M. H., “Dynamics of a Rectangular Plate Resting on an Elastic Foundation and Partially in Contact with a Quiescent Fluid”, Journal of Sound and Vibration, Vol. 317, pp. 308-328, (2008).
[29] Kerboua, Y., Lakis, A. A., Thomas, M. and Marcouiller, L. “Vibration Analysis of Rectangular Plates Coupled with Fluid”, Applied Mathematical Modelling, Vol. 32, pp. 2570-2586, (2008).
[30] Cho, Seung, D., Kim, B.H., Vladimir, N., and Choi, T.M., ”Natural Vibration Analysis of Rectangular Bottom Plate Structures in Contact with Fluid”, Ocean Engineering. Vol. 103, pp. 171-179, (2015).
[31] Khorshid, K., and Farhadi, S, “Free Vibration Analysis of a Laminated Composite Rectangular Plate in Contact with a Bounded Fluid”, Composite Structures, Vol. 104, pp. 176-186, (2013).
[32] Ergin, A., and B. Uğurlu. “Linear Vibration Analysis of Cantilever Plates Partially Submerged in Fluid”, Journal of Fluids and Structures, Vol. 17, pp. 927-939, (2003).
[33] Shahbaztabar, A., and Rahbar Ranji, A., “Effects of In-plane Loads on Free Vibration of Symmetrically Cross-ply Laminated Plates Resting on Pasternak Foundation and Coupled with Fluid”, Ocean Engineering, Vol. 115, pp. 196-209, (2016).
[34] Khorshidi, K., and Bakhsheshy, A., “Free Vibration Analysis of a Functionally Graded Rectangular Plate in Contact with a Bounded Fluid”, Acta Mechanica, Vol. 226. pp. 3401-3423, (2015).
[35] Shafiee, A.A., Daneshmand, F., Askari, E., and Mahzoon, M., “Dynamic Behavior of a Functionally Graded Plate Resting on Winkler Elastic Foundation and in Contact with Fluid”, Struct. Eng. Mech. Vol. 50, pp. 53-71, (2014).
[36] Hosseini-Hashemi, Sh., Karimi, M., and Rokni, H., “Natural Frequencies of Rectangular Mindlin Plates Coupled with Stationary Fluid”, Applied Mathematical Modelling. Vol. 36, pp. 764-778, (2012).
[37] Shahbaztabar, A., and Rahbar Ranji, A., “Free Vibration Analysis of Functionally Graded Plates on Two-parameter Elastic Supports and in Contact with Stationary Fluid”, Journal of Offshore Mechanics and Arctic Engineering, Vol. 140, pp. 021302, (2018).
[38] Yousefzadeh, Sh, Jafari, A.A., and Mohammadzadeh, A., “Effect of Hydrostatic Pressure on Vibrating Functionally Graded Circular Plate Coupled with Bounded Fluid”, Applied Mathematical Modelling, Vol. 60, pp. 435-446, (2018).
[39] Chang, T.P., and Liu, M.F., “Vibration Analysis of Rectangular Composite Plates in Contact with Fluid”, Vol. 1, pp. 101-120, (2001).
[40] Qing, W.Y., and Yang, Z., “Nonlinear Vibrations of Moving Functionally Graded Plates Containing Porosities and Contacting with Liquid: Internal Resonance”, Nonlinear Dynamics. Vol. 90, pp. 1461-1480, (2017).
[41] Reddy, J.N., “Mechanics of Laminated Composite Plates and Shells: Theory and Analysis”, CRC Press, (2004).
[42] Chia, C.Y., “Nonlinear Analysis of Plates”, McGraw-Hill International Book Company, (1980).
[43] Nayfeh, A.H., and Mook, D.T, “Nonlinear Oscillation”, John Wiley & Sons, Inc, (1995).
[44] He, J.H., “Modified Lindstedt–Poincare Methods for Some Strongly Non-linear Oscillations: Part I: Expansion of a Constant”, International Journal of Non-Linear Mechanics. Vol. 37, pp. 309-314, (2002).