Nonlinear Free Vibration Analysis of Bi-directional Functionally Graded Rectangular Plates

Authors

1 Department of mechanical engineering, K. N. Toosi university of technology, Tehran, Iran

2 Professor of KNT University

Abstract

In the present study, nonlinear free vibration analysis of bi-directional functionally graded simply supported rectangular plates is investigated analytically for the first time. For this purpose, with the aid of Hamilton’s principle and von Karman nonlinear strain-displacement relations, the partial differential equations of motion are developed. Afterward, the nonlinear partial differential equations are transformed into the time-dependent nonlinear ordinary differential equations by applying the Galerkin method. The nonlinear equation of motion is then solved analytically by the Modified Lindstedt-Poincare method to determine the the plate nonlinear frequencies. The volume fraction distribution is assumed to be continuously graded in both the length and width directions of the plate. Finally, the effects of some system parameters such as the vibration amplitude, FG indexes and aspect ratio on the nonlinear frequency are discussed in detail. To validate the analysis, the results of this paper are compared with both the published data and numerical method, and good agreements are found.

Keywords


[1] Zhang DG, Zhou YH (2008) A theoretical analysis of FGM thin plates based on physical neutral surface. Comp Mat Sci 44(2): 716-720.
[2] Abrate S (2008) Functionally graded plates behave like homogeneous plates. Comp Part B: Eng 39(1): 151–158.
[3] Najafizadeh MM, Ayalvar A (2006) Investigation of free vibrations of gunctionally graded rectangular plate using first-order shear deformation theory. Ir Soci Mech Eng 7(1): 52- 68. (In Persian)
[4] Khorshidi K, Onsorinezhad S (2017) Exact free vibration analysis of sector plates coupled with piezoelectric layers using first-order shear deformation plate theory. Journal of Solid and Fluid Mechanics 6(4): 125-138. (In Persian)
[5]  Khorshidi K, Bakhsheshi A, Ghadirian H (2017) The study of the effects of thermal environment on free vibration analysis of two dimensional functionally graded rectangular plates on pasternak elastic foundation. Journal of Solid and Fluid Mechanics 6(3): 137-147. (In Persian)
[6]  Hosseini Hashemi SH, Akhavan H, Fadaee M (2012) Exact closed-form free vibration analysis of moderately thick rectangular functionally graded plates with two bonded piezoelectric layers. Modares Mechanical Engineering 11(3): 57-74. (In Persian) 
[7]  Zhao X, Lee YY, Liew KM (2009) Free vibration analysis of functionally graded plates using the element-free kp-Ritz method. J Sound Vib 319(3-5): 918-939.
[8]  Farzam A, Hassani B (2019) Size-dependent analysis of FG microplates with temperature-dependent material properties using modified strain gradient theory and isogeometric approach. Comp Part B: Eng.
[9]  Gupta A, Talha M, Singh BN (2016) Vibration characteristics of functionally graded material plate with various boundary constraints using higher order shear deformation theory. Comp Part B: Eng 94(1): 64-74.
[10] Khorshidi K, Siahpush A, Fallah A (2017) Electro-Mechanical free vibrations analysis of composite rectangular piezoelectric nanoplate using modified shear deformation theories. J Sci Tech Comp 4(1): 151-160. (In Persian)
[11] Khorshidi K, Asgari T, Fallah A (2015) Free vibrations analysis of functionally graded rectangular nano-plates based on nonlocal exponential shear deformation theory. Mech Ad Comp Struct 4(2): 79-93.
[12] Khorshidi K,  Bakhsheshy A (2015) Free vibration analysis of a functionally graded rectangular plate in contact with a bounded fluid. Acta Mech 266(10): 3401-3423.
[13] Ghaheri A, Nosier A (2015) Nonlinear forced vibrations of thin circular functionally graded plates. J Sci Tech Comp 1(2): 1- 10. (In Persian)
[14] Wang YQ, Zu JW (2017) Large-amplitude vibration of sigmoid functionally graded thin plates with porosities,” Thin-Walled Struct 119(1): 911-924.
[15] Yazdi AA (2013) Homotopy perturbation method for nonlinear vibration analysis of functionally graded plate. J Vib Acoust 135(2): 12-21.
[16] Lotfavar A, Rafiei Pour H, Hamze Shalamdari S, Mohammadi T (2015) Nonlinear vibration analysis of laminated composite plates using approximate and analytical methods. Ir soc Mech Eng 1(17): 16-39. (In Persian)
[17] Woo J, Meguid SA, Ong LS (2006) Nonlinear free vibration behavior of functionally graded plates. J Sound Vib 289(3): 595-611.
[18] Malekzadeh P, Monajjemzadeh SM (2015) Nonlinear response of functionally graded plates under moving load. Thin-Walled Struct 96(1): 120-129.
[19] Duc ND, Cong PH (2015) Nonlinear vibration of thick FGM plates on elastic foundation subjected to thermal and mechanical loads using the first-order shear deformation plate theory. Cogent Eng 2(1): 1045222.
[20] Fung CP, Chen CS (2006) Imperfection sensitivity in the nonlinear vibration of functionally graded plates. Eur J Mech A/Solids 25(3): 425-461.
[21] Şimşek M (2015) Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions. Comp Struct 133: 968-978.
[22] Tang Y, Ding Q (2019) Nonlinear vibration analysis of a bi-directional functionally graded beam under hygro-thermal loads. Comp Struct 111076.
[23] Fariborz J, Batra RC (2019) Free vibration of bi-directional functionally graded material circular beams using shear deformation theory employing logarithmic function of radius. Comp Struct 210: 217-230.
[24] Rajasekaran S, Khaniki HB (2019) Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part I: Theoretical formulations. Thin-Walled Struct 141: 627-645.
[25] Tang Y, Lv X, Yang T (2019) Bi-directional functionally graded beams: asymmetric modes and nonlinear free vibration. Comp Part B: Eng. 156: 319-331.
[26] Rajasekaran S, Khaniki HB (2019) Size-dependent forced vibration of non-uniform bi-directional functionally graded beams embedded in variable elastic environment carrying a moving harmonic mass. App Math Modelling 72: 129-154.
[27] Chen M, Jin G, Ma X, Zhang Y, Ye T, Liu Z (2018) Vibration analysis for sector cylindrical shells with bi-directional functionally graded materials and elastically restrained edges. Comp Part B: Eng 153: 346-363.
[28] Chen M, Jin G, Ma X, Zhang Y, Ye T, Liu Z (2018) Vibration analysis for sector cylindrical shells with bi-directional functionally graded materials and elastically restrained edges. Comp Part B: Eng 153: 346-363.
[29] Lieu QX, Lee D, Kang J, Lee J (2019) NURBS-based modeling and analysis for free vibration and buckling problems of in-plane bi-directional functionally graded plates. Mech Ad Mat Struct 26(12): 1064-1080.
[30] Lieu QX, Lee S, Kang J, Lee J (2018) Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis. Comp Struct 192: 434-451.
[31] Farzam A, Hassani B (2019) Isogeometric analysis of in-plane functionally graded porous microplates using modified couple stress theory. Aerosp Sci Technol 91: 508-524.
[32] Kumar Y, Lal R (2013) Prediction of frequencies of free axisymmetric vibration of two-directional functionally graded annular plates on Winkler foundation. Eur J Mech A/Solids 42: 219-228.
[33] Shariyat M, Alipour MM (2011) Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials resting on elastic foundations. Arch App Mech 81(9): 1289-1306.
[34] Alipour MM, Shariyat M, Shaban M (2010) A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations. Int J Mech Mat Design 6(4): 293-304.
[35] Aragh BS, Hedayati H, Farahani EB, Hedayati M (2011) A novel 2-D six-parameter power-law distribution for free vibration and vibrational displacements of two-dimensional functionally graded fiber-reinforced curved panels. Eur J Mech A/Solids 30(6): 865-883.
[36] Kumar Y (2015) Free vibration of two-directional functionally graded annular plates using Chebyshev collocation technique and differential quadrature method. Int J Struct Stab Dy 15(06): 450086.
[37] Lal R, Ahlawat N (2017) Buckling and vibrations of two-directional functionally graded circular plates subjected to hydrostatic in-plane force. J Vib Con 23(13): 2111-2127.
[38] Lal R, Ahlawat N (2019) Buckling and vibrations of two-directional FGM Mindlin circular plates under hydrostatic peripheral loading. Mech Ad Mat Struct 26(3): 199-214.
[39] Shariyat M, Alipour MM (2013) A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations. App Math Modelling 37(5): 3063-3076.
[40] Tahouneh V, Naei MH (2014) A novel 2-D six-parameter power-law distribution for three-dimensional dynamic analysis of thick multi-directional functionally graded rectangular plates resting on a two-parameter elastic foundation. Mecca 49(1): 91-109.
[41] Tahouneh V, Yas, MH (2013) Semianalytical solution for three-dimensional vibration analysis of thick multidirectional functionally graded annular sector plates under various boundary conditions. J Eng Mech 140(1): 31-46.
[42] Yas MH, Moloudi N (2015) Three-dimensional free vibration analysis of multi-directional functionally graded piezoelectric annular plates on elastic foundations via state space based differential quadrature method. App Math Mech 36(4): 439-464.
[43] Reddy JN (2006) Theory and analysis of elastic plates and shell. CRC press.
[44] Chia CY (1980) Nonlinear analysis of plates. McGraw-Hill International Book Company.
[45] Reddy JN (2004) Mechanics of laminated composite plates and shells: Theory and analysis. CRC press.
[46] Nayfeh AH, Mook DT (1995) Nonlinear oscillation. John Wiley & Sons, Inc.
[47] He JH (2002) Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant. Int J Non Mech 37: 309-314.