Nonlinear Investigation of Magnetic Influence on Dynamic Behaviour of Non-Homogeneous Varying Thickness Circular Plates Resting on Elastic Foundations

Subject Areas : Engineering

S.A Salawu ¹ , G.M Sobamowo ² , O.M Sadiq ³

1 - Department of civil and Environmental Engineering, University of Lagos, Akoka, Nigeria
2 - Department of Mechanical Engineering, University of Lagos, Akoka, Lagos, Nigeria
3 - Department of civil and Environmental Engineering, University of Lagos, Akoka, Nigeria

Received: 2021-07-30 Accepted : 2021-10-05 Published : 2021-12-30

Keywords: Nonlinear vibration, Optimal homotopy asymptotic method, Three-parameter foundations, Variable thickness circular plates, Non-Homogeneous,

Abstract :

In this work, a nonlinear investigation of non-homogeneous varying thickness circular plates resting on elastic foundations under the influence of the magnetic fieldis investigated. The non-homogeneity of the circular plates’ material is presumed to occur due to linear and parabolic changes in Young’s modulus likewise the density along the radial direction in a unique manner. The geometric Von Kármán equations are used in modelling the governing differential equations. The transverse deflection is approximated using an assumed single term mode shape while the central deflection in form of Duffing’s equation is obtained using the Galerkin method. Subsequently, the semi-analytical solutions are provided using the Optimal Homotopy Asymptotic Method (OHAM), the analytical solutions are used for parametric investigation. The results in this work are in good harmony with past results in the literature. From the results, it is realized that the nonlinear frequency of the circular plate increases with an increase in the linear elastic foundation. Also, the results showed that clamped edge and simply supported edge condition produced the same hardening nonlinearity. However, varying taper and non-homogeneity lower the nonlinear frequency ratio. Also, maximum deflection occurs when excitation force is zero, and attenuation of deflection is observed due to the presence of a magnetic field, varying thickness, homogeneity, and elastic foundation. It is anticipated that the discoveries from this research will boost the design of structures subjected to vibration.

References:

[1] Zhiming Y., 1998, Nonlinear vibration of circular plate with exponential varying thickness, Journal of Shanghai University 2(1): 27-33.
[2] Wattanasakulpong N., Charoensuk J., 2015, Vibration characteristics of stepped beams made of FGM using differential transform method, Meccanica 50(4): 1089-1101.
[3] Chakravorty J.G., 1980, Bending of symmetrically loaded circular plate of variable thickness, Indian Journal of Pure and Applied Mathematics 11(2): 258-267.
[4] Hu Y.D., Wang T., 2016, Nonlinear free vibration of a rotating circular plate under the static load in magnetic field, Nonlinear Dynamics 85: 1825-1835.
[5] Wang C.M., Wang C.Y., Reddy J.N., 2004, Exact Solution for Buckling of Structural Members, CRC Press.
[6] Shi-Chao Y., Lin-Quan Y., Bal-Jian T., 2017, A novel higher-order shear and normal deformable plate theory for the static-free vibration and buckling analysis of functionally graded plate, Mathematical Problems in Engineering Article 2017: ID 6879508.
[7] Shishesaz M., Zakipour A., Jafarzadeh A., 2016, Magneto-elastic analysis of annular FGM plate based on classical plate theory using GDQ method, Latin America Journal of Solids Structures 13(14): 2436-2462.
[8] Banerjee B., 1983, Large deflection of circular plates of variable thickness, Journalof Solids and Structures 19(2): 179-182.
[9] Shariyat M., Jafari A.A., Alipour M.M., 2013, Investigation of the thickness variability and material heterogeneity effects for free vibration of the viscoelastic circular plates, Acta Mechanica Solida Sinica 26(1): 83-98.
[10] Gupta U.S., Sharma S., Singhal P., 2014, Effect of two-parameter foundation on free transverse vibration of non-homogenous orthotropic rectangular plate of linearly varying thickness, Journal of Engineering and Applied Science 6(2): 32-51.
[11] Malekzadeh P., 2019, Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundation, Composite structure 89(3): 367-373.
[12] Kamal K., Durvasula S., 1983, Bending of circular plate on elastic foundation, Journal of Engineering Mechanics 109: 1293-1298.
[13] Yazdi A.A., 2013, Homotopy perturbation method for nonlinear vibration analysis of functionally graded plate, Journal of Vibration and Acoustic 135(2): 021012.
[14] Shama S., Lai R., Singh N., 2015, Effects of non-homogeneity on asymmetric vibration of non-uniform circular plates, Journal of Vibration and Control 23: 1635-1644.
[15] Gupta A.K., Kumar L., 2010, Vibration of non-homogeneous visco-elastic circular plate of linearly varying thickness in steady-state temperature field, Journal of Theoretical and Applied Mechanics 48(1): 255-266.
[16] Math Y., Varma K.K., Mahrenholtz D., 1986, Nonlinear dynamic response of rectangular plates on linear elastic foundation, Computers & Structures 4: 391-399.
[17] Dumir P.C., 1986, Non-Linear vibration and post-buckling of isotropic thin circular plates on elastic foundations, Journal of Sound and Vibration 107(2): 253-263.
[18] Touzé C., Thomas O., Chaigne A., 2002, Asymmetric non-linear forced vibrations of free-edge circular plates, Journal of Sound and Vibration 258(4): 649-676.
[19] Bhoyar S., Varghese V., Khalsa L., 2020, Thermoelastic large deflection bending analysis of elliptical plate resting on elastic foundations, Waves in Random and Complex Media, DOI:10.1080/17455030.2020.1822563.
[20] Bhad P., Varghese V., Khalsa L., 2017, A modified approach for the thermoelastic large deflection in the elliptical plate, Archive of Applied Mechanics 87: 767-781.
[21] Dastjerdi Sh., Yazdanparast L., 2018, New method for large deflection analysis of an elliptic plate weakened by an eccentric circular hole, Journal of Solid Mechanics 10(3): 561-570.
[22] Dastjerdi Sh., Jabbarzadeh M., 2016, Nonlocal bending analysis of bilayer annular/circular nano plates based on first order shear deformation theory, Journal of Solid Mechanics 8(3): 645-661.
[23] Dastjerdi Sh., Jabbarzadeh M., 2016, Non-Local thermo-elastic buckling analysis of multi-layer annular/circular nano-plates based on first and third order shear deformation theories using DQ method, Journal of Solid Mechanics 8(4): 859-874.
[24] Dastjerdi Sh., Jabbarzadeh M., Aliabadi S., 2016, Nonlinear static analysis of single layer annular/circular graphene sheets embedded in Winkler–Pasternak elastic matrix based on non-local theory of eringen, Ain Shams Engineering Journal 7(2): 873-884.
[25] Dastjerdi Sh., Beni Y.T., 2019, A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment, Mechanics Based Design of Structures and Machines 47(4): 453-478.
[26] Yazdi A.A., 2016, Assessment of homotopy perturbation method for study the forced nonlinear vibration of orthotropic circular plate on elastic foundation, Latin America Journal of Solids and Structures 13(2): 243-256.
[27] Zhong X., Liao S., 2016, Analytic solutions of von karman plate under arbitrary uniform pressure(1): Equations in differential form, Studies in Applied Mathematics 138(4): 10.1111
[28] Wu T.Y., Liu G.R., 2001, Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature, International Journal of Solids and Structures 38: 7967-7980.
[29] Herisanu N., Marinca V., Dordea T., Madescu G., 2008, A new analytical approach to nonlinear vibration of an electrical machine, Proceedings of the Romanian Academy - Series A 9(3): 229-236.
[30] Vasile M., Nicolae H., 2011, Nonlinear Dynamical Systems in Engineering, New York, Springer-Verlag Berlin Heidelberg.
[31] Marinca V., Herisanu N., Nemes L., 2008, Optimal homotopy asymptotic method with application to thin-film flow, Central European Journal of Physics 6(3): 648-653.
[32] Haterbouch M., Benamar R., 2003, The effects of large vibration amplitudes on the axisymmetric mode shapes and natural frequencies of clamped thin isotropic circular plates, part I: iterative and explicit analytical solution for nonlinear transverse vibrations, Journal of Sound and Vibration 265: 123-154.
[33] Civalek O., 2007, Nonlinear analysis of thin rectangular plates on Winkler-Pasternak elastic foundations by DSC-HDQ Methods, Applied Mathematical Modelling 31: 606-624.
[34] Hu Y., Wang T., 2015, Nonlinear resonance of the rotating circular plate under static loads in magnetic field, Chinese Jornal of Mechanical Engineering 28: 1277-1284.