Investigation of Von Karman rectangular plates nonlinear vibration : Optimal Ho-motopy Analysis Method

Kazemi, Mohammad Ali; Fallahian, Milad; Jafari, Seyed Sajad; Khaledian, Amir

doi:10.48301/jear.2025.475824.1041

Investigation of Von Karman rectangular plates nonlinear vibration : Optimal Ho-motopy Analysis Method

Document Type : Original Article

Authors

Mohammad Ali Kazemi ¹

Milad Fallahian ²

seyed sajad jafari ³

Amir Khaledian ⁴

¹ Assistant Professor Department of Mechanical Engineering, Technical and Vocational University (TVU), Tehran, Iran

² Young Researchers & Elite Club, Hamedan Branch, Islamic Azad University, Hamedan, Iran

³ Assistant Professor Department of Mechanical Engineering, Hamedan University of Technology, Hamedan, Iran

⁴ Department of Electrical Engineering, Technical and Vocational University (TVU), Tehran, Iran

10.48301/jear.2025.475824.1041

Abstract

Investigation of Von Karman rectangular plates nonlinear vibration : Optimal Ho-motopy Analysis Method:

In the present study, the Optimal Homotopy Analysis Method (OHAM) is employed to solve the motion equation for a rec-tangular isotropic plate in the presence of the shear deformation and rotary inertia effects based on the Von Karman theo-ry. Suitable agreement between OHAM solution and previously published studies was observed as well as numerical results obtained by bvp function from Maple in the especial cases. The effects of some system parameters such as and initial amplitude on the amplitude oscillation have been checked and studied and also different mode shapes of oscillation for different values of physical parameters are illustrated.

Keywords: Nonlinear vibration; Rectangular plate; Time function; Optimal Homotopy Analysis Method (OHAM)

To accelerate solution convergence, HAM with two auxiliary parameters was applied to investigate the nonlinear vibration of a rectangular plate. The second auxiliary parameter increases the rate of convergence . In addition, the system parameters effects on the amplitude of oscillation are displayed and different mode shapes of the oscillation for various plate parameters are illustrated

Keywords

Nonlinear vibration

Rectangular plate

Time function

Optimal Homotopy Analysis Method (OHAM)

Subjects

Mechanical Engineering

[1] Rashidi, M., Shooshtari, A., & Bég, O. A. (2012). Homotopy perturbation study of nonlinear vibration of Von Karman rectangular plates. Computers & Structures, 106, 46–55.https://doi.org/10.1016/j.compstruc.2012.04.004

[2] Mahabadi, R. K., Shakeri, M., & Daneshpazhooh, M. (2016). Free Vibration of Laminated Composite Plate with Shape Memory Alloy Fibers. Latin American Journal of Solids and Structures, 13(2), 314–330. https://doi.org/10.1590/1679-78252162

[3] Yazdi, A. A. (2016). Assessment of Homotopy Perturbation Method for Study the Forced Nonlinear Vibration of Orthotropic Circular Plate on Elastic Foundation. Latin American Journal of Solids and Structures, 13(2), 243–256. https://doi.org/10.1590/1679-78252436

[4] Wang, Y.-G., Song, H.-F., Lin, W.-H., & Wang, J.-K. (2015). Large amplitude free vibration of micro/nano beams based on nonlocal thermal elasticity theory. Latin American Journal of Solids and Structures, 12(10), 1918–1933. https://doi.org/10.1590/1679-78251904

[5] Huang, X.-L., & Shen, H.-S. (2005). Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators. International Journal of Mechanical Sciences, 47(2), 187–208. https://doi.org/10.1016/j.ijmecsci.2005.01.003

[6] Singha, M. K., & Daripa, R. (2007). Nonlinear vibration of symmetrically laminated composite skew plates by finite element method. International Journal of Non-Linear Mechanics, 42(9), 1144–1152. https://doi.org/10.1016/j.ijnonlinmec.2007.08.001

[7] Huang, X.-L., & Shen, H.-S. (2004). Nonlinear vibration and dynamic response of functionally graded plates in thermal environments. International Journal of Solids and Structures, 41(9–10), 2403–2427. https://doi.org/10.1016/j.ijsolstr.2003.11.012

[8] Kang, J.-H. (2003). Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation. Computers and Structures, 81(16), 1663–1675. https://doi.org/10.1016/s0045-7949(03)00168-8

[9] Amabili, M. (2004). Nonlinear vibrations of rectangular plates with different boundary conditions: theory and experiments. Computers and Structures, 82(31–32), 2587–2605. https://doi.org/10.1016/j.compstruc.2004.03.077

[10] Ye, Z. (1999). Application of Maple V to the nonlinear vibration analysis of circular plate with variable thickness. Computers and Structures, 71(5), 481–488. https://doi.org/10.1016/s0045-7949(98)00302-2

[11] Shen, L., Shen, H.-S., & Zhang, C.-L. (2010). Nonlocal plate model for nonlinear vibration of single layer graphene sheets in thermal environments. Computational Materials Science, 48(3), 680–685. https://doi.org/10.1016/j.commatsci.2010.03.006

[12] Belalia, S. A., & Houmat, A. (2012). Nonlinear free vibration of functionally graded shear deformable sector plates by a curved triangular p-element. European Journal of Mechanics - A/Solids, 35(0), 1–9. https://doi.org/10.1016/j.euromechsol.2012.01.004

[13] Leung, A. Y. T., & Zhu, B. (2004). Geometric nonlinear vibration of clamped Mindlin plates by analytically integrated trapezoidal p-element. Thin-Walled Structures, 42(7), 931–945. https://doi.org/10.1016/j.tws.2004.03.010

[14] Malekzadeh, P. (2007). A differential quadrature nonlinear free vibration analysis of laminated composite skew thin plates. Thin-Walled Structures, 45(2), 237–250. https://doi.org/10.1016/j.tws.2007.01.011

[15] Nejati, M., Fard, K. M., Eslampanah, A., & Jafari, S. S. (2017). Free vibration analysis of reinforced composite functionally graded plates with steady state thermal conditions. Latin American Journal of Solids and Structures, 14(5), 886–905. https://doi.org/10.1590/1679-78253705

[16] Yas, M. H., Nejati, M., & Jafari, S. S. (2017). Free Vibration of Functionally Graded Variable Thickness Carbon Nanotube Annular Plates [Research]. Journal of Computational Methods In Engineering, 35(2), 131–158. https://doi.org/10.18869/acadpub.jcme.35.2.131

[17] Singh, B. N., & Lal, A. (2010). Stochastic analysis of laminated composite plates on elastic foundation: The cases of post-buckling behavior and nonlinear free vibration. International Journal of Pressure Vessels and Piping, 87(10), 559–574. https://doi.org/10.1016/j.ijpvp.2010.07.013

[18] Abdulkerim, S., Dafnis, A., & Riemerdes, H.-G. (2019). Experimental Investigation of Nonlinear Vibration of a Thin Rectangular Plate. International Journal of Applied Mechanics, 11(06), 1950059. https://doi.org/10.1142/s1758825119500595

[19] Chen, C.-S. (2007). The nonlinear vibration of an initially stressed laminated plate. Composites Part B: Engineering, 38(4), 437–447. https://doi.org/10.1016/j.compositesb.2006.09.002

[20] Feli, S., Karami, L., & Jafari, S. S. (2019). Analytical modeling of low velocity impact on carbon nanotube-reinforced composite (CNTRC) plates. Mechanics of Advanced Materials and Structures, 26(5), 394–406. https://doi.org/10.1080/15376494.2017.1400613

[21] Lo'pez-Puente, J., Zaera, R., & Navarro, C. (2007). An analytical model for high velocity impacts on thin CFRPs woven laminated plates. International Journal of Solids and Structures, 44, 2837–2851. https://doi.org/10.1016/j.ijsolstr.2006.08.022

[22] Liao, S. J. (2004). Beyond perturbation: introduction to the homotopy analysis method. Chapman & Hall/CRC. http://books.google.com/books?id=AlUdMFes_JUC

[23] Liao, S. J. (2004). On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147(2), 499–513. https://doi.org/10.1016/S0096-3003(02)00790-7

[24] Marinca, V., Herişanu, N., & Nemeş, I. (2008). Optimal homotopy asymptotic method with application to thin film flow. Open Physics, 6(3), 648–653. https://doi.org/10.2478/s11534-008-0061-x

[25] Yongqiang, L., Feng, L., & Dawei, Z. (2010). Geometrically nonlinear free vibrations of the symmetric rectangular honeycomb sandwich panels with simply supported boundaries. Composite Structures, 92(5), 1110–1119. https://doi.org/10.1016/j.compstruct.2009.10.012

[26] M. Fooladi, S.R. Abaspour, A. Kimiaeifar, & M. Rahimpour. (2009). On the Analytical Solution of Kirchhoff Simplified Model for Beam by using of Homotopy Analysis Method. World Applied Sciences Journal, 6, 297–302. https://doi.org/10.1590/1979-78276437

[27] Jafari, S. S., Rashidi, M. M., & Johnson, S. (2016). Analytical approximation of nonlinear vibration of euler-bernoulli beams. Latin American Journal of Solids and Structures, an ABCM Journal, 13(7), 1250–1264. https://doi.org/10.1590/1679-78252437

[28] Ajam, H., Jafari, S. S., & Freidoonimehr, N. Analytical approximation of MHD nano-fluid flow induced by a stretching permeable surface using Buongiorno’s model. Ain Shams Engineering Journal. http://dx.doi.org/10.1016/j.asej.2016.03.006

[29] Jafari, S. S., & Freidoonimehr, N. (2015). Second law of thermodynamics analysis of hydro-magnetic nano-fluid slip flow over a stretching permeable surface [journal article]. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 37(4), 1245–1256. https://doi.org/10.1007/s40430-014-0250-z

[30] Hoshyar, H., Ganji, D., Borran, A., & Falahati, M. (2015). Flow behavior of unsteady incompressible Newtonian fluid flow between two parallel plates via homotopy analysis method. Latin American Journal of Solids and Structures, 12(10), 1859–1869. https://doi.org/10.1590/1679-78251807

[31] Bayat, M., Pakar, I., & Domairry, G. (2012). Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review. Latin American Journal of Solids and Structures, 9(2), 1–93. https://doi.org/10.1590/S1679-78252012000200003

[32] Kazemi, M. A., Jafari, S. S., Musavi, S. M., & Nejati, M. (2018). Analytical solution of convective heat transfer of a quiescent fluid over a nonlinearly stretching surface using Homotopy Analysis Method. Results in Physics, 10, 164–172. https://doi.org/10.1016/j.rinp.2018.05.036

[33] Masoumnezhad, M., Kazemi, M., Askari, N., Taheri, M. H., & Ghamati, M. (2021). Semi-Analytical Solution of Unsteady Newtonian Fluid Flow and Heat Transfer between two Oscillation Plate under the Influence of a Magnetic Field. Karafan Journal, 18(1), 35–62. https://doi.org/10.48301/kssa.2021.131037

[34] Askari, N., Salmani, H., Taheri, M. H., Masoumnezhad, M., & Kazemi, M. A. (2021). Heat transfer of water–graphene oxide nanofluid magnetohydrodynamic flow through a channel in the presence of the induced magnetic field. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 235(11), 1966–1978. https://doi.org/10.1177/0954406220948902

[35] Aliakbar, V., Alizadeh-Pahlavan, A., & Sadeghy, K. (2009). The influence of thermal radiation on MHD flow of Maxwellian fluids above stretching sheets. Communications in Nonlinear Science and Numerical Simulation, 14(3), 779–794. http://dx.doi.org/10.1016/j.cnsns.2007.12.003

[36] Alizadeh-Pahlavan, A., Aliakbar, V., Vakili-Farahani, F., & Sadeghy, K. (2009). MHD flows of UCM fluids above porous stretching sheets using two-auxiliary-parameter homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation, 14(2), 473–488. https://doi.org/10.1016/j.cnsns.2007.09.011

[37] Nourbakhsh, S. H., Pasha Zanoosi, A. A., & Shateri, A. R. (2011). Analytical solution for off-centered stagnation flow towards a rotating disc problem by homotopy analysis method with two auxiliary parameters. Communications in Nonlinear Science and Numerical Simulation, 16(7), 2772–2787. https://doi.org/10.1016/j.cnsns.2010.10.018

[38] Chia, C. (1980). Nonlinear analysis of plates. McGraw- Hill. https://www.abebooks.com/9780070107465/Nonlinear-Analysis-Plates-Chia-Chuen-Yuan-0070107467/plp