### Modified Multi-level Residue Harmonic Balance Method for Solving Nonlinear Vibration Problem of Beam Resting on Nonlinear Elastic Foundation

Document Type : Research Paper

Authors

1 Department of Mathematics, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh

2 Department of Civil Engineering, Rajshahi University of Engineering & Technology, Rajshahi-6204, Bangladesh

Abstract

Nonlinear vibration behavior of beam is an important issue of structural engineering. In this study, a mathematical modeling of a forced nonlinear vibration of Euler-Bernoulli beam resting on nonlinear elastic foundation is presented. The nonlinear vibration behavior of the beam is investigated by using a modified multi-level residue harmonic balance method. The main advantage of the method is that only one nonlinear algebraic equation is generated at each solution level. The computational time of using the new method is much less than that spent on solving the set nonlinear algebraic equations generated in the classical harmonic balance method. Besides the new method can generate higher-level nonlinear solutions neglected by previous multi-level residue harmonic balance methods. The results obtained from the proposed method compared with those obtained by a classical harmonic balance method to verify the accuracy of the method which shows good agreement between the proposed and classical harmonic balance method. Besides, the effect of various parameters such as excitation magnitude, linear and nonlinear foundation stiffness, shearing stiffness etc. on the nonlinear vibration behaviors are examined

Keywords

Main Subjects

[1] Ansari, R., Pourashraf, T., Gholami, R., An exact solution for the nonlinear forced vibration of functionally graded nanobeams in thermal environment based on surface elasticity theory, Thin-Walled Structures, 93, 2015, pp. 169-176.
[2] Almitani, K. H., Buckling Behaviors of Symmetric and Antisymmetric Functionally Graded Beams, Journal of Applied and Computational Mechanics, 4(3), 2018, pp. 115-124.
[3] Eltaly, B., Structural performance of notch damaged steel beams repaired with composite materials, International Journal of Advanced Structural Engineering, 8(2), 2016, pp. 119–131.
[4] Lee, Y. Y., Su, R. K. L., Ng, C. F., Hui, C. K., The effect of modal energy transfer on the sound radiation and vibration of a curved panel: Theory and experiment, Journal of Sound and Vibration, 324, 2009, pp. 1003-1015.
[5] Meziane, M. A. A., Abdelaziz, H. H., Tounsi, A., An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions, Journal of Sandwich Structures & Materials, 16(3), 2014, pp. 293-318.
[6] Mahapatra, T. R., Kar, V. R., Panda, S. K., Large amplitude vibration analysis of laminated composite spherical panels under hygrothermal environment, International Journal of Structural Stability and Dynamics, 16(3), 2016, 1450105.
[7] Mahapatra, T. R., Kar, V. R., Panda, S. K., Nonlinear free vibration analysis of laminated composite doubly curved shell panel in hygrothermal environment, Journal of Sandwich Structures & Materials, 17(5), 2015, pp. 511-545.
[8] Panda, S. K., Mahapatra, T. R., Nonlinear finite element analysis of laminated composite spherical shell vibration under uniform thermal loading, Meccanica, 49(1), 2014, pp. 191-213.
[9] Mehar, K., Panda, S. K., Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field, Composite Structures, 143, 2016, pp. 336-346.
[10] Mehar, K., Panda, S. K., Dehengia, A., Kar, V. R., Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment, Journal of Sandwich Structures & Materials, 18(2), 2015, pp. 151-173.
[11] Hadji, L., Khelifa, Z., Abbes, A. B. E., A New Higher Order Shear Deformation Model for Functionally Graded Beams, KSCE Journal of Civil Engineering, 20(6), 2015, pp. 1-7.
[12] Ghasemi, A. R., Taheri-Behrooz, F., Farahani, S. M. N., Mohandes, M., Nonlinear free vibration of an Euler-Bernoulli composite beam undergoing finite strain subjected to different boundary conditions, Journal of Vibration and Control, 22(3), 2016, pp. 799–811.
[13] Reddy, J. N., Khodabakhshi, P., A unified beam theory with strain gradient effect and the von Karman nonlinearity, ZAMM Zeitschrift für Angewandte Mathematik und Mechanik, 97(1), 2017, pp. 70-91.
[14] Chen, W., Wu, Z., A new higher-order shear deformation theory and refined beam element of composite laminates, Acta Mechanica Sinica, 21, 2005, pp. 65–69.
[15] Reddy, J. N., Mahaffey, P., Generalized beam theories accounting for von Kármán nonlinear strains with application to buckling, Journal of Coupled Systems and Multiscale Dynamics, 1(1), 2013, pp. 120-134.
[16] Panda, S. K., Singh, B. N., Nonlinear free vibration of spherical shell panel using higher order shear deformation theory–a finite element approach, International Journal of Pressure Vessels and Piping, 86 (6), 2009, pp. 373-383.
[17] Singh, V. K., Panda, S. K., Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels, Thin-Walled Structures, 85, 2014, pp. 341-349.
[18] Kar, V. R., Panda, S. K., Nonlinear free vibration of functionally graded doubly curved shear deformable panels using finite element method, Journal of Vibration and Control, 22(7), 2016, pp. 1935-1949.
[19] Panda, S. K., Singh, B. N., Nonlinear free vibration analysis of thermally post-buckled composite spherical shell panel, International Journal of Mechanics and Materials in Design, 6(2), 2010, pp. 175-188.
[20] Panda, S. K., Singh, B. N., Non-linear free vibration analysis of laminated composite cylindrical/hyperboloid shell panels based on higher-order shear deformation theory using non-linear finite-element method, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 222(7), 2008, pp. 993-1006.
[21] Mehar, K., Panda, S. K., Bui, T. Q., Mahapatra, T. R., Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM, Journal of Thermal Stresses, 40(7), 2017, pp. 899-916.
[22] Mahapatra, T. R., Panda, S. K., Nonlinear free vibration analysis of laminated composite spherical shell panel under elevated hygrothermal environment: A micromechanical approach, Aerospace Science and Technology, 49, 2016, pp. 276-288.
[23] Kar, V. R., Panda, S. K., Geometrical nonlinear free vibration analysis of FGM spherical panel under nonlinear thermal loading with TD and TID properties, Journal of Thermal Stresses, 39(8), 2016, pp. 942-959.
[24] Kar, V. R., Panda, S. K., Free vibration responses of temperature dependent functionally graded curved panels under thermal environment, Latin American Journal of Solids and Structures, 12(11), 2015, pp. 2006-2024.
[25] Panda, S. K., Singh, B. N., Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel embedded with SMA fibers, Nonlinear Dynamics, 74(1-2), 2013, pp. 395-418.
[26] Hirwani, C. K., Panda, S. K., Numerical nonlinear frequency analysis of pre-damaged curved layered composite structure using higher-order finite element method, International Journal of Non-Linear Mechanics, 102, 2018, pp. 14-24.
[27] Hirwani, C. K., Mahapatra, T. R., Panda, S. K., Sahoo, S. S., Singh, V. K., Patle, B. K., Nonlinear free vibration analysis of laminated carbon/epoxy curved panels, Defence Science Journal, 67(2), 2017, pp. 207-2018.
[28] Kaci, A., Houari, M. S. A., Bousahla, A. A., Tounsi, A., Mahmoud, S. R., Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory, Structural Engineering and Mechanics, 65(5), 2018, pp. 621-631.
[29] Bourada, M., Kaci, A., Houari, M. S. A., Tounsi, A., A new simple shear and normal deformations theory for functionally graded beams, Steel and Composite Structures, 18(2), 2015, pp. 409–423.
[30] Bounouara, F., Benrahou, K. H., Belkorissat, I., Tounsoi, A., A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation, Steel and Composite Structures, 20(2), 2016, pp. 227-249.
[31] Bellifa, H., Benrahou, K.H., Bousahla, A. A., Tounsi, A., Mahmoud, S. R., A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams, Structural Engineering and Mechanics, 62(6), 2017, pp. 695 - 702.
[32] Bouafia, K., Kaci, A., Houari, M. S. A., Benzair, A., Tounsi, A., A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams, Smart Structures and Systems, 19(2), 2017, pp. 115-126.
[33] Mouffoki, A., Bedia, E. A. A., Houari, M. S. A., Tounsi, A., Hassan, S. , Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory, Smart Structures and Systems, 20(3), 2017, pp. 369-383.
[34] Attia, A. Bousahla, A. A., Tounsi, A., Mahmoud, S. R., Alwabli, A. S., A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations, Structural Engineering and Mechanics, 65(4), 2018, pp. 453-464.
[35] Azrar, L., Benamar, R., White, R. G., A semi-analytical approach to the nonlinear dynamic response problem of beams at large vibration amplitudes, Part II: multimode approach to the steady state forced periodic response, Journal of Sound and Vibration,255(1), 2002, pp. 1–41.
[36] Panda, S. K., Singh, B. N., Large amplitude free vibration analysis of thermally post-buckled composite doubly curved panel using nonlinear FEM, Finite Elements in Analysis and Design, 47(4), 2011, pp. 378-386.
[37] Hirwani, C. K., Patil, R. K., Panda, S. K., Mahapatra, S. S., Mandal, S. K., Srivastava, L., Buragohain, M. K., Experimental and numerical analysis of free vibration of delaminated curved panel, Aerospace Science and Technology, 54, 2016, pp. 353-370.
[38] Wickert, J. A., Non-linear vibration of a traveling tensioned beam, International Journal of Non-Linear Mechanics, 27, 1992, pp. 503–517.
[39] Nayfeh, A. H., Introduction to Perturbation Techniques, John Wylie, 1993.
[40] Fooladi, M., Abaspour, S. R., Kimiaeifar, A., Rahimpour, M., (2009) On the Analytical Solution of Kirchhoff Simplified Modelfor Beam by using of Homotopy Analysis Method, World Applied Sciences Journal, 6, 2009, pp. 297-302.
[41] Liao, S. J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
[42] Moeenfard, H., Mojahedi, M., Ahmadian, M. T., A homotopy perturbation analysis of nonlinear free vibration of Timoshenko micro beams, Journal of Mechanical Science and Technology, 25, 2011, pp. 557-565.
[43] Younesian, D., Saadatnia, Z., Askari, H., Analytical solutions for free oscillations of beams on nonlinear elastic foundations using the variational iteration method, Journal of Theoretical and Applied Mechanics, 50, 2012, pp. 639-652.
[44] Mickens, R. E., A generalization of the method of harmonic balance, Journal of Sound and Vibration, 111, 1986, pp. 515-518.
[45] Rahman, M. S., Haque, M. E., Shanta, S. S., Harmonic Balance Solution of Nonlinear Differential Equation (Non-Conservative), Journal of Advances in Vibration Engineering, 9(4), 2010, pp. 345-356.
[46] Lee, Y. Y., Analytic Solution for Nonlinear Multimode Beam Vibration Using a Modified Harmonic Balance Approach and Vieta’s Substitution, Shock and Vibration, 2015, Article ID 3462643, 6 pages.
[47] Foda, M. A., Analysis of large amplitude free vibrations of beams using the KBM method, Journal of Engineering and Applied Science, 42, 1995, pp. 125-138.
[48] Coskun, I., Engin, H., Non-linear vibrations of a beam on an elastic foundation, Journal of Sound and Vibration,223(3), 1999, pp. 335-354.
[49] Peng, J. S., Liu, Y., Yang, J., A semi-analytical method for nonlinear vibration of Euler-Bernoulli beams with general boundary conditions,Mathematical Problems in Engineering, 2010, Article ID 591786, 17 pages.
[50] Pirbodaghi, T., Ahmadian, M. T., Fesanghary, M., On the homotopy analysis method for non-linear vibration of beams, Mechanics Research Communications, 36, 2009, pp. 143–148.
[49] Sedighi, H. M., Shirazi, K. H., Zare, J., An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method, International Journal of Non-Linear Mechanics, 47, 2012, pp. 777-784.
[52] Motallebi, A. A., Poorjamshidian, M., Sheikh, J., Vibration analysis of a nonlinear beam under axial force by homotopy analysis method, Journal of Solid Mechanics, 6, 2014, pp. 289-298.
[53] Sedighi, H. M., Daneshmand, F., Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term, Journal of Applied and Computational Mechanics, 1(1), 2015, pp. 1-9.
[54] Baghani, M., Jafari-Talookolaei, R. A., Salarieh, H., Large amplitudes free vibrations and post-buckling analysis of unsymmetrically laminated composite beams on nonlinear elastic foundation, Applied Mathematical Modelling, 35, 2011, pp. 130–138.
[55] Fallah, A., Aghdam, M. M., Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation, European Journal of Mechanics A/Solids, 30, 2011, pp. 571-583.
[56] Yaghoobi, H., Torabi, M., An Analytical Approach to Large Amplitude Vibration and Post-Buckling of Functionally Graded Beams Rest on Non-Linear Elastic Foundation, Journal of Theoretical and Applied Mechanics, 51(1), 2013, pp. 39-52.
[57] Yaghoobi, H., Torabi, M., Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation, Applied Mathematical Modelling, 37, 2013, 8324–8340.
[58] Kanani, A. S., Niknam, H., Ohadi, A. R., Aghdam, M. M., Effect of nonlinear elastic foundation on large amplitude free and forced vibration of functionally graded beam, Composite Structures, 115, 2014, pp. 60–68
[59] Hasan, A. S. M. Z., Lee, Y. Y., Leung, A. Y. T., The multi-level residue harmonic balance solutions of multi-mode nonlinearly vibrating beams on an elastic foundation, Journal of Vibration and Control, 22(14), 2016, pp. 3218-3235.
[60] Rahman, M. S., Lee, Y. Y., New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem, Journal of Sound and Vibration,406, 2017, pp. 295-327.