### A Study on the Motion of Mass "m" in a Model for Buckling of a Column Using HAFF and EBM

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#### Abstract

He's Amplitude-Frequency Formulation (HAFF) and Energy Balance Method (EBM) are applied to obtain an approximate solution of three practical cases in terms of a nonlinear oscillation system. After finding an approximate solution of the nonlinear equation and comparing the obtained results by Numerical solution the effectiveness of the proposed methods with respect to the whole range of involved parameters is revealed as well as the excellent agreement with the approximate frequencies and periodic solutions with the exact ones. It is predicted that HAFF and EBM can be found widely applicable in engineering. *Copyright © 2020 Praise Worthy Prize - All rights reserved.*

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S. Natsiavas, On the dynamics of oscillators with bi-linear damping and stiffness, International Journal of Non-Linear Mechanics,vol. 25 n. 5, 1990,pp. 535.

L. Zhang, X.H. Ding, Solution of the prey–predator problem by He’s parameterized perturbation methods, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10 n. 9,2009,pp. 1223–1226.

M.Kazemi-Sangsereki,D.D. Ganji, M. Gorji-Bandpy, M. Mostofi, Homotopy Perturbation Method for a non-Linear Unforced and Undamped Duffing Equation, (2011) International review of Mechanical Engineering (IREME), 5 (3), pp. 436–441.

N.H. Sweilam, M.M. Khader, Application of He’s parameter-expansion method for the nonlinear differential equations, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10 n. 2,2009,pp. 265–272.

A.O. Gilchrist, The free oscillations of conservative quasilinear systems with two-degrees-of freedom, International Journal of Mechanical Science, vol. 3,1961,pp. 286.

G.J. Efstathiades, Combination tones in single mode motion of a class of non-linear systems with two-degrees-of-freedom, Journal of Sound and Vibration, vol. 34,1974,pp. 379.

G. Chen, Applications of a generalized Galerkin’s method to non-linear oscillations of two degree-of-freedom systems, Journal of Sound and Vibration, 119,1987,pp. 225.

L.N. Zhang, L. Xu, Determination of the limit cycle by He’s parameter-expansion for oscillators in a u(3)/(1 + u(2)),potential, ZeitschriftfürNaturforschung A, vol. 62 n. 7-8,2007,pp. 396–398.

L. Cveticanin, Vibrations of a coupled two-degree-of-freedom system, Journal of Sound and Vibration,vol. 247,2001,pp. 279.

K. Shin, M.J. Brennan, J.E. Oh, C.J. Harris, Analysis of disk brake noise using of a two degree-of-freedom model, Journal of Sound and Vibration, vol. 245 n. 5,2002,pp. 837.

L. Cveticanin, The motion of a two-mass system with non-linear connection, Journal of Sound and Vibration, vol. 252, 2002,pp. 361.

J.H. He, Homotopy perturbation method for bifurcation on nonlinear problems, International Journal of Non-Linear Sciences and Numerical Simulation, vol. 6,2005,pp. 207.

M.R. Isvandzibaei, M.Setareh, Analysis T.S.D.T Theory for a Functionally Graded Cylindrical Shell under Free Vibration with Effects Free-Free Boundary Conditions, (2011) International Review of Mechanical Engineering (IREME), 5 (4), pp. 664 - 671.

A. Beléndez, C. Pascual, S. Gallego, M. Ortuno, C. Neipp, Application of a modified He’s homotopy perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator, Physics Letters A, vol. 371,2007,pp. 421.

S.S. Ganji, S. Karimpour, D.D. Ganji, Z.Z. Ganji, Periodic solution for strongly nonlinear vibration systems by energy balance method, ActaApplicandaeMathematicae, vol. 106 n. 1, 2009,pp. 79-92.

B. Hervé, J.-J. Sinou, H. Mahé, L. Jézéquel, Estimation of the Non-linear Limit Cycles of Autonomous Mechanical Systems Based on a Constrained Non-linear Approach, (2007) International Review of Mechanical Engineering (IREME), 1 (4) ,pp. 356 - 362.

A. Barari, M. Omidvar, A.R. Ghotbi, D.D. Ganji, Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations, ActaApplicandaeMathematicae, vol. 104 n. 2,2008,pp. 161–171.

M.G. Sfahani, A. Barari, M. Omidvar, S.S. Ganji, G. Domairry, Dynamic response of inextensible beams by improved energy balance method, Proceedings of the Institution of Mechanical Engineers, Part K Journal of multi-body Dynamics, vol. 225 n. 1, 2010,pp. 66-73.

M. Momeni, N. Jamshidi, A. Barari, D.D. Ganji, Application of He’s energy balance method to duffing harmonic oscillators, International Journal of Computer Mathematics, vol. 88 n. 1,2011,pp. 135–144.

S.M. Hashemi, A. Roach, Free Vibration of Helical Springs Using A Dynamic Finite Element Mesh Reduction Technique, (2008) International Review of Mechanical Engineering (IREME), 2 (3), pp. 435–448.

S.S. Ganji, A. Barari, L.B. Ibsen, G. Domairry, Differential transform method for mathematical modeling of jamming transition problem in traffic congestion flow, Central European Journal of Operations Research, vol. 20 n. 1,2010,pp. 87-100.

L.B. Ibsen, A. Barari, A. Kimiaeifar, Analysis of highly nonlinear oscillation systems using He’s max–min method and comparison with homotopy analysis and energy balance methods, Sadhana,vol. 35 n. 4,2010,pp. 1–16.

S.S. Ganji, D.D. Ganji, A.G. Davodi, S. Karimpour, Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max–min approach, Applied Mathematical Modelling, vol. 34 n. 9,2010,pp. 2676–2684.

M.G. Sfahani, S.S. Ganji, A. Barari, H. Mirgolbabaei, G. Domairry, Analytical solutions to nonlinear conservative oscillator with fifth-order non-linearity, Earthquake Engineering and Engineering Vibration, vol. 9 n. 3,2010,pp. 367–374.

S. Berger, P. Ragot, JJ. Sinou, E. Aubry, Model of Chatter Vibrations and Stability Analysis of a Non-linear Wiper System, (2008) International Review of Mechanical Engineering (IREME), 2 (3), pp. 349–356.

J.H. He, An improved amplitude-frequency formulation for nonlinear oscillators. International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9 n. 2,2008,pp. 211.

W.H. Jefferys, Some dynamical systems of two degrees of freedom in Clestial mechanics, The Astronomical Journal, vol. 71,1966, pp. 306.

S.M. Megahed, A.K. Abd El-Razik, Vibration control of two degrees of freedom system using variable inertia vibration absorbers: modeling and simulation, Journal of Sound and Vibration, vol. 329 n. 23,2010,pp. 4841–4865.

T.C. Harris, Periodic motions of arbitrarily long periods in non-linear spring-mass systems, International Journal of Non-Linear Mechanics, vol. 5,1970,pp. 491–500.

G.W. Luo, J.N. Yu, J.G. Zhang, Periodic-impact motions and bifurcations of a dual component system, Nonlinear Analysis: Real World Applications, vol. 7, 2006,pp. 813–828.

G.W. Luo, X.H. Lv, L. Ma, Periodic-impact motions and bifurcations in dynamics of a plastic impact oscillator with a frictional slider, European Journal of Mechanics: A/Solids, vol. 27,2008,pp. 1088–1107.

SHA. HashemiKachapi, R.V. Dukkipati, S.G. HashemiKachapi, S.Mey. HashemiKachapi, S.Meh. HashemiKachapi, S.K. HashemiKachapi, Analysis of the nonlinear vibration of a two-mass-spring system with linear and nonlinear stiffness, Nonlinear Analysis: Real World Applications, vol. 11 n. 3,2010,pp. 1431–1441.

E.V. Abrarova, Orbital steady motions of a system of two bodies with an elastic connection, Aerospace Science and Technology, vol. 8,1998,pp. 525–535.

F.L. Chernousko, The optimum rectilinear motion of a two-mass system, Journal of Applied Mathematics and Mechanics, vol. 66 n. 1,2002, pp. 1–7.

A.H. Nayfeh, D.T. Mook, Nonlinear Oscillations(Wiley-Interscience, New York, NY, USA, 1979).

H. Pashaei, D.D. Ganji, M. Akbarzade, Applications of the energy balance method for strongly nonlinear oscillators, Progress in Electromagnetic Research M, vol. 2,2008,pp. 47–56.

P. Hagedorn, Nonlinear Oscillations translated by Wolfram Stadler(Clarendon Press, Oxford, 1981).

J.H. He, Varational iteration method-some recent results and new interpretations, Journal of Computational and Applid Mathematics, vol. 207 n. 1,2007,pp. 3–17.

J.H. He, Preliminary report on the energy balance for nonlinear oscillations, Mechanics Research Communications, vol. 29,2002,pp. 107–111.

S.S. Ganji, M. GhalamiSfahani, S.M. ModaresTonekaboni, A.K. Moosavi, D.D. Ganji, Higher-order solutions of coupled systems using the parameter expansion method, Mathematical Problems in Engineering, 2009.

S.S. Ganji, A.Barari ,D.D. Ganji, Approximate analysis of two-mass-spring systems and buckling of a column, Computers and mathematics with applications, vol. 61 n. 4,2011,pp. 1088-1095.

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