A Modified Energy Balance Method to Obtain Higher-order Approximations to the Oscillators with Cubic and Harmonic Restoring Force

Document Type: Research Paper

Authors

1 Department of Mathematics, Rajshahi University of Engineering and Technology, Rajshahi-6204, BANGLADESH

2 Department of Mathematics, Faculty of Science, Sohag University, Sohag, 82524, EGYPT

3 Department of Mathematics, Ege University, Bornova-─░zmir, TURKEY

4 Graduate School of Science and Technology, Gunma University, Kiryu 376-8515, JAPAN

Abstract

This article analyzes a strongly nonlinear oscillator with cubic and harmonic restoring force and proposes an efficient analytical technique based on the modified energy balance method (MEBM). The proposed method incorporates higher-order approximations. After applying the proposed MEBM, a set of complicated higher-order nonlinear algebraic equations are obtained. Higher-order nonlinear algebraic equations are cumbersome to investigate especially in the case of a large initial oscillation amplitude. This limitation is overcome in the proposed method by using the iterative procedure based on the homotopy perturbation method. The approximated results agree well with the numerically obtained exact solutions. These third-order approximate solutions are found to be almost the same as exact solutions, which cannot be found using the existing energy balance method. Highly accurate result and simple solution procedure are advantages of this proposed method, which could be applied to other nonlinear oscillatory problems arising in nonlinear science and engineering.

Keywords

Main Subjects

[1] Leissa AW, Vibration of plates. Acoustical Society of America, Washington DC, 1993.

[2] Chopra AK, Dynamic of structures, theory and application to earthquake engineering. Prentice-Hall, New Jersey, 1995.

[3] Elmas N, Boyaci H, A new perturbation technique in solution of nonlinear differential equations by using variable transformation. Appl Math Comp, 227 (2014) 422-427.

[4] El-Naggar AM, Ismail GM, Analytical solution of strongly nonlinear Duffing Oscillators. Alex Engg J, 55 (2016) 1581-1585.

[5] Yao S, Cheng Z, The homotopy perturbation method for a nonlinear oscillator with a damping. J Low Freq Noise Vib Active Control, (2019) DOI: https://doi.org/10.1177/1461348419836344.

[6] Alam MS, Yeasmin IA, Ahamed MS, Generalization of the modified Lindstedt-Poincare method for solving some strong nonlinear oscillators. Ain Shams Engg J, 10 (2019) 195-201.

[7] Suleman M, Wu Q, Comparative solution of nonlinear quintic cubic oscillator using modified homotopy perturbation method. Ad Math Phy, 5 (2015) 932905.

[8] Razzak MA, Alam MZ, Sharif MN, Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems. Res Phy, 8 (2018) 231-238.

[9] Marinca V, Herisanu N, An optimal iteration method for strongly nonlinear oscillators. J Appl Math, 11 (2012) 906341.

[10] Sedighi HM, Shirazi KH, Attarzadeh MA, A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches. Acta Astronaut, 91 (2013) 245-250.

[11] Sedighi HM, Shirazi KH, Dynamic pull-in instability of double-sided actuated nano-torsional switches. Acta Mech Solida Sin, 28(1) (2015) 91-101.

[12] Akbari M, Ganji DD, Ahmadi A, Kachapi SHH, Analysing the nonlinear vibrational wave differential equation for the simplified model of tower cranes by algebraic method. Front Mech Engg, 9(1) (2014) 58-70.

[13] Akbari MR, Nimafar M, Ganji DD, Chalmiani HK, Investigation on non-linear vibration in arched beam for bridges construction via AGM method. Appl Math Comput, 298 (2017) 95-110.

[14] Beléndez A, Hernández A, Beléndez T et al, Solutions for conservative nonlinear oscillators using an approximate method based on chebyshev series expansion of the restoring force. Act Phy Pol A, 130(3) (2016) 667-678.

[15] Nhat LA, Using differentiation matrices for pseudospectral method solve Duffing Oscillator. J Non Sci Appl, 11 (2018) 1331-1336.

[16] Wang Q, Shi X, Li Z, A short remark on Ren-Hu’s modification of He’s frequency-amplitude formulation and the temperature oscillation in a polar bear hair. J Low Freq Noise Vib Active Control, (2019) DOI: 10.1177/1461348419831478.

[17] Daeichin M, Ahmadpoor MA, Askari H, Yildirim A, Rational energy balance method to nonlinear oscillators with cubic term. Asian-European J Math, 6(2) (2013) 1350019.

[18] Yazdi MK, Tehrani PH, Rational variational approaches to strong nonlinear oscillations. Int J Appl Comp Math, 3(2) (2017) 757-771.

[19] Shui X, Wang S, Closed-form numerical formulae for solutions of strongly nonlinear oscillators. Int J Non Mech, 103 (2018) 12-22.

[20] Hoang T, Duhamel D, Foret G et al, Frequency dependent iteration method for forced nonlinear oscillators. Appl Math Mod, 42 (2017) 441-448.

[21] Javidi M, Iterative methods to nonlinear equations. Appl Math Comput, 193 (2007) 360-365.

[22] Razzak MA, A simple new iterative method for solving strongly nonlinear oscillator systems having a rational and an irrational force. Alex Engg J, 57 (2018) 1099-1107.

[23] Yazdi MK, Tehrani PH, Frequency analysis of nonlinear oscillations via the global error minimization. Non Engg, 5(2) (2016) 87-92.

[24] Mickens RE, A generalization of the method of harmonic balance. J Sound Vib, 111 (1986) 115-518.

[25] Chowdhury MSH, Hosen MA, Ali MY, Ismail AF, An analytical technique to obtain higher-order approximate periods for nonlinear oscillator. IIUM Engg J, 19(2) (2018) 182-191.

[26] Hosen MA, Chowdhury MSH, A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force. Res Phy, 5 (2015) 111-114.

[27] Hosen MA, Rahman MS, Alam MS, Amin MR, An analytical technique for solving a class of strongly nonlinear conservative systems. Appl Math Comput, 218 (2012) 5474-5486.

[28] Belendez A, Gimeno E, Alvarez ML et al, A novel rational harmonic balance approach for periodic solutions of conservative nonlinear oscillators. Int J Non Sci Num Sim, 10(1) (2009) 13-26.

[29] Akbarzade M, Farshidianfar A, Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method. Appl Math Mod, 45 (2017) 393-404.

[30] Rahman MS, Lee YY, New modified multi-level residue harmonic balance method for solving nonlinearly vibrating double-beam problem. J Sound Vib, 406 (2017) 295-327.

[31] Younesian D, Esmailzadeh E, Askari H, Vibration analysis of oscillators with generalized inertial and geometrical nonlinearities. In: Dai L., Jazar R. (eds) Nonlinear Approaches in Engineering Applications. Springer, 2018.

[32] Sedighi HM, Shirazi KH, Noghrehabadi A, Application of recent powerful analytical approaches on the non-linear vibration of cantilever beams. Int J Nonlinear Sci Numer Simul, 13(7-8) (2012) 487-494.

[33] Sedighi HM, Size-dependent dynamic pull-in instability of vibrating electrically actuated microbeams based on the strain gradient elasticity theory. Acta Astronaut, 95 (2014) 111-123.

[34] Sedighi HM, Shirazi KH, Vibrations of micro-beams actuated by an electric field via Parameter Expansion Method. Acta Astronaut, 85 (2013) 19-24.

[35] Junfeng L, Li M, The VIM-Pade technique for strongly nonlinear oscillators with cubic and harmonic restoring force. J Low Freq Noise Vib Active Control, (2018) DOI: 10.1177/1461348418813612.

[36] He JH, Preliminary report on the energy balance for nonlinear oscillations. Mech Res Comm, 29 (2002) 107-111.

[37] Hosen MA, Chowdhury MSH, Ali MY, Ismail AF, An analytical approximation technique for the duffing oscillator based on the energy balance method. Italian J Pur Appl Math, 37 (2017) 455-466.

[38] Askari H, Saadatnia Z, Esmilzadeh E, Younesian D, Multi-frequency excitation of stiffened triangular plates for large amplitude oscillations. J Sound Vib, 333 (2014) 5817-5835.

[39] Molla MHU, Alam MS, Higher accuracy analytical approximations to nonlinear oscillators with discontinuity by energy balance method. Res Phy, 7 (2017) 2104-2110.

[40] Koudahoun LH, Kpomahou YJF, Adjaï DKK, Periodic solutions for nonlinear oscillations in elastic structures via energy balance method. (2016), http://vixra.org/abs/1611.0214.