The Rank Upgrading Technique for a Harmonic Restoring Force ‎of Nonlinear Oscillators

Document Type : Research Paper


1 Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

2 Department of Mathematics, Faculty of Applied Science, Umm Al-Qura University, Makkah, Saudi Arabia


An enhanced analytical technique for nonlinear oscillators having a harmonic restoring force is proposed. The approach is passed on the change of the auxiliary operator by another suitable one leads to obtain a periodic solution. The fundamental idea of the new approach is based on obtaining an alternative equation free of the harmonic restoring forces. This method is a modification of the homotopy perturbation method. The approach allows not only an actual periodic solution but also the frequency of the problem as a function of the amplitude of oscillation. Three nonlinear oscillators including restoring force, the simple pendulum motion, the cubic Duffing oscillator, the Sine-Gordon equation are offered to clarify the effectiveness and usefulness of the proposed technique. This approach allows an effective mathematical approach to noise and uncertain properties of nonlinear vibrations arising in physics and engineering.


Main Subjects

[1] Jin, X., Liu, M.N., Pan, F., Li, Y., Fan J., Low frequency of a deforming capillary vibration, part 1: mathematical model, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4), 2019, 1676-1680.
[2] He, J.H., Jin, X., A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube, Mathematical Methods in the Applied Sciences, 2020, 1-8,
[3] Liu, P., He, J.H., Geometrical potential: an explanation on of nanofibers wettability, Thermal Science, 22, 2018, 33-38.
[4] Zhou, C.J., Tian, D., He, J.H., What factors affect lotus effect?, Thermal Science, 22, 2018, 1737-1743.
[5] Li, X.X., He, J.H., Nanoscale adhesion and attachment oscillation under the geometric potential, part 1: the formation mechanism of nanofiber membrane in the electrospinning, Results in Physics, 12, 2019, 1405-1410.
[6] He, J.H., Variational principle and periodic solution of the Kundu–Mukherjee–Naskar equation, Results in Physics, 17, 2020, 103031.
[7] Fan, J., Zhang, Y., Liu, Y., et al., Explanation of the cell orientation in a nanofiber membrane by the geometric potential theory, Results in Physics, 15, 2019, 102537.
[8] Yang Z.P., Dou F., Yu T., et al., On the cross-section of shaped fibers in the dry spinning process: physical explanation by the geometric potential theory, Results in Physics, 14, 2019, 102347.
[9] Tian, D., Li, X.X., He, J.H., Geometrical potential and nanofiber membrane's highly selective adsorption property, Adsorption Science and Technology, 37(5-6), 2019, 367-388.
[10] Yuste, S.B., Sánchez, Á.M., A weighted mean-square method of Cubication for nonlinear oscillators, Journal of Sound and Vibration, ‎‎134(3), 1989, 423–433.
[11] Yuste, S.B., Cubication of nonlinear oscillators using the principle of harmonic balance, International Journal of Non-Linear Mechanics, ‎‎27(3), 2002, 347–356.
[12] Sinha, S.C., Srinivasan, P., A weighted mean square method of linearization in non-linear oscillations, Journal of Sound and Vibration, ‎‎1971, 16, 139–148.
[13] 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, Acta Physica Polonica A, 130(3), 2016, 667-678.
[14] Starossek, U., Exact analytical solutions for forced cubic restoring force oscillator, Nonlinear Dynamics, 83, 2016, 2349–2359.
[15] Marinca, V., Herisanu, N., An optimal iteration method for strongly nonlinear oscillators, Journal of Applied Mathematics, 11, 2012, ‎‎906341.
[16] Hosen, M.A., Chowdhury, M.S.H., A new reliable analytical solution for strongly nonlinear oscillator with cubic and harmonic restoring force, Results in Physics, 5, 2015, 111-114.
[17] Junfeng, L., Li, M., The VIM-Pade technique for strongly nonlinear oscillators with cubic and harmonic restoring force, Journal of Low Frequency Noise, Vibration and Active Control, 38 (3-4), 2018, 1272-1278.
[18] Akbari, M.R., Nimafar, M., Ganji, D.D., Chalmiani, H.K., Investigation on non-linear vibration in arched beam for bridges construction via AGM method, Applied Mathematics and Computation, 298, 2017, 95-110.
[19] Nhat, L.A., Using differentiation matrices for pseudo spectral method solve Duffing Oscillator, Journal of Nonlinear Sciences and Applications, 11, 2018, 1331-1336.
[20] 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, Journal of Low Frequency Noise, Vibration and Active Control, 2019,
[21] Yazdi, M.K., Tehrani, P.H., Rational variational approaches to strong nonlinear oscillations, International Journal of Applied and Computational Mathematics, 3(2), 2017, 757-771.
[22] Shui, X., Wang, S., Closed-form numerical formulae for solutions of strongly nonlinear oscillators, International Journal of Non-Linear Mechanics, 103, 2018, 12-22.
[23] Hoang., T., Duhamel, D., Foret, G., et al., Frequency dependent iteration method for forced nonlinear oscillators, Applied Mathematical Modelling, 42, 2017, 441- 448.
[24] Javidi, M., Iterative methods to nonlinear equations, Applied Mathematics and Computation, 193, 2007, 360-365.
[25] Razzak, M.A., A simple new iterative method for solving strongly nonlinear oscillator systems having a rational and an irrational force, Alexandria Engineering Journal, 57, 2018, 1099-1107.
[26] Mickens, R.E., A generalization of the method of harmonic balance, Journal of Sound and Vibration, 111, 1986, 115-518.
[27] Chowdhury, M.S.H., Hosen, M.A., Ali, M.Y., Ismail, F.A., An analytical technique to obtain higher-order approximate periods for nonlinear oscillator, IIUM Engineering Journal, 19(2), 2018, 182-191
[28] Akbarzade, M., Farshidianfar, A. , Nonlinear transversely vibrating beams by the improved energy balance method and the global residue harmonic balance method, Applied Mathematical Modelling, 45, 2017, 393-404.
[29] Haller, E., Hart, R., Mark, M.J., et al., Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons, Nature, ‎‎466, 2010, 597-601.
[30] Sedighi, H.M., Shirazi, K.H., Bifurcation analysis in hunting dynamical behavior in a railway bogie: Using novel exact equivalent functions for discontinuous nonlinearities, Scientia Iranica, 19(6), 2012, 1493-1501.
[31] Sedighi, H.M., Shirazi, K.H., Asymptotic approach for nonlinear vibrating beams with saturation type boundary condition, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 227(11), 2013, 2479-2486.
[32] Sedighi, H.M., Size-dependent dynamic pull-in instability of vibrating electrically actuated micro-beams based on the strain gradient elasticity theory, Acta Astronautica, 95(1), 2014, 111-123.
[33] Sedighi, H.M., Daneshmand, F., Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He’s iteration perturbation method, Journal of Mechanical Science and Technology, 28, 2014, 3459–3469.
[34] Schweigler, T., Kasper, V., Erne, S., et al., Experimental characterization of a quantum many-body system via higher-order correlations, Nature, 545, 2017, 323-335.
[35] Jing, D., Hatami, M., Peristaltic Carreau-Yasuda nanofluid flow and mixed heat transfer analysis in an asymmetric vertical and tapered wavy wall channel, Reports in Mechanical Engineering, 1(1), 2020, 128-140.
[36] Mahmudov, N.I., Huseynov, I.T., Aliev, N.A., Aliev, F.A., Analytical approach to a class of Bagley-Torvik equations, TWMS Journal of Pure and Applied Mathematics, 11(2), 2020, 238-258.
[37] He, J.H., El-Dib, Y.O., The reducing rank method to solve third-order Duffing equation with the homotopy perturbation, Numerical Methods for Partial Differential Equations, 2020, 1–9,
[38] Yao, S., Cheng, Z., The homotopy perturbation method for a nonlinear oscillator with a damping, Journal of Low Frequency Noise, Vibration and Active Control, 38, (3-4), 2019, 1110-1112.
[39] El-Dib, Y.O., Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple-scales method, Pramana - Journal of Physics, 92(1), 2019, 7.
[40] Alam, M.S., Yeasmin, I.A., Ahamed, M.S., Generalization of the modified Lindstedt-Poincare method for solving some strong nonlinear oscillators, Ain Shams Engineering Journal, 10, 2019, 195-201.
[41] El-Dib, Y.O., Homotopy perturbation method with rank upgrading technique for the superior nonlinear oscillation, Mathematics and Computers in Simulation, 182, 2021, 555–565.
[42] Razzak, M.A., Alam, M.Z., Sharif, M.N., Modified multiple time scale method for solving strongly nonlinear damped forced vibration systems, Results in Physics, 8, 2018, 231-238.
[43] El-Dib, Y.O., Moatimid, G.M., Stability Configuration of a Rocking Rigid Rod over a Circular Surface Using the Homotopy Perturbation Method and Laplace Transform, Arabian Journal for Science and Engineering, 44, 2019, 6581–6591.
[44] El-Dib, Y.O., Multiple scales homotopy perturbation method for nonlinear oscillators, Nonlinear Science Letters A, 8, 2017, 352–364.
[45] Filobello-Nino, U., Vazquez-Leal, H., Jimenez-Fernandez, V.M., et al., Enhanced classical perturbation method, Nonlinear Science Letters A, 9, 2018, 172–185.
[46] Li, X.X., He, C.H., Homotopy perturbation method coupled with the enhanced perturbation method, Journal of Low Frequency Noise, Vibration and Active Control, 38 (3-4), 2018, 1399-1403.
[47] He, J.H., El-Dib, Y.O., Periodic property of the time-fractional Kundu–Mukherjee–Naskar equation, Results in Physics, 19, 2020, 103345.
[48] He, J.H., Homotopy Perturbation Method with an Auxiliary Term, Abstract and Applied Analysis, 2012, 857612.
[49] El-Dib, Y.O., Multi-homotopy perturbation technique for solving nonlinear partial differential equation with Laplace transforms, Nonlinear Science Letters A, 9(4), 2018, 349-359.
[50] Yu, D.N., He, J.H., Garcıa, A.G., Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4), 2018, 1540-1554.
[51] Shen, Y., El-Dib, Y.O., A periodic solution of the fractional sine-Gordon equation arising in architectural engineering, Journal of Low Frequency Noise, Vibration and Active Control, 2020, 1461348420917565.
[52] He, J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 1999, 257–262.
[53] He, J.H., El-Dib, Y.O., Homotopy perturbation method for Fangzhu oscillator, Journal of Mathematical Chemistry, 58, 2020, 2245–2253.
[54] Nayfeh, A.H., Perturbation methods, Wiley, New York, USA, 1973.
[55] Tabor, T., The Sine-Gordon Equation An Introduction, Wiley, New York, USA, 1989.
[56] Sun, Y., New Travelling Wave Solutions for Sine-Gordon Equation, Journal of Applied Mathematics, 2014, 841416, 4 pages.
[57] Zarmi, Y., Nonlinear quantum-mechanical system associated with Sine-Gordon equation in (1+2) dimensions, Journal of Mathematical Physics, 55(10), 2014, 103510.