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The stability conditions of the cubic damping Van der Pol-Duffing oscillator using the HPM with the frequency-expansion technology

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Języki publikacji
EN
Abstrakty
EN
In this paper, we perform the frequency-expansion formula for the nonlinear cubic damping van der Pol’s equation, and the nonlinear frequency is derived. Stability conditions are performed, for the first time ever, by the nonlinear frequency technology and for the nonlinear oscillator. In terms of the van der Pol’s coefficients the stability conditions have been performed. Further, the stability conditions are performed in the case of the complex damping coefficients. Moreover, the study has been extended to include the influence of a forcing van der Pol’ oscillator. Stability conditions have been derived at each resonance case. Redoing the perturbation theory for the van der Pol oscillator illustrates more of a resonance formulation such as sub-harmonic resonance and super-harmonic resonance. More approximate nonlinear dispersion relations of quartic and quintic forms in the squaring of the extended frequency are derived, respectively.
Rocznik
Strony
31--44
Opis fizyczny
Bibliogr. 28 poz.
Twórcy
autor
  • Department of Mathematics, Faculty of Education Ain Shams University, Roxy, Cairo, Egypt
Bibliografia
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  • [4] Ju, P., & Xue, X. (2015). Global residue harmonic balance method for large-amplitude oscillations of a nonlinear system. J. Appl. Math. Model., 39, 449-454.
  • [5] He, J.H. (2010). Hamiltonian approach to nonlinear oscillators. Phys. Lett. A, 374, 2312-2314.
  • [6] Liao, S.J., & Cheung, A.T. (1998). Application of homotopy analysis method in nonlinear oscillations. ASME J. Appl. Mech., 65, 914-922.
  • [7] He, J.H. (2008). Max-min approach to nonlinear oscillators. Int. J. Nonlinear Sci. Numer. Simul., 9, 207-210.
  • [8] Akbarzede, M., Langari J., & Ganji D.D. (2011). A coupled homotopy variational method and variational formulation applied to nonlinear oscillators with and without discontinuities. ASME J. Vib. Acoust., 133, 044501.
  • [9] Guo, Z., & Leung, A.Y.T. (2010). The iterative homotopy harmonic balance method for conservative Helmholtz-Duffing oscillators. J. Appl. Math. Comp., 215, 3163-3169.
  • [10] Luo, A.C.J., & Yu, B. (2015). Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. J. Vib. Cont., 21(5), 896-906.
  • [11] He, J.H. (2006). Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern Phys. B, 20, 1141-1199.
  • [12] He, J.H. (2008). An improved amplitude frequency formulation for nonlinear oscillators. Int. J. Nonlinear Sci. Numer. Simul., 9(2), 211-212.
  • [13] García, A.G. (2017). An amplitude-period formula for a second order nonlinear oscillator. Nonlinear Sci. Lett. A, 8(4), 340-347.
  • [14] Shou, H.D., & He, J.H. (2007). Application of parameter-expanding method to strongly nonlinear oscillators. Int. J. Nonlinear Sci. Numer. Simul., 8, 121-124.
  • [15] He, J.H. (2010). A note on the homotopy perturbation method. Thermal Science, 14(2), 565-568.
  • [16] He, J.H. (2014). Homotopy perturbation method with two expanding parameters. Indian Journal of Physics, 88(2), 193-196.
  • [17] He, J.H. (2012). Homotopy perturbation method with an auxiliary term. Abstract and Applied Analysis, DOI: 10.1155/2012/857612.
  • [18] El-Dib, Y.O. (2018). Stability of a strongly displacement time-delayed Duffing oscillator by the multiple-scales-homotopy perturbation method. J. of Appl. Compu. Mech., 4(4), 260-274.
  • [19] Wu, Y., & He, J.H. (2018). Homotopy perturbation method for nonlinear oscillators with coordinate dependent mass. Results in Physics, 10, 270-271.
  • [20] Ebenezer, B., Asamoah, J.K.K., Khan I., et al. (2017). The multi-step homotopy analysis method for a modified epidemiological model for measles disease. Nonlinear Sci. Lett. A, 8(3), 320-332.
  • [21] El-Dib, Y.O. (2017). Homotopy perturbation for excited nonlinear equations. Science and Engineering Applications, 2(1), 96-108.
  • [22] El-Dib, Y.O. (2017). Multiple scales homotopy perturbation method for nonlinear oscillators. Nonlinear Sci. Lett. A, 8(4), 352-364.
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  • [24] El-Dib, Y.O., & Moatimid, G.M. (2018). On the coupling of the homotopy perturbation and Frobenius method for exact solutions of singular nonlinear differential equations. Nonlinear Sci. Lett. A, 9(3), available online July 5, 220-230.
  • [25] Domany, E., & Gendelman, O.V. (2013). Dynamic responses and mitigation of limit cycle oscillations in Van der Pol-Duffing oscillator with nonlinear energy sink. J. Sound Vib., 332(21), 5489-5507.
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Uwagi
Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-260006a6-412a-4f9a-af33-43be0618d6a9
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