---

Journal of Applied Nonlinear Dynamics 11(4) (2022) 949--960 | DOI:10.5890/JAND.2022.12.012

Serge Bruno Yamgou\'{e}$^{1}$, Bonaventure Nana$^{1}$, Fran\c{c}ois Beceau Pelap$^{2}$

$^{1}$ Department of Physics, Higher Teacher Training College Bambili, The University of Bamenda, P.O. Box 39 Bamenda, Cameroon

$^{2}$ Unit'{e} de Recherche de M'{e}canique et de Mod'{e}lisation des Syst`{e}mes Physiques (UR-2MSP), Facult'{e} des Sciences, de Dschang, BP 69, Dschang, Cameroun

Download Full Text PDF

 

Abstract

In this paper, we apply a nonlinear time transformation to find accurate analytical solutions of a generalized Duffing-harmonic oscillator having a rational form for the restoring force. The transformation considered here has the interesting and useful feature that it leads to fully explicit solutions, in contrast to most such transformations. Its combination with the single term harmonic balance is found to be equivalent to the cubication method. The accuracy of the results can be improved by using the Newton linearization technique when looking for an-harmonic approximate solutions. It is observed that this process not only increases the accuracy of the solutions, but it can also enlarge the range of validity of the approximations in the space of parameters as compared to the method of cubication.

References

1. [1]	Nayfeh, A.H. and Mook, D.T. (2008), Nonlinear Oscillation, John Wiley: New York.

2. [2]	Amore, P. and Aranda, A. (2003), Presenting a new method for the solution of nonlinear problems, Physics Letters A, 316(3), 218-225.

3. [3]	Amore, P., Aranda, A., and Fern{a}ndez, F.M. (2005), Comparison of alternative improved perturbative methods for nonlinear oscillations, Physics Letters A, 340(1), 201-208.

4. [4]	He, J.-H. (1999), Variational iteration method--a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics, 34(4), 699-708.

5. [5]	He, J.-H. (2003), Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation, 135(1), 73-79.

6. [6]	He, J.-H. (2010), Hamiltonian approach to nonlinear oscillators, Physics Letters A, 374(23), 2312-2314.

7. [7]	Navarro, H.A. and Cveticanin, L. (2016), Amplitude-frequency relationship obtained using hamiltonian approach for oscillators with sum of non-integer order nonlinearities, Applied Mathematics and Computation, 291, 162-171.

8. [8]	Mickens, R.E. (1984), Comments on the method of harmonic balances, Journal of Sound and Vibration, 94(3), 456-460.

9. [9]	Wu, B.S., Sun, W.P., and Lim, C.W. (2006), An analytical approximate technique for a class of strongly non-linear oscillators, International Journal of Non-Linear Mechanics, 41(6), 766-774.

10. [10]	Lim, C.W. and Wu, B.S. (2005), Accurate higher-order approximations to frequencies of nonlinear oscillators with fractional powers, Journal of Sound and Vibration, 281(3), 1157-1162.

11. [11]	Lim, C.W. and Wu, B.S. (2003), A new analytical approach to the duffing-harmonic oscillator, Physics Letters A, 311, 365-373.

12. [12]	Yamgou{e}, S.B. and Kofan{e}, T.C. (2007), Application of the krylov--bogoliubov--mitropolsky method to weakly damped strongly non-linear planar hamiltonian systems, International Journal of Non-Linear Mechanics, 42(10), 1240-1247.

13. [13]	Wu, B.S. and Sun, W.P. (2011), Construction of approximate analytical solutions to strongly nonlinear damped oscillators, Archive of Applied Mechanics, 81(8), 1017-1030.

14. [14]	Luo, A.J.C., Xu, Y. and Chen, Z. (2016), On periodic motions in the first-order nonlinear systems, ASME International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers Digital Collection, 81(8), 1017-1030.

15. [15]	Xu, Y., Luo, A.J.C., and Chen, Z. (2017), Analytical solutions of periodic motions in 1-dimensional nonlinear systems, Chaos, Solitons and Fractals, 97, 1-10.

16. [16]	Wang, H. and Chung, K.-W. (2012), Analytical solutions of a generalized duffing-harmonic oscillator by a nonlinear time transformation method, Physics Letters A, 376(12), 1118-1124.

17. [17]	Cao, Y.Y., Chung, K.-W. and Xu, J. (2011), A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method, Nonlinear Dynamics, 64(3), 221-236.

18. [18]	Xu, J., Chan, H.S.Y. and Chung, K.-W. and (1996), Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method, Nonlinear Dynamics, 11(3), 213-233.

19. [19]	Abramowitz, M. and Stegun, I.A. (1972), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover: New York.

20. [20]	Gradshteyn, I.S. and Ryzhik, I.M. (2007), Table of Integrals, Series, and Products (seventh edition), Academic Press: San Diego.

21. [21]	Mickens, R.E. (2002), Fourier representations for periodic solutions of odd-parity systems, Journal of Sound and Vibration, 258(2), 398-401.

22. [22]	Mickens, R.E. and Semwogerere, D. (1996), Fourier analysis of a rational harmonic balance approximation for periodic solutions, Journal of Sound and Vibration, 195(3), 528-550.

23. [23]	Yamgou{e}, S.B., Bogning, J.R. and Kenfack-Jiotsa, A. (2010), Rational harmonic balance-based approximate solutions to nonlinear single-degree-of-freedom oscillator equations, Physica Scripta, 81(3), 035003.

24. [24]	Febbo, M. (2011), Harmonic response of a class of finite extensibility nonlinear oscillators, Physica Scripta, 81(6), 065009.

25. [25]	Yamgou{e}, S.B., Nana, B. and Lekeufack, O.T. (2015), Improvement of harmonic balance using jacobian elliptic functions, Journal of Applied Mathematics and Physics, 3, 680-690.