Approximate solution of a nonlinear oscillator of mass attached to a stretched elastic wire

Md. Zahangir Alam, Ismot Ara Yeasmin, Nazmul Sharif

Abstract

This paper presents the application of the harmonic balance method to solve a nonlinear oscillator governing the oscillations of a mass attached to a stretched elastic wire. The oscillator, characterized by a restoring force parameter , serves as a model for confined systems where relativistic quantum effects are significant, including nuclear and quark models of matter and particle momentum. The method is also applicable to the study of massive photon propagation in space-time and quantum shell correction estimation. The proposed procedure is effective across the entire range of the parameter , showing excellent agreement with numerical simulations for both approximate frequencies and periodic solutions. Additionally, this approach is simplified by its avoidance of elliptic integrals, enhancing its utility and accessibility for researchers in the field.

Conclusion

A new analytical approach of harmonic balance method is presented for determining the approximate frequencies as well as the approximate periodic solutions of strongly nonlinear oscillator govern by a mass attached to a stretched elastic wire. The main advantage of this paper is its simplicity. Also, the results obtained by present method are nicely coincidence with the results corresponding to numerical one for both small and large values of amplitudes. Though the present approach is exemplified for a strongly nonlinear oscillator of a mass attached to a stretched elastic wire, it is also applicable for parallel strongly nonlinear oscillators for example Duffing-harmonic oscillator and relativistic oscillator.

References

[1] J. B. Marion, “Classical Dynamics of Particles and System,” Harcourt Brace Jovanovich, San Diego, CA, 1970.

[2] N. N. Krylov and N. N. Bogoliubov, “Introduction to Nonlinear Mechanics,” Princeton University Press, New Jersey, 1947.

[3] N. N. Bogolyubov and Yu. A. Mitropolskii, “Asymptotic methods in the theory of nonlinear oscillations,” Gordan and Breach 1961.

[4] A. H. Nayfeh, “Perturbation Method,” New York: John Wiley & Sons, 1973.

[5] A. H. Nayfeh and D. T. Mook, “Nonlinear Oscillations,” John Wiley & Sons, New York, 1979.

[6] A. H. Nayfeh, “Introduction to Perturbation Techniques,” John Wiley & Sons, New York, 1981.

[7] P. Amore and A. Aranda, “Improved Lindstedt-poincare method for the solution of nonlinear problems, ” Journal of Sound and Vibration, vol. 283, no. 3–5, pp. 1115–1136, May 2005.

[8] J. Awrejcewicz, I. V. Andrianov, and L. I. Manevitch, “Asymptotic Approaches in Nonlinear Dynamics, New trends and applications,” Springer, Berlin, Heidelberg, pp. 14–19, 1998.

[9] Y. K. Cheung, S. H. Mook and S. L. Lau, “A modified Lindstedt-Poincare method for certain strongly nonlinear oscillators, ” International Journal of Non-Linear Mechanics, vol. 26, no. 3-4, pp. 367-378, 1991.

[10] J. H. He, “Modified Lindstedt-Poincare methods for some strongly nonlinear oscillations-I: expansion of a constant, ” International Journal of Non-Linear Mechanics, vol. 37, no. 2, pp. 309-314, March 2002.

[11] T. Ozis and A. Yildirim, “Determination of periodic solution for a force by He’s modified Lindstedt-Poincare method, ” Journal of Sound and Vibration, vol. 301 pp. 415–419, 2007.

[12] A. Belendez, C. Pascual, S. Gallego, M. Ortuno and C. Neipp, “Application of a modified He’s homotopy perturbation method to obtain higher-order approximation of an force nonlinear oscillator,” Physics Letters A. vol. 371, no. 5-6, pp. 421–426, 2007.

[13] J. H. He, “Some asymptotic methods for strongly nonlinear equations, ” International journal of Modern physics B, vol. 20, no. 10, pp. 1141–1199, 2006.

[14] M. S. Alam, I. A. Yeasmin and M. S. Ahamed, “Generalization of the modified Lindstedt-Poincare method for solving some strong nonlinear oscillators, ” Ain Shams Engineering Journal, vol. 10, no. 1, pp. 195-201, March 2019.

[15] A. Beléndez, A. Hernández, T. Beléndez, A. Márquez and C. Neipp, “Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators,” Chaos, Solitons & Fractals, vol. 37, no. 3, pp. 770-780, August 2008.

[16] J. H. He, “Homotopy perturbation technique, ” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262, August 1999.

[17] N. Anjum N and J. H. He, “Homotopy perturbation method for N/MEMS oscillators, ” Mathematical Methods in the Applied Sciences, pp. 1-15, June 2020. Doi: org/ https://doi.org/10.1002/mma.6583

[18] J. H. He, “Homotopy perturbation method for bifurcation on nonlinear problems, ” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 207-208, June 2005.

[19] A. Belendez, T. Belendez, C. Neipp, A. Hernandez and M. L. Alvarez, “Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method,” Chaos, Solitons and Fractals, vol. 39, no. 2, pp. 746-764, January 2009.

[20] Haque, B. I., Alam, M., & Rahman, M. M. (2013). Modified solutions of some oscillators by iteration procedure. Journal of the Egyptian Mathematical Society, 21(2), 142-147.

[21] Ikramul Haque, B., Asifuzzaman, M., & Kamrul Hasan, M. (2017). Improvement of analytical solution to the inverse truly nonlinear oscillator by extended iterative method. International Conference on Mathematics and Computing.

[22] Haque, B. I., & Flora, S. A. (2020). On the analytical approximation of the quadratic non-linear oscillator by modified extended iteration method. Applied Mathematics and Nonlinear Sciences, 6(1), 527-536.

[23] Haque, B. I., & Hossain, M. A. (2021). An Effective Solution of the Cube‐Root Truly Nonlinear Oscillator: Extended Iteration Procedure. International Journal of Differential Equations, 2021(1), 7819209.

[24] Hossain, M. A., & Haque, B. I. (2022). An Analytic Solution for the Helmholtz-Duffing Oscillator by Modified Mickens’ Extended Iteration Procedure. International Conference on Mathematics and Computing,

[25] Ali, M. I., Haque, B., & Hossain, M. (2024). Haque’s approach with mickens’ iteration method to find a modified analytical solution of nonlinear jerk oscillator containing displacement time velocity and time acceleration. Journal of Umm Al-Qura University for Applied Sciences, 1-8.

[26] Hossain, M. A., & Haque, B. I. (2024). An improved Mickens’ solution for nonlinear vibrations. Alexandria Engineering Journal, 95, 352-362.

[27] R. E. Mickens, “Oscillation in Planar Dynamic Systems,” World Scientific, Singapore, 1996.

[28] J. C. West, “Analytical Techniques for Nonlinear Control Systems,” English Univ. Press, London, 1960.

[29] R. E. Mickens, “Comments on the method of harmonic balance, ” Journal of Sound and Vibration, vol. 94, pp. 456-460, 1984.

[30] R. E. Mickens, “A generalization of the method of harmonic balance, ” Journal of Sound and Vibration, vol. 111, pp. 515-518, December 1986.

[31] C. W. Lim and B. S.Wu, “A new analytical approach to the Duffing-harmonic oscillator,” Physics Letters A, vol. 311, no. 4-5, pp. 365-373, May 2003.

[32] B. S. Wu, W. P. Sun, and C.W. Lim, “An analytical approximate technique for a class of strongly nonlinear oscillators, ” International Journal of Non-Linear Mechanics, vol. 41, no. 6-7 pp. 766-774, July- September 2006.

[33] M. S. Alam, M. E. Haque and M.B. Hossian, “A new analytical technique to find periodic solutions of nonlinear systems,” International Journal of Non-Linear Mechanics, vol. 42, no. 8, pp. 1035-1045, October 2007.

[34] M. Alal Hosen, M. S. Rahman, M .S. Alam and M. R. Amin, “A new analytical technique for solving a class of strongly nonlinear conservative systems,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5474-5486, January 2012.

[35] M. A. Razzak, “A new analytical approach to investigate the strongly nonlinear oscillators, ” Alexandria Engineering Journal, vol. 55, No. 2, pp. 1827-1834, June 2016.

[36] L. M. Milne-Thomson, “Elliptic integrals,” In: Abramowitz M, Stegun IA, editors, Handbook of mathematical functions, New York: Dover Publications, Inc.; 1972.

[37] J. B. Marion, “Classical dynamics of particles and systems,” San Diego, CA: Harcourt Brace Jovanovich; 1970.

[38] W. P. Sun, B. S. and C. W. Lim, “Approximate analytical solutions for oscillation of a mass attached to a stretched elastic wire, ” Journal of Sound and Vibration, vol. 300, no. 3-5, pp. 1042-1047, March 2007.

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