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Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms

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Abstract

This paper is focused on solving the generalized second-order strongly nonlinear differential equation \({\ddot{x}+\sum_i {c_i^2 }x \left| x \right|^{i-1}=0}\) which describes the motion of a conservative oscillator with restoring force of series type with integer and noninteger displacement functions. The approximate analytical solution procedures are modified versions of the simple solution approach, the energy balance method, and the frequency–amplitude formulation including the Petrov–Galerkin approach. For the case where the linear term is dominant in comparison with the other series terms of the restoring force, the perturbation method based on the solution of the linear differential equation is applied. If the dominant term is nonlinear and the additional terms in the restoring force are small, the perturbation method based on the approximate solution of the pure nonlinear differential equation is introduced. Using the aforementioned methods, the frequency–amplitude relations in the first approximation are obtained. The suggested solution methods are compared and their advantages and disadvantages discussed. A numerical example is considered, where the restoring force of the oscillator contains a linear and also a noninteger order term (i = 5/3). The analytically obtained results are compared with numerical results as well as with some approximate analytical results for special cases from the literature.

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References

  1. Nayfeh AH (1973) Perturbation methods. Wiley, New York

    MATH  Google Scholar 

  2. Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York

    MATH  Google Scholar 

  3. Bogoliubov NN, Mitropolski YuA (1974) Asimptoticheskie metodi v teorii nelinejnih kolebanij. Nauka, Moscow

    Google Scholar 

  4. Moiseev NN (1981) Asimptoticheskie metodi nelinejnoj mehaniki. Nauka, Moscow

    Google Scholar 

  5. Cveticanin L (2009) The approximate solving methods for the cubic Duffing equation based on the Jacobi elliptic functions. Int J Nonlin Sci Numer Simul 10: 1491–1516

    Article  Google Scholar 

  6. Cveticanin L (2011) Analysis techniques for the various forms of the Duffing equation. In: Kovacic I, Brennan M (eds) Duffing’s equation: non-linear oscillators and their behaviour. Wiley, Singapore

    Google Scholar 

  7. Liao SJ (1992) On the proposed homotopy analysis technique for nonlinear problems and its applications. Ph.D. dissertation, Shanghai Jio Tong University, Shanghai, China

  8. Liao SJ (2003) Beyond perturbation: introduction to homotopy analysis method. Chapman & Hall, Boca Raton

    Book  Google Scholar 

  9. Liao S, Tan Y (2007) A general approach to obtain series solutions of nonlinear differential equations. Stud Appl Math 119: 297–354

    Article  MathSciNet  Google Scholar 

  10. He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178: 257–262

    Article  ADS  MATH  Google Scholar 

  11. He JH (2003) Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 135: 73–79

    Article  MathSciNet  MATH  Google Scholar 

  12. Cveticanin L (2006) Homotopy-perturbation method for pure nonlinear differential equation. Chaos Solitons Fractals 30: 1221–1230

    Article  ADS  MATH  Google Scholar 

  13. Yildirim T, Yildirim A (2007) A comparative study of He’s homotopy perturbation method for determining frequency-amplitude relation of a nonlinear oscillator with discontinuities. Int J Nonlin Sci Numer Simul 8: 243–248

    Article  Google Scholar 

  14. Xu L (2007) He’s parameter-expanding methods for strongly nonlinear oscillators. J Comput Appl Math 207: 148–154

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Yildirim T, Yildirim A (2009) Generating the periodic solutions for forcing van der Pol oscillators by the iteration perturbation method. Nonlin Anal 10: 1984–1989

    Article  MATH  Google Scholar 

  16. Younesian D, Kalami-Yazdi M, Askari H, Saadatnia Z (2010) Frequency analysis of higher-order Duffing oscillator using homotopy and iteration-perturbation techniques. In: 18th annual international conference on mechanical engineering–ISME2010, Sharif University of Technology, Tehran, Iran

  17. Yildirim A (2010) He’s homotopy perturbation method for nonlinear differential-difference equations. Int J Comput Math 87(5): 992–996

    Article  MathSciNet  MATH  Google Scholar 

  18. Abbasbandy S (2006) The application of the homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360: 109–113

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Abbasbandy S (2007) The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equations. Phys Lett A 361: 478–483

    Article  ADS  MATH  Google Scholar 

  20. Sajid M, Hayat T (2008) Comparison of HAM and HPM methods in nonlinear heat conduction equations. Nonlin Anal 9: 2296–2301

    Article  MathSciNet  MATH  Google Scholar 

  21. Liang S, Jeffrey DJ (2009) Comparison of homotopy analysis method and homotopy perturbation method through an evolution equation. Commun Nonlin Sci Numer Simul 14: 4057–4064

    Article  MathSciNet  MATH  Google Scholar 

  22. Herisanu N, Marinca V (2010) Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method. Comput Math Appl 60: 1507–1615

    Article  MathSciNet  Google Scholar 

  23. Reddy JN (1984) An introduction to the finite element method. McGraw-Hill, New York

    Google Scholar 

  24. He JH (2002) Preliminary report on the energy balance for nonlinear oscillations. Mech Res Commun 29(2–3): 107–111

    Article  MATH  Google Scholar 

  25. He JH (2004) Solution of nonlinear equations by an ancient Chinese algorithm. Appl Math Comput 151: 293–297

    Article  MathSciNet  MATH  Google Scholar 

  26. Ren ZF, He JH (2009) A simple approach to nonlinear oscillators. Phys Lett A 373: 3749–3752

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Mickens RE (2010) Truly nonlinear oscillations: harmonic balance, parametric expansions, iteration, and averaging methods. World Scientific, Singapore

    Book  Google Scholar 

  28. Askari H, Kalami-Yazdi M, Saadatnia Z (2010) Frequency analysis of nonlinear oscillators with rational restoring force via He’s energy balance method and He’s variational approach. Nonlin Sci Lett A 1: 425–430

    Google Scholar 

  29. Younesian D, Askari H, Saadatnia Z, Kalami-Yazdi M (2010) Frequency analysis of strongly nonlinear generalized Duffing oscillators using He’s frequency–amplitude formulation and He’s energy balance method. Comput Math Appl 59: 3222–3228

    Article  MathSciNet  MATH  Google Scholar 

  30. He JH (2010) Hamiltonian approach to nonlinear oscillators. Phys Lett A 374: 2312–2314

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Mickens RE (2001) Oscillations in a x 4/3 potential. J Sound Vib 246: 375–378

    Article  MathSciNet  ADS  MATH  Google Scholar 

  32. Mickens RE (2002a) Analysis of non-linear oscillators having non-polynomial elastic terms. J Sound Vib 255: 789–792

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Mickens RE (2002b) A study of nonlinear oscillations in systems having non-polynomial elastic force functions. Recent Res Dev Sound Vib 1: 241–251

    Google Scholar 

  34. Mickens RE (2006) Iteration method solutions for conservative and limit-cycle x 1/3 force oscillators. J Sound Vib 292: 964–968

    Article  MathSciNet  ADS  MATH  Google Scholar 

  35. Cveticanin L (2009) Oscillator with fraction order restoring force. J Sound Vib 320: 1064–1077

    Article  ADS  Google Scholar 

  36. Cveticanin L, Kovacic I, Rakaric Z (2010) Asymptotic methods for vibrations of the pure non-integer order oscillator. Comput Math Appl 60: 2616–2628

    Article  MathSciNet  MATH  Google Scholar 

  37. Cveticanin L (2011) Pure odd-order oscillators with constant excitation. J Sound Vib 330: 976–986

    Article  ADS  Google Scholar 

  38. Cveticanin L, Zukovic M (2009) Melnikov’s criteria and chaos in systems with fractional order deflection. J Sound Vib 326: 768–779

    Article  ADS  Google Scholar 

  39. Kovacic I, Rakaric Z, Cveticanin L (2010) A non-simultaneous variational approach for the oscillators with fractional-order power nonlinearities. Appl Math Comput 217: 3944–3954

    Article  MathSciNet  MATH  Google Scholar 

  40. Yildirim A (2010) Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt-Poincaré method. Meccanica 45: 1–6

    Article  MathSciNet  MATH  Google Scholar 

  41. Cveticanin L, Kalami-Yazdi M, Saadatnia Z, Askari H (2010) Application of Hamiltonian approach to the generalized nonlinear oscillator with fractional power. Int J Nonlin Sci Numer Simul 11(12): 997–1002

    Article  Google Scholar 

  42. Gradshtein IS, Rjizhik IM (1971) Tablici integralov, summ, rjadov i proizvedenij. Nauka, Moscow

    Google Scholar 

  43. Mickens RE (2004) Mathematical methods for the natural and engineering sciences. World Scientific, Singapore

    MATH  Google Scholar 

  44. Krylov N, Bogolubov N (1943) Introduction to nonlinear mechanics. Princeton University Press, Princeton

    Google Scholar 

  45. Abramowitz M, Stegun IA (1979) Handbook of mathematical functions with formulas, graphs and mathematical tables. Nauka, Moscow, (in Russian)

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Authors and Affiliations

  1. Faculty of Technical Sciences, University of Novi Sad, Trg D. Obradovica 6, 21000, Novi Sad, Serbia

    L. Cveticanin

  2. Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, 16846, Narmak, Tehran, Iran

    M. Kalami-Yazdi

  3. School of Railway Engineering, Iran University of Science and Technology, 16846, Narmak, Tehran, Iran

    H. Askari

Authors
  1. L. Cveticanin
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  2. M. Kalami-Yazdi
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  3. H. Askari
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Correspondence to L. Cveticanin.

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Cveticanin, L., Kalami-Yazdi, M. & Askari, H. Analytical approximations to the solutions for a generalized oscillator with strong nonlinear terms. J Eng Math 77, 211–223 (2012). https://doi.org/10.1007/s10665-012-9542-4

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  • DOI: https://doi.org/10.1007/s10665-012-9542-4

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