Stability Configuration of a Rocking Rigid Rod over a Circular Surface Using the Homotopy Perturbation Method and Laplace Transform

  • Yusry O. El-Dib
  • Galal M. Moatimid
Research Article - Physics


The current paper is concerned with the motion of a rocking uniform rigid rod, without slipping, over a rigid circular surface. The governing equation of motion resulted in a highly nonlinear second-order ordinary differential equation. This nonlinear equation has no natural frequency. A coupling the homotopy perturbation method and Laplace transform is adopted to obtain an approximate solution of the equation of motion. In addition, He’s transformation method is used to obtain a periodic solution. Stability conditions are derived by making use of a nonlinear frequency analysis. Numerical calculations are achieved to investigate the governed perturbed solutions as well as the stability picture. It is found that the radius of the length of the rigid rod as well as the radius of the circular surface has a stabilizing influence. In contrast, the initial angular velocity has a destabilizing effect.


Homotopy perturbation method Laplace transform Frequency analysis Stability analysis Rocking rigid rod 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



Funding was provided by National Brain Tumor Society (US).


  1. 1.
    Nayfeh, A.H.; Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)zbMATHGoogle Scholar
  2. 2.
    Nayfeh, A.H.: Perturbation Methods. Wiley, New York (1973)zbMATHGoogle Scholar
  3. 3.
    Awrejcewicz, J.; Andrianov, I.V.; Manevitch, L.I.: Asymptotic Approaches in Nonlinear Dynamics (New Trends and Applications). Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  4. 4.
    El-Dib, Y.O.; Moatimid, G.M.: On the coupling of the homotopy perturbation and Frobenius method for exact solutions of singular nonlinear differential equations. Nonlinear Sci. Lett. A 9(3), 220–230 (2018)Google Scholar
  5. 5.
    He, J.H.: Modified Lindstedt–Poincare methods for some strongly nonlinear oscillations: part I—expansion of a constant. Int. J. Nonlinear Mech. 37(2), 309–314 (2002)CrossRefzbMATHGoogle Scholar
  6. 6.
    He, J.H.: Max–min approach to nonlinear oscillator. Int. J. Nonlinear Sci. Numer. Simul. 9(2), 207–210 (2008)Google Scholar
  7. 7.
    Ganji, S.S.; Ganji, D.D.; Karimpour, S.; Babazadeh, H.: Applications of He’s homotopy perturbation method to obtain second-order approximations of the coupled two-degree-of-freedom systems. Int. J. Nonlinear Sci. Numer. Simul. 10(3), 303–312 (2009)CrossRefGoogle Scholar
  8. 8.
    Yildirim, A.: Determination of periodic solutions for nonlinear oscillators with fractional powers by He’s modified Lindstedt–Poincare method. Meccanica 45(1), 1–6 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    El-Dib, Y.O.: Homotopy perturbation for excited nonlinear equations. Sci. Eng. Appl. 2(1), 96–108 (2017)Google Scholar
  10. 10.
    El-Dib, Y.O.: Multiple scales homotopy perturbation method for nonlinear oscillators. Nonlinear Sci. Lett. A 8(4), 352–364 (2017)Google Scholar
  11. 11.
    El-Dib, Y.O.: Periodic solution and stability behavior for nonlinear oscillator having a cubic nonlinearity time-delayed. Int. Annuls Sci. 5(1), 12–25 (2018)CrossRefGoogle Scholar
  12. 12.
    Mondal, Md.M.H.; Molla, Md.H.U.; Abdur Razzak, Md.; Alam, M.S.: A new analytical approach for solving quadratic nonlinear oscillators. Alex. Eng. J. 56, 629–634 (2017)Google Scholar
  13. 13.
    Wu, B.S.; Lim, C.W.; He, L.H.: A new method for approximate analytical solutions to nonlinear oscillations of natural systems. Nonlinear Dyn. 32, 1–13 (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ganji, D.D.; Karimpour, S.; Ganji, S.S.: Approximate analytical solutions to nonlinear oscillations of non-natural systems using He’s energy balance method. Prog. Electromagn. Res. M 5, 43–54 (2008)CrossRefGoogle Scholar
  15. 15.
    Barari, A.; Kimiaeifar, A.; Nejad, M.G.; Motevalli, M.; Sfahani, M.G.: A closed form solution for nonlinear oscillators frequencies using amplitude–frequency formulation. Shock Vib. 19, 1415–1426 (2012)CrossRefGoogle Scholar
  16. 16.
    Abul-Ez, M.; Ismail, G.M.; El-Moshneb, M.M.: Analytical solutions for free vibration of strongly nonlinear oscillators. Inf. Sci. Lett. 4(2), 101–105 (2015)Google Scholar
  17. 17.
    Ganji, S.S.; Ganji, D.D.; Babazadeh, H.; Karimpour, S.: Variational approach method for nonlinear oscillations of the motion of a rigid rocking back and forth and cubic-quintic Duffing oscillators. Prog. Electromagn. Res. M 4, 23–32 (2008)CrossRefGoogle Scholar
  18. 18.
    Khan, Y.; Wu, Q.; Askari, H.; Saadatnia, Z.; Kalami-Yazdi, M.: Nonlinear vibration analysis of a rigid rod on a circular surface via Hamiltonian approach. Math. Comput. Appl. 15(5), 974–977 (2010)zbMATHGoogle Scholar
  19. 19.
    Ghasemi, S.E.; Zolfagharian, A.; Ganji, D.D.: Study on motion of rigid rod on a circular surface using MHPM. Propuls. Power Res. 3(3), 159–164 (2014)CrossRefGoogle Scholar
  20. 20.
    Hosen, Md.A.: Approximate solutions of the equation of motion’s of the rigid rod which rocks on a circular surface without slipping. Ain Shams Eng. J. 5, 895–899 (2014)Google Scholar
  21. 21.
    Ali, F.; Sheikh, N.A.; Khan, I.; Saqib, M.: Solutions with Wright function for time fractional free convection flow of Casson fluid. Arab. J. Sci. Eng. 42, 2565–2572 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Goldstein, H.; Poole, C.; Safko, J.: Classical Mechanics, 3rd edn. Addison Wesley, New York (2000)zbMATHGoogle Scholar
  23. 23.
    El-Dib, Y.O.: Stability of a strongly displacement time-delayed Duffing oscillator by the multiple scales-homotopy perturbation method. J. Appl. Comput. Mech. 4(4), 260–274 (2018)Google Scholar
  24. 24.
    Kurtz, D.C.: A sufficient condition for all the roots of a polynomial to be real. Am. Math. Mon. 99(3), 259–263 (1992)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationAin Shams UniversityRoxy, CairoEgypt

Personalised recommendations