Application of Adaptation HAM for Nonlinear Oscillator Typified as A Mass Attached to A Stretched Elastic Wire

A. Sami Bataineh

Abstract


This paper applies the adaptation of homotopy analysis method (AHAM) for the first time to obtained the periodic solutions for the oscillation of a mass attached to a stretched elastic wire. The AHAM approach can be applied directly to the governing equation without rewrite it in a form that does not contain the square-root expression. More precisely, with the help of the homotopy polynomials procedure the nonlinear term of the problem can be decompose as a series of polynomials to overcomes the difficulty arising in calculating complicated integrals. A comparative study between AHAM and other existing solutions obtained by several authors is conducted to demonstrate the simplicity and the efficiency of AHAM. The approximate frequency and periodic solution for both small and large amplitude of oscillations show a good agreement with the numerical solution.

Keywords


Homotopy analysis method; Homotopy polynomials; Nonlinear oscillation; Periodic solutions

Full Text:

PDF

References


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 39 (2009), 746 – 764.

S. Durmaz and M.O. Kaya, High-order energy balance method to nonlinear oscillators, Jour. Appl. Math. 2012, 7, Article ID 518684, http://dx.doi.org/10.1155/2012/518684.

S. Durmaz, S.A. Demirbag and M.O. Kaya, High order He’s energy balance method based on collocation method, Int. Jour. Nonlinear Sci. Numer. Simul. 11 (2010), 1 – 5.

D.D. Ganji, N.R. Malidarreh and M. Akbarzade, Comparison of energy balance period with exact period for arising nonlinear oscillator equations, Acta Appl. Math. 108 (2009), 353–362.

J.H. He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations, Part I: expansion of a constant, Int. Jour. Nonlinear Mech. 37 (2) (2002), 309 – 314.

J.H. He, Modified Lindstedt–Poincare methods for some strongly non-linear oscillations, Part II: a new transformation, Int. Jour. Nonlinear Mech. 37 (2) (2002), 315 – 320.

J.H. He, New interpretation of homotopy perturbation method. Int. Jour. Mod. Phys. B 20 (18) (2006), 2561 – 2568.

Y. Khan and A. Mirzabeigy, Improved accuracy of He’s energy balance method for analysis of conservative nonlinear oscillator, Neural Comput. Appl. 25 (2014), 889 – 895.

S.J. Liao and I. Pop, Explicit analytic solution for similarity boundary layer equations., Int. Jour. Heat Mass Transfer 47 (2004), 75 – 78.

S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton, Chapman and Hall (2003).

S.J. Liao, A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. Heat. Mass. Transfer. 48 (2005), 2529 – 3259.

S.J. Liao, An approximate solution technique which does not depend upon small parameters (Part 2): an application in fluid mechanics, Int. Jour. Nonlinear Mech. 32 (1997), 815 – 822.

S.J. Liao, An approximate solution technique which does not depend upon small parameters: a special example, Int. Jour. Nonlinear Mech. 30 (3) (1995), 371 – 380.

S.J. Liao, An explicit totally analytic approximation of Blasius viscous flow problems, Int. Jour. Nonlinear Mech. 34 (1999), 759 – 778.

S.J. Liao, Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Comput. 169 (2005), 1186 – 1194.

S.J. Liao, On the homotopy anaylsis method for nonlinear problems, Appl. Math. Comput. 147 (2004), 499 – 513.

R.E. Mickens, Oscillation in Planar Dynamic Systems, World Scientific, Singapore (1996).

L.M.Milne-Thomson, Elliptic integrals, in: M. Abramowitz and I.A. Stegun (Eds.), Handbook of Mathematical Functions, Dover Publications, New York (1972).

Z. Odibat and A. Sami Bataineh, An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials, Mathematical Methods in the Applied Sciences 38 (5) (2015), 991 – 1000.

Md. Abdur Razzak and Md. Mashiar Rahman, Application of new novel energy balance method to strongly nonlinear oscillator systems, Results in Physics 5 (2015), 304 – 308.

W.P. Sun, B.S. Wu and C.W. Lim, Approximate analytical solution for oscillation of a mass attached to stretched elastic wire, Jour. Sound Vibration 300 (2007), 1042 – 1047.

L. Xu, Application of He’s parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire, Phys. Lett. A 368 (2007), 259 – 262.

L. Xu, Determination of limit cycle by He’s parameter-expanding method for strongly nonlinear oscillators, Jour. Sound Vibr. 302 (2007), 178 – 84.

L. Zhao, He’s frequency-amplitude formulation for nonlinear oscillators with an irrational force, Comput. Math. Appl. 58 (2009), 2477 – 2479.


Refbacks

  • There are currently no refbacks.


eISSN 0975-8607; pISSN 0976-5905