Analytical solutions to nonlinear conservative oscillator with fifth-order nonlinearity

This paper describes analytical and numerical methods to analyze the steady state periodic response of an oscillator with symmetric elastic and inertia nonlinearity. A new implementation of the homotopy perturbation method (HPM) and an ancient Chinese method called the max-min approach are presented to obtain an approximate solution. The major concern is to assess the accuracy of these approximate methods in predicting the system response within a certain range of system parameters by examining their ability to establish an actual (numerical) solution. Therefore, the analytical results are compared with the numerical results to illustrate the effectiveness and convenience of the proposed methods.

[1]  Davood Domiri Ganji,et al.  Variational Iteration Method and Homotopy-Perturbation Method for Solving Different Types of Wave Equations , 2008 .

[2]  鈴木 増雄.  A. H. Nayfeh and D. T. Mook: Nonlinear Oscillations, John Wiley, New York and Chichester, 1979, xiv+704ページ, 23.5×16.5cm, 10,150円. , 1980 .

[3]  Davood Domiri Ganji,et al.  New class of solutions for water infiltration problems in unsaturated soils , 2010 .

[4]  Mehdi Dehghan.  The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications , 2004, Int. J. Comput. Math..

[5]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[6]  Ji-Huan He,et al.  Nonlinear oscillator with discontinuity by parameter-expansion method , 2008 .

[7]  Ji-Huan He,et al.  Max-Min Approach to Nonlinear Oscillators , 2008 .

[8]  S. E. Mirmohammadsadeghi,et al.  APPLICATION OF He's VARIATIONAL ITERATION METHOD AND ADOMIAN'S DECOMPOSITION METHOD TO PROCHHAMMER CHREE EQUATION , 2009 .

[9]  Baisheng Wu,et al.  A new analytical approach to the Duffing-harmonic oscillator , 2003 .

[10]  S. S. Ganji,et al.  Application of amplitude–frequency formulation to nonlinear oscillation system of the motion of a rigid rod rocking back , 2009 .

[11]  M. N. Hamdan,et al.  ON THE LARGE AMPLITUDE FREE VIBRATIONS OF A RESTRAINED UNIFORM BEAM CARRYING AN INTERMEDIATE LUMPED MASS , 1997 .

[12]  D. Ganji,et al.  Solving Ratio-Dependent Predator-Prey System with Constant Effort Harvesting Using Homotopy Perturbation Method , 2008 .

[13]  Ji-Huan He,et al.  Construction of solitary solution and compacton-like solution by variational iteration method , 2006 .

[14]  Davood Domiri Ganji,et al.  Applications of He's Homotopy Perturbation Method to Obtain Second-order Approximations of the Coupled Two-Degree-of-Freedom Systems , 2009 .

[15]  Vimal Singh,et al.  Perturbation methods , 1991 .

[16]  Ji-Huan He,et al.  Determination of limit cycles for strongly nonlinear oscillators. , 2003, Physical review letters.

[17]  Ji-Huan He.  Homotopy Perturbation Method for Bifurcation of Nonlinear Problems , 2005 .

[18]  Ji-Huan He.  New interpretation of homotopy perturbation method , 2006 .

[19]  M. Dehghan,et al.  The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics , 2008 .

[20]  S. S. Ganji,et al.  HES ENERGY BALANCE AND HES VARIATIONAL METHODS FOR NONLINEAR OSCILLATIONS IN ENGINEERING , 2009 .

[21]  Davood Domiri Ganji,et al.  Application of Homotopy Perturbation Method and Variational Iteration Method to Nonlinear Oscillator Differential Equations , 2008 .

[22]  Ji-Huan He.  Some new approaches to Duffing equation with strongly and high order nonlinearity (II) parametrized perturbation technique , 1999 .

[23]  S. Çatal.  Solution of free vibration equations of beam on elastic soil by using differential transform method , 2008 .

[24]  Nobukatsu Okuizumi,et al.  Multiple time scale analysis of hysteretic systems subjected to harmonic excitation , 2004 .

[25]  Ji-Huan He,et al.  Addendum:. New Interpretation of Homotopy Perturbation Method , 2006 .

[26]  Ali Konuralp.  The steady temperature distributions with different types of nonlinearities , 2009, Comput. Math. Appl..

[27]  Ji-Huan He,et al.  Modified Lindstedt–Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant , 2002 .

[28]  R. Pletcher,et al.  Computational Fluid Mechanics and Heat Transfer. By D. A ANDERSON, J. C. TANNEHILL and R. H. PLETCHER. Hemisphere, 1984. 599 pp. $39.95. , 1986, Journal of Fluid Mechanics.

[29]  Augusto Beléndez,et al.  Application of a modified He's homotopy perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator , 2007 .

[30]  J. Ramos.  An artificial parameter–Linstedt–Poincaré method for oscillators with smooth odd nonlinearities , 2009 .

[31]  Lan Xu,et al.  Variational approach to solitons of nonlinear dispersive K(m, n) equations , 2008 .

[32]  Ahmet Yildirim,et al.  Determination of periodic solution for a u1/3 force by He's modified Lindstedt–Poincaré method , 2007 .

[33]  S. K. Lai,et al.  Newton–harmonic balancing approach for accurate solutions to nonlinear cubic–quintic Duffing oscillators , 2009 .

[34]  Ji-Huan He.  Variational approach for nonlinear oscillators , 2007 .