Analytical Solutions for Autonomous Conservative Nonlinear Oscillator

This paper adapts the Energy Balance Method (EBM) and Frequency Amplitude Formulation to solve the free vibrations of a conservative oscillator with inertia and static cubic non-linearities. Case studies on the effects of the time response are presented. The results that obtained from the EBM and FAF are then compared with those from the numerical solution in order to verify the accuracy of the proposed method.

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

[2]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[3]  Ji-Huan He,et al.  Preliminary report on the energy balance for nonlinear oscillations , 2002 .

[4]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[5]  Chao-Kuang Chen,et al.  Application of the differential transformation method to a non-linear conservative system , 2004, Appl. Math. Comput..

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

[7]  Ebrahim Esmailzadeh,et al.  Existence of Periodic Solutions for the Generalized Form of Mathieu Equation , 2005 .

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

[9]  Ahmet Yildirim,et al.  Determination of the frequency-amplitude relation for a Duffing-harmonic oscillator by the energy balance method , 2007, Comput. Math. Appl..

[10]  Ji-Huan He.  Variational iteration method—Some recent results and new interpretations , 2007 .

[11]  Davood Domiri Ganji,et al.  The application of He's variational iteration method to nonlinear equations arising in heat transfer , 2007 .

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

[13]  Augusto Beléndez,et al.  Higher order analytical approximate solutions to the nonlinear pendulum by He's homotopy method , 2008 .

[14]  Jie Fan,et al.  He's frequency-amplitude formulation for the Duffing harmonic oscillator , 2009, Comput. Math. Appl..

[15]  Ji-Huan He,et al.  An elementary introduction to the homotopy perturbation method , 2009, Comput. Math. Appl..

[16]  Ling Zhao,et al.  He's frequency-amplitude formulation for nonlinear oscillators with an irrational force , 2009, Comput. Math. Appl..

[17]  M. G. Sfahani,et al.  Application of AFF and HPM to the systems of strongly nonlinear oscillation , 2010 .

[18]  F. Austin,et al.  THE VARIATIONAL ITERATION METHOD WHICH SHOULD BE FOLLOWED , 2010 .

[19]  Ji-Huan He,et al.  Hamiltonian approach to nonlinear oscillators , 2010 .

[20]  Ahmet Yildirim,et al.  Analytical approach to Fokker–Planck equation with space- and time-fractional derivatives by means of the homotopy perturbation method , 2010 .

[21]  Davood Younesian,et al.  Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method , 2010, Comput. Math. Appl..

[22]  Hossein Jafari,et al.  Fractional variational iteration method via modified Riemann–Liouville derivative , 2011 .

[23]  Yasir Khan,et al.  Analytical approach to two-dimensional viscous flow with a shrinking sheet via variational iteration algorithm-II , 2011 .

[24]  Y. Khan,et al.  Application of modified Laplace decomposition method for solving boundary layer equation , 2011 .