In this paper, He's Frequency-Amplitude Formulation (HFAF) and He's Energy Balance Method (HEBM) are employed to solve the generalized Duffing equation in the form of x@?+x+@a"3x^3+@a"5x^5+@a"7x^7+...+@a"nx^n=0. For any arbitrary power of n, a frequency analysis is carried out and the relationship between the natural frequency and the initial amplitude is obtained in analytical form. Accuracy and validity of the proposed techniques are then verified by comparing the numerical results obtained based on the HFAF, HEBM and exact integration method. Numerical simulations are extended for even very strong nonlinearities and very good correlations are achieved between the numerical results.

Frequency analysis of strongly nonlinear generalized Duffing oscillators using He's frequency-amplitude formulation and He's energy balance method

This review features a survey of some recent developments in asymptotic techniques and new developments, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the achieved approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modified perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to over-come the shortcomings.In this review we have applied different powerful analytical methods to solve high nonlinear problems in engineering vibrations. Some patterns are given to illustrate the effectiveness and convenience of the methodologies.

http://www.scielo.br/pdf/lajss/v9n2/a03v9n2.pdf

Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: a review

Approximate periodic solutions for the Helmholtz-Duffing oscillator are obtained in this paper. He's Energy Balance Method (HEBM) and He's Frequency Amplitude Formulation (HFAF) are adopted as the solution methods. Oscillation natural frequencies are analytically analyzed. Error analysis is carried out and accuracy of the solution methods is evaluated.

Approximate periodic solutions for the Helmholtz-Duffing equation

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.

/pdf/analytical-solutions-for-autonomous-conservative-nonlinear-4z81fqcf3l.pdf

Analytical Solutions for Autonomous Conservative Nonlinear Oscillator

Iterative homotopy harmonic balancing approach for conservative oscillator with strong odd-nonlinearity

He's frequency-amplitude formulation is used to solve the Duffing harmonic oscillator problem The solution procedure is simple, and the result obtained is valid for the whole solution domain with high accuracy

Jie Fan

Papers

He's frequency-amplitude formulation for the Duffing harmonic oscillator