Elsevier

Applied Mathematical Modelling

Volume 49, September 2017, Pages 243-254
Applied Mathematical Modelling

Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

https://doi.org/10.1016/j.apm.2017.05.004Get rights and content
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Highlights

  • This paper focuses on a strongly nonlinear conservative symmetric oscillator.

  • A new method for constructing analytical approximate solutions to the oscillator is presented.

  • Exploring the full efficiency and capability of a second-order Taylor expansion and the harmonic balance method.

  • With one single iteration, explicit yet very brief and accurate analytical approximate solutions have been derived.

Abstract

A new accurate iterative and asymptotic method is introduced to construct analytical approximate solutions to strongly nonlinear conservative symmetric oscillators. The method is based on applying a second-order expansion with the harmonic balance method and it excludes the requirement of solving a set of coupled nonlinear algebraic equations. Newton's iteration or the linearized model may be readily deduced by considering only the first-order terms in the model. According to this iterative approach, only Fourier series expansions of restoring force function, its first- and second-order derivatives for each iteration are required. It is concluded here that by only one single iteration, very brief and yet accurate analytical approximate solutions can be attained. Three physical examples are solved and accurate solutions are presented to illustrate the physics of the system and the effectiveness of the proposed asymptotic method.

Keywords

Analytical approximation
Conservative symmetric oscillator
Harmonic balance
Nonlinear dynamic system
Second-order Newton iteration

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