Analytical approximate solutions for conservative nonlinear oscillators by modified rational harmonic balance method

Abstract

An analytical approximate technique for conservative nonlinear oscillators is proposed. This method is a modification of the generalized harmonic balance method in which analytical approximate solutions have a rational form. This approach gives us not only a truly periodic solution but also the frequency of motion as a function of the amplitude of oscillation. Three truly nonlinear oscillators including the cubic Duffing oscillator, fractional-power restoring force and anti-symmetric quadratic nonlinear oscillators are presented to illustrate the usefulness and effectiveness of the proposed technique. We find that this method works very well for the cubic oscillator, and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed. For the second-order approximation, we have shown that the relative error in the analytical approximate frequency is as low as 0.0046%. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This has allowed us to compare the coefficients for the different harmonic terms in these solutions. For the other two nonlinear oscillators considered, the relative errors in the analytical approximate frequencies are 0.098 and 0.066%, respectively. The most significant features of this method are its simplicity and its excellent accuracy for the whole range of oscillation amplitude values, and the results reveal that this technique is very effective and convenient for solving conservative truly nonlinear oscillatory systems.

Keywords:

• truly nonlinear oscillators
• approximate solutions
• generalized harmonic balance method

2000 AMS Subject Classifications :

• 34C15
• 34C25
• 39A11
• 81Q15
• 34L30

Acknowledgements

This work was supported by the ‘Ministerio de Educación y Ciencia’, Spain, under project FIS2005-05881-C02-02. The authors express their gratitude to the reviewers for their useful suggestions and for their comments, which significantly improved the original manuscript.