Rational harmonic balance based method for conservative nonlinear oscillators: application to the Duffing equation

A second-order modified rational harmonic balance method is used approximately solve the nonlinear differential equation that governs the oscillations of a conservative autonomous system with one degree of freedom. The Duffing oscillator is analyze to illustrate the usefulness and effectiveness of the proposed technique. We find that this method works very well for this 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 0.0055% when A tends to infinity. 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.