Double Fourier harmonic balance method for nonlinear oscillators by means of Bessel series

The standard harmonic balance method consists in expanding the displacement of an oscillator as a Fourier cosine series in time. A key modification is proposed here, in which the conservative force is additionally expanded as a Fourier sine series in space. As a result, the steady-state oscillation frequency can be expressed in terms of a Bessel series, and the sums of many such series are known or can be developed. The method is illustrated for five different physical situations, including a ball rolling inside a V-shaped ramp, an electron attracted to a charged filament, a large-amplitude pendulum, and a Duffing oscillator. As an example of the results, the predicted period of a simple pendulum swinging between −90° and +90° is found to be only 0.4% larger than the exact value. Even better, the predicted frequency for the V-ramp case turns out to be exact.