Exact solutions of the autonomous Duffing equation and their computation. 2nd Report. Some improvements of accuracy.

Abstract

In the previous paper, exact solutions of the autonomous Duffing equation were obtained in terms of the Fourier series making use of the so-called q-expansion of the elliptic functions. A characteristic parameter which plays a central role in calculating the Fourier coefficients, must be determined as a root of the modulus equation essentially by a trial method. As for some algorithms as well as approximation formulae proposed in the previous Saper, a few extensions of the discussions are presented here to improve the accuracy. A new trial algorithm and a revised approximation formula are proposed for a case of ultra low frequency (strongly nonlinear square wave and pulse train), and a new expression of the Fourier coefficient is also discussed, which is suitable in particular for the case of quasi-linear oscillation. Some salient features of the solution are illustrated for the case of a snap-through spring system in a full swing as well as in half swing modes.