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Transient analysis of nonlinear non-autonomous second order systems using Jacobian elliptic functions Barkham, Peter George Douglas

Abstract

A method is presented for determining approximate solutions to a class of grossly nonlinear, non-autonomous second order differential equations characterized by [formula omitted] with the restriction that resonance effects be negligible. Solutions are developed in terms of the Jacobian elliptic functions, and may be related directly to the degree, of non-linearity in the differential equation. An integral error definition, which can be applied to any particular differential equation, is used to portray regions of validity of the approximate solution in terms of equation parameters. In practice the approximate solution is shown to be of greater accuracy than would be expected from the error analysis, and use of the error diagram leads to a pessimistic estimate of solution accuracy. Two autonomous equations are considered to facilitate comparison between the elliptic function approximation and that obtained from the method of Kryloff and Bogoliuboff. The elliptic function solution is shown to be accurate even for heavily damped nonlinear autonomous equations, when the quasi-linear approximation of Kyrloff-Bogoliuboff cannot with validity be applied. Four examples are chosen, from the fields of astrophysics, mechanics, circuit theory and control systems to illustrate, some areas to which the general approximation method relates.

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