Approximate analytical solutions to nonlinear differential equations governing the behaviour of some oscillating marine systems using the Beecham-Titchener method

Abstract

The Beecham-Titchener (B-T) method is a technique for obtaining approximate analytical solutions of nonlinear differential equations representing the oscillatory behaviour of underdamped dynamical systems. The method closely parallels the better-known method of Kryloff and Bogoliuboff (K-B) but appears less restrictive in its application and is generally superior in its prediction of the frequency behaviour of nonlinear systems. Simpson has extended the B-T method to cope with nonlinearities that are even functions of, for example, position and velocity. These types of nonlinearities are common in the marine dynamics area (it is shown, in the appendix, how to deal with such nonlinearities using the K-B method). In this paper the B-T method is applied to some simple nonlinear mooring and ship-rolling problems and, where appropriate, a comparison is made of its efficacy with that of the K-B technique. So far as the author is aware the applications described here are the first outside of the flight-dynamics area.