A new type of non-linear approximation with application to the duffing equation

A new type of trial solution which differs from the usual linear combination of approximating functions is considered. It involves modifying the approximating functions with “form functions;” functions containing undetermined parameters appearing non-linearly, the proper choice of which provide a closer approximation to the large local curvatures which appear in some non-linear problems. In this paper the “form function” approximation is demonstrated for steady-state solutions of the Duffing equation. This equation arises in the problem of non-linear vibration of buckled beams and plates. It is shown that the stability behavior of these steady-state solutions is governed by a Hill equation. It is found that the “form function” approximation gives noticeably better numerical results than, for example, those given by the harmonic balance method. The method also provides additional insight into the non-linear behavior, particularly in the low frequency response region.