Exact Solutions for Free Vibration in a Single-Degree-of-Freedom System Having a Spring with an Arbitrary Cubic Function. Introductory Remarks.

Abstract

For a single-degree-of-freedom system having a spring function (in addition to a linear term) with a cubic term (Duffing oscillator) or a quadratic term (Helmholtz oscillator), the exact solutions of the free vibration are expressed in terms of Jacobian elliptic functions. However, for a system having a spring function with all terms from a constant (0th order) up to a cubic (3rd order), the most generalized 3rd-order polynomial including an asymmetrical spring, the situation has not always been clear since Duffing's early days. A certain transform was proposed to convert the relevant system into a standard (symmetrical) Duffing system previously, but the algorithms were poor and some difficulties were found in the computation In this paper, revised algorithms are discussed to realize accurate numeration. It also becomes possible to obtain the solution for a given natural frequency as well as for a given initial rest amplitude. As an example the result for an asymmetrical soft spring system is demonstrated.