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

Abstract

For a single-degree-of-freedom system having an arbitrary cubic spring function, exact solutions of the free oscillations are solved in terms of Jacobian elliptic functions by employing a bilinear transformation. This fact has been shown in principle, but ad hoc algorithms are needed to obtain accurate numerical expressions of the exact solution using a normal (constant-digit digital) computer. The original system is reduced to a modified Duffing equation accompanied by a constant term. The reduced system is classified into three types, viz. oscillators with asymmetrical soft, hard, or snap-through springs. The case of an asymmetrical soft spring was treated in the previous papers. In this paper, as a continuation of the initial studies, a detailed discussion of algorithms for accurate numeration and the dynamic properties of the free oscillator having a one-dimensional asymmetrical hard spring is presented. Algorithms are developed especially for some critical cases.