Coupled purely nonlinear oscillators: normal modes and exact solutions for free and forced responses

Abstract

This article is concerned first with free vibrations in a chain of two-mass oscillators with purely nonlinear springs whose power of nonlinearity can be any real number higher than unity. Similar normal modes are obtained by uncoupling the equations of motion. The corresponding time responses are determined in exact analytical forms in terms of Ateb and Jacobi elliptic functions. The external excitation is designed so as to uncouple the equations of motion again and to determine the exact solution for forced vibrations. Then, the case of a three-mass chain is investigated, and similar normal modes are found for the nonlinearity of interest. In addition, these novel results are extended to a continuous system—a rod exhibiting longitudinal vibrations when the material of the rod is characterized by a purely nonlinear stress–strain relationship.