Nonlinear Modal Interaction Analysis for a Three Degree-of-Freedom System with Cubic Nonlinearities

Abstract

The majority of work in the literature on modal interaction is based on two degree-of-freedom nonlinear systems with cubic nonlinearities. In this paper we consider a three degree-of-freedom system with nonlinear springs containing cubic nonlinear terms. First the undamped, unforced case is considered. Specifically the modal interaction case that occurs when all the underlying linear modal frequencies are close is considered (i.e. $$\omega _{n1}:\omega _{n2}:\omega _{n3} \simeq 1: 1: 1$$ ). In the case considered, due to the symmetry of the system, the first mode is linear and not coupled with the other two modes. The analysis is carried out by using a normal form transformation to obtain the nonlinear backbone curves of the undamped, unforced response. In addition, the frequency response function (FRF) of the corresponding lightly damped and harmonically forced system obtained from the continuation software AUTO-07p is compared with the backbone curves to show its validity for predicting the nonlinear resonant frequency and amplitude. A comparison of the results gives an insight into how modal interactions in the forced-damped response can be predicted using just the backbone curves, and how this might be applied to predict resonant responses of multi-modal nonlinear systems more generally.