Numerically Assessing the Relative Significance of Nonlinear Normal Modes to Forced Responses

Nonlinear normal modes, which describe the unforced, undamped responses of nonlinear systems, are often used for understanding the dynamic behaviour of nonlinear structures in engineering. As such, a key property of nonlinear normal modes (NNMs) is that they relate to the dynamic behaviour of the system when forcing and damping are applied. Previous work has shown that an extremely large number of NNMs may be predicted for relatively simple systems; however only a small number of these NNMs have a meaningful influence on the forced and damped dynamics. As such, a method for determining which NNMs are significant (i.e., strongly relate to the forced dynamics) and which are not is crucial for the use of NNMs as a tool for understanding nonlinear systems. As shown in the literature, NNMs only relate to the forced dynamics in the presence of a mechanism for energy transfer between resonating components of the system. Whilst this provides an understanding of the mechanism, computing which NNMs exhibit this property requires detailed analytical analysis. In this paper, we discuss a numerical approach for determining the degree of significance, with respect to the forced responses, of the NNMs. Such an approach removes the need for analytical investigation and provides an efficient and practical approach to determining the significance of the NNMs.