On the dynamic properties of statistically-independent nonlinear normal modes

Much attention has been given to the study of nonlinear normal modes (NNMs), a nonlinear extension to the eminently useful framework for the analysis of linear dynamics provided by linear modal analysis (LMA). In the literature, several approaches have gained traction, with each able to preserve a subset of the useful properties of LMA. A recently-proposed framework (Worden and Green, 2017) casts nonlinear modal analysis as a problem in machine learning, viewing the NNM as directions in a latent modal coordinate space within which the modal dynamics are statistically uncorrelated. Thus far, the performance of this framework has been measured in a largely qualitative way. This paper presents, for the first time, an exploration into the underlying dynamics of the statistically-independent NNMs using techniques from nonlinear system identification (NLSI) and higher-order frequency-response functions (HFRFs).

In this work, the statistically-uncorrelated NNMs are found for two simulated nonlinear cubic-stiffness systems using a recently-proposed neural-network based approach. NLSI models are fitted to both physical and modal displacements and the HFRFs of these models are compared to theoretical values. In particular, it is found for both systems that the modal decompositions here permit an independent single-input single-output (SISO) representation that can be projected back onto the original displacements with low error. It is also shown via the HFRFs that the underlying linear natural frequencies of the modal dynamics lie very close to the underlying linear natural frequencies of the nonlinear systems, indicating that a true nonlinear decomposition has been identified.