Normalising Flows and Nonlinear Normal Modes

In the context of dynamic decoupling problems, engineering dynamics has long held modal analysis as an exemplar. The method allows the exact decomposition of linear multi-degree-of-freedom (MDOF) systems into single-degree-of-freedom (SDOF) oscillators, thus simplifying complex dynamic problems considerably. However, modal analysis is very much a linear theory; if applied to nonlinear systems, the decoupling property (among others) is lost. This unfortunate situation has led to numerous attempts to formulate workable nonlinear versions of the theory. The current paper extends previous work by the authors in using machine learning methods to learn nonlinear modal transformations on measured data, based on the premise that any latent modal variables should be statistically independent. Unlike previous work, the transformation here exploits the recent development of normalising flows in constructing the required transformations. The new approach is shown to overcome a number of the problems in the original approach when demonstrated on a simulated nonlinear system.