Summary
The recent theory of spectral submanifolds (SSMs) has emerged as a general, mathematically rigorous approach to reducing very high dimensional mechanical systems to very low dimensional models. Successful applications of SSM-reduction range from geometrically nonlinear finite-element models in structural dynamics through experimental fluid-structure interactions to model-predictive control in soft robotics. Open-source codes with growing libraries of worked examples are now available to carry out equation-driven SSM reduction (SSMTool) and data-driven SSM reduction (SSMLearn). As the mathematical theory of SSMs has only been developed in detail for smooth physical systems, its applicability to non-smooth systems is still the subject of ongoing research. We contribute to this development by applying data-driven SSM reduction to learn a low-dimensional reduced model for the simulated, unforced dynamics of a thin-walled jointed mechanical assembly. In this benchmark problem, frictional contact occurring at the interfaces of the assembled components is a non-smooth effect that can lead to a highly nonlinear dynamic response. Recent results by the joints research community show that predictive equation-driven reduced order modeling for this benchmark problem remains a challenge. Here, we discuss the construction of a very low dimensional, nonlinear reduced model for this problem by applying SSMLearn to numerical data obtained from a small number of unforced finite-element simulations. We then use this SSM-reduced model to make predictions for the forced response of the assembly, which we subsequently verify by direct finite-element simulations of the forced structure.
