A Geometric Approach to Nonlinear Mechanical Vibrations: from Analytic to Data-driven Methods

Abstract

This doctoral thesis devises analytical and data-driven methods for the analysis of nonlinear vibrations in mechanical systems, potentially with a large number of degrees of freedom. Modern challenges in engineering require deeper understanding of nonlinear oscillations in mechanical systems, as well as extracting data-driven models for their predictions. In the first part of this thesis, the focus is set on analytical models, with forced-damped nonlinear mechanical systems viewed as small perturbations from their energy-preserving counterpart. Indeed, weakly damped mechanical systems under small periodic forcing tend to showcase periodic response in a close vicinity of some periodic orbits of their conservative limit. Specifically, amplitude frequency plots for the conservative limit have frequently been observed, both numerically and experimentally, to serve as backbone curves for the near-resonance peaks of the forced response. A systematic mathematical analysis is then derived, allowing to predict which members of conservative periodic orbit families will survive in the forced-damped response. Moreover, the method is not limited to predicting existence, but can also forecast stability type of vibrations in the forced response. Not only does this method provide a rigorous analytical tool, but it also finds precise mathematical conditions under which approximate numerical and experimental approaches, such as energy balance and force appropriation, are justified. The second part of this thesis looks at oscillatory dynamics from a data-driven perspective. The objective is to determine reduced-order models from trajectory data of dynamical systems. Based on the theory of spectral submanifolds, a method is developed for simultaneous dimensionality reduction and identification of the dynamics in normal form. In contrast with other data-driven modeling techniques, the normal form of the dynamics offers valuable insights and is capable of predictions when small perturbations, such as external forcing, are added to systems. Moreover, there are, in principle, no restrictions of dimensionality or constraints on the states observed in the trajectory data. The algorithm based on this approach automatically detects the appropriate normal form for a given set of trajectories, thereby providing an intelligent, unsupervised learning strategy for dynamical systems. The accuracy and the validity of the method is demonstrated on different examples, featuring data from numerical simulations and physical experiments.