Continuation of nonlinear normal modes using reduced-order models based on generalized characteristic value decomposition

Abstract Over the past two decades, data-driven reduced-order modeling (ROM) strategies have gained significant traction in the nonlinear dynamics community. Currently, several challenges in physical interpretation and data availability remain overlooked in current methodologies. This work proposes a novel ROM methodology based on a newly proposed generalized characteristic value decomposition (GCVD) to address these obstacles. The GCVD-ROM approach proposes a new perspective toward data-driven ROMs via characterization of the dynamics before any ROM considerations are made. In doing so, a significant degree of versatility is inherited in the GCVD-ROM strategy, allowing our models to reproduce the full-scale dynamics in different regions of the parameter space at the cost of a single training data set. Our approach utilizes computationally efficient free-decay data sets alongside a windowed-decomposition scheme, allowing us to extract energy-dependent modal structures for use in model-order reduction. This is accomplished using the physically insightful characteristic values provided by the GCVD, which are shown to be directly related to the system poles at a particular response amplitude. This natural metric, paired with a resonance tracking scheme, allows us to address the difficulties associated with physical interpretation and data availability without sacrificing the convenient aspects of linear projection-based model order reduction. A computational framework for the continuation and bifurcation analysis using linear projection-based ROMs is also presented, permitting us to deploy rigorous analysis and bifurcation studies to verify that our ROMs reproduce the intrinsic complexity of full-scale systems. A detailed walk-through of the GCVD-ROM approach is demonstrated on a simple system where important practical considerations and implementation details are discussed using a concrete example. The discretized von Kármán beam and shallow arch partial differential equations are also used to explore complicated scenarios involving modal coupling across disparate time scales and internal resonances.