Data-driven Nonlinear Model Reduction to Spectral Submanifolds in Mechanical Systems

TL;DR

This paper presents a data-driven method for reducing complex nonlinear mechanical systems to low-dimensional spectral submanifolds, enabling accurate predictions of system responses under external forcing.

Contribution

It introduces a novel spectral submanifold-based reduction technique that handles nonlinearizable systems with multiple steady states using observational data.

Findings

Successfully reduced nonlinear systems to low-dimensional models

Accurately predicted responses under external forcing

Validated on synthetic and experimental vibration data

Abstract

While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing non-linearizable systems with multiple coexisting steady states have been unavailable. In this paper, we review such a data-driven nonlinear model reduction methodology based on spectral submanifolds (SSMs). As input, this approach takes observations of unforced nonlinear oscillations to construct normal forms of the dynamics reduced to very low dimensional invariant manifolds. These normal forms capture amplitude-dependent properties and are accurate enough to provide predictions for non-linearizable system response under the additions of external forcing. We illustrate these results on examples from structural vibrations, featuring both synthetic and experimental data.