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

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.

References in corpus (17)

• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Deep learning for universal linear embeddings of nonlinear dynamics
• Data-driven discovery of coordinates and governing equations
• Applied Koopmanism
• Koopman invariant subspaces and finite linear representations of nonlinear dynamical systems for control
• Data-Driven POD-Galerkin Reduced Order Model for Turbulent Flows
• Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds
• Construction of Reduced Order Models for Fluid Flows Using Deep Feedforward Neural Networks
• How to Compute Invariant Manifolds and their Reduced Dynamics in High-Dimensional Finite-Element Models
• Automated Computation of Autonomous Spectral Submanifolds for Nonlinear Modal Analysis
• Exact Nonlinear Model Reduction for a von Karman beam: Slow-Fast Decomposition and Spectral Submanifolds
• Explicit Backbone Curves from Spectral Submanifolds of Forced-Damped Nonlinear Mechanical Systems
• Koopman mode expansions between simple invariant solutions
• Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations
• Koopman analysis of Burgers equation
• How do Conservative Backbone Curves Perturb into Forced Responses? A Melnikov Function Analysis
• Stability of Forced-Damped Response in Mechanical Systems from a Melnikov Analysis

Cited by in corpus (5)

• Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds
• Nonlinear Model Reduction to Fractional and Mixed-Mode Spectral Submanifolds
• Data-Assisted Non-Intrusive Model Reduction for Forced Nonlinear Finite Elements Models
• Data-driven modeling of the regular and chaotic dynamics of an inverted flag from experiments
• Dynamics-based machine learning of transitions in Couette flow