Data-driven Nonlinear Modal Analysis with Physics-constrained Deep Learning: Numerical and Experimental Study

2025-03-11

AI Summary

Identifying intrinsic modal coordinates for nonlinear dynamical systems remains challenging due to the absence of closed-form models and inherent nonlinearity. To address this, we propose a physics-informed, data-driven method that directly learns the nonlinear normal mode (NNM) transformation from vibration response data—enabling modal decomposition, reconstruction, and prediction without explicit governing equations. This work introduces, for the first time, physics-informed neural networks (PINNs) into NNM identification, embedding fundamental physical constraints such as energy conservation and symmetry to overcome reliance on analytical system models. The approach successfully isolates the first two NNMs of a nonlinear beam in both numerical simulations and free-vibration experiments, accurately capturing hallmark nonlinear phenomena—including amplitude-dependent frequency hardening and distorted mode shapes. Reconstruction and prediction errors remain below 5%, demonstrating high fidelity and generalizability.

Technology Category

• Machine Learning: Multimodal Learning
• Natural Language Processing: Language Grounding & Multi-modal NLP
• Computer Vision: Multi-modal Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

To fully understand, analyze, and determine the behavior of dynamical systems, it is crucial to identify their intrinsic modal coordinates. In nonlinear dynamical systems, this task is challenging as the modal transformation based on the superposition principle that works well for linear systems is no longer applicable. To understand the nonlinear dynamics of a system, one of the main approaches is to use the framework of Nonlinear Normal Modes (NNMs) which attempts to provide an in-depth representation. In this research, we examine the effectiveness of NNMs in characterizing nonlinear dynamical systems. Given the difficulty of obtaining closed-form models or equations for these real-world systems, we present a data-driven framework that combines physics and deep learning to the nonlinear modal transformation function of NNMs from response data only. We assess the framework's ability to represent the system by analyzing its mode decomposition, reconstruction, and prediction accuracy using a nonlinear beam as an example. Initially, we perform numerical simulations on a nonlinear beam at different energy levels in both linear and nonlinear scenarios. Afterward, using experimental vibration data of a nonlinear beam, we isolate the first two NNMs. It is observed that the NNMs' frequency values increase as the excitation level of energy increases, and the configuration plots become more twisted (more nonlinear). In the experiment, the framework successfully decomposed the first two NNMs of the nonlinear beam using experimental free vibration data and captured the dynamics of the structure via prediction and reconstruction of some physical points of the beam.

Problem

• Identify intrinsic modal coordinates in nonlinear dynamical systems.
• Develop a data-driven framework combining physics and deep learning.
• Assess framework's accuracy in mode decomposition and prediction.

Innovation

• Data-driven framework combines physics and deep learning
• Nonlinear Normal Modes (NNMs) characterize nonlinear dynamics
• Experimental vibration data isolates NNMs effectively