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

TL;DR

This work tackles nonlinear dynamical systems where linear modal analysis fails to capture intrinsic modes. It introduces a data-driven, physics-constrained autoencoder that learns nonlinear modal transformations and modal dynamics directly from response data. The method is validated on a numerical and an experimental nonlinear beam, demonstrating NNMs isolation, mode decomposition, reconstruction, and short-horizon prediction, with energy-dependent frequency shifts and increasing nonlinearity in configurations. This approach enables physics-informed, data-driven modal analysis without explicit governing equations and has potential for broader nonlinear structural dynamics applications.

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is challenging as the modal transformation based on the superposition principle that works well for linear systems is no longer applicable.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.

Figures (12)

• Figure 1: Our physics-constrained deep autoencoder architecture: a The framework includes a deep autoencoder that transforms system states $z=(x,y)$ into intrinsic coordinates $(p,q)$ or $\varphi$ through the function ${\mathrm{\varphi}} = \vartheta \left( {\mathrm{z}} \right)$. The autoencoder then decodes the intrinsic coordinates back to the original coordinates using ${\mathrm{z}} = {\vartheta ^{ - 1}}\left( {\mathrm{\varphi}} \right)$. Additional physics-based constraints can be applied to the intrinsic coordinates to convert them to desired modal coordinates. b A dynamics block ($G$) is also implemented, which advances intrinsic coordinates over time and ensures that encoding the next original coordinates is equivalent to advancing the current intrinsic coordinates. c By combining the encoder, dynamics block, and decoder in the appropriate order, intrinsic coordinates can be determined for predicting future states. It is important to note that the decoder is not the exact inverse of the encoder, but it is approximated as closely as possible through a reconstruction loss function.
• Figure 2: Nonlinear beam: a The finite element model of a nonlinear beam. Nonlinear spring is shown as $k_nl$ and rotational spring is shown in the junction of main and thin beam as $k_r$. b The experimental nonlinear beam. Shaker is located at position 2 and we measure the acceleration of points 3 and 7.
• Figure 3: First three natural frequencies and corresponding mode shapes of the nonlinear beam
• Figure 4: Nonlinear mode isolation of first NNM of a weakly nonlinear beam. a First NNM for different energy levels. b Second NNM for different energy levels. c Configuration plots for three different energy levels (marked in plot a). d Configuration plots of three different frequencies at a specific level of energy (as marked in plot b)
• Figure 5: Nonlinear mode isolation of the first NNM of a highly nonlinear beam. a First NNM for different energy levels. b NNM isolation at a specific energy level (extracted from plot a) c Configuration plots for three different energy levels (marked in plot a). d Configuration plots of three different frequencies at a specific level of energy (as marked in plot b)