Data-driven modeling of mechanical systems on spectral submanifolds in theory and practice
OPEN ACCESS
Loading...
Author / Producer
Date
2024
Publication Type
Doctoral Thesis
ETH Bibliography
yes
Citations
Altmetric
OPEN ACCESS
Data
Rights / License
Abstract
In this thesis, we discuss how to apply spectral submanifold (SSM) theory to obtain low-dimensional models of smooth dynamical systems from data. To this end, we build on the recent SSMLearn algorithm, which identifies the geometry of an SSM in an observable space and its reduced dynamics in the normal form. In particular, we validate many of the assumptions on which data-driven SSM modeling is based and show how to expand its applicability to larger datasets, higher-dimensional models, and forced systems. We demonstrate these improvements on examples of mechanical systems ranging from nonlinear beam vibrations to tank sloshing.
In the first part, we address the computations involved in SSMLearn. For the geometry identification, we develop an alternative method based on singular value decomposition. In the reduced dynamics, we investigate computing the normal form analytically rather than via optimization, and summarize this choice as a tradeoff between numerical stability and the model's range of validity. We introduce an alternative algorithm, fastSSM, that builds on a vastly simplified implementation that speeds up computations by several orders of magnitude without observing any significant loss in accuracy.
In the second part, we look into the reconstruction of invariant manifolds from scalar data with delay embedding. We show that this crucial first step to SSM identification obeys a consistent structure at leading order, and prove that the eigenvectors of a delay-embedded differentiable dynamical system at a fixed point are given by the Vandermonde matrix related to the eigenvalues and the timelag. Subsequently, we show how to exploit this structure to improve data-driven models. We also extend our theory to systems subject to small general time-dependent forcing and prove that the application of periodic forcing to SSMs in delay-embedded spaces is justified.
In the third part, we study linear and nonlinear resonances and develop a method for learning SSMs directly from periodically forced systems. By passing to the Poincaré map, we demonstrate that fastSSM can identify quasiperiodic forced response due to an internal resonance. Next, we develop an example proving the occurrence of subharmonic forced response as a result of a nonlinear resonance in the system's integrable limit. Finally, we apply our method to experiment data to explain three-periodic forced response in tank sloshing.
Permanent link
Publication status
published
External links
Editor
Contributors
Examiner : Haller, George
Examiner : Eriten, Melih
Book title
Journal / series
Volume
Pages / Article No.
Publisher
ETH Zurich
Event
Edition / version
Methods
Software
Geographic location
Date collected
Date created
Subject
Nonlinear dynamics; Model order reduction; Spectral submanifolds; Invariant manifolds; Machine learning
Organisational unit
03973 - Haller, George / Haller, George