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Global Description of Flutter Dynamics via Koopman Theory

Jiwoo Song, Daning Huang

TL;DR

The paper tackles nonlinear aeroelastic flutter by adopting a global linearization approach based on Koopman theory. It introduces the Extended Koopman Bilinear Form (EKBF) to capture higher-order dependence on flutter parameters and demonstrates that EKBF can interpolate and extrapolate principal eigenvalues, predict flutter boundaries, and reveal nonlinear stability features via isostables and isochrons. The methodology is validated on a 2D academic model and a nonlinear panel flutter problem, showing robustness to noise and advantages over traditional AR approaches. The work provides a unified framework that connects Koopman eigenstructure with pre-flutter and post-flutter behavior, yielding practical tools for flutter analysis and prediction with actionable filtering and lifting strategies.

Abstract

This paper presents a novel parametrization approach for aeroelastic systems utilizing Koopman theory, specifically leveraging the Koopman Bilinear Form (KBF) model. To address the limitations of linear parametric dependence in the KBF model, we introduce the Extended KBF (EKBF) model, which enables a global linear representation of aeroelastic dynamics while capturing stronger nonlinear dependence on, e.g., the flutter parameter. The effectiveness of the proposed methodology is demonstrated through two case studies: a 2D academic example and a panel flutter problem. Results show that EKBF effectively interpolates and extrapolates principal eigenvalues, capturing flutter mechanisms, and accurately predicting the flutter boundary even when the data is corrupted by noise. Furthermore, parameterized isostable and isochron identified by EKBF provides valuable insights into the nonlinear flutter system.

Global Description of Flutter Dynamics via Koopman Theory

TL;DR

The paper tackles nonlinear aeroelastic flutter by adopting a global linearization approach based on Koopman theory. It introduces the Extended Koopman Bilinear Form (EKBF) to capture higher-order dependence on flutter parameters and demonstrates that EKBF can interpolate and extrapolate principal eigenvalues, predict flutter boundaries, and reveal nonlinear stability features via isostables and isochrons. The methodology is validated on a 2D academic model and a nonlinear panel flutter problem, showing robustness to noise and advantages over traditional AR approaches. The work provides a unified framework that connects Koopman eigenstructure with pre-flutter and post-flutter behavior, yielding practical tools for flutter analysis and prediction with actionable filtering and lifting strategies.

Abstract

This paper presents a novel parametrization approach for aeroelastic systems utilizing Koopman theory, specifically leveraging the Koopman Bilinear Form (KBF) model. To address the limitations of linear parametric dependence in the KBF model, we introduce the Extended KBF (EKBF) model, which enables a global linear representation of aeroelastic dynamics while capturing stronger nonlinear dependence on, e.g., the flutter parameter. The effectiveness of the proposed methodology is demonstrated through two case studies: a 2D academic example and a panel flutter problem. Results show that EKBF effectively interpolates and extrapolates principal eigenvalues, capturing flutter mechanisms, and accurately predicting the flutter boundary even when the data is corrupted by noise. Furthermore, parameterized isostable and isochron identified by EKBF provides valuable insights into the nonlinear flutter system.

Paper Structure

This paper contains 28 sections, 41 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Mathematical Formulation
  3. Global Linearization of Parametric Nonlinear Systems via Koopman Theory
  4. Koopman Bilinear Form
  5. Autonomous case
  6. Lattice structure of the Koopman eigenvalues
  7. Koopman bilinear form
  8. Extended KBF Model
  9. Stability Analysis Using Koopman Theory
  10. General Concept
  11. Fixed Point
  12. Analytical Example, Part I
  13. Limit Cycle
  14. Analytical Example, Part II
  15. Flutter Prediction Algorithm via EKBF Model
  16. ...and 13 more sections

Figures (25)

  • Figure 1: A 2D Hopf bifurcation example in the pre-bifurcation regime with stable fixed-point.
  • Figure 2: Isostable for different system parameter $\mu$ in the pre-bifurcation regime.
  • Figure 3: A 2D Hopf bifurcation example in the post-bifurcation regime with stable limit cycle.
  • Figure 4: Isocrhon for different system parameter $\mu$ in the post-bifurcation regime.
  • Figure 5: Figure describing the finite dimensional Koopman operator for nonlinear dynamical systems.
  • ...and 20 more figures