Effects of Internal Resonance and Damping on Koopman Modes

TL;DR

This paper explores how internal resonance and damping influence Koopman modes in coupled nonlinear oscillators, validating the Koopman eigenfunction approach for analyzing nonlinear normal modes and examining its limitations.

Contribution

It introduces a quantitative measure for internal resonance and assesses the robustness of Koopman-based analysis in systems with damping and nonlinear coupling.

Findings

Koopman eigenfunctions effectively describe NNMs in coupled Duffing oscillators.

Internal resonance significantly affects the accuracy of Koopman mode analysis.

Damping impacts the robustness of Koopman-based NNM approximations.

Abstract

This study investigates the nonlinear normal modes (NNMs) of a system comprising of two coupled Duffing oscillators, with one oscillator being grounded and with the coupling being both linear and nonlinear. The study utilizes the eigenfunctions of the Koopman operator and validates their connection with the Shaw-Piere invariant manifold framework for NNMs. Furthermore, the study delves into the impact of internal resonance and dissipation on the accuracy of this framework by defining a continuous quantitative measure for internal resonance. The applicability and robustness of the framework for the systems which are very similar qualitatively to that of an ENO, are also observed and discussed about the limitations of the approximation technique.