Global complexity effects due to local damping in a nonlinear system in 1:3 internal resonance

TL;DR

This paper reveals that even weak local damping in a nonlinear 1:3 internal resonance system causes global asynchronous motion and mode complexity, challenging the assumption that near-resonant responses follow autonomous dynamics.

Contribution

It demonstrates analytically and numerically how local damping induces global asynchronous effects and mode complexity in a nonlinear resonant system, extending the understanding of damping impacts.

Findings

Maximum mode complexity with 90° phase difference achieved

Global asynchronous motion caused by local damping

Extension of mode complexity and localization definitions to quasi-periodic motions

Abstract

It is well-known that nonlinearity may lead to localization effects and coupling of internally resonant modes. However, research focused primarily on conservative systems commonly assumes that the near-resonant forced response closely follows the autonomous dynamics. Our results for even a simple system of two coupled oscillators with a cubic spring clearly contradict this common belief. We demonstrate analytically and numerically global effects of a weak local damping source in a harmonically forced nonlinear system under condition of 1:3 internal resonance: The global motion becomes asynchronous, i.e., mode complexity is introduced with a non-trivial phase difference between the modal oscillations. In particular, we show that a maximum mode complexity with a phase difference of $90^\circ$ is attained in a multi-harmonic sense. This corresponds to a transition from generalized standing to traveling waves in the system's modal space. We further demonstrate that the localization is crucially affected by the system's damping. Finally, we propose an extension of the definition of mode complexity and mode localization to nonlinear quasi-periodic motions, and illustrate their application to a quasi-periodic regime in the forced response.