Multistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators

TL;DR

This paper investigates how weakly nonlinear, cyclically symmetric structures modeled as forced Duffing oscillators can exhibit multiple localized states due to nonlinear stiffness effects, revealing a snaking bifurcation pattern.

Contribution

It demonstrates that localization in symmetric structures can be caused by nonlinearities alone, without mistuning, and identifies isolated solution branches during the transition to nonlinear behavior.

Findings

Multiple localized solutions exist in the weakly nonlinear regime.

Localization is driven by nonlinear stiffness hardening at high excitation levels.

The bifurcation diagram exhibits a snaking pattern similar to phenomena in physics.

Abstract

Many engineering structures are composed of weakly coupled sectors assembled in a cyclic and ideally symmetric configuration, which can be simplified as forced Duffing oscillators. In this paper, we study the emergence of localized states in the weakly nonlinear regime. We show that multiple spatially localized solutions may exist, and the resulting bifurcation diagram strongly resembles the snaking pattern observed in a variety of fields in physics, such as optics and fluid dynamics. Moreover, in the transition from the linear to the nonlinear behaviour isolated branches of solutions are identified. Localization is caused by the hardening effect introduced by the nonlinear stiffness, and occurs at large excitation levels. Contrary to the case of mistuning, the presented localization mechanism is triggered by the nonlinearities and arises in perfectly homogeneous systems.