Subcritical Hopf Bifurcation-Based Oscillator for Nonresonant Vibration Stabilization

Abstract

A physical system exhibiting subcritical Hopf bifurcation followed by a fold bifurcation of the limit cycles presents three operational regimes: a single limit cycle around an unstable equilibrium, a pair of stable and unstable limit cycles around a stable equilibrium, and a globally stable equilibrium. In this study, we investigate whether such an oscillator can be utilized to nonresonantly suppress single-mode instability in structural systems. To this end, we analyze a system comprising a negatively damped linear primary oscillator coupled with a lightweight secondary oscillator capable of subcritical Hopf responses. The secondary oscillator is modelled using a modified van der Pol oscillator exhibiting the stated three regimes of operations. For the single limit cycle regime, prior work (Tandel et al., Nonlinear Dynamics, 2023) provided a criterion for conditional stabilization, defining the threshold of instability in the primary system for given secondary oscillator parameters. The present study demonstrates that the double limit cycle regime offers conditional stabilization over a significantly larger instability range, leveraging the benefits of the subcritical Hopf dynamics. Finally, the globally stable equilibrium regime offers unconditional stabilization, though its stabilization limit remains equivalent to that using the single limit cycle regime. Hence, a secondary vibration absorber with a stable equilibrium surrounded by an unstable limit cycle which is further surrounded by a larger amplitude stable limit cycle is beneficial as it offers conditional stabilization of an unstable primary system with an arbitrary degree of instability.