Abstract
Helmholtz resonator (HR) is widely used to control thermoacoustic instabilities, and its interaction with combustion systems has been studied. However, modal interaction across a range of coupling strengths remains to be discussed. This study introduces a subsystem-based analytical framework to clarify the modal interaction mechanisms between HR and thermoacoustic systems across varying coupling intensities. By treating the resonator as another dynamical subsystem, the interaction is expressed as a coupled characteristic equation that provides enhanced physical clarity. We identify three distinct situations: independent weak coupling, a critical hybridization window near the exceptional point (EP), and a strong-coupling regime where modes transform into hybrid eigenmodes. Findings indicate that while the EP serves as a localized milestone for mode merging, the global optimal suppression in the strong-coupling regime is governed by the global roots of the coupled system's characteristic function. Consequently, minor modifications to boundary conditions can lead to a significant enhancement in suppression effectiveness. Furthermore, parametric investigations show that optimal design often deviates from conventional empirical rules, such as pressure-antinode placement or perfect frequency matching, and is instead dictated by the modal phase-matching conditions of the coupled system. This work provides theoretical insights into the design of resonators, facilitating their application in complex thermoacoustic environments.