This paper deals with passive stabilization of thermoacoustic dynamics in a Rijke tube using a Helmholtz resonator. Thermoacoustic instabilities result from the dynamic coupling between the heat release and pressure in a chamber. Helmholtz resonators are used akin to vibration absorbers to suppress unwanted pressure oscillations in such structures and prevent instabilities. The first contribution of the paper is a state-space representation of the thermoacoustic dynamics for the resonator-mounted Rijke tube. This relationship happens to be in the class of linear time invariant, neutral multiple time delay systems (LTI-NMTDS). Then, benefiting from the cluster treatment of characteristic roots (CTCR) paradigm, we investigate the effect of resonator location on suppression of thermoacoustic instability. CTCR is a mathematical tool that determines the stability of LTI-NMTDS exhaustively and nonconservatively in the parameter space of the system. This capability provides a novel tool for the futuristic design concepts of combustors. These analytically obtained findings are also supported with experimental results from a laboratory-scale Rijke tube. In addition, a conceptual case study is presented where the stabilizing contributions of the resonator to the dynamics are investigated under strong thermoacoustic coupling.

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