This study addresses the crucial issue of thermoacoustic instability, an undesirable phenomenon that severely impacts combustion chambers. The mathematical dynamic model of the Rijke tube has been reassessed as a linear time-invariant multi-time-delay system (LTI-MTDS). To analyze its stability characteristics, a precise approximation of the quasi-polynomial was performed. Subsequently, a new theoretical framework based on the Dixon discriminant method was established to derive the critical frequency range. The curves that represent the stability transition points are clearly depicted through comprehensive frequency scanning. This method reveals important information about the stable operating regions of the tube within the parameter space, including delays and geometric dimensions.
Asadi JafariMHZarastvandMZhouJ (2024) Doubly curved truss core composite shell system for broadband diffuse acoustic insulation. Journal of Vibration and Control30(17–18): 4035–4051. DOI: 10.1177/10775463231206229.
2.
BellowsBD (2006) Characterization of Nonlinear Heat Release-Acoustic Interactions in Gas Turbine Combustors. Atlanta, GA: Georgia Institute of Technology.
3.
BredaDMasetSVermiglioR (2005) Pseudospectral differencing methods for characteristic roots of delay differential equations. SIAM Journal on Scientific Computing27(2): 482–495.
4.
CaiJGaoQLiuY, et al. (2024) Generalized Dixon resultant for strong delay-independent stability of linear systems with multiple delays. IEEE Transactions on Automatic Control69(4): 2697–2704. DOI: 10.1109/TAC.2023.3337691.
5.
CandelS (2002) Combustion dynamics and control: progress and challenges. Proceedings of the Combustion Institute29(1): 1–28.
6.
Cepeda-GomezROlgacN (2011) Consensus analysis with large and multiple communication delays using spectral delay space concept. International Journal of Control84(12): 1996–2007.
7.
DixonAL (1908) The eliminant of three quantics in two independent variables. Proceedings of the London Mathematical Society6(49.69): 209.
8.
DowlingAP (1997) Nonlinear self-excited oscillations of a ducted flame. Journal of Fluid Mechanics346: 271–290.
9.
DowlingAPMorgansAS (2005) Feedback control of combustion oscillations. Annual Review of Fluid Mechanics37: 151–182.
10.
EngelborghsKLuzyaninaTRooseD (2002) Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL. ACM Transactions on Mathematical Software28(1): 1–21.
11.
GaoQOlgacN (2016) Bounds of imaginary spectra of LTI systems in the domain of two of the multiple time delays. Automatica72: 235–241.
12.
GaoQOlgacN (2017) Stability analysis for LTI systems with multiple time delays using the bounds of its imaginary spectra. Systems & Control Letters102: 112–118.
13.
GelbertGMoeckJPPaschereitC, et al. (2012) Feedback control of unstable thermoacoustic modes in an annular Rijke tube. Control Engineering Practice20(8): 770–782.
14.
GuKNiculescuSI (2006) 4 stability analysis of time-delay systems: a Lyapunov approach. In Advanced Topics in Control Systems Theory: Lecture Notes from FAP 2005. Berlin: Springer, 139–170.
15.
HuangJMaXCheH, et al. (2019) Further result on interval observer design for discrete-time switched systems and application to circuit systems. IEEE Transactions on Circuits and Systems II: Express Briefs67(11): 2542–2546.
16.
HuangJMaXZhaoX, et al. (2020) Interval observer design method for asynchronous switched systems. IET Control Theory & Applications14(8): 1082–1090.
17.
KapurDSaxenaTYangL (1994) Algebraic and geometric reasoning using Dixon resultants. Proceedings of the international symposium on symbolic and algebraic computation. New York, NY, USA: Association for Computing Machinery, 99–107.
18.
KashinathKWaughICJuniperMP (2014) Nonlinear self-excited thermoacoustic oscillations of a ducted premixed flame: bifurcations and routes to chaos. Journal of Fluid Mechanics761: 399–430.
19.
KopitzJPolifkeW (2008) CFD-based application of the Nyquist criterion to thermo-acoustic instabilities. Journal of Computational Physics227(14): 6754–6778.
20.
KrebsWKredietHPortilloE, et al. (2013) Comparison of nonlinear to linear thermoacoustic stability analysis of a gas turbine combustion system. Journal of Engineering for Gas Turbines & Power135(8): 081503.
21.
OgataK (2004) Fluid systems and thermal systems. In: OgataK (ed) System Dynamics. 4th edition. New Jersey, USA: Pearson Prentice Hall.
22.
OlgacNZalluhogluUKammerAS (2014) Predicting thermoacoustic instability: a novel analytical approach and its experimental validation. Journal of Propulsion and Power30(4): 1005–1015.
23.
OlgacNCepeda-GomezRZalluhogluU, et al. (2015) Parametric investigation of thermoacoustic instability (TAI) in a Rijke tube: a time-delay perspective. International Journal of Spray and Combustion Dynamics7(1): 39–68.
24.
RekasiusZV (1980) A stability test for systems with delays. Joint Automatic Control Conference17: 39.
25.
SelimefendigilFSujithRIPolifkeW (2011) Identification of heat transfer dynamics for non-modal analysis of thermoacoustic stability. Applied Mathematics and Computation217(11): 5134–5150.
26.
SipahiROlgacN (2006) Kernel and offspring concepts for the stability robustness of multiple time delayed systems (MTDS). Journal of Dynamic Systems, Measurement, and Control129(3): 245–251.
27.
TalebitootiRZarastvandM (2018) The effect of nature of porous material on diffuse field acoustic transmission of the sandwich aerospace composite doubly curved shell. Aerospace Science and Technology78: 157–170.
28.
VyhlidalTZitekP (2009) Mapping based algorithm for large-scale computation of quasi-polynomial zeros. IEEE Transactions on Automatic Control54(1): 171–177.
29.
YuanCSongSGaoQ, et al. (2020) A novel frequency-domain approach for the exact range of imaginary spectra and the stability analysis of LTI systems with two delays. IEEE Access8: 36595–36601.
30.
ZalluhogluUOlgacN (2016) A study of Helmholtz resonators to stabilize thermoacoustically driven pressure oscillations. Journal of the Acoustical Society of America139(4): 1962–1973.
31.
ZarastvandMGhassabiMTalebitootiR (2021) A review approach for sound propagation prediction of plate constructions. Archives of Computational Methods in Engineering28: 2817–2843.
32.
ZarastvandMGhassabiMTalebitootiR (2022) Prediction of acoustic wave transmission features of the multilayered plate constructions: a review. Journal of Sandwich Structures & Materials24(1): 218–293.
