Analytical Solutions to a Rijke Tube System with Periodic Excitations Through a Semi-analytical Approach

Conference paper
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 622)


Thermoacoustic instability problems are widely existed in many real-world applications such as gas turbines and rocket motors. A Rijke tube is a typical thermoacoustic system, and it is difficult to analyze such a system due to the nonlinearity and time delay. In this paper, a set of nonlinear ordinary differential equations with time delay which represent a Rijke tube system will be studied. The state space of such a tube system is consisted of velocity and pressure, and the periodic motion can be discretized based on an implicit midpoint scheme. Through Newton-Raphson method, the node points on the periodic motion will be solved, and the analytical solution of such a periodic motion for Rijke tube system can be recovered using a set of Fourier representations. According to the theory of discrete maps, the stability of the periodic motion will be obtained. Finally, specific system parameters will be adopted in order to carry out numerical studies to show different periodic motions for such a tube system. The analytical bifurcations which show how period-1 motion involves to period-m motion and then becomes chaos will be demonstrated. With such a technique, some interesting nonlinear phenomena will be explained analytically, which will be of great help to understand and control such a Rijke tube system.


Thermoacoustic Periodic motions Chaos Semi-analytic Bifurcation 



This work was sponsored by Shanghai Sailing Program (Grant No. 19YF1421600).


  1. 1.
    Rijke PLLXXI (1859) Notice of a new method of causing a vibration of the air contained in a tube open at both ends. Phil Mag 17(116):419–422CrossRefGoogle Scholar
  2. 2.
    Rayleigh (1878) The explanation of certain acoustic phenomena. Nature 18:319–321Google Scholar
  3. 3.
    Friedlander MM, Smith TJB, Powell A (1964) Experiments on the Rijke-tube phenomenon. J Acoust Soc Am 36(9):1737–1738ADSCrossRefGoogle Scholar
  4. 4.
    Bisio G, Rubatto G (1999) Sondhauss and Rijke oscillations-thermodynamic analysis, possible applications and analogies. Energy 24(2):117–131CrossRefGoogle Scholar
  5. 5.
    Balasubramanian K, Sujith RI (2008) Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys Fluids 20(4):044103ADSCrossRefGoogle Scholar
  6. 6.
    Subramanian P, Mariappan S, Sujith S, Wahi P (2010) Bifurcation analysis of thermoacoustic instability in a horizontal Rijke tube. Int J Spray Combust Dyn 2(4):325–356CrossRefGoogle Scholar
  7. 7.
    Mariappan S, Sujith RI (2011) Modelling nonlinear thermoacoustic instability in an electrically heated Rijke tube. J Fluid Mech 680:511–533ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Orchini A, Rigas G, Juniper MP (2016) Weakly nonlinear analysis of thermoacoustic bifurcations in the Rijke tube. J Fluid Mech 805:523–550ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Huang JZ, Luo ACJ (2015) Periodic motions and bifurcation trees in a buckled, nonlinear Jeffcott rotor system. Int J Bifurcat Chaos 25(1):1550002MathSciNetCrossRefGoogle Scholar
  10. 10.
    Luo ACJ, Jin HX (2014) Period-m motions to chaos in a periodically forced, Duffing oscillator with a time-delayed displacement. Int J Bifurcat Chaos 24(10):1450126MathSciNetCrossRefGoogle Scholar
  11. 11.
    Yu B, Luo ACJ (2018) Periodic motions and limit cycles of linear cable galloping. Int J Dyn Control 6:41–78MathSciNetCrossRefGoogle Scholar
  12. 12.
    Luo ACJ (2015) Discretization and implicit mapping dynamics. Springer, BerlinCrossRefGoogle Scholar
  13. 13.
    Wang DH, Huang JZ (2016) Periodic motions and chaos for a damped mobile piston system in a high pressure gas cylinder with P control. Chaos, Solitons Fractals 95:168–178ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Guo Y, Luo ACJ (2017) Complete bifurcation trees of a parametrically driving pendulum. J Vib Test Syst Dyn 1(2):93–134Google Scholar
  15. 15.
    Luo ACJ, Xing SY (2017) Time-delay effects on periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator. J Vib Test Syst Dyn 1(1):73–91Google Scholar
  16. 16.
    Matveev KI, Culick FEC (2003) A model for combustion instability involving vortex shedding. Combust Sci Technol 175(6):1059–1083CrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Shanghai Jiao Tong UniversityShanghaiChina

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