Advertisement

Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 521–534 | Cite as

Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure

  • B. Bergeot
  • A. Almeida
  • C. Vergez
  • B. Gazengel
Original Paper

Abstract

Reed instruments are modeled as self-sustained oscillators driven by the pressure inside the mouth of the musician. A set of nonlinear equations connects the control parameters (mouth pressure, lip force) to the system output, hereby considered as the mouthpiece pressure. Clarinets can then be studied as dynamical systems; their steady behavior being dictated uniquely by the values of the control parameters. Considering the resonator as a lossless straight cylinder is a dramatic yet common simplification that allows for simulations using nonlinear iterative maps.

This paper investigates analytically the effect of a linearly increasing blowing pressure on the behavior of this simplified clarinet model. When the control parameter varies, results from the so-called dynamic bifurcation theory are required to properly analyze the system. This study highlights the phenomenon of bifurcation delay and defines a new quantity, the dynamic oscillation threshold. A theoretical estimation of the dynamic oscillation threshold is proposed and compared with numerical simulations.

Keywords

Musical acoustics Clarinet-like instruments Iterated maps Dynamic bifurcation Bifurcation delay Transient processes 

Notes

Acknowledgements

We wish to thank Mr. Jean Kergomard for his valuable comments on the manuscript.

This work was done within the framework of the project SDNS-AIMV “Systèmes Dynamiques Non-Stationnaires—Application aux Instruments à Vent” financed by Agence Nationale de la Recherche (ANR).

References

  1. 1.
    Atig, M., Dalmont, J.P., Gilbert, J.: Saturation mechanism in clarinet-like instruments, the effect of the localised nonlinear losses. Appl. Acoust. 65(12), 1133–1154 (2004) CrossRefGoogle Scholar
  2. 2.
    Baesens, C.: Slow sweep through a period-doubling cascade: delayed bifurcations and renormalisation. Physica D 53, 319–375 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Baesens, C.: Gevrey series and dynamic bifurcations for analytic slow-fast mappings. Nonlinearity 8, 179–201 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bender, C., Orszag, S.: Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, New York (1987) Google Scholar
  5. 5.
    Bergeot, B., Vergez, C., Almeida, A., Gazengel, B.: Measurement of attack transients in a clarinet driven by a ramp-like varying pressure. In: 11ème Congrès Français d’Acoustique and 2012 Annual IOA Meeting, Nantes, France, April 23rd–27th 2012 Google Scholar
  6. 6.
    Chaigne, A., Kergomard, J.: Instruments à anche. In: Acoustique des Instruments de Musique, pp. 400–468. Belin, Paris (2008). Chap. 9 Google Scholar
  7. 7.
    Dalmont, J., Gilbert, J., Kergomard, J., Ollivier, S.: An analytical prediction of the oscillation and extinction thresholds of a clarinet. J. Acoust. Soc. Am. 118(5), 3294–3305 (2005) CrossRefGoogle Scholar
  8. 8.
    Feigenbaum, M.J.: The universal metric properties of nonlinear transformations. J. Stat. Phys. 21(6), 669–706 (1979) MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ferrand, D., Vergez, C., Silva, F.: Seuils d’oscillation de la clarinette: validité de la représentation excitateur-résonateur. In: 10ème Congrès Français d’Acoustique, Lyon, France, April 12nd–16th 2010 Google Scholar
  10. 10.
    Fruchard, A.: Canards et râteaux. Ann. Inst. Fourier 42(4), 825–855 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Fruchard, A.: Sur l’équation aux différences affine du premier ordre unidimensionnelle. Ann. Inst. Fourier 46(1), 139–181 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Fruchard, A., Schäfke, R.: Bifurcation delay and difference equations. Nonlinearity 16, 2199–2220 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fruchard, A., Schäfke, R.: Sur le retard à la bifurcation. In: International Conference in Honor of Claude Lobry (2007). http://intranet.inria.fr/international/arima/009/pdf/arima00925.pdf Google Scholar
  14. 14.
    Hirschberg, A.: Aero-acoustics of wind instruments. In: Hirschberg, A., Kergomard, J., Weinreich, G. (eds.) Mechanics of Musical Instruments. CISM Courses and Lectures, vol. 335, pp. 291–361. Springer, Berlin (1995). Chap. 7 Google Scholar
  15. 15.
    Hirschberg, A., de Laar, R.W.A.V., Maurires, J.P., Wijnands, A.P.J., Dane, H.J., Kruijswijk, S.G., Houtsma, A.J.M.: A quasi-stationary model of air flow in the reed channel of single-reed woodwind instruments. Acustica 70, 146–154 (1990) Google Scholar
  16. 16.
    Kapral, R., Mandel, P.: Bifurcation structure of the nonautonomous quadratic map. Phys. Rev. A 32(2), 1076–1081 (1985) CrossRefGoogle Scholar
  17. 17.
    Kergomard, J.: Elementary considerations on reed-instrument oscillations. In: Hirschberg, A., Kergomard, J., Weinreich, G. (eds.) Mechanics of Musical Instruments. CISM Courses and Lectures, vol. 335, pp. 229–290. Springer, Berlin (1995). Chap. 6 Google Scholar
  18. 18.
    Kergomard, J., Dalmont, J.P., Gilbert, J., Guillemain, P.: Period doubling on cylindrical reed instruments. In: Proceeding of the Joint Congress CFA/DAGA 04. Société Française d’Acoustique—Deutsche Gesellschaft für Akustik, pp. 113–114. Strasbourg, France (2004) Google Scholar
  19. 19.
    Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory vol. 112, 3rd edn. p. 136. Springer, Berlin (2004). Chap. 4 zbMATHCrossRefGoogle Scholar
  20. 20.
    Maganza, C., Caussé, R., Laloë, F.: Bifurcations, period doublings and chaos in clarinet-like systems. Europhys. Lett. 1(6), 295 (1986) CrossRefGoogle Scholar
  21. 21.
    Mcintyre, M.E., Schumacher, R.T., Woodhouse, J.: On the oscillations of musical instruments. J. Acoust. Soc. Am. 74(5), 1325–1345 (1983) CrossRefGoogle Scholar
  22. 22.
    Ollivier, S., Dalmont, J.P., Kergomard, J.: Idealized models of reed woodwinds. Part 2: On the stability of two-step oscillations. Acta Acust. united Acust. Acta. Acust. Acust. 91, 166–179 (2005) Google Scholar
  23. 23.
    Taillard, P., Kergomard, J., Laloë, F.: Iterated maps for clarinet-like systems. Nonlinear Dyn. 62, 253–271 (2010) zbMATHCrossRefGoogle Scholar
  24. 24.
    Tredicce, J.R., Lippi, G., Mandel, P., Charasse, B., Chevalier, A., Picqué, B.: Critical slowing down at a bifurcation. Am. J. Phys. 72(6), 799–809 (2004) CrossRefGoogle Scholar
  25. 25.
    Wilson, T., Beavers, G.: Operating modes of the clarinet. J. Acoust. Soc. Am. 56(2), 653–658 (1974) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • B. Bergeot
    • 1
  • A. Almeida
    • 1
  • C. Vergez
    • 2
  • B. Gazengel
    • 1
  1. 1.Laboratoire d’Acoustique de l’Université du Maine (LAUM-CNRS UMR 6613)Le Mans Cedex 9France
  2. 2.Laboratoire de Mécanique et Acoustique (LMA-CNRS UPR7051)Marseille Cedex 20France

Personalised recommendations