Abstract
An autonomous fourth order model of vocal fold vibrations is proposed. Each fold is represented by a lower and upper mass, and the aerodynamic forces are derived from a modified Bernoulli equation. The model exhibits many features of normal phonation in a wide parameter region. At the borderlines of this region coexistence of limit cycles, period-doubling and chaos are observed. Implications for an understanding of pathological voices are discussed.
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References
van den Berg, J., Zantema, J. T., and Doornenbal, P., ‘On the air resistance and the Bernoulli effect of the human larynx’, J. Acoust. Soc. Am. 29, 1957, 626–631.
Ishizaka, K. and Flanagan, J. L., ‘Synthesis of voiced sounds from a two-mass model of the vocal cords’, Bell. Syst. Techn. 51, 1972, 1233–1268.
Titze, I. R. (ed.), Vocal Fold Physiology: New Frontiers in Basic Science, Singular Publ. Group, San Diego, 1993.
Mende, W., Herzel, H., and Wermke, K., ‘Bifurcations and chaos in newborn infant cries’, Phys. Lett. A 145, 1990, 418–424.
Herzel, H. and Wendler, J., ‘Evidence of chaos in phonatory samples’, in Proceed, EUROSPEECH, Genova, 1991, 263–266.
Titze, I. R. and Talkin, D. T., ‘A theoretical study of the effects of various laryngeal configurations on the acoustics of phonation’, J. Acoust. Soc. Am. 66, 1979, 60–74.
Liljenkrants, J., ‘Numerical simulations of glottal flow’, in Proceed. EUROSPEECH, Genova, 1991, 255–258.
Berry, M., Herzel, H., Titze, I., and Krischer, K., ‘Interpretation of biochemical simulations of normal and chaotic vocal fold oscillations with empirical eigenfunctions’, J. Acoust. Soc. Am., in press.
Stevens, K. N., ‘Physics of laryngeal behaviour and larynx modes’, Phonetica 34, 1977, 264–279.
Baer, T., ‘Investigation of the phonatory mechanism’, ASHA Report 11, 1981, 38–47.
Iijima, H., Miki, N., and Nagai, N., ‘Glottal impedance based on a finite element analysis of two-dimensional unsteady viscous flow in a static glottis’, IEEE Trans. Signal Proc. 40, 1992, 2125–2135.
Cranen, B. and Boves, L., ‘On subglottal formant analysis’, J. Acoust. Soc. Am. 81, 1987 734–746.
Hirano, M., ‘Structure and vibratory behaviour of the vocal folds’, in Dynamic Aspects of Speech Production, M. Sawashima and F. S. Cooper (eds.), University of Tokyo Press, Tokyo, 1977, 13–30.
Awrejcewicz, J., ‘Numerical analysis of the oscillations of human vocal cords’, Nonlinear Dynamics 2, 1991, 35–52.
Kubicek, M. and Marek, M., Computational Methods in Bifurcation Theory and Dissipative Structures, Springer, New York, 1983.
Knudsen, C., Feldberg, R., and Jaschinski, A., ‘Non-linear dynamic phenomena in the behaviour of a rallway wheelset model’, Nonlinear Dynamics 2, 1991, 389–404.
Titze, I. R., ‘The human vocal cords: A mathematical model. Part II’, Phonetica 28, 1973, 129–170.
Smith, M. E., Berke, G. S., Gerratt, B. R., and Kreiman, J., ‘Laryngeal paralyses: Theoretical considerations and effects on laryngeal vibration’, J. Speech Hear. Res. 35, 1992, 545–554.
Herzel, H., Steinecke, I., Mende, W., and Wermke, K., ‘Chaos and bifurcations during voiced speechls, in Complexity. Chaos and Biological Evolution, E. Mosekilde and L. Mosekilde (eds.), Plenum Press, New York, 1991, 41–50.
Kelman, A. W., ‘Vibratory pattern of the vocal folds’, Folia Phoniatr 33, 1981, 73–99.
Ramig, L. A., Scherer, R. C., Titze, I. R., and Ringel, S. P., ‘Acoustic analysis of voices of patients with neurologic disease: Rational and preliminary data, Ann. Otol. Rhinol, Laryngol. 97, 1988, 164–172.
Ishizaka, K. and Isshiki, N., ‘Computer simulation of pathological vocal-cord vibration’, J. Acoust. Soc. Am. 60, 1976, 1193–1198.
Herzel, H., Berry, D., Titze, I., and Saleh, M., ‘Analysis of vocal disorders with methods from nonlinear dynamics’, J. Speech Hear. Res., in press.
Wong, D., Ito, M. R., Cox, N. B., and Titze, I. R., ‘Observation of perturbation in a lumped-clement model of the vocal folds with application to some pathological cases’, J. Acoust. Soc. Am. 89, 1991, 383–394.
Ebeling, W., Engel-Herbert, H., and Herzel, H., Selbstorganisation in der Zeit, Akademic-Verlag, 1990.
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Herzel, H., Knudsen, C. Bifurcations in a vocal fold model. Nonlinear Dyn 7, 53–64 (1995). https://doi.org/10.1007/BF00045125
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DOI: https://doi.org/10.1007/BF00045125