Minimal blowing pressure allowing periodic oscillations in a simplied reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation
Résumé
A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplied model of a reed-like instrument having two quasi-harmonic resonances, represented by a 4 dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the dierent solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the rst acoustic resonance (rst register) or the second acoustic resonance (second register) are successfully identied by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The eect of the inharmonicity between the rst two impedance peaks is observed both when the amplitude of the rst is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.
Domaines
Acoustique [physics.class-ph]
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