How well can linear stability analysis predict the behavior of an outward valve brass instrument model ?
Résumé
A physical model of brass instrument is considered in this paper : a one degree-of-freedom outward-striking valve for the lips, non-linearly coupled to a modal representation of the air column. It is studied through Linear Stability Analysis (LSA) of the equilibrium solution. This approach provides the threshold blowing pressure value, at which instability occurs, and the instability frequency value. The relevance of the results of this method is theoretically limited to the neighbourhood of the equilibrium solution. This paper checks the efficiency of LSA to understand the behaviour of the model computed through time-domain simulations. As expected, a good agreement is observed between LSA and numerical simulations of the complete nonlinear model around the oscillation threshold. For blowing pressures far above the oscillation threshold, the picture is more contrasted. In most of the cases tested, a periodic regime coherent with the LSA results is observed, but over-blowing, quasi-periodicity and period-doubling also occur. Interestingly, LSA predicts the production of the pedal note by a trombone, for which only nonlinear hypotheses have been previously proposed. LSA also predicts the production of a saxhorn note which, although known to musicians, has barely been documented.
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