Numerical continuation of a physical model of brass instruments: Application to trumpet comparisons.

The system formed by a trumpet player and his/her instrument can be seen as a non-linear dynamic system and modeled by physical equations. Numerical tools can then be used to study these models and clarify the influence of the model parameters. The acoustic input impedance, for instance, is strongly dependent on the geometry of the air column and is therefore of primary interest for a musical instrument maker. In this study, a method of continuation of periodic solutions based on the combination of the Harmonic Balance Method (HBM) and the Asymptotic Numerical Method (ANM) is applied to a physical model of brass instruments. It allows the study of the evolution of the system where one parameter of the model (static mouth pressure) varies. This method is used to compare different B♭ trumpets on the basis of two descriptors (hysteresis behavior and dynamic range) computed from the continuation outputs. Results show that this methodology enables the differentiation of instruments in the space of the calculated descriptors. Calculations for different values of the lip parameters are also performed to confirm that the obtained categorization is independent of variations of lip parameters.

Oscillation threshold of a clarinet model: a numerical continuation approach.

This paper focuses on the oscillation threshold of single reed instruments. Several characteristics such as blowing pressure at threshold, regime selection, and playing frequency are known to change radically when taking into account the reed dynamics and the flow induced by the reed motion. Previous works have shown interesting tendencies, using analytical expressions with simplified models. In the present study, a more elaborated physical model is considered. The influence of several parameters, depending on the reed properties, the design of the instrument or the control operated by the player, are studied. Previous results on the influence of the reed resonance frequency are confirmed. New results concerning the simultaneous influence of two model parameters on oscillation threshold, regime selection and playing frequency are presented and discussed. The authors use a numerical continuation approach. Numerical continuation consists in following a given solution of a set of equations when a parameter varies. Considering the instrument as a dynamical system, the oscillation threshold problem is formulated as a path following of Hopf bifurcations, generalizing the usual approach of the characteristic equation, as used in previous works. The proposed numerical approach proves to be useful for the study of musical instruments. It is complementary to analytical analysis and direct time-domain or frequency-domain simulations since it allows to derive information that is hardly reachable through simulation, without the approximations needed for analytical approach.