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Numerical continuation using the Asymptotic Numerical Method (ANM), together with the Harmonic Balance Method (HBM), makes it possible to follow the periodic solutions of non-linear dynamical systems such as physical models of wind instruments. This has been recently applied to practical problems such as the categorization of musical instruments from the calculated bifurcation diagrams [V. Fréour et al. Journal of the Acoustical Society of America 148 (2020) https://doi.org/10.1121/10.0001603]. Nevertheless, one problem often encountered concerns the uncertainty on some parameters of the model (reed parameters in particular), the values of which are set almost arbitrarily because they are too difficult to measure experimentally. In this work we propose a novel approach where constraints, defined from experimental measurements, are added to the system. This operation allows uncertain parameters of the model to be relaxed and the continuation of the periodic solution with constraints to be performed. It is thus possible to quantify the variations of the relaxed parameters along the solution branch. The application of this technique to a physical model of a trumpet is presented in this paper, with constraints derived from experimental measurements on a trumpet player.

The lowest fingerings of the saxophone can lead to several different regimes, depending on the musician’s control and the characteristics of the instrument. This is explored in this paper through a physical model of saxophone. The harmonic balance method shows that for many combinations of musician control parameters, several regimes are stable. Time-domain synthesis is used to show how different regimes can be selected through initial conditions and the initial evolution (rising time) of the blowing pressure, which is explained by studying the attraction basin of each stable regime. These considerations are then applied to study how the produced regimes are affected by properties of the resonator. The inharmonicity between the first two resonances is varied in order to find the value leading to the best suppression of unwanted overblowing. Overlooking multistability in this description can lead to biased conclusions. Results for all the lowest fingerings show that a slightly positive inharmonicity, close to that measured on a saxophone, leads to first register oscillations for the greatest range of control parameters. A perfect harmonicity (integer ratio between the first two resonances) decreases first register production, which adds nuance to one of Benade’s guidelines for understanding sound production. Thus, this study provides some a posteriori insight into empirical design choices relative to the saxophone.

Musical instruments display a wealth of dynamics, from equilibria (where no sound is produced) to a wide diversity of periodic and non-periodic sound regimes. We focus here on two types of flute-like instruments, namely a recorder and a pre-hispanic Chilean flute. A recent experimental study showed that they both produce quasiperiodic sound regimes which are avoided or played on purpose depending on the instrument. We investigate the generic model of sound production in flute-like musical instruments, a system of neutral delay-differential equations. Using time-domain simulations, we show that it produces stable quasiperiodic oscillations in good agreement with experimental observations. A numerical bifurcation analysis is performed, where both the delay time (related to a control parameter) and the detuning between the resonance frequencies of the instrument – a key parameter for instrument makers – are considered as bifurcation parameters. This demonstrates that the large detuning that is characteristic of prehispanic Chilean flutes plays a crucial role in the emergence of stable quasiperiodic oscillations. [ABSTRACT FROM AUTHOR]

Periodic solutions of systems with friction are difficult to investigate because of the non-smooth nature of friction laws. This paper examines periodic solutions and most notably stick–slip, on a simple one-degree-of-freedom system (mass, spring, damper, and belt), with Coulomb's friction law, and with a regularized friction law (i.e. the friction coefficient becomes a function of relative speed, with a stiffness parameter). With Coulomb's law, the stick–slip solution is constructed step by step, which gives a usable existence condition. With the regularized law, the Asymptotic Numerical Method and the Harmonic Balance Method provide bifurcation diagrams with respect to the belt speed or normal force, and for several values of the regularization parameter. Formulations from the Coulomb case give the means of a comparison between regularized solutions and a standard reference. With an appropriate definition, regularized stick–slip motion exists, its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force. [ABSTRACT FROM AUTHOR]