Flute-like musical instruments: a toy model investigated through numerical continuation

Soizic Terrien (1) , Christophe Vergez (1) , Benoît Fabre (2)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 DALEMBERT - Institut Jean Le Rond d'Alembert

Self-sustained musical instruments (bowed string, woodwind and brass instruments) can be modeled by nonlinear dynamical systems. Among these instruments, flutes and flue organ pipes present the particularity to be modeled as a delay dynamical system. In this paper, such a system, a toy model of flute-like instruments, is studied using numerical continuation. Equilibrium and periodic solutions are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches, as well as "jumps" between periodic solution branches. The influence of a second model parameter (namely the inharmonicity) on the behaviour of the system is addressed. It is shown that harmonicity plays a key role in the presence of hysteresis or quasi-periodic regime. Throughout the paper, experimental results on a real instrument are presented to illustrate various phenomena, and allow some qualitative comparisons with numerical results.

Mots clés: musical acoustics; flute-like instruments; numerical continuation; periodic solutions; nonlinear delay dynamical systems

Domaines: Acoustique [physics.class-ph]; Acoustique [physics.class-ph]

Soizic Terrien, Christophe Vergez, Benoît Fabre. Flute-like musical instruments: a toy model investigated through numerical continuation. 2012. ⟨hal-00721873v1⟩