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

TL;DR

This paper investigates a simplified delay-based model of flute-like instruments using numerical continuation to analyze equilibrium and periodic solutions, revealing how parameters like blowing pressure and inharmonicity influence sound production and hysteresis.

Contribution

It introduces a toy delay dynamical system model for flute-like instruments and applies numerical continuation to study its solutions and parameter effects, including experimental validation.

Findings

Harmonicity affects hysteresis and quasi-periodic regimes.

Amplitude and frequency vary along solution branches with blowing pressure.

Experimental results qualitatively match numerical predictions.

Abstract

Self-sustained musical instruments (bowed string, woodwind and brass instruments) can be modeled by nonlinear lumped 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.