Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure

TL;DR

This paper analyzes how a linearly increasing blowing pressure affects the oscillation threshold in a simplified clarinet model, using dynamic bifurcation theory to predict bifurcation delay and the dynamic oscillation threshold.

Contribution

It introduces a theoretical estimation of the dynamic oscillation threshold in a clarinet model with time-varying blowing pressure, validated by numerical simulations.

Findings

Identification of bifurcation delay phenomenon

Development of a theoretical estimate for the dynamic oscillation threshold

Validation of the model through numerical simulations

Abstract

Reed instruments are modeled as self-sustained oscillators driven by the pressure inside the mouth of the musician. A set of nonlinear equations connects the control parameters (mouth pressure, lip force) to the system output, hereby considered as the mouthpiece pressure. Clarinets can then be studied as dynamical systems, their steady behavior being dictated uniquely by the values of the control parameters. Considering the resonator as a lossless straight cylinder is a dramatic yet common simplification that allows for simulations using nonlinear iterative maps. In this paper, we investigate analytically the effect of a time-varying blowing pressure on the behavior of this simplified clarinet model. When the control parameter varies, results from the so-called dynamic bifurcation theory are required to properly analyze the system. This study highlights the phenomenon of bifurcation delay and defines a new quantity, the dynamic oscillation threshold. A theoretical estimation of the dynamic oscillation threshold is proposed and compared with numerical simulations.