Nonlinear modes of clarinet-like musical instruments

TL;DR

This paper applies nonlinear mode analysis to a model of clarinet-like instruments, enabling the study of transient behavior, limit cycles, and stability without explicit ODE integration.

Contribution

It introduces a nonlinear modal framework for woodwind instruments, providing a new way to analyze their dynamic behavior and stability.

Findings

The method captures transient and steady-state behaviors.

It determines frequency and damping functions without explicit ODE integration.

The approach is demonstrated on a three-mode model.

Abstract

The concept of nonlinear modes is applied in order to analyze the behavior of a model of woodwind reed instruments. Using a modal expansion of the impedance of the instrument, and by projecting the equation for the acoustic pressure on the normal modes of the air column, a system of second order ordinary differential equations is obtained. The equations are coupled through the nonlinear relation describing the volume flow of air through the reed channel in response to the pressure difference across the reed. The system is treated using an amplitude-phase formulation for nonlinear modes, where the frequency and damping functions, as well as the invariant manifolds in the phase space, are unknowns to be determined. The formulation gives, without explicit integration of the underlying ordinary differential equation, access to the transient, the limit cycle, its period and stability. The process is illustrated for a model reduced to three normal modes of the air column.