Numerical tools for musical instruments acoustics: analysing nonlinear physical models using continuation of periodic solutions

Sami Karkar (1) , Christophe Vergez (2) , Bruno Cochelin (3)

1 LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 Sons
3 M&S - Matériaux et Structures

We propose a new approach based on numerical continuation and bifurcation analysis for the study of physical models of instruments that produce self- sustained oscillation. Numerical continuation consists in following how a given solution of a set of equations is modified when one (or several) parameter of these equations are allowed to vary. Several physical models (clarinet, saxophone, and violin) are formulated as nonlinear dynamical systems, whose periodic solutions are directly obtained using the harmonic balance method. This method yields a set of nonlinear algebraic equations. The solution of this system, which represent a periodic solution of the instrument, is then followed using a numerical continuation tool when a control parameter (e.g. the blowing pressure) varies. This approach enables us to compute the whole periodic regime of the instruments, without any additional simplification of the models, thus giving access to characteristics such as playing frequency, sound level, as well as sound spectrum as a functions of the blowing pressure.

Mots clés: physical models; nonlinear dynamical systems; numerical continuation; periodic solutions

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

Sami Karkar, Christophe Vergez, Bruno Cochelin. Numerical tools for musical instruments acoustics: analysing nonlinear physical models using continuation of periodic solutions. Acoustics 2012, Apr 2012, Nantes, France. ⟨hal-00810847⟩