Parameter identification of a physical model of brass instruments by constrained continuation

Abstract

Numerical continuation using the Asymptotic Numerical Method (ANM), together with the Harmonic Balance Method (HBM), makes it possible to follow the periodic solutions of non-linear dynamical systems such as physical models of wind instruments. This has been recently applied to practical problems such as the categorization of musical instruments from the calculated bifurcation diagrams [V. Fréour et al. Journal of the Acoustical Society of America 148 (2020) https://doi.org/10.1121/10.0001603]. Nevertheless, one problem often encountered concerns the uncertainty on some parameters of the model (reed parameters in particular), the values of which are set almost arbitrarily because they are too difficult to measure experimentally. In this work we propose a novel approach where constraints, defined from experimental measurements, are added to the system. This operation allows uncertain parameters of the model to be relaxed and the continuation of the periodic solution with constraints to be performed. It is thus possible to quantify the variations of the relaxed parameters along the solution branch. The application of this technique to a physical model of a trumpet is presented in this paper, with constraints derived from experimental measurements on a trumpet player.

Asymptotic numerical method for hyperelasticity and elastoplasticity: a review

“…HBM can be applied directly to a PDE [138], but most authors have solved ODEs [19,139,140] or ODEs deduced from PDEs by a discretization technique as the finite-element method. A wide application field has been considered: dynamics of structures as beams, plates and shells [141][142][143][144][145][146], electromechanical systems [147][148][149] or musical acoustics [21,150,151].…”

Identification of lip-parameters associated to different trumpets using constrained continuation

“…The physical model considered is based on four equations: two coupled mechanical equations to represent the dynamics of a two d.o.f. lip model as presented in [1], a nonlinear flow equation and a modal representation of the air-column to represent the nonlinear characteristics and the linear air-column resonator respectively, as detailed in [2]. The choice of the two d.o.f lip model for the lips is motivated by some limitations observed in previous work with the simple one d.o.f.…”

“…The choice of the two d.o.f lip model for the lips is motivated by some limitations observed in previous work with the simple one d.o.f. lip model, where constrained continuation could be successfully performed, but the exact value of the target fundamental frequency could not be reached [2]. Indeed a drawback of the one d.o.f.…”

Brass player’s mask parameters obtained by inverse method

“…Concerning brass instruments, the body of literature is more reduced, although many articles deserve to be cited and will serve as reference for the present work (see [11] for a table summarizing known values). Most notably, [12] who was one of the first to give a complete set of parameters, [13] who built an asymptotic state observer, [14] who used simulated annealing with a cost function depending on playing frequency only, [3] with data coming from high speed camera and [15] using bifurcation diagrams.…”