Contribution to harmonic balance calculations of self-sustained periodic oscillations with focus on single-reed instruments

Abstract

The harmonic balance method (HBM) was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying any periodic solution, whether stable or unstable, without care of particular initial conditions in time. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, HARMBAL, has been developed based on the HBM. The method as well as convergence improvements and continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion (square signals) of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring.

Calculation of the Steady-State Oscillations of a Flute Model Using the Orthogonal Collocation Method

“…Widely used in many scientific fields, methods dedicated to computation of periodic solutions of nonlinear dynamical systems have demonstrated, in several previous works [10,11,12,16,17], their benefits for the study of musical instruments. However, no available software (such as AUTO [8], Manlab [18], or DDE-Biftool [19]) is suitable for neutral delayed systems, involved in the modeling of flute-like instruments.…”

“…Particularly, different studies have dealt with both computation and continuation of periodic solutions and bifurcation analysis of reed instruments with the software AUTO [16], Harmbal [11,12] or Manlab [17,18]. These software differ in numerical methods they use, both for the calculation of periodic solutions (harmonic balance and collocation for Manlab, collocation for AUTO), and for the continuation algorithm (predictorcorrector method for AUTO, asymptotic numerical method for Manlab).…”

“…This technique has then been used in various works focusing on single reed musical instruments [11,12,13] and brass instruments [14,15]. Recent studies [16,17] have then demonstrated the benefits of combining this technique with numerical continuation methods to understand the behaviour of clarinet-like instruments.…”

Hybrid coupling method to nonlinear acoustic source and linear duct system using parameter identification of the input impedance in fluid machinery

“…Then, the measured input impedance is estimated using rational fractional polynomial method based on Eqs. (7)- (9). The denominator and numerator in the rational fractional polynomial method must have equal polynomial orders to make a relatively stable and accurate approximated model as illustrated in Figs.…”

Influencias en la percepción sonora y en la interpretación del rebajado de la lengüeta del oboe.

“…Ricardo da Silva et al (2007) estudiaron la relación en el clarinete entre la interacción del fluido-estructura durante los regímenes de flujo dinámico, y del comportamiento transitorio y de estado estable del sistema en función de la geometría de la boquilla. Farner et al (2005) utilizaron el modelo HBM para los casos de cañas simples y Smyth y Abel (2010) realizaron un estudio acústico sobre las lengüetas. Estos estudios se centran en el comportamiento físico de la lengüeta y sus puntos y modos de vibración, pero sin aportar unos datos acústicos concretos que permitan caracterizar a los diferentes tipos de lengüetas.…”