Minimal blowing pressure allowing periodic oscillations in a simplified reed musical instrument model: Bouasse-Benade prescription assessed through numerical continuation

Abstract

A reed instrument model with N acoustical modes can be described as a 2N dimensional autonomous nonlinear dynamical system. Here, a simplified model of a reed-like instrument having two quasi-harmonic resonances, represented by a four dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams of equilibria and periodic solutions are explored with respect to the blowing mouth pressure, with focus on amplitude and frequency evolutions along the different solution branches. Equilibria and periodic regimes are connected through Hopf bifurcations, which are found to be direct or inverse depending on the physical parameters values. Emerging periodic regimes mainly supported by either the first acoustic resonance (first register) or the second acoustic resonance (second register) are successfully identified by the model. An additional periodic branch is also found to emerge from the branch of the second register through a period-doubling bifurcation. The evolution of the oscillation frequency along each branch of the periodic regimes is also predicted by the continuation method. Stability along each branch is computed as well. Some of the results are interpreted in terms of the ease of playing of the reed instrument. The effect of the inharmonicity between the first two impedance peaks is observed both when the amplitude of the first is greater than the second, as well as the inverse case. In both cases, the blowing pressure that results in periodic oscillations has a lowest value when the two resonances are harmonic, a theoretical illustration of the Bouasse-Benade prescription.

Minimal blowing pressure allowing periodic oscillations in a model of bass brass instruments

“…The behaviour of an oscillatory solution of the system with respect to a control parameter such as p m can then be assessed by plotting bifurcation diagrams, which are shown in Section 4. This approach is implemented in several softwares, such as AUTO [11][12][13][14][15] which is used in this publication. However, continuation methods require the system to be written in the form dX dt ¼ F ðXÞ, with certain smoothness properties on F. Therefore, some work has yet to be done on the equations presented in Section 2.1, which is done in the following.…”

Peak-picking identification technique for modal expansion of input impedance of brass instruments

“…function fminsearch in Matlab or scipy.optimize.minimize with Python). Solving equation (7) in a least squares sense within the cost function is computationally inexpensive and straightforward, since it is a linear system of equations. It can be done for instance using mldivide in Matlab or linalg.lstsq with Python.…”

“…Unsurprisingly, this action consists in removing one of the poles from the modal basis. Immediately after this operation, the amplitudes associated with the remaining poles are then updated using the least squares solution of equation (7).…”

“…The input impedance of a wind instrument can be mathematically represented using a modal expansion [1]. By carrying out modal identification from a measured input impedance, it is therefore possible to model the resonator of an existing instrument by a set of ordinary differential equations which can be directly plugged into numerical methods for analysing the emergence of self-sustained oscillations, such as time-domain simulations [2], linear stability analysis [3][4][5] or numerical continuation [6][7][8]. This approach is facilitated by the fact that well-established experimental devices and procedures exist for measuring the input impedance of wind instruments [9,10].…”

“…This approach has been adopted in most of the above-mentionned studies (e.g. [4,6,7]). In a recent paper dedicated to modal analysis of wind instruments [14], Taillard et al also propose a two-step method, which combines the Least Squares Complex Exponential (LSCE) technique to identify the poles with a fitting procedure to obtain the residues.…”