15 Juillet – Thesis defense - Gaby Abou Haidar

14 h30 In videoconferencing

Application of fractional order system in the acoustics domain: Modeling and synthesis of a wind instrument.

This work focuses on the identification, modeling and synthesis of fractional order derivative systems applied in the field of acoustics through the viscous-thermal losses in the flute musical instrument and the control of the air flow in the artificial mouth.
Initially, the modeling of the flute resonator is proposed as a function of the length L and of the position x between 0 (input of the resonator) and L (output of the resonator). Then, a system approach makes it possible to decompose the input impedance into subsystems, thus facilitating the analysis, particularly when moving from a semi-infinite system to a finite system. In addition, extending the fractional expression by taking into account the viscous-thermal losses, where the order m is usually equal to 0.5, makes it possible to make a first analysis of the influence of order m, ranging between 0 (conservative system) and 1 (capacitive system), on the frequency response of the resonator impedance.
Secondly, the study of the coupling between the nonlinear exciter and the resonator, in the usual case where the order m is equal to 0.5 is presented. Based on a bibliographic synthesis, a nonlinear model used in the literature is developed. Then, after showing the poor digital conditioning of such a model, a solution is proposed allowing the development of a digital simulator programmed using MATLAB / Simulink. As the field of study consists of the use of a constant pressure at the mouthpiece bounded by a minimum value of 400 Pa and a maximum value of 1000 Pa, a detailed analysis of the temporal responses highlights the presence of three phases during which the variations of the pressure at the resonator, of the acoustic velocity and of the lateral displacement of the air jet remain small, around zero.
Next, the design and implementation of an artificial mouth is necessary to control the pressure at the input of the mouthpiece. Thus, a first simulator was developed on the basis of a bibliographic synthesis concerning artificial mouths used in musical acoustics. This first simulator provided a good understanding of the operation of such a system, thus facilitating the choice and sizing of the components of the regulation loop. Following the implementation of the artificial mouth, modeling work on this experimental device led to a second simulator. The latter was the subject of a registration based on comparisons between experimental and simulation results.
Finally, a study of the influence of the order m (between 0 and 1, around its nominal value m0 = 0.5) on the stationary periodic regime is proposed in simulation. Thus, from the extension of the fractional model, it is possible with a single high-level parameter, the order m in this case, to easily vary in numerical simulation, the visco-thermal losses, while from an experimental point of view, it would be necessary to manufacture and test a large number of resonators with different dimensions, roughness and materials.

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