Organisation et Contenu de la Thèse

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Convection speed

Avant-Propos

Cette thèse s'inscrit dans le cadre d'une collaboration internationale entre Lebanese German University (LGU), American University of Science and Technology (AUST) et l'Université de Bordeaux (UBx), au sein de l'équipe CRONE du Groupe Automatique du laboratoire IMS, UMR 5218 du CNRS.

Cette collaboration a débuté en 2008 (figure 1) avec Clovis FRANCIS, aujourd'hui Professeur à l'Université Libanaise et membre du Conseil National de la Recherche Scientifique du Liban (CNRS-L). En 13 ans, cette collaboration internationale a donné lieu à 4 thèses soutenues, 1 thèse en cours (objet de ce mémoire), une prochaine thèse (septembre 2021), une HDR en cours de préparation, 13 publications dans des Revues Internationales avec Comité de Lecture (RICL), 33 communications dans des Conférences Internationales avec Comité de Lecture (CICL), 5 chapitres d'ouvrages, 2 ouvrages de synthèse, … Les travaux présentés dans ce mémoire de thèse concernent les Systèmes à Dérivées Non Entières (SDNE) dans le domaine de l'acoustique avec l'étude des pertes viscothermiques dans un instrument à vent.

Contrairement aux travaux menés, par exemple, à l'IRCAM (Institut de Recherche et Coordination Acoustique/Musique) ou dans des laboratoires d'acoustique, les travaux présentés dans ce mémoire de thèse n'abordent pas le sujet sous l'angle de l'acoustique musicale, mais sous l'angle de l'Automatique, c'est-à-dire l'étude de la dynamique des systèmes complexes. La flûte à bec ne constitue qu'un support dont le domaine d'étude défini dans ce mémoire est beaucoup plus restreint que celui envisageable avec un instrument à vent.

2015 2021

Début de la collaboration

Thèse Ainsi, le chapitre 1, intitulé: « Etude des pertes viscothermiques au sein du résonateur d'un instrument à vent », est consacré, dans un premier temps, à la modélisation du résonateur seul en tant que tube acoustique (tous les trous permettant de faire varier la longueur apparente sont supposés fermés), modélisation qui aboutit à l'impédance générale Y(x,s,L) (débit/pression) du résonateur en fonction, notamment, de la longueur L et de la position x comprise entre 0 (entrée du résonateur) et L (sortie du résonateur). Ensuite, une approche système (au sens de l'Automatique) permet de décomposer l'impédance d'entrée Yin(s,L) = Y(0,s,L) en sous-systèmes, facilitant ainsi l'analyse notamment lors du passage d'un système semi-infini à un système fini. De plus, l'introduction d'une extension de l'expression fractionnaire utilisée pour la prise en compte des pertes viscothermiques, où l'ordre m est habituellement égal à 0.5, permet de faire une première analyse de l'influence de l'ordre m, compris en 0 (système conservatif) et 1, sur la réponse fréquentielle de l'impédance du résonateur. Enfin, ce chapitre se termine par la présentation d'une méthode permettant de passer de la forme fractionnaire de l'impédance à ses formes rationnelles cascade et parallèle indispensables pour la simulation temporelle.

Le chapitre 2, intitulé: « Etude du couplage entre l'excitateur non linéaire et le résonateur d'un instrument à vent », est dédié, dans le cas habituel où l'ordre m est égal à 0.5, à l'étude du couplage entre l'excitateur non linéaire et le résonateur. Sur la base d'une synthèse bibliographique, un modèle non linéaire fréquemment utilisé dans la littérature est développé. Ensuite, après avoir insisté sur le mauvais conditionnement numérique d'un tel modèle, une solution est proposée permettant la mise en oeuvre d'un simulateur numérique programmé sous MatLab/Simulink. Pour un domaine d'étude défini par une pression constante à l'entrée du bec de flûte bornée par une valeur minimale de 400 Pa et une valeur maximale de 1000 Pa, une analyse détaillée des réponses temporelles simulées met en évidence la présence de trois phases durant lesquelles les variations -de la pression à l'entrée du résonateur, -de la vitesse acoustique -et du déplacement latéral du jet d'air restent petites autour de zéro. Ce constat permet le développement de manière légitime et réaliste de deux modèles linéarisés, l'un pour l'analyse de la phase de démarrage de la simulation, l'autre pour celle de la phase des auto-oscillations (régime périodique stationnaire), facilitant ainsi la compréhension des phénomènes mis en jeu.

Le chapitre 3, intitulé: « Conception et implémentation d'une bouche artificielle pour un instrument à vent », se focalise sur la démarche de conception et sur l'implémentation d'une bouche artificielle nécessaire pour contrôler la pression à l'entrée du bec de flûte. Durant la phase de conception, un premier simulateur a été développé sur la base d'une synthèse bibliographique concernant les bouches artificielles utilisées en acoustique musicale. Ce premier simulateur a permis de bien comprendre le fonctionnement d'un tel système, facilitant ainsi le choix et le dimensionnement des composants de la boucle de régulation (servovalve de débit, capteurs de débit et de pression, volume de la capacité pneumatique,…). A l'issue de la réalisation de la bouche artificielle, un travail de modélisation de ce dispositif expérimental a conduit à un deuxième simulateur. Ce dernier a fait l'objet d'un recalage à partir de comparaisons entre des résultats expérimentaux et de simulation. Ainsi, pour une flûte à bec bien réelle (en plastique dont la longueur du résonateur L = 30 cm, son rayon moyen r = 5 mm, l'ordre fractionnaire associé aux pertes viscothermiques m = 0.5, avec une fenêtre de l'excitateur dont les dimensions sont 1 cm x 0.4 cm, une hauteur du canal du bec h = 1 mm,…), le simulateur reproduit avec une bonne précision les conditions d'étude spécifiées au chapitre 2. Ce comportement bien réel (reproduit en simulation) est considéré au chapitre 4 comme le comportement nominal de référence.

Enfin, l'objectif du chapitre 4, intitulé: « Analyse de l'influence de l'ordre fractionnaire sur le régime périodique stationnaire », est d'étudier l'influence de l'ordre m compris entre 0 et 1, autour de sa valeur nominale m0 = 0.5, sur le régime périodique stationnaire. Ce chapitre commence par présenter l'organisation du simulateur global développé sous MatLab/Simulink à partir des travaux présentés dans les trois premiers chapitres. Ensuite, une analyse dans le domaine fréquentiel de l'influence de l'ordre m est développée, d'abord dans le cas de l'impédance du résonateur établie au chapitre 1, puis dans le cas de la fonction de transfert en boucle ouverte définie au chapitre 2. Cette analyse se poursuit dans le domaine temporel avec les réponses issues du simulateur global en reprenant les scénarios des deux exemples du chapitre 2. Ainsi, à partir de l'extension du modèle fractionnaire proposée au chapitre 1, il est possible avec un seul paramètre de haut niveau, l'ordre m en l'occurrence, de faire varier facilement les pertes viscothermiques, alors que d'un point de vue expérimental, il faudrait fabriquer et tester un nombre important de résonateurs avec des dimensions, des rugosités et des matériaux différents. This study would not have been possible without the financial support provided by the University of Bordeaux -IMS Laboratory that were related to conference and accommodation fees, different types of precise sensors and actuators, in addition to travel fees.

Xavier MOREAU

A sincere thanks also goes to the American University of Science and Technology -Zahle Branch that supported me with all the laboratory resources and facilities during my four years' research process.

I would also like to express the deepest appreciation to the President of the American University of Science and Technology Dr. Hiam Sakr for her encouragement and continuous support.

Finally, I would like to express gratitude to my dear mother, sister, and brothers for their constant moral and financial support.

Motivation

Music has always been the source of creativity and imagination for most thinkers and a relaxing tool for stress and depression. The greatest minds and thinkers like Albert Einstein, Mozart, and Frank Lloyd Wright all had something in common in that they were constantly exploring their imagination and creativity through music to make discoveries, innovations and many other different fields. Music and sound synthesis are two faces for the same coin. Sound synthesis started with the wish to generate any kind of music (or sound) using mathematical techniques, and by research improvement, it became possible to generate or transform any sound conceivable. The available sound synthesis techniques are capable of a perfect reproduction of sound but not actually a perfect generation of a sound; this is why model-based digital instruments should be available. The main advantage behind modelbased synthesis is the more control over the physical variables (which influence the sound reproduction) it gives for the player.

This work aims to model and control a flute wind musical instrument for better numerical performance. To have a completely automated system, the mouth and the flute musical instrument parts will be modeled and implemented. So, based on the delivered air pressure by the artificial mouth, a certain frequency will be generated from the instrument.

To achieve the preset goal, the physical implementation of the artificial mouth is first established. Then, a study for every part of the flute will be conducted, starting by the resonator and then the exciter. This resonator is of a particular importance as it has been shown that the air flow within its boundaries is subject to visco-thermal losses that are being modeled by a fractional transfer function of order 0.5.

The last stage of this work will be to confront the already presented theoretical study with the experimental measurements and the implementation of a system that most closely resembles the real system. Note that this Ph.D. thesis will be the first block in a much larger project that will aim to reach the final objective already listed. Thus, the outcome of this work is not to have a complete autonomous system as a first step but to be able to model this complicated system and to generate sound while controlling the mouth pressure and/or flow. 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c) ..........

List of Figures

General Overview

-Introduction

The flute wind musical instrument family contains a large number of instruments types, which may seem from a first approach very different from each other, whether from the produced sounds perspectives, the geometries of the instrument, the used materials, or the contexts in which they are applied. This family of instruments includes indeed, in a nonexhaustive manner, the notched flutes such as Latin American quena or Japanese shakuhachi, the pan flutes, the globular flutes such as the ocarina, as well as the transverse flutes, the recorders and the mouth organ pipes. In addition, a general overview of all wind instruments will be presented in more details in the following section.

The diversity of these instruments is illustrated -among other criteria -by the relative influence of instrument and musician factor on the geometry of the instrument, and therefore on the sound production. For some instruments, such as the recorder or the mouth organ pipes, a large part of the geometry (jet formation channel, distance between the channel output and the bevel), is fixed by the instrument builder at the time of construction. However, in other flute types such as the quena or the transverse, the instrumentalist/musician has complete freedom to adapt these geometric parameters while playing music since the channel is formed by the musician's lips. This freedom gives the music player more autonomy, but makes it more difficult to control the instrument. Despite this diversity, the instruments of the flute family share a mechanism of production of the common sound, involving the oscillation of an air jet around a bevel. A state of the art of the different wind musical instruments, the sound formation, the main components of the flute and the problem statement will be discussed in the following sections of this general overview.

-Wind Musical Instrument

In this section, the human voice will be presented as it is the main source for the generation of sound. The voice is one of the most colorful interfaces for musical expression. It is certainly the most personal of all instruments (each voice is unique) -and the only one capable of presenting both words and musical sound simultaneously. Voices are classified by their tone-color, register and range. One can identify the following groups (FactMonster, 2001):

• soprano: highest female voice;

• mezzo-soprano: rich female voice mixing soprano and alto colors;

• contralto or "alto": lowest female voice;

• countertenor: a very high male voice;

• tenor: high male voice;

• baritone: rich male voice mixing tenor and bass colors;

• bass: very low male voice; Hereafter, the different families of the wind musical instruments will be presented. For each family, the corresponding instruments will be shown along with their properties.

-Woodwind Family

The woodwind family as in Figure 3 is composed of wooden instruments that the player must blow into to create a musical sound. Most members of the modern woodwind family are "reed" instruments (a piece of wooden reed attached to its mouthpiece that adds character to the sound). The complete woodwind family became a standard part of the orchestra by the early 1800's. Common woodwind instruments include [START_REF] Factmonster | Families of Musical Instruments[END_REF]) :

• the piccolo (very high) and various-sized flutes;

• the oboe and the "English" Horn (a tenor oboe);

• the bassoon and contrabassoon (very low bassoon);

• various-sized clarinets;

• various-sized saxophones;

• various-sized recorders (ancestors to the modern flute family);

The standard range categorization of this family is the flute (soprano), the oboe (alto), the clarinet (tenor) and the bassoon (bass).

-Brass Family

The brass family in Figure 4 is composed from powerful metallic instruments that must be blown into by the player to produce a musical sound. The tone-color (timbre) of most brass instruments can be altered by the use of various types of mutes which are inserted into the large end of the instrument. Until the invention of the valve, brass instruments could only produce a limited number of pitches, which lessened their usefulness to composers. Instrument builders experimented with various valves that would avail more pitch varieties to these instruments.

Many modern brass instruments have a system of three valves that can be combined in various combinations to produce different pitches. The modern "rotary" valve was invented in the early romantic period (c. 1830) -an invention that made the brass family more responsive and reliable. As a result, romantic composers made greater use of the brass family. The most common types of brass instruments include [START_REF] Factmonster | Families of Musical Instruments[END_REF]):

• various-sized trumpets (use valves to change pitch);

• various-sized trombones (use a slide instead of valves to change pitch);

• the "French Horn" (use valves to change pitch);

• the Tuba-a very low brass instrument (uses valves to change pitch);

The standard range categorization of this family is the trumpet (soprano), the alto trombone (alto), the tenor trombone that is equivalent to the upper french horn (tenor), the bass trombone that is equivalent to the low french horn (bass).

-Flute Musical Instrument

The first published stage in the modeling and functioning of flute-like instruments probably came from two different works conducted by Mersenne and Bernoulli and they consisted of the analysis of the resonance frequencies of the bore. This approach is still followed today and is very useful for the understanding of several construction properties mainly from the resonator's position, diameter, height of the chimney and tone holes, shape corrections, etc… refer to Figure 5 [START_REF] Mersenne | Harmonie Universelle Contenant la théorie et la pratique de la musique[END_REF] [START_REF] Bernoulli | Sur le son et sur les tons des tuyaux d'orgue differemment construits[END_REF]. The next stage in modeling consists of describing the source mechanisms. Helmholtz and Rayleigh were the first to propose such a description. Helmholtz described them as flow sources or a monopole while Rayleigh argued that since the sources act at an open end of the bore, they should be described as a dipole [START_REF] Rayleigh | The theory of sound[END_REF].

In fact, close to the open end of the resonator there are large velocity fluctuations but the pressure fluctuations are small. Helmholtz rapidly changed his opinion, as he modified the comments on the second version of his book. Surprisingly, the same discussion reappeared this century in the sixties between Elder Coltman and Fletcher. Like in many of his works, Coltman's pragmatic, careful and clever approach unquestionably resolved the discussion [START_REF] Coltman | Sounding mechanism of the flute and organ pipe[END_REF] [START_REF] Elder | On the mechanism of sound production in organ pipes[END_REF].

At the beginning of the sixties, the works by Powell on the edge-tone suggested a feedback loop model. The different components of the model were analyzed as independent blocks that were then connected together. The analysis of the loop gain fixed the phase and magnitude necessary for the loop to oscillate, and predicted the conditions and frequency for a linear model of oscillations. However, the actual amplitude of the oscillation can only be predicted by a non-linear model. His approach has inspired many other descriptions based on loop systems [START_REF] Fletcher | Air flow and sound generation in musical wind instruments[END_REF].

An important period in the development of flute models came with the work of Mc Intyre et al., who were the first to propose a time simulation from a set of simplified equations derived from physical models. Their models were so neat and elegant that many researchers were persuaded that the secrets of the flute functioning had been found and captured in a physical model. Unfortunately, as it turned out, the model was better suited to the violin and clarinet but the flute doesn't even oscillate with the values published. After fine tuning the parameters and adding nonphysical elements, like a DC blocking filter and feeding the system with white noise, the model produces sound but is impossible to relate the quantities in the model with physical variables [START_REF] Mcintyre | On the oscillations of musical instruments[END_REF].

A more rigorous approach considering fluid mechanics and aero-acoustics has been followed by Howe for the flute and by Crighton and Elder for the edge-tones. In parallel, a group centered in Eindhoven, Netherlands adopted an intermediate path inspired by the work of different researchers in the field. Their approach lies between the rigorous formulation and the simplified description, producing good quality sound synthesis through models that use lumped elements with a sufficient amount of aero-acoustics [START_REF] Elder | On the mechanism of sound production in organ pipes[END_REF] [START_REF] Howe | Contribution to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute[END_REF].

Modern flute musical instruments are usually depicted from old versions of this instrument with some additions on the shape, size, and other effects. Although the normal old versions of flutes are more likely preferred by musicians, modern flutes are being used numerically with the help of embedded systems and computers adding new features to the way of playing and to the quality of sound [START_REF] Heller | An Augmented Flute for Beginners[END_REF]. However, what is no doubt with is that playing a typical old musical flute naturally will produce the perfect sound associated with feelings and that what numerical flutes are not able to do so far.

The flute, as shown in the Figure 6, is mainly composed from three main parts: the exciter which contains the mouth piece and the whistle (exciter), the resonator that contains the holes, and the horn which act as a reflector of the output waves. Figure 7 shows a block diagram representation of this musical instrument. The system is composed from the musician's ear that serves as a sensor to form the loop and to act accordingly on the flow delivered by its mouth towards the flute exciter. The radiation at the output of this system is generated at the flute horn level. Flutes are generally made from plastic, steel, or bamboo. Each type of material has a different effect on the quality of sound produced. This is due to the different types of materials used and to the visco-thermal effect of each material.

There are several types of flutes some are named according to the culture it is being used in (Chinese, Western, Indian, Irish, etc.), and others according to its shape and style (modern D, classical C, modern B, classical D, etc.). Every type has its own number of holes and their sizes and its own different shape and thus a different number of notes. For example, classical C flute has 6 holes and due to its shape, it can offer 8 notes. As for the classical D flute, it has the same number of holes (6 holes) but with only 6 notes because the D foot is smaller than the C foot; same number of holes, but different shape and size, give different number of notes. Classical flared flute has a larger number of holes (8 holes) and thus can offer more notes (23 notes) whereas baroque flute has 7 holes (less than classical flared flute) but yet can provide more notes due to its shape and size (32 notes to be precise). Figure 8 shows an illustration of these three types of flutes.

-Sound Formation in Flutes

When the musician blows into the instrument, the overpressure inside his mouth creates a flow inside the channel, which can be a part of the instrument (refer to Figure 5 -case of the recorders), or constituted by the lips of the musician, like in the flute. At the output of the channel, the viscosity of the air prevents the flow from following the geometry of the boundaries: the flow takes off from the flute boundaries, and gives rise to a jet of air. Due to this unstable nature, the fewer disturbances to which the jet is subjected are amplified over time.

In the case of flutes, the jet created at the output of the channel is convicted inside the window excitement, until meeting the bevel. This jet-bevel interaction causes a hydrodynamic return on the jet, a mechanism which is notably at the origin of bevel sounds [START_REF] Powell | On the edgetone[END_REF] [START_REF] Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF]. If this phenomenon is negligible once the steady state of the instrument is reached [START_REF] Verge | Aeroacoustics of confined jets: with applications to the physical modeling of recorder-like instruments[END_REF], on the other hand, it plays a role in the initialization of the oscillation of the jet, by providing it with its first disturbances. Due to the natural instability of the air jet, disturbances are amplified naturally along this jet, from the exit of the channel to the bevel, and cause its oscillation on both sides of the bevel. This oscillation causes a flow injection alternately inside and outside of the instrument, and is the aero-acoustic source of pressure which supplies energy to the resonator [START_REF] Chaigne | Acoustique des instruments de musique[END_REF]. The acoustic waves thus created in the resonator propagate and reflect (mainly) at the first open hole. The superposition of the outward and return waves created a standing wave system in the resonator, which disturbs the jet back to the channel output level, thus closing the self-oscillation loop. This initial disturbance jet, amplified and convicted from the channel output to the bevel, maintains the oscillation of the jet, and therefore the sound production mechanism.

Unlike percussion and struck or plucked string instruments, in which sound production is based on a phenomenon of free oscillations, the instruments of the flute family form with the reed instruments, strings rubbed, and brass instruments, the family of self-oscillating instruments. The self-oscillation mechanism, characterized by the ability of the instrument to produce an oscillation (in this case a wave acoustic) from a continuous source of energy -or at least varying slowly from acoustic variables -provided by the musician, is inseparable from non-linear nature of the instrument. A classic approach in musical acoustics consists of representing the instrument, as in Figure 9, by a nonlinear excitation system coupled to a passive linear resonant system [START_REF] Mcintyre | On the oscillations of musical instruments[END_REF] [START_REF] Rayleigh | The theory of sound[END_REF].

-Acoustic Resonator of Wind Instruments

The acoustic resonator in Figure 9 is a passive linear resonator that has been widely studied for several centuries. By its linear nature, it is also considered the easiest element to describe (at least taking into account a certain number of approximations). As in all wind instruments, the flute resonator is formed by the air column contained inside the body of the instrument. This is delimited by the hole in the instrument, that is to say the geometric shape of the resonator.

The geometrical characteristics of this air column, such as the conditions at the ends, the size and position of the different note holes, or the variations in the section of the bore, condition the propagation of acoustic waves inside the resonator, and therefore the resonator's response to the excitement provided.

As can be seen in Figure 5, which provides an overview of the main elements of a recorder, a large part of the instruments in the flute family have two open ends, which distinguish them from the most other wind instruments.

-Flute Exciter

If the propagation phenomena inside the resonator still pose many questions, the characteristics of the exciter and its coupling with the resonator (for which it is the energy source) appears even more complex due to its non-linear nature.

The peculiarity of flutes lies precisely in the nature of the exciter where it involves the vibration of a solid body for all other wind instruments -reed blade for reed instruments, instrumentalist's lips for brass instruments -it is made up of flutes from the oscillation of a naturally unstable air jet around a bevel (refer to Figure 5).

In flutes, the self-oscillation mechanism is therefore based on a synchronization of the oscillation jet and acoustic waves: the same way as a person who wants to maintain the oscillations of a swing must restore energy to it during each period of oscillation (or a multiple of this period), the jet-bevel system must excite the resonator at the periodicity of the sound field.

-Closed-loop system

Generic modeling of self-oscillating instruments by the looped system given in Figure 9 can then be very specific to the flute family instruments. The diagram represented in Figure 10 underlines the principal physical phenomena retained in the state of the art model of this family of instruments.

-Literature Review

The production of a note by a musical instrument results from a complex coupling between the instrument and the musician, where to obtain the desired sound requires the instrumentalist a constant adaptation of the parameters which are at his disposal. Experimentally, it is then difficult to ask an instrumentalist to set all these control parameters.

In fact, the system as shown in Figure 10 depends on two main entities, the musician and the flute instrument. As the ear and the musical experience of the musician plays a major role in controlling the air flow blowing in the exciter of the flute, the objective of this work will be to model all the parts of the system and then find some physical equivalence in order to be able to get the required frequency output precisely based on a closed loop control system.

Many researchers have been interested in the field of wind musical instruments [START_REF] Mignot | Réalisation en guides d'ondes numériques stables d'un modèle acoustique réaliste pour la simulation en temps réel d'instruments à vent[END_REF] [START_REF] Lefebvre | Computational Acoustic Methods for the Design of Woodwind Instruments[END_REF]. They have studied this field from many perspectives. Some tried to model wind instruments mathematically as will be demonstrated in the coming discussions, others tried to control the blowing part, and few went deeply in discussing the visco-thermal losses and their effects on the musical output.

The work of Mignot dealt with the physical modelling of cylindrical acoustic tubes for digital simulation in real-time. The main application is the sound synthesis of wind instruments, with a realistic model, a modular method and a low-cost digital implementation. The acoustic model of "Webster-Lokshin", used in this work, is a unidimensional model which takes into account the "curvature" of the profile and the visco-thermal losses at the wall. With this acoustic model, a framework for simulation which is compatible with the "Waveguides" approach was obtained: a tube is then represented by a system with delays and closed loops, involving several sub-systems without internal delay [START_REF] Mignot | Réalisation en guides d'ondes numériques stables d'un modèle acoustique réaliste pour la simulation en temps réel d'instruments à vent[END_REF]. Figure 11 shows a representation of the flute using the tools that Mignot has developed.

Digital waveguide synthesis models are computational physical models for certain classes of musical instruments (string, winds, brasses, etc.) which are made up of delay lines, digital filters, and often nonlinear elements. Digital waveguide models typically share the following characteristics:

• Sampled acoustic traveling waves; • Follow geometry and physical properties of a desired acoustic system; • Efficient for nearly lossless distributed wave media (strings, tubes, rods, membranes, plates, vocal tract); • Losses and dispersion are consolidated at sparse points along each waveguide [START_REF] Smith | A Basic Introduction to Digital Waveguide Synthesis[END_REF]. The structure shown in Figure 11 disregards the effects of sections (representing the coupling between two adjacent blocks), slopes at each extremity of the section, curvature and loss, and isolates wave propagators through the tube. Thus, it does not pretend to replace them, but allows to extend these models to a higher degree of refinement, considering the curvature and visco-thermal losses of the tube. Finally, the number of sections is imposed by the sampling frequency.

Another work presented a number of methods for the computational analysis of woodwind instruments which is the Transmission-Matrix Method (TMM) for the calculation of the input impedance. The latter is an approach based on the Finite Element Method (FEM) and it is applied to the determination of the transmission-matrix parameters of woodwind instrument tone holes, from which new formulas are developed that extend the range of validity of current theories. This approach is applied as well to tone holes on a conical bore, and, as a result, the tone hole transmission matrix parameters developed on a cylindrical bore are equally valid for use on a conical bore. A boundary condition for the approximation of the boundary layer losses for use with the FEM was developed, and it enables the simulation of complete woodwind instruments. The comparison of the simulations of instruments with many open or closed tone holes with calculations using the TMM reveal discrepancies that are most likely attributable to internal or external tone hole interactions. The maximal error is found to be smaller than 10 cents (Musical intervals are often expressed in cents, a unit of pitch based upon the equal tempered octave such that one tempered semitone is equal to 100 cents) [START_REF] Lefebvre | Computational Acoustic Methods for the Design of Woodwind Instruments[END_REF]. Figure 12 shows an example of the different musical notes with their corresponding frequencies and the active holes that produces them. Moreover, preliminary studies on an artificial mouth adapted to the recorder have highlighted phenomena still unknown on this instrument, which raise many questions.

First phenomenon: for some fingerings, sounds are detected for a lower feed pressure than that usually used, sometimes less than the oscillation threshold of the desired note. These oscillations systems recall the "Aeolian sounds" observed on certain organ pipes or on the cross-section, which appear for very low values of the supply pressure. They can be heard at the organ during the extinction, and are sometimes used musically at the peak crossbred, their existence on the record was only highlighted recently, and their differences with the systems of "usual" oscillations remain rather poorly known.

Second phenomenon: it seems to exist, especially at the level of registry changes, a great diversity of behaviors according to the fingering used. We can observe for some fingerings a bifurcation towards a higher note (and, starting from this new note, a bifurcation towards the initial note when one feels less and less strong). If we reproduce the same experience on other fingerings, such as that of B flat, we observe a transition to a quasiperiodic system, corresponding audibly to a "rolling" note [START_REF] Terrien | ANALYSE DE DEUX PHÉNOMÈNES OBSERVÉS SUR BOUCHE ARTIFICIELLE[END_REF].

Added to that, Matignon classified the wind instrument into three main blocks which are the cup, pipe, bell, and radiation as shown in Figure 13. This design was represented in the form of blocks of delay lines which allowed each block to be modelled as a separate waveguide element, thus allowed to a more advanced sound synthesis technique [START_REF] Matignon | Viscoelastic materials and viscothermal losses[END_REF]. Moreover, the musical instruments of the brass family are complex physical systems. They involve mechanical phenomena of deformable solids (muscles, tissues), fluid mechanics (jet, turbulence) including acoustic propagation. From a dynamic systems perspective, they correspond to self-oscillating non-linear and chaotic systems. In addition, their control is delicate and requires a long learning from the musician. For all these reasons, the modeling of these instruments, their analysis, simulation, inversion input / output determination of a command providing a target sound and as well as control are still active research topics, as is the case of an artificial mouth coupled to an acoustic system as shown in Figure 14. In order to carry out experiments that are calibrated and reproducible, the robotic design of an artificial mouth dedicated to brass instruments was initiated at IRCAM (Institut de Recherche et Coordination Acoustique/Musique). This work aims to propose models and simulations for the mouth system, (lips, instrument) while developing its robotic version, in order to consider the creation exploitation and control laws. An approach well adapted to the modeling of complex physical systems is based on the "Hamiltonian formulation with ports": it leads to well-balanced energy systems and naturally preserves the passivity. Thus, this work falls within the following three scientific fields: musical acoustics, robotics, and control [START_REF] Lopes | Approche passive pour la modélisation, la simulation et l'étude d'un banc de test robotisé pour les instruments de type cuivre[END_REF]. Current modeling of the flute family instruments allows a good prediction of their operation in transient and steady state. They allow to interpret a large part of the phenomena observed when the control parameters of the instrument vary slowly with respect to the time response of the instrument. The musician, however, constantly varies these parameters to articulate the musical discourse. During sudden spikes in particular, an instant variation in the supply pressure induce a gradual oscillation of the instrument. The birth of sound plays an important role in the perception of sound, the control of these sudden spikes is therefore an essential point in learning the concept of making flute instruments. The works presented focus on the study of sudden transients in flute instruments. The study is restricted to instruments whose jet geometry is fixed by the manufacturer.

At the end of this section, we have to mention the recent work presenting a comprehensive study of visco-thermal effects in quasi-2D artificial structures which are designed to exhibit double-negative behavior using a digital waveguide approach. The building units consist of structured cylinders made of a rigid material having air cavities penetrating deeply into the waveguide. The reported experimental characterization of these types of metamaterials is unable to demonstrate any features confirming double-negative effects. Extensive numerical simulations based on the boundary-element method were treated, which has been improved and adapted to tackle visco-thermal losses in these metamaterial structures. These structures are very efficient in absorbing the energy of acoustic waves traveling through them. For frequencies within the first passband, the absorbed energy reaches values as high as 80% of the amount of absorbance being directly proportional to the reciprocal of the group velocity in the band. The visco-thermal losses, which are enhanced due to the extremely low value of the group velocity inside this narrowband region, in addition to the possibility of reducing the relevance of losses by applying a scaling factor to the dimensions of the initial samples were treated. So, the study lead to the conclusion that final based metamaterial structures are not able to exhibit the predicted double-negative behavior because of the strong dissipation associated with visco-thermal losses. A large amount of losses has also been reported in other rigid-based structures with embedded resonances, where the lack of a significant signal in the transmitted energy made them unfeasible for developing practical devices. These results may indicate that viscothermal losses are relevant to any rigid-based double negative metamaterials. The rigid-based metamaterial structures could become interesting alternatives to conventional absorbers in particular situations, e.g., when treating low frequencies or when the excitation is narrow banded, such as damped and low-frequency resonances in room acoustics [START_REF] Henríquez | Viscothermal Losses in Double-Negative Acoustic Metamaterials[END_REF].

Chapter 1-Study of Viscothermal Losses Within the

Resonator of a Wind Musical Instrument

-Introduction

Historically, the first efficient and inexpensive simulations introduced for simple models (conservative plane waves) were based on signal processing tools: the so-called digital waveguide formalism [START_REF] Tassart | Modélisation, simulation et analyse des instruments à vent avec retards fractionnaires[END_REF] [START_REF] Helie | Modélisation physique d'instruments de musique en systèmes dynamiques et inversion[END_REF] [START_REF] Helie | Simulation en guides d'ondes numériques stables pour des tubes acoustiques à profil convexe[END_REF] and more specifically, a factored form introduced by Kelly-Lochbaum [START_REF] Matignon | Waveguide modeling of lossy flared acoustic pipes: derivation of Kelly-Lochbaum structure for real-time simulations[END_REF]. The initial idea rested on the factoring of the alembertian of the equation of plane waves into two transport operators who each govern decoupled "round trip" progressive waves, from which we could derive a form in efficient delay system for the simulation [START_REF] Mignot | Réalisation en guides d'ondes numériques stables d'un modèle acoustique réaliste pour la simulation en temps réel d'instruments à vent[END_REF] [START_REF] Helie | Ondes découplées et ondes progressives pour les problèmes mono-dimensionnels d'acoustique linéaire[END_REF].

Furthermore, 3D and 2D models with realistic boundary conditions are far too complex to be considered, especially in real-time sound synthesis. They can be effectively reduced to a 1D wave equation including a term that models the tube profile. It is about the equation of the pavilions also called model of Webster. A more elaborated version of this conservative model includes the effect of visco-thermal losses due to the boundary layers in the vicinity of the walls. This dissipative model, known as de Webster-Lokshin 1D [START_REF] Lokshin | Fundamental solutions of the wave equation with retarded time[END_REF]) [START_REF] Haddar | A Webster-Lokshin model for waves with viscothermal losses and impedance boundary conditions: strong solutions[END_REF] [START_REF] Haddar | Analyse théorique et numérique du modèle de Webster Lokshin[END_REF], includes a term which involves a fractional derivation in time of order 3/2. This operator plays a crucial role from a perceptual point of view on sound realization [START_REF] Vigué | Continuation of periodic solutions for systems with fractional dérivatives[END_REF].

The objective of this chapter is, first, to establish a knowledge model from partial differential equations which define the Webster-Lokshin model of an acoustic tube of constant radius r. Thus, a conventional resolution in the operational field leads to the analytical expression of the acoustic impedance and admittance of the tube as a function of the position x, its length L and its radius r. This working methodology will be similar to the one used for the modelling of the diffusive phenomenon in a semi-infinite homogeneous tube already proposed in previous works [START_REF] Assaf | Modélisation des phénomènes de diffusion thermique dans un milieu fini homogène en vue de l'analyse, de la synthèse et de la validation de commandes robustes[END_REF]. Then, a system vision is proposed aiming to causally decompose the whole model into sub-models in order to facilitate the analysis in the frequency domain. In addition, the introduction of uncertainty at the fractional order level makes it possible to study its influence on visco-thermal losses. Finally, for the time domain simulation in Chapters 2 and 4, two rational forms composed of an integrator and N secondorder cells, one in cascade and the other in parallel, are introduced in as a role model. The parametric values of the cascaded rational form are then determined using the Frequency Domain System Identification (FDSI) module of the CRONE Toolbox, with the frequency response of the knowledge model as target. As for the parametric values of the parallel form, they are obtained by the decomposition of the cascade form into simple elements.

-Modelling

-System Definition

Consider an acoustic tube of length L and constant radius r subjected to an acoustic flow (also called volume flow) Qv(t) with When an acoustic wave propagates in the air, this sets the particles of the fluid in motion which vibrate at a speed v(t) around their equilibrium position. The acoustic flow Qv(t) then measures the flow [in m 3 /s] of this speed through a surface and present it as a scalar quantity [START_REF] Blanc | Production de son par couplage écoulement/résonateur : étude des paramètres de facture des flûtes par expérimentations et simulations numériques d'écoulements[END_REF] [START_REF] Ducasse | Modélisation d'instruments de musique pour la synthèse sonore : application aux instruments à vent[END_REF] (Ségoufin C. , 2000) [START_REF] Ducasse | Modélisation et simulation dans le domaine temporel d'instruments à vent à anche simple en situation de jeu : méthodes et modèles[END_REF] [START_REF] Terrien | Instrument de la famille des flûtes : analyse des transitions entre régimes[END_REF].

The acoustic impedance Zac (also called specific acoustic impedance, because it is an intensive quantity) of a medium is defined in steady state by the ratio between the acoustic pressure [in Pa] and the speed [in m/s] of the associated particle. When the medium is air, Zac is equal to the product between the density of air, ρa, and the speed of sound in air, ca, thus Zac = ρa ca. These two parameters depend also on the air temperature Ta. For more illustration, Table 1.1 gives the values of the speed of the sound ca, the density ρa and the characteristic acoustic impedance Zac as a function of the temperature Ta of the air. The model used in this work is that of Webster-Lokshin [START_REF] Haddar | A Webster-Lokshin model for waves with viscothermal losses and impedance boundary conditions: strong solutions[END_REF]. It is a model with mono-spatial dependence which characterizes the linear propagation of acoustic waves in tubes with axial symmetry. This model takes also into account visco-

0 L x x r Q v (t)
thermal losses at the wall boundaries with the assumption of wide tubes [START_REF] Mignot | Réalisation en guides d'ondes numériques stables d'un modèle acoustique réaliste pour la simulation en temps réel d'instruments à vent[END_REF]. Thus, in an axisymmetric tube of constant section S = π r 2 , the acoustic pressure P(x,t,L) and the acoustic flow Qv(x,t,L) are governed by the equation of the pavilions, also called Webster-Lokshin, and Euler equation, leading to system (1.1):

                                    0 , , , , 0 , ; 0 , 0 , , , , 2 , , 2 2 2 / 3 2 / 3 2 2 L t x P x L t x Q t S t L x L t x P x r L t x P t c r L t x P t c r v a a a r  , (1.1)
where  is a parameter associated with visco-thermal losses. More precisely,  is given by the relation:

  h v l l K r K 1 with , 0 0       , (1.2)
where v l and lh represent the characteristic lengths of viscous ( v l = 4 x 10 -8 m) and thermal (lh = 6 x 10 -8 m) effects,  being the ratio of specific heats.

The phenomenon of visco-thermal losses is a dissipative effect at the wall of the tube, which is due to the viscosity of the air and to the thermal conduction [START_REF] Mignot | Réalisation en guides d'ondes numériques stables d'un modèle acoustique réaliste pour la simulation en temps réel d'instruments à vent[END_REF]) [START_REF] Boutin | Modèle de propagation acoustique dans un tuyau cylindrique à paroi poreuse[END_REF]. For the case of wind musical instruments resonators, the assumption of wide tubes is used. This hypothesis is expressed by the following relation:

r ≫ max [ rv = (lv λ) 0.5 ; rh = (lh λ) 0.5 ],
(1.3)

where λ = ca/f represents the wavelength (in m) and f the frequency (in Hz).

Thus, for a speed of the sound ca and a frequency fmin corresponding to the lower limit of the frequency domain of study of the model, it is possible to determine the minimum value of the radius rmin of the acoustic tube below which the model is not valid.

As an illustration where lv = 4 x 10 -8 m, lh = 6 x 10 -8 m and ca = 346.3 m/s (at a constant temperature Ta = 25°C), Figure 1.2 presents the curves of rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f.

If we consider that the frequency domain of study of the Webster-Lokshin model is that of the frequencies audible by the human ear, frequencies ranging between 20 Hz (most serious frequency) and 20 000 Hz (most acute frequency), then fmin = 20 Hz. For this value of fmin, the model is valid for acoustic tube radius greater than 1 mm (in next section, entitled 1.2.3 -Frequency Response Analysis, r = 5 mm).

Resolution in the Symbolic Domain

Under the assumption of zero initial conditions, the Laplace transformation applied to the system (1.1) leads to:

                                                        0 , , , , 0 , , 2 2 2 2 / 3 2 L s x P x L s x Q s S L s x P r x c s c s v a a a r  , (1.4) with       L t x P L s x P , , TL , ,  and       L t x Q L s x Q v v , , TL , , 
, s being the Laplace variable and TL its transformation. Solving the Webster-Lokshin equation [START_REF] Haddar | A Webster-Lokshin model for waves with viscothermal losses and impedance boundary conditions: strong solutions[END_REF] gives the solution  

L s x P , ,
in the general form:

          s x s x r s B r s A L s x P      e e , , , (1.5)
where A(s) and B(s) are rational functions of s which depend on the boundary conditions, and (j) = j k(ω), k(ω) being a standard complex wave number. (s) is given in the Laplace domain by [START_REF] Mignot | From a model of lossy frared pipes to a general framework for simulation of waveguides[END_REF]:

𝛤(𝑠) = √ ( 𝑠 𝑐 𝑎 ) 2 + 2𝜀 ( 𝑠 𝑐 𝑎 ) 3 2 ⁄ , with 𝑅 𝑒 (𝛤(𝑠)) ≥ 0 if 𝜀 ≥ 0 . (1.6)
The expression of the solutions of  

L s x Q v , ,
is deduced in two ways:

1-Using the Euler equation in the Laplace domain (second equation of the system (1.4)), that is:

    L s x P x s S L s x Q a v , , 1 , ,     r , (1.7) 2-By introducing the general solution of   L s x P , ,
in relation (1.7), that is:

                        s x s x a v r s B r s A x s S L s x Q e e 1 , , r
. (1.8) Finally, the solution of   ,, v Q x s L is expressed in relation (1.9):               s x a s x a v s B s s r S s A s s r S L s x Q                         e 1 e 1 , , r r .
(1.9)

Taking into account the boundary conditions makes it possible to determine the two unknowns A(s) and B(s), and finally the impedance

      L s x Q L s x P L s x Z v , , / , , , , 
of the finite medium of length L.

As an example, consider a zero impedance at x = L, that is Z (L, s, L) = 0, which leads to

  0 , ,  L s L P
. According to relation (1.5), we obtain:

        0 e e      s L s L r s B r s A , (1.10) from which we deduce that       s L s A s B    2 e
.

(1.11)

Then, replacing B(s) by its expression (1.11) in the relation

(1.5) of   L s x P , , :             s x L s x r s A L s x P      2 e 1 e , ,
.

(1.12)

In the same way, by replacing B(s) by its expression (1.10) in the relation (1.8) of

  L s x Q v , ,
, we obtain:

              s x L s x a v s A s s r S L s x Q                2 e 1 e 1 , , r . (1.13) Finally, the impedance       L s x Q L s x P L s x Z v , , / , , , , 
is given by the relationship of equations (1.11) and (1.12), that is:

                        s x L s x a s x L s x s A s s r S r s A L s x Z                   
                             s x L s x L a s s S L s x Z 2 2 e 1 e 1 , , r , (1.15) knowing that tanh(y) = -(1 -e 2y )/(1 + e 2y
), tanh(.) being the tangent hyperbolic function,

          s x L s s S L s x Z a     tanh , , r
.

(1.16)

From the perspective of a system approach, the function (s) defined by the relation (1.5) is rewritten by putting the term s/ca into factor, that is:

  2 / 1 2 1                  s c c s s a a  , (1.17) or again, in canonical form,     , 0 1/ , , 0.5 1 with 2 2 m rm m a m r m a rm s m s K s m c r s c                             , (1.18)
where ωr,m is a transitional frequency (in rad/s). Note that in the theoretical case where the system is conservative, that is to say ɛ = 0, the function Г(s

) (relation (1.17)) is reduced to Г(s) = s/ca. By replacing Г(s) of relation (1.18) in relation (1.14),   L s x Z , ,
can be expressed as follows:

                                                                m m r m m r a m m r m m r a a s s c s x L s s S c L s x Z , , , , 1 tanh 1 , ,     r , (1.19)
or again, by introducing the characteristic acoustic impedance Zac = ra ca and the transitional

frequency ωLx = ca/(L-x) (in rad/s),                                                              m m r m m r x L m m r m m r ac s s s s s S Z L s x Z , , , , , 1 tanh 1 , ,     
.

(1.20)

Thus, from the analytical expression of the impedance Z(x,s,L) (1.20), knowing the flow

  L s x Q v , ,
at any point x of the acoustic tube of length L makes it possible to deduce the pressure   L s x P , , [START_REF] Hélie | Estimation paramétrique de la perce d'un instrument à vent à partir de la mesure de son impédance d'entrée[END_REF].

Remarks

At x = L, 1/Lx = 0 hence, knowing that tanh(0) = 0, we verify that   0 , ,  L s L Z . At x = 0, the input impedance     L s Z L s Z in , , 0 , 
of the finite medium of length L is given by:

𝑍 𝑖𝑛 (𝑠, 𝐿) = 𝜌 𝑎 𝑆 𝑠 𝛤(𝑠)
tanh(𝐿 𝛤(𝑠)) .

(1.21)

Always at x = 0, but for a semi-infinite medium (

  L ), knowing that 1 ) tanh( lim    y y the input impedance        , , 0 , s Z s Z in becomes:       s s S L s x Z s Z a L in        r , , 0 lim , . (1.22)
Finally, in the theoretical case of a purely conservative system (ɛ = 0) the acoustic impedance Z(x, s, L), noted in this case Z0(x, s, L), of a finite medium is reduced to

           Lx ac s S Z L s x Z  tanh , , 0 , (1.23)
and that of a semi-infinite medium, denoted Z0(x,s, 

), at

    cst S Z L s x Z s x Z ac L       , , lim , , 0 .
(1.24)

Note that the conservative case, although purely theoretical, allows by comparison to better observe the effect of visco-thermal losses.

To conclude this paragraph concerning the resolution in the symbolic domain, the study of asymptotic behaviors of Z(x,s,L), that is

  1 0 , , 0 lim 1 1 , , lim     m s x L m m r ac s s S Z L s x Z   (1.25) and   lim , , ac s Z Z x s L cst S   , (1.26)
highlights that Z(x,s,L) tends towards a behavior of the following types:

fractional derivative of order m+1, i.e. 1.5 with m = 0.5, when s tends to zero;

proportional, whose gain value is fixed by Zac/S, when s tends to infinity.

-Frequency Response Analysis

In stationary harmonic system, the frequency response Z(x,jω,L) is given by: where the transitional frequencies r,m and L,x have the following expressions:

                                                             m m
, 1/ 0 , 4 a rm m a Lx c r mK c Lx                  
.

(1.28) Thus, showing that r,m decreases when the radius r increases and that, on the contrary, L,x increases when the position x considered moves away from the origin and approaches the end of the acoustic tube, L.

For illustration, and in order to plot the frequency response Z(x,jω,L) in stationary harmonic system, let us consider an acoustic tube whose nominal dimensions are as follows:

radius r = 5 x 10 -3 m (value perfectly in accordance with the domain of validity of the Webster-Lokshin model);

-length L = 0.3 m;
temperature of 25°C;

density ρa = 1.184 kg/m 3 and a speed of sound in the air ca = 346.3 m/s (refer to Table 1.1).

Note that these values of r and L representing the resonator of a recorder lead to the values of the transitional frequencies ωr,m = 4.92 rad/s (0.78 Hz) and ωL,x = 1154 rad/s (184 Hz) with x = 0, thus showing for such an instrument that ωr,m << ωL,x. Although an in-depth analysis of the frequency response of the model is developed in the following paragraph, the observation of these diagrams over the range of audible frequencies leads to a remark which concerns the existence of two very distinct behaviors:

-The first on the frequency range [20; L,x /2] Hz with a derivative behavior of order 1, highlighting the absence of the fractional behavior on this frequency range which exists for frequencies lower than r,m /2Hz ; -The second on the frequency range [L,x /220 000] Hz with an alternation of resonances and anti-resonances.

-System Approach

As a reminder, the resonator is only a part of our simulator which includes a non-linear exciter (Chapter 2), an artificial mouth (Chapter 3), and the simulation of the whole system (Chapter 4).

From a causal point of view, the input of the resonator at x = 0 is defined by the pressure at the output of the nonlinear exciter. This is the reason why the system approach developed in this paragraph considers the admittance Y(x,s,L) = Z -1 (x,s,L) and not the impedance Z(x,s,L). More specifically, it is the input admittance at x = 0, denoted Yin(s,L) = Y(0,s,L). Note that this consideration of the admittance Yin(jω,L) leads to an integrative behavior for frequencies lower than L,x (derivative for Z(x,s,L)), thus respecting integral causality, a fundamental notion in a system approach.

In addition, the global admittance Y(x,s,L) is broken down into a cascade of local transfer functions of which all the parameters, as well as all the input and output variables will have a physical meaning. This decomposition then facilitates the frequency analysis of the Webster-Lokshin model, thus reaching a reduced model to be implemented in the simulator.

Decomposition of Admittance Y (x, s, L) into Subsystems

The admittance Y(x,s,L) = Z -1 (x,s,L) of an acoustic tube of length L at a point x between 0 and L is therefore defined by the expression:

                                                                  m m r m m r x L m m r m m r ac v s s s s s Z S L s x P L s x Q L s x Y , , , , , 1 tanh 1 1 , , , , , ,      , (1.29)
relation that can be expressed as follows,

      L s x T s I H L s x Y m , , , , 0  , (1.30) by taking a a ac c S Z S H r   0 , (1.31)   m m r m m r m s s s I                   , , 1   , (1.32) and       L s x F L s x T , , tanh 1 , ,  , (1.33)
where

    s I s L s x F m x L          , , ,  .
(1.34)

For the following, the concept of

acoustic admittance       L s x P L s x Q L s x Y v , , / , , , ,  is replaced by the concept of transfer function   L s x H , ,
defined between the pressure source    

s x P s P in , 0  
at the input of the tube at x = 0 and the flow  

L s x Q v , ,
at any point x of the tube of length L (semi finite) and of constant radius r, that is:

          L s x T s I H s P L s x Q L s x H m in v , , , , , , 0   .
(1.35) At x = 0, for this finite medium, the admittance of input Yin(s,L) has the expression:

            L s T s I H s P L s Q L s H L s Y m in v in , , 0 , , 0 , , 0 , 0    . (1.36) At x = 0, but for a semi-infinite medium (   L ), the admittance of input Yin(s,  ) is reduced to:           s I H s P s Q s H s Y m in v in 0 , , 0 , , 0 ,       .
(1.37)

Figure 1.4 presents the block diagrams associated with this system approach where the different transfer functions are defined by:

                  0 , , , 1 0, , ,, 1 ,, 0, , tanh in in ac m rm v m m in rm v v m Lx Qs S H cst P s Z s Qs Is Qs s Q x s L T x s L Qs s Is                                            
.

(1.38)

Note that the quantity

  s P H in 0 is homogeneous at a flow, noted   s Q in , corresponding
to the conversion of the pressure source applied at x=0 (Dirichlet condition) into an equivalent source of flow always applied at x=0 (Neumann condition) [START_REF] Assaf | Modélisation des phénomènes de diffusion thermique dans un milieu fini homogène en vue de l'analyse, de la synthèse et de la validation de commandes robustes[END_REF].

          L j x T j I H j P L j x Q L j x H m in v , , , , , , 0        , (1.39) with T(x,s,L)   s Q in   s P in    , , 0 s Q v   L s x P , , H 0 I m (s)   L s x Q v , , Z(x,s,L)   L s x H , , T(0,s,L)   s Q in   s P in    , , 0 s Q v H 0 I m (s)   L s Q v , , 0     L s H L s Y in , , 0 ,    s Q in   s P in    , , 0 s Q v H 0 I m (s)        , , 0 , s H s Y in       m m r m m r in v m j j j Q j Q j I                     , , 1 , , 0        (1.40) and           L j x F j Q L j x Q L j x T v v , , tanh 1 , , 0 , , , ,        , (1.41)
where

        j I j L j x F m x L          , , ,
.

(1.42)

The remaining part of this paragraph is dedicated to a detailed analysis of the frequency responses Im(j), F(x,jL) and T(x,jL) of each subsystem, then of the frequency response H(x,jL) of the overall system.

-Analysis of Im(jω)

The analysis of Im (jω) highlights two behaviors whose transition zone is fixed by the transitional frequency ωr,m, these behaviors are:

-for ω << ωr,m, a fractional integrative behavior of order m/2 = 0.25, Figure 1.6 presents the same frequency response of Im(j) but only on the range [20; 20,000] Hz of the frequencies audible by the human ear, and this with a gain diagram in linear-linear scale and a phase diagram with the frequency axis also in linear scale. The observation of this answer makes it possible to affirm, for this representative example of a recorder, that the unit proportional behavior is dominant, that is:

                                                   4 arg 1 , 2 / , 2 / , , , , ,                 m j I j I j j j j I m m m r m m m r m m r m m r m m r m r ; (1.43) -for ω >> ωr,m, unit proportional behavior,                                   0 arg 1 1 1 , , , , ,           
      1 , , 0 , rad/s 20 2           j Q j Q j I in v m
.

(1.45)

Thus, for the area of study considered in this work, area defined by the range [20; 20,000] Hz of the audible frequencies, the transfer Im(s) can be reduced to the unit which

leads to     s Q s Q in v   , , 0
, leading to a reduction in the block diagrams of Figure 1.4.

The direct consequence is that in the case of a semi-infinite medium at x = 0 (Figure 1.4.c), the fractional integrative behavior has no influence on the range of audible frequencies.

-Analysis of F(0,jω,L)

Knowing that in the case of a recorder ωr,m << ωL,x, the analysis of F(0,jω,L) again highlights two behaviors whose transition zone is fixed by the transitional frequency ωr,m, that is:

-for ω << ωr,m, a fractional derivative behavior of order (1-m/2) = 0.75,

      2 / 1 , 2 / , , , , , , 1 , , , m x L m m r m m r m m r x L m r j j j j L j x F m r                                                      , (1.46)
hence the module and the argument are defined as follows:

                               2 2 1 , , arg , , 2 / 1 , 2 / ,       m L j x F L j x F m x L m m r ;
(1.47) -for ωr << ω, a derivative behavior of order 1,

                                                      2 , , arg , , 1 , , , , , , , , , ,  
                L j x F L j x F j j j j L j x F x L x L m m

-Analysis of T(0,jω,L)

The analysis of T(x,jω,L) highlights three behaviors whose transition zones are fixed by the transitional frequencies ωr,m and ωLx :

-for ω << ωLx, an integrative behavior with two different orders according to the frequency range. Indeed,

        L j x F L j x F L j x T x L x L , , 1 , , tanh 1 , , , , ,             , (1.49)
with, for ω << ωr,m, an orderly fractional integrative behavior -(1-m/2) = -0.75, that is

                                                   2 2 1 , , arg 1 , , 1 , , 1 , 2 / 1 2 / , 2 / 1 2 / , , ,               m L j x T L j x T j L j x F m m r x L m m r x L m r m r
(1.50) and, for ωr,m << ω, a derivative behavior of order 1, that is

                         2 , , arg , , , , 1 , , , , ,  
          L j x T L j x T j L j x F x L x L m r m r ;
(1.51) -for ωL,x << ω, a behavior composed of an alternation of anti-resonances and resonances, giving the expression of T(x,jω,L), ; ωLx/2π = 184] Hz, the responses of 1/F(0,jω,L) (in red) and T(0,jω,L) (in blue) overlap where:

                                              m m r m m r x L x L j j j L j x T , , , , 1 tanh 1 , , ,          . (1.52)
-a fractional integration behavior of order -0.75 over the range [10 -4 ; ωr/2π = 0.784] Hz is observed; -an integrative behavior of order 1 over the range [ωr,m/2π = 0.784 ; ωLx/2π = 184] Hz is observed.

Beyond 184 Hz, the frequency response T(0,jω,L) (in blue) clearly presents an alternation of anti-resonances and resonances introduced by the hyperbolic tangent function.

-Analysis of H(x,jω,L)

The analysis of H(x,jω,L) highlights three behaviors whose transition zones are fixed by the transitional frequencies ωr and ωLx:

-for ω << ωr << ωLx, an orderly fractional integrative behavior -(1-m/2) = -0.75, that is

                                                   2 2 1 , , arg , , , , , 2 / 1 0 2 / , , 2 / 1 0 2 / , , , ,               m L j x H H L j x H j H L j x H m m m r x L m m m r x L m r m r ;(1.53)
-for ωr,m << ω << ωLx, a derivative behavior of order 1, that is:

                           2 , , arg , , , , , , 0 , 0 , , , ,               L j x H H L j x H j H L j x H x L x L x L m r x L m r ; (1.54)
-for ωLx << ω, a behavior composed of an alternation of anti-resonances and resonances, that is:

                                               m m r m m r x L x L j j j H L j x H x L , , , 0 , 1 tanh , , , ,            .
(1.55) Figure 1.9 presents at x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and at x = 3L/4 (ωLx/2π = 735 Hz) the Bode diagrams of H(0,jω,L) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) on the range [20; 4000] Hz of the audible and achievable frequencies with a recorder.

Over the range [20; ωLx/2π] Hz, the three responses of H(x,jω,L) present an integration behavior of order 1. The fractional integration behavior of order -0.75 does not appear over this range as it is present within a much lower frequency (0.784 Hz). Beyond ωLx, the three responses present a succession of alternation of anti-resonances and resonances introduced by the hyperbolic tangent (tanh) function. In the journal (Abou Haidar, Moreau, & Abi Zeid Daou, 2021), the authors show that the influence of the order m is essentially located:

-for gain diagrams, at the peaks of resonances and anti-resonances, quantifiable effects using quality factors for anti-resonances and for resonances illustrating well the phenomenon of dissipation associated with visco-thermal losses; -for phase diagrams, at the crossing points at 0° with a local slope which is important as the order is small where the slope becomes infinite for m = 0 (purely conservative case).

Moreover, note that the farther the position x moves away from the origin, the higher the transitional frequency ωLx pushes the anti-resonance and resonance frequencies towards the high frequencies.

In addition, the position x has no influence on the transitional frequency ωr,m, the fractional integrative behavior of order -0.75 still does not appear on this frequency range. In the Webster-Lokshin model, the fractional order m has the value 0.5. The objective of this paragraph and the rest of this chapter is to analyze the influence of the order m on the behavior of the resonator by considering that m belongs to the interval [0; 1] with a nominal value m0 = 0.5, a consideration which facilitates the introduction of the concept of parametric uncertainty (additive or multiplicative) at the fractional order level. Thus, by generalizing the expression of the parameter  = K0/r (relation (1.2)) associated with visco-thermal losses in the Webster-Lokshin model at  = 2 m K0/r (relation (1.18) which for m = 0.5 gives the same expression), the analytical link is naturally established between visco-thermal losses and fractional order. Thus, the fractional order occurs only in the presence of visco-thermal losses. In the theoretical case of a purely conservative system, the parameter  is zero which is equivalent to m = 0, taking into account the relation (1.17). In this case, the expression of the acoustic transfer H (x, s, L), denoted as H0 (x, s, L), of a finite medium is reduced to:

           x L s H L s x H , 0 0 tanh , ,  .
(1.56)

Figure 1.10 shows the Bode diagrams at x = 0 of H(0,jL) for different values of the fractional order over the range [20; 4000] Hz of the audible and achievable frequencies with a recorder (Figure 1.10).

In order to magnify the different curves in Figure 1.9 for better observation, Figure 1.11 presents the reduced frequency responses H(0,jL)/H0 with the frequency axis on a linear scale over the range [20; 1000] Hz.

The observation of these frequency responses shows that the influence of the order m is essentially located:

-for gain diagrams, at the peaks of resonances and anti-resonances, quantifiable effects using quality factors Qzi for anti-resonances and Qpi for resonances illustrating well the phenomenon of dissipation associated with visco-thermal losses; -for phase diagrams, at the crossing points at 0° with a local slope which is all the more important as the order is small, slope which becomes infinite for m = 0 (purely conservative case).

-From the Simplified Fractional Model to its Rational Forms

For the area of study defined by the range [20; 20,000] Hz of the audible frequencies, the analysis presented in the previous paragraph shows that the frequency response Im(j), is:

  m m r m m r m j j j I                   , , 1      ,
(1.57) can be reduced to the unit (see paragraph 1.3.2.1). This is the reason why for this field of study the frequency response H (x, j, L) defined, as a reminder, by

          L j x T j I H j P L j x Q L j x H m in v , , , , , , 0        , (1.58) is simplified and noted   L j x H , , ~
, that is:

    L j x T H L j x H , , , , ~0    , (1.59)
with, always as a reminder,

                                                   j I j j I j H L j x H L j x T m x L m x L , , 0 cotanh tanh 1 , , , ,
.

(1.60) It is important to note that this simplification does not affect T(x,j,L) because the absence of Im(j) in relation (1.59) corresponds to the conservative case where  and m are zero (see paragraph 1.3.3 relation (1.55)). Figure 1.12 shows the block diagrams associated with this simplification in the field of study. In general, the time simulation of fractional models often requires the use of rational models [START_REF] Assaf | Modélisation des phénomènes de diffusion thermique dans un milieu fini homogène en vue de l'analyse, de la synthèse et de la validation de commandes robustes[END_REF]. Thus, the fractional form defined by the relation (1.58) can be put in a rational form of N cells in cascade, noted

  L s x H c N , , ~,
, that is:

                                         N i pi pi pi zi zi zi c N s s s s s A L s x H 1 2 2 0 , 2 1 2 1 , , ~      , (1.61) or in a rational form of N cells in parallel, noted   L s x H p N , , ~, ,                                  N i pi pi pi i i p N s s B s A s A L s x H 1 2 0 , 2 1 , , ~   , (1.62) T(x,s,L)   s P in    , , 0 s Q v   L s x P , , H 0   L s x Q v , , Z(x,s,L)   L s x H , , ~T(0,s,L)   s P in    , , 0 s Q v H 0   L s Q v , , 0   L s H , , 0   s P in    , , 0 s Q v H 0    , , 0 ~s H
with A0 = H0 L,x where the zi and zi represent the frequencies and the damping factors associated with the anti-resonances, pi and pi the frequencies and the damping factors associated with the resonances while changing from the cascade form (1.60) to the parallel form (1.61) by decomposing into simple elements. Note that the parallel rational form facilitates the return to the time domain by inverse Laplace transform and that it is often associated with a decomposition in a modal space [START_REF] Debut | Deux études d'un instrument de musique de type clarinette : analyse des fréquences propres du résonateur et calcul des auto-oscillations par décomposition modale[END_REF].

From a theoretical point of view, zi corresponds to the roots of the numerator of T(x,j,L), that is:

0 1 cosh , , ,                                           m m r m m r x L j j j       , (1.63)
and the pi correspond to the roots of the denominator of T(x,j,L), that is:

0 1 sinh , , ,                                           m m r m m r x L j j j       . (1.64)
From a practical point of view, finding these roots by analytical resolution is complex, if not impossible [START_REF] Fabrice | Émergence des auto-oscillations dans un instrument de musique à anche simple[END_REF]. On the other hand, the search by numerical resolution does not pose any particular problem. For example, it is possible to use the fact that the alternation of zi and pi appears clearly on the phase from when passing at 0 °, from -90 ° to + 90 ° for zi, and from + 90 ° to -90 ° for pi.

In the context of the work of this thesis, the rational forms of N cells in cascade and in parallel are considered as behavior models whose numerical values of the parameters are obtained using an optimal approach aiming to minimize the difference between the target frequency response defined by the fractional form  

L s x H , , ~
and the frequency response of the rational cascade form

  L s x H c N , , ~,
. This digital procedure is available in the Frequency Domain System Identification (FDSI) module of the CRONE Toolbox [START_REF] Malti | CRONE Toolbox for system identification using fractional differentiation models[END_REF].

As an illustration, let us take the acoustic tube used as an example throughout this chapter: r = 5x10 -3 m, L = 0.3 m, ra = 1.184 kg/m 3 , ca = 346.3 m/s, L,x = 1,154 rad/s at x = 0, H0 = 19.15x10 -8 m 3 s -1 Pa -1 and A0 = 22.11x10 -5 rad/s. Figure 1.13 shows two screenshots from the CRONE Toolbox before optimization (Figure 1 the lowest values (left) to the most important (right). Note that in this graphical interface, the term "cell" corresponds to a polynomial (numerator or denominator). Thus, with each addition in the "Transfer function" menu (in purple at the bottom) a column in blue for the numerator and in yellow for the denominator appears. The first line "Cell Frequency" gives the value in rad/s of zi or pi, the second "Cell Order" the highest order of the polynomial (here +2 for the numerator and -2 for the denominator), the third "Local Order" is equal to +1 (for the numerator) and -1 (for the denominator) insofar as these are explicit forms [START_REF] Malti | CRONE Toolbox for system identification using fractional differentiation models[END_REF], and finally the fourth "Damping Factor" gives the value of zi or pi. All these values in the columns can be modified by clicking in the corresponding box. Thus, in the case of the resonator, all the values of zi or pi are initialized to 0.01.

This first stage of the procedure therefore makes it possible to fix the structure of the behavior model, as well as the initial values of its parameters. The second step is an optimization step launched using the "Tools" menu (in orange at the bottom right). For the example of illustration, the result appears in Figure 1.13.b in particular the optimal values of zi, pi, zi and pi. In summary, the cascade form defined by the relation (1.60) comprises N = 4 cells, that is 2N = 8 "cells" as defined in Figure 1.13, to which we must add the integrating cell A0/s, either in total N + 1 = 5 cells, or else 2N + 1 = 9 "cells".

Remark

In linear systems dynamics, in the general case of a polynomial of order 2 having 1 pair of conjugate complex roots, the damping factor  and the resonance factor Q associated with this pair are linked by a relation of the form :

1 2 1     Q . (1.65)
In instrument acoustics (Chaigne & Kergomard, 2013), and in particular in the specific case of resonators of wind instruments, the damping factors are very small compared to the unit. This is the reason why the relation (1.64) is reduced to:

Q Q 2 1 2 1 , 1 0          . (1.66)
Note that in instrument acoustics, the term quality factor is used in place of the resonance factor. Thus, in many works, the taking into account of the visco-thermal losses is made directly using the parallel rational form defined by the relation (1.62) in which the pi are replaced by the corresponding Qpi (relation (1.66 )) (Chaigne & Kergomard, 2013) without going through fractional models. Thus, the procedure presented in this paragraph, and illustrated in the nominal case m = 0.5, must be repeated for each value of m considered over the interval [0; 1].

-Conclusion

The structure and progression of this chapter are organized in a didactic way so that readers with no idea about visco-thermal losses in wind instruments can understand the dynamic behavior of an acoustic tube of constant radius. From the two partial differential equations which define the Webster-Lokshin model, a classical resolution in the operational domain leads to the analytical expression of the acoustic impedance and admittance of the function tube of position x, its length L and its radius r. Moreover, a system vision is proposed aiming to causally decompose the global model into sub-models, thus facilitating analysis in the frequency domain. One of the conclusions of this frequency analysis is that the fractional model can be simplified over the range [20; 20,000] Hz of the audible frequencies. In Addition, the introduction of an uncertainty at the level of the fractional order (whose value considered as nominal is that of the initial Webster-Lokshin model, namely m0 = 0.5) allows to study the influence of the order m when this varies between 0 (conservative case) and 1. Although the fractional order behavior (fractional integrator) is only present for very low non-audible frequencies (less than 1 Hz), the influence of the fractional order m does appear at resonances and anti-resonances (in the audible frequencies), illustrating well the phenomenon of dissipation associated with viscothermal losses. The simulation in the time domain of fractional models requires the establishment of rational forms. Thus, two rational forms composed of an integrator and N second-order cells, one in cascade and the other in parallel, are introduced. The parameters of the cascade form are then determined using the Frequency Domain System Identification (FDSI) module of the CRONE Toolbox. As for the parameters of the parallel form, they are obtained by a decomposition into simple elements of the cascade form.

More generally in the fractional model, this study of visco-thermal losses within the resonator of a wind instrument leads to a finding similar to that already made in other fields. Indeed, the main interest of the fractional form resides in the parametric parsimony, that is to say the capacity which the integral-differential operator of non-integer order has to model the greatest number dynamic phenomena with a minimum of parameters. Thus, the study of parametric sensitivity, in particular in the frequency domain, is simpler.

In the case of visco-thermal losses treated in this chapter, the study of the sensitivity is reduced to the only parameter m. The same parametric sensitivity study in the frequency domain with one of the two rational forms presented here is much more complex because this form has 4N parameters (4 parameters for each of the N cells, i.e. 16 for the example of illustration presented corresponding at the only value m = 0.5). The transition to one of the rational forms is then carried out only for time simulation.

Chapter 2-Study of the Nonlinear Exciter of a Wind

Musical Instrument

-Introduction

The self-oscillation mechanism is characterized by the ability of the instrument to produce an acoustic wave from a stationary or quasi-stationary source of energy with respect to the acoustic variables. This mechanism is highly associated with the non-linear nature of the instrument. A classic approach in musical acoustics (Kergomard & Chaigne, 2013) represents the instrument (Figure 2.1) by a nonlinear excitation system coupled to a passive linear resonant system (Chapter 1). Due to its non-linear nature, modeling the exciter and its coupling with the resonator that is the energy source is very complex. Indeed, the phenomenon results from the oscillation of a naturally unstable air jet around a bevel (Figure 2.2). Thus, the self-oscillation mechanism relies on a synchronization of the oscillation of the jet and the acoustic waves. It is therefore necessary that the jet-bevel system excites the resonator at the periodicity of the acoustic field.

In order to fully understand the phenomena involved and to be able to reproduce them by numerical simulation, this chapter proposes, first of all, a modeling by the proper diagram and parameters of the various essential elements of the system, then by recalling the mechanism of sound production. Then, from a bibliographic synthesis, a complete nonlinear model frequently used in the literature is developed. Moreover, after focusing on the poor digital conditioning of such a model, a solution is proposed in order to be able to develop a digital simulator programmed in MatLab/Simulink. A scenario is then defined for several values of the pressure at the input of the mouthpiece of the flute. This scenario is defined by respecting, not only the domain of validity of the nonlinear model, but also the values of pressure achievable experimentally by the artificial mouth presented in chapter 3. Thus, for this field of study, a detailed analysis of the time responses simulated is proposed. This analysis makes it possible, in particular, to observe that the variations of a certain number of physical quantities (pressure at the input of the resonator, acoustic speed, etc.) are small and around zero. On the basis of this observation, a linearization is developed leading to two linearized models, one for the analysis of the start-up phase of the simulation (phase 1), the other for that of the self-oscillations (phase 3). Finally, the last paragraph analyzes the conditions of self-oscillation of such a system.

-Modelling

-Schematic and Configuration of the Exciter

-Sound Production Mechanism: Reminder

From a causal point of view, the generation of a pressure input Pm(t) at the mouthpiece of the flute (imposed by the musician or by an artificial mouth device as defined in chapter 3) gives rise to a flow in the spout channel (Figure 2.2). At the exit of the channel, the viscosity of the air prevents the flow from following the geometry of the walls [START_REF] Terrien | Instruments de la famille des flûtes: analyse des transitions entre régimes[END_REF]. The consequence is that the flow takes off from the walls and gives rise to an air jet whose speed at x = 0 is denoted Uj(x = 0, t). This air jet is unstable in nature (Kergomard & Chaigne, 2013). Indeed, if in the vicinity of the origin at x = 0 the flow is well ordered, eddies appear when the flow moves away from the starting point. Due to this unstable nature, the slightest disturbance to the jet is amplified over time (Kergomard & Chaigne, 2013).

The jet created at the channel output flows inside the excitation window. It meets the bevel, the point of which is located at a distance w from the channel output and laterally offset by x0 with respect to the longitudinal axis y of the flow. This jet-bevel interaction causes a hydrodynamic return on the jet. This phenomenon plays an essential role in the initiation of the oscillation of the jet by providing it with its first disturbances. Due to the natural instability of the air jet, disturbances naturally amplify along this jet, from the exit of the channel to the bevel, and cause it to oscillate on either side of the bevel. This oscillation causes a flow injection alternately inside and outside the instrument. It forms the aeroacoustic source of pressure which supplies energy to the resonator (Kergomard & Chaigne, 2013).

The acoustic waves thus created in the resonator (chapter 1) propagate and are reflected (mainly) at the level of the first open hole. The superposition of the outward wave and the return wave gives rise to a system of standing waves in the resonator, which disturbs the return jet at the channel output, thus closing the self-oscillation loop. This initial disturbance of the jet maintains its oscillation and therefore the sound production mechanism.

-From a Phenomenon with Distributed Parameters to a Model with Localized Parameters

The models of sound production in flutes are based on the loop system in (Figure 2.3) and involve the modeling of each of the three main phenomena: -the instability of the jet; -aero-acoustic sources related to the jet-bevel interaction; -the propagation of acoustic waves in the resonator. The separation of the three main phenomena in operation and the fact that their interactions are supposed to be localized create a characteristic and an important limitation of this representation. This approach may seem counter-intuitive, in the sense that the boundaries between these different elements remain relatively blurred. For example, where to locate the sources of flow resulting from the oscillation of the jet? However, the phenomena involve different scales and therefore different hypotheses. Thus, for low frequencies, corresponding to the very first acoustic resonances, the area of development of the unstable air jet is small compared to the wavelengths of the acoustic waves. The jet can then be studied under the hypothesis of incompressibility, a hypothesis which is of course no longer compatible with the description of the propagation in the resonator. As discussed by Fabre and Hirschberg [START_REF] Hirschberg | Physical modeling of flue instruments : A review of lumped models[END_REF], it is precisely because the different phenomena are described under different assumptions (which may at first appear paradoxical) that such a representation by blocks with localized interactions is possible. For each of the blocks highlighted in the functional diagram of (Figure 2.3), the physical phenomena in action, as well as their models retained are presented in the continuation of this paragraph. The complete model thus obtained is undoubtedly the most used one in the literature [START_REF] Hirschberg | Physical modeling of flue instruments : A review of lumped models[END_REF].

-Receptivity: Initial Disturbance of the Jet

Once the self-oscillations have been established, the disturbance of the jet is brought about by the acoustic field present in the resonator. Due to the non-viscous fluid hypothesis adopted for the description of the jet, this phenomenon, called receptivity, is considered localized at the point of separation of the jet, i.e. at the channel output at x = 0 (Kergomard & Chaigne, 2013). Different receptivity models are proposed in the literature, in particular by Fabre and Hirschberg [START_REF] Hirschberg | Physical modeling of flue instruments : A review of lumped models[END_REF] and by [START_REF] Blanc | Modeling the receptivity of an air jet to transverse acoustic disturbance with application to musical instruments[END_REF].

In the rest of this chapter, the model used is the one proposed by de la Cuadra [START_REF] Cuadra | The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments[END_REF] based on Schlieren visualizations of a jet subjected to a transverse acoustic field. One of the hypotheses which reduces the domain of validity of this model is that the pressure

 

  e m m m

P t P p t 

in the musician's mouth or the artificial mouth (see Chapter 3) is assumed to be stationary (or quasi-stationary), an assumption which leads to a central jet speed

    0, 0, e j j j U x t U u x t    
at the exit of the channel also stationary. This is the reason why in the rest of this chapter, we consider that:

    0, e mm e jj P t P U x t U        , ( 2.1)
where, according to Bernoulli's law in stationary system at the channel exit [START_REF] Cuadra | The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments[END_REF] ,

2 e e m j P U r  , (2.2)
rbeing the density of air.

Thus, for this domain of validity, the initial disturbance of the jet at x = 0 at the exit of the channel is represented by a transverse displacement 0(x = 0, t), noted 0(t), linked to the acoustic speed vac(t) of the resonator always at x = 0 by a relation of the form:

    0 ac e j h t v t U   .
(2.3)

-Jet Instability: Amplification and Convection of Disturbances

As noted above, the air jet that propagates between the channel output and the bevel is naturally unstable. This instability of the jet, known under the name of Kelvin-Helmholtz instability [START_REF] Reid | Hydrodynamic stability[END_REF], results from the natural instability of shear layers animated at different speeds, that is to say of the interfaces between the jet and the external environment at rest. The natural instability of the jet results in amplification and convection of the slightest disturbance. Rayleigh [START_REF] Rayleigh | The theory of sound[END_REF] first proposed a linear description of this phenomenon in the case of small perturbations of a semi-infinite jet in the form of an exponential amplification of the perturbations with the convection distance.

In the flute family instruments, the jet develops inside an excitation window limited at one end by the channel at x = 0 and at the other by the bevel with which it interacts at x = w. An experimental study by de la Cuadra [START_REF] Cuadra | The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments[END_REF] showed that the exponential amplification of the initial disturbance 0(t) with the convection distance x remained a reasonable approximation. In the case of the flute, at the level of the bevel at x = w, we therefore obtain the transverse displacement (w, t) of the disturbance whose expression is given by:

      0 , exp i x w t t w        , ( 2.4)
where, still according to de la Cuadra [START_REF] Cuadra | The sound of oscillating air jets: Physics, modeling and simulation in flute-like instruments[END_REF], the estimate of the amplification factor i as a function of the height h of the channel is of the order:

0.4 i h   . (2.5)
The delay  introduced in equation ( 2.4) is related to the convection time of the initial disturbance 0(t) along the jet, from the exit of the channel at x = 0 to the wedge at x = w.

Various experimental results [START_REF] Cuadra | Visualization and analysis of jet oscillation under transverse acoustic perturbation[END_REF] [START_REF] Nolle | Sinuous instability of a planar jet: propagation parameters and acoustic excitation[END_REF], as well as the theoretical work of Rayleigh [START_REF] Rayleigh | The theory of sound[END_REF] (2.6)

-Jet-Bevel Interaction: Aero-Acoustic Sources

In wind instruments, the oscillation of the jet around the bevel acts as an exciter, producing the acoustic energy transmitted to the resonator. While historically this source was first described by Helmholtz using a monopole, the dipole modeling, initiated by Rayleigh [START_REF] Rayleigh | The theory of sound[END_REF], is now used. The jet-drive model, initially proposed by Coltman [START_REF] Coltman | Jet drive mechanisms in edge tones and organ pipes[END_REF], then by Verge [START_REF] Verge | Aeroacoustics of confined jets: with applications to the physical modeling of recorder-like instruments[END_REF] [START_REF] Verge | Sound production in recorder-like instruments[END_REF]) is based on this principle of representing the source term by a force. Figure 2.4 schematically illustrates the behavior of the jet. (2.7)

Since this equivalent distance d is small compared to the acoustic wavelengths, the mass of air between the two flow injection points can be considered incompressible. The alternating sloshing of this air mass, induced by the oscillation of the jet, exerts a force on the acoustic field, which is modeled by a pressure difference, noted psrc(t), whose expression is given by [START_REF] Terrien | Instruments de la famille des flûtes: analyse des transitions entre régimes[END_REF]: F., 1994) and Ségoufin (Ségoufin C. , 2000), by:

    0 , tanh
2 5 b h  , (2.9)
and where tanh(.) is the hyperbolic tangent function.

Moreover, the excitation window of the flute between the channel output at x = 0 and the bevel at x = w, from the point of view of the resonator, constitutes an open end, but also a constriction. In fact, its section is most of the time smaller than the section of the bore of the instrument. The acoustic speeds observed in this zone can therefore be high and most often cannot be neglected in relation to the speed of the jet. The presence of a sharp edge (the bevel), combined with the effects of viscosity, results in a separation of the flow at this point (Kergomard & Chaigne, 2013). This then causes detachments of vortices at the level of the bevel, as demonstrated experimentally by Fabre and Hirschberg (B Fabre, 1996).

Taking into account this phenomenon linked to energy loss in the instrument models appears to be decisive for correctly describing the mechanisms of saturation of the oscillation amplitude (B Fabre, 1996) (R Auvray, 2012). If different models have been proposed to account for this phenomenon [START_REF] Fabre | Vortex shedding in steady oscillation of a flue organ pipe[END_REF], an approach often retained (mainly because of its simplicity (Kergomard & Chaigne, 2013)) consists in modeling it by an additional pressure difference plos(t) between the two sides of the bevel, namely:

      2 sign 2 ac los ac vc vt p t v t r          , (2.10)
where vc is a factor corresponding to an effect of vena contracta estimated at 0.6 in the case of a sharp arrest [START_REF] Falkovich | Fluid mechanics: A short course for physicists[END_REF] and where sign represents the sign function.

Finally, the aero-acoustic pressure source p(t) at the bevel is given by [START_REF] Terrien | Instruments de la famille des flûtes: analyse des transitions entre régimes[END_REF]:

      src los p t p t p t     
.

(2.11)

-Complete Model

The complete model of the instrument retained within the framework of this study is finally composed of a system of three equations, each one being linked to a given element of the looped system shown in (Figure 2.5), namely:

                      2 0 -1 exp , ,
  r  r                                               
, (2.12) where S and L represent the section and the length of the resonator, Yin(s,L) its input admittance (defined in Chapter 1) and P(s) = TL{p(t)}, TL designating the Laplace Transform.

-Limits of the Model: Reminder of the Validated Domain

The capacity of the complete model to qualitatively reproduce many phenomena observed experimentally has been demonstrated in various works [START_REF] Auvray | Regime change and oscillation thresholds in recorder-like instruments[END_REF] [START_REF] S Terrien | Numerical resolution of a physical model of flute-like instruments : comparison between different approaches[END_REF]. However, like any model, this one is only valid in an area that must be taken into account when interpreting the results.

The linear description of the jet instability is theoretically valid only for small amplitudes of transverse deflection (x,t). When the value of (x,t) becomes close to the thickness of the jet, the jet rolls up on itself giving rise to a series of vortices which can be modeled as a Von Karman alley (Kergomard & Chaigne, 2013). The model described by the system of equations (2.12) does not take this nonlinear phenomenon into account.

On the other hand, when the jet speed e j U becomes sufficiently high, a transition to a turbulent state can be observed. As a reminder [START_REF] Terrien | Instruments de la famille des flûtes: analyse des transitions entre régimes[END_REF], the transition zone from a laminar (linear) flow to a turbulent (nonlinear) flow is located for a Reynolds number between 2500 and 3000, knowing that the Reynolds number represents the ratio between inertial forces and viscous forces. According to Fabre (Kergomard & Chaigne, 2013), in the case of a turbulent flow at the outlet of the channel, the modeling should take into account various elements that are not described in this model.

Finally, the model is well suited [START_REF] Terrien | Instruments de la famille des flûtes: analyse des transitions entre régimes[END_REF] for:

-low values of the number of Strouhal defined by: .13) where f0 represents the self-oscillation frequency in Hz of the waves within the resonator; -jets with a thin thickness compared to the length w;

0 e j w f U   , ( 2
-rather large hydrodynamic wavelengths.

-Numerical Simulation

-Digital Packaging Problem

The complete model, as defined by the system of equations ( 2.12), is poorly conditioned from a numerical point of view. Indeed, the presence of a derivation in the second equation of the system (2.12) reveals a causality problem. Moreover, assuming that the derivation can be programmed without posing a numerical problem, if the initial conditions associated with the variables (w,t), vac(t) and p(t) are zero (which is the case physically as long as the pressure Pm(t) at the mouthpiece input is zero), then there is no generating process to initiate the start of a transient system which must then make it possible to reach a steady periodic system imposed by the self-oscillation conditions. Finally, the presence of the flow speed Uj(t) at the denominator of the first equation of the system (2.12) excludes any zero initial value of this speed at the start of the digital simulation, and therefore of the pressure Pm(t) at the entry of the mouthpiece of the flute, a situation however very physically realistic.

The solution adopted in this thesis concerning the derivation is the use of a frequency truncated differentiator whose expression D(s) in the symbolic domain is defined by:

  1 h n c s Ds s       , ( 2

.14)

c being the cutoff angular frequency (in rad/s) and nh the order of the low-pass filter. The useful effect of the low-pass filter is to prevent D(s) from tending to infinity as s tends to infinity (i.e. the first instants taking into account the applicable time-frequency duality to this linear operator). The parasitic phase shift effect is as important as the order nh. Faced with this dilemma, a good compromise is nh = 2 which leads to:

  (2.15) This value of nh must be combined with a cut-off angular frequency c compared to the largest natural un-damped angular frequency pi of the rational form of the impedance Yin(s,L) as defined in Chapter 1.

It is important to note that the choice of this operator makes it possible, not only to solve the numerical problem related to the derivation, but also to introduce a generator process allowing, in the presence of zero initial conditions, to initially trigger the start of a transient system then leading to a stationary periodic system (self-oscillation conditions). The following paragraph illustrates and then demonstrates this result.

-Simulator Developed using MatLab / Simulink

The complete model in the presence of operator D(s) is programmed in MatLab / Simulink with the following parametric values: The initial conditions of the various simulations carried out are: -phase 1: start-up with a convergent transient system; -phase 2: divergent transient system;

           
                    , ( 2
phase 3: stationary periodic system.

Among these 13 tests, two (corresponding to the limits of the membership interval of e m P

) are presented and analyzed by way of illustration:

-example 1:

400 Pa For each of these two examples, three families of time responses are proposed for: -the total duration of the simulation from t0 to tmax, with t0 = 0 s and tmax = 1 s; -phase 1 from t0 to t1; -phase 3 from t2 to tmax = 1 s. shows the time responses of the pressure p(t) at the input of the resonator ((a) and (b) ), the acoustic velocity vac(t) ((c) and (d)) and the transverse displacement (w,t) of the jet ((e) and (f)). Observing the results allows us to make the following remarks:

-the duration of phase 1 depends on the value of the delay  and therefore on the value of e m P . Thus, the lower the delay  (that is, the greater the e m P pressure), the shorter phase 1.

Focusing on

-Linearized Models

In order to facilitate the analysis of phase 1 (start of the simulation) and phase 3 (selfoscillation conditions), a linearization is developed leading to two linearized models, one for the analysis of the phase 1, the other for that of phase 3. Note that in phase 3, that is to say for time t >> c = 1/c = 3.18 x10 -6 s (time constant of the low pass filter, see Remark 2.1) the two linearized models exhibit the same behavior.

Thus, the observation of the simulations presented in the previous paragraph allows us to affirm that the variations of (w,t), vac(t) and p(t) are indeed around zero. This is the reason why the linearization of the system of equations (2.12) is done around this value.

More precisely, a first non-linearity appears in expression (2.8) of psrc(t) with the function tanh(.) and a second non-linearity appears in expression (2.10) of plos(t) with the acoustic speed vac(t) squared. The expansion limited to the first order of the function tanh(.)

around  = 0 leads to:

    2 0 0 0 0 1 tanh , tanh tanh , x w t x b x wt b b b                             , (2.17) form relation     0 01 0 , tanh , w t x c c w t b          , (2.18) taking 0 0 2 0 1 tanh 1 tanh x c b x b c b                      , (2.19)
c0 and c1 being constants.

As for the first-order limited expansion of [vac(t)] 2 around 0, its expression is given by: .20) From these developments limited to the first order, two linear models are proposed: -a consideration in the time domain of the derivation operator d/dt, called the theoretical linearized model; -the other considering in the operational field the derivative operator D(s), called the programmed linearized model.

    2 0 0 2 0 0 ac ac ac v v t v t         . ( 2

-Theoretical Linearized Model

The derivative with respect to time of relation (2.17) K1 being a constant.

Finally, the linearization of the system of equations ( 2.12) around 0 leads to: under the assumption of zero initial conditions, is then given by:

                0 1 -1 , , 1 TL , a c i c n a Y w t K v t d p t K w t dt v t P s S sL               , ( 2
              0 1 , , 1 , i ac n s ac w s K e V s P s s Y s L K w s V s P s S               , (2.26) where       , TL , w s w t   .
This theoretical linearized model is used in section 2.5 for the analysis of the autooscillation conditions (phase 3).

Programmed Linearized Model

As specified in paragraph 2.3.1 -Numerical conditioning problem, the solution adopted to solve the causality problem lies in the use of a derivation operator D(s) as defined by the relation (2.14) with nh = 2 and c = 2 50 000 rad/s. Thus, the application in the symbolic domain of this derivation operator D(s) to relation (2.17) leads to an expression of the pressure P(s)of the form: By distributing the operator D(s), we show that the pressure P(s) (relation (2.27)) is composed of the sum of two terms, P0(s) and P1(s), that is: .30) where it is important to note that the two constants C0 and C1 are proportional to the value of the stationary speed Thus, in the first moments of phase 1 where the variations are of low amplitude, the responses of the linearized model (curve 3 in blue) are identical to those of the nonlinear model (curve 2 in red). These answers clearly illustrate the legitimacy of the analysis made in the first moments using the linearized model which leads to the analytical expression (2.32) of the pressure signal (curve 1 in green). Observation of these responses allows us to make the following remarks:

      * 0 11 , c P s K D s c w s s        , ( 2
      01 P s P s P s      , (2.28) taking,           0 0 11 , C P s D s s P s C D s w s         , (2.29) with, * 0 0 1 0 * 1 1 1 1 e d j e d j b C c K c U w b C c K c U w r r                 , ( 2
      1 1 0 0 0 1 1! 1 n h c h h t n c n h c p t C TL C t e n s                      . ( 2
-the pressure angular frequency p0(t) as defined by relation (2.32) clearly appears from t = 0 s to t = 3x10 -5 s with a maximum amplitude of 310 Pa (Figure 2.13.a). This pressure angular frequency at the input of the resonator is at the origin of the appearance of the response of the acoustic speed vac(t) (Figure 2.13.c) and of the transverse displacement (w,t) of the jet in x = w (Figure 2.13.e); -from t = 3x10 -5 s to t = 4x10 -4 s, the pressure p0(t) is zero (the pressure angular frequency is over, Figure 2.13.b), while the transient system of the acoustic speed vac(t) (Figure 2.13.d) and the transverse displacement (w,t) is not finished (Figure 2.13.f); -from t = 4x10 -4 s (value of the order of the delay  = 0.4 ms), under the effect of the looping of the system, a transient system associated with the pressure p0(t) appears (Figure 2.13 .b) leading to an additional transient system for the acoustic speed vac(t) (Figure 2.13.d) and the The remainder of this paragraph is devoted to the detailed analysis of the stationary periodic system present during phase 3.

-Analysis of Auto-Oscillation Conditions

-Hydrodynamic Modes of the Jet and Acoustic Modes of the Resonator

The sound production mechanism is linked to the coupling between the jet and the acoustic resonator. Different playing modes exist according to the hydrodynamic k and acoustic i modes involved. Indeed, like the acoustic resonator (which has several resonance modes), the jet can oscillate on different hydrodynamic modes, shown in Figure 2.15. The normal functioning of the instrument corresponds to the coupling of an acoustic mode i with the first hydrodynamic mode k = 1 of the jet (M P [START_REF] Verge | Sound production in recorderlike instruments. I. dimensionless amplitude of the internal acoustic field[END_REF]. In this case, we observe between the channel output and the bevel about 1/2 hydrodynamic wavelength. The coupling of this hydrodynamic mode with each of the acoustic modes of the resonator gives rise to the registers of the instrument. Thus, the i th register corresponds to the coupling of this hydrodynamic mode k = 1 of the jet with the i th acoustic mode of the resonator. The appearance of one register or another is linked to the choice of parameters controlled by the musician and set by the instrument maker.

(a) (b) (c)

In certain particular cases, the coupling of an acoustic mode with a hydrodynamic mode of order k greater than 1 is observed, and gives rise to a wind system (Kergomard & Chaigne, 2013) [START_REF] Meissner | Aerodynamically excited acoustic oscillations in cavity resonator exposed to an air jet[END_REF] [START_REF] Terrien | Flute-like musical instruments: a toy model investigated through numerical continuation[END_REF]. For a wind system linked to the k th hydrodynamic mode of the jet (with k > 1), we then observe (2k -1) / 2 wavelength along the jet (Figure 2.10). For the k th hydrodynamic mode, this condition on the wavelength is written: .33) where  is the wavelength whose expression is given by  = Thus, for wind systems (k > 1), the delay  is greater than the oscillation period T0, while in normal operation (k = 1)  < T0. In a musical context, the wind systems are obtained for very low values of the pressure Pm(t).

21 2 k w    , ( 2

-Self-Oscillation Conditions

The analysis of the self-oscillation conditions of the loop system (Figure 2.5) sheds light on the mechanisms of appearance of these different types of systems (we can also refer to [START_REF] Terrien | Flute-like musical instruments: a toy model investigated through numerical continuation[END_REF] for more information. details). Such a looped system can enter self-oscillation if the frequency response (j) of the open-loop transfer of the linearized system is equal to 1, i.e. if it satisfies the two self-oscillation conditions [START_REF] Fletcher | Autonomous vibration of simple pressure-controlled valves in gas flows[END_REF]:

-its modulus, u = |(ju)|, is equal to 1, u being the angular frequency at unity gain, -its phase, u = arg[(ju)], is zero or equal to a relative whole multiple n of 2 nZ  . In steady harmonic system, the system of equations (2.26) becomes: .36) from which we easily deduce the expression of (j), that is: .37) with 0 a constant whose expression is given by:   The gain and the phase of (j) therefore ultimately have the following expression: From a physical point of view, the value of n is related to the hydrodynamic mode of the jet. Indeed, from the relation concerning the phase () of the open loop given by relation (2.40), we can verify posteriori that for a given steady harmonic system, the self-oscillation threshold is located at a frequency f0 lower than the resonant frequency. Thus, at 0 = 2f0, it is possible to make the approximation. The self-oscillation condition on the phase of the open loop, namely Then, once the wind system can no longer be obtained, it possible to check the oscillation conditions e j U for registers of higher orders (i.e. for resonance modes of larger order). We go from the first register, to the second, then to the third etc.

              0 1 , , , 1 j ac ac in w j K e V j P j K j w j V j j P j S YL                       , ( 2
    0 , j in j j e j YL         , ( 2
            0 arg a 2 , r , g in in jj j YL Y j L                          . ( 2
  0 arg 2 , in Y j L   then reduces to: 0 2 π n π   , ( 2
These findings correspond to experimental observations (Kergomard & Chaigne, 2013). In recorders, wind sounds have long been considered non-existent, due to the low value of the w/h ratio (Kergomard & Chaigne, 2013). The development of an artificial mouth regulated in pressure (see Chapter 3) makes it possible to demonstrate the appearance of these systems in recorders for particularly low pressure e m P values which are not used by the musician in normal music playing conditions.

For the two examples of illustrations presented in this chapter, the observation of (Figure 2.16) makes it possible to affirm that the self-oscillation conditions in the two cases are obtained for n = 0 (normal operation), that is to say u = 0, which targets frequencies that satisfy the condition: Taking into account the very low values of the damping factors of the input admittance Yin(j,L) of the resonator (see Remark 2.2), the anti-resonance and resonance frequencies are easily identified in (Figure 2.17). They correspond to the intersection of the horizontal dotted line at 90°with the blue phase curve in the areas: -positive slopes for the anti-resonance frequencies; -negative slopes for the resonant frequencies. Thus, still observing (Figure 2.17), it is interesting to note that the two intersections of the black line and the three intersections of the green line with the areas of positive slopes of the blue phase curve correspond to highly unstable operating points. (around anti-resonance frequencies) [START_REF] Auvray | Fréquence de jeu des instruments à embouchure de flutes[END_REF]. Conversely, the two intersections of the black line and the three intersections of the green line with the areas of negative slopes of the blue phase curve correspond to operating points where ranges of playing frequencies (registers) are accessible (around resonant frequencies).

  , arg 2 in YL j      . ( 2
This phase analysis must be completed by an analysis of the gain () of the open loop.

If for a given playing frequency the latter is strictly greater than unity, then a perturbation (w,t) at the operational frequency is amplified. When this disturbance becomes too great, the non-linear phenomena (saturation of the tanh(.) function) provide the conditions for selfoscillation.

-Conclusion

The different elements of the sound production mechanism recalled in this chapter make it possible to capture the main phenomena observed, in particular, in recorder-type instruments. From the complete nonlinear model frequently used in the literature, a solution to improve its digital conditioning is proposed, allowing its programming using MatLab / Simulink. For the defined field of study, a detailed analysis of the simulated time responses highlights the presence of three phases during which the variations of the pressure at the input of the resonator, of the acoustic speed and of the lateral displacement of the air jet remain small around zero. The assumption of small variations being thus perfectly legitimate and realistic, two linearized models, one for the analysis of phase 1 (start of the simulation), the other for that of phase 3 (self-oscillations) are used to further understand the phenomena involved.

However, one should be aware of the limitations of this model when interpreting the results and comparing them with experimental data. This is the subject of Chapters 3 and 4.

Chapter 3-Design and Implementation of an Artificial

Mouth for Wind Musical Instrument

-Introduction

Blowing machines, so-called artificial mouths, are used in musical acoustics when studying wind instruments, at least since 80 years [START_REF] Ferrand | Blowing machine for wind musical instrument: toward a real-time[END_REF]. As proposed in the literature review [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth : the case of recorder-like musical instruments[END_REF] [START_REF] Takanishi | Toward understanding the nature of musical performance and interaction with wind instrument-playing humanoids[END_REF] [START_REF] Maki-Patola | Physics-based modeling of musical instruments for interactive virtual reality[END_REF] [START_REF] Hamilton | Coretet: A Dynamic Virtual Musical Instrument for the Twenty-First Century[END_REF] [START_REF] Paine | New Musical Instrument Design Considerations[END_REF] [START_REF] Chatziioannou | An artificial blowing machine to investigate single-reed woodwind instruments under controlled articulation conditions[END_REF], the objective of controlling the pressure inside the artificial mouth is considered in many applications:

(R. Saar, 2014) (V.
-to ensure quasi-static variation in order to analyze experimental bifurcation diagrams; -to reproduce typical signals (Heaviside step function, sinus, ramps…) to compare the dynamics of the real instrument to the one obtained through numerical simulations; -to mimic the time evolutions recorded on real musicians in order to analyse the strategies discovered over the years of practice.

The objective of this chapter is the study of visco-thermal losses in a wind musical instrument from a hardware-in-the-loop simulation platform. The hardware part of the platform is made up of an automatic blowing machine connected to the mouthpiece of a wind instrument [START_REF] Abou Haidar | Modelling and Identification of the Musicians Blowing Part and the Flute Musical Instrument[END_REF].

The software part of the platform is composed of resonator numerical model of the wind instrument including visco-thermal losses based on fractional model and the Control System (CS) used for regulating the pressure inside the artificial mouth. The first part of the project was the Computer Aided Design (CAD) of the platform. Thus, a digital simulator was developed with MatLab/Simulink software based on wind instruments artificial mouths found in several publications [START_REF] Ferrand | Blowing machine for wind musical instrument: toward a real-time[END_REF] [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth : the case of recorder-like musical instruments[END_REF]. The second part, presented in this chapter, is based on the realization of the test bench from the first part. The objective is to adjust the dynamic behavior of the numerical simulator to the real dynamic behaviour of test bench. The third and last part consists in modelling and analysing the visco-thermal losses present in the resonator of the wind instrument from fractional models [START_REF] Andréa | Spectral and time-domain consequences of an integrodifferential perturbation of the wave PDE[END_REF].

-System Description

The test bench is presented in Figure 3.1. A servo-valve is connected to an air compressor through a pressure reducer. The maximum pressure available is around 6 bars, and the pressure reducer (with its manometer) is used to adjust the pressure P1 upstream the servovalve. The servo-valve is connected at the entrance of the artificial mouth whose internal volume V = 343 cm 3 is the place where the air pressure Pm must be controlled. The artificial mouth blows into the mouthpiece of a recorder flute. A MatLab/Simulink/LabVIEW program is used in order to control the air pressure Pm. Added to that, a flow meter, a temperature and a pressure transducers are used in order to characterize the behavior of the different parts of this system.

Remark

For the rest of the chapter, the following notation is adopted for a variable X(t):

    t x X t X e   , (3.1)
where X e is a constant value fixed by a given operating point and x(t) the fluctuation around

X e . Moreover,   t X ~ represents a measurement of X(t) and   t X ˆ
an estimate of X(t).

-Specifications for the Control

In the classical use of an artificial mouth, the pressure

    t p P t P m e m m  
upstream the mouthpiece of the recorder flute (inside the artificial mouth) is tuned by hand through the pressure reducer and its manometer. When the compressed air is produced by a compressor such as that presented in Figure 3.1, the pressure    

t p P t P e 1 1 1  
upstream the servo-valve, fluctuates because the compressor tank being in need to be recharged once its pressure becomes below a certain level. This is the reason why it is difficult to manually control the pressure. Thus, the purpose of automatic pressure control inside the artificial mouth is to increase the accuracy by rejecting the pressure fluctuation p1(t) considered as a disturbance, while satisfying robust tracking of the reference pressure Pref(t) [START_REF] Ferrand | Blowing machine for wind musical instrument: toward a real-time[END_REF]. To recall, the final objective of this work project is the study of visco-thermal losses in a wind instrument based on fractional model. In order to facilitate the analysis of this complex problem, the reference pressure is chosen such as:

      t f P P t p P t P ref e ref ref e ref ref 0 0 2 cos      .
(

For all these reasons, the architecture of the control system presented in Figure 3.4 consists of a Pref(t) reference generator, a Uff(t) feedforward control and a Ufb(t) feedback control, the robust controller of which is designed with the CRONE methodology [START_REF] Oustaloup | La dérivation non entière : Théorie, synthèse et applications[END_REF] [START_REF] Oustaloup | La commande CRONE[END_REF] (P. [START_REF] Lanusse | CRONE control system design toolbox for the control engineering community: tutorial and case study[END_REF] [START_REF] Lanusse | De la commande CRONE de première génération à la commande CRONE de troisième generation[END_REF] [START_REF] Lanusse | CRONE Control System Design, a CRONE toolbox for Matlab[END_REF].

-System Modelling and Validation

As presented in Figure 3.3, the experimental setup is divided into two parts:

-the first one consists of a voltage-current amplifier, a servo-valve connected to an air compressor via a pressure reducer. The inputs of this part are P1(t) the pressure upstream the servo-valve and Uc(t) = Uff(t) + Ufb(t) the control signal generated from MatLab/ Simulink/ LabVIEW, whereas the output is Qsv(t) the flow rate delivered by the servovalve. This last device presents a nonlinear behavior; -the second part is the artificial mouth and the mouthpiece of the recorder flute. The input of this part is Qsv(t) the flow rate from the servo-valve, whereas the output is

  t P m ~ the pressure measured within the artificial mouth,       t n t P t P m m   ~
where n(t) is the measurement noise.

-Modelling and Validation of the Servo-Valve

The servo-valve is designed by Bürkert firm (ref. Bürkert 2871). Based on previous works [START_REF] Ferrand | Blowing machine for wind musical instrument: toward a real-time[END_REF] [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth : the case of recorder-like musical instruments[END_REF][START_REF] Abou Haidar | Robust Control of an Artificial Mouth for a Wind Musical Instrument[END_REF]) (G. Abou Haidar, 2019), it has been shown that the servovalve's behavior can be divided in two parts: a nonlinear static part and a linear dynamic part. The linear dynamic part between Qstat and Qsv(t) is represented by a second order transfer function Hsv(s) with a unit static gain [START_REF] Ferrand | Blowing machine for wind musical instrument: toward a real-time[END_REF] [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth : the case of recorder-like musical instruments[END_REF]

    2 0 0 / / 2 1 1    s s s H sv    , ( 3.3)
where  = 0.3 and ω0 = 2π 240 rad/s.

Nonlinear Static Part

Q stat

Static volume flow of servo-valve [m 3 /s]

Linear

Dynamic

Part

Q sv (t) U c (t) Control voltage [V] Dynamic volume flow of servo-valve [m 3 /s] P 1 (t)
As already mentioned, the output flow rate Qsv = f(Uc, P1) depends on the control voltage signal Uc (that may vary between 0 and 10V) and the pressure P1 upstream the servo-(max[P1] = 6 bar). , the flow rate Qsv(t, Uc, P1) can be written as follow:

      1 1 1 , , , , , P U t q P U Q P U t Q c sv e e c e sv c sv   , (3.4)
where

          t p P U K t u P U K P U t q e e c qp c e e c qu c sv 1 1 1 1 , , , ,   , ( 3
P U K u Q P U K 1 1 1 1 1 , , ,       . ( 3.6)
The estimated value qu K ˆ of the static gain Kqu is obtained from:

              1 min 1 max 1 .V l.mn 1 max 0 , max 1 1 P U U P U Q P U Q P K c sv c sv qu         , ( 3.7)
with Umax = max[Uc] = 10 V.

      B P A P K qu     bar 1 .V l.mn 1 1 1 ˆ, (3.8) with       1 - 1 - 1 -1 -1 .V l.mn 4 . 3 .bar .V l.mn 99 . 0 B A .
(3.9) The non-linear static part and the linear dynamic part of the servo-valve are integrated in a digital simulator programmed with MATLAB / Simulink.

-Choosing an Operating Point

The limits of the servo-valve operating domain being estimated, the choice of an operating point

  e e c P U O 1 ; 
and the amplitude of the variations around this point is essential in order to avoid the risks of saturation of the flow rate Qsv(t).

After observing the pressure P1(t) during numerous tests, the fluctuations p1(t) are considered to be limited between -1 and + 1bar. This is the reason why we chose and Umax = 10 V, so:

      V 10 V 55 . 7     t U t U t U fb ff c , (3.10) with     0 0 2 cos f U U t U c e c ff
                       V 55 . 7 V 8 min V 10 V 9 max min 0 max 0 U U U t U U U U t U c e c ff c e c ff , (3.12)
and the variation range of the feedback control signal Ufb is given by:

                             V 45 . 0 min min V 1 max max

-Modelling and Validation of the Artificial Mouth

The artificial mouth is implemented using a cubic box whose input is the flow rate Qsv(t) coming from the servo-valve and the output is the pressure Pm(t) inside the box. The expression of the pressure Pm(t) is given by the state equation of perfect gases:

    t M V T r t P m  , (3.14)
where r = 287 J.kg -1 .K -1 , the thermodynamic constant of air; -T =293.5°K, the temperature of air inside the box; -M(t) is the mass of the air inside the box of volume V. This value depends on the variation of the flow between the input (represented by Qsv(t)) and the output (represented by Qmp(t)). As small variations are considered, the air density ρ is considered to be constant. Thus, the expression of M(t) is given by: .15) where the expression of Qmp(t) is given by Bernoulli law [START_REF] Ferrand | High-precision regulation of a pressure controlled artificial mouth : the case of recorder-like musical instruments[END_REF]:

          0 0 M d Q Q t M t mp sv        r , ( 3
    t P t Q mp mp    , (3.16)
where α is a coefficient estimated from measurements and ΔPmp(t) = Pm(t) -Patm (Patm is the relative atmospheric pressure which is zero by definition, so ΔPmp(t) = Pm(t)). By introducing in relation (3.15) the pressure Pm(t) from relation (3.14), namely:

          0 0 M V T r d Q Q V T r t P t mp sv m        r , (3.17)
we obtain .18) where Cam is the pneumatic capacity associated with the volume V of the artificial mouth given by:

          0 1 0 m t mp sv am m P d Q Q C t P        , ( 3
T r V C am r  , ( 3.19)
and Pm(0) the initial value of the pressure Pm(t) given by , and for a quasi-static variation

    0 0 M V T r P m  . ( 3
    t Q t Q sv mp  .
Figure 3.9 shows the plot of Qsv(t) versus Pm(t) 0.5 for a quasi-static variation. The linear fitting (in red) of the measurements (x) leads to an estimate  ˆ given by: From relations (3.16), (3.17) and (3.20), the behavior of the artificial mouth is integrated in the digital simulator programmed with MATLAB / Simulink.

-Linearized Model of the Plant

In order to design the CRONE controller in frequency domain, a linearized model of the plant is derived around the operating point O. Thus, the pressure Pm(t) and the flow rate Qmp(t) can be expressed as follows:

                 t q Q t Q t p P t P
q sv (t) q c (t)  t d 0 .  am C 1 + - q mp (t) p m (t)   dt t dp m mp R 1 The transfer function H1(s) between       t p s P m m LT  and       t q s Q sv sv LT 
, where LT represents Laplace Transform, is given by: as follow:

      1 0 1 / 1 1  s H s C R R s Q s P s H am mp mp sv m      , ( 3
         1 2 0 0 1 0 1 / 1 / / 2 1 ,     s s s P G P s G     , (3.27)
where G0(P1) = Kqu(P1) H0. (3.28) For the design of the robust control law, the main parameter's values used are:

                          
rad/s 8 . 17

-Controller Design

The feedforward part is based on the nominal inverse static gain of the linearized model (G0_nom = G0(5 bar)) used for the design of the feedback control law, namely:

    t P G t U ref nom ff 1 _ 0   .
(3.31)

-User Specifications

The user specifications of the control system defined from a preliminary work [START_REF] Ferrand | Blowing machine for wind musical instrument: toward a real-time[END_REF] [START_REF] Lanusse | De la commande CRONE de première génération à la commande CRONE de troisième generation[END_REF] are the following:

-a phase margin MØ > 40°; -an open-loop gain crossover frequency ωu  2π 10 rad/s; -a steady-state error equal to 0; -a variation range of Ufb given by

V 1 V 45 . 0    fb U .

-CRONE Control-System Design (CSD) Methodology

The CRONE CSD methodology is a frequency-domain approach developed since the eighties [START_REF] Oustaloup | La dérivation non entière : Théorie, synthèse et applications[END_REF] [START_REF] Oustaloup | La commande CRONE[END_REF] [START_REF] Lanusse | De la commande CRONE de première génération à la commande CRONE de troisième generation[END_REF]. It is based on the common unity-feedback configuration presented in Figure 4. Three CRONE CSD methods have been developed, each one of them denotes a generation of CRONE design. The general

                            / 1 / 1 / 1 / / 1 ) ( 0 . (3.32)
The first part of the above equation (3.32) represents the behavior at low frequencies with an integer order nl, the second represents the behavior at middle frequencies with non-integer order n varying between 1 and 2 around ωu, and the last represents the behavior at high frequencies with an integer order nh. As for the gain β0, it is defined by [START_REF] Lanusse | De la commande CRONE de première génération à la commande CRONE de troisième generation[END_REF]:

              2 / 2 2 / 2 0 / 1 / 1 / n n h u n n l u n l u h l l             . (3.33) With MØ = 45°, ωu = 2π10 rad/s,   Pa/V 5 . 412 ; 325 0  G
and in accordance with the methodology described in [START_REF] Oustaloup | La commande CRONE[END_REF] When the nominal open-loop transfer is determined, the fractional controller CF(s) is defined by its frequency response:

          j G j j C nom nom F /  . (3.35)
The synthesis of the ideal frequency response CF(j) consists of identifying a rational frequency response CR(j) given by: (3.36) where B(jω) and A(jω) are polynomials of specified integer degrees n B and n A . All the frequency-domain system identification techniques can be used [START_REF] Oustaloup | La commande CRONE[END_REF]. Figure 3.12 presents the Bode plot of the controller CR(jω). As one can observe, the phase margin MØ (Figure 3.13.b) and the resonant peaks QT of T(jω) (Figure 3.14.a) and QS of S(jω) (Figure 3.14.b) remain constant for all the cases thus showing the robustness of stability degree [START_REF] Oustaloup | La commande CRONE[END_REF].

         j A j B j C R /  ,

-Time Domain

The reference pressure Pref(t) is chosen such as: It is important to note that the gain of the feedforward part is calculated only for the nominal case P1 = 5 bar and that it is not adjusted when P1 varies upstream the servo-valve. We observe that the robust feedback and feedforward control system ensures a good pressure tracking (b) (d), not only for the nominal case (P1 = 5 bar), but also for the minimal (P1 = 4 bar) and maximal (P1 = 6 bar) cases. Without the robust feedback control system (a) (c), pressure tracking is less effective. In all cases, the control signal Uc(t) (e) (f) remains within the variation range defined by Umin and Umax.

                          e ref

-Conclusion

The first important work of this chapter was to mathematically describe, model, and simulate the linear and nonlinear parts of the servo valve and the artificial mouth and try to linearize their modeled equations. Moreover, understanding the influence of pressure P1 on the limits of the servo-valve operating range is essential in order to be then able to define a strategy for automatic pressure Pm control inside the artificial mouth. The second important step is the design of the control architecture for a robust control of the pressure Pm. The application of the CRONE system design methodology was able to achieve the target set with very good dynamic performance and respecting the linear operating range of the servo valve.

Chapter 4-Analysis of the Influence of Fractional

Order on the Stationary Periodic System

-Introduction

This chapter 4 uses the developments of chapters 1, 2 and 3 in order to analyze the influence of the fractional order m on the stationary periodic system. As a reminder, Figure 4.1 illustrates the progression and sequencing of the chapters of this thesis. Thus, from the extension of the fractional model proposed in chapter 2, 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 dimensions (length L and radius r), roughness (or surface condition) and different materials (wood, plastic, etc.). The artificial mouth simulator has undergone a readjustment with respect to the test bench developed specifically for this study. The regulation loop makes it possible, in particular, to limit the sensitivity of the pressure Pm(t) within the pneumatic capacity to variations in the supply pressure P1(t) of the servo valve. As part of the analysis of the influence of fractional order on the stationary periodic system, the reference pressure Pref(t) is a constant equal to 400 Pa (example 1) and 1000 Pa (example 2).

The exciter simulator, on the other hand, is the one developed in Chapter 2. The fractional order is present in the acoustic tube model studied in Chapter 1 as a resonator. In addition, the expression (4.1) of Yin(s) can be put in a form YN(s) composed of an integrator of order 1 in cascade with N fractional cells of order 2, namely:

    lim in N N Y s Y s   , (4.5) with   2 2 0 2 2 1 12 12 m N zi zi zi N m i pi pi pi s s Y Ys s s s                                                            , ( 4
                          , ( 4.8)
where czi and cpi are constants resulting from the procedure allowing to pass from Yin(s) to YN(s). This procedure, presented in appendix A, is composed of 4 steps:

-Step 1: finite expansion of the function tanh(z) into a continuous fraction in z;

-Step 2: reduction of the continuous fraction to a ratio of two polynomials in z;

-Step 3: factorization of each polynomial in z;

-Step 4: rewriting Yin(s) in the form YN(s), namely an integrator of order 1 in cascade with a product of N fractional cells of order 2.

From a numerical point of view, there is no limitation on the choice of the number of cells N, the precision being the better the number N is high. Theoretically, N must be infinite. Thus, any finite value of N causes an edge effect on the last cell linked to the truncation. This is the reason why we must choose an expansion of order N + 1 to have good precision with N cells.

From an analytical point of view, the search for the roots of the polynomials obtained in step 2 (research necessary for factorization at the level of step 3) limits N to 2. Taking into account the edge effects, the analytical developments do not can only be used for the first cell which corresponds to the first mode of the resonator. However, the analysis of expressions (4.6) and (4.8) shows that each cell of rank i is completely defined as soon as the two constants czi and cpi are known. Based on this observation, a specific method is proposed in 3 steps.

Step 1 consists of determining the angular frequencies zi and pi solutions of the equation: (4.9) knowing that the phase curve alternates between -90 ° and + 90 ° (Figure 4.4), and that the zi correspond to solutions for which the slope of the phase curve of Yin(j) is positive (anti-resonance), i.e.: Given the complexity of the expression of Yin(j), the search for the roots is done numerically at the scale of each cell of rank i.

  arg 0 in Yj   ,
Then, for a given angular frequency L, the constants czi and cpi are calculated in step 2 in accordance with relations (4.8), namely:

and zi zi L pi pi L c c                . (4.12)
Finally, for a given order m, step 3 makes it possible to calculate the damping factors zi and pi of YN(s) in accordance with relations (4.8), namely:

    0 0 2 2 m zi zi m pi pi KL m m rc KL m m rc                     . ( 4

Remark

The denominator Den(s) of YN(s) is defined by:

    1 N i i Den s s Den s    , (4.14) with   2 2 12 m i pi pi pi s s Den s                        . ( 4

.15)

The roots of Den(s) are solution of Den(s) = 0. Apart from the obvious root s = 0, the search for the roots is done by solving .16) which amounts to looking for the roots of each Deni(s) polynomial.

  1 0 N i i Den s    , ( 4
The interested reader will find in the thesis [START_REF] Ivanova | Identification de systèmes multivariables par modèle non entier en utilisant la méthode des sous-espaces[END_REF] developments concerning, in particular, the search for the roots of a polynomial of the form of Deni(s). and that the ratio pi/zi depends on cpi, czi and m, that is to say: Like the recursive factors  and  used by Oustaloup [START_REF] Oustaloup | La dérivation non entière : Théorie, synthèse et applications[END_REF] to establish the link between non-integer derivation and frequency recursion, ratios i and i (whose values depend here on rank i) are introduced:

1 1 1 1 i i i i pi pi i zi zi zz i pp c c c c                        . ( 4

.19)

Taking into account relations (4.18) and ( 4.19), the ratio of the damping factors zi and pi of the same cell of rank i is then written only as a function of i and of the order m, that is to say: (4.20) and the passage relations of the parameters of the cell of rank i and those of the cell of rank i + 1, are given by -the resonance frequencies pi and anti-resonance zi are independent of the order m and increase with rank i;

1 1 i i p m z i     ,
    1 1 1 1 1 1 ; 1 1 1 ; 1 i i i i i i i i z z ii m z z ii p p ii m p p ii                                . ( 4
-the damping factors depend on the order m and decrease with rank i for a fixed m. Moreover, the expressions (4.13) of z1 and p1 clearly show that for a given rank i and an order m, the damping factors are all the greater (and therefore the visco-thermal losses) as the length L of the acoustic tube is large and its radius r is small. As a reminder, its frequency response (j) in steady harmonic system resulting from the linearization carried out in Chapter 3 has the expression:

    0 in j j j e j Y         , ( 4.22)
with 0 a constant whose expression is given by:   In addition, for this linearized loop system within the scope of study of this thesis work, the two auto-oscillation conditions are: and

            0 arg arg 2 in in jj j Y Y j                          . ( 4
        0 1 arg arg 0 2 n u u u u uu i u n u i Y Y jj jj                             , ( 4

.25)

u being the angular frequency at unity gain.

In fact, as already specified in Chapter 3, the stationary periodic system observed in the time domain (see following paragraph), in particular in example 1 (Pm = 400 Pa) and example 2 (Pm = 1000 Pa), does not result from these self-oscillation conditions. This system is the result of an unstable behavior (in closed loop) which associated with the nonlinear phenomenon of saturation of the tanh(.) function leads to this steady periodic system. These conditions of stationary periodic system (in closed loop) result in open loop by a gain  strictly greater than the unit with the angular frequency  for which the phase  is zero, that is to say:

If at  = ,             0 0 0 0 0 0 0 0 0 arg arg 0 2 1 in in Y jj jj Y                               , ( 4
.26)

then the system is unstable (closed loop).

For the five values of the order m (0; 0.25; 0.5; 0.75; 1), Figures 4.12 The observation of the frequency responses (j) of the open loop, in particular in the Black-Nichols plane with Figure 4.13, makes it possible to affirm that for the two values of the pressure Pm = 400 Pa and 1000 Pa, and whatever the value of m between 0 and 1, at the angular frequency 0 for which the phase of the open loop is equal to 0° (horizontal line in blue on the phase diagrams of Figures 4.12.a and 4.12. b), the open loop gain value is greater than 1 (i.e. 0 dB).

As a reminder, the analysis of the stationary periodic system presented in Chapter 3 shows that for a constant pressure Pm at the entrance of the mouthpiece (that is to say at the level of the artificial mouth) belonging to a very precise interval, the angular frequency 0 (for which the phase 0 is zero) is very close to the angular frequency pi of the mode i concerned. To illustrate graphically the search for the value of 0, the condition 0 = 0 is rewritten in the form:

    0 0 0 0 arg 2 in j Y           , ( 4

-Time Domain Illustration of the Influence of Fractional Order

As a reminder (see Chapter 3), the time simulations carried out at Pm = cst reveal three phases:

phase 1: start-up with a convergent transient system;

phase 2: divergent transient system;

phase 3: stationary periodic system.

The objective of this paragraph is to illustrate through time simulation the influence of order m on the periodic system of phase 3.

Thus, for this periodic system of phase 3, Figures 4.17 -transverse displacement (w, t) of the jet (Figure 4.17), -the pressure p(t) at the input of the resonator (Figure 4.18), -the acoustic speed vac(t) (Figure 4.19).

To help the reader to identify the variables (w,t), p(t) and vac(t), the functional diagram of the recorder is recalled at the beginning of each of these figures with the concerned variable in red.

Observation of all of these time responses leads to the following remarks.

For a given pressure Pm and for the five values of the order m (0; 0.25; 0.5; 0.75; 1), the amplitudes of the time responses are all maximum for m = 0 (conservative case) and minimum for an order m included between 0.5 and 0.75.

Although one cannot analytically establish a link between open-loop behavior and that observed in closed-loop due to the non-linear character, a trend emerges regarding the influence of the order m on the amplitudes of the time responses. Indeed, we observe in the time domain a similarity with the influence of the order m on the open loop gain (0) (Figure 4.16). An analysis based on the method of the first harmonic or a decomposition in series of Volterra, for example, is a prospect which should make it possible to confirm this tendency.

-Conclusion

One of the most important contributions of this chapter is related to the impedance of the resonator with the development of a method in several stages allowing to pass from the form Yin(s) (relation (4.1)) from Chapter 2 to the form YN(s) (relation (4.5)). Indeed, even if the form Yin(s) remains the reference for the plot of the frequency responses, the form YN(s) makes it possible to reveal in an explicit way the parameters zi, zi, pi, pi which characterize the resonator modes.

The step of determining the constants czi and cpi (relations (4.12)) makes it possible to define each cell of rank i from the 4 parameters z1, z1, p1, p1 of the cell of rank 1 (obtained using an analytical development: Appendix A) and the 2 factors i and i (relations (4.19)) of the cell of rank i (obtained numerically). This method, allowing to switch from the form Yin(s) to the form YN(s), facilitates the analysis of the influence of the order m. Thus, it is demonstrated that the order m:

-has no influence on the angular frequencies zi and pi, -has an influence on the damping factors zi and pi.

More precisely, for a given rank i and for a variation of the order of m from 0 to 1, the damping factors are zero for m = 0 (conservative case), go through a maximum, then decrease. In addition, for a fixed order m, the damping factors decrease with rank i.

Note that the expressions (4.8) of zi and pi clearly show that for a given rank i and an order m, the damping factors are all the greater (and therefore the visco-thermal losses) as the length L of the acoustic tube is large and its radius r is small.

Finally, the analysis of the influence of the order m on the conditions of existence of the stationary periodic system observed in the time domain is an important contribution which completes the first analysis presented in Chapter 3. Indeed, these conditions are reflected in open loop by a gain  strictly greater than the unit with the angular frequency  for which the phase  (of the open loop) is null. Thus, for a given order m, the value of the open loop gain (0) (greater than unity) increases with the pressure Pm at the mouthpiece input (i.e. in the artificial mouth) and for a given pressure Pm, the curves of (0) as a function of the order m have a different local minimum depending on the value of Pm.

General Conclusion and Perspective

The interpretation and discussion of the main results presented in this thesis are the subject of the general conclusion.

Chapter 1 constitutes the first essential modeling step for the reader unfamiliar with wind instruments. Indeed, due to its structuring and its didactic progression, it allows to "soak up" the dynamic behavior of an acoustic tube of constant radius including the effect of viscothermal losses due to the boundary layers in the vicinity of the walls. The knowledge model used in this work, called Webster-Lokshin 1D, is mono-spatial dependent. It characterizes the linear propagation of acoustic waves in axially symmetrical tubes taking into account viscothermal losses. Thus, in an axisymmetric tube of constant section, the sound pressure and the sound flow rate are governed by the horn equation, known as Webster-Lokshin, and the Euler equation. This dissipative model includes a term that involves a fractional derivation of order m = 0.5. The classical resolution in the operational domain of the system of partial differential equations leads to the analytical expression of the acoustic impedance and admittance of the tube as a function of its length L, of its radius r and of the position x of the tube point considered.

One of the contributions of this chapter is the introduction of an extension of the fractional expression used to take into account visco-thermal losses where the order m is usually equal to 0.5. This extension makes it possible to consider an uncertainty of the order m, whose domain bounds are 0 (conservative system) and 1. Another contribution is the introduction of a system vision aiming to causally decompose the global model into sub models, to facilitate analysis in the frequency domain. One of the conclusions of this frequency analysis is that the fractional model can be simplified over the audible frequency range [20; 20,000]. Finally, for the simulation in the time domain, two rational forms composed of an integrator of order one and N cells of the second order, one in cascade and the other in parallel, are introduced as a model of behavior. The parametric values of the rational cascade form are determined using the Frequency Domain System Identification (FDSI) module of the CRONE Toolbox with the frequency response of the knowledge model as target. As for the parametric values of the parallel form, they are obtained by a decomposition into simple elements of the cascade form.

Chapter 2 falls within the framework of the dynamics of complex systems. Indeed, the self-oscillation mechanism, characterized by the ability of the instrument to produce an acoustic wave from a stationary or quasi-stationary energy source with respect to the acoustic variables, is inseparable from the non-linear nature of the exciter and its coupling with the resonator. This phenomenon results from the oscillation of a naturally unstable air jet around a bevel. Thus, the self-oscillation mechanism relies on a synchronization of the oscillation of the jet and the acoustic waves. The jet-bevel system must therefore excite the resonator at the periodicity of the acoustic field.

From a bibliographic synthesis, a complete nonlinear model frequently used in the literature is developed in this chapter. Faced with the poor digital conditioning of this model, a solution is proposed in order to be able to develop a digital simulator programmed under MatLab / Simulink. This solution constitutes a first contribution in this chapter. A scenario is then defined while respecting, not only the domain of validity of the nonlinear model, but also the values of pressure attainable experimentally by the artificial mouth presented in chapter 4. The detailed analysis of the simulated time responses makes it possible, in particular, to observe that the variations of a certain number of physical quantities (pressure at the input of the resonator, acoustic speed, etc.) are small and around zero. This observation allows us to legitimize the realism of the hypothesis of small variations. Thus, two linearized models, one for the analysis of phase 1 (start of the simulation), the other for that of phase 3 (self-oscillations) are developed and used to better understand the phenomena with the selfoscillation conditions.

Chapter 3 presents the model-based design approach of an artificial mouth, its realization and the experimental characterization of each of its components, in order to adjust the nonlinear models integrated in the digital simulator associated with this part. The use of an artificial mouth, in the context of comparative studies of the complex dynamic phenomena present in wind instruments, is necessary to ensure reproducibility of test conditions that are difficult to achieve with a human being. This three-step engineering approach is essential to fully understand the non-linear behavior of each component, define the limits of their operating range and establish the influence of different physical quantities on the pressure at the mouthpiece input. Note that this last point is essential to understand the need for a pressure regulation loop. Thus, the first step in this approach was the development of a first simulator on the basis of a bibliographic synthesis concerning artificial mouths used in musical acoustics. This first simulator made it possible to fully understand the operation of such a system, thus facilitating the choice and sizing of the components of the regulation loop (the servo valve, pressure and flow sensors, temperature sensors, the volume of the pneumatic capacity, etc.), as well as the summary of the pressure control (feedforward and feedback parts). The second step was the realization of the test bench with the assembly of the various parts and their tuning. Finally, the third step consisted of a second iteration, not only of modeling applied precisely to the experimental device produced, but also of synthesis of the pressure control. This step lead to a second digital simulator. The latter was the subject of a retiming from comparisons between experimental and simulation results. Thus, for a real recorder (made of plastic with a length of the resonator L = 30 cm, an average radius r = 5 mm, a fractional order associated with visco-thermal losses m = 0.5, ...), the simulator reproduces with a good precision the study conditions specified in chapters 2 and 3. This very real behavior (reproduced in simulation) is considered in chapter 4 as the nominal reference behavior.

Finally, Chapter 4 brings together the developments and results of the first three chapters to deepen the analysis of the influence of the fractional order m on the mechanism leading to the stationary periodic system.

One of the strong points of this chapter is the development of a method allowing to pass from the initial fractional form of the Yin(s) impedance from chapter 1, where the modes of the resonator appear implicitly through the periodicity from the tanh(.) function, to another fractional form YN(s) where, on the contrary, the modes of the resonator appear explicitly through the product of N fractional cells of order 2. Indeed, even if the form Yin(s) remains the reference for the plot of the frequency responses, the form YN(s) makes it possible to reveal in an explicit way the parameters zi, zi, pi, pi which characterize the modes of the resonator. This method facilitates the analysis by showing that the order m has no influence on the angular frequencies zi and pi, but that it does have an influence on the damping factors zi and pi. For a given rank i and for a variation of the order of m from 0 to 1, the damping factors are zero for m = 0 (conservative case), go through a maximum, then decrease. In addition, for a fixed order m, the damping factors decrease with rank i. Finally, the analysis of the influence of the order m on the conditions of existence of the stationary periodic system observed in the time domain using the complete simulator (artificial mouth + nonlinear exciter + resonator) is an important contribution. which completes the first analysis made in chapter 2. Indeed, these conditions result in open loop by a gain  strictly greater than the unit with the angular frequency  for which the phase  (of the open loop) is zero. Thus, for a given order m, the value of the open loop gain  (greater than unity) increases with the pressure Pm at the entrance to the mouthpiece (i.e. in the artificial mouth) and for a given pressure Pm, the curves of () as a function of the order m have a different local minimum depending on the value of Pm.

More generally in terms of fractional models, this study of the influence of order m on visco-thermal losses leads to a finding similar to that already made in other fields. Indeed, the main interest of the fractional form lies in the parametric parsimony, that is to say the capacity possessed by the integro-differential operator of non-integer order to model with a minimum of parameters the greatest number of dynamic phenomena.

The medium-term outlook falls directly within the continuity of the work in progress, that is to say within the framework of the dynamics of complex systems.

From a modeling point of view, a first perspective concerns the introduction of the extension of the fractional expression used to take account of visco-thermal losses in the more general framework of the pavilion equation as encountered in the literature of wind instruments. Indeed, in this case, the radius of the acoustic tube depends on the longitudinal position x of the point considered within the resonator. In addition, its apparent length may vary depending on the number of closed holes. Thus, taking into account a fractional order m between 0 and 1, associated with variations in radius and apparent length, should make it possible to extend the domain of validity of the model developed in Chapter 1.

At the same time, the increase in the field of study defined in chapters 2 and 4 by considering a pressure Pm at the entrance to the mouthpiece, no longer constant, but variable (like that generated by a musician) constitutes a second perspective. To achieve this goal, it is necessary to solve the problem of causality due to the presence of a shunt in the model used to describe the interaction between the air jet and the bevel. The solution proposed in this thesis, namely the use of a frequency bounded derivation operator, only makes it possible to better condition the model numerically for a constant pressure Pm from the initial moment of the simulation.

From a systems theory point of view, an interesting perspective is the definition of a fractional resonator with one degree of freedom (1 dof). The first step will consist in defining its structure from the decomposition into simple elements of the product of the N fractional cells of order 2 present in the expression of the acoustic impedance YN(s) introduced in chapter 4. Then, the structure being perfectly defined, the study of the fundamental properties of this fractional resonator with 1 dof will constitute the second stage. The progression proposed within the framework of systems whose dynamics are described by differential equations of integer orders, with first the study of systems with 1 dof, then that of systems with q dof which in the modal space are in the form of q 1-DOF systems, will be generalized to SDNE (Non-Integer Derivative Systems which include the dynamic systems of integer orders).

Finally, from an experimental point of view at the level of the artificial mouth, the integration of components designed specifically for the medical sector, in particular those used in automatic respiratory assistance devices (servo valves, sensors, etc.), should make it possible to improve the performance of the test bench. -Step 4: rewriting Yin(s) in the form YN(s), namely an integrator of order 1 in cascade with a product of N fractional cells of order 2.

                                                         
Step 1

The finite expansion of the function tanh(z) into a continuous fraction in z is given by [START_REF] Wyman | An essay on continued fractions[END_REF]   (A.9)

Step 2

The reduction of the continued fraction defined by relation (A.8) leads to an expression in the form of a product of the variable z and a ratio of two polynomials in z voluntarily limited to an order of 4/4 for the analytical developments, namely: (A.12)

     
Step 3

Factoring each polynomial in z requires finding their roots. To do this, we perform the change of variable by setting X = z 2 , hence the system to be solved: (A.13)

Solving this system made up of two quadruple equations in X does not present any particular difficulty. Each polynomial in X has two distinct real roots, namely:

-for the numerator: .20)

                                                                                                                                         2 2 , , , , 1 1 15 2
                                                                                                                                                                    
                                                            , ( A

Figure 1 -

 1 Figure 1 -Chronogramme illustrant 13 ans de collaboration avec le Liban

Figure 2 -

 2 Figure 2 -Illustration de la progression et de la logique d'enchainement des chapitres du mémoire de thèse

Figure 1 -

 1 Figure 1 -Chronogramme illustrant 13 ans de collaboration avec le Liban .......................................... Figure 2 -Illustration de la progression et de la logique d'enchainement des chapitres du mémoire de thèse...................................................................................................................................................Figure 3 -Common Woodwind Instruments ........................................................................................ Figure 4 -Common Brass Instruments ................................................................................................. Figure 5 -Overall Flute System ............................................................................................................ Figure 6 -Traditional flute musical instrument .................................................................................... Figure 7 -Artificial loop based on the ear of the musician .................................................................. Figure 8 -Types of flutes: (a) Class D, (b) Classical Flared and (c) Baroque ......................................... Figure 9 -Block diagram of the mechanism of sound production in auto-oscillatory musical instruments ........................................................................................................................................... Figure 10 -Closed-Loop system of the self-oscillation mechanism of flute family instruments ......... Figure 11 -Example of the musical wind instrument array ................................................................. Figure 12 -Diagram of Keefe's flute ..................................................................................................... Figure 13 -Trumpet-like instrument-Decomposition into elementary subsystems ........................... Figure 14 -Diagram of copper type instrument operation coupled to the musician (first resonance) .............................................................................................................................................................. Figure 1.1 -One-dimensional description of an acoustic tube of radius r = constant and of finite length L subjected to an acoustic flow Qv(t) with x = 0 ........................................................................ Figure 1.2 -Curves rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f ............................................................................................................................................ Figure 1.3 -Bode Diagrams of Z(0,jω,L) on the range [20 ; 20 000] Hz of the frequencies audible by the human ear (a) and on the range [20 ; 4 000] Hz of frequencies attainable with a recorder (b) .... Figure 1.4 -Block diagrams associated with the system approach: Whatever x is between 0 an L (a), .............................................................................................................................................................. Figure 1.5 -Bode diagrams of Im (jω) on the range [10 -4 ; 10 4 ] Hz ........................................................ Figure 1.6 -Frequency response of Im(jω) on the range [20; 20,000] Hz of the audible frequencies . Figure 1.7 -Bode diagrams of F(0, jω,L) on the range [10 -4 ; 10 4 ] Hz .................................................... Figure 1.8 -Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) on the range [10 -4 ; 10 4 ] Hz (a) and on the range [20; 4000] Hz of audible and achievable frequencies with a recorder (b) .......... Figure 1.9 -Bode diagrams in x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and in x = 3L/4 (ωLx/2π = 735 Hz) of x = 3L/4 (ωLx/2π = 735 Hz) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) in the range [20; 4000] Hz of audible and achievable frequencies .......................................... Figure 1.10 -Bode diagrams at x = 0 of H(0, jω,L) for different values of the fractional order m on the range [20; 4000] Hz of audible and achievable frequencies with a recorder ....................................... Figure 1.11 -Reduced frequency responses H(0, jω,L) / H0 with the frequency axis on a linear scale over the range [150; 750] Hz ................................................................................................................ Figure 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c) .......... Figure 1.13 -Screenshots from the CRONE Toolbox before optimization (a) and after optimization (b) at x = 0 in the nominal case m = 0.5 .............................................................................................

Figure 3 -

 3 Figure 1 -Chronogramme illustrant 13 ans de collaboration avec le Liban .......................................... Figure 2 -Illustration de la progression et de la logique d'enchainement des chapitres du mémoire de thèse...................................................................................................................................................Figure 3 -Common Woodwind Instruments ........................................................................................ Figure 4 -Common Brass Instruments ................................................................................................. Figure 5 -Overall Flute System ............................................................................................................ Figure 6 -Traditional flute musical instrument .................................................................................... Figure 7 -Artificial loop based on the ear of the musician .................................................................. Figure 8 -Types of flutes: (a) Class D, (b) Classical Flared and (c) Baroque ......................................... Figure 9 -Block diagram of the mechanism of sound production in auto-oscillatory musical instruments ........................................................................................................................................... Figure 10 -Closed-Loop system of the self-oscillation mechanism of flute family instruments ......... Figure 11 -Example of the musical wind instrument array ................................................................. Figure 12 -Diagram of Keefe's flute ..................................................................................................... Figure 13 -Trumpet-like instrument-Decomposition into elementary subsystems ........................... Figure 14 -Diagram of copper type instrument operation coupled to the musician (first resonance) .............................................................................................................................................................. Figure 1.1 -One-dimensional description of an acoustic tube of radius r = constant and of finite length L subjected to an acoustic flow Qv(t) with x = 0 ........................................................................ Figure 1.2 -Curves rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f ............................................................................................................................................ Figure 1.3 -Bode Diagrams of Z(0,jω,L) on the range [20 ; 20 000] Hz of the frequencies audible by the human ear (a) and on the range [20 ; 4 000] Hz of frequencies attainable with a recorder (b) .... Figure 1.4 -Block diagrams associated with the system approach: Whatever x is between 0 an L (a), .............................................................................................................................................................. Figure 1.5 -Bode diagrams of Im (jω) on the range [10 -4 ; 10 4 ] Hz ........................................................ Figure 1.6 -Frequency response of Im(jω) on the range [20; 20,000] Hz of the audible frequencies . Figure 1.7 -Bode diagrams of F(0, jω,L) on the range [10 -4 ; 10 4 ] Hz .................................................... Figure 1.8 -Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) on the range [10 -4 ; 10 4 ] Hz (a) and on the range [20; 4000] Hz of audible and achievable frequencies with a recorder (b) .......... Figure 1.9 -Bode diagrams in x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and in x = 3L/4 (ωLx/2π = 735 Hz) of x = 3L/4 (ωLx/2π = 735 Hz) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) in the range [20; 4000] Hz of audible and achievable frequencies .......................................... Figure 1.10 -Bode diagrams at x = 0 of H(0, jω,L) for different values of the fractional order m on the range [20; 4000] Hz of audible and achievable frequencies with a recorder ....................................... Figure 1.11 -Reduced frequency responses H(0, jω,L) / H0 with the frequency axis on a linear scale over the range [150; 750] Hz ................................................................................................................ Figure 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c) .......... Figure 1.13 -Screenshots from the CRONE Toolbox before optimization (a) and after optimization (b) at x = 0 in the nominal case m = 0.5 .............................................................................................

Figure 4 -

 4 Figure 1 -Chronogramme illustrant 13 ans de collaboration avec le Liban .......................................... Figure 2 -Illustration de la progression et de la logique d'enchainement des chapitres du mémoire de thèse...................................................................................................................................................Figure 3 -Common Woodwind Instruments ........................................................................................ Figure 4 -Common Brass Instruments ................................................................................................. Figure 5 -Overall Flute System ............................................................................................................ Figure 6 -Traditional flute musical instrument .................................................................................... Figure 7 -Artificial loop based on the ear of the musician .................................................................. Figure 8 -Types of flutes: (a) Class D, (b) Classical Flared and (c) Baroque ......................................... Figure 9 -Block diagram of the mechanism of sound production in auto-oscillatory musical instruments ........................................................................................................................................... Figure 10 -Closed-Loop system of the self-oscillation mechanism of flute family instruments ......... Figure 11 -Example of the musical wind instrument array ................................................................. Figure 12 -Diagram of Keefe's flute ..................................................................................................... Figure 13 -Trumpet-like instrument-Decomposition into elementary subsystems ........................... Figure 14 -Diagram of copper type instrument operation coupled to the musician (first resonance) .............................................................................................................................................................. Figure 1.1 -One-dimensional description of an acoustic tube of radius r = constant and of finite length L subjected to an acoustic flow Qv(t) with x = 0 ........................................................................ Figure 1.2 -Curves rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f ............................................................................................................................................ Figure 1.3 -Bode Diagrams of Z(0,jω,L) on the range [20 ; 20 000] Hz of the frequencies audible by the human ear (a) and on the range [20 ; 4 000] Hz of frequencies attainable with a recorder (b) .... Figure 1.4 -Block diagrams associated with the system approach: Whatever x is between 0 an L (a), .............................................................................................................................................................. Figure 1.5 -Bode diagrams of Im (jω) on the range [10 -4 ; 10 4 ] Hz ........................................................ Figure 1.6 -Frequency response of Im(jω) on the range [20; 20,000] Hz of the audible frequencies . Figure 1.7 -Bode diagrams of F(0, jω,L) on the range [10 -4 ; 10 4 ] Hz .................................................... Figure 1.8 -Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) on the range [10 -4 ; 10 4 ] Hz (a) and on the range [20; 4000] Hz of audible and achievable frequencies with a recorder (b) .......... Figure 1.9 -Bode diagrams in x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and in x = 3L/4 (ωLx/2π = 735 Hz) of x = 3L/4 (ωLx/2π = 735 Hz) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) in the range [20; 4000] Hz of audible and achievable frequencies .......................................... Figure 1.10 -Bode diagrams at x = 0 of H(0, jω,L) for different values of the fractional order m on the range [20; 4000] Hz of audible and achievable frequencies with a recorder ....................................... Figure 1.11 -Reduced frequency responses H(0, jω,L) / H0 with the frequency axis on a linear scale over the range [150; 750] Hz ................................................................................................................ Figure 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c) .......... Figure 1.13 -Screenshots from the CRONE Toolbox before optimization (a) and after optimization (b) at x = 0 in the nominal case m = 0.5 .............................................................................................

Figure

  Figure 1 -Chronogramme illustrant 13 ans de collaboration avec le Liban .......................................... Figure 2 -Illustration de la progression et de la logique d'enchainement des chapitres du mémoire de thèse...................................................................................................................................................Figure 3 -Common Woodwind Instruments ........................................................................................ Figure 4 -Common Brass Instruments ................................................................................................. Figure 5 -Overall Flute System ............................................................................................................ Figure 6 -Traditional flute musical instrument .................................................................................... Figure 7 -Artificial loop based on the ear of the musician .................................................................. Figure 8 -Types of flutes: (a) Class D, (b) Classical Flared and (c) Baroque ......................................... Figure 9 -Block diagram of the mechanism of sound production in auto-oscillatory musical instruments ........................................................................................................................................... Figure 10 -Closed-Loop system of the self-oscillation mechanism of flute family instruments ......... Figure 11 -Example of the musical wind instrument array ................................................................. Figure 12 -Diagram of Keefe's flute ..................................................................................................... Figure 13 -Trumpet-like instrument-Decomposition into elementary subsystems ........................... Figure 14 -Diagram of copper type instrument operation coupled to the musician (first resonance) .............................................................................................................................................................. Figure 1.1 -One-dimensional description of an acoustic tube of radius r = constant and of finite length L subjected to an acoustic flow Qv(t) with x = 0 ........................................................................ Figure 1.2 -Curves rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f ............................................................................................................................................ Figure 1.3 -Bode Diagrams of Z(0,jω,L) on the range [20 ; 20 000] Hz of the frequencies audible by the human ear (a) and on the range [20 ; 4 000] Hz of frequencies attainable with a recorder (b) .... Figure 1.4 -Block diagrams associated with the system approach: Whatever x is between 0 an L (a), .............................................................................................................................................................. Figure 1.5 -Bode diagrams of Im (jω) on the range [10 -4 ; 10 4 ] Hz ........................................................ Figure 1.6 -Frequency response of Im(jω) on the range [20; 20,000] Hz of the audible frequencies . Figure 1.7 -Bode diagrams of F(0, jω,L) on the range [10 -4 ; 10 4 ] Hz .................................................... Figure 1.8 -Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) on the range [10 -4 ; 10 4 ] Hz (a) and on the range [20; 4000] Hz of audible and achievable frequencies with a recorder (b) .......... Figure 1.9 -Bode diagrams in x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and in x = 3L/4 (ωLx/2π = 735 Hz) of x = 3L/4 (ωLx/2π = 735 Hz) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) in the range [20; 4000] Hz of audible and achievable frequencies .......................................... Figure 1.10 -Bode diagrams at x = 0 of H(0, jω,L) for different values of the fractional order m on the range [20; 4000] Hz of audible and achievable frequencies with a recorder ....................................... Figure 1.11 -Reduced frequency responses H(0, jω,L) / H0 with the frequency axis on a linear scale over the range [150; 750] Hz ................................................................................................................ Figure 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c) .......... Figure 1.13 -Screenshots from the CRONE Toolbox before optimization (a) and after optimization (b) at x = 0 in the nominal case m = 0.5 .............................................................................................

  Figure 1 -Chronogramme illustrant 13 ans de collaboration avec le Liban .......................................... Figure 2 -Illustration de la progression et de la logique d'enchainement des chapitres du mémoire de thèse...................................................................................................................................................Figure 3 -Common Woodwind Instruments ........................................................................................ Figure 4 -Common Brass Instruments ................................................................................................. Figure 5 -Overall Flute System ............................................................................................................ Figure 6 -Traditional flute musical instrument .................................................................................... Figure 7 -Artificial loop based on the ear of the musician .................................................................. Figure 8 -Types of flutes: (a) Class D, (b) Classical Flared and (c) Baroque ......................................... Figure 9 -Block diagram of the mechanism of sound production in auto-oscillatory musical instruments ........................................................................................................................................... Figure 10 -Closed-Loop system of the self-oscillation mechanism of flute family instruments ......... Figure 11 -Example of the musical wind instrument array ................................................................. Figure 12 -Diagram of Keefe's flute ..................................................................................................... Figure 13 -Trumpet-like instrument-Decomposition into elementary subsystems ........................... Figure 14 -Diagram of copper type instrument operation coupled to the musician (first resonance) .............................................................................................................................................................. Figure 1.1 -One-dimensional description of an acoustic tube of radius r = constant and of finite length L subjected to an acoustic flow Qv(t) with x = 0 ........................................................................ Figure 1.2 -Curves rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f ............................................................................................................................................ Figure 1.3 -Bode Diagrams of Z(0,jω,L) on the range [20 ; 20 000] Hz of the frequencies audible by the human ear (a) and on the range [20 ; 4 000] Hz of frequencies attainable with a recorder (b) .... Figure 1.4 -Block diagrams associated with the system approach: Whatever x is between 0 an L (a), .............................................................................................................................................................. Figure 1.5 -Bode diagrams of Im (jω) on the range [10 -4 ; 10 4 ] Hz ........................................................ Figure 1.6 -Frequency response of Im(jω) on the range [20; 20,000] Hz of the audible frequencies . Figure 1.7 -Bode diagrams of F(0, jω,L) on the range [10 -4 ; 10 4 ] Hz .................................................... Figure 1.8 -Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) on the range [10 -4 ; 10 4 ] Hz (a) and on the range [20; 4000] Hz of audible and achievable frequencies with a recorder (b) .......... Figure 1.9 -Bode diagrams in x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and in x = 3L/4 (ωLx/2π = 735 Hz) of x = 3L/4 (ωLx/2π = 735 Hz) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) in the range [20; 4000] Hz of audible and achievable frequencies .......................................... Figure 1.10 -Bode diagrams at x = 0 of H(0, jω,L) for different values of the fractional order m on the range [20; 4000] Hz of audible and achievable frequencies with a recorder ....................................... Figure 1.11 -Reduced frequency responses H(0, jω,L) / H0 with the frequency axis on a linear scale over the range [150; 750] Hz ................................................................................................................ Figure 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c) .......... Figure 1.13 -Screenshots from the CRONE Toolbox before optimization (a) and after optimization (b) at x = 0 in the nominal case m = 0.5 .............................................................................................

Figure 1 . 14 -Figure 2 . 2 -

 11422 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 3 -

 23 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 4 -

 24 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 5 -

 25 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 6 -

 26 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 7 -

 27 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 8 -

 28 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 . 9 -

 29 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 .

 2 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 2 .

 2 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 3 . 1 -Figure 3 . 2 -Figure 3 . 3 -Figure 3 . 4 -

 31323334 Figure 1.14 -Bode diagrams of the response 𝑯𝑵, 𝑪(𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯𝑵, 𝑷(𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz ................... Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments .............................................................................................................................. Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: ................................................................Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000].......................................................................................Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources ............................................................................. Figure 2.5 -Functional diagram of the complete model ...................................................................... Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz. .....................................................................................................................Figure 2.7 -Parametric configuration retained for the Simulink solver ..............................................Figure 2.8 -Time responses obtained for a pressure = 400 Pa (on the left) and = 1000 Pa (on the right) concerning: -the pressure p(t) at the input of the resonator (a) and (b); -the acoustic speed vac(t) (c) and (d); -the transverse displacement  (w, t) of the jet (e) and (f) ......................................Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c) .................................................................................................................................Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c) transverse displacements 0(t) (in blue), 0(t-) (in green) and (w,t) (in red) (d) pressures p(t) (in black), psrc(t) (in red) and plos(t) (in blue) ..........................................................................................Figure 2.11 -Time responses obtained for a pressure = 1000 Pa ........................................................Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: ....................................................................................................... Figure 2.13-Time responses obtained for a pressure = 400 Pa with the linearized model (in blue) and the non-linear model (in red) during the first moments of phase 1 (start-up) over a time interval [0; 5]x10 -5 sec (left) and [0; 5]x10 -4 s (right): ......................................................................................... Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure ...................................................................Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: .................Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green) ....................................................................................................................................................Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for .............................Figure 3.1 -Test bench photos .............................................................................................................Figure 3.2 -Scheme of the experimental setup ...................................................................................Figure 3.3 -Block diagram of the experimental setup .........................................................................Figure 3.4 -Architecture of the Control System (CS) ........................................................................... Figure 3.5 -Block diagram of the servo-valve ......................................................................................Figure 3.6 -Static operating domain obtained from measurements ..................................................Figure 3.7 -Variation of 𝑲𝒒𝒖versus 𝑷𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓 .........................................................................

Figure 3 . 8 -

 38 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 .

 4 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 2 -

 42 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 3 -

 43 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 4 -

 44 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 5 -

 45 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 6 -

 46 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 7 -

 47 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 8 -

 48 Figure 3.8 -Operating point O and linear static operating domain ..................................................... Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation....................................................... Figure 3.10 -Block diagram of the artificial mouth linearized model .................................................. Figure 3.11 -Bode plots of G(j) for three values of the pressure: ................................................... Figure 3.12 -Bode plot of CR(j) ........................................................................................................ Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) .................................... Figure 3.14 -Bode plots of T(jω) (a), of S(jω) (b), of CS(jω) (c) and GS(jω) (d) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red) ............................................................................................................................................................ Figure 3.15 -Time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases: ...................................................................................................................................................Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation ............................................................................................................................................................Figure 4.2 -Functional diagram of the entire simulator .....................................................................Figure 4.3 -Plot of ωr,m as a function of the order m .........................................................................Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2.....................................................................................................................Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4 .....................................................................................................................Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1 ..........................................................................................Figure 4.7 -Bode plots of the admittance Yin(s) of the resonator for different values of the fractional order m over the range [100; 2000] Hz ..............................................................................................Figure 4.8 -Frequency responses Yin(jω) of the resonator admittance for different values of the fractional order m with the linear scale frequency axis in the range [100; 1400] Hz .........................

Figure 4 . 9 -

 49 Figure 4.9 -Distribution of anti-resonance ωzi (o) and resonance ωpi (x) frequencies according to the first 4 modes of the admittance Yin(s) of the resonator ....................................................................

Figure 4 .

 4 Figure 4.10 -Distribution of damping factors ωzi (a) and ωpi (b)according to the first 4 ranks of the YN(s) admittance of the resonator for 5 values of the order m (0; 0.25; 0.5; 0.75; 1) ........................

Figure 4 .

 4 Figure 4.11 -Distribution of the constants czi(o) and cpi(x) (a); αi (o), ηi (x) and αi (o)*ηi (x) (b) according to the first 4 ranks of the admittance YN(s) of the resonator.............................................Figure 4.12 -Bode diagrams of the frequency response β(jω) in open loop.....................................Figure 4.13 -Frequency responses β(jω) in open loop in the Black-Nichols plane ...........................Figure 4.14 -Zoom of the frequency responses β(jω) in open loop in the Black-Nichols plane .......

Figure 4 .

 4 12 -Bode diagrams of the frequency response β(jω) in open loop.....................................

Figure 4 .

 4 13 -Frequency responses β(jω) in open loop in the Black-Nichols plane ...........................

Figure 4 .

 4 14 -Zoom of the frequency responses β(jω) in open loop in the Black-Nichols plane .......

Figure 4 .

 4 Figure 4.15 -For Pm = 400 Pa (a) and for Pm = 1000 Pa (b), plots of the line τ ω and the frequency responses π/2 + arg(Yin(jω)) for 5 values of the order m ( 0; 0.25; 0.5; 0.75; 1). ...............................Figure 4.16 -For Pm = 400 Pa ( ) and for Pm = 1000 Pa ( ), variation of the open loop gain β(ω0) in dB scale (a) and in linear scale (b) as a function of the order m ...........................................

Figure 4 .

 4 16 -For Pm = 400 Pa ( ) and for Pm = 1000 Pa ( ), variation of the open loop gain β(ω0) in dB scale (a) and in linear scale (b) as a function of the order m ...........................................

Figure 4 .

 4 Figure 4.17 -Time responses of the transverse displacement η(w, t) of the jet (a) for 5 values of the order m (0; 0.25; 0.5; 0.75; 1), for a pressure Pm = 400 Pa (b) and Pm = 1000 Pa (c) .......................... Figure 4.18 -Time responses of the pressure Δp(t) at the input of the resonator (a) for 5 values of the order m (0; 0.25; 0.5; 0.75; 1), for a pressure Pm = 400 Pa (b) and Pm = 1000 Pa (c) ....................

Figure 4 .

 4 Figure 4.19 -Time responses of the acoustic velocity vac(t) (a) for 5 values of the order m (0; 0.25; 0.5; 0.75; 1), for a pressure Pm = 400 Pa (b) and Pm = 1000 Pa (c) ...................................................... 132

Figure 3 -

 3 Figure 3 -Common Woodwind Instruments

Figure 4 -

 4 Figure 4 -Common Brass Instruments

Figure 5 -

 5 Figure 5 -Overall Flute System

Figure 6 -

 6 Figure 6 -Traditional flute musical instrument

Figure 7 -

 7 Figure 7 -Artificial loop based on the ear of the musician

Figure 8 -

 8 Figure 8 -Types of flutes: (a) Class D, (b) Classical Flared and (c) Baroque

Figure 9 -

 9 Figure 9 -Block diagram of the mechanism of sound production in auto-oscillatory musical instruments

Figure 10 -

 10 Figure 10 -Closed-Loop system of the self-oscillation mechanism of flute family instruments

Figure 11 -

 11 Figure 11 -Example of the musical wind instrument array

Figure 12 -

 12 Figure 12 -Diagram of Keefe's flute

Figure 13 -

 13 Figure 13 -Trumpet-like instrument-Decomposition into elementary subsystems

Figure 14 -

 14 Figure 14 -Diagram of copper type instrument operation coupled to the musician (first resonance)

Figure 1 . 1 -

 11 Figure 1.1 -One-dimensional description of an acoustic tube of radius r = constant and of finite length L subjected to an acoustic flow Qv(t) with x = 0

Table 1. 1 -

 1 Values of the speed of sound ca, the density ρa and the characteristic acoustic impedance Zac as a

Figure 1 . 2 -

 12 Figure 1.2 -Curves rv(f) = (lv ca/f) 0.5 (in red) and rh(f) = (lh ca/f) 0.5 (in blue) with respect to the frequency f

Figure 1 .

 1 Figure 1.3 presents in x = 0 the Bode diagrams of Z(0,j,L) on the range [20; 20,000] Hz of the audible frequencies by the human ear (Figure 1.3.a) and on the range [20; 4000] Hz of the frequencies attainable with a recorder (Figure 1.3.b).

  Figure 1.3 -Bode Diagrams of Z(0,jω,L) on the range [20 ; 20 000] Hz of the frequencies audible by the human ear (a) and on the range [20 ; 4 000] Hz of frequencies attainable with a recorder (b)

Figure 1 . 4 -1. 3 . 2 -

 1432 Figure 1.4 -Block diagrams associated with the system approach: Whatever x is between 0 an L (a), at x=0 for the finite system L (b), at x=0 for a semi finite system (c)

  an illustration, let us take the acoustic tube whose nominal dimensions are fixed by a radius r = 5 x 10 -3 m and a length L = 0.3 m at a temperature of 25 °C, with ra = 1.184 kg/m 3 and ca = 346.3 m/s. In this case, and as a reminder, the numerical value of the transitional frequency r,m (relation (1.28)) and 4.92 rad/s (0.784 Hz).

Figure 1 .

 1 Figure 1.5 presents the Bode diagrams of the frequency response Im(j) over the range [10 -4 ; 10 4 ] Hz. The two behaviors appear clearly: -for  << r,m, a gain diagram with a straight line with slope p = -m/2*20 dB/dec = -5 dB/dec and a phase diagram with a horizontal line at -m/2*90° = -22.5°; -for  >> r,m, a gain diagram with a horizontal line at 0 dB and a phase diagram with a horizontal line at 0°.

Figure 1

 1 Figure 1.5 -Bode diagrams of Im (jω) on the range [10 -4 ; 10 4 ] Hz

  Figure 1.7 presents at x = 0 the Bode diagrams of the frequency response F(0,jω,L) on the range [10 -4 ; 10 4 ] Hz. The two behaviors appear clearly: for ω << ωr,m, a gain diagram with a straight line with p1 = (1-m/2)*20 dB/dec = 15 dB/dec and a phase diagram with a horizontal line at (1-m/2)*90° = 67.5° ; -for ω >> ωr,m, a gain diagram with a straight line with slope p2 = 20 dB/dec of this a phase diagram with a horizontal straight line at 90 °.

Figure 1 . 7 -

 17 Figure 1.7 -Bode diagrams of F(0, jω,L) on the range [10 -4 ; 10 4 ] Hz

Figure 1 .

 1 Figure 1.8 shows the Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) over the range [10 -4 ; 10 4 ] Hz (Figure 1.8.a) and on the range [20; 4000] Hz of the audible and achievable frequencies with a recorder (Figure 1.8.b).

  Figure 1.8 -Bode diagrams of 1/F(0,jω,L) (in red) and of T(0,jω,L) (in blue) on the range [10 -4 ; 10 4 ] Hz (a) and on the range [20; 4000] Hz of audible and achievable frequencies with a recorder (b)

Figure 1 . 9 -

 19 Figure 1.9 -Bode diagrams in x = 0 (ωLx/2π = 184 Hz), in x = L/2 (ωLx/2π = 368 Hz) and in x = 3L/4 (ωLx/2π = 735 Hz) of x = 3L/4 (ωLx/2π = 735 Hz) (in black), of H(L/2,jω,L) (in blue) and of H(3L/4,jω,L) (in red) in the range [20; 4000] Hz of audible and achievable frequencies

Figure 1 .Figure 1

 11 Figure 1.10 -Bode diagrams at x = 0 of H(0, jω,L) for different values of the fractional order m on the range [20; 4000] Hz of audible and achievable frequencies with a recorder

Figure 1

 1 Figure 1.12 -Block diagrams associated with the simplified model: whatever x is between 0 and L (a), at x = 0 for the finite system of length L (b) and at x = 0 for a semi-infinite system (c)

  .13.a) and after optimization (Figure 1.13.b) at x = 0 in the nominal case m = 0.5.The procedure consists, in a first step, from generating the target frequency response form which appears in red (Data) in Figure1.13.a. Then, cells are added one after the other by clicking on the "Add New Cell" command in the "Cells actions" menu (in green at the top right), then by positioning the mouse cursor on the phase diagram at point considered where the cutting phase of  

  (gain and phase) obtained before optimization (Figure1.13.a) and after optimization (Figure1.13.b). As for the green lines, these are asymptotic lines.

Figure 1

 1 Figure 1.13 -Screenshots from the CRONE Toolbox before optimization (a) and after optimization (b) at x = 0 in the nominal case m = 0.5

  pi and Qpi of N = 4 cells of the parallel form (relation (1.61)) to which cell A0/s is added.



  of the parallel form (in green) on the range [100: 2000] Hz where we observe the perfect superposition of the two curves.

Figure 1 .

 1 Figure 1.14 -Bode diagrams of the response 𝑯 ̃𝑵,𝑪 (𝒋𝝎, 𝒔, 𝑳) of the cascade form (in blue) and of the response 𝑯 ̃𝑵,𝑷 (𝒋𝝎, 𝒔, 𝑳) of the parallel form (in green) over the range [100: 2000] Hz

Figure 2 . 1 -

 21 Figure 2.1 -Schematic representation of the mechanism of sound production in self-oscillating musical instruments

Figure 2 . 2 (

 22 Figure 2.2 (which is a more detailed representation of the flute shown in Figure 1.3 in Chapter 1) shows a cross section of a recorder, as well as a schematic representation of its exciting mechanism (consisting of the interaction of an air jet with a bevel) where the notations that appear in the figure are defined by: -1: outlet of the channel; 2: bevel; 3: resonator (air column); 4: exciter; -  ;; R o x y  : local reference associated with the jet; -h: height of the spout channel; -w: distance between the outlet of the channel and the tip of the bevel; -x0: lateral offset of the tip of the bevel from the longitudinal y axis of the channel;

Figure 2 . 2 -

 22 Figure 2.2 -Cross section of a recorder and schematic representation of its excitation mechanism consisting of the interaction of an air jet with a bevel with: 1: spout channel output; 2: bevel; 3: resonator (air column); 4: exciter

Figure 2 . 3 -

 23 Figure 2.3 -Functional diagram of the loop system describing the self-oscillation mechanism of instruments of the flute family [Fabre, 2000]

  have shown that in a steady state the convection speed e v C of transverse disturbances along a jet is linked to the difference in speed between the jet and the external environment is in general 0

Figure 2 . 4 -

 24 Figure 2.4 -Schematic representation of the behavior of the jet, according to Fabre in [Chaigne, 2013]: (a) disturbance of the jet at the outlet of the channel by the acoustic field present in the resonator; (b) convection and amplification of the disturbance (c) oscillation of the jet around the bevel which gives rise to the aero-acoustic sources

  b represents the estimated half-thickness of the jet, according to Verge (M P Verge B.

Figure 2 . 5 -

 25 Figure 2.5 -Functional diagram of the complete model

  r=1.184 kg/m 3 : density of air at 25°C;  = 15.6x10 -6 m 2 /s: kinematic viscosity of air at 25°C; h = 10 -3 m: height of the spout channel; w = 4.25x10 -3 m: distance between the outlet of the nozzle channel and the tip of the bevel; b = 0.4x10 -3 m: half-thickness of the jet; d = 3.7x10 -3 m: equivalent distance between the 2 sources of flow Qin and Qout at the level of the bevel vc = 0.6: effect factor vena contracta; x0 =0.1x10 -3 m: lateral offset of the tip of the bevel from the longitudinal axis of the channel; L = 0.3 m: length of the resonator tube; r = 5x10 -3 m: radius of the resonator tube; S = 7.854x10 -5 m 2 : section of the resonator tube; nh = 2: order of the low-pass filter of the operator D(s); c = 2x50 000 rad/s: cutoff angular frequency of the low-pass filter of the operator D(s).As a reminder (see Chapter 1), Figure2.6 shows the frequency response of the input admittance Yin(s,L) of the resonator on the range [100; 2000] Hz and Table2.1 gives the numerical values of the parameters Ai, Bi, pi,  pi and Qpi of its parallel rational form for the N = 4 first modes of the resonator.

Figure 2 . 6 -

 26 Figure 2.6 -Frequency response of the input admittance Yin(s, L) of the resonator considered over the range [100; 2000] Hz.

Figure 2 .

 2 Figure 2.7 shows the parametric configuration chosen for the Simulink solver.

Figure 2 . 7 -

 27 Figure 2.7 -Parametric configuration retained for the Simulink solver

  .16) and this for 13 different values of the pressure e m P between 400 Pa and 1000 Pa, with a step of 50 Pa. For each of its simulations, three phases appear:

Figure 2

 2 Figure 2.8, for a pressure e m P = 400 Pa (on the left) and e m P = 1000 Pa (on the right),

Figure 2 . 9 -Figure 2 .

 292 Figure 2.9 -Time responses obtained in phase 1 for a pressure = 400 Pa concerning: -the pressure p(t) at the input of the resonator (a); -the acoustic speed vac(t) (b); -the transverse displacement  (w, t) of the jet (c)

  the jet (mm) The observation of these responses leads to the following remarks: -Figures 2.10.a and 2.11.a: the pressure p(t) (in green) at the input of the resonator (relation (2.11)), which oscillates around 0, has a low amplitude compared to the pressure e m P (in blue); -Figures 2.10.b and 2.11.b: at the level of the channel output at x = 0, same remark for the acoustic speed vac(t) (in green) which oscillates around 0 and has a low amplitude compared to flow velocity e j U (in blue); -Figures 2.10.c and 2.11.c: the instability of the jet (which results in an amplification of its transverse displacement (x,t) (relation (2.4)) appears clearly by comparing the amplitudes of (x=w,t) (in red) and (x=0,t-) = 0(t-) (in green); -Figures 2.10.d and 2.11.d: the pressure plos(t) (in blue) linked to the loss of energy at the level of the bevel (relation (2.10)) remains low compared to the pressure p(t) (in black) at the input of the resonator (relation (2.11)).

Figure 2 .

 2 Figure 2.10 -Time responses obtained for a pressure = 400 Pa: (a) pressure (in blue) and pressure p(t) at the input of the resonator (in green); (b) speed (in blue) and acoustic speed vac(t) (in green); (c)

  . The Laplace transform of the system (2.25),

  , p0(t) = TL -1 {P0(s)} represents the step response of the operator D(s) to a step of amplitude C0 (proportional to also represents the impulse response of the low-pass filter of order nh = 2 multiplied by the constant C0, that is

  and for the first moments of phase 1, Figure2.12 presents the time responses of the pressure p0(t) obtained with the relation (2.32) (curve 1 in green), the nonlinear model (curve 2 in red) and the linearized model (curve 3 in blue).

Figure 2 .

 2 Figure 2.12 -For a pressure equal to 400 Pa and for the first moments of phase 1, time responses of the pressure p0(t) obtained with: -relation (2.30) (curve 1 in green); -the nonlinear model (curve 2 in red); -the linearized model (curve 3 in blue)

Figure 2 .

 2 Figure 2.14 shows the variation in flow velocity e j U and the variation in frequency f0

Figure 2 .

 2 Figure 2.14 -Variations in the flow speed of the jet at the outlet of the channel (a) and the frequency f0 of the self-oscillations (b) as a function of the pressure

Figure 2 .

 2 Figure 2.15 -Schematic representation of the different hydrodynamic modes of the jet: the first hydrodynamic mode k = 1 (a) corresponds to the case where one observes 1/2 wavelength between the outlet of the channel and the bevel;

  with T0 = 1/f0 the period of the self-oscillations. Taking into account the expression for the delay  = w

Figure 2 .

 2 .40) It is to highlight that: -the gain  of the open loop does not depend on the flow speed e j U at the channel output and therefore on the pressure e m P at the mouthpiece input; -the phase  of the open loop depends on the speed of flow e 16 shows the frequency responses of the gain  and of the phase  for Examples 1 (in black) and 2 (in green) defined in paragraph 2.3.

Figure 2 .

 2 Figure 2.16 -Frequency responses of gain  and phase  for examples 1 (in black) and 2 (in green)

  value of n is directly related to the hydrodynamic mode k of the jet. If the instrument is in normal operation (n = 0) then k = 1. All cases where |n| > 1 correspond to the wind systems necessarily obtained for low values of e m P and e j U , therefore large values of .

  .46) Graphically, the solution frequencies of this equality are located at the intersections between the phase of the input admittance Yin(j,L) increased by /2 and the phase lines  whose slopes  depend on the speed flow e of arg(Yin(j,L)+90° (in blue) and of  for examples 1 (

Figure 2 .

 2 Figure 2.17 -Frequency responses of argYin (j, L) + 90 ° (in blue) and of * for example 1 (Pm = 400 Pa, Uj = 26 m/s, in black) and example 2 (Pm = 1000 Pa, Uj = 41 m / s, in green) defined in paragraph 2.3

Figure 3 .

 3 Figure 3.2 presents the scheme of the experimental setup and Figure 3.3 the block diagram associated.

Figure 3 Figure 3 . 3 -

 333 Figure 3.1 -Test bench photos

Figure 3 . 4 -

 34 Figure 3.4 -Architecture of the Control System (CS)

Figure 3 .

 3 Figure 3.5 shows the block diagram of the servo-valve. In order to express the output of the nonlinear part, a static flow Qstat is introduced.

Figure 3 . 5 -

 35 Figure 3.5 -Block diagram of the servo-valve

Figure 3 .

 3 Figure 3.6 shows the static operating domain obtained from measurements for  V 10 ; 0  c U and

Figure 3 . 6 -

 36 Figure 3.6 -Static operating domain obtained from measurements

Figure 3

 3 Figure 3.7 shows the variation of qu K ˆ versus

Figure 3 . 7 -

 37 Figure 3.7 -Variation of 𝑲 ̂𝒒𝒖 versus 𝑷 𝟏 ∈ [𝟏; 𝟔]𝒃𝒂𝒓

  variation range of P1, the variation range of Uc without saturation of Qsv has as limits

  range of the feed-forward control signal Uff is given by:

  Figure 3.8 shows the chosen operating point

Figure 3 . 8 -

 38 Figure 3.8 -Operating point O and linear static operating domain

Figure 3 . 9 -

 39 Figure 3.9 -Plot of Qsv(t) vs sqrt(Pm(t)) for a quasi-static variation

Figure 3 .

 3 Figure 3.10 shows the block diagram of the artificial mouth linearized model.

Figure 3 .

 3 Figure 3.10 -Block diagram of the artificial mouth linearized model

  complete linearized model used for the design of the control law is represented by the transfer function G(s,P1) between  

Figure 3 .

 3 Figure 3.11 presents the Bode plots of G(jω) for three values of the pressure: P1 = 4 bar (in blue), P1 = 5 bar (in black) and P1 = 6 bar (in red).

Figure 3 .

 3 Figure 3.11 -Bode plots of G(j) for three values of the pressure: P1= 4 bar (in blue), P1 =5 bar (in black), P1 =6 bar (in red)

  nominal open-loop transfer function βnom(s) of the second generation CRONE control is defined by:

Figure 3 .Figure 3 .

 33 Figure 3.12 -Bode plot of CR(j)

Figure 3 .Figure 3 .

 33 Figure 3.13 -Bode plots (a) and Nichols loci (b) of β(jω) obtained with the CRONE controller for the three cases: min (4 bar in blue), nom (5 bar in black) and max (6 bar in red)

Figure 3 .

 3 Figure 3.15 presents time responses of Pref(t) and Pm(t) (a) (b), of error signal ɛ(t) = Pref(t) -Pm(t) (c) (d), and of control signal Uc(t) (e) (f) obtained without feedback (a) (c) (e) and with feedback (b) (d) (f) for the three cases (min, nom, max).

Figure 4 . 1 -

 41 Figure 4.1 -Illustration of the progression and sequencing of the chapters of the thesis dissertation

4 . 2 -

 42 This chapter therefore begins by recalling the organization of the global simulator developed under MatLab / Simulink from the work presented in the first three chapters. Then, an analysis in the frequency domain of the influence of the order m is developed, first in the case of the impedance of the resonator established in Chapter 2, then in the case of the defined open-loop transfer function in Chapter 3. This analysis continues in the time domain with the responses from the global simulator by using the scenarios of the two examples of Chapter 3. Finally, the main contributions of this chapter are recalled in the conclusion. Simulator of the Artificial Mouth -Recorder Set: Reminder

Figure 4 .

 4 Figure 4.2 shows the block diagram of the global simulator used in this chapter. It consists of: -the artificial mouth simulator (Chapter 3); -the exciter simulator (Chapter 2).

Figure 4 . 2 -where

 42 Figure 4.2 -Functional diagram of the entire simulator

Figure 4 . 3 -

 43 Figure 4.3 -Plot of ωr,m as a function of the order m

Figure 4 .

 4 Figure 4.4 shows on the range[100; 2000] Hz for m = 0.5 the frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 2. The integrative behavior of order 1 present at low frequencies is perfectly reproduced by the equation (4.6), as well as the behavior associated with anti-resonance and resonance of the first mode. Beyond that, the side effect linked to the truncation (N = 2) appears clearly.

Figure 4 . 4 -

 44 Figure 4.4 -In the range [100; 2000] Hz and for m = 0.5, frequency responses of Yin(s) (in black) and of YN (s) (in red) for N = 2

  pi to the solutions for which the slope of the phase curve of Yin(j) is negative (resonance), namely:

Figure 4 .

 4 Figure 4.5 shows on the range [100; 4000] Hz for m = 0.75 the frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4, minimum value to cover the field of study defined within the framework of this thesis work. Comparing the responses in Figure4.4 allows us to appreciate the effectiveness of the proposed method. Indeed, the behavior associated with the anti-resonance and the resonance of the first three modes is perfectly reproduced. Beyond that, the side effect linked to the truncation (N = 4) appears clearly.

Figure 4 . 5 -

 45 Figure 4.5 -In the range[100; 4000] Hz and for m = 0.75, frequency responses of Yin(s) (in black) and of YN(s) (in red) for N = 4

  Note that the ratio pi/zi only depends on cpi and czi namely:

  )

Figure 4 .

 4 Figure 4.6 presents the variation of the damping factors z1 and p1 of rank 1 (1 st mode of the resonator) according to the order m ranging between 0 and 1. These plots are obtained from the analytical relations (4.13).

Figure 4 . 6 -

 46 Figure 4.6 -Variation of the damping factors ζz1 and ζp1 of the first mode of the resonator as a function of the order m between 0 and 1

Figures 4 .Figure 4 . 7 -Figure 4 . 8 -Figure 4 .Figure 4 . 2 -

 44748442 Figures 4.7 and 4.8 show the frequency responses of the input admittance Yin(s) of the resonator for different values of the fractional order m between 0 and 1, with the frequency axis scaled:

   of the open loop does not depend on the flow speed e j U at the output of the channel and therefore on the pressure Pm = cst at the input of the mouthpiece; -on the other hand, the phase  of the open loop depends on the speed of flow e

  and 4.13 show the frequency responses (j) of the open-loop transfer function in the planes: -from Bode (Figure 4.12); -from Black-Nichols (Figure 4.13), for -Pm = 400 Pa (Figures 4.12.a and 4.13.a); -Pm = 1000 Pa (Figure 4.12.b and 4.13.b).

Figure 4 .

 4 Figure 4.14 is a zoom of Figure 4.13 in the Black-Nichols plane around the point (0°, 0 dB), or (0°, 1), thus allowing to better visualize the conditions (4.26).

Figure 4

 4 Figure 4.13 -Frequency responses β(jω) in open loop in the Black-Nichols plane for Pm = 400 Pa (a) and for Pm = 1000 Pa (b), for 5 values of the order m (0; 0.25; 0.5; 0.75; 1)

Figure 4 .

 4 Figure 4.14 -Zoom of the frequency responses β(jω) in open loop in the Black-Nichols plane around the point (0°; 0 dB) for Pm = 400 Pa (a) and for Pm = 1000 Pa (b), for 5 values of the order m (0; 0.25; 0.5; 0.75; 1)

  .27) the solution  resulting from the intersection of the plots associated with the expressions located to the right and to the left of the equal sign of relation (4.27). Thus, Figure 4.15 presents the frequency responses of /2 + arg(Yin(j)) for five values of the order m, as well as the line associated with the expression  for: example 1 where Pm = 400 Pa (right in blue Figure 4.15.a) and where mode 1 is concerned (p1 = 3 639 rad/s, is 579.5 Hz), the solution being 0 = 3 577 rad/s (570 Hz); example 2 where Pm = 1000 Pa (right in purple Figure 4.15.b) and where mode 2 is concerned (p2 = 7 218 rad/s, is 1 149 Hz), the solution being 0 = 7 096.4 rad/s (1 130 Hz). Finally, Figure 4.16 shows for the two examples Pm = 400 Pa ( ) and Pm = 1 000 Pa ( ), the variation of the open loop gain (0) in dB scale (Figure 4.16.a) and in linear scale (Figure 4.16.b) as a function of the fractional order m between 0 and 1 on a linear scale.Observation of these plots leads to two remarks:-for a given order m, the value of the open loop gain (0) greater than unity increases with the pressure Pm;-for a given pressure Pm, the curves of (0) as a function of the order m show a different local minimum depending on the value of Pm.

Figure 4 .

 4 Figure 4.15 -For Pm = 400 Pa (a) and for Pm = 1000 Pa (b), plots of the line τ ω and the frequency responses π/2 + arg(Yin(jω)) for 5 values of the order m ( 0; 0.25; 0.5; 0.75; 1). The solution ω0 results from the intersection of these two paths which is marked in the figure by a small rectangle

  to 4.19 show for: -the two pressures Pm = 400 Pa (example 1) and Pm = 1000 Pa (example 2) -and the five values of the order m (0; 0.25; 0.5; 0.75; 1), time responses:

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.17 -Time responses of the transverse displacement η(w, t) of the jet (a) for 5 values of the order m (0; 0.25; 0.5; 0.75; 1), for a pressure Pm = 400 Pa (b) and Pm = 1000 Pa (c)

  A.1) of Yin(s) can be put in a form YN(s) composed of an integrator of order 1 in cascade with N fractional cells of order 2, namely:

- 2 :

 2 czi and cpi are constants resulting from the procedure allowing to pass from Yin(s) to YN(s) This procedure consists of 4 steps: -Step 1: finite expansion of the function tanh(z) into a continuous fraction in z; Step reduction of the continuous fraction to a ratio of two polynomials in z; -Step 3: factorization of each polynomial in z;

  expression (A.10) of tanh(z) is rewritten as: the expression (A.17) of z(s) in the relation (A.16) of tanh(z) leads to:

  the expression (A.1) of Yin(s), where Im(s) is defined by: the relation (A.18) in the expression (A.1) allows (after some manipulations and simplifications) the transition to the form YN(s) for N = 2, that is:

  My great and sincere thanks go to my supervisors Prof. Xavier Moreau and Dr. Roy Abi Zeid Daou, who provided valuable and continuous support and guidance during the course of this research project. I would like to thank my committee members Professor Valerie Budinger, Professor Clovis Francis and Professor Rachid Malti for reviewing and reporting my thesis.

	Achnowledgment
	Directeur de thèse

Table 1 .

 1 2 summarizes the final numerical values of the parameters zi, zi, Qzi, pi, pi and Qpi of the N = 4 cells of the cascade form (relation (1.60)) to which we must not forget the cell number zero, namely the integrator A0/s.As for Table1.3, it gives the numerical values of the parameters Ai, Bi, pi, 

Table 1 .2 -Final numerical values of the parameters ωzi, zi, Qzi, ωpi, pi and Qpi of the N = 4 cells of the cascade form

 1

	N	 zi (rad/s)	 zi	Q zi	 pi (rad/s)	 pi	Q pi
			-3			-3
	1	1780		27.78	3580		38.76
			18 10			12.9 10
			-3			-3
	2	5380		47.62	7190		54.35
			10.5 10			9.2 10
	3	8990	8.12 10 -3	61.8	10800	7.47 10 -3	66.93
	4	12600	6.92 10 -3	72.25	14500	6.37 10 -3	78.5

Table 1 .3 -Numerical values of parameters Ai, Bi, ωpi, pi and Qpi of N = 4 cells of the parallel form

 1

	N	A i	B i	 pi (rad/s)	 pi	Q pi
		-12	-10		-3
						38.76
	1	36.02 10	16.63 10	3580	12.9 10
		-13	-11		-3
						54.35
	2	98.24 10	64.99 10	7190	9.2 10
	3	53.38 10 -13	43.07 10 -11	10800	7.47 10 -3	66.93
	4	58.98 10 -13	54.47 10 -11	14500	6.37 10 -3	78.5

Figure 1.14 presents the Bode diagrams of the response

Table 2 .1 -Numerical values of the parameters Ai, Bi, pi,  pi and Qpi of the parallel rational form of the input admittance Yin(s,L) for the N = 4 first modes of the resonator

 2 For more illustration, table 2.2 gives the numerical values of this ratio for the first 4 modes of the resonator considered in table 2.1.

	N	A i	B i		 pi (rad/s)	 pi			Q pi
	1	36.02 10	-12 16.63 10	-10	2 570	12.9 10	-3	38.76
	2	98.24 10	-13 64.99 10	-11	2 1145	9.2 10	-3	54.35
	3	53.38 10 -13 43.07 10 -11 2 1720	7.47 10 -3	66.93
	4	58.98 10 -13 54.47 10 -11 2 2309	6.37 10 -3	78.5
	Remark 2.1
	Remark 2.2
	In the dynamics of linear systems, in the general case of a polynomial of order 2 having
	1 pair of conjugate complex roots, the natural undamped angular frequency n and the
	resonance angular frequency r are linked by a relation of the form	rn  	12 	2

With a ratio c/p4 = 21.6 the cut-off angular frequency c = 2 x 50,000 rad/s of the low-pass filter of the operator D(s) is sufficiently large compared to the largest natural nondamped angular frequency p4 = 2 x 2,309 rad/s of the rational form of the impedance Yin(s,L). Thus for times t >> c = 1/c = 3.18 x 10 -6 s, the two differentiation operators, D(s) and s have the same behavior.



where  represents the associated damping factor. In acoustic instruments, and in particular in the particular case of wind instrument resonators, the damping factors pi are very small compared to unity (see table

2.1)

. This is the reason why the natural un-damped angular frequencies pi and the resonance angular frequencies ri associated with each of the i  is close to unity.

Table 2 .2 -Numerical values of the ratio for the first 4 modes of the resonator considered

 2

	N	 pi	 ri /  ni
		-3
			0.9998
	1	12.9 10
	2	9.2 10

  gives:

	  t  src p	2    0 1 tanh x b b             e dj bU w r   	d dt		  , w t	,	(2.22)
	or again, after simplification by b,
		  t  src p	1 K	d dt		  , w t	,	(2.23)
	taking,
	2 KU 0 1 1 tanh d x wb  r             	e j	,	(2.24)
	  , w t x 0 dd wt   2 0 1 tanh tanh , x b dt b b dt                        	.	(2.21)
	Thus, the expression (2.8) of psrc(t) linearized around 0 is given by:

  [START_REF] Abou Haidar | Robust Control of an Artificial Mouth for a Wind Musical Instrument[END_REF] (Abou Haidar, Abi Zeid Daou, & Moreau, 2019):

  , the parameter's values of the open-loop transfer function are:

	    l l n	 	79 . 5 , 2 n	 rad/s , , 5 . 1	 h h n	  2 20	444	rad/s	,		0		92 . 32	SI	.	(3.34)

Table 4 .1 -Values of ωr,m as a function of the order m

 4

	m		0	0.25	0.5	0.75	1
	 r,m	rad/s Hz	0 0	4.4 10 -3 7 10 -4	4.92 0.783	34.88 5.55	82.55 13.14

Table 4 .

 4 2 specifies the different parametric values which characterize the cells of ranks 1 to 4 within the framework of the work of this thesis.

Table 4 .2 -Parametric values which characterize the cells of ranks 1 to 4 in the framework of the work of this thesis

 4

	Rank i	1	2	3	4
	 zi (rad/s)	1814	5412	9030	12660
	 pi (rad/s)	3639	7218	10860	14460
	c zi	1.572	4.688	7.823	10.967
	c pi	3.153	6.253	9.408	12.527
	 i	2	1.334	1.203	1.142
	 i	1.487	1.251	1.166
	 i  i	2.983	1.668	1.402

81 | P a g e transverse displacement (w,t) (Figure 2.13.f), transient system which is superimposed on that still in course resulting from the initial pressure angular frequency.

In addition to Figures 4.7 and 4.8,Figures 4.9 This appendix presents the details of the analytical developments allowing to pass from the initial form of the fractional impedance Yin(s) of chapter 1 to its form YN(s) of chapter 4.

Thus, as a reminder and for the field of study defined within the framework of this thesis work, the input admittance Yin(s) of the acoustic tube (resonator) of length L and radius r has the expression: