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International audience; This paper investigates the dynamic behavior of a simplified single reed instrument model subject to a stochastic forcing of white noise type when one of its bifurcation parameters (the dimensionless blowing pressure) increases linearly over time and crosses the Hopf bifurcation point of its trivial equilibrium position. The stochastic slow dynamics of the model is first obtained by means of the stochastic averaging method. The resulting averaged system reduces to a non-autonomous one-dimensional Itô stochastic differential equation governing the time evolution of the mouthpiece pressure amplitude. Under relevant approximations the latter is solved analytically treating separately cases where noise can be ignored and cases where it cannot. From that, two analytical expressions of the bifurcation parameter value for which the mouthpiece pressure amplitude gets its initial value back are deduced. These special values of the bifurcation parameter characterize the effective appearance of sound in the instrument and are called deterministic dynamic bifurcation point if the noise can be neglected and stochastic dynamic bifurcation point otherwise. Finally, for illustration and validation purposes, the analytical results are compared with direct numerical integration of the model in both deterministic and stochastic situations. In each considered case, a good agreement is observed between theoretical results and numerical simulations, which validates the proposed analysis.

International audience; This article develops the design of a sound synthesis model of a woodwind instrument by modal decomposition of the input impedance, taking into account viscothermal losses as well as localized nonlinear losses at the end of the resonator. This formalism has already been applied by Diab et al. (2022) to the study of forced systems. It is now implemented for self-oscillating systems. The employed method extends the denition of the input impedance to the nonlinear domain by adding a dependance on the RMS acoustic velocity at a geometric discontinuity. The poles and residuals resulting from the modal decomposition are interpolated as a function of this velocity. Thus, the pressure-ow relation dened by the resonator is completed by new equations which account for the dependence with the velocity at the end of the tube. To assess the ability of the model to reproduce a real phenomenon, comparisons with the experimental results of Atig et al. (2004) and Dalmont et al. (2007) were carried out. Simulations show that the model reproduces these experimental results qualitatively and quantitatively.; Cet article élabore la conception d'un modèle de synthèse sonore d'un instrument de la famille des bois par décomposition modale de l'impédance d'entrée, en tenant compte des pertes viscothermiques ainsi que des pertes non linéaires localisées à l'extrémité du résonateur. Ce formalisme a déjà été appliqué par Diab et al. (2022) à l'étude des systèmes forcés. Il est maintenant mis en œuvre pour les systèmes auto-oscillants. La méthode employée étend la définition de l'impédance d'entrée au domaine non linéaire, en ajoutant une dépendance à la vitesse acoustique RMS au niveau d'une discontinuité géométrique. Les pôles et les résidus résultant de la décomposition modale sont interpolés en fonction de cette vitesse. Ainsi, la relation pression-débit définie par le résonateur est complétée par de nouvelles équations qui tiennent compte de la dépendance avec la vitesse à l'extrémité du tube. Pour évaluer la capacité du modèle à reproduire un phénomène réel, des comparaisons avec les résultats expérimentaux de Atig et al. (2004) et Dalmont et al. (2007) ont été effectuées. Les simulations montrent que le modèle reproduit qualitativement et quantitativement ces résultats expérimentaux.

International audience; The dynamics of flexible beams in confined flows has been a subject of research for many years, as its fundamental behaviour is found in many applications, from energy harvesters to musical instruments. Most studies are concerned solely with the conditions for linear stability and do not explore the ensuing nonlinear behaviour of the system. This is particularly delicate as fluttering beams in confined flows are known to often result in dynamics with intermittent impacts between the beam and the side-walls. Here we present a nonlinear analytical resolution to a simplified 1-D model, based on a modal beam and bulk-flow equations. The model accounts for dissipation through distributed frictional and localised head-loss terms. The latter are imposed at the boundary conditions and aims to describe the complex effects occurring outside the domain (turbulence, vortex shedding, etc.). The present analytical resolution leads to a compact system for linear stability analysis, but also to a nonlinear formulation of the fluid-structure interaction. The inclusion of a regularized contact model allows for the computation of the full nonlinear dynamics, including intermittent impacts. Linear stability results are compared to previously published results using 2-D CFD models, and the relative merits of the model are discussed. A variety of limit cycles, (1) with and without impacts, (2) in symmetric and asymmetric configurations and (3) with impacts both at the beam tip and along its length, are shown to illustrate the diversity of dynamics encountered. Moreover, we show that, at large flow velocities, particular model configurations can lead to aperiodic dynamics, a phenomenon reported in several experimental observations. To the authors knowledge, the proposed formulation presents, for the first time, a framework for the comprehensive understanding of the nonlinear dynamics associated with flexible beams in confined axial flow.

The ghost note is a natural note which can be played exclusively on bass brass instruments with a predominantly-expanding bore profile such as tubas, euphoniums or saxhorns. It stands between the pedal note-the lowest natural note playable, or first regime-and the instrument's second regime. However, if the interval between the pedal note and the second regime remains close to an octave regardless of the instrument, the interval between the pedal note and the ghost note vary from a minor third to a perfect fourth. References about this note are very scarce, and it is not commonly known among tuba players.This study shows that an elementary brass model describing the player coupled to the instrument is capable of bringing both the ghost and the pedal note to light. Here, we adopt a dynamical systems point of view and perform a bifurcation analysis using a software of numerical continuation. The numerical results provided in terms of frequency intervals between pedal note and ghost note are compared with frequency intervals experimentally inferred from recordings of seven different types of tuba, each of them being played by two professional tuba players., Comment: arXiv admin note: text overlap with arXiv:2112.08751

Woodwind tonehole's linear behavior is characterized by two complex quantities: the series and shunt acoustic impedances. A method to determine experimentally these two quantities is presented for the case of open toneholes. It is based on two input impedance measurements. The method can be applied to clarinet-like instruments, and can be used for undercut toneholes as well as toneholes with pads above their output, under the condition that a symmetry axis exists. The robustness of the method proposed is explored numerically through the simulation of the experiment when considering geometrical and measurement uncertainties. Experimental results confirm the relevance of the method proposed to estimate the shunt impedance. Even the effect of small changes in the hole's geometry, such as those induced by undercutting, are characterized experimentally. The main effect of undercutting is shown to be a decrease in the tonehole's acoustic mass, in agreement with theoretical considerations based on the shape of the tonehole. Investigation on the effects of pads will be studied in a further work. Experimental results also reveal that losses in toneholes are significantly higher than those predicted by the theory. Therefore, the method is suitable for the experimental determination of the shunt impedance, but it is not convenient for the characterization of the series impedance.

International audience; This paper describes a generic Taylor series based continuation method, the so-called Asymptotic Numerical Method, to compute the bifurcation diagrams of nonlinear systems. The key point of this approach is the quadratic recast of the equations as it allows to treat in the same way a wide range of dynamical systems and their solutions. Implicit Differential-Algebraic Equations, forced or autonomous, possibly with time-delay or fractional order derivatives are handled in the same framework. The static, periodic and quasi-periodic solutions can be continued as well as transient solutions.

International audience; Self-sustained musical instruments are complex nonlinear dynamical systems that are known to produce a wealth of dynamical regimes. This includes different kinds of non-periodic sounds, which are either played on purpose or avoided depending on the cultural and musical context. We investigate non-periodic sounds produced by two types of flute-like instruments, namely, an alto recorder and traditional pan-like flutes from Central Chile. We adopt a nonlinear dynamics point of view to characterize the multiphonics produced by the alto recorder and the sonidos rajados produced by the Chilean flutes. Our results unveil the common quasiperiodic nature of the two types of sound regimes and suggest that they result from a similar physical sound production mechanism. This paves the way for a better control of non-periodic sound regimes by the instrument makers.

The lowest fingerings of the saxophone can lead to several different regimes, depending on the musician’s control and the characteristics of the instrument. This is explored in this paper through a physical model of saxophone. The harmonic balance method shows that for many combinations of musician control parameters, several regimes are stable. Time-domain synthesis is used to show how different regimes can be selected through initial conditions and the initial evolution (rising time) of the blowing pressure, which is explained by studying the attraction basin of each stable regime. These considerations are then applied to study how the produced regimes are affected by properties of the resonator. The inharmonicity between the first two resonances is varied in order to find the value leading to the best suppression of unwanted overblowing. Overlooking multistability in this description can lead to biased conclusions. Results for all the lowest fingerings show that a slightly positive inharmonicity, close to that measured on a saxophone, leads to first register oscillations for the greatest range of control parameters. A perfect harmonicity (integer ratio between the first two resonances) decreases first register production, which adds nuance to one of Benade’s guidelines for understanding sound production. Thus, this study provides some a posteriori insight into empirical design choices relative to the saxophone.

Computational optimization algorithms coupled with acoustic models of wind instruments provide instrument makers with an opportunity to explore new designs. Specifically, they enable the automatic discovery of geometries exhibiting desired resonance characteristics. In this paper, the design optimization of woodwind instruments with complex geometrical features (e.g., non-cylindrical bore profile and side holes with various radii and chimney heights) is investigated. Optimal geometric designs are searched so that their acoustic input impedance has peaks with specific target frequencies and amplitudes. However, woodwind instruments exhibit complex input impedance whose features, such as resonances, might have non-smooth evolution with respect to design variables, thus hampering gradient-based optimization. For this reason, this paper introduces new formulations of the impedance characteristics (resonance frequencies and amplitudes) using a regularized unwrapped angle of the reflection function. The approach is applied to an illustrative instrument subjected to geometric constraints similar to the ones encountered by manufacturers (a key-less pentatonic clarinet with two-registers). Three optimization problems are considered, demonstrating a strategy to simultaneously adjust several impedance characteristics on all fingerings.

International audience; The system formed by a trumpet player and his/her instrument can be seen as a non-linear dynamical system, and modeled by physical equations. Numerical tools can then be used to study these models and clarify the influence of the model parameters. The acoustic input impedance, for instance, is strongly dependent on the geometry of the air column and is therefore of primary interest for a musical instrument maker. In this study, a method of continuation of periodic solutions based on the combination of the Harmonic Balance Method (HBM) and the Asymptotic Numerical Method (ANM), is applied to a physical model of brass instruments. It allows the study of the evolution of the system where one parameter of the model (static mouth pressure) varies. This method is used to compare different B trumpets on the basis of two descriptors (hysteresis behavior and dynamic range) computed from the continuation outputs. Results show that this methodology enables to differentiate instruments in the space of the calculated descriptors. Calculations for different values of the lip parameters are also performed to confirm that the obtained categorization is independent of variations of lip parameters.

International audience; Since the founding theory established by G. Floquet more than a hundred years ago, computing the stability of periodic solutions has given rise to various numerical methods, mostly depending on the way the periodic solutions are themselves determined , either in the time domain or in the frequency domain. In this paper, we address the stability analysis of branches of periodic solutions that are computed by combining a pure Harmonic Balance Method (HBM) with an Asymptotic Numerical Method (ANM). HBM is a frequency domain method for determining periodic solutions under the form of Fourier series and ANM is continuation technique that relies on high order Taylor series expansion of the solutions branches with respect to a path parameter. It is well established now that this HBM-ANM combination is efficient and reliable, provided that the system of ODE is first of all recasted with quadratic nonlinearities, allowing an easy manipulation of both the Taylor and the Fourier series. In this context, Hill's method, a frequency domain version of Floquet theory, is revisited so as to become a by-product of the HBM applied to a quadratic system, allowing the stability analysis to be implemented in an elegant way and with good computing performances. The different types of stability changes of periodic solutions are all explored and illustrated through several academic examples, including systems that are autonomous or not, conservative or not, free or forced.

International audience; A saxophone mouthpiece fitted with sensors is used to observe the oscillation of a saxophone reed, as well as the internal acoustic pressure, allowing to identify qualitatively different oscillating regimes. In addition to the standard two-step regime, where the reed channel successively opens and closes once during an oscillation cycle, the experimental results show regimes featuring two closures of the reed channel per cycle, as well as inverted regimes, where the reed closure episode is longer than the open episode. These regimes are well-known on bowed string instruments and some were already described on the Uilleann pipes. A simple saxophone model using measured input impedance is studied with the harmonic balance method, and is shown to reproduce the same two-step regimes. The experiment shows qualitative agreement with the simulation: in both cases, the various regimes appear in the same order as the blowing pressure is increased. Similar results are obtained with other values of the reed opening control parameter, as well as another fingering.

International audience; This paper explores the analogy between a saxophone resonator and a bicylindrical resonator, sometimes called transverse saxophone or cylindrical saxophone. The dimensions of a bicylindrical resonator are optimized numerically to approximate a saxophone impedance. The target is the impedance measured on an usual saxophone. A classical gradient-based non-linear least-square fit function is used. Several cost functions corresponding to distances to the target impedance are assessed, according to their influence on the optimal geometry. Compromises appear between the frequency regions depending on the cost function. It is shown that the chosen cost functions are differentiable and locally convex. The convexity region contains the initial geometrical dimensions obtained by crude approximation of the first resonance frequency of the target. One optimal geometry is submitted to further analysis using descriptors of the impedance. Its deviations from the target saxophone are put into perspective with the discrepancies between the target saxophone and a saxophone from a different manufacture. Descriptors such as harmonicity or impedance peak ratio set the bicylindrical resonator apart from saxophone resonators, despite a good agreement of the resonance frequencies. Therefore, a reed instrument with a bicylindrical resonator could be tuned to produce the same notes as a saxophone, but due to differences in the intrinsic characteristics of the resonator, it should be considered not as a saxophone but as a distinct instrument.

International audience; In this work we apply and compare two numerical path continuation algorithms for solving algebraic equations arising when applying the Harmonic Balance Method to compute periodic regimes of nonlinear dynamical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach. This algorithm can be applied directly also to non-analytic nonlinearities. The second algorithm relies on a high-order Taylor series expansion of the solution path (the so-called Asymptotic Numerical Method) and can be formulated entirely in the frequency domain. The series expansion can be viewed as a high-order predictor equipped with inherent error estimation capabilities, which permits to avoid correction steps. The second algorithm is limited to analytic nonlinearities, and typically additional variables need to be introduced to cast the equation system into a form that permits the efficient computation of the required high- order derivatives. We apply the algorithms to selected vibration problems involving mechanical systems with polynomial stiffness, dry friction and unilateral contact nonlinearities. We assess the influence of the algorithmic parameters of both methods to draw a picture of their differences and similarities. We analyze the computational performance in detail, to identify bottlenecks of the two methods.

International audience; This article presents an extension of the Asymptotic Numerical Method combined with the Harmonic Balance Method to the continuation of periodic orbits of Delay Differential Equations. The equations can be forced or autonomous and possibly of neutral type. The approach developed in this paper requires the system of equations to be written in a quadratic formalism which is detailed. The method is applied to various systems, from Van der Pol and Duffing os-cillators to toy models of clarinet and saxophone. The Harmonic Balance Method is ascertained from a comparison to standards time-integration solvers. Bifurcation diagrams are drawn which are sometimes intricate, showing the robustness of this method.

International audience; This paper is concerned with a Taylor series based continuation algorithm, ie, the so-called Asymptotic Numerical Method (ANM). It describes a generic continuation procedure that apply the ANM principle at best, that is to say, that presents a high level of genericity without paying the price of this genericity by low computing performances. The way to quadratically recast a system of equation is now part of the method itself, and the way to handle elementary transcendental function is detailed with great attention. A sparse tensorial formalism is introduced for the internal representation of the system, which, when combines with a block condensation technique, provides a good computational efficiency of the ANM. Three examples are developed to show the performance and the versatility of the implementation of the continuation tool. Its robustness and its accuracy are explored. Finally, the potentiality of this method for complex non linear finite element analysis is enlightened by treating 2D elasticity problem with geometrical nonlinearities.

International audience; Spectra of musical instruments exhibit formants or anti-formants which are important characteristics of the sounds produced. In the present paper, it is shown that anti-formants exist in the spectrum of the mouthpiece pressure of saxophones. Their frequencies are not far but slightly higher than the natural frequencies of the truncated part of the cone. To determine these frequencies, a first step is the numerical determination of the playing frequency by using a simple oscillation model. An analytical analysis exhibits the role of the inharmonicity due to the cone truncation and the mouthpiece. A second step is the study of the input impedance values at the harmonics of the playing frequency. As a result, the consideration of the playing frequency for each note explains why the anti-formants are wider than those resulting from a Helmholtz motion observed for a bowed string. Finally numerical results for the mouthpiece spectrum are compared to experiments for three saxophones (soprano, alto and baritone). It is shown that when scaled by the length of the missing cone, the anti-formant frequencies in the mouthpiece are very similar for the three instruments. The frequencies given by the model are close to the natural frequencies of the missing cone length, but slightly higher. Finally, the numerical computation shows that anti-formants and formants might be found in the radiated pressure.

International audience; This paper addresses the numerical computation of periodic solutions of nonlinear differential systems involving fractional derivatives. For this purpose, the Harmonic Balance Method and the Asymptotic Numerical Method are combined, generalizing an approach largely followed in non-fractional systems. This enables to perform the continuation of periodic solutions of fractional systems with respect to a system parameter or to the fractional order. In the particular case of a constant fractional order, the results are validated by a successful comparison with an alternative formulation based on the diffusive representation of fractional operators. The new numerical strategy presented here allows to simulate phenomena still lacking from theoretical foundations. For example, the numerical experiments proposed here lead to a bifurcation similar to the Hopf bifurcation, well known in the case of non-fractional systems. Throughout this article, the Weyl derivative is used; its link with the classic Caputo derivative is elucidated.

International audience; A reed-like instrument having two quasi-harmonic resonances, represented by a 4-dimensional dynamical system, is studied using the continuation and bifurcation software AUTO. Bifurcation diagrams are explored with respect to the blowing pressure, with focus on amplitude and frequency evolutions along the different solution branches.

International audience; Sound production on a " coaxial saxophone " is investigated experimentally. The coaxial saxophone is a variant of the cylindrical saxophone made up of two tubes mounted in parallel, which can be seen as a low-frequency analogy of a truncated conical resonator with a mouthpiece. Initially developed for the purposes of theoretical analysis, an experimental verification of the analogy between conical and cylindrical saxophones has never been reported. The present paper explains why the volume of the cylindrical saxophone mouthpiece limits the achievement of a good playability. To limit the mouthpiece volume, a coaxial alignment of pipes is proposed and a prototype of coaxial saxophone is built. An impedance model of coaxial resonator is proposed and validated by comparison with experimental data. Sound production is also studied through experiments with a blowing machine. The playability of the prototype is then assessed and proven for several values of the blowing pressure, of the embouchure parameter, and of the instrument's geometrical parameters.

International audience; In this work we apply and compare two algorithms for setting up Harmonic Balance equations and numerical continuation of the solution path for harmonically driven mechanical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach (AFT-PreCo). The second algorithm relies on a high-order Taylor series expansion of the solution path (asymptotic numerical method) and classical Harmonic Balance formulated entirely in the frequency domain (cHB-ANM). We conclude that the cHB-ANM is suited for smooth nonlinearities, for instance geometrically nonlinear finite element models. Here, cHB-ANM avoids aliasing errors and convinces with a numerically robust adjustment of the continuation step length and a continuous representation of the solution path. For non-smooth nonlinearities such as stick-slip friction or unilateral constraints, AFT-PreCo is better suited, and convinces with high numerical robustness and efficiency.

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

International audience; Time-periodic solutions of dynamical systems can be looked for using a discretization method. This paper tests the Harmonic Balance Method (HBM) on a one-degree-of-freedom system (mass, damper, spring, belt) with a regularized friction law. Its relative error is computed with respect to the number of discretization unknowns. Despite the widespread idea that frequency methods are hardly applicable to friction problems, the HBM compares well with a classical time-domain method for this nonlinear system. The main conclusion of this article is that the HBM, without any specific optimization, is well suited for regularized friction.

International audience; Brass players use a variety of mutes to change the sound of their instrument for artistic expression. However, mutes can also modify the intonation and the playability of the muted instrument. An example is the use of a straight mute on a trombone, which makes it very difficult to play stable pedal notes.Previous studies have shown that using a straight mute establishes a parasitic acoustic resonance in the trombone. To cancel this modification, an active control device was developed and integrated into a mute, with satisfying experimental results [Meurisse et al., 2015]. With this device, the perturbed pedal notes can easily be played again.This paper investigates the ability of a physical model of brass instrument to reproduce the behaviour of the trombone pedal Bb without mute, or with an "active" or a "passive" straight mute. Linear stability analysis and time-domain simulations are used to analyse the behaviour of the model in the parameter range corresponding to the pedal note. Numerical results are compared for different models of instruments: a trombone, a trombone with a straight (passive) mute, an a trombone with an active mute. It is shown that the simple physical model considered behaves similarly to what is experienced with real instruments: the playing of the pedal note is perturbed with a passive mute, whereas the model of trombone with the experimental active mute gives results very similar to those obtained with an open trombone.

International audience; A physical model of brass instrument is considered in this paper : a one degree-of-freedom outward-striking valve for the lips, non-linearly coupled to a modal representation of the air column. It is studied through Linear Stability Analysis (LSA) of the equilibrium solution. This approach provides the threshold blowing pressure value, at which instability occurs, and the instability frequency value. The relevance of the results of this method is theoretically limited to the neighbourhood of the equilibrium solution. This paper checks the efficiency of LSA to understand the behaviour of the model computed through time-domain simulations. As expected, a good agreement is observed between LSA and numerical simulations of the complete nonlinear model around the oscillation threshold. For blowing pressures far above the oscillation threshold, the picture is more contrasted. In most of the cases tested, a periodic regime coherent with the LSA results is observed, but over-blowing, quasi-periodicity and period-doubling also occur. Interestingly, LSA predicts the production of the pedal note by a trombone, for which only nonlinear hypotheses have been previously proposed. LSA also predicts the production of a saxhorn note which, although known to musicians, has barely been documented.

International audience; Periodic solutions of systems with friction are difficult to investigate because of the irregular nature of friction laws. This paper examines periodic solutions and most notably stick-slip, on a simple one-degre-of-freedom system (mass, spring, damper, belt), with Coulomb's friction law, and with a regularized friction law (i.e. the friction coefficient becomes a function of relative speed, with a stiffness parameter). With Coulomb's law, the stick-slip solution is constructed step by step, which gives a usable existence condition. With the regularized law, the Asymptotic Numerical Method and the Harmonic Balance Method provide bifurcation diagrams with respect to the belt speed or normal force, and for several values of the regularization parameter. Formulations from the Coulomb case give the means of a comparison between regularized solutions and a standard reference. With an appropriate definition, regularized stick-slip motion exists, its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force.

International audience; A time-domain numerical modeling of brass instruments is proposed. On one hand, outgoing and incoming waves in the resonator are described by the Menguy-Gilbert model, which incorporates three key issues: nonlinear wave propagation, viscothermal losses, and a variable section. The non-linear propagation is simulated by a TVD scheme well-suited to non-smooth waves. The fractional derivatives induced by the viscothermal losses are replaced by a set of local-in-time memory variables. A splitting strategy is followed to couple optimally these dedicated methods. On the other hand, the exciter is described by a one-mass model for the lips. The Newmark method is used to integrate the nonlinear ordinary differential equation so-obtained. At each time step, a coupling is performed between the pressure in the tube and the displacement of the lips. Finally, an extensive set of validation tests is successfully completed. In particular, self-sustained oscillations of the lips are simulated by taking into account the nonlinear wave propagation in the tube. Simulations clearly indicate that the nonlinear wave propagation has a major influence on the timbre of the sound, as expected. Moreover, simulations also highlight an influence on playing frequencies, time envelopes and on the playability of the low frequencies in the case of a variable lips tension.

International audience; Single-reed instruments can produce multiphonic sounds when they generate quasi-periodic oscillations. The aim of this article is to identify a minimal model of a single reed-instrument producing quasi-periodic oscillations. To better understand the influence of model parameters on the production of quasi-periodic regimes, the mapping between parameters and quasi-periodic regimes is explicitly identified using a support vector machine (SVM) classifier. SVMs enable the construction of boundaries between quasi-periodic and periodic regimes that are explicitly defined in terms of the parameters. Results and conclusions obtained from the numerical model are compared to published experiments related to the the production of quasi-periodic oscillations with an alto saxophone. This qualitative comparison highlights the influence of key parameters on the production of multiphonic sounds.

This paper presents an analysis of the effects of noise and precision on a simplified model of the clarinet driven by a variable control parameter. When the control parameter is varied the clarinet model undergoes a dynamic bifurcation. A consequence of this is the phenomenon of bifurcation delay: the bifurcation point is shifted from the static oscillation threshold to an higher value called dynamic oscillation threshold. In a previous work [8], the dynamic oscillation threshold is obtained analytically. In the present article, the sensitivity of the dynamic threshold on precision is analyzed as a stochastic variable introduced in the model. A new theoretical expression is given for the dynamic thresholds in presence of the stochastic variable, providing a fair prediction of the thresholds found in finite-precision simulations. These dynamic thresholds are found to depend on the increase rate and are independent on the initial value of the parameter, both in simulations and in theory., 14 pages

Reed instruments are modeled as self-sustained oscillators driven by the pressure inside the mouth of the musician. A set of nonlinear equations connects the control parameters (mouth pressure, lip force) to the system output, hereby considered as the mouthpiece pressure. Clarinets can then be studied as dynamical systems, their steady behavior being dictated uniquely by the values of the control parameters. Considering the resonator as a lossless straight cylinder is a dramatic yet common simplification that allows for simulations using nonlinear iterative maps. In this paper, we investigate analytically the effect of a time-varying blowing pressure on the behavior of this simplified clarinet model. When the control parameter varies, results from the so-called dynamic bifurcation theory are required to properly analyze the system. This study highlights the phenomenon of bifurcation delay and defines a new quantity, the dynamic oscillation threshold. A theoretical estimation of the dynamic oscillation threshold is proposed and compared with numerical simulations., 14 pages

International audience; Nonlinear systems can have periodic solutions evolving with the parameters of the system. Studying this evolution (numerical continuation of solutions) uncovers sought-after regimes in musical acoustics : many musical instruments rely on auto-oscillation, that is, the excitation of a nonlinear system coupled with a linear resonator, where some parameters may be adjusted by the player. Periodic solutions can be approximated as truncated Fourier series (Harmonic Balance Method) ; the period is one of the unknowns. Several stable or unstable solutions can be found for the same playing parameters thanks to continuation. An important challenge is the continuation of quasi-periodic solutions, also called multiphonic sounds by musicians. Depending on the context, these oscillation regimes are considered pleasant (jazz or contemporary music for instance) or unpleasant (classical music). We developed a method based on double Fourier series, coupled with a continuation technique. The two base frequencies are unknowns and incommensurable. The system is reformulated as quadratic in order to allow straight interface with previous work on periodic harmonic balance. This method is illustrated on simple models relevant to musical acoustics, though the method can be applied to many nonlinear problems, without a priori knowledge of the solutions.

L’usage de sourdines est courant pour modifier le timbre des instruments de la famille des cuivres. Cependant, ces sourdines peuvent également modifier la jouabilité de l’instrument. Ainsi, il devient plus difficile de jouer certaines notes pédales d’un trombone équipé d’une sourdine sèche,voire impossible à la nuance pianissimo.Des mesures précédentes ont montré que l’utilisation d’une sourdine sèche provoque principalement l’apparition d’un mode de résonance supplémentaire dans l’instrument, qui se traduit par un pic d’impédance d’entrée supplémentaire, de faible amplitude, entre le premier et le secondpic. Un dispositif de contrôle actif, ajouté à une sourdine sèche permet de supprimer ce pic d’impédance parasite sans modifier le timbre de l’instrument. De plus l’expérience a montré qu’un trombone équipé de cette sourdine permet aux trombonistes de jouer le Sib pédale sansdifficulté particulière.Ce travail propose une analyse de la dégradation de la jouabilité du Sib pédale par un trombone équipé de sourdines sèches passive et active. L’approche consiste à réaliser une analyse de stabilité linéaire de la solution non oscillante d’un modèle de trombone. Le modèle retenu couple une valve en dehors à un degré de liberté au résonateur de l’instrument représenté par la décomposition modale de son impédance d’entrée. Il est ainsi possible de réaliser l’analyse de stabilité du trombone avec sourdine sèche, avec sourdine active « allumée » ou « éteinte », et d’évaluer la pertinence du modèle physique utilisé pour interpréter les résultats expérimentaux décrits par les publications antérieures.

International audience; Properties of small amplitude oscillations of the single reed woodwind instruments near the oscillation threshold are investigated. Analytical formulae with explicit dependence on the physical parameters of the instrument and instrumentist allow to determine the bifurcation point, the nature of the bifurcation, the amplitude of the first harmonics and the oscillation frequency. The model, which takes the reed dynamics into account, is entirely expressed in the frequency domain and its behaviour is analysed. An application to a model of saxophone is proposed and compared with numerical results obtained through the harmonic balance technique and a time domain simulation, showing excellent agreements.

Combinig the harmonic balance method (HBM) and a continuation method is a well-known technique to follow the periodic solutions of dynamical systems when a control parameter is varied. However, since deriving the algebraic system containing the Fourier coefficients can be a highly cumbersome procedure, the classical HBM is often limited to polynomial (quadratic and cubic) nonlinearities and/or a few harmonics. Several variations on the classical HBM, such as the incremental HBM or the alternating frequency/time domain HBM, have been presented in the literature to overcome this shortcoming. Here, we present an alternative approach that can be applied to a very large class of dynamical systems (autonomous or forced) with smooth equations. The main idea is to systematically recast the dynamical system in quadratic polynomial form before applying the HBM. Once the equations have been rendered quadratic, it becomes obvious to derive the algebraic system and solve it by the so-called ANM continuation technique. Several classical examples are presented to illustrate the use of this numerical approach., Comment: PACS numbers: 02.30.Mv, 02.30.Nw, 02.30.Px, 02.60.-x, 02.70.-c

International audience; This article uses a basic model of a reed instrument , known as the lossless Raman model, to determine analytically the envelope of the sound produced by the clarinet when the mouth pressure is increased gradually to start a note from silence. Using results from dynamic bifur-cation theory, a prediction of the amplitude of the sound as a function of time is given based on a few parameters quantifying the time evolution of mouth pressure. As in previous uses of this model, the predictions are expected to be qualitatively consistent with simulations using the Raman model, and observations of real instruments. Model simulations for slowly variable parameters require very high precisions of computation. Similarly, any real system, even if close to the model would be affected by noise. In order to describe the influence of noise, a modified model is developed that includes a stochastic variation of the parameters. Both ideal and stochastic models are shown to attain a minimal amplitude at the static oscillation threshold. Beyond this point, the amplitude of the oscillations increases exponentially, although some time is required before the oscillations can be observed at the " dynamic oscillation threshold ". The effect of a sudden interruption of the growth of the mouth pressure is also studied, showing that it usually triggers a faster growth of the oscillations.

International audience; The aim of this work is to highlight experimentally how inharmonicity of the bore resonance frequencies of an alto saxophone influence the nature of the oscillation regimes. A variable volume branching from the neck of an alto sax at an appropriate position allows to change the frequency of the first resonance independently from the second. A blowing machine with artificial lips is used to make the saxophone play while controlling independently the control parameters : the blowing pressure and an embouchure parameter. Values of these parameters are estimated experimentally through the measurement of the nonlinear characteristics linking the mean air flow blown into the instrument to the static pressure difference across the reed. Experiments with different values of the control parameters as well as of the inharmonicity produce different kinds of oscillation regimes. These regimes are categorized through the analysis of the pressure signal inside the mouthpiece. The resulting maps demonstrate that the emergence of quasi-periodic regimes, and their extent, depend on the level of inharmonicity, but also on the values of the control parameters. Periodic regimes playable by choosing appropriate values of the control parameters also differ according to the level of inharmonicity, a higher inharmonicity facilitating the emergence of the third register.

International audience; Since they correspond to a jump from a given note to another one, the mouth pressure thresholds leading to regime changes are particularly important quantities in flute-like instruments. In this paper, a comparison of such thresholds between an artificial mouth, an experienced flutist and a non player is provided. It highlights the ability of the experienced player to considerabily shift regime change thresholds, and thus to enlarge its control in terms of nuances and spectrum. Based on recent works on other wind instruments and on the theory of dynamic bifurcations, the hypothe- sis is tested experimentally and numerically that the dynamics of the blowing pressure influences regime change thresholds. The results highlight the strong influence of this parameter on thresholds, suggesting its wide use by experienced musicians. Starting from these observations and from an analysis of a physical model of flute-like instruments, involving numerical continuation methods and Floquet stability analysis, a phenomenological modelling of regime change is proposed and validated. It allows to predict the regime change thresholds in the dynamic case, in which time variations of the blowing pressure are taken into account.

International audience; A continued fraction expansion to the immittances defining viscothermal wave propagation in acylindrical tube has been presented recently in this journal, intended as a starting point for timedomain numerical method design. Though the approximation has the great benefit of passivity, orpositive realness under truncation, its convergence is slow leading to approximations of high orderin practice. Other passive structures, when combined with optimisation methods, can lead to goodaccuracy over a wide frequency range, and for relatively low order.

Several factors explain the difference between the resonance frequencies of the resonator of reed instruments and their playing frequency: the flow rate due to the reed displacement, the reed dynamics, the resonator inharmonicity, and the temperature gradient. For conical instruments, the truncation of the cone entails non-negligible inharmonicity effects. This is true even if the inharmonicity is reduced by a proper choice of the mouthpiece volume. The playing frequency was calculated by using a minimum, ab initio model of self-sustained oscillations, and compared to analytical results. Furthermore, the “reactive power rule” (Boutillon, 1989) provides a link between the pressure spectrum and inharmonicity. Compared to the natural frequency of the complete cone, which is a well known approximation, the playing frequency is slightly higher: typical discrepancies are 50 cents for short truncated cones (higher notes) and 10 cents for long truncated cones (lower notes). This discrepancy, due to inharmonicity,...

This paper proposes a methodology to map the various acoustic regimes of wind instruments. The maps can be generated in a multi-dimensional space consisting of design, control parameters, and initial conditions. The bound- aries of the maps are obtained explicitly in terms of the parameters using a support vector machine (SVM) classifier as well as a dedicated adaptive sam- pling scheme. The approach is demonstrated on a simplified clarinet model for which several maps are generated based on different criteria. Examples of computation of the probability of occurrence of a specific acoustic regime are also provided. In addition, the approach is demonstrated on a design optimization example for optimal intonation.

International audience; Recent works on simplified clarinet models using results from dynamic bifurcation theory have allowed to predict the evolution of the amplitude of sound (the amplitude envelope) for a gradual increase of the blowing pressure. The unrealistic model that predicted the amplitudes to attain very small values, far below the precision of a computer, was later corrected by the addition of stochastic noise to the model. The two models are useful in explaining and understanding why the oscillations appear with a delay relative to the threshold of oscillation that is predicted by purely steady-state models. Both the model of the instrument and that of the noise are extremely simplistic, raising the question of its applicability to real instruments. These models can however be made gradually more complex by introducing more realistic details in the reed or in the resonator, and applying parameter profiles with more complex shapes or noise amplitudes. This presentation shows the differences encountered in the time-evolution of the acoustic wave simulated using two models of different complexity, one with an instantaneous reflection function, another with dispersion. The article explores to which extent can the dynamic predictive model be used to describe the time evolution of more realistic models, and hopefully that of the real instrument.

Simple models of clarinet instruments based on iterated maps have been used in the past to successfully estimate the threshold of oscillation of this instrument as a function of a constant blowing pressure. However, when the blowing pressure gradually increases through time, the oscillations appear at a much higher value, called dynamic oscillation threshold, than what is predicted in the static case. This is known as bifurcation delay, a phenomenon studied in [1,2] for a clarinet model. In particular the dynamic oscillation threshold is predicted analytically when the blowing pressure is linearly increased. However, the mouth pressure cannot grow indefinitely. During a note attack, after an increasing phase, the musician stabilizes the mouth pressure. In the present work, the analytical prediction of the dynamic oscillation threshold is extended to a situations in which the mouth pressure approaches a steady state pressure according to an exponential time profile. The predictions still show a good agreement with simulation of the simple clarinet-model. This situation is compared in terms of dynamic oscillation bifurcation., International Symposium on Musical Acoustics, Le Mans : France (2014)

International audience; In a self-sustained instrument, the sound is maintained as long as energy is supplied to the oscillating system, and the control parameters are kept in a range that permits a periodic regime. Stopping a note is thus a voluntary action, in opposition to what can happen in plucked strings, membranophones, xylophones etc. In a reed instrument two basic methods of stopping the note can be identified, although usually the musician will use a combination of these. One is to stop the oscillation of the reed by using the tongue, or less likely by using a stronger biting force that can close the reed. The other is to cut the supply in air from the lungs, or to reduce it below a critical level that is called the threshold of oscillation. These two methods have very different consequences on the characteristics of the sound of the note extinction. Using simulations and experiments, this presentation identifies the main characteristics of each of the extinction methods. Whenever possible, an investigation into the causes of the effects identified in the description are justified with basic models or further experiments.

International audience; In this paper we exploit the method of numerical continuation combined with orthogonal collocation to determine the steady-state oscillations of a model of flute-like instruments that is formulated as a nonlinear neutral delay differential equation. The delay term in the model causes additional complications in the analysis of the behaviour of the model, in contrast to models of other wind instruments which are formulated as ordinary differential equations. Fortunately, numerical continuation provides bifurcation diagrams that show branches of stable and unstable static and periodic solutions of the model and their connections at bifurcations, thus enabling an in depth analysis of the global dynamics. Furthermore, it allows us to predict the thresholds of the different registers of the flute-like instrument and thus to explain the classical phenomenon of register change and the associated hysteresis.

The high-order purely frequency-based harmonic balance method (HBM) presented by Cochelin and Vergez (2009) [1] and extended by Karkar et al. (2013) [2] now allows to follow the periodic solutions of regularized non-smooth systems (stiff systems). This paper compares its convergence property to a reference method in applied mathematics: orthogonal collocation with piecewise polynomials. A first test is conducted on a nonlinear smooth 2 degree-of-freedom spring mass system, showing better convergence of the HBM. The second test is conducted on a one degree-of-freedom vibro-impact system with a very stiff regularization of the impact law. The HBM continuation of the nonlinear mode was found to be very robust, even with a very large number of harmonics. Surprisingly, the HBM was found to have a better convergence than the collocation method for this vibro-impact system. (C) 2014 Elsevier Ltd. All rights reserved.

Using an artificial mouth with an accurate pressure control, the onset of the pressure oscillations inside the mouthpiece of a simplified clarinet is studied experimentally. Two time profiles are used for the blowing pressure: in a first set of experiments the pressure is increased at constant rates, then decreased at the same rate. In a second set of experiments the pressure rises at a constant rate and is then kept constant for an arbitrary period of time. In both cases the experiments are repeated for different increase rates. Numerical simulations using a simplified clarinet model blown with a constantly increasing mouth pressure are compared to the oscillating pressure obtained inside the mouthpiece. Both show that the beginning of the oscillations appears at a higher pressure values than the theoretical static threshold pressure, a manifestation of bifurcation delay. Experiments performed using an interrupted increase in mouth pressure show that the beginning of the oscillation occurs close to the stop in the increase of the pressure. Experimental results also highlight that the speed of the onset transient of the sound is roughly the same, independently of the duration of the increase phase of the blowing pressure., Comment: 14 pages

International audience; Nonlinear Energy Sinks (NES) are used to passively reduce the amplitude of vibrations. This reduction is made possible by introducing a nonlinearly stiffening behavior in the NES, which might lead to an irreversible transfer of energy between the main system (e.g., a building) and the NES. However, this irreversible transfer, and therefore the efficiency of the NES, is strongly dependent on the design parameters of the NES. In fact, the efficiency of the NES might be so sensitive to changes in design parameters and other factors (e.g., initial conditions) that it is dis-continuous, switching from efficiency to inefficiency for a small perturbation of parameters. For this reason, this work introduces a novel technique for the optimization under uncertainty of NES. The approach is based on a support vector machine classi-fier, which is insensitive to discontinuities and allows one to efficiently propagate uncertainties. This enables one to efficiently solve an optimization under uncertainty problem. The various techniques presented in this paper are applied to an analytical NES example.

International audience; The stationary behavior of flute-like instruments is fairly well understood. Models and experimental studies allow to predict and to understand the influences of the principal parameters (flow velocity, position of the edge, etc) on the sound if these parameters stay constant in time. Depending on the instrument, these parameters can be fixed by the flute maker or by the musician. In musical playing, the musician plays on them to act on the sound. Some parameters can vary rapidly, like during the attack transients. The response of the instruments to these variations is crucial to determine their quality, in musical use. The target of this study is to understand the influences of these parameters on the characteristics of attack transients. The study presented is based on measurements on an actual recorder in musical context. Parameters of attack transient for acoustic and musician control are extracted from the data. Relations between these parameters are searched by taking into account the characteristics of the instruments. This study is a first step in the understanding of the possibilities of the musicians' control and of the physical limitations.