Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure : influence of noise

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.Comment: 14 page

Analytical Determination of the Attack Transient in a Clarinet With Time-Varying Blowing Pressure

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

Naissance des oscillations dans les instruments de type clarinette à paramètre de contrôle variable.

This research is a contribution to the study of attack transients in clarinet-like instruments. The main objective is to understand the behavior of the instrument when the mouth pressure is increased slowly through time at a constant rate.Although previous research proves that oscillations can appear at a value of the static oscillation threshold, numerical simulations and in vitro experiments show that for gradual increases of the mouth pressure, the audible sound generally appears when mouth pressure reaches a much higher value, called the dynamic oscillation threshold. This phenomenon is referred to as bifurcation delay in this work.A major part of this work follows an analytical approach, using the foundations of dynamic bifurcation theory to study the bifurcation delay in the simple "Raman" clarinet model. The properties of the dynamic oscillation threshold are related to indicators of the time variation of the mouth pressure such as its initial value and its slope. One of the remarkable features of the bifurcation delay is its strong dependence on noise, including that arising from round-off errors of the computer. The properties of the dynamic threshold are different according to whether the noise can be ignored or not.Additionally, an artificial mouth is used on a clarinet-like instrument to show that the bifurcation delay is not only a numerical phenomenon. Experimental observations performed on a clarinet-like instrument blown by an artificial mouth prove that bifurcation delay exists also on real-life systems. These observations show that the properties of the bifurcation delay observed in low-precision simulations are similar to experimental ones.Ce travail de recherche est une contribution à l'étude des transitoires d'attaque dans les instruments de type clarinette. L'objectif est d'analyser le comportement de l'instrument en réponse à une variation lente et linéaire de la pression dans la bouche du musicien.Dans des simulations numériques ou des expériences in vitro, lorsque la pression dans la bouche du musicien varie lentement et linéairement dans le temps, on observe en général l'apparition du son lorsque la pression dans la bouche atteint une valeur, appelée seuil d’oscillation dynamique, supérieure au seuil d'oscillation statique. L'apport principal de ce travail est d'interpréter ce phénomène par la présence d'un retard à la bifurcation.L'approche analytique est privilégiée. La contribution majeure de ce doctorat est de comprendre les fondements de la théorie de la bifurcation dynamique afin d’étudier le retard à la bifurcation dans le modèle de clarinette dit "de Raman". Les propriétés du seuil dynamique d’oscillation sont ainsi reliées aux caractéristiques de la variation temporelle de la pression dans la bouche (sa valeur initiale et sa pente). L'une des caractéristiques notoires du retard à la bifurcation est sa grande dépendance au bruit, même s’il provient des erreurs d’arrondi de l’ordinateur. Les propriétés du seuil dynamique changent selon que le bruit peut être ignoré ou non.Nous montrons ensuite expérimentalement à l'aide d’une bouche artificielle et d'une clarinette de laboratoire que le retard à la bifurcation n'est pas qu'un phénomène numérique. Il est ainsi mis en évidence expérimentalement et ses propriétés sont également étudiés et comparées avec celles obtenues dans le cas numérique

Attack transients in clarinet models with different complexity - a comparative view

International audienceRecent 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

Passive suppression of helicopter ground resonance instability by means of a strongly nonlinear absorber

International audienceIn this paper, we study a problem of passive suppression of helicopter Ground Resonance (GR) using a single degree freedom Nonlinear Energy Sink (NES). GR is a dynamic instability involving the coupling of the blades motion in the rotational plane (i.e. the lag motion) and the helicopter fuselage motion. A reduced linear system reproducing GR instability is used. It is obtained using successively Coleman transformation and binormal transformation. The analysis of the steadystate responses of this model is performed when a NES is attached on the helicopter fuselage. The NES involves an essential cubic restoring force and a linear damping force. The analysis is achieved applying complexification-averaging method. The resulting slow-flow model is finally analyzed using multiple scale approach. Four steady-state responses corresponding to complete suppression, partial suppression through strongly modulated response, partial suppression through periodic response and no suppression of the GR are highlighted. An algorithm based on simple criterions is developed to predict these steady-state response regimes. Numerical simulations of the complete system confirm this analysis of the slow-flow dynamics. A parametric analysis of the influence of the NES damping coefficient and the rotor speed on the response regime is finally proposed

Prediction of the dynamic oscillation threshold in a clarinet model with a linearly increasing blowing pressure

14 pagesInternational audienceReed 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

Prediction of the dynamic oscillation threshold of a clarinet model: comparison between analytical predictions and simulation results

International audienceSimple 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 than what is predicted in the static case. This is known as bifurcation delay, a phenomenon studied in [1] for a clarinet model. In numerical simulations the bifurcation delay showed a strong sensitivity to numerical precision

Sparse polynomial chaos expansion for stability analysis of a clutch system with uncertain parameters

In the transmission systems of vehicles, unforced vibrations can be observed during the sliding phase of clutch engagement. These vibrations are due to frictional forces and may generate noise. Several studies have shown that the stability of such friction systems is highly sensitive to parameters (e.g. friction law, damping) which lead to significant dispersion. Therefore, uncertain parameters must be considered in the stability analysis of a clutch system. In several studies of the literature, the usual generalized polynomial chaos (PC) expansion has been used to study the stability of a clutch system using non-intrusive techniques. However, non-intrusive techniques require a number of model evaluations (i.e. the computational cost) which can become prohibitive when the studied system has a large number of uncertain parameters. To remedy this problem, in this work, we use the sparse polynomial chaos expansion which has been recently developed in reliability domain. The method is compared to the reference Monte Carlo method and with the usual PC expansion in the context of the stability analysis of a clutch system. The results show that the use of the sparse PC allows a remarkable reduction of the computational cost by ensuring a high accuracy compared with the usual PC expansion

Response of an artificially blown clarinet to different blowing pressure profiles

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 page

Vibrations non linéaires

MasterCe cours est donné aux étudiants de 5ème année de l'option Ingénierie Mécanique et Conception de l'INSA Centre Val de Loire et du Master de Mécanique en région Centre Val de Loire (Master mention Mécanique, Génie Civil, Matériaux, Structures). Il s'adresse en général aux étudiants en école d'ingénieur ou en Master de Mécanique désireux d'obtenir un cours introductif aux vibrations non linéaires.Le Chapitre 1 permet de mettre en contexte le cours en présentant d'abord quelques généralités sur non-linéarités en mécanique. Il présente ensuite, à l'aide d'exemples, certaines propriétés propres aux systèmes vibratoires non linéaires qui seront étudier en détails dans ce cours. Le Chapitre 2 détaille les techniques qui serviront à l'analyse des systèmes vibratoires non linéaires effectuées dans les chapitres qui suivent. Le forçage harmonique de l'oscillateur de Duffing est étudié dans le Chapitre 3. La résonance primaire est examinée en Section 3.2 et les résonances secondaires (super-harmonique et sous-harmonique) en Section 3.3. Le chapitre 4 est consacré à l'analyse de systèmes auto-oscillants. Le cas d'un amortissement négatif induit par le frottement est étudié en Section 4.1 et celui de l'instabilité aéroélastique d'une aile d'avion en Section 4.2