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

Bergeot, Baptiste; Almeida, André; Vergez, Christophe; Gazengel, Bruno

doi:10.1007/s11071-013-0991-8

Cited by 5 publications

(16 citation statements)

References 19 publications

(64 reference statements)

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“…This result is surprising at first glance since the consequences of the nonlinear propagation are expected to vanish at the oscillation threshold where b (u ± ) 2 2 ≪ a|u ± | in (14b). However, the behavior of dynamic bifurcations thresholds can be counterintuitive, even when considering small perturbations [47]. The so-called bifurcation delay observed here is around 0.2 s, which corresponds to a pressure difference around 800 Pa.…”

Section: Resultsmentioning

confidence: 69%

Time-Domain Numerical Modeling of Brass Instruments Including Nonlinear Wave Propagation, Viscothermal Losses, and Lips Vibration

Berjamin¹,

Lombard²,

Vergez³

et al. 2017

Acta Acustica united with Acustica

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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 nonlinear 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.

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Section: Resultsmentioning

confidence: 69%

Time-Domain Numerical Modeling of Brass Instruments Including Nonlinear Wave Propagation, Viscothermal Losses, and Lips Vibration

Berjamin¹,

Lombard²,

Vergez³

et al. 2017

Acta Acustica united with Acustica

Self Cite

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“…An analytical study to explain this observation has already been carried out on a discrete-time model of the clarinet 1 [5,6]. Considering a blowing pressure linearly increasing over time, the following results have been obtained in these works, which are typical of what is known in the field of dynamical bifurcation of discrete-time systems.…”

Section: Introductionmentioning

confidence: 71%

Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments

Bergeot

Vergez

2022

Nonlinear Dyn

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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.

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“…For a constantly increasing parameter, the dynamic oscillation threshold γ dt [13,16] gives the approximate value of the mouth pressure parameter for which an audible sound appears, or in other terms, the distance from the invariant w curve becomes "macroscopic". When the linear growth of the mouth pressure is suddenly stopped at n = M and then kept constant at a value γ M , two situations must be distinguished:…”

Section: Discussionmentioning

confidence: 99%

“…Analytical reasoning [13] based on dynamic bifurcation theory [14,15] predicts a delay in the threshold of oscillation for a linearly increasing mouth pressure, but the exact value of mouth pressure at which it occurs is only valid for simulations performed with very high precision. The threshold observed with normal precision simulations can only be explained with a modified theory [16] using stochastic perturbations [14].…”

Section: Introductionmentioning

confidence: 99%

“…In section 2, the model of the clarinet used in this work is briefly presented, as well as some of its known properties. The remaining of this section provides a brief overview of the key concepts that are needed for the present article (most of these concepts are described with more details in two articles by the authors [13,16]). Section 3 describes the calculation of the envelope of the oscillations relative to the invariant curve, firstly in an ideal case with infinite precision, then with limited precision or noise (section 3.3).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Almeida¹,

Bergeot²,

Vergez³

et al. 2015

Acta Acustica united with Acustica

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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 bifurcation 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.

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

Cited by 5 publications

References 19 publications

Time-Domain Numerical Modeling of Brass Instruments Including Nonlinear Wave Propagation, Viscothermal Losses, and Lips Vibration

Time-Domain Numerical Modeling of Brass Instruments Including Nonlinear Wave Propagation, Viscothermal Losses, and Lips Vibration

Analytical prediction of delayed Hopf bifurcations in a simplified stochastic model of reed musical instruments

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

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