arXiv subjects

Jean Kergomard

Publications and source records attributed to Jean Kergomard.

18 recordsLinked to original sources

arXiv2016

An Analytical Prediction of the Bifurcation Scheme of a Clarinet-Like Instrument: Effects of Resonator Losses

The understanding of the relationship between excitation parameters andoscillation regimes is a classical topic concerning bowed stringinstruments. The paper aims to study the case of reed woodwinds and attemptsto find consequences on the ease of playing.In the minimum model of clarinet-like instruments, three parameters areconsidered: i) the mouth pressure, ii) the reed opening at rest, iii) thelength of the resonator \ assumed to be cylindrical. Recently asupplementary parameter was added: the loss parameter of the resonator(using the \textquotedblleft Raman model\textquotedblright, that considersresonator losses to be independent of frequency). This allowed explainingthe extinction of sound when the mouth pressure becomes very large. Thepresent paper presents an extension of the paper by Dalmont et al (JASA,2005), searching for a diagram of oscillation regimes with respect to thereed opening and the loss parameter. An alternative method is used, whichallows easier generalization and simplifies the calculation. The emphasis is done on the emergencebifurcation: for very strong losses, it can be inverse, similarly to theextinction one for weak losses. The main part of the calculations areanalytical, giving clear dependence of the parameters. An attempt to deducemusical consequences for the player is given.

physics.class-ph↗

arXiv2016

Propagation of acoustic waves in two waveguides coupled by perforations. II. Application to periodic lattices of finite length

The paper deals with the generic problem of two waveguides coupled by perforations, which can be perforated tube mufflers without or with partitions, possibly with absorbing materials. Other examples are ducts with branched resonators of honeycomb cavities , which can be coupled or not, and splitter silencers. Assuming low frequencies, only one mode is considered in each guide. The propagation in the two waveguides can be very different, thanks e.g. to the presence of constrictions. The model is a discrete, periodic one, based upon 4th-order impedance matrices and their diagonalization. All the calculation is analytical, thanks to the partition of the matrices in 2nd-order matrices, and allows the treatment of a very wide types of problems. Several aspects are investigated: the local or non-local character of the reaction of one guide to the other; the definition of a coupling coefficient; the effect of finite size when a lattice with n cells in inserted into an infinite guide; the relationship between the Insertion Loss and the dispersion. The assumptions are as follows: linear acoustics, no mean flow, rigid wall. However the effect of the series impedance of the perforations, which is generally ignored, is taken into account, and is dis- cussed. When there are no losses, it is shown that, for symmetry reasons, the cutoff frequencies depend on either the series impedance or the shunt admittance , and are the eigenfrequencies of the cells of the lattice, with zero-pressure or zero-velocity at the ends of the cells.

physics.class-ph↗

arXiv2014

Statistical Estimation of Mechanical Parameters of Clarinet Reeds Using Experimental and Numerical Approaches

A set of 55 clarinet reeds is observed by holography, collecting 2 series of measurements made under 2 different moisture contents, from which the resonance frequencies of the 15 first modes are deduced. A statistical analysis of the results reveals good correlations, but also significant differences between both series. Within a given series, flexural modes are not strongly correlated. A Principal Component Analysis (PCA) shows that the measurements of each series can be described with 3 factors capturing more than $90\%$ of the variance: the first is linked with transverse modes, the second with flexural modes of high order and the third with the first flexural mode. A forth factor is necessary to take into account the individual sensitivity to moisture content. Numerical 3D simulations are conducted by Finite Element Method, based on a given reed shape and an orthotropic model. A sensitivity analysis revels that, besides the density, the theoretical frequencies depend mainly on 2 parameters: $E_L$ and $G_{LT}$. An approximate analytical formula is proposed to calculate the resonance frequencies as a function of these 2 parameters. The discrepancy between the observed frequencies and those calculated with the analytical formula suggests that the elastic moduli of the measured reeds are frequency dependent. A viscoelastic model is then developed, whose parameters are computed as a linear combination from 4 orthogonal components, using a standard least squares fitting procedure and leading to an objective characterization of the material properties of the cane \textit{Arundo donax}.

physics.class-ph↗

arXiv2014

Transients in porous media: asymptotic time-domain Green functions and limits of current frequency-domain models

Time domain responses of porous media have been studied by some authors, but generally the possible descriptions have been given in the frequency domain. The aim of this paper, limited to materials with rigid skeleton considered as equivalent fluids, is to compare the descriptions by Johnson-Allard ($JA$%) as well as by Pride-Lafarge ($PL$) with i) some analytical, approximate formulas, based upon asymptotic high frequency expansion ; ii) the exact formula by Zwikker and Kosten for the case of cylindrical pores. The paper starts with a short summary of the statement of the different general full frequency models ($JA$ and $PL).$ The Green function in the time domain is shown to exhibit interesting properties of materials. In particular the maximum response depends on one dimensionless parameter only, which is denoted $ξ$ and is the ratio of the travelled distance to the product of the \textquotedblleft frozen\textquotedblright\ sound speed and a characteristic viscous relaxation time. The distance $ξ$ is related to a time domain Stokes number. The numerical computation of the Green function is done by FFT, with some precautions, because of the importance of the higher frequencies on the response shape. The $PL$ description is shown to be the best full frequency general model, but some discrepancies with the exact model appear at short times or short distances. When the distance $ξ$ increases from zero, the asymptotic expansion shows that the maximum of the Green function decreases first as $1/ξ^{2}$, then exponentially.

cond-mat.soft↗

arXiv2013

External Tonehole Interactions in Woodwind Instruments

The classical Transfer-Matrix Method (TMM) is often used to calculate the input impedance of woodwind instruments. However, the TMM ignores the possible influence of the radiated sound from toneholes on other open holes. In this paper a method is proposed to account for external tonehole interactions. We describe the Transfer-Matrix Method with external Interaction (TMMI) and then compare results using this approach with the Finite Element Method (FEM) and TMM, as well as with experimental data. It is found that the external tonehole interactions increase the amount of radiated energy, reduce slightly the lower resonance frequencies, and modify significantly the response near and above the tonehole lattice cutoff frequency. In an appendix, a simple perturbation of the TMM to account for external interactions is investigated, though it is found to be inadequate at low frequencies and for holes spaced far apart.

physics.class-ph↗

arXiv2013

The logical clarinet: numerical optimization of the geometry of woodwind instruments

The tone hole geometry of a clarinet is optimized numerically. The instrument is modeled as a network of one dimensional transmission line elements. For each (non-fork) fingering, we first calculate the resonance frequencies of the input impedance peaks, and compare them with the frequencies of a mathematically even chromatic scale (equal temperament). A least square algorithm is then used to minimize the differences and to derive the geometry of the instrument. Various situations are studied, with and without dedicated register hole and/or enlargement of the bore. With a dedicated register hole, the differences can remain less than 10 musical cents throughout the whole usual range of a clarinet. The positions, diameters and lengths of the chimneys vary regularly over the whole length of the instrument, in contrast with usual clarinets. Nevertheless, we recover one usual feature of instruments, namely that gradually larger tone holes occur when the distance to the reed increases. A fully chromatic prototype instrument has been built to check these calculations, and tested experimentally with an artificial blowing machine, providing good agreement with the numerical predictions.

physics.pop-ph↗

arXiv2011

On the cutoff frequency of clarinet-like instruments. Geometrical versus acoustical regularity

A characteristic of woodwind instruments is the cutoff frequency of their tone-hole lattice. Benade proposed a practical definition using the measurement of the input impedance, for which at least two frequency bands appear. The first one is a stop band, while the second one is a pass band. The value of this frequency, which is a global quantity, depends on the whole geometry of the instrument, but is rather independent of the fingering. This seems to justify the consideration of a woodwind with several open holes as a periodic lattice. However the holes on a clarinet are very irregular. The paper investigates the question of the acoustical regularity: an acoustically regular lattice of tone holes is defined as a lattice built with T-shaped cells of equal eigenfrequencies. Then the paper discusses the possibility of division of a real lattice into cells of equal eigenfrequencies. It is shown that it is not straightforward but possible, explaining the apparent paradox of Benade's theory. When considering the open holes from the input of the instrument to its output, the spacings between holes are enlarged together with their radii: this explains the relative constancy of the eigenfrequencies.

physics.class-ph↗

arXiv2010

Iterated maps for clarinet-like systems

The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime for string instruments. In this article, we study in more detail the properties of the corresponding non-linear iterations, with emphasis on the geometrical constructions that can be used to classify the various solutions (for instance with or without reed beating) as well as on the periodicity windows that occur within the chaotic region. In particular, we find a regime where period tripling occurs and examine the conditions for intermittency. We also show that, while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument, the graph of the high order iterates directly gives visible information on the oscillation regime (characterization of the number of period doubligs, chaotic behaviour, etc.).

nlin.CD↗

arXiv2010

Complex resonance frequencies of a finite, circular radiating duct with an infinite flange

Radiation by solid or fluid bodies can be characterized by resonance modes. They are complex, as well as resonance frequencies, because of the energy loss due to radiation. For ducts, they can be computed from the knowledge of the radiation impedance matrix. For the case of a flanged duct of finite length radiating on one side in an infinite medium, the expression of this matrix was given by Zorumski, using a decomposition in duct modes. In order to calculate the resonance frequencies, the formulation used in Zorumski's theory must be modified as it is not valid for complex frequencies. The analytical development of the Green's function in free space used by Zorumski depends on the integrals of Bessel functions which become divergent for complex frequencies. This paper proposes first a development of the Green's function which is valid for all frequencies. Results are applied to the calculation of the complex resonance frequencies of a flanged duct, by using a formulation of the internal pressure based upon cascade impedance matrices. Several series of resonance modes are found, each series being shown to be related to a dominant duct mode. Influence of higher order duct modes and the results for several fluid densities is presented and discussed.

physics.class-ph↗

arXiv2009

Synthèse sonore de la clarinette avec modèle de résonateur à trous latéraux optimisé sur des impédances d'entrée d'une clarinette réelle

Sound synthesis can be developed using physics-based modelling of the musical instruments. These methods need the description of the various parts of sound production mechanism, in particular a model of the acoustic resonator that have to be simple enough due to the real-time constraint. However, in the case of the clarinet, the bore is quite not a cylindrical tube. An implementation of a resonator including a open-tone holes lattice is presented in this paper.

physics.class-ph↗

arXiv2008

Approximation of the acoustic radiation impedance of a cylindrical pipe

Useful approximation formulae for radiation impedance are given for the reflection coefficients of both infinitely flanged and unflanged rigid-walled cylindrical ducts. The expressions guarantee that simple but necessary physical and mathematical principles are met, like hermitian symmetry for the reflection coefficient (identical behaviour of positive and negative frequencies) and causality for the impulse response. A non causal but more accurate expression is also proposed that is suitable for frequency-domain applications. The formulae are obtained by analytical and numerical fitting to reference results from Levine & Schwinger for the unflanged case and extracted from the radiation impedance matrix given by Zorumski for the infinite flanged case.

physics.class-ph↗

arXiv2008

Interaction of reed and acoustic resonator in clarinetlike systems

Sound emergence in clarinetlike instruments is investigated in terms of instability of the static regime. Various models of reed-bore coupling are considered, from the pioneering work of Wilson and Beavers ["Operating modes of the clarinet", J. Acoust. Soc. Am. 56, 653--658 (1974)] to more recent modeling including viscothermal bore losses and vena contracta at the reed inlet. The pressure threshold above which these models may oscillate as well as the frequency of oscillation at threshold are calculated. In addition to Wilson and Beavers' previous conclusions concerning the role of the reed damping in the selection of the register the instrument will play on, the influence of the reed motion induced flow is also emphasized, particularly its effect on playing frequencies, contributing to reduce discrepancies between Wilson and Beavers' experimental results and theory, despite discrepancies still remain concerning the pressure threshold. Finally, analytical approximations of the oscillating solution based on Fourier series expansion are obtained in the vicinity of the threshold of oscillation. This allows to emphasize the conditions which determine the nature of the bifurcation (direct or inverse) through which the note may emerge, with therefore important consequences on the musical playing performances.

physics.class-ph↗

arXiv2008

Seuils d'instabilité d'un instrument de musique à anche simple : approche modale

Many musical instruments, as for example woodwind instruments, flute or violins, are self-sustained oscillating systems, i.e. musician enacts as a continuous energy source to drive an oscillation in the passive resonator, the body of the instrument, by means of a nonlinear coupling. For single reed instruments like clarinet, there exists a minimal value of mouth pressure beyond which sound can appear. This paper deals with the analysis of this oscillation threshold, calculated using a modal decomposition of the resonator, in order to have a better comprehension of how reed characteristics, such as its strength and its damping, may influence the attack transient of notes.

physics.class-ph↗

arXiv2007

Oscillation thresholds for "strinking outwards" reeds coupled to a resonator

This paper considers a "striking outwards" reed coupled to a resonator. This expression, due to Helmholtz, is not discussed here : it corresponds to the most common model of a lip-type valve, when the valve is assumed to be a one degree of freedom oscillator. The presented work is an extension of the works done by Wilson and Beavers (1974), Tarnopolsky (2000). The range of the playing frequencies is investigated. The first results are analytical : when no losses are present in the resonator, it is proven that the ratio between the threshold frequency and the reed resonance frequency is found to be necessarily within the interval between unity and the square root of 3. This is a musical sixth. Actually the interval is largely smaller, and this is in accordance with e.g. the results by Cullen et al.. The smallest blowing pressure is found to be directly related to the quality factor of the reed. Numerical results confirm these statements, and are discussed in comparison with previous ones by Cullen et al (2000).

physics.class-ph↗

arXiv2007

Simulation of Single Reed Instruments Oscillations Based on Modal Decomposition of Bore and Reed Dynamics

This paper investigates the sound production in a system made of a bore coupled with a reed valve. Extending previous work (Debut, 2004), the input impedance of the bore is projected on the modes of the air column. The acoustic pressure is therefore calculated as the sum of modal components. The airrrﬂow blown into the bore is modulated by reed motion, assuming the reed to be a single degree of freedom oscillator. Calculation of self-sustained oscillations controlled by time-varying mouth pressure and player's embouchure parameter is performed using ODE solvers. Results emphasize the par ticipation of the whole set of components in the mode locking process. Another impor tant feature is the mutual innnﬂuence of reed and bore resonance during growing blowing pressure transients, oscillation threshold being altered by the reed natural frequency and the reed damping. Steady-state oscillations are also investigated and compared with results given by harmonic balance method and by digital sound synthesis.

physics.class-ph↗

arXiv2006

Resonance modes in a 1D medium with two purely resistive boundaries: calculation methods, orthogonality and completeness

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for "resonance modes" or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the 1D propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then the method of separation of variables (space and time) leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or/and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode.

physics.class-ph↗

arXiv2005

Contribution to harmonic balance calculations of periodic oscillation for self-sustained musical instruments with focus on single-reed instruments

The harmonic balance method (HBM) was originally developed for finding periodic solutions of electronical and mechanical systems under a periodic force, but has later been adapted to self-sustained musical instruments. Unlike time-domain methods, this frequency-domain method does not capture transients and so is not adapted for sound synthesis. However, its independence of time makes it very useful for studying every periodic solution of the model, whether stable or unstable without care of initial conditions. A computer program for solving general problems involving nonlinearly coupled exciter and resonator, Harmbal, has been developed based on the HBM. The method as well as convergence improvements and continuations facilities are thorougly presented and discussed in the present paper. Application of the method is demonstrated on various problems related to a common model of the clarinet: a reed modelled as a simple spring with and without mass and damping, a nonlinear coupling and a cubic simplification of it, and a cylindrical bore with or without dissipation and dispersion as well as a bore formed as a stepped cone.

physics.class-ph↗

arXiv2004

Analysis and optimisation of the tuning of the twelfths for a clarinet resonator

Even if the tuning between the first and second register of a clarinet has been optimized by instrument makers, the lowest twelfths remain slightly too large (inharmonicity). In this article, we study the problem from two different points of view. First, we systematically review various physical reasons why this inharmonicity may take place, and the effect of different bore perturbations inserted in cylindrical instruments. Applications to a real clarinet resonator and comparisons with impedance measurements are then presented. A commonly accepted idea is that the register hole is the dominant cause for this inharmonicity: it is natural to expect that opening this hole will raise the resonance frequencies of the instrument, except for the note for which the hole is at the pressure node. We show that the real situation is actually more complicated because other effects, such as open holes or bore taper and bell, introduce resonance shifts that are comparable but with opposite sign, so that a relatively good overall compensation takes place. The origin of the observed inharmonicity in playing frequencies is therefore different. In a second part, we use an elementary model of the clarinet in order to isolate the effect of the register hole: a perfect cylindrical tube without closed holes. Optimization techniques are then used to calculate an optimum location for the register hole; the result turns out to be close to the location chosen by clarinet makers. Finally, attempts are made numerically to improve the situation by introducing small perturbations in the higher part of the cylindrical resonator, but no satisfactory improvement is obtained.

physics.class-ph↗