Taillard, Pierre-André
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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: EL and GLT . 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 Arundo donax.
Scraping technique for clarinet reeds derived from a static bending
Reed scraping is an art mastered only by a few clarinettists. The published empirical methods do not agree on how to determine where to scrape. In order to understand the problematic more clearly, a purely mechanical and simplified approach is attempted. The reed is modelled by finite elements and a simulation of free static bending shows that the longitudinal flexural stress dominates the problem. By targeting a stress field on different areas of the reed, it is observed that the strain is mainly longitudinal and only slightly lateral, due to the strong anisotropy of the cane (Arundo donax). Thus, a bending moment imposed on the axis of the reed causes almost no translation of the edges of the reed. A local decrease in thickness causes a localized increase in curvature, when the stress field remains constant. With this result, a series of ClaripatchTM was developed to impose a localized decrease in curvature on the mouthpiece’s lay. Using these patches and assuming some hypotheses based on observations, simulations and on the viscoelastic properties of the reed, one can deduce how to scrape it, according to the musical preferences of the clarinettist.
Influence of the reed flow on the intonation of the clarinet
The playing frequency of a reed instrument is generally mainly imposed by the resonator. Nevertheless, numerous other factors have an influence on the playing frequency. Among those, the volume velocity pulsated by the reed must be taken into account in the models as it lowers significantly the playing frequency. At first order, this volume velocity can be taken into account by adding an equivalent volume of about 1.cm3 at the entrance of the pipe. However, the concept of equivalent volume is theoretically only valid for linear vibrations i.e., for the clarinet: the non beating reed regime. In the beating regime, this volume is likely to vary quite a lot with the mouth pressure which can lead to intonation problems. In addition, this volume depends much on the reed opening, but experiments performed with an artificial mouth show that for usual openings (0.3-0.7mm) the intonation remains surprisingly stable. This suggests that the reed and mouthpiece makers know how to control this problem by a well chosen design of the lay. A counter-example is obtained by using a ClaripatchTM especially designed so that intonation is particularly difficult to control.