Numerical and experimental modal analysis of the reed and pipe of a clarinet

A modal computation of a complete clarinet is presented by the association of finite-element models of the reed and of part of the pipe with a lumped-element model of the rest of the pipe. In the first part, we compare modal computations of the reed and the air inside the mouthpiece and barrel with measurements performed by holographic interferometry. In the second part, the complete clarinet is modeled by adjoining a series of lumped elements for the remaining part of the pipe. The parameters of the lumped-resonator model are determined from acoustic impedance measurements. Computed eigenmodes of the whole system show that modal patterns of the reed differ significantly whether it is alone or coupled to air. Some modes exhibit mostly reed motion and a small contribution of the acoustic pressure inside the pipe. Resonance frequencies measured on a clarinet with the mouthpiece replaced by the cylinder of equal volume differ significantly from the computed eigenfrequencies of the clarinet taking the actual shape of the mouthpiece into account and from those including the (linear) dynamics of the reed. This suggests revisiting the customary quality index based on the alignment of the peaks of the input acoustical impedance curve.     

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 such as bore flare, open and closed holes, taper, temperature gradients, visco-thermal effects, etc. 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 shift the position of the resonances of the instrument to higher frequencies, except of course for the note for which the hole is exactly 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. This is checked by experimental and theoretical studies of the acoustical impedance curves of a clarinet. The origin of the observed inharmonicity in playing frequencies is therefore different, maybe related to the reed or the vocal tract. 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 (without taking into account the use of the register hole as a B flat tone 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.     

The metre

A musical wind instrument transforms a constant pressure input from the player's mouth into a fluctuating pressure output in the form of a radiating sound wave. In reed woodwind and brass instruments, this transformation is achieved through a nonlinear coupling between two vibrating systems: the flow control valve formed by the mechanical reed or the lips of the player, and the air column contained by the pipe. Although the basic physics of reed wind instruments was developed by Helmholtz in the nineteenth century, the application of ideas from the modern theory of nonlinear dynamics has led to recent advances in our understanding of some musically important features of wind instrument behaviour. As a first step, the nonlinear aspects of the musical oscillator can be considered to be concentrated in the flow control valve; the air column can be treated as a linear vibrating system, with a set of natural modes of vibration corresponding to the standing waves in the pipe. Recent models based on these assumptions have had reasonable success in predicting the threshold blowing pressure and sounding frequency of a clarinet, as well as explaining at least qualitatively the way in which the timbre of the sound varies with blowing pressure. The situation is more complicated for brass instruments, in which the player's lips provide the flow valve. Experiments using artificial lips have been important in permitting systematic studies of the coupling between lips and air column; the detailed nature of this coupling is still not fully understood. In addition, the assumption of linearity in the air column vibratory system sometimes breaks down for brass instruments. Nonlinear effects in the propagation of high amplitude sound waves can lead to the development of shock waves in trumpets and trombones, with important musical consequences.     

An analytical prediction of the oscillation and extinction thresholds of a clarinet

This paper investigates the dynamic range of the clarinet from the oscillation threshold to the extinction at high pressure level. The use of an elementary model for the reed-mouthpiece valve effect combined with a simplified model of the pipe assuming frequency independent losses (Raman's model) allows an analytical calculation of the oscillations and their stability analysis. The different thresholds are shown to depend on parameters related to embouchure parameters and to the absorption coefficient in the pipe. Their values determine the dynamic range of the fundamental oscillations and the bifurcation scheme at the extinction.     

Toward an estimation of the clarinet reed pulse from instrument performance

In this work, a technique is presented for estimating the reed pulse from the pressure signal recorded at the bell of a clarinet during performance. The reed pulse is a term given to the typically periodic sequence of bore input pressure pulses, a signal related to the volume flow through a vibrating reed by the characteristic impedance of the aperture to the bore. The problem is similar to extracting glottal pulse sequence from recorded speech; however, because the glottis and instrument reeds have very different masses and opening areas, the source-filter model used in speech processing is not applicable. Here, the reed instrument is modeled as a pressure-controlled valve coupled to a bi-directional waveguide, with the output pressure approximated as a linear time invariant transformation of the product of reed volume flow and the characteristic impedance of the bore. By noting that pressure waves will make two round trips from the mouthpiece to the bell and back for each reed pulse, yielding a distinct positive and negative lobe in the running autocorrelation period of the recorded signal, the round-trip attenuation experienced by pressure waves in the instrument is estimated and used to invert the implied waveguide, producing reed pulse estimates.     

Oscillation and extinction thresholds of the clarinet: comparison of analytical results and experiments

In the context of a simplified model of the clarinet in which the losses are assumed to be frequency independent the analytic expressions of the various thresholds have been calculated in a previous paper [Dalmont et al., J. Acoust. Soc. Am. 118, 32.94-3305 (2005)]. The present work is a quantitative comparison between "theoretical" values of the thresholds and their experimental values measured by using an artificial mouth. It is shown that the "Raman" model, providing that nonlinear losses are taken into account, is reliable and able to predict the values of thresholds.     

Quasistatic nonlinear characteristics of double-reed instruments

This article proposes a characterization of the double reed in quasistatic regimes. The nonlinear relation between the pressure drop, deltap, in the double reed and the volume flow crossing it, q, is measured for slow variations of these variables. The volume flow is determined from the pressure drop in a diaphragm replacing the instrument's bore. Measurements are compared to other experimental results on reed instrument exciters and to physical models, revealing that clarinet, oboe, and bassoon quasistatic behavior relies on similar working principles. Differences in the experimental results are interpreted in terms of pressure recovery due to the conical diffuser role of the downstream part of double-reed mouthpieces (the staple).     

Contribution to harmonic balance calculations of self-sustained periodic oscillations 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 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 any periodic solution, whether stable or unstable, without care of particular initial conditions in time. 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 continuation facilities are thoroughly presented and discussed in the present paper. Applications of the method are demonstrated, especially on problems with severe difficulties of convergence: the Helmholtz motion (square signals) of single-reed instruments when no losses are taken into account, the reed being modeled as a simple spring.     

Reed vibration in lingual organ pipes without the resonators

Vibrations of plucked and blown reeds of lingual organ pipes without the resonators have been investigated. Three rather surprising phenomena are observed: the frequency of the reed plucked by hand is shifted upwards for large-amplitude plucking, the blown frequency is significantly higher than the plucked one, and peaks halfway between the harmonics of the fundamental frequency appear in the spectrum of the reed velocity. The dependence of the plucked frequency on the length of the reed reveals that the vibrating length at small vibrations is 3 mm shorter than the apparent free length. The frequency shift for large-amplitude plucking is explained by the periodic change of the vibrating length during the oscillation. Reed vibrations of the blown pipe can be described by a physical model based on the assumption of air flow between the reed and the shallot. Aerodynamic effects may generate and sustain the oscillation of the reed without acoustic feedback. The appearance of subharmonics is explained by taking into account the periodic modulation of the stress in the reed material by the sound field. Therefore, a parametric instability appears in the differential equation of vibration, leading to the appearance of subharmonics.     

Vibration characteristics of pipe organ reed tongues and the effect of the shallot, resonator, and reed curvature

Pipe organ reed pipes sound when a fixed-free curved brass reed mounted on a shallot connected to a resonator is forced to vibrate by an impressed static air pressure. Five sets of experiments were performed in order to investigate the influence of the most important parameters which could affect the tone of a reed pipe. First, the phase difference between the pressure variation in the shallot and the boot, and its relationship to the motion of the reed tongue were analyzed to compare their phases and their spectra. Next, the frequency dependence of the reed on three basic parameters (reed thickness, its vibrating length, and the imposed static air pressure) was investigated in an attempt to determine an empirical equation for the frequency. For each trial, two of the variables were kept constant while the third was altered in order to construct an equation giving frequency as a function of the three variables. Third, experiments were conducted using three different types of shallots: the American (or English) style, the French style, and the German style. The results show that for each shallot, the frequency increases linearly with thickness and linearly with air pressure (over the normal operating range of the reed). For each of the shallots, frequency varies inversely with length when the other variables are held constant. The effect on the reed spectrum of using the three different types of shallot was also investigated, as was the effect of reducing the interior volume of each type. Progressively filling the shallot interior generally decreases the frequency of the vibrating reed. The effect of the resonators on frequency and spectrum was studied because the resonator is an integral part of the resulting tone; virtually every reed stop has some type of resonator. The resonator tends to raise the Q of the impedance peaks and reduce the fundamental frequency. Finally, the influence of the type and degree of curvature on reed vibration was briefly examined; increasing the reed curvature tends to decrease the vibration frequency and increase the sound intensity by creating a richer spectrum.     

点簧簧片振动特性数值分析

簧片是笙等中国传统簧管类乐器的核心部件之一.通过点簧工艺可调整簧片振动频率,点簧簧片的差异对整体乐器的声音效果具有重要影响.针对点簧簧片的振动特性问题,对其建立非均匀截面并具有质量负载的振动模型,采用有限元方法计算点簧簧片的固有频率.大量算例分析了点簧质量、位置以及簧片的边界条件对振动频率的影响,研究发现:若点簧质量不...     

An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—Application to bolted structures

Aeronautical structures are commonly assembled with bolted joints in which friction phenomena, in combination with slapping in the joint, provide damping on the dynamic behavior. Some models, mostly nonlinear, have consequently been developed and the harmonic balance method (HBM) is adapted to compute nonlinear response functions in the frequency domain. The basic idea is to develop the response as Fourier series and to solve equations linking Fourier coefficients. One specific HBM feature is that response accuracy improves as the number of harmonics increases, at the expense of larger computational time. Thus this paper presents an original adaptive HBM which adjusts the number of retained harmonics for a given precision and for each frequency value. The new proposed algorithm is based on the observation of the relative variation of an approximate strain energy for two consecutive numbers of harmonics. The developed criterion takes the advantage of being calculated from Fourier coefficients avoiding time integration and is also expressed in a condensation case. However, the convergence of the strain energy has to be smooth on tested harmonics and this constitutes a limitation of the method. Condensation and continuation methods are used to accelerate calculation. An application case is selected to illustrate the efficiency of the method and is composed of an asymmetrical two cantilever beam system linked by a bolted joint represented by a nonlinear LuGre model. The practice of adaptive HBM shows that, for a given value of the criterion, the number of harmonics increases on resonances indicating that nonlinear effects are predominant. For each frequency value, convergence of approximate strain energy is observed. Emergence of third and fifth harmonics is noticed near resonances both on vibratory responses and on approximate strain energy. Parametric studies are carried out by varying the excitation force amplitude and the threshold value of the adaptive algorithm. Maximal amplitudes of vibration and frequency response functions are plotted for three different points of the structure. Nonlinear effects become more predominant for higher force amplitudes and consequently the number of retained harmonics is increased.     

Vibration characteristic analysis of buried pipes using the wave propagation approach

A theoretical analysis for the free vibration of simply supported buried pipes has been investigated using the wave propagation approach. The pipe modeled as a thin cylindrical shell of linear homogeneous isotropic elastic material buried in a linear isotropic homogeneous elastic medium of infinite extent. The vibrations of the pipe are examined by using Flüggle shell equation. The natural frequencies are obtained for the pipes surrounded by vacuo or elastic medium. The results are compared with those available in the literature and agreement is found with them. It is found that the free vibration frequency of the pipe does not appear for some of the axial or circular vibration modes and the real natural frequencies of the pipe are significantly dependent on the rigidity of the surrounding medium.     

Acoustic modes induced by flow in a pipe with two closed side-branches

An air flow in a pipe with two closed side-branches can induce high pressure pulsations in a pipe system. This phenomenon has been investigated and the results are reported in this paper. A simple theoretical model based on a wave transmission was used to determine a resonance condition associated with an acoustic coupling between branches. In the model a plane wave approximation and an impedance representation of a branch were applied. The experiment was carried out in a pipe system with a relatively large distance between branches compared to branch lengths. A frequency and a pressure of pulsations were measured in a wide range of length of downstream branch. A support for the theory is provided by a favorable comparison between experimental data and calculated resonant frequencies of the system.     

How do clarinet players adjust the resonances of their vocal tracts for different playing effects?

In a simple model, the reed of the clarinet is mechanically loaded by the series combination of the acoustical impedances of the instrument itself and of the player's airway. Here we measure the complex impedance spectrum of players' airways using an impedance head adapted to fit inside a clarinet mouthpiece. A direct current shunt with high acoustical resistance allows players to blow normally, so the players can simulate the tract condition under playing conditions. The reproducibility of the results suggest that the players' "muscle memory" is reliable for this task. Most players use a single, highly stable vocal tract configuration over most of the playing range, except for the altissimo register. However, this "normal" configuration varies substantially among musicians. All musicians change the configuration, often drastically for "special effects" such as glissandi and slurs: the tongue is lowered and the impedance magnitude reduced when the player intends to lower the pitch or to slur downwards, and vice versa.     

Study of airfoil gust response alleviation using an electro-magnetic dry friction damper. Part 2: Experiment

In Part 1 of this work, a theoretical simulation study of the non-linear gust response of a three degree-of-freedom typical airfoil section with a control surface using an electro-magnetic dry friction damper is presented. For validation of this theoretical model, an electro-magnetic dry friction damper has been designed and an experimental investigation of the gust response has been carried out in a wind tunnel. Results for both periodic and linear frequency sweep gust excitations have been computed and measured. The fair to good quantitative agreement between theory and experiment verifies that the present electro-magnetic dry friction damper can be used to alleviate the gust response, especially for the plunge and pitch responses. It also shows that the present theoretical method can be successfully applied to determine the non-linear gust response when an electro-magnetic dry friction damper is used in the linear aeroelastic system.     

Optical characterization of the polymer embedded alloyed bimetallic nanoparticles

A theoretical approach for the calculation of the bimetallic nanoparticles absorption spectra has been developed as an extension of the Mie theory in which nanoparticle dielectric function is found by the weighted linear combination of the dielectric functions for particles made of the corresponding pure metals. In the frame work of the theoretical model an expression for the resonance light absorption frequency were derived taking into account the interband transitions in the dielectric functions. We propose a simple method for the on-line monitoring of the bimetallic nanoparticles composition based on the measurement of the absorption peak position. Elaborated theoretical approach was used to investigate the polymer embedded Ag/Au nanoparticles which were prepared by reducing gold and silver salts (HAuCl4 and AgNO3, respectively) by ethylene glycol in presence of poly(vinyl pyrrolidone) (PVP) at room temperature. Calculated absorption spectra for the Ag/Au nanoscopic systems showed good agreement with the experimental data. Temporal evolution of the Ag/Au nanoparticles has also been investigated by this approach.     

传统笙的物理模型及声音合成方法

为研究传统笙的物理结构与音色之间的关系,提出了传统笙的物理模型及声音合成方法。在簧舌振动模型的基础上,结合指孔和音窗的传输矩阵计算方法,建立了笙管的等效电路,并将二者结合,得到笙的完整物理模型。为了验证模型的有效性,设计了一套笙的实验系统,实现了对笙管发声过程中特征量的多通道同步测量。通过对比实验和模型仿真结果的时频域特点以及与音色相关的特征量,分析了模型的性能.结果表明,该模型可较好地模拟传统笙管的发声过程,合成笙样本能够较准确的反映真实笙样本的音色特征。     

利用激光超声研究功能梯度材料中声表面波的传播特性

利用有限元的方法建立了脉冲激光在功能材料中激发声表面波的理论模型,根据该理论模型分别分析了脉冲激光在不同空间调制下所激发的声表面波(SAWs)在功能梯度材料表面的传播特性,以及不同力学参量变化对所激发的声表面波传播特性的影响。模拟结果表明:线源激发的宽带声表面波在功能梯度材料表面传播时,声表面波会出现明显的色散效应;当使用光栅调制的脉冲激光在功能梯度材料上激发声表面波时,与均匀材料中所激发的声信号相比,激发频率发生相应的频移现象。该模型为功能梯度材料的非接触表征提供了理论基础。