Modelling and validation of the non-linear elastic stress-strain behaviour of multi-layer silicone composites.

Multi-layer silicone composites are commonly used to mold deformable silicone vocal folds replicas. Nevertheless, so far the stress-strain characterisation of such composite specimens is limited to their effective Young's modulus (up to 40 kPa) characterising the elastic low-strain range, i.e. up to about 0.3. Therefore, in this work, the characterisation is extended to account for the non-linear strain range. Stress-strain curves on 6 single-layer and 34 multi-layer silicone specimens, with different layer stacking (serial, parallel, combined or arbitrary), are measured at room temperature using uni-axial tensile tests for strains up to 1.36, which amounts to about 4.5 times the extent of the linear low-strain range. Cubic polynomial and exponential two-parameter relationships are shown to provide accurate continuous fits (coefficient of determination R2≥99%) of the measured stress-strain data. It is then shown that the parameters can be a priori modelled as a constant or as a linear function of the effective low-strain Young's modulus for strains up to 1.55, i.e. 5 times the low-strain range. These a priori modelled parameter are confirmed by approximations of the best fit parameters for all assessed specimens as a function of the low-strain Young's modulus. Thus, the continuous stress-strain behaviour up to 1.55 can be predicted analytically from the effective low-strain Young's modulus either using the modelled parameters (R2≥85%) or the approximations of the best fit parameter sets (R2≥94%). Accurate stress-strain predictions are particularly useful for the design of composites with different composition and stacking. In addition, analytical expressions of the linear high-strain Young's modulus and the linear high-strain onset, again as a function of the effective low-strain Young's modulus, are formulated as well.

Experimental study on the quasi-steady approximation of glottal flows.

To examine the quasi-steady approximation of the glottal flow, widely used in the modeling of vocal fold oscillations, intraglottal pressure distributions were measured in a scaled-up static vocal fold model under time-varying flow conditions. The left and right vocal folds were slightly open and set to a symmetric and oblique configuration with a divergence angle. To realize time-varying flow conditions, the flow rate was sinusoidally modulated with a frequency of 2 and 10 Hz, which correspond to 112.5 and 562.5 Hz, respectively, in real life. Measurements of the intraglottal pressures under both steady and time-varying flows revealed that the pressure profiles of the time-varying flow conditions are non-distinguishable from those of the steady flow conditions as far as they have the same subglottal pressure as an input pressure. The air-jet separation point was also non-distinguishable between the steady and the time-varying flow conditions. Our study therefore suggests that the time-varying glottal flow can be approximated as a series of steady flow states with a matching subglottal pressure in the range of normal vocalization frequencies. Since the glottal closure was not taken into account in the present experiment, our argument is valid except for such a critical situation.

Influence of level difference due to vocal folds angular asymmetry on auto-oscillating replicas.

Dysphonia is often caused by level difference between left and right vocal folds, which are positioned on different angles with respect to the transverse plane, resulting in angular asymmetry. Unilateral vocal fold paralysis may cause such angular asymmetry. In this case, the normal vocal fold is located on the transverse plane, whereas the paralyzed vocal fold is rotated in the sagittal plane as its posterior edge is moved up to the superior direction. The effect of such angular asymmetry (up to 25°) between the left and right vocal fold on the auto-oscillation is experimentally studied using mechanical replicas. For all replicas, it is observed that, as full contact between vocal folds is lost, increase of angular asymmetry results in a decrease of the signal-to-noise ratio, an increase of the total harmonic distortion rate, and an increase of the oscillation threshold pressure. These general tendencies are in agreement with clinical findings reported for vertical level difference during phonation. In analogy to the preceding experimental study in which vocal folds are spaced in parallel with a vertical trade-off, a formula is proposed to describe the oscillation threshold as a function of angular asymmetry.

Multimodal radiation impedance of a waveguide with arbitrary cross-sectional shape terminated in an infinite baffle.

Simulations of waveguide acoustics require a description of the boundary condition at the open end. For problems involving higher order transverse modes, it is often described by a multimodal radiation impedance matrix. Expressions for the computation of this matrix for an infinite flange condition are available only for circular and rectangular cross-sectional shapes. Thus, a general expression valid for arbitrary cross-sectional shapes is of interest. Such an expression is proposed, validated against known cases, and applied to an arbitrary cross-section shape. The solution is shown to be computationally efficient.

Threshold of oscillation of a vocal fold replica with unilateral surface growths.

Among vocal fold diseases, the presence of a surface growth is often encountered and can be considered a public health issue. While more energy is required to achieve phonation than in healthy cases, this situation can lead to a wide range of voice perturbations, from a change of voice quality to aphonia. The present study aims at providing finer comprehension of the physical phenomena underlying this type of pathological phonation process. A vocal fold replica is used to perform measurements of mechanical responses of each vocal fold as well as of the subglottal pressure in both healthy and pathological configurations. Besides these physical measurements, a theoretical model is derived, using the one-mass-delayed model involving asymmetry of mass and geometry in order to simulate pressure signals. The theoretical model parameters are determined according to mechanical measurements on the replica. Results from measurements and simulations show that this unique vocal fold replica behaves in a manner comparable to clinical observations. The energy required to produce sound increases in the presence of a growth as well as with the size of the growth. Further investigation tends to show that the contact of the growth on the opposite vocal fold, considered as additional damping, plays a critical role.

Influence of vocal tract geometry simplifications on the numerical simulation of vowel sounds.

For many years, the vocal tract shape has been approximated by one-dimensional (1D) area functions to study the production of voice. More recently, 3D approaches allow one to deal with the complex 3D vocal tract, although area-based 3D geometries of circular cross-section are still in use. However, little is known about the influence of performing such a simplification, and some alternatives may exist between these two extreme options. To this aim, several vocal tract geometry simplifications for vowels [ɑ], [i], and [u] are investigated in this work. Six cases are considered, consisting of realistic, elliptical, and circular cross-sections interpolated through a bent or straight midline. For frequencies below 4-5 kHz, the influence of bending and cross-sectional shape has been found weak, while above these values simplified bent vocal tracts with realistic cross-sections are necessary to correctly emulate higher-order mode propagation. To perform this study, the finite element method (FEM) has been used. FEM results have also been compared to a 3D multimodal method and to a classical 1D frequency domain model.

Influence of lips on the production of vowels based on finite element simulations and experiments.

Three-dimensional (3-D) numerical approaches for voice production are currently being investigated and developed. Radiation losses produced when sound waves emanate from the mouth aperture are one of the key aspects to be modeled. When doing so, the lips are usually removed from the vocal tract geometry in order to impose a radiation impedance on a closed cross-section, which speeds up the numerical simulations compared to free-field radiation solutions. However, lips may play a significant role. In this work, the lips' effects on vowel sounds are investigated by using 3-D vocal tract geometries generated from magnetic resonance imaging. To this aim, two configurations for the vocal tract exit are considered: with lips and without lips. The acoustic behavior of each is analyzed and compared by means of time-domain finite element simulations that allow free-field wave propagation and experiments performed using 3-D-printed mechanical replicas. The results show that the lips should be included in order to correctly model vocal tract acoustics not only at high frequencies, as commonly accepted, but also in the low frequency range below 4 kHz, where plane wave propagation occurs.

Modeling of aerodynamic interaction between vocal folds and vocal tract during production of a vowel-voiceless plosive-vowel sequence.

The context of this study is the physical modeling of speech production. The objective is, by using a mechanical replica of the vocal tract, to test quantitatively an aerodynamic model of the interaction between the vocal folds and the vocal tract during the production of a vowel-voiceless plosive-vowel sequence. The first step is to realize acoustic and aerodynamic measurements on a speaker during the production of an /apa/ sequence. The aperture and width of the lips are also derived from a high-speed video recording of the subject's face. Theoretical models to describe the flow through the lips and the effect of an expansion of the supraglottal cavity are proposed and validated by comparison with measurements made using a self-oscillating replica of the phonatory system. Finally, using these models, numerical simulations of an /apa/ sequence are performed using the measured lip parameters as the only time-varying input parameters. The results of these simulations suggest that the realization of an occlusion of the vocal tract produces a passive increase in glottal area associated with a voice offset and that the expansion of the supraglottal cavity is responsible for the extension of the phonation up to 40 ms after closure of the lips.

Self-entrainment of the right and left vocal fold oscillators.

This article presents an analysis of entrained oscillations of the right and left vocal folds in the presence of asymmetries. A simple one-mass model is proposed for each vocal fold. A stiffness asymmetry and open glottis oscillations are considered first, and regions of oscillation are determined by a stability analysis and an averaging technique. The results show that the subglottal threshold pressure for 1:1 entrainment increases with the asymmetry. Within that region, both folds oscillate with the same amplitude and with the lax fold delayed in time with regard to the tense fold. At large asymmetries, a region involving several different phase entrainments or toroidal regimes at constant threshold pressure appears. The effect of vocal fold collisions and asymmetry in the damping coefficients of the oscillators are explored next by means of numerical analyses. It is shown that the damping asymmetry expands the 1:1 entrainment region at low subglottal pressures across the whole asymmetry range. In the expanded region, the oscillator with the lowest natural frequency is dominant and the other oscillator has a large phase advance and small amplitude. The theoretical results are finally compared with data collected from a mechanical replica of the vocal folds.

Effects of higher order propagation modes in vocal tract like geometries.

In this paper, a multimodal theory accounting for higher order acoustical propagation modes is presented as an extension to the classical plane wave theory. This theoretical development is validated against experiments on vocal tract replicas, obtained using a 3D printer and finite element simulations. Simplified vocal tract geometries of increasing complexity are used to investigate the influence of some geometrical parameters on the acoustical properties of the vocal tract. It is shown that the higher order modes can produce additional resonances and anti-resonances and can also strongly affect the radiated sound. These effects appear to be dependent on the eccentricity and the cross-sectional shape of the geometries. Finally, the comparison between the simulations and the experiments points out the importance of taking visco-thermal losses into account to increase the accuracy of the resonance bandwidths prediction.

Influence of glottal cross-section shape on phonation onset.

Phonation models commonly rely on the assumption of a two-dimensional glottal geometry to assess kinetic and viscous flow losses. In this paper, the glottal cross-section shape is taken into account in the flow model in order to capture its influence on vocal folds oscillation. For the assessed cross-section shapes (rectangular, elliptical, or circular segment) the minimum pressure threshold enabling to sustain vocal folds oscillation is altered for constriction degrees smaller than 75%. The discrepancy between cross-section shapes increases as the constriction degree decreases.

The influence of glottal cross-section shape on theoretical flow models.

Physical and mathematical phonation models commonly rely on a quasi-one-dimensional flow model. The assumption of quasi-one-dimensional flow through a glottis with fixed length is analyzed for different cross-section shapes: Circle, rectangle, ellipse, and circular segment. A simplified flow model is formulated which accounts for kinetic losses, viscosity, and cross-section shape. It is seen that the cross-section shape cannot be neglected since it alters boundary layer development and hence the viscous contribution to the pressure drop across the glottis. The commonly applied quasi-one-dimensional flow model is shown to be inaccurate, indicating the potential benefit of taking into account the cross-section shape.

Effect of source-tract acoustical coupling on the oscillation onset of the vocal folds.

This paper analyzes the interaction between the vocal folds and vocal tract at phonation onset due to the acoustical coupling between both systems. Data collected from a mechanical replica of the vocal folds show that changes in vocal tract length induce fluctuations in the oscillation threshold values of both subglottal pressure and frequency. Frequency jumps and maxima of the threshold pressure occur when the oscillation frequency is slightly above a vocal tract resonance. Both the downstream and upstream vocal tracts may produce those same effects. A simple mathematical model is next proposed, based on a lumped description of tissue mechanics, quasi-steady flow and one-dimensional acoustics. The model shows that the frequency jumps are produced by saddle-node bifurcations between limit cycles forming a classical pattern of a cusp catastrophe. The transition from a low frequency oscillation to a high frequency one may be achieved through two different paths: in case of a large acoustical coupling (narrow vocal tract) or high subglottal pressure, the bifurcations are crossed, which causes a frequency jump with a hysteresis loop. By reducing the acoustical coupling (wide vocal tract) or the subglottal pressure, a path around the bifurcations may be followed with a smooth frequency variation.

Experimental validation of flow models for a rigid vocal tract replica.

Flow through the vocal tract is studied through an in vitro rigid replica for different geometrical configurations and steady flow conditions with bulk Reynolds numbers Re<15,000. The vocal tract geometry is approximated by two consecutive obstacles, representing "tongue" and "tooth," in a rectangular channel of fixed length. For the upstream tongue obstacle with fixed constriction degree (81%) the streamwise position is varied and for the downstream obstacle the constriction degree is varied from 0% up to 96%. Different upstream pressures are considered for each geometrical configuration. Point pressure measurements at three fixed locations along the channel are experimentally assessed. In addition, the volume airflow rate is measured. The pressure distribution is estimated with a one-dimensional flow model, and the effects of different corrections to a laminar irrotational flow are assessed. The model outcome is validated against experimental data. Depending on the geometrical configuration, the best model accuracy is obtained by accounting for viscosity (needed for constriction degrees at the tooth that are small, i.e.,≤58%, or very large, i.e., ≥96%), a sudden constriction (large gap between both constrictions), or a bending geometry (narrow gap between both constrictions). Best overall model errors vary between 4% and 30% for all assessed geometrical configurations in cases where a tongue obstacle is present.

A lumped mucosal wave model of the vocal folds revisited: recent extensions and oscillation hysteresis.

This paper examines an updated version of a lumped mucosal wave model of the vocal fold oscillation during phonation. Threshold values of the subglottal pressure and the mean (DC) glottal airflow for the oscillation onset are determined. Depending on the nonlinear characteristics of the model, an oscillation hysteresis phenomenon may occur, with different values for the oscillation onset and offset threshold. The threshold values depend on the oscillation frequency, but the occurrence of the hysteresis is independent of it. The results are tested against pressure data collected from a mechanical replica of the vocal folds, and oral airflow data collected from speakers producing intervocalic /h/. In the human speech data, observed differences between voice onset and offset may be attributed to variations in voice pitch, with a very small or inexistent hysteresis phenomenon.

Vocal fold and ventricular fold vibration in period-doubling phonation: physiological description and aerodynamic modeling.

Occurrences of period-doubling are found in human phonation, in particular for pathological and some singing phonations such as Sardinian A Tenore Bassu vocal performance. The combined vibration of the vocal folds and the ventricular folds has been observed during the production of such low pitch bass-type sound. The present study aims to characterize the physiological correlates of this acoustical production and to provide a better understanding of the physical interaction between ventricular fold vibration and vocal fold self-sustained oscillation. The vibratory properties of the vocal folds and the ventricular folds during phonation produced by a professional singer are analyzed by means of acoustical and electroglottographic signals and by synchronized glottal images obtained by high-speed cinematography. The periodic variation in glottal cycle duration and the effect of ventricular fold closing on glottal closing time are demonstrated. Using the detected glottal and ventricular areas, the aerodynamic behavior of the laryngeal system is simulated using a simplified physical modeling previously validated in vitro using a larynx replica. An estimate of the ventricular aperture extracted from the in vivo data allows a theoretical prediction of the glottal aperture. The in vivo measurements of the glottal aperture are then compared to the simulated estimations.

Experimental validation of quasi-one-dimensional and two-dimensional steady glottal flow models.

Physical modelling of phonation requires a mechanical description of the vocal fold coupled to a description of the flow within the glottis. In this study, an in-vitro set-up, allowing to reproduce flow conditions comparable to those of human glottal flow is used to systematically verify and discuss the relevance of the pressure and flow-rate predictions of several laminar flow models. The obtained results show that all the considered flow models underestimate the measured flow-rates and that flow-rates predicted with the one-dimensional model are most accurate. On the contrary, flow models based on boundary-layer theory and on the two-dimensional numerical resolution of Navier-Stokes equations yield most accurate pressure predictions. The influence of flow separation on the predictions is discussed since these two models can estimate relevant flow separation positions whereas this phenomenon is treated in a simplified ad-hoc way in the one-dimensional flow modelling. Laminar flow models appear to be unsuitable to describe the flow downstream of the glottal constriction. Therefore, the use of flow models taking into account three-dimensional effects as well as turbulence is motivated.

Validation of theoretical models of phonation threshold pressure with data from a vocal fold mechanical replica.

This paper analyzes the capability of a mucosal wave model of the vocal fold to predict values of phonation threshold lung pressure. Equations derived from the model are fitted to pressure data collected from a mechanical replica of the vocal folds. The results show that a recent extension of the model to include an arbitrary delay of the mucosal wave in its travel along the glottal channel provides a better approximation to the data than the original version of the model, which assumed a small delay. They also show that modeling the vocal tract as a simple inertive load, as has been proposed in recent analytical studies of phonation, fails to capture the effect of the vocal tract on the phonation threshold pressure with reasonable accuracy.

Modelling the human pharyngeal airway: validation of numerical simulations using in vitro experiments.

In the presented study, a numerical model which predicts the flow-induced collapse within the pharyngeal airway is validated using in vitro measurements. Theoretical simplifications were considered to limit the computation time. Systematic comparisons between simulations and measurements were performed on an in vitro replica, which reflects asymmetries of the geometry and of the tissue properties at the base of the tongue and in pathological conditions (strong initial obstruction). First, partial obstruction is observed and predicted. Moreover, the prediction accuracy of the numerical model is of 4.2% concerning the deformation (mean quadratic error on the constriction area). It shows the ability of the assumptions and method to predict accurately and quickly a fluid-structure interaction.

Influence of a constriction in the near field of the vocal folds: physical modeling and experimental validation.

The involvement of the ventricular folds is often observed in human phonation and, in particular, in pathological and or some throat-singing phonation. This study aims to explore and model the possible aerodynamic interaction between the ventricular and vocal folds using suitable in vitro setups allowing steady and unsteady flow conditions. The two experimental setups consist of a rigid and a self-oscillating vocal-fold replica, coupled to a downstream rigid ventricular-fold replica in both cases. A theoretical flow modeling is proposed to quantify the aerodynamic impact of the ventricular folds on the pressure distribution and thereby on the vocal-fold vibrations. The mechanical behavior of the vocal folds is simulated by a distributed model accounting for this impact. The influence of the ventricular constriction is measured in both flow conditions and compared to the model outcome. This study objectively evaluates the additional pressure drop implied by the presence of a ventricular constriction in the larynx. It is demonstrated that such constriction can either facilitate or impede the glottal vibrations depending on the laryngeal geometrical configuration. The relevance of using static or dynamic vocal-fold replicas is discussed.