Glottal flow through a two-mass model: comparison of Navier-Stokes solutions with simplified models

A new numerical model of the vocal folds is presented based on the well-known two-mass models of the vocal folds. The two-mass model is coupled to a model of glottal airflow based on the incompressible Navier-Stokes equations. Glottal waves are produced using different initial glottal gaps and different subglottal pressures. Fundamental frequency, glottal peak flow, and closed phase of the glottal waves have been compared with values known from the literature. The phonation threshold pressure was determined for different initial glottal gaps. The phonation threshold pressure obtained using the flow model with Navier-Stokes equations corresponds better to values determined in normal phonation than the phonation threshold pressure obtained using the flow model based on the Bernoulli equation. Using the Navier-Stokes equations, an increase of the subglottal pressure causes the fundamental frequency and the glottal peak flow to increase, whereas the fundamental frequency in the Bernoulli-based model does not change with increasing pressure.     

A theoretical model of the pressure field arising from asymmetric intraglottal flows applied to a two-mass model of the vocal folds

A theoretical flow solution is presented for predicting the pressure distribution along the vocal fold walls arising from asymmetric flow that forms during the closing phases of speech. The resultant wall jet was analyzed using boundary layer methods in a non-inertial reference frame attached to the moving wall. A solution for the near-wall velocity profiles on the flow wall was developed based on a Falkner-Skan similarity solution and it was demonstrated that the pressure distribution along the flow wall is imposed by the velocity in the inviscid core of the wall jet. The method was validated with experimental velocity data from 7.5 times life-size vocal fold models, acquired for varying flow rates and glottal divergence angles. The solution for the asymmetric pressures was incorporated into a widely used two-mass model of vocal fold oscillation with a coupled acoustical model of sound propagation. Asymmetric pressure loading was found to facilitate glottal closure, which yielded only slightly higher values of maximum flow declination rate and radiated sound, and a small decrease in the slope of the spectral tilt. While the impact on symmetrically tensioned vocal folds was small, results indicate the effect becomes more significant for asymmetrically tensioned vocal folds.     

Two-dimensional model of vocal fold vibration for sound synthesis of voice and soprano singing

Voiced sounds were simulated with a computer model of the vocal fold composed of a single mass vibrating both parallel and perpendicular to the airflow. Similarities with the two-mass model are found in the amplitudes of the glottal area and the glottal volume flow velocity, the variation in the volume flow waveform with the vocal tract shape, and the dependence of the oscillation amplitude upon the average opening area of the glottis, among other similar features. A few dissimilarities are also found in the more symmetric glottal and volume flow waveforms in the rising and falling phases. The major improvement of the present model over the two-mass model is that it yields a smooth transition between oscillations with an inductive load and a capacitive load of the vocal tract with no sudden jumps in the vibration frequency. Self-excitation is possible both below and above the first formant frequency of the vocal tract. By taking advantage of the wider continuous frequency range, the two-dimensional model can successfully be applied to the sound synthesis of a high-pitched soprano singing, where the fundamental frequency sometimes exceeds the first formant frequency.     

Pressure-flow relationships during phonation in the canine larynx

Measurements of air pressure and flow were made using an in vivo canine model of the larynx. Subglottic pressures at varying flow rates were taken during phonation induced by laryngeal nerve stimulation. Results showed that during constant vocal fold stiffness, subglottic pressure rose slightly with increased air flow. The larynx in the in vivo canine model exhibited a flow-dependent decrease in laryngeal airway resistance. Increasing flow rate was associated with an increase in frequency of phonation and open quotient, as measured glottographically. Results from this experiment were compared with a theoretical two-mass model of the larynx and other theoretical models of phonation. The influence of aerodynamic forces on glottal vibration is explained by increased lateral excursion of the vocal folds during the open interval and shortening of the closed interval during the glottal cycle.     

Vocal intensity in speakers and singers.

Vocal intensity is studied as a function of fundamental frequency and lung pressure. A combination of analytical and empirical models is used to predict sound pressure levels from glottal waveforms of five professional tenors and twenty five normal control subjects. The glottal waveforms were obtained by inverse filtering the mouth flow. Empirical models describe features of the glottal flow waveform (peak flow, peak flow derivative, open quotient, and speed quotient) in terms of lung pressure and phonation threshold pressure, a key variable that incorporates the Fo dependence of many of the features of the glottal flow. The analytical model describes the contributions to sound pressure levels SPL by the vocal tract. Results show that SPL increases with Fo at a rate of 8-9 dB/octave provided that lung pressure is raised proportional to phonation threshold pressure. The SPL also increases at a rate of 8-9 dB per doubling of excess pressure over threshold, a new quantity that assumes considerable importance in vocal intensity calculations. For the same excess pressure over threshold, the professional tenors produced 10-12 dB greater intensity than the male nonsingers, primarily because their peak airflow was much higher for the same pressure. A simple set of rules is devised for predicting SPL from source waveforms.     

利用声带动力学模型参数反演方法进行病变嗓音分类

提出一种声带动力学模型参数反演方法,从发声机理角度对声带病变嗓音进行有效区分。依据声带生理组织和伯努利定律构建声带动力学模型,确定模型优化参数向量,耦合声门气流获取模型声门波;利用迭代自适应逆滤波算法获得实际嗓音声门波作为目标声门波;采用遗传优化算法提出通过匹配目标和模型声门波特征参数实现模型参数反演。实验结果表明,表征声门波的各时频域参数匹配相对误差不超过2%;依据反演所获模型参数提出去除声门下压影响的平均归一化缩放系数,克服声带非对称性特征在区分病变嗓音方面的不足,实现病理嗓音的全面有效区分。      

On subglottal formant analysis

When subglottal pressure signals which are recorded during normal speech production are spectrally analyzed, the frequency of the first spectral maximum appears to deviate appreciably from the first resonance frequency which has been reported in the literature and which stems from measurements of the acoustic impedance of the subglottal system. It is postulated that this is caused by the spectrum of the excitation function. This hypothesis is corroborated by a modeling study. Using an extended version of the well-known two-mass model of the vocal folds that can account for a glottal leak, it is shown that under realistic physiological assumptions glottal flow waveforms are generated whose spectral properties cause a downward shift of the location of the first spectral maximum in the subglottal pressure signals. The order of magnitude of this effect is investigated for different glottal settings and with a subglottal system that is modeled according to the impedance measurements reported in the literature. The outcomes of this modeling study show that the location of the first spectral maximum of the subglottal pressure may deviate appreciably from the natural frequency of the subglottal system. As a consequence, however, the comfortable assumption that in normal speech the glottal excitation function is constant and zero during the "closed glottis interval" has to be called into question.     

Just noticeable differences forglottal flow waveform characteristics

This study was primarily motivated by the need to establish the correspondence between auditory abilities and laryngeal function. Just noticeable differences (JNDs) were obtained for the open quotient and speed quotient of the glottal flow waveform. The quotients were synthesized for both the glottal flow alone, and for the output pressure signal after the glottal flow signal was applied to the synthesis vocal tract for the vowel /a/. Six adult men and five adult women, all teachers of singing, participated as listeners. An adaptive auditory listening procedure was used to estimate JNDs for the four types of stimuli. The group average JND values were as follows. For the standard open quotient value of .6000, JND = 0.0264 (SD = .010) for the glottal flow and JND = 0.0344 (SD = .020) for the output pressure. For the open quotient, there was no statistically significant difference between genders or between the types of signals. For the standard speed quotient value of 2.000, JND = 0.154 (SD = .043) for the glottal flow and JND = 0.319 (SD = .167) for the output pressure. For the speed quotient, there was no statistically significant difference between genders, but the difference between types of stimulus (glottal flow versus output pressure) was significant (p <.006). The variance among the JND values was significantly larger for the output pressure stimuli compared to the glottal flow stimuli for both the open quotient and the speed quotient.     

Theoretical simulation and experimental validation of inverse quasi-one-dimensional steady and unsteady glottal flow models

In physical modeling of phonation, the pressure drop along the glottal constriction is classically assessed with the glottal geometry and the subglottal pressure as known input parameters. Application of physical modeling to study phonation abnormalities and pathologies requires input parameters related to in vivo measurable quantities commonly corresponding to the physical model output parameters. Therefore, the current research presents the inversion of some popular simplified flow models in order to estimate the subglottal pressure, the glottal constriction area, or the separation coefficient inherent to the simplified flow modeling for steady and unsteady flow conditions. The inverse models are firstly validated against direct simulations and secondly against in vitro measurements performed for different configurations of rigid vocal fold replicas mounted in a suitable experimental setup. The influence of the pressure corrections related to viscosity and flow unsteadiness on the flow modeling is quantified. The inversion of one-dimensional glottal flow models including the major viscous effects can predict the main flow quantities with respect to the in vitro measurements. However, the inverse model accuracy is strongly dependent on the pertinence of the direct flow modeling. The choice of the separation coefficient is preponderant to obtain pressure predictions relevant to the experimental data.     

Voice production model integrating boundary-layer analysis of glottal flow and source-filter coupling

A voice production model is created in this work by considering essential aerodynamic and acoustic phenomena in human voice production. A precise flow analysis is performed based on a boundary-layer approximation and the viscous-inviscid interaction between the boundary layer and the core flow. This flow analysis can supply information on the separation point of the glottal flow and the thickness of the boundary layer, both of which strongly depend on the glottal configuration and yield an effective prediction of the flow behavior. When the flow analysis is combined with the modified two-mass model of the vocal fold [Pelorson et al. (1994). J. Acoust. Soc. Am. 96, 3416-3431], the resulting acoustic wave travels through the vocal tract and a pressure change develops in the vicinity of the glottis. This change can affect the glottal flow and the motion of the vocal folds, causing source-filter coupling. The property of the acoustic feedback is explicitly expressed in the frequency domain by using an acoustic tube model, allowing a clear interpretation of the coupling. Numerical experiments show that the vocal-tract input impedance and frequency responses representing the source-filter coupling have dominant peaks corresponding to the fourth and fifth formants. Results of time-domain simulations also suggest the importance of these high-frequency peaks in voice production.     

Effect of inferior surface angle on the self-oscillation of a computational vocal fold model

Geometry of the human vocal folds strongly influences their oscillatory motion. While the effect of intraglottal geometry on phonation has been widely investigated, the study of the geometry of the inferior surface of the vocal folds has been limited. In this study the way in which the inferior vocal fold surface angle affects vocal fold vibration was explored using a two-dimensional, self-oscillating finite element vocal fold model. The geometry was parameterized to create models with five different inferior surface angles. Four of the five models exhibited self-sustained oscillations. Comparisons of model motion showed increased vertical displacement and decreased glottal width amplitude with decreasing inferior surface angle. In addition, glottal width and air flow rate waveforms changed as the inferior surface angle was varied. Structural, rather than aerodynamic, effects are shown to be the cause of the changes in model response as the inferior surface angle was varied. Supporting data including glottal pressure distribution, average intraglottal pressure, energy transfer, and flow separation point locations are discussed, and suggestions for future research are given.     

Phonation thresholds as a function of laryngeal size in a two-mass model of the vocal folds

This letter analyzes the oscillation onset-offset conditions of the vocal folds as a function of laryngeal size. A version of the two-mass model of the vocal folds is used, coupled to a two-tube approximation of the vocal tract in configuration for the vowel /a/. The standard male configurations of the laryngeal and vocal tract models are used as reference, and their dimensions are scaled using a single factor. Simulations of the vocal fold oscillation and oral output are produced for varying values of the scaling factor. The results show that the oscillation threshold conditions become more restricted for smaller laryngeal sizes, such as those appropriate for females and children.     

Computational Simulations of Vocal Fold Vibration: Bernoulli Versus Navier–Stokes

The use of the mechanical energy (ME) equation for fluid flow, an extension of the Bernoulli equation, to predict the aerodynamic loading on a two-dimensional finite element vocal fold model is examined. Three steady, one-dimensional ME flow models, incorporating different methods of flow separation point prediction, were compared. For two models, determination of the flow separation point was based on fixed ratios of the glottal area at separation to the minimum glottal area; for the third model, the separation point determination was based on fluid mechanics boundary layer theory. Results of flow rate, separation point, and intraglottal pressure distribution were compared with those of an unsteady, two-dimensional, finite element Navier-Stokes model. Cases were considered with a rigid glottal profile as well as with a vibrating vocal fold. For small glottal widths, the three ME flow models yielded good predictions of flow rate and intraglottal pressure distribution, but poor predictions of separation location. For larger orifice widths, the ME models were poor predictors of flow rate and intraglottal pressure, but they satisfactorily predicted separation location. For the vibrating vocal fold case, all models resulted in similar predictions of mean intraglottal pressure, maximum orifice area, and vibration frequency, but vastly different predictions of separation location and maximum flow rate.     

Integrated aerodynamic measurements

The myoelastic-aerodynamic model of phonation implies that aerodynamic factors are crucial to the evaluation of voice function. Subglottal pressure and mean flow rate represent the vocal power source. If they can be related to the magnitude of the radiated sound power, they may provide an index of vocal efficiency. Methods of evaluating the aerodynamic characteristics associated with the ventilatory and laryngeal systems are critically discussed, and normative aerodynamic values for use in diagnosis and physiologic investigations are presented. Measurements having excellent time resolution of the glottal flow wave and of pressure changes in the vicinity of the larynx itself demonstrate the importance of study vocal tract acoustics.     

Physiologic and acoustic differences between male and female voices

Comparison is drawn between male and female larynges on the basis of overall size, vocal fold membranous length, elastic properties of tissue, and prephonatory glottal shape. Two scale factors are proposed that are useful for explaining differences in fundamental frequency, sound power, mean airflow, and glottal efficiency. Fundamental frequency is scaled primarily according to the membranous length of the vocal folds (scale factor of 1.6), whereas mean airflow, sound power, glottal efficiency, and amplitude of vibration include another scale factor (1.2) that relates to overall larynx size. Some explanations are given for observed sex differences in glottographic waveforms. In particular, the simulated (computer-modeled) vocal fold contact area is used to infer male-female differences in the shape of the glottis. The female glottis appears to converge more linearly (from bottom to top) than the male glottis, primarily because of medial surface bulging of the male vocal folds.     

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.     

Vocal fold bulging effects on phonation using a biophysical computer model

Glottal adduction is a primary laryngeal variable that helps to determine glottal configuration and phonatory output. Greater adduction of the vocal folds can be produced by narrowing the gap between the vocal processes or by bulging the medial surface of the vocal folds. This study examined phonatory effects due to changing the degree of bulging using a computational model. Bulging was modeled as a quadratic surface and was related to active muscle stress. Results indicated that bulging had a significant effect on glottal flow resistance, maximum glottal width and area, and mean glottal volume velocity. The results are discussed relative to clinical issues of hyperfunction.     

Vocal quality factors: analysis, synthesis, and perception.

The purpose of this study was to examine several factors of vocal quality that might be affected by changes in vocal fold vibratory patterns. Four voice types were examined: modal, vocal fry, falsetto, and breathy. Three categories of analysis techniques were developed to extract source-related features from speech and electroglottographic (EGG) signals. Four factors were found to be important for characterizing the glottal excitations for the four voice types: the glottal pulse width, the glottal pulse skewness, the abruptness of glottal closure, and the turbulent noise component. The significance of these factors for voice synthesis was studied and a new voice source model that accounted for certain physiological aspects of vocal fold motion was developed and tested using speech synthesis. Perceptual listening tests were conducted to evaluate the auditory effects of the source model parameters upon synthesized speech. The effects of the spectral slope of the source excitation, the shape of the glottal excitation pulse, and the characteristics of the turbulent noise source were considered. Applications for these research results include synthesis of natural sounding speech, synthesis and modeling of vocal disorders, and the development of speaker independent (or adaptive) speech recognition systems.     

Polyps and paralysis phonation classification with nonlinear dynamics model

In order to provide the basis for parameter selection of vocal diseases classification,a nonlinear dynamic modeling method is proposed.A biomechanical model of vocal cords with polyp or paralysis,which couples to glottal airflow to produce laryngeal sound source,is introduced.And then the fundamental frequency and its perturbation parameters are solved.Poincare section and bifurcation diagram are applied to nonlinear analysis of model vibration.By changing the pathological parameters or subglottal pressure,the changes of fundamental frequency and Lyapunov exponents are analyzed.The simulation results show that,vocal cord paralysis reduces the fundamental frequency,and the chaos occurs only within a certain pressure range;while vocal cord with a polyp don't reduce the fundamental frequency,chaos distributes throughout the entire range of pressure.Therefore this study is helpful for classification of polyp and paralysis by the acoustic diagnoses.