Fig. 1. Schematic of the glottal space.

Fig. 2. Two-degree-of-freedom model.

Fig. 3. Schema of the model for the closed glottis when the vocal-fold-shaped element is loaded by a resulting aerodynamic force F_A and the Hertz contact force F_H.

Fig. 4. Stability map for the model with the first natural frequency and second natural frequency: - - -■- - -, ; ——, ; —•—, .

Fig. 5. Calculated critical values (flutter boundaries) for loosing the stability of the vocal fold model. The subglottal pressure P_sub.crit and flow rate Q_crit for various glottal half-gaps g=0.2–1 mm and for various values of natural frequencies: —♦—, , ; ——, , ; —■—, , .

Fig. 6. Example of numerical simulation of the self-oscillations: (a) and (b) phase-plane diagrams for displacement w₁(t) and w₂(t), respectively; (c) and (d) displacements w₁(t) andw₂(t), respectively, with marked impact duration within each period of vibration; (e) glottal area S(t); and (f) glottal pressure . The input parameters are the same as in Fig. 4, and the air-flow velocity is , glottal half-gap , prescribed natural frequencies , , time step of numerical integration , mean flow rate and pressure in lungs .

Fig. 7. Spectrum of glottal pressure ; all parameters as in Fig. 6.

Fig. 8. Phases (animation) of the vocal-fold motion during one oscillation cycle; all parameters as in Fig. 6.

Fig. 9. Fundamental frequency F₀ for self-oscillations versus oncoming flow velocity U₀ for various natural frequencies f₁, f₂ of the vocal-fold model: , , ; ■, , ; •, , ; and for several glottal half-gaps g: —, ; - - -, ; ---, .

Fig. 10. Open quotient OQ for self-oscillations versus flow volume velocity Q for , and the glottal half-gap .

Fig. 11. Maximum impact stress during self-oscillations versus pressure in lungs for , and the glottal half-gaps: ♦ ; •, .

The comparison of the expected values P_th, Eq. (32) with the computed values P_sub,crit from Eq. (24) for phonation thresholds at different fundamental frequencies F₀

The glottal half-gap value of was used for computation.