Characteristics of phonation onset in a two-layer vocal fold model

Abstract

Characteristics of phonation onset were investigated in a two-layer body-cover continuum model of the vocal folds as a function of the biomechanical and geometric properties of the vocal folds. The analysis showed that an increase in either the body or cover stiffness generally increased the phonation threshold pressure and phonation onset frequency, although the effectiveness of varying body or cover stiffness as a pitch control mechanism varied depending on the body-cover stiffness ratio. Increasing body-cover stiffness ratio reduced the vibration amplitude of the body layer, and the vocal fold motion was gradually restricted to the medial surface, resulting in more effective flow modulation and higher sound production efficiency. The fluid-structure interaction induced synchronization of more than one group of eigenmodes so that two or more eigenmodes may be simultaneously destabilized toward phonation onset. At certain conditions, a slight change in vocal fold stiffness or geometry may cause phonation onset to occur as eigenmode synchronization due to a different pair of eigenmodes, leading to sudden changes in phonation onset frequency, vocal fold vibration pattern, and sound production efficiency. Although observed in a linear stability analysis, a similar mechanism may also play a role in register changes at finite-amplitude oscillations.

Figure 1

The two-dimensional vocal fold model and the glottal channel. The coupled vocal fold-flow system was assumed to be symmetric about the glottal channel centerline, and only the left half of the system was considered in this study. T is the thicknesses of the medial surface of the vocal fold in the flow direction; D_b and D_c are the depths of the vocal fold body and cover layers at the center of the medial surface, respectively; g₀ is the minimum prephonatory glottal half-width of the glottal channel at rest. The divergence angle α is the angle formed by the medial surface of the vocal fold with the z-axis. Other parameters include the thickness of the cover layer at the base of the vocal fold, t, the rounding fillet radius, r, and the entrance and exit angles of the body and cover layers. The dashed line indicates the glottal channel centerline.

Figure 2

(a) Phonation threshold pressure P_th, (b) phonation onset frequency F₀, (c) prephonatory minimum glottal half-width g, and (d) amplitude of radiated acoustic pressure p_a as functions of body-cover stiffness ratio E_b∕E_c for five different glottal channel divergence angles: ◇: −10, ◻: −5, ○: 0, ▽: 5, and △: 10. Also shown in (c) is the minimum glottal half-width at rest (solid line). Model parameters are given in Eq. 5.

Figure 3

The vocal fold geometry during one oscillation cycle. E_b∕E_c=1, α=0 and other model parameters are given in Eq. 5. The first frame in time is shown in the leftmost plot in the first row, and the last frame is shown in the rightmost plot in the last row. The thin lines correspond to the mean deformed vocal fold geometry at onset as obtained from solving the steady-state problem.

Figure 4

The vocal fold geometry during one oscillation cycle. E_b∕E_c=100, α=0, and other model parameters are given in Eq. 5. The first frame in time is shown in the leftmost plot in the first row, and the last frame is shown in the rightmost plot in the last row. The thin lines correspond to the mean deformed vocal fold geometry at onset as obtained from solving the steady-state problem.

Figure 5

The spatiotemporal plot of the medial-lateral (left) and superior-inferior (right) components of the vocal fold surface displacement for E_b∕E_c=1 (top) and E_b∕E_c=100 (bottom). α=0 and other model parameters are given in Eq. 5. For each case, the two components were normalized by the maximum value of the two components along the surface.

Figure 6

The amplitudes of the medial-lateral (thick solid lines) and superior-inferior components (dashed lines) of the vocal fold surface displacement along the flow direction for E_b∕E_c=1 (gray lines) and E_b∕E_c=100 (dark lines). α=0 and other model parameters are given in Eq. 5. For each case, the FSI eigenmode was normalized so that the vibrational energy was 1. The thin solid line denotes the vocal fold surface.

Figure 7

The spatiotemporal plot of the medial-lateral (left) and superior-inferior (right) components of the vocal fold surface displacement for α=−5 (top) and α=5 (bottom). E_b∕E_c=1 and other model parameters are given in Eq. 5. For each case, the two components were normalized by the maximum value of the two components along the surface.

Figure 8

The frequencies (top) and growth rates (bottom) of the first six eigenvalues (○: first; ◻: second; ◇: third; ▽: fourth; ∗: fifth; △: sixth) of the coupled fluid-structure system as a function of the subglottal pressure for E_b∕E_c=6 (left) and E_b∕E_c=7 (right). α=−5 and other model parameters are given in Eq. 5. The vertical line indicates the point of onset. Despite a slight change in the body-cover stiffness ratio E_b∕E_c, onset occurred as a different eigenmode was destabilized.

Figure 9

The vocal fold geometry during one oscillation cycle. E_b∕E_c=6, α=−5 and other model parameters are given in Eq. 5. The first frame in time is shown in the leftmost plot in the first row, and the last frame is shown in the rightmost plot in the last row. The thin lines correspond to the mean deformed vocal fold geometry at onset as obtained from solving the steady-state problem.

Figure 10

The vocal fold geometry during one oscillation cycle. E_b∕E_c=7, α=−5, and other model parameters are given in Eq. 5. The first frame in time is shown in the leftmost plot in the first row, and the last frame is shown in the rightmost plot in the last row. The thin lines correspond to the mean deformed vocal fold geometry at onset as obtained from solving the steady-state problem.

Figure 11

The spatiotemporal plot of the medial-lateral (left) and superior-inferior (right) components of the vocal fold surface displacement for E_b∕E_c=6 (top) and E_b∕E_c=7 (bottom). α=−5 and other model parameters are given in Eq. 5. For each case, the two components were normalized by the maximum value of the two components along the surface.

Figure 12

The amplitudes of the medial-lateral (thick solid lines) and superior-inferior components (dashed lines) of the vocal fold surface displacement along the flow direction for E_b∕E_c=6 (gray lines) and E_b∕E_c=7 (dark lines). α=−5 and other model parameters are given in Eq. 5. For each case, the FSI eigenmode was normalized so that the vibrational energy was 1. The thin solid line denotes the vocal fold surface.

Figure 13

(a) Phonation threshold pressure P_th, (b) phonation onset frequency F₀, (c) prephonatory minimum glottal half-width g, and (d) amplitude of radiated acoustic pressure p_a as functions of body-cover stiffness ratio E_b∕E_c for five different glottal channel divergence angles: ◇: −10, ◻: −5, ○: 0, ▽: 5, and △: 10. Also shown in (c) is the minimum glottal half-width at rest (solid line). T=1, D_b=6D_c=1.2, and other parameters are given in Eq. 5.

Figure 14

The left column shows the (a) phonation threshold pressure and (b) phonation onset frequency as functions of structural loss factor (E_b=10, g₀=0.03). The right column shows the (c) phonation threshold pressure and (d) phonation onset frequency as functions of minimum glottal half-width at rest (E_b=10, σ=0.4).

Figure 15

(a) Constant F₀ contours and (b) constant P_th contours in the E_c−E_b space. T=3 mm and ρ_c=1000 kg∕m³. In (a), the four solid lines indicate F₀ of 100, 150, 200, and 250 Hz. In (b), the four solid lines indicate P_th of 100, 200, 300, and 400 Pa. The two dashed lines indicate constant body-cover stiffness ratio of E_b∕E_c=1 and E_b∕E_c=100. F₀ can be effectively controlled by changing the body stiffness at small body-cover stiffness ratios (E_b∕E_c<10) and by changing the cover stiffness for large body-cover stiffness ratios (E_b∕E_c>10).