Linear eigenmode analysis

In this section, the results of linear eigenmode analysis of the vocal membrane model are described and briefly discussed. The influence of the introduced control parameters on eigenfrequencies and amplitudes and phase angles of the eigenmodes was studied.

The ranges of the tuning parameter q, where entrained oscillations of the full model could be expected, depended on the mass ratio q_m, the resting angle θ₀, and the vocal membrane height d₃. Comparable eigenfrequencies were expected to be related to frequency locking, toroidal oscillations and chaos in the full non-linear model. Eigenmode analysis allowed to classify oscillatory modes as “vocal fold modes” and “vocal membrane modes”. These different modes could facilitate the definition of different registers of the vocal membrane model. The study of eigenmode amplitudes and phase relations, depending on the geometrical and dy-namical control parameters d₃, θ₀, and q_m, could help to qualitatively understand the phonation onset behavior of the full nonlinear vocal membrane model.

For the study of the eigenfrequencies and eigenmodes of the vocal membrane model, the linearized homogeneous equations of motion without driving forces could be written as: A complex exponential was chosen as an ansatz for the solution of the linearized homogeneous vocal membrane equations:

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tuning parameter q 5 10 15

tuning parameter q 5 10 15 tuning parameter q

Figure 5.11: Variation of eigenfrequencies and relative damping times with mass ratio q_m where d₃ = 0.05 cm and θ₀ = 0 degree, otherwise standard parameter values were used.

where A₁, A₂, A_θ ∈ C were complex amplitudes, and λ ∈ C was the complex frequency .

The solvability condition for the algebraic set of linear equations in the com-plex amplitudes lead to the characteristic polynom, a 6th order polynom for the complex frequency λ. The complex solutions λi = γi± iωi, i = 1, 2, 3 lead to the eigenfrequencies f_i = _2π^ωi and the damping time τ_i = 1/γ_i. The ratio of the damping time τ_i to the eigenperiod 1/f_i, termed the relative damping time τ_i^rel = f_i/γ_i, was used in the following analysis. The moduli |Ai| and the differences of the phase angles ϕ_i, i = 1, 2, θ of the complex amplitudes A_i, i = 1, 2, θ were calculated with the solutions λ_i of the characteristic polynom.

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Eigenmode 2 Eigenmode 3 Eigenmode 1

Figure 5.12: Typical shape of eigenmodes of the vocal membrane model

Influence of mass ratio q_m

In Fig. 5.11 for four different mass ratios q_m ∈ {0.1, 2, 3, 100}, the eigenfrequencies and the relative damping times as functions of the tuning parameter q are shown.

It was observed that for small mass ratios (e.g., q_m = 0.1) the first eigenfrequency f₁ shifted towards higher frequencies. The relative damping time τ₁ is mostly smaller than one.

This eigenmode had a large vocal membrane component whereas the ampli-tudes of the two lower masses were very small. The phase relations of the first eigenmode are shown in Fig. 5.12, where all masses of the model oscillated typically out-of-phase. This mode is termed “vocal membrane mode”. The second eigen-frequency was characterized by large eigenmode amplitudes of the lower mass m₁ and the vocal membrane. Fig. 5.12 shows typical phase relations of the second mode. Out-of-phase oscillation of the lower and upper mass and in-phase oscil-lation of the upper mass and the vocal membrane mass characterized the second mode (Fig. 5.14). The third eigenfrequency corresponded to large eigenmode am-plitudes of the upper mass m2 and the vocal membrane. As shown in Fig. 5.12, the third eigenmode consisted of in-phase oscillations of all masses of the vocal membrane model (Fig. 5.14). The second and third mode were called “vocal fold modes”. They could be subdivided further into “glottal wave mode” for the second eigenmode and “vocal fold bulk mode” for the third eigenmode.

For small mass ratios q_m = 0.1, the variation of the first and the third eigenfre-quencies, f1, and f3, with increasing tuning parameter q were small in comparison to the variation of the second eigenfrequency f₂. Frequency locking at comparable frequencies was expected only at high values of q, approx. q ≥ 13.

As the mass ratio increased, the distance between the first eigenfrequency f₁ and the third eigenfrequency f₃ decreased, until f₁(q) and f₃(q) intersected. For increasing mass ratio, the damping time τ₁ increased; whereas τ₃ decreased. The region for entrained oscillations of the full nonlinear model could be expected to be in the interval 6 ≤ q ≤ 10.

For large mass ratios (e.g., q_m = 100), f₁(q) was nearly constant and ap-proached the value corresponding to the fixed angular frequency ω₃ = 2π 1000Hz.

The relative damping time τ₁ became the largest damping time. The interaction with eigenmodes 2 and 3 vanished as the third eigenfrequency became a linear function of the tuning parameter. The relative damping time τ₃ became the small-est damping time. From the theory of coupled oscillators it could be expected that mode entrainment could occur in the intervals 6 ≤ q ≤ 10 and 12 ≤ q ≤ 15.

Varying the mass ratio q_m, the second eigenfrequency f₂ showed no significant changes. The effect on the damping time τ₂ was also very small. Furthermore, no significant effect of q_m on the eigenmodes could be observed.

Within the framework of coupled oscillators (Berg´e et al., 1984), the mass ratio q_mcan be expected to be a crucial control parameter influencing the frequency ratio of the coupled oscillators. It has a significant influence on the distance between eigenfrequencies as a function of the tuning parameter q. The mass ratio q_m varies the ranges of the tuning parameter q where entrained oscillations of the full nonlinear model can be expected.

Furthermore, an increasing mass ratio was shown to increase the relative damp-ing time of the vocal membrane mode. This could affect the onset behavior of self-sustained vocal membrane oscillations as viscous energy dissipation is reduced. It could also be crucial for only weakly damped high frequency oscillations during low frequency oscillation of the vocal fold masses. This effect could be crucial for the generation of pulsed echolocation calls in bats (Fattu and Suthers, 1981; Suthers and Fattu, 1973).

Influence of resting angle θ₀

In Fig. 5.13, the variation of the eigenfrequencies and the relative damping time with increasing resting angle θ₀ is shown. The chosen mass ratio q_m = 2.0 allowed comparison with the eigenfrequencies and relative damping times for θ₀ = 0 degree, shown in Fig. 5.11.

As the resting angle increased, a decrease of the distance between the first and the third eigenfrequency could be observed. At θ₀ = 30 degrees, f₁(q) and f₃(q) intersected. For large θ₀ (e.g., θ₀ = 60 degrees), f₁(q) and f₃(q) were nearly linear functions of the tuning parameter q.

The relative damping time τ₁of the vocal membrane mode was shifted to higher values for increasing resting angles θ₀. The relative damping time τ₃ was lowered;

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tuning parameter q 5 10 15

tuning parameter q

Figure 5.13: Variation of eigenfrequencies and relative damping times with resting angle θ₀[degrees] where d₃ = 0.05 cm and q_m = 2.0, otherwise standard parameter values were used.

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tuning parameter q 5 10 15

tuning parameter q

Figure 5.14: Eigenvectors of the vocal membrane model for d₃ = 0.05 cm, q_m = 2.0, θ0 = 0 degree, otherwise standard parameter values. Note that due to the definition of the vocal membrane angle θ, a zero phase difference ϕ_Aθ − ϕ_A2 = 0 between the upper mass and the vocal membrane corresponds to, e.g., an increase of the glottal area a2 and a decrease of the vocal membrane area avm.

whereas the relative damping time τ₂ remained uneffected by varying θ₀.

No significant effect of varying resting angles θ₀ on the eigenmodes could be observed. They remained similar to the eigenmodes shown in Fig. 5.14.

Similar to the influence of the mass ratio q_m, the resting angle θ₀ could be expected to influence the ranges of the tuning parameter q, where entrained oscil-lations of the full nonlinear vocal membrane model could occur. Increasing resting angles also decreased the damping of the vocal membrane mode. For self-sustained vocal membrane oscillations and pulsed oscillations (e.g., pulsed echolocation calls in bats) the resting angle could facilitate these modes of vibration.

0

tuning parameter q 5 10 15

tuning parameter q

Figure 5.15: Eigenvectors of the vocal membrane model for d₃ = 0.50 cm, q_m = 2.0, θ₀ = 0 degree. Otherwise standard parameter values were used.

Influence of vocal membrane height d₃

In Figs 5.14 and 5.15, the eigenmodes for two different vocal membrane heights, d₃ = 0.05 cm and d₃ = 0.5 cm, are shown. The eigenfrequencies and relative damping times for these cases were similar to the results shown in Fig. 5.11.

For small vocal membrane height d₃ (e.g., d₃ = 0.05 cm for Fig. 5.14), the first eigenmode (“vocal membrane mode”) was characterized by a large vocal membrane component. The components of the lower and upper masses were about two orders of magnitudes smaller. As a function of the tuning parameter q, the two lower masses changed from oscillating out-of-phase for small q to vibrating in-phase at larger q. The upper mass m2 and the vocal membrane m3 showed in-phase oscillations for the shown range of tuning parameter variations. Note that due to the definition of the vocal membrane angle θ (Fig. 5.8), in-phase oscillation of the upper mass and the vocal membrane corresponded to out-of-phase oscillation of the glottal area a₂ and the area at the tip of the vocal membrane a_vm.

For larger vocal membrane heights d₃ (e.g., d₃ = 0.5 cm for Fig. 5.15) the eigen-mode component of the upper mass was significantly increased. The eigeneigen-mode component of the lower mass remained small as for small vocal membrane heights.

No significant effect on the phase differences could be observed.

The second eigenmode changed more drastically with increasing vocal mem-brane height. For small vocal memmem-brane height, the components of the lower mass m₁ and the vocal membrane m₃ were dominant. The component of the vocal mem-brane was about one order of magnitude larger than the lower mass component.

The component of the upper mass m₂ was about two orders of magnitudes smaller than the vocal membrane component. For larger vocal membrane heights the com-ponents of the lower mass and the vocal membrane were of similar magnitude. The magnitude of the upper mass was significantly increased.

For small vocal membrane heights and for small tuning parameter values q, all model masses vibrated out-of-phase. At higher values of q, the phase difference between the lower and upper mass approached −120 degrees whereas the phase difference between the upper mass and the vocal membrane approached 0 degree.

At larger vocal membrane heights, the phase differences as functions of the tuning parameter q were more complex. In particular, for large values of q, both phase differences approached about −30 degrees.

Increasing d₃, only the amplitudes of the upper mass and the vocal membrane component of the third eigenmode changed significantly. This mode consisted of large components for the upper mass and the vocal membrane. The phase differences showed no changes with varying vocal membrane height. The phase difference ϕ₂ − ϕ₁ remained a flat function of q. Ranging between −30 degrees and 0 degree, it indicated in-phase oscillations of the lower and upper mass. The phase difference ϕ_θ − ϕ₂ varied smoothly between −150 degrees and 90 degrees.

This corresponded mainly to out-of-phase oscillations of the upper mass and the vocal membrane.

The vocal membrane height d₃had no effect on the eigenfrequencies and relative damping times. Observe that d₃ effected only the shape of the eigenmodes, i.e.

the complex amplitudes important for the phase differences between the lower and upper masses and the vocal membrane plate.

These observations suggested that d₃ was an important parameter controlling the oscillation onset behavior of the full nonlinear model. In the two-mass model, efficient energy transfer from air flow to model mass vibrations is known to depend on the phase difference between the lower and upper mass. The vocal membrane height could be a crucial geometrical control parameter for the onset of oscillations of the full nonlinear vocal membrane model.