Acoust. Sci. & Tech. 25, 6 (2004)
Dynamics of a scaled two-mass model of the vocal folds for men and women
Jorge C. Lucero
Department Mathematics, University of Brasilia,
Brasilia DF 70910–900, Brazil
( Received 19 April 2004, Accepted for publication 7 June 2004 )
Keywords: Two-mass model, vocal folds, phonation, voice, oscillation hysteresis
PACS number: 43.70.Aj, 43.70.Bk [DOI: 10.1250/ast.25.478]
1.
Introduction
The purpose of this work is to analyze how the oscillatory
behavior of the vocal folds changes according to laryngeal
size, in cases of phonation by men and women. Since air
viscosity terms of glottal aerodynamics depend on the crosssectional areas of airways, then larger losses of the airflow
energy would occur in smaller larynges, which in turn would
leave less energy to fuel the oscillation. Thus, variations of the
oscillation dynamics as functions of those dimensions might
be expected. Such variations might influence the strategies for
controlling voicing onset and offset during speech by women
vs. men [1], and might be important to understand the
development of motor control of the larynx in children [2].
A modified two-mass model of the larynx and two-tube
model of vocal tract, in male and female configurations, are
used to simulate voice production. Past research has indicated
that low-dimensional models, though simpler than the human
larynx, still capture many significant aspects of vocal fold
vibration for speech (e.g. [3],). The next sections will present
the models and simulation results of male and female voices.
2.
Model
The larynx is modeled using a modified version of the
two-mass model [4]. For the damping forces, instead of the
usual linear term r x_, where r is a damping coefficient and x is
the tissue displacement, a nonlinear characteristics of the form
rð1 þ jxjÞx_ is adopted, where is a suitable coefficient. This
nonlinear characteristics is required to limit the amplitude of
the vocal fold oscillation [1,2]. Also, it reproduces experimental data which shows that the time constant of relaxation
curves of vocal fold tissue increases with the level of strain
[5]. In the present work, a value of ¼ 150 was used.
The glottal aerodynamics is modeled using an updated
version of Ishizaka and Flanagan’s equations [4]. Following
Pelorson et al. [6], we assume no flow contraction at the
glottal entry, and the formation of a free jet inside the glottis
when the glottal channel has a divergent shape. Downstream
the point of flow separation from the glottal walls, the flow
losses all its energy due to turbulence.
The vocal tract was represented by a standard two-tube
configuration for vowel /a/ [3,4]. Its equations were derived
using a transmission line analogy, terminated in a radiation
load of a circular piston in an infinite baffle.
A male configuration of the models was defined by using
e-mail: lucero@mat.unb.br
478
their standard parameters. For their female configuration, the
dimensions of the larynx and vocal tract were reduced by a
scaling factor. According to [7], the size relation between
male and female larynges is in the range 1.2–1.6, depending
on the dimension. For simplicity, a single scaling factor for all
dimensions was adopted, with a value ¼ 1:4. Masses were
accordingly computed dividing by 3 , to compensate the
volume reduction. The tissue stiffness coefficients were
reduced by a factor , to keep a constant tissue elasticity
modulus between men and women.
3.
Results
Simulations of voice records were obtained by numerical
solution of the equations of the model.
Figure 1 shows simulations of oral airflow, when the
glottal width is changed following an abduction-adduction
gesture, as in the production of utterance /aha/. Subglottal
pressure and Q factor are kept constant at Ps ¼ 600 Pa and
Q ¼ 1, and x0 is smoothly increased from 0.01 cm to 0.09 cm,
and then decreased to the initial value.
Comparing the male and female results, we see that the
male flow has a larger amplitude, and lower fundamental
frequency (approximately 110 Hz vs. 200 Hz in the female
flow), as expected. In the female case, the glottal pulses stop
at the peak abduction, and restart at the end of the following
adduction. This is a clear oscillation hysteresis phenomenon,
in which the vocal fold oscillation stops and then starts at
different values of the glottal width. It is a common nonlinear
phenomenon in cases of flow-induced oscillations, and has
been observed in several experimental studies of phonation
(see a detailed discussion in [8]). In the male case, there is no
interruption of the glottal pulses.
Figure 2 shows the RMS amplitude of the AC component
of the oral airflow [2], when varying one of the control
parameters while keeping the other two fixed at the previous
reference values. In case of the male configuration, the AC
amplitude increase almost linearly with the subglottal pressure. There is threshold pressure value around 200 Pa, which
must be overcome for the start of the vocal fold oscillation.
When the glottal half width is increased abducting the vocal
folds, the vocal fold oscillation becomes more difficult, and so
the AC amplitude decreases. There is a threshold value of
0.34 cm, above which the oscillation is no longer possible.
Considering finally increases of the Q factor, we see that it
causes the decrease of the AC amplitude. As the Q factor is
larger, the vocal fold stiffness is larger, and so the oscillation
J. C. LUCERO: SCALED TWO-MASS MODEL OF THE VOCAL FOLDS
respectively, smaller than the male values. These results,
together with the larger threshold pressure, implies that the
female larynx has a more restricted region of vocal fold
oscillation. Thus, voice would be harder to sustain in women
than in men. Particularly, the smaller threshold for the glottal
width would explain the larger occurrence of devoicing in
glottal abduction-adduction gestures for producing /h/ in
running speech by women, compared to men [9]. As the vocal
folds abduct, they easily reach the oscillation offset threshold
in women, whereas men would still require more effort.
3
Flow (cm /s)
1000
500
0
3
Flow (cm /s)
1000
500
0
0
0.05
0.1
Time (s)
0.15
0.2
Fig. 1 Simulated oral airflow in a glottal abductionadduction gesture. Top: male model. Bottom: female
model.
Flow (cm3/s)
200
150
100
50
0
0
200
Flow (cm3/s)
200
800
1000
100
50
0
0.05
0.1
200
0.15
0.2
0.25
0.3
Glottal half−width (cm)
0.35
0.4
150
100
50
0
0
1
2
3
Conclusion
In general, the results show the capability of the two-mass
model of the vocal folds to reproduce their oscillatory
behavior at phonation by men and women. A hysteresis
effect is present at voice onset-offset, with threshold conditions to start the vocal fold oscillation more severe than
those to stop it. The oscillation threshold conditions are more
restricted for the smaller female larynges, in agreement with
reported experimental results. This effect seems consequence
of increased losses for air viscosity, due to smaller crosssectional areas of airways.
Acknowledgments
This work was supported by CNPq (Brazil).
References
150
0
Flow (cm3/s)
400
600
Subglottal pressure (Pa)
4.
4
5
Q factor
Fig. 2 RMS amplitude of the AC component of the oral
airflow when varying the subglottal pressure (top),
glottal half-width (middle), and Q factor (bottom).
Circles: male model. Crosses: female model.
is difficulted. Here too, there is a threshold value of 4.0, which
sets an upper limit for the oscillation. The results for the
female configuration are qualitatively similar to the male case.
The threshold value of the subglottal pressure is around
230 Pa, larger than the male value, and the thresholds for the
glottal half width and Q factor are 0.18 cm and 3.3,
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