Biomedical Engineering Reference
In-Depth Information
denotes the elastic coupling constant between the two masses; and
B
1
,
2
are
the linear dissipation constants. Unlike the case for the flapping model, a
collision between labia is modeled here by
G
1
,
2
, an additional restoring force:
G
i
=
g
i
a
i
during collision
,
0
(6.14)
otherwise
.
Here
a
i
=2
L
g
(
x
0
+
x
i
) denotes the lower (
i
= 1) and upper (
i
= 2) labial
areas, which take negative values during a collision.
L
g
is the length of the
glottis, and
x
0
is the labial rest position. Finally, the forces
F
1
,
2
are defined
by
F
1
,
2
=
L
g
d
1
,
2
P
1
,
2
,where
d
1
,
2
are the thicknesses of parts 1 and 2, and
P
1
,
2
are the pressures acting on masses 1 and 2. Using Bernoulli's law, when
the glottis is open we can write
U
a
1
2
,
ρ
2
P
1
=
P
s
−
(6.15)
P
2
=
P
0
,
(6.16)
where
U
=
a
min
2(
P
s
−
P
0
)
/ρ
is the velocity of air through the glottis (
a
min
is the minimal labial area between
a
1
and
a
2
; it is set to 0 during a collision).
These equations reflect the fact that the pressure between the upper labia is
always the pressure at the entrance of the tract
P
0
, and the pressure between
the lower labia can be either
P
0
when the profile is divergent or a larger
quantity (between
P
0
and
P
s
) when the profile is convergent.
Once these rules have been stated, the important questions are: Will the
masses begin to oscillate for realistic values of the pressure? Are the fre-
quencies of the oscillations comparable to the expected ones? When this
model was studied by Ishizaka and Flannagan [Ishizaka et al. 1972] in the
framework of an analysis of the human voice, surprisingly good results were
obtained. The model was afterwards studied extensively by many authors.
(In any case, no one considers a theory to be more than a set of partial
truths, ready to be abandoned for another one that explains both whatever
could be explained by the previous theory plus new phenomena which could
not . . . In fact, there is nothing as frustrating as a theory that is too suc-
cessful...!) A detailed analysis of this model performed by Steinecke and
coworkers [Steinecke and Herzel 1995] has shown that for sublabial pressures
larger than a certain critical value, oscillations are established which are very
similar to the ones predicted by our first model, that of the flapping mo-
tion. Similar in what sense? That the motions of the upper and lower parts
are highly correlated, the upper part always oscillating with a certain phase
difference with respect to the lower part.
6.3.3 Asymmetries
The old model in Chap. 4 could not account for the existence of subharmonics.
In terms of sonograms, what the old model cannot reproduce is the lines
