Biomedical Engineering Reference
In-Depth Information
k
(high frequency)
Labia
Oscillations
at rest
(low frequency)
p
Fig. 5.3. The parameter space of a simple model of sound production. The axes
are the restitution constant ( K ) and the pressure ( P ). The shaded region denotes
the region of the parameter space in which oscillations take place. If, initially, the
system operates with parameters outside the shaded region, the labia will not move.
If the parameters are slowly changed towards the shaded region, oscillations will
start as we enter it (modified after [Laje et al. 2002])
example belongs to the class of problems that we must understand in order
to address the issue of labial oscillations. The appearence of oscillations with
a well-defined frequency is known as a Hopf bifurcation . In the case of labial
oscillations, this frequency depends on the value of the restitution coe cient
of the labia when the oscillations begin.
In Fig. 5.3, we show the states of the syringeal labia for a range of values
of the pressure and muscle tension. The pressure and the tension constitute
what we call the parameter space of the system. To the left of the verti-
cal dashed line, the pressure is not large enough to establish oscillations, and
therefore the labia remain at rest. In contrast, as the pressure is increased and
the vertical line is passed, oscillations are started (shaded region). Depend-
ing on the tension as the critical pressure is passed, oscillations of different
frequencies are started. The turning off of the oscillations when the pressure
is decreased is described in terms of an inverse Hopf bifurcation.
The model described in the previous chapter to describe the dynamics of
the midpoint of a labium also assumed a restricted number of active modes,
as in the example where we discussed the deformation of a plate. In the case
of the labial motion, we assumed a lateral mode and a wavelike mode in order
to allow the transfer of energy from the airflow to the labia. By assuming a
small number of modes, we restrict the level of complexity that the solutions
of the model can display. We have to add this assumption to our list, in order
to explore the possible consequences of removing it. As with other effects,
we shall deal with this question in Chap. 6. Now, we shall continue with our
description of the simple, low-dimensional model presented in Chap. 4 for
the midpoint of a labium x :
 
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