Biomedical Engineering Reference
In-Depth Information
induce larger and larger oscillations; we would be delivering energy to the
system.
This description in terms of external forces that compensate the friction is
somewhat abstract. We are anxious to understand how these elements could
be present in the avian vocal organ, but we still need a few more elements in
order to be able to give an adequate description of the mechanisms used to
produce birdsong.
4.2 Nonlinear Oscillators
4.2.1 Bounding Motions
In the previous section, we discussed the fact that if we could somehow ex-
ert a force proportional to the velocity of a body, we could compensate (and
eventually overcome) the effect of the friction force. In terms of the discussion
in our previous section, this could be achieved if
β>B
. It is particularly per-
tinent for our problem to understand what happens if the oscillating system
has to move within a bounded region of space. At a certain distance from
the rest position, for example, there could be walls of some sort. If this is
the case, our previous description will remain adequate as long as the body
under analysis does not touch the walls. However, if the external force pro-
portional to the velocity that we exert on our oscillating body overcomes the
friction, there will be a net force proportional to the velocity of the body, in
the same its direction. Therefore the system will oscillate with progressively
larger amplitude. Eventually, the system will reach the walls, and a new ef-
fect will have to be considered. What happens when the system touches the
walls?
The effect of this collision can be modeled in terms of an additional fric-
tion, since energy is lost. How can we describe this phenomenon? The usual
procedure in physics is to extend what we know works well for a given range
of variables to encompass a new range of variables. This enriches both the
description and our understanding of the phenomenon under study.
4.2.2 An Additional Dissipation
So far, we have described energy dissipation in a system in terms of friction:
a force proportional to the velocity and opposing the system's motion. Now,
we want to propose an additional dissipative force that acts only when the
body reaches the walls. In this way, the total dissipation is no longer a func-
tion of
only
the velocity of the body, but a function of the position as well
[Gardner et al. 2001]. We shall not discuss details of this function, which de-
pends on many factors, such as the rigidity of the walls. What is important is
that it is a function of the position and therefore the dissipation is no longer
a function of only one of the variables of the problem (the velocity). Recall
