Biomedical Engineering Reference
In-Depth Information
be described in terms of harmonic oscillations is a pendulum performing small
oscillations (like a child on a swing) [Titze 1994]. A child sitting passively on
a swing after an initial push will stop after a few oscillations. What force are
we forgetting in our discussion?
4.1.2 Energy Losses
We have omitted a force which we can hardly avoid in our daily experience:
the friction that occurs during the motion of an object. This is a force that
is zero if the mass whose motion we are studying is at rest, but acts on any
body that is moving, with an intensity that is proportional to its velocity. The
direction of this force is reactive: if the velocity points in a given direction,
the force will point in the opposite direction. The constant of proportionality
between the friction force and the velocity (let us call it
B
) expresses how
strong the friction is (for example, the friction experienced by a hockey puck
sliding on ice is different from what it would be if it was sliding on a wooden
floor, although in both cases there is a force proportional (and opposed) to
the velocity of the puck). Mathematically speaking, the friction force can be
written as
B
˙
x,
(4.3)
where
x
=
dx/dt
is the velocity. In the case of the swing, the friction due to
the air interacting with the various oscillating components, and that due to
the chains interacting with the structure to which they are attached will be
responsible for the loss of the energy initially provided to the system, and
will be responsible for the stopping of the motion.
Our understanding of the nature of the agent responsible for stopping
our autonomous oscillator gives us a hint of how to counteract the effect
of friction. If we want a child on a swing to keep on performing periodic
oscillations, we have to push him/her periodically. But this strategy will not
work if we push the child as he/she approaches us (exerting on the child a
force opposite to the direction of his/her velocity); such a force is precisely
what eventually stops the motion. What is appropriate to do is to wait until
the child gets to the position of maximal departure from equilibrium, and
only push when he/she begins to move away from us. The external force
that we are exerting will not cancel the friction at every instant; rather, we
are trying to compensate the losses that have taken place in a cycle with
the energy provided in each push. If we could push at every instant with
a force exactly equal to the friction we could effectively cancel the friction
and achieve a perfect harmonic oscillator (amazing everyone around us
...
).
In this way, we exert a force in the same direction as the velocity, that is,
F
friction
=
−
F
gain
=
β
˙
x.
(4.4)
If we could exert a force proportional to the velocity at every instant, in the
same direction as the velocity, and more intense than the friction, we would
