Up to this point we have only examined the special case in which the net
force
on an oscillating particle is always proportional to the displacement of
the
particle. Oftentimes, however, there are other forces in addition to
this
restoring
force,
which
create
more complex oscillations. Though much of the study of this motion lies
in the
realm of differential equations, we will give at least an introductory
treatment
to the topic.

Damped Harmonic Motion

In most real physical situations, an oscillation cannot go on
indefinitely.
Forces such as friction and air resistance eventually dissipate energy
and
decrease both the speed and amplitude of oscillation until the system is
at rest
at its equilibrium point. The most common dissipative force encountered
is a
damping force, which is proportional to the velocity of the object,
and
always acts in a direction opposite the velocity. In the case of the
pendulum,
air resistance always works against the motion of the pendulum,
counteracting
the gravitational force, shown below.

We denote the force as F_{d}, and relate it to the velocity of the
object:
F_{d} = - bv, where b is a positive constant of proportionality,
dependent on
the
system. Recall that we generated the differential equation for simple
harmonic
motion using Newton's Second
Law:

- kx = m

We must add our damping force to the left side of this equation:

- kx - b = m

Unfortunately generating a solution to this equation requires more
advanced
mathematics than just calculus. We will simply state the final solution
and
discuss its implications. The position of the damped oscillating
particle
is
given by:

x = x_{m}e^{-bt/2m}cos(σ^{â≤}t)

Where

σ^{â≤} =

Clearly this equation is a complicated one, so let's take it apart piece
by
piece. The most notable change from our simple harmonic equation is
the
presence of the exponential function, e^{-bt/2m}. This function
gradually
decreases the amplitude of the oscillation until it reaches zero. We
still have
our cosine function, though we must calculate a new angular
frequency.
As we can
tell
by
our equation for σ^{â≤}, this frequency is smaller than with
simple
harmonic motion--the damping causes the particle to slow down,
decreasing
the
frequency and increasing the period. Shown below is a graph of typical
damped
harmonic motion:
We can see from the graph that the motion is a superposition of an
exponential
function and a sinusoidal function. The exponential function, on both
the
positive and negative sides, acts as a limit for the amplitude of the
sinusoidal
function, resulting in a gradual decrease of oscillation. Another
important
concept from the graph is that the period of the oscillation does not
change,
even though the amplitude is constantly decreasing. This property
allows
grandfather clocks to work: the pendulum of the clock is subject to
frictional
forces, gradually decreasing the amplitude of the oscillation but, since
the
period remains the same, it can still accurately measure the passage of
time.

The study of damped harmonic motion could be a chapter in and of itself;
we have
simply given an overview of the concepts that give rise to this complex
motion.

Resonance

The second example of complex harmonic motion we will examine is that of
forced
oscillations and resonance. Up to this point we have only looked at
natural
oscillations: cases in which a body is displaced and then released,
subject only
to natural restoring and frictional forces. In many cases, however, an
independent force acts on the system to drive the oscillation. Consider
a
mass
spring system in which the mass oscillates on the spring (as usual) but
the wall
to which the spring is attached oscillates at a different frequency, as
shown
below:

Usually the frequency of the external force (in this case the wall)
differs from
the frequency of the natural oscillation of the system. As such, the
motion is
quite complex, and can sometimes be chaotic. Considering the
complexity,
we
will omit the equations governing this motion, and simply examine the
special
case of resonance in forced oscillations.