Undamped Forced Vibrations
We begin with the undamped case: . As we have done in the
Constant Coefficients: Complex Roots page, we look for a
particular solution of the form
where
. Moreover, for that equation we would
take a trial solution
(in the case where
is not a root of the characteristic equation, i.e.
).
We get
where
is the natural angular frequency of the spring-mass system
(i.e. the homogeneous system has solutions with angular frequency
).
Thus our particular solution is
When
, i.e.
the natural frequency of the system is higher than the frequency of the external
force, we find that
is in phase with the external force. Here is a picture
of this: the external force is provided by moving the support of the spring
up and down (the top curve), and the response
is the bottom curve.
In this case
and
. Note that the mass
is up when the support is up, and down when the support is down.
Click on picture for animation.
On the other hand, when
, i.e. the natural frequency
is lower than the frequency of the external force, the coefficient of
is negative, which means that
is 180 degrees out
of phase with the external force. Here is a picture of this, with
and
.
Note that the mass
is down when the support is up, and up when the support is down.
Click on picture for animation.
Of course, this is only one particular solution. For the general solution,
we must add the general solution of the homogeneous equation, obtaining
For example, consider the initial conditions ,
. A solution satisfying these initial conditions is
. We can rewrite this using a trigonometric identity:
Click on picture for animation
The effect is that of an angular frequency that is the average of
and
, with an amplitude that varies because of the
low-frequency factor
. This phenomenon is known as ``beats'': in an audio signal
you hear the signal grow louder and softer, dying out when
is a multiple of
and sounding loudest when it is an
odd multiple of
. This can be used to tune stringed instruments
such as guitars.
Resonance
Our formula for the particular solution doesn't work in the case
, for which we would have
in the denominator.
Instead, our methods for ``Case 2'' tell us to take a trial solution
for
of the form
. We then get
so
our particular solution is
The picture shows our spring-mass system, starting from rest, with the support moving up and down at the resonant frequency with amplitude 0.2. Eventually the mass collides with the support of the spring.
Click on picture for animation.
Damped Forced Vibrations
Now suppose the damping constant is positive.
Again we take
with
, and use a trial solution
. This should work since the roots of the
characteristic equation will all have negative real parts. We get
For example, if ,
and
, we have
. Suppose
and
.
We get
so
,
,
so
.
Again, this is one particular solution. It is called the steady-state
solution, because it represents a vibration with constant amplitude.
A solution of
the homogeneous equation, which we add to get any other solution, is
called a transient: because of the damping, these contain a factor
and go to 0 as
. Thus whatever
initial conditions you start with, if you wait long enough the solution
will be very close to the steady state solution.
In our example, which is underdamped,
and
.
Consider the initial conditions
. The steady-state
solution has
and
, so the
transient will have
and
.
The transient is
. Here is the solution:
Click on picture for animation.
The above example had
, which in an undamped
system would result in
being 180 degrees or
radians
out of phase with the external force. With damping,
if
we will have
and
, so that
. If
we
have
and again
, so
.
Thus damping increases
when
(so that the maximum of the response comes after the maximum of the
force) and decreases
it when
(so that the maximum of the response
comes before the minimum of the force). When
we have
and
.
For any given and
the amplitude of the steady
state solution will be less than the amplitude of the corresponding solution
of the undamped system, and as the damping constant
increases
the amplitude will decrease (because
is an increasing function
of
). If you keep
,
and
fixed and allow
(and thus
) to vary, it's easy to see that the maximum amplitude
will occur when
.
On the other hand, if you fix ,
and
and
allow
to vary, it's not so easy to see where the maximum
amplitude will occur. To maximize the amplitude we want to minimize
. We may as well minimize
. Letting
for convenience, we have
so
Thus we have seen three effects of damping on forced vibrations: