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By Giuseppe Tenti and annotated by Douglas Wilhelm Harder.

# Free Oscillations

If the forcing function is zero, the the differential equation becomes
homogeneous:

From our previous observations, we note that the characteristic equation is and
the roots of the characteristic are:

and the response to the homogeneous system are

respectively.

We will now discuss the physical interpretation of these three classes of responses.

## Over Damping

If the friction is stronger than the elastic force, that is , the response quickly converges to zero. The response drops monotonically
to zero without any oscillations. Figure 1 shows such an example.

Figure 1. An over damped system.

## Critically Damping

If the effect of the friction equals the elastic force, the response decays to zero as quickly as is possible without
oscillating. In this case, it is called *critically damped* and is shown in Figure 2.

Figure 2. A critically damped system.

## Under Damping

Finally, if the elastic force is greater than the effect of the friction (), the response
decays to zero but oscillates around that response. This is called underdamping and is shown in Figure 3.

Figure 3. An under damped system.

# Mechanical Oscillator

The differential equation

may be written in the form

from which we may determine that:

and .

Suppose that the mass is and the spring
constant is . Immediately, we note that
the natural frequency is . If there was no friction (), the mass would naturally oscillate
with a period of .

and therefore the damping parameter is

.

In each of these cases, we will move the object to a distance of and we will
release the object, and therefore the initial velocity is

## Over Damping

Suppose the friction coefficient is greater than 1.2, for example, , in which
case we have that
and therefore we see the response shown in Figures 4 and 5.
This is the response .

Figure 4. The response to an over damped system.

Figure 5. The movement of the mass following the response in Figure 4.

Note, with sufficiently strong initial conditions (e.g., and ), it may happen that the response may initially
change sign, but after this point, it will decay monotonically to zero.

## Critically Damped

Suppose the friction coefficient equals 1.2, that is, , in which case we have
and therefore we see the response shown in Figures 6 and 7.
This is the response .

Figure 6. The response to a critically damped system.

Figure 7. The movement of the mass following the response in Figure 6.

Note, with sufficiently strong initial conditions (e.g., and ), it may happen that the response may initially
change sign, but after this point, it will decay monotonically to zero.

## Under Damped

Suppose the friction coefficient equals 0.3, that is, , in which case we have
and therefore we see the response shown in Figures 8 and 9.
This is the response .

Figure 8. The response to an under damped system.

Figure 9. The movement of the mass following the response in Figure 8.

We can also write this as .