Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 576

So far, we have been studying a physically nonexistent situation in which
the restorative force is the only force present, and the spring will oscillate
forever. In reality, there are usually at least two more interesting forces.
The first of these forces is an external force, sometimes called the driving
force, that acts as the “input” to the system and causes the motion to begin
in the first place. The other force is the friction that any real spring experi-
ences, which eventually causes the motion to cease. The general term used
to describe any effect that tends to reduce the amplitude of an oscillatory
system is damping, and we call oscillation where the amplitude decays over
time damped oscillation. Damping forces are particularly important for our
purposes, so let's discuss them in more detail.
The most common model for the damping force is a simple one that
acts proportional to velocity but in the opposite direction, similar to the
friction law. (Unlike the friction laws from the previous section, we don't
have any of the business concerning the normal force.) The force is simply

f d = −c x,
where f d indicates the instantaneous magnitude and direction of the damp-
ing force, x is the instantaneous velocity, and c is a constant that describes
the viscosity, roughness, etc.
The damping force has an extremely simple form, but just as with the
restorative force, things get interesting when we study the motion over
time. Qualitatively, we can make some basic predictions about how damped
oscillation of a spring would differ from undamped oscillation of the same
spring. The more obvious prediction is that we would expect the amplitude
of oscillation to decay over time, meaning the maximum displacement at
the crest of each cycle is a bit less than the previous one. Like the force
of friction, damping tends to remove energy from the system. The second
observation is only slightly less obvious: since damping in general slows the
velocity of the mass on the end of the spring, we would expect the frequency
of oscillation to be reduced compared to undamped oscillation. Those two
intuitive predictions turn out to be correct, although, of course, to be more
specific we will need to analyze the math.
Combining the restorative and damping forces, the net force can be
written as

Damping force

f net = f r + f d = −kx − c x.
To derive the equation of motion, we will need accelerations, not forces.
Applying Newton's second law and dividing both sides by the mass, we
have

x = f net
m

= − k

m x − c

m x,

(12.11)