Mechanics 2: Linear and Rotational Dynamics - 3D Math Primer for Graphics and Game Development - page 577

Next we rewrite this in terms of two new quantities. The first quantity,
ω 0 , is the undamped angular frequency and is not really new. It is identical
to the ω = k/m introduced earlier; we are adding the zero subscript just

to emphasize that it is the frequency that would occur without the damping
rather than the actual frequency. (Remember, our prediction is that the
actual frequency will be slower in some way.)
The second quantity is called the damping ratio, not to be confused with
the damping coe cient c. The damping ratio is traditionally denoted by ζ,
the Greek letter zeta, which looks weird and takes some practice to write
by hand. The damping ratio is related to the damping coe cient, mass,
and undamped angular frequency by the formula

c

mk = c

2mω 0 .
In just a moment, when we explain the qualitative meaning of ζ, the utility
in using this arbitrary formula will become apparent.
Substituting the undamped frequency ω 0 and damping ratio ζ into
Equation (12.11), we have

ζ = √

Damping ratio

2

Differential equation for
damped harmonic
oscillation

x = −ω 0 x − 2ζω 0 x. (12.12)

Readers with training in differential equations should recognize Equa-
tion (12.12) as a second-order linear homogenous differential equation with
constant coe cients, which is one of the very nicest differential equations
we could hope for, meaning we can actually solve it with pencil and paper.
Readers without this training shouldn't worry, because it won't be needed to
understand the answer, to which we now fast forward, skipping the deriva-
tion. There are three distinct cases: underdamping, critical damping, and
overdamping.
When 0 ≤ ζ < 1, we say that the system is underdamped. In this case,
as we have been predicting, the motion will continue to oscillate indefinitely
with an amplitude that decays exponentially over time. The equation that
describes this motion is

Kinematic equation for
underdamped system

−ζω 0 t ,

x(t) = (k 1 cos(ω d t) + k 2 sin(ω d t))e (12.13)

where ω d is the actual frequency of the damped oscillation and is related
to the undamped frequency ω 0 by

Damped angular
frequency ω d = ω 0 1 − ζ 2 . (12.14)

The constants k 1 and k 2 are determined by the initial position and velocity:

k 2 = ζω 0 x(0) + x(0)
ω d

k 1 = x(0), .