Game Development Reference
In-Depth Information
Using ζ = 0 produces undamped oscillation, and Equation (12.13) is equiv-
alent to Equation (12.10).
Your common sense tells you that as we increase the damping ratio, the
frequency of oscillation decreases; consulting Equation (12.14), we see that
at ζ = 1 the frequency completely vanishes. At this threshold, known as
critical damping, the behavior of the system changes qualitatively. The sys-
tem no longer oscillates, but instead decays exponentially. The kinematic
equation in this situation is
Equation of motion at
critical damping
−ω
0
t
,
x(t) = (k
1
+ k
2
t)e
(12.15)
where k
1
and k
2
are again determined by the initial conditions:
k
1
= x(0),
k
2
= ω
0
x(0) + x(0).
Critical damping is just the right amount such that the system decays
as quickly as possible without oscillation. If the damping is decreased, the
system is underdamped, as previously described, and will oscillate. If the
damping is increased, the system is overdamped; it will not oscillate, and the
rate of decay will be slower than the rate at critical damping. Figure 12.6
shows how the damping value affects the behavior of a system.
Now that we've reviewed the classic equations that may be found in
any physics textbook or on wikipedia.org, let's say a few words about how
spring-damper systems are used in video games as control systems. In
Figure 12.6.
Undamped, underdamped, and critically damped systems.
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