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In-Depth Information
already in motion tend to be bouncing away from their plane of contact at a micro-
scopic scale. Small bounces preclude many of the surface interactions, both chem-
ical and mechanical, that cause friction.
We model kinetic friction (Figure 35.25) as a force
F
Normal
F
kinetic
friction
v
||
F
kf
(
t
,
y
,
v
)=
−μ
k
||
F
n
||
(35.44)
ˆ
∙
v
||
F
g
...butwemustbecarefulinsimulationtoneverreversedirectionofvelocityina
time step due to frictional forces. This constraint must be enforced by the integra-
tor or at least with knowledge of the integrator's time steps.
Drag
is the frictional force on a moving object (Figure 35.26) due to the sur-
rounding fluid medium, which is often water or air. Lord Raleigh's model of drag
is a force,
Figure 35.25: Kinetic friction
has magnitude proportional to
the normal force magnitude and
direction opposite velocity (in the
plane of the surface).
1
2
ρ||
F
d
(
t
,
y
,
v
)=
−
v
||
aC
d
v
,
(35.45)
where
a
is the surface area of the object presented along the direction of motion;
C
d
is the drag coefficient, which depends on the shape of the object, its roughness,
and the chemical composition of the materials; and
is the density of the fluid.
As with other frictional forces, we must be careful that the drag force does not
reverse the direction of motion during a time step.
Drag force leads to two particularly interesting macroscopic phenomena. The
first is
terminal velocity.
An object falling through a medium does not experience
continual net acceleration. For example, a skydiver's velocity levels off during
freefall. This occurs because at some point
F
g
=
ρ
x
F
drag
−
F
d
, since
F
d
is proportional
to velocity but
F
g
is constant on the object.
Lift
is another interesting phenomenon. An airfoil moving through a fluid
experiences a net upward force, which allows airplanes and birds to rise in the
absence of updrafts. Bernoulli's principle describes how lift arises from variations
in drag force over the entire surface. As was the case with a car tire, friction is
paradoxically
enabling
motion in this case—just not motion in the direction that
the friction opposed.
Figure 35.26: Drag forces are
caused by friction between an
object and the surrounding fluid,
and by the pressure built up in
the fluid by the object's relative
motion and friction within the
fluid. Drag forces are hard to
model accurately and efficiently
because the fluid's behavior is
complex and highly dependent on
the object's shape at all scales.
35.6.4.6 Other Forces
There are of course other fundamental real-world forces: for example, magnetic,
strong and weak nuclear, and electrical. These can be adopted from a physics text-
book in the same manner as the ones described here. There are also nonfundamen-
tal forces that are useful modeling techniques, such as tension and compression,
that can be found in any mechanical engineering text.
Nonphysical forces, such as tractor and repulsor beams in a science fiction
context, can be described using arbitrary functions. Any force desired can be
inserted into the simulation framework, since Newton's first law of motion reduces
it to an acceleration that can be inserted into the integrator.
35.6.5.1 Collision Detection
For particles with very small cross sections, collision detection is equivalent to ray
tracing. We can ignore the collisions between particles because the probability
of those collisions occurring is vanishingly small. The path of a particle over a
