Six Degree of Freedom Effects on Bullet Flight
As described in the opening paragraphs
of Section 3.0, a flying bullet has six degrees of dynamic freedom
(6 DOF), three translational degrees of freedom and three rotational
degrees of freedom. All sporting bullets, except round balls from
smoothbore black powder guns, are spin-stabilized during flight.
For a flying bullet that is well stabilized (gyroscopically stabilized),
trajectories calculated with a three degree of freedom (3 DOF) model
of bullet flight are almost exactly correct. The 3 DOF model treats
only the three translational degrees of freedom of the bullet (e.g.,
bullet position in downrange, vertical and crossrange coordinates),
producing a trajectory based on a simplified bullet model that is
a point mass with a ballistic coefficient. The three degrees of
freedom of rotational motions of the bullet cause only very small
variations in the trajectory calculated from the 3 DOF model. Fundamentally,
this is because the spinning bullet is so well stabilized that the
rotational motions, other than the spinning motion, are very tiny.
However, there are at least four, possibly
five, effects caused by rotational motions of the bullet that are
small but observable in small arms trajectories. These are:
|The small rotation downward
(aerodynamic pitch) of the nose of a bullet as it flies along
an arced trajectory, so that the longitudinal axis of the bullet
stays almost exactly parallel to the velocity vector throughout
the trajectory. This motion is caused by a very small aerodynamic
sideforce on the bullet resulting from a yaw angle known as
the yaw of repose. This angle is true aerodynamic
yaw. The nose of the bullet is pointed very slightly to the
right of the trajectory plane for a bullet of right hand (RH)
spin, or very slightly to the left of the trajectory plane for
a bullet of left hand (LH) spin.
|A small crossrange deflection
of the bullet, to the right for RH spin or to the left for LH
spin. This crossrange deflection is caused by the tiny aerodynamic
sideforce on the bullet resulting from the yaw of repose.
|The bullet turning horizontally
to the right or left to follow a crosswind, or turning upward
or downward to follow a vertical wind. This turning motion causes
a large crossrange deflection of the bullet to follow a crosswind,
or a large vertical deflection of the bullet to follow a vertical
wind. These bullet deflections were described in Section 3.2.
|A small vertical deflection
(upward or downward) of the bullet together with the large crossrange
deflection, resulting from a crosswind. This small vertical
deflection is caused by a tiny aerodynamic lift force, or negative
lift force, on the bullet, which is necessary to make the bullet
turn to follow the crosswind.
|A small horizontal deflection
of the bullet (right or left) together with the large vertical
deflection, resulting from a vertical wind. This small horizontal
deflection is caused by a tiny aerodynamic sideforce on the
bullet, which is necessary to make the bullet turn upward or
downward to follow the vertical wind.
These effects seem very strange.
For example, it does not seem correct that a small horizontal
sideforce would cause a bullet to rotate downward to keep
the bullet longitudinal axis tangent to the arc of the trajectory
as the bullet flies, although we can easily imagine that such
a sideforce would deflect the bullet horizontally as it flies.
These effects truly do happen, but
in general they are observable only at longer ranges of 300
yards or more. This is for two reasons. First, as described
in Section 2.4, a bullet exits the muzzle with some ballistic
yaw, generally an angle on the order of one degree. This initial
yaw causes the bullet to precess, or cone about the velocity
vector. As the bullet flies, this coning motion damps out
or damps to some minimum value over the first 200 yards or
so. This motion is, of course, a 6 DOF effect, and it is initially
much larger than the small effects that we will describe here.
The second reason is that the small effects grow with range
distance (or flight time). They are overwhelmed by the coning
motion at short ranges, but they become observable at longer
ranges when the coning motion damps out.
To explain the causes of the small
6 DOF effects, we need to delve into a branch of physics,
specifically into the dynamics of rigid, spin-stabilized bodies.
Because spin-stabilized bullets are very simple rigid bodies,
we can do this without using advanced mathematics. Readers
who are familiar with rigid body dynamics will be able to
understand the following explanation with no trouble. Readers
who have never studied this branch of physics will need to
accept a number of statements on faith, but they will be able
to follow the logic of the explanation and understand the
causes of the effects. It is important to understand that
we cannot quantify the effects, that is, we will not be able
to calculate how far a bullet will deflect as a result of
the rotational motions. The deflections depend on dynamic
properties of bullets that simply are not known and cannot
be measured with testing facilities available to commercial
bullet manufacturers. These properties have been measured
for a few bullets in military testing facilities, and the
results of such tests verify that the 6 DOF effects on bullet
trajectories are indeed small variations on the 3 DOF trajectories
for bullets used in small arms.