The remaining part of this article is dedicated to anomalies or - probably a better expression - unexpected bullet behavior, which a shooter may be confronted with.
In their experiments, Giles and Leeming found "no evidence of purely end over end tumbling" that the unspun Nato bullets flew "near to broadside-on" and "were simply slewing sideways following launch and then damping rapidly to an equilibrium position featuring a large angle of yaw". The authors were even able to estimate this stationary yaw angle on a theoretical base by applying the crossflow analogy and found that the statically unstable Nato bullet flies base forward with a stationary yaw angle of approximately 127°.
shows the yawing motion of a .38 Special Wadcutter bullet fired
from a revolver. The bullet is statically stable, the static stability
factor of more than four indicates that it even has too much spin. Obviously,
the bullet is dynamically unstable. The slow modal arm oscillation continuously
increases. The maximum yaw angle increases approximately by a factor of
three, from a value of five degrees at the muzzle to 15° for a traveling
distance of 8000 calibers (240 feet=73 m).
A similar conclusion, with the exception of the magnitude of the yaw
angle, can be drawn for the .32 ACP FMJ RN bullet (7.65 mm
Browning), fired from a pistol. This is shown in the figure .
The question is justified, whether these instabilities may have a significant practical effect. As far as short ranges within a few thousand calibers are considered, a dynamic bullet instability may hardly be detected, except when applying highly sophisticated measuring techniques. If a bullet vastly exceeds that range, the yaw angle gets a considerable magnitude, the drag increases, and accuracy suffers. Most probably, shot-to-shot variations will become enormous and the trajectories become unpredictable.
The two drawings in the figure
display the velocity-vs.-time curve of a standard 7.62 x 51 Nato bullet,
fired at almost 40°. The measurement has been taken by a long-range Doppler
radar tracking system, which is capable to follow the bending trajectory
from the muzzle to the impact (see acknowledgements).
At first sight, everything seems to be normal. The bullet's velocity considerably decreases close to the muzzle, and after a total flight time of almost 30 seconds, the bullet impacts at a distance of more than 2.5 kilometers.
A closer examination of the velocity-vs.-time curve, starting at 14
seconds after launching, clearly displays an oscillating behavior.
A zoomed sector of the velocity-vs.-time curve is shown in the lower drawing
of the figure .
An evaluation shows that the frequency of this velocity oscillation increases from approximately one revolution per second at 20 seconds of flight time to almost two revolutions per second at 28 seconds.
It is beyond doubt that the Doppler radar measurements are not erroneous. On the other hand, we have not met an aerodynamic force which could be responsible for accelerating and decelerating a bullet to cause an oscillating velocity.
This experimental observation can be explained by the dynamic instability of the 7.62 x 51 Nato bullet at low velocities.
We have learned from a previous figure that the 7.62x51 Nato bullet is statically and dynamically stable close to the muzzle. Thus, the yawing motion will be damped, and after a certain traveling distance, the yaw, with the exception of the small yaw of repose, will practically be zero.
However, if this has come true, the bullet's velocity has been considerably
retarded and it moves at a subsonic velocity. As a consequence, the flowfield
has changed tremendously. It has been found by an experimental investigation
of the BRL (see reference [6]) that one of the consequences
of the flowfield change is the displacement of the center of pressure of
the Magnus force. For supersonic velocities, this point is located behind
the CG, but moves in front of the CG for subsonic velocities. As
seen previously (see figure ),
the Magnus moment thus turns to be a strong destabilizing moment,
and as a consequence, the bullet becomes dynamically unstable.
The slow mode oscillation, also called precession, will no longer be damped and slowly increases. However, the bullet still has excessive static stability and thus the gyroscopic effect continues to take place.
As a consequence, the bullet's longitudinal axis moves on a cone's surface
with the trajectory being the axis of the cone. As this oscillation is
undamped, the opening angle of the cone continuously increases. The figure
schematically shows the coning motion of the Nato bullet on the
descending branch of the trajectory.
Keeping this in mind, the experimental findings of the Doppler radar velocity measurement can now easily be interpreted. It should be remembered that the Doppler radar technique is only capable to measure the radial velocity of an object in the radar beam. This means that the Doppler analyzer only detects velocity components, which either approach or withdraw from the antenna.
For a bullet on the descending path of the trajectory, describing a
coning motion, the velocity of the body, the axis of which withdraws
from the antenna adds to the radial velocity of the CG. On the other
hand, the velocity of the body, the axis of which approaches the
antenna subtracts from the radial velocity of the CG. This explains
the oscillating nature of the measured Doppler signal. It results from
a superposition of the radial velocity of the CG and the slow mode oscillation
of the bullet's longitudinal axis. This is schematically shown in the figure .
What is most amazing is the fact that the antenna detects all this for a tiny little bullet, flying far away at a distance of more than two kilometers.
Obviously, the dynamic instability of the Nato bullet has a tremendous effect on its trajectory. As the yaw increases, the drag increases, the bullet's velocity is far more retarded and the range decreases. It has been observed that for the studied brand of Nato bullets, instabilities were not reproducible and thus the ranges, even when firing with almost the same muzzle velocity and almost the same angle of departure vary enormously, simply by chance.
As a further consequence, exterior ballistic calculations (see reference [3]) based either on the point mass model or the modified point mass model will not be able to predict accurately the trajectory of such an unstable bullet.
