# Anomalies

All of the examples shown so far may lead to the conclusion that instabilities do not occur in reality and many of the theoretical aspects, which we considered at the beginning of this paper, are of no practical significance.

The remaining part of this article is dedicated to anomalies or better unexpected bullet behavior, which one may be confronted with in forensic case work.

## Statically unstable bullet

As already said, static instability is normally nothing one has to look after, when dealing with well-designed bullets fired from small arms. However, the results of an experimental investigation of Giles and Leeming (see reference [5]) for 7.62 x 51 Nato bullets, fired from a smooth bore barrel, can be extended, at least to some degree, to the post ricochet flight of unstable projectiles.

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°.

## Dynamically unstable bullet

### Close to the muzzle

The figure 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 (7.65 mm Browning) FMJ RN bullet, 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.

### At a long distance

Finally, we will examine a bullet, which is stable close to the muzzle, but looses dynamic stability after having traveled a considerable distance.

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 (see the upper drawing of the figure), 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.