4.0 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.