The shadowgraphs have shown that the flowfield in the vicinity of a bullet most generally consists of laminar and turbulent regions. The flowfield depends in particular on the velocity at which the bullet moves, the shape of the bullet and the roughness of its surface, just to mention the most important factors. The flowfield obviously changes tremendously, as the velocity drops below the speed of sound, which is about 1115 ft/s (340 m/s) at standard atmospheric conditions.
The mathematical equations, by means of which the flowfield parameters (for example pressure and flowfield velocity at each location) might be determined are well known to the physicist as the NavierStokes equations. However, having equations and having useful solutions for these equations are entirely different matters. With the help of powerful computers, numeric and practically useful solutions to these equations have been found up to now only for very specific configurations.
Because of these computational restrictions, ballisticians all over the world consider bullet motion in the atmosphere by disregarding the specific characteristics of the flowfield and apply a simplified viewpoint: the flowfield is characterized by the forces and moments affecting the body. Generally those forces and moments must be determined experimentally, which is done by shooting experiments and through wind tunnel tests.
Generally, a body moving through the atmosphere is affected by a variety of forces. Some of those forces are mass forces, which apply at the CG (center of gravity) of the body and depend on the body mass and the mass distribution. A second group of forces is called aerodynamic forces. These forces result from the interaction of the flowfield with the bullet and depend on the shape and surface roughness of the body. Some aerodynamic forces depend on either yaw or spin or both. A summary of the most important forces affecting a bullet's motion through the atmosphere is shown in the table below. As an example, another table gives the magnitude of forces for a typical military bullet.
Table: Forces, affecting a bullet's movement
through the air
Forces 

Remarks  



Mass Forces  
Gravity 


responsible for bending of trajectory 
Coriolis Force 


usually very small 
Centrifugal Force 


small; usually included in gravity 
Aerodynamic Forces  
Drag 


major aerodynamic force 
Lift (Crosswind Force) 


responsible for side drift 
Magnus 


very important for stability 
Pitch Damping 


usually very small, important for stability 
Transversal Magnus 


usually very small 
As we intend to study the movement of bullets on earth, we have to consider its rotation. However, Newton's equations of motion are only valid in an inertial reference system  which either rests or moves with constant speed. As soon as we consider bullet motion in a reference frame bound to the rotating earth, we have to deal with an accelerated reference frame. But we can compensate for that  and still use Newton's equations of motion  by adding two additional forces: The centrifugal force and the Coriolis force.
For such a bullet, the pressure differences at the bullet's surface result in a force, which is called the wind force. The wind force seems to apply at the center of pressure of the wind force (CPW), which, for spinstabilized bullets, is located in front of the CG. The location of the CPW is by no means stationary and shifts as the flowfield changes. The figure schematically shows the wind force F_{1}, which applies at its center of pressure CPW.
As shown in another figure , it is possible to add two forces to the wind force, having the same magnitude as the wind force but opposite directions. If one let those two forces attack at the CG, these two forces obviously do not have any effect on the bullet as they mutually neutralize.
Now let us consider the two forces F_{1} and F_{2}. It can be shown that this couple is a free vector, which is called the aerodynamic moment of the wind force or, for short, the overturning momentM_{W}. The overturning moment tries to rotate the bullet around an axis, which passes through the CG and is perpendicular to the bullet's axis of form, just as indicated in the figure .
Summary: The wind force, which applies at the center of pressure, can be replaced by a force of the same magnitude and direction plus a moment. The force applies at the CG, the moment turns the bullet about an axis running through the CG.
This is a general rule of classical mechanics (see any elementary physics textbook) and applies for any force that operates at a point different from the CG of a rigid body.
You may proceed one step further and split the force, which applies at the CG, into a force which is antiparallel to the direction of movement of the CG plus a force, which is perpendicular to this direction. The first force is said to be the drag force F_{D} or simply drag, the other force is the lift force F_{L} or lift for short. The name lift suggests an upward directed force, which is true for a climbing airplane, but which is generally not true for a bullet. The direction of the lift force depends on the orientation of the yaw angle. Thus a better word for lift force could be crosswind force, an expression which can be found in some ballistic textbooks.
Obviously, in the absence of yaw, the wind force reduces to the drag force.
So far, we have explained the forces, how the wind force and the overturning moment are generated, but we haven' t yet dealt with their effects.
Drag and lift apply at the CG and simply affect the motion of the CG. Of course, the drag retards this motion. The effects of the lift force will be met later.
Obviously, the overturning moment tends to increase the yaw angle, and one could expect that the bullet starts tumbling and become unstable. This indeed can be observed when firing bullets from an unrifled barrel. However, at this point, as we consider spinning projectiles, the gyroscopic effect comes into the play, causing an unbelievable effect.
The gyroscopic effect can be explained and derived from general rules of physics and can be verified by applying mathematics. For the moment we simply have to accept what can be observed: due to the gyroscopic effect, the bullet' s longitudinal axis moves aside towards the direction of the overturning moment, as indicated by the arrow in the figure .
As the global outcome of the gyroscopic effect, the bullet's axis of symmetry thus would move on a cone's surface, with the velocity vector indicating the axis of the cone. This movement is often called precession. However, a more recent nomenclature defines this motion as the slow mode oscillation.
To complicate everything even more, the true motion of a spinstabilized bullet is much more complex. An additional fast oscillation is superimposed on the slow oscillation. However, we will return to this point later.
With respect to the figure , we are looking at a bullet from the rear. Suppose that the bullet has righthanded twist, as indicated by the two arrows. We additionally assume the presence of an angle of yaw d. The bullet's longitudinal axis should be inclined to the left, just as indicated in the previous drawings.
Due to this inclination, the flowfield velocity has a component perpendicular to the bullet's axis of symmetry, which we call vn.
However, because of the bullet's spin, the flowfield turns out to become asymmetric. Molecules of the air stream adhere to the bullet's surface. Air stream velocity and the rotational velocity of the body add at point B and subtract at point A. Thus one can observe a lower flowfield velocity at A and a higher streaming velocity at B. However, according to Bernoulli's rule (see elementary physics textbook), a higher streaming velocity corresponds with a lower pressure and a lower velocity with a higher pressure. Thus, there is a pressure difference, which results in a downward (only in this diagram!) directed force, which is called theMagnus force F_{M} (Heinrich Gustav Magnus, *1802, died 1870; German physicist).
This explains, why the Magnus force, as far as flying bullets are concerned, requires spin as well as an angle of yaw, otherwise this force vanishes.
If one considers the whole surface of a bullet, one finds a total Magnus force, which applies at its instantaneous center of pressure CPM (see figure ). The center of pressure of the Magnus force varies as a function of the flowfield structure and can be located behind, as well as in front of the CG. The magnitude of the Magnus force is considerably smaller than the magnitude of the wind force. However, the associated moment, the discussion of which follows, is of considerable importance for bullet stability.
You can repeat the steps that were followed after the discussion of the wind force. Again, you can substitute the Magnus force applying at its CP by an equivalent force, applying at the CG, plus a moment, which is said to be the Magnus moment M_{M}. This moment tends to turn the body about an axis perpendicular to its axis of symmetry, just as shown in the figure .
However, the gyroscopic effect also applies for the Magnus force. Remember that due to the gyroscopic effect, the bullet's nose moves into the direction of the associated moment. With respect to the conditions shown in the figure , the Magnus force thus would have a stabilizing effect, as it tends to decrease the yaw angle, because the bullet's axis will be moved opposite to the direction of the yaw angle.
A similar examination shows that the Magnus force has a destabilizing effect and increases the yaw angle, if its center of pressure is located in front of the CG. Later, this observation will become very important, as we will meet a dynamically unstable bullet, the instability of which is caused by this effect.
The yawing motion of a spinstabilized bullet, resulting from the sum of all aerodynamic moments can be modeled as a superposition of a fast and a slow mode oscillation and can most easily be explained and understood by means of a two arms model (see reference [4]).
Imagine looking at the bullet from the rear as shown in the figure . Let the slow mode arm CG to A rotate about the CG with the slow mode frequency. Consequently point A moves on a circle around the center of gravity.
Let the fast mode arm A to T rotate about A with the fast mode frequency. Then T moves on a circle around point A. T is the bullet's tip and the connecting line of CG and T is the bullet's longitudinal axis.
This simple model adequately describes the yawing motion, if one additionally considers that the fast mode frequency exceeds the slow mode frequency, and the arm lengths of the slow mode and the fast mode are, for a stable bullet, continuously shortened.
With respect to the figure imagine looking at a bullet approaching an observer's eyes. Then the bullet's tip moves on a spirallike (also described as helical) path as indicated in the drawing, while the CG remains attached to the center of the circle. The bullet's tip periodically returns back to the tangent to the trajectory. If this occurs, the yaw angle becomes a minimum.