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## Forces and momentsHint: If you are not familiar with the physical concept of forces
and moments, reference is made to any elementary physics textbook (chapter
classical mechanics, forces, moments).
The shadowgraphs have shown that the flowfield in the vicinity of a
bullet most generally consists of 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 Navier-Stokes 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: Generally, a body moving through the atmosphere is affected by a variety
of forces. Some of those forces are
## Mass forcesThe most simple "ballistic model", considering only theforce of gravity,
was invented by Galileo Galilei (1590). A discussion of this important
force can be found in any elementary physics book.
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 ## Wind force and overturning momentLet us consider the most general case of a bullet having ayaw angle
d.
By saying so, the ballistician means that the direction of motion of the
bullet' s CG deviates from the direction into which the bullet's axis of
symmetry points. Innumerable experimental observations have shown that
an initial yaw angle at the muzzle of a gun is essentially unavoidable
and is caused by perturbations such as barrel vibrations and muzzle blast
disturbances.
For such a bullet, the pressure differences at the bullet's surface
result in a force, which is called the 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.
It can be shown that this couple is a free vector, which is called the
_{2}aerodynamic
moment of the wind force or, for short, the overturning momentM.
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.
_{W}
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 drag, the other force is thelift
force F
or _{L}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 cross-wind
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 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 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 To complicate everything even more, the true motion of a spin-stabilized bullet is much more complex. An additional fast oscillation is superimposed on the slow oscillation. However, we will return to this point later. ## Spin damping momentSkin friction at the projectile's surface retards its spinning motion. However, the angular velocity of the rotating bullet is much less damped by the spin damping moment than the translational velocity, which is reduced due to the action of the drag force. As will be shown later, this is the reason why bullet's, which are gyroscopically stable at the muzzle will remain gyroscopically stable for the rest of their flight.## Magnus force and Magnus momentGenerally, the wind force is the dominant aerodynamic force. However, there are numerous other smaller forces but we want to consider only theMagnus
force, which turns out to be very important for bullet stability.
With respect to the figure, we are looking at a bullet from the rear. Suppose that the bullet has right-handed 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 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 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 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 A similar examination shows that the Magnus force has a ## Two arms model of yawing motionWe have now finished discussing the most important forces and aerodynamic moments affecting a bullet's motion, but so far we haven't seen what the resulting movement looks like. For the moment we are not interested in the trajectory itself (the translational movement of the body), but we want to concentrate on the body's rotation about the CG.The yawing motion of a spin-stabilized bullet, resulting from the sum
of all aerodynamic moments can be modeled as a superposition of a 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 With respect to the figure imagine looking at a bullet approaching an observer's eyes. Then the bullet's tip moves on a spiral-like (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. |

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