The Education Forum

# Ballistics

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Greetings All:

For those who wish a bit of hard science on the subject I will be posting a series of articles on the subject of Ballistics which if carefully studied and digested will almost certainly demonstrate the inherent impossibility of the JFK Kill-Shot Scenerio as proposed by the Warren Commision, if one applies the science to that scenerio.

Forces and Moments:

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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, to mention just 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 atmosphere conditions.

The mathematical equations, by means of which the flowfield parameters (for

example pressure and flowfield velocity at each location) could be

determined are well known to the physicist. However, with the help of

powerful computers, numeric solutions to these equations have been found up

to now for very specific configurations only.

Because of these 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, and

this is done by shooting experiments and 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. However, the following discussion will be restricted

only to drag, lift and the Magnus force.

Table: Forces, affecting a bullet's movement through the air

Forces Requires Remarks

Yaw Spin

Mass Forces

Gravity (1) N N responsible for bending of trajectory

Coriolis Force (2) N N usually very small

Centrifugal Force N N small; usually included in gravity

Aerodynamic Forces

Drag N N major aerodynamic force

Lift (3) (Cross-wind Y N responsible for side drift

Force)

Magnus Y Y very important for stability

Pitch Damping Y Y usually very small, important for stability

Transversal Magnus Y Y usually very small

Remarks:

(1) The acceleration of gravity depends on the degree of latitude.

(2) The Coriolis force and the centrifugal force automatically arise from

the fact that the earth is not resting, but rotates about an axis.

(3) The name lift suggests an upward directed force, which is true for a

climbing airplane. However, the direction of the lift force depends on the

orientation of the yaw angle (see later). Thus a better word for lift force

could be cross-wind force.

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Wind Force and Overturning Moment:

Now let us consider the most general case of a bullet having a yaw angle.

By saying so, the ballistican 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 is principally unavoidable and is caused by perturbations

like 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 wind force. The wind force seems to apply

at the center of pressure of the wind force (CPW), which, for

spin-stabilized 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 shows the wind force F1, which applies at its center

of pressure CPW.

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 forcesobviously do not have any effect on the bullet as

they mutually neutralize.

Now let us consider the two forces F1 and F2. 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 moment MW. 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.

We want to summarize: The wind force, which applies at the center of

pressure, can be substituted 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 attacks 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 FD or simply drag, the other force is the lift

force FL or lift for short. Obviously, in the absence of a yaw angle, the

wind force reduces to the drag.

So far, we have explained the forces, which compose the wind force and the

overturning moment, but we haven't 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 scene, 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 believe what can be observed: due to the gyroscopic effect, the

bullet' s longitudinal axis moves aside into the direction of the

overturning moment, just 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 spin-stabilized

bullet is much more complex. A fast oscillation superposes the slow

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Magnus Force and Magnus Moment:

Generally, the wind force is the dominant aerodynamic force. However, there

are numerous other smaller forces but we want to consider only the Magnus

force, which turns out to be very important for bullet stability.

With respect to the figure, we imagine to look at a bullet from the rear.

Suppose that the bullet has right-handed twist, We additionally assume

the presence of an angle of yaw. The bullet's longitudinal axis should be

inclined to the left.

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, 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 directed force, which is said to be

the Magnus force FM (Heinrich Gustav Magnus, *1802, 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 center of pressure CPM. 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 MM. This moment tends to turn the body about an axis

perpendicular to its axis of symmetry.

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. 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.

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Two Arms Model of Yawing Motion:

We have now finished to discuss the most important forces and aerodynamic

moments affecting a bullet's motion, but so far we haven't seen how 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 fast and a

slow mode oscillation and can most easily be explained and understood by

means of a two arms model.

Imagine to look at the bullet from the rear. 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 to look at a bullet approaching an

observer's eyes. Then the bullet's tip moves on a spiral-like (also

described as helical), 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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End of Part One:

Respectfully: 