Ballistics theory and design of guns and ammunition rar

This is because the center of mass of the bullet is behind the geometric center, causing the back to come forward at impact, and therefore tumbling through soft tissue, creating much greater damage, according to the "energy dump" theory. Testing done by HKPro in indicated that the 4. All sizes in millimeters mm. The common rifling twist rate for this cartridge is mm 1 in 6.

According to the official C. The Belgian 5. The German Army version of the 4. At the stated effective range of m the DM11 projectile will be travelling at approximately Mach 1. DM11 cartridge technical data: [11] [ dead link ].

The Action 4. The cartridge is designed for the MP7. This ammunition is optimized for energy transfer in soft targets and should offer decent penetration performance on hard and combined targets like car doors or glass and body armour.

At the stated effective range of m the 2 g Action projectile will be travelling at approximately Mach 1. Action Law Enforcement cartridge technical data: [14]. The Ball 4. This ammunition is optimized for energy transfer in soft targets and should offer good precision. At the stated effective range of m the 2. Ball cartridge technical data: [16].

VBR produces a 4. Military Wiki Explore. Popular pages. Grant Richard Winters Harry Welsh. Project maintenance. Explore Wikis Community Central.

Register Don't have an account? The tapers shown facilitate the removal rearward of the spent cartridge case that hugs the chamber wall. Such turbulence contributes to the erosion of the bore surfaces. It has been found that much of the erosive wear in high performance guns and howitzers can be ameliorated by the introduction of a cool liquid, gaseous, or particulate layer between the hot propellant gases and the bore.

The grains commonly have seven perforations for progressive burning. In high performance rounds, vibrating the case to help settle the grains maximizes the loading density of the charge. Tank munitions are often loaded with perforated stick propellant.

Supplementary granular propellant is occasionally added to the stick bundles to further boost the charge mass and increase the progressivity of burning. Howitzer separate-loaded charges are made up of bagged increments that are ignited by the last increment loaded, the base-pad igniter.

A primer in the breechblock sets off the igniter. With bagged propelling charges, since there is no cartridge case present, it is extremely important that all of the material be combusted. If burning materials are present and a fresh charge is inserted into the bore, the propellant may ignite and cause serious injury to the gun crew.

However, severe operational and performance problems have prevented their adoption. Attempts to get around these so-called Taylor instabilities have had some success with regenerative pressurized systems that atomize the pumped-in liquids, ignite this cloud, and avoid the pressure wave unpredictability of an ignited bulk of liquid. This concept, even though it has shown promise, still may not be able to overcome the poor low temperature properties of the liquid propellants.

Two other concepts of gun propulsion should be mentioned. These are the use of electromagnetically generated force to propel a projectile down a gun and the idea of using a low molecular weight gas to propel the projectile—the light gas gun. Production of gas from a grain depends on the evolution of the total surface of the grain as the burning proceeds.

If the surface area increases with time, the grain is considered progressive. If the total surface remains constant over time the grain is neutral, and if the surface decreases with time the grain is considered regressive. The perforations in the grain affect the surface area and therefore the burning characteristics. In cylindrical grains, the number of perforations are usually one of the numbers in the sequence: 1, 7, 19, and The various types of grains are shown in Figure 4.

The web, D, that is the smallest thickness of propellant between any two surfaces is one of the major parameters in interior ballistic computations.

The axial load on a projectile, however, is for the most part only present during acceleration in the tube, is a function of time, and occurs whether the projectile is spinning or not.

The setback load and if spinning the centrifugal and torsional loads must all be superimposed on the projectile to determine its state of stress. From Equations 4. These forces are proportional to the moment of inertia of the sections ahead of or behind the application of the 0 torque load, Izz.

This pressure can be greater than the gas base pressure on the projectile. The pressure required to cause this stress is called the interface pressure. An excellent example of this relationship is contained in Ref. We have the forces on the projectile structure but now must translate these into stresses that allow us to determine how much design margin is present.

Since projectiles may be made of a variety of materials, specialized criteria may have to be used on each material. This full procedure is somewhat complicated and beyond the scope of this book, but we will attempt to describe the basics through an examination of a simple M1, high explosive projectile structure depicted in Figure 4.

Assume a thick-walled cylinder as shown for stress calculations where S1j—Longitudinal stress at the jth location S2j—Hoop stress at the jth location S3j—Radial stress at the jth location t11—Longitudinal shear at the base t2j—Torsional shear stress at the jth location It is helpful to recap here all of the loads on an element of projectile wall material at a generalized location such as point A in the diagram. This element of material is. Compressed in the axial direction due to the axial acceleration.

Loaded in tension in the hoop direction because of the wall mass being pulled radially outward due to the spin. Loaded in shear due to the rotating band accelerating the projectile in an angular direction. Note that when including mass forward of a particular section, we must include all mass transmitting loads to the section, e. The pressures applied to our model of the M1 projectile are shown in Figure 4. For convenience, these have been tabulated in Table 4.

Equation 4. The drawback is that a base of this type requires a skirted boat tail which is more expensive to manufacture but saves considerable weight. In Equation 4. We must also account for the shear stresses which are most severe at location 1. Wherever these calculations are done on the shell, the proper Izz and the proper inner and outer diameters must be used.

While this is usually very accurate and saves a good deal of time, there are instances when one would like to check the answers through a hand calculation. Projectile OD—4. Projectile ID average —2. Projectile base intrusion into cartridge case— Total length of explosive column— Fill surface area— in.

The properties of the section ahead of the location of interest are provided in Figure 4. We shall determine the stress tensor at the location shown. We shall assume the projectile obturates perfectly and that there is no friction between the projectile and the tube.

We shall use Equation 4. We put negative sign in the above equation to denote compressive stress because only the axial component loads the inner wall in compression. For instance, in some projectiles with poorly designed rotating bands, leaking of the propellant gases known as blow-by causes the exterior of the projectile to be pressurized.

This load must be considered because it has been known to collapse projectiles in development. Another point is that, while it is common to check a projectile at peak acceleration, the spin rate at this location is not a maximum. Maximum spin occurs at the exit of the muzzle of the weapon where the velocity is the highest.

It is always good practice to check a projectile for maximum spin with no axial acceleration to simulate this. Problem 4 A high explosive projectile is to be designed for a mm cannon using a 12 in. Derive the expression to calculate the torque on the projectile that achieves this acceleration if the torque is applied at the OD of the shell. Calculate the value of the torque assuming the density of steel is 0. Assume the explosive sticks completely to the interior wall.

What is the spin rate at the muzzle in Hz? Answer: The design requirements are for the ring to have an ID of 4 in. How thick does the ring have to be? The properties of brass are as follows: Yield strength of 15, psi and density of approximately 0. In some instances, the thread form is not the usual continuous spiral associated with a normal thread, but a series of discontinuous grooves that exhibit the cross-sectional form of the buttress.

In this section, we will discuss a true thread with lead-ins and partial thread shapes, but we will assume that the basic analysis will apply to buttress grooves as well. Buttress threads are designed to maximize the load carrying capability in one direction of a threaded joint. First number is the major diameter of the thread here it is in inches.

Second number is how many threads per inch. LH means left handed there will be no callout if the threads are right-hand twist or if the thread is a groove and not a continuous spiral.

Major diameter is the largest diameter of the thread form. Minor diameter is the smallest diameter of the thread form. We use buttress threads for several reasons: most important is to improve the directional loading characteristics of the thread; also to allow for a more repeatable, controllable shear during an expulsion event, i.

If thread slip occurs, the threads can either dilate or contract elastically and the joint can pop apart with little or no apparent damage to the threads. When we design for strength, we typically calculate the strength based on the shear area at the pitch diameter in the weaker material. This, of course, translates to half the length of engagement of the threads.

This is acceptable because we usually use conservative properties and add a safety factor to account for material variations and tolerances. We must always base our calculations on the weaker material if the design is to be robust. When designing to actually fail the threads, however, we need to be more exact in our analysis and take everything such as actual material property variation and tolerancing into account or our answers will be wrong.

We will proceed in this analysis in meticulous detail, initially, as a cantilevered beam subjected to compressive and tensile stresses caused by contact forces and bending moments. We consider the thread form as a short, tapered, cantilever beam and assume that failure will occur as a result of a combination of stresses and that combined bending and compressive stress precipitate the failure.

It is this combined load that will cause failure of the material. These are the diameters of the loading i. The loading is further described by Figure 4. If we assume the contact is frictionless, the average normal stress is simply the total axial force, F, divided by the projected area, A. We have assumed that the normal stress is constant over the contact area. This gives us a negative value because the stress is compressive.

Figure 4. Since an axial loading is what shears the threads, we need to project the components of this force along the axis of the projectile i. We will use the yield strength as this material strength because at that point in failure the geometry of the part is changing. Experience has shown that once this begins to happen the part is in the process of failing anyway and will not recover.

We will now combine Equations 4. The procedure now involves solving both Equations 4. The lowest value in either equation is then the force and location at which the joint will fail. It is recommended that these solutions be performed with the aid of a computerized numerical calculation program such as MathCAD. If the joint were designed to survive, it is generally best to ignore the additional strength afforded by partial threads and base the design margin on the calculation method above.

Discarding sabots have been in general use since the Second World War and are still popular. As stated previously, velocity is proportional to the square root of the pressure achieved in the tube, the area of the bore, the length of travel, and inversely proportional to the square root of the projectile weight. Therefore, we must design as light a sabot as feasible so that we can maintain a very dense, small diameter subprojectile usually an armor penetrator. The combination of the full bore area, a dense, streamlined sub-projectile, and a lightweight sabot has the overall effect of generating unusually high velocities, a characteristic essential for kinetic energy armor penetration.

There are many requirements for a successful sabot:. It must seal the propellant gases behind the projectile obturate. It must support the sub-projectile during travel in the bore to provide stable motion called providing a suitable wheelbase. It must transfer the pressure load from the propellant gases to the sub-projectile. The discarded sabot parts must also fall reliably within a danger area in front of the weapon so as not to injure troops nearby. It must be minimally parasitic, i. These are formidable requirements that necessitate great ingenuity on the part of the designers.

The problem has been solved in a variety of ways. Such armor defeating munitions were highly effective against the tank armor of the times and pot-type, saboted, kinetic energy penetrators were adopted in tank cannon around the world. This type of munition is now in the arsenals of all nations. The design of the ring sabot begins with the stress analysis of the shear traction between the sabot inner diameter and the penetrator outer diameter.

This analysis is crucial for determining the mass of the ring and thus the parasitic weight of the sabot. We will follow the work of Drysdale [6] throughout this development. The essential parameters of the computation are shown in Figure 4. If the penetrator were weaker for some reason, the sabot length would depend upon that material.

Analysis of these surfaces can be rather complicated but is similar to standard or buttress thread design practice. In both of these expressions, se is the equivalent stress as discussed in Section 4. If we substitute Equation 4. Early sabots were saddle shaped Figure 4.

These had points of high shear stress concentrations near the ends. These sabots had an excellent wheel base the distance between the forward and aft bourrelets which prevented balloting in the tube and provided good accuracy. Single- and double-ramp sabots have come into use because of the favorable weight reduction that can be obtained with this design. Axial distance the sabot to the penetrator and have the added advantage of maintaining an almost constant shear stress between the sabot and the penetrator.

The double-ramp sabot is shown in Figure 4. Detailed studies have shown that a higher order nonlinear curved ramp yields a constant shear stress under load.

Figures 4. If we examine Figure 4. Details of this derivation are found in Ref. Now we need to relate szp to szs by applying the assumption of strain compatibility, i. Source: Based on analysis from Drysdale, W. Two of the basic types of sabots have been shown in Figures 4. The double ramp also incorporates a front air scoop to facilitate discard in the air stream as well as providing additional bourrelets riding surface in the tube.

A great deal of work on the effect of sabot design parameters has been accomplished at the U. Army research laboratory. A treatment of the effect of sabot stiffness on how clean a projectile launch is can be found in Ref.

References 1. Budynas, R. Boresi, A. Montgomery, R. Pangburn, D. Drysdale, W. Plostins, P. Further Reading Barber, J. Ugural, A. The design of gun systems is so complex that it is best dealt with as a text in its own right. We begin with an introduction to fatigue so that some understanding of the basic principles in gun design can be developed.

We then proceed to discuss some introductory concepts in tube design, gun dynamics, and muzzle devices.

The reader is directed to the references for a more in-depth treatment of these topics. Fatigue is the term used for a mechanical part that undergoes cyclic loading and fails suddenly. Unlike a component that is simply overstressed and fails because the yield or ultimate strength is exceeded, a part that is subject to fatigue failure has been subjected to many small loads that stress the component below the yield strength.

Damage begins to accumulate through various mechanisms such as micro-crack growth or slipping along macroscopic boundaries. A simple example of fatigue is one where you take a metal paper clip and bend it If one repeats this multiple times with the same paper clip, it will eventually break. A projectile usually undergoes one cycle of loading so fatigue is normally not an issue.

Gun tubes, however, undergo thousands of cycles and fatigue is a major consideration in their design. This is determined by every maintenance crew by periodically checking the internal condition of the bore of the weapon. The endurance of a material is the ability of the material to survive multiple cycles of loading. This ability of a material is depicted graphically in Figure 5. An S—N diagram plots the allowable stress in the material against the number of cycles required by the designer.

Jennifer Cordes of Picatinny Arsenal when she explains the nature of fatigue to new engineers or visitors. Some materials have an endurance limit. Figure 5. Aluminums are notorious for not having an endurance limit. There are many contributing factors to the endurance of a component. Three of thesefactors which we have already touched upon are the material of the part, the number of loading cycles, and the stress level of each load cycle.

Every reference that deals with this subject has a different twist no pun intended to the governing equation. References [1] and [2] are excellent treatments of this behavior. Unfortunately, they are all subject to interpretation and vary with each material and even from reference to reference.

Problem 1 It is desired to construct a mm gun for a pressure of 43, psi. The chamber diameter has been chosen to be 3. Assume that the tube is not autofrettaged and the endurance limit S0n for is 0. Assume the chamber is open ended as a conservative measure.

Answer: in. The recoil force is estimated to be lbf. There are two failure points: a weld on the barrel and two 10—32 screws connecting the receiver to the barrel.

The projectile designer needs to know the maximum pressure on the base of the moving projectile during its time in the tube, known as the single base maximum pressure. Once this single pressure induced stress is accommodated, the designer can move on to other considerations. The tube designer, on the other hand, must know the maximum pressure exerted on the tube at every axial location in the bore as the projectile transits the tube.

These are known as the station maximum pressures in tube design. We use the projectile and charge combination which applies the most stress to the weapon usually this is the heaviest projectile and the biggest charge.

But FEA will become more important as the codes develop and weight of the weapon becomes more of an issue. Another major consideration in tube design is the degradation of material strength with temperature.

In tube artillery or tank cannons, the temperatures developed can become high enough to begin to affect the material properties in an adverse way.

The maximum pressure of this curve gives the computed maximum pressure CMP , which is the nominal pressure for the gun. However, because of the stochastic nature of a gun launch, the designer will add psi to the CMP. This is the rated maximum pressure RMP for the weapon. This experimentally determined number is the normal operating pressure NOP for the weapon and should replace the CMP as soon as it is available and accepted.

This is the elastic strength pressure ESP for the tube. A good example of how these concepts are applied can be found in Ref. When we examine the travel of the most stressful projectile down the tube, a point is reached, xmax, where the breech pressure is at maximum, pB max. If we substitute Equations 5. This is depicted in Figure 5. If we consider that a good gun steel of , psi yield strength is used, the allowable internal pressure should be kept lower than , psi.

For modern, high velocity cannons, this restriction had to be overcome and the autofrettaging process described below has been used with marked success. The design concept began around and has been in use since, but is now considered obsolete.

When an internal pressure is applied, the stresses on the inner cylinders are relieved by the pressure and then put into tension as the pressure is increased. Autofrettage is a method of prestressing a tube to improve its load carrying capability as well as its fatigue life. The procedure consists of plastically deforming the interior of the gun tube toward the outside diameter. Regions of the interior wall will now exceed the yield point, but the exterior will not have yielded.

When the load is removed, the outer layers of the material attempt to return to their unstressed state but cannot because of the plastically deformed portion of the wall. Thus, an equilibrium condition is attained where the outer wall regions remain in tension and the inner wall regions are in compression.

The process is physically accomplished by either pressurizing the interior of the tube with water above its elastic limit or by pulling an oversized mandrel through the tube to force the yielding. To further insure that the OD never goes plastic, tubes are sometimes autofrettaged in containers that act as an outer jacket during manufacture. Problem 3 The gun in Problem 1 is sized to a in. Assuming the material behaves elasticperfectly-plastic: 1.

Approximately to what radial distance does the compressive layer extend into the tube wall? Answer: Approximately 0. A layer outside the plastic region would be stressed to some level below yield and would return elastically down this curve sqq A layer inside the plastic region would be stressed to some level above yield and would return elastically down this curve This is the residual tensile stress at some outer layer Expansion strain This is the residual compressive stress at some inner layer FIGURE 5.

We will discuss the recoil response in terms of the forces and motions and the response known as gun jump. We will not attempt to discuss recoil abatement or mounting techniques. Recoil is generated on the gun by the reaction of its moveable parts to the impulse of the gas pressure both while the projectile is in the tube and while the propelling gases are being exhausted after the projectile exits.

After projectile exits, we usually assume that the pressure decays linearly with time. This period is called the gas exhaust aftereffect and is shown in Figure 5. This depends essentially on the balance of momentum between the projectile and its propelling gases and the mass of the gun.

It is caused by the momentum exchange of the mass of gas still exhausting from the tube after the projectile is long gone. We look for an estimate of the length of this motion, Ran, by examining the impulse of the gas. In real weapons, this never occurs. These forces need to be added to the above analysis to make it more accurate. The effect of a muzzle brake should be added as well. This couple causes a rotation of the gun that usually results in muzzle rise.

This contributes to projectile jump but is by no means the sole cause of it. The gun is an elastic body, so that when the propelling charge is ignited many complicated structural reactions take place. Stress and pressure waves are set up in the chamber and in the unpressurized portion of the bore, loading the tube in a highly transient fashion.

These phenomena are highly complicated and we will not discuss them further here. Sometimes increased accuracy results from shot to shot because of reduced weapon movement. Muzzle devices have also been devised to limit muzzle climb.

Muzzle brakes consist of surfaces placed perpendicular to the bore axis such that impinging gases exert a net forward thrust on the weapon. This thrust is accomplished through conservation of momentum principles.

Best design practice is to divert gases to the sides of the weapon because rearward diversion could affect an exposed gun crew. Downward diversion could kick up excessive debris and without a balancing upward diversion would strain operating gun mechanisms.

There are generally two types of muzzle brakes: closed and open. The chief purpose of these brakes is to mitigate the recoil. If the weapon is already horizontal and the venting thrust has a large vertical component, this can be a substantial loading. It diverts the blast load so that it is carried by the trunnions.

Unfortunately, at high quadrant elevations, it ducts the blast toward the crew, which is not good. Simple silencer These all are affected by the presence of the expanding gas shock wave. Various designs of suppressors have been developed and they fall into two basic types Figure 5. Smoke generated from a weapon is usually made up of a solid—liquid—gas mixture and is composed of metal or metal oxide particles from the cartridge case and its components, the projectile and the tube.

Also present are water vapor or condensate liquid and chemical elements such as carbon, copper, lead, zinc, antimony, iron, titanium, aluminum, potassium, chlorine, sodium, sulfur, and other particulates. These components in themselves obscure vision, but they may also combine with the atmosphere to allow water vapor there to condense on the particles.

Air temperature and relative humidity affect the density and longevity of the smoke as well. Electrostatic types are primarily used in a laboratory environment. Mechanical types work by robbing momentum from the particles. These suppressors work by forcing the propellant gas to pass through non-straight channels similar to pores.

The impingement of the particles robs them of momentum. The downside to this is that the suppressor adds weight to the tube at the muzzle end, adds cost, and requires frequent maintenance. There are three basic types of mechanical smoke suppressors: 1.

Increasing perforation density toward the muzzle which allows particles that would normally build up closer to the muzzle to be spread more evenly in the device because pressure drops in the axial direction. This allows some axial impingement and also helps spread out the heavier particles. Devices used to reduce noise, which are sometimes referred to as silencers, attempt to reduce the report of the weapon. In a closed-land vehicle, ship or aircraft, there is frequently a differential in air pressure between the interior and the exterior environment.

This impairs sight and breathing or burning particles introduced into the compartment could ignite ready ammunition. Flashback could occur when un-reacted propellant gas combines with the air in the compartment similar to the events at the muzzle.

In a ship mounting, ventilators are usually installed which mechanically push the muzzle gases out after shot exit. This equipment is rather large and is not practical in a land vehicle or aircraft. We design bore evacuators or bore scavengers to deal with this problem in land vehicles and aircraft. This method is simply to mount a chamber on the outside of the tube with ports that connect directly into the tube bore. These ports are designed so that they discharge in the direction of the muzzle.

When the projectile passes the open ports, gas pressure builds up in the evacuation chamber. Once shot exit occurs, the pressure in the tube eventually drops below the evacuator chamber pressure. When this occurs, the gas trapped in the evacuator rushes out of the muzzle, dragging with it the majority of the residual gas in the tube. This generates a partial vacuum so that when the breech is opened fresh air is pulled in from the compartment.

These actions are shown in Figures 5. We shall step through the muzzle exit process in the order in which the events occur.

As a projectile begins to move down the gun tube, it compresses the air ahead of it. The gun tube acts like a shock tube in which a near-planar shock forms. When this shock exits the muzzle, it forms a spherical shock wave as seen in Figure 5. It can occur regardless of the presence of the precursors. Several microseconds after the precursor shock appears, but before the projectile emerges, the so-called barrel shock and Mach cone form. This bottle-shaped structure is referred to as the shock bottle.

One important concept to keep in mind is that pressure acts in all directions—it is a point function. This is shown in Figure 5. After formation of the shock bottle but still before shot ejection, gases are still jetting out of the muzzle.

An annular vortex is formed as the gas at the center of the jet continues to rush out while gas near the outer boundary is being robbed of momentum forming a vortex. This vortex progresses downrange and will eventually approach the precursor shock. When the projectile obturator uncorks from the muzzle, there is more room for highpressure gases to escape. These gases may still be reacting and expand at a rate which results in them moving faster than the projectile.

In many instances, they are supersonic with respect to the projectile and a base shock forms. The result is a bulge of the propellant gases through the precursor shock both preceding and following the projectile. The Mach disk retreats toward the muzzle and the shock bottle recedes.

Deutschman, A. Norton, R. Smith, D. Germershausen, R. Gay, H. Further Reading Headquarters, U. Headquarters, U. Klingenberg, G. The study of these motions and the progress of the projectile to its target are the subject of this part of the book.

Then we will proceed to introduce the force due to the pressure of the air, but still considering the projectile as a mass concentrated at a point. Finally, we will consider the projectile as a three-dimensional body acted upon by the air, its spin, and gravity.

Since this text is intended to have a broad scope, some of the material is not derived in detail. The reader is encouraged to seek the more detailed treatments in the references noted throughout each section.

These principles will be extended in this section with a view toward an exterior ballistician—commonly called an aero-ballistician. These terms are commonly used in the military by gunners and researchers alike. First is the so-called map range. Gunners of large caliber weapons and mortars take great care in assuring that the sights on the weapon are leveled in the direction depicted as well as the plane out of the paper.

The line of site and angle of site yes, they are spelled that way in much of the literature are what the gunner uses to aim at the target. As you can see, they only assist in determination of the pointing of the weapon and the relative height of the target. An important feature of this diagram is the line of departure. You have probably noticed that it is not collinear with the elevation of the weapon i.

The reality is that a projectile almost never leaves the bore of a gun aligned with the bore—we shall discuss this in detail later. For now, we will simply state that this is due to the dynamics of the projectile and gun as well as aerodynamic effects. It should be noted that Figure 6. The out-of-plane angular position of the projectile at muzzle exit is known as lateral or azimuthal jump. This will combine vectorially with the vertical jump that is depicted to give a resultant jump vector.

The aerodynamics and ballistics literature are quite diverse and terminology is far from consistent. In this text, we shall use the coordinate system of Ref. The primary difference between this scheme and those of, say, Refs.

This makes sense to the authors with up being a more intuitive positive direction. We shall adhere to the broad scope of this text by including only what is necessary for a basic understanding of ballistics.

We mentioned the yaw and pitch of the projectile earlier in this section. The projectile geometry in an arbitrary state of yaw is depicted in Figure 6.

This illustration shows the projectile yawed and pitched to some angle, at, relative to the velocity vector. Thus, the velocity vector is everywhere tangent to the trajectory curve. The inset shows the decomposition of the angle between the projectile axis of symmetry, x OB , and the velocity vector, V OA.

Most projectiles have at least trigonal symmetry. This is symmetry about three planes through the projectile long axis, apart. Because of symmetry, it is common to vectorially combine the yaw and pitch of the projectile into one term which we simply call total yaw, at. Later, when we discuss advanced topics it will be necessary to once again separate them. An examination of Figure 6.

Since the drag is generated by the motion of the projectile through the air, it is naturally directed opposite to the velocity vector as illustrated in Figure 6. There are, in general, two types of drag: pressure drag and skin friction drag.

This is because nature can only act on the surface area of the projectile in two ways: normal to the surface and along it. A third type of drag called wave drag is a form of pressure drag generated by a shock wave formed when the local velocity along the surface of the projectile reaches Mach 1. For now, knowing that the drag force is opposite to the velocity vector and that its scalar magnitude, as depicted on the far RHS in Equation 6.

This is discussed more elegantly in Ref. This dynamic pressure is multiplied by a reference area, S. In every case we shall examine, this reference area is based on the projectile circular cross-section.

Also, as we shall soon see, moments require a length scale as well. In all of these instances, we shall use the projectile diameter as the reference length. This phenomenon is accounted for by a moment applied to the projectile called the spin-damping moment.

If a left-hand twist were involved, the spin vector, p and the spin-damping moment vector would be reversed. This rolling moment is depicted in Figure 6. This is depicted in Figure 6.

This is obvious even for the linear case since d appears in Equation 6. In those cases, expressions such as Cx and CN are used for drag and lift, respectively. An example of this is provided in Ref. At this point, we must discuss two quantities known as center of pressure CP and center of gravity CG sometimes called center of mass.

The CG is the location on the projectile where all of the mass can be concentrated so that for an analysis, the gravitational vector will operate at this point. The CP is the point through which a vector can be drawn, i. Figure 6.

In many cases, the Magnus force is small and is usually neglected with respect to the other forces acting on the projectile. In contrast, the moment developed because of this force is considerable. The CP for the lift force and the CP for the Magnus force are usually not the same, thus the moments will act through differing moment arms. The reason for this is the different physics that give rise to the different phenomena.

Pitch damping is the tendency of a projectile to cease its pitching motion due to air resistance. If we mount the projectile such that the bearing is transverse to the long axis and spin it, we will still have the viscous action slowing the projectile down; however, this will be overwhelmed by the pressure forces that retard the motion and the projectile will spin down much faster.

This combination of forces is called pitch damping. This is described in eloquent detail in Ref. With this assumption, Figure 6. We will not go any further here as this text is meant to be most general. McCoy, R. Murphy, C. McShane, E. Nicolaides, J. Nielsen, J. Further Reading Bull, G. Two vectors are considered equal if both their magnitude and direction are identical. However, this does not mean that they have to originate at the same point, i.

A scalar is simply a numerical quantity a magnitude. When a scalar and a vector are multiplied in any order they form a vector. In all of the above expressions, note that i, j, and k are the unit vectors in the x, y, and z coordinate directions, respectively. Consider two vectors A and B shown in Figure 7. Here en is a unit vector normal to the plane made by vectors A and B. This is depicted in Figure 7. We can also invoke Equation 7.

Thus, we can rewrite Equation 7. Let us consider a vector, A, dependent upon a scalar variable, u, as shown in Figure 7. We will now examine the kinematics of a particle. Kinematics is the study of the motion of particles and rigid bodies without regard to the forces which generate the motion. Particle kinematics assumes that a point can represent the body. The rotations of the particle itself are neglected making this a three degree of freedom DOF model.

If the particle, P, moves along a trajectory, T, its instantaneous velocity is always in a direction tangent to the trajectory and its magnitude is the speed at which it moves along the curve. Thus, the tip of this vector, r, traces out the trajectory Figure 7.

This introduces the concept of curvilinear motion with radial coordinates, r, u. Again, since eu is a unit vector, the only thing that changes with time is its direction. Here we note from Figure 7. On the left is the body rotating in space. On the right is a view of this same body looking down the axis toward O. We will now look at the rotation in terms of the vector kinematic equations for point, P.

Now we can write the velocity, v, in terms of radial and circumferential components as we discussed earlier. Thus, from Equation 7. On the left is the body rotating at angular velocity, v. On the left is the body rotating at angular velocity, v, and accelerating with angular acceleration, a.

We can then write the acceleration, a, in terms of radial and circumferential components as discussed earlier. We need to recall Equation 7. We can break down any planar motion of the rigid body into a translation and a rotation about some point. Let us choose point A in Figure 7. Equation 7. Thus, we can write Equation 7. The coordinate system in this case moves with the body but does not rotate allowing the body to rotate relative to the moving coordinate system. We choose point A in Figure 7.

At the instant considered, point A has a position, rA, a velocity, vA, and an acceleration aA, while the x—y axes and the body are rotating with angular velocity, v, and accelerating with angular acceleration, a. Since we are looking only at motion in the x—y plane, the z component is nonexistent. The second pair of terms of Equation 7. Finally, for the last term of Equation 7. First, let us review the generalized velocity and acceleration equations we derived.

Draw the situation. Determine the drag force vector. Determine the lift force vector. Determine the overturning moment vector. Determine the magnus moment vector. Greenwood, D. Colley, S. Hibbeler, R. Each section builds upon the previous one so that we recommend even seasoned veterans progress in numerical order. As we progress, we shall add in the atmosphere but neglect dynamics, atmospheric perturbations, and earth rotation. Nothing could be further from the truth. In many instances, some of the complications only slightly affect the solution and a ballistician is well placed to assume them away.

Some of these common situations will be pointed out as they arise. Notice that unlike the generalized trajectory shown previously in Figure 6. But from Equations 8. The maximum ordinate is achieved when the y-component of the velocity is 0.

By differentiating Equation 8. Further evidence of the symmetry may be seen by examining the angle of fall.

If we differentiate Equation 8. For any given launch velocity, V0, maximum range in a vacuum is achieved with an initial launch angle of The trajectory envelope is a curve that bounds all possible trajectories that attempt to reach all ranges from zero to the maximum range possible for the given launch velocity [1].

We shall now mathematically describe this curve. We know from Equation 8. These conditions yield a repeated root. The other instances a repeated root occurs are whenever the trajectory touches the trajectory envelope. This occurs only once at any given elevation. If the roots of this equation are complex conjugates, the range in question cannot be achieved with the given muzzle velocity. We can solve for all of the double roots to obtain the equation of the trajectory envelope. Equation 8.

In particular, if we rewrite Equation 8. Problem 1 A target is located at 20 km. Answer: It can be hit. Assuming a vacuum trajectory: 1. The test consists of a U. The projectile weighs 96 lbm. Using a vacuum trajectory, calculate and plot the trajectory envelope for the test.

We do this so that projectile dynamics do not enter yet into the equations of motion. We are essentially still dealing with a spherical, nonrotating cannon ball. Since there is no angle of yaw, the lift and drag forces due to yaw and the Magnus force due to spin are also negligibly small. These will be discussed in detail later. Thus, only the projectile drag forces base, wave, and skin-friction are working to slow the projectile down and gravity is pulling it toward the earth.

We can separate the velocity, acceleration, and gravitational vectors into components along the coordinate axes, so that they will be convenient to work with.

It is often convenient to use distance as the independent variable. By making a common transformation of variables to allow distance along the trajectory to be the independent variable instead of time, we can improve our ability to work with these expressions. By dividing both equations by Vx, we obtain Equations 8. For these equations, an analytic solution does exist. To do this, we must integrate over time the velocities we have found in Equations 8.

Separating the variables and substituting Equation 8. Figure 8. With computer power nowadays, we usually solve or approximate the exact solutions numerically, doing the quadratures by breaking the area under the curve into quadrilaterals and summing the areas. Essentially, we are linearizing the problem when we do this. These equations are details in Ref. The details of the derivations are again available in Ref. You can see this in. It used an 8-mm cartridge called the balle D with a bullet mass of grains and a diameter of 0.

If an infantryman is looking at a target at yards, what angle will the sight have relative to the tube assuming they used standard met in the design? Answer: About Comment on the validity of this method with respect to 2 above. Problem 6 British 0. The bullets mass is grains. The weapon was used by British units assigned to bolster the Italians in the Alps during the First World War Italy came in on the Allied side because they wanted the Tyrol region from Austria more than they wanted the Nice region from France.

The k1 value for this case is 0. Problem 8 A U. Answer: The weapon must be aimed 0. In the basic equations, we have neglected any change in air density with a change in altitude since the effect is small.

We also have assumed the equations could be solved in closed form. A diagram of the problem is shown in Figure 8. We can only solve the exact equations using numerical methods.

We can alter Equation 8. Thus, we shall set Wy equal to zero from now on. If we make this substitution into Equations 8. Now we have already solved differential Equations 8. It is an exact solution for a constant crosswind. Another interesting point is seen from examination of Equation 8.

If Vx is always equal to the initial x velocity, no matter how hard the wind blows, the projectile will not be affected. Thus, a rocket motor that maintains the initial x velocity could make the projectile insensitive to wind, a concept called automet.

Note also that if the thrust is greater than the initial velocity, the projectile will actually move into the wind. We consider next the effect of a variable crosswind. A simple way to model this effect on a projectile is to superimpose solutions for constant crosswinds over incremental distances and piece the resultant trajectory together. This technique of superposition works only with linear phenomena. However, since Equation 8. An alternative approach would be to apply Equation 8.

To do this, we shall rewrite Equation 8. These tedious calculations are best done with a computer program for small intervals of time. We will now examine the effects of a constant range wind, both head-on and a tailwind. We make the initial assumption that there is no crosswind, i. The second term is the effect of the range wind on it. If we examine Equation 8. This is important because if we had a table of velocities versus range for the no-wind case, we could then tabulate the effect of range wind.

If we now look at the y-velocity, we can operate on Equation 8. We use them when the angle of departure and angle of fall are both below 5.

We solved the crosswind equations assuming constant and variable crosswinds and introduced the classic lag rule.

With variable crosswinds, we saw it is fairly accurate to piece the trajectory together using locally constant values for the crosswind.

We have solved the range wind equations assuming only constant range wind. We could treat variable range wind in a manner similar to variable crosswinds, but the difference in results is usually not worth the added effort.