Modern Exterior Ballistics is a comprehensive text covering the basic free flight dynamics of symmetric projectiles. The book provides a historical perspective of. Modern Exterior Ballistics - The Launch and Flight Dynamics of Symmetric Projectiles 2nd ed. - R. McCoy (Schiffer, ) caite.info - Ebook download as PDF File. Abstract (en). An aiming system for use with a weapon is provided and may include a processor, at least one sensor in communication with the processor, and.
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Modern Exterior Ballistics same form as Bashforth's. Mayevski's firings covered the velocity range from to feet per second. Colonel Hojel of Holland. PDF | The contemporary method we are introducing to solve the exterior ballistics problems is based on the expansion in series applied in an. Audiobook MODERN EXTERIOR BALLISTICS 2ND ED For any device Download here: caite.info?book=
Even if it does exist it must be quite insignificant compared with the gyroscopic and Coriolis drifts. It ranges from 0. Archived from the original on 2 August To plan for projectile drop and compensate properly, one must understand parabolic shaped trajectories. At the very base, or heel of a projectile or bullet, there is a 0. If you click on the link below, many browsers will just open the file, as they recognize this standard format.
In the Scandinavian ammunition manufacturer Nammo Lapua Oy released a 6 DoF calculation model based ballistic free software named Lapua Ballistics.
The software is distributed as a mobile app only and available for Android and iOS devices. For the precise establishment of drag or air resistance effects on projectiles, Doppler radar measurements are required. Weibel e or Infinition BR Doppler radars are used by governments, professional ballisticians, defence forces and a few ammunition manufacturers to obtain real-world data of the flight behavior of projectiles of their interest. Correctly established state of the art Doppler radar measurements can determine the flight behavior of projectiles as small as airgun pellets in three-dimensional space to within a few millimetres accuracy.
The gathered data regarding the projectile deceleration can be derived and expressed in several ways, such as ballistic coefficients BC or drag coefficients C d. Because a spinning projectile experiences both precession and nutation about its center of gravity as it flies, further data reduction of doppler radar measurements is required to separate yaw induced drag and lift coefficients from the zero yaw drag coefficient, in order to make measurements fully applicable to 6-dof trajectory analysis.
Doppler radar measurement results for a lathe-turned monolithic solid. The initial rise in the BC value is attributed to a projectile's always present yaw and precession out of the bore. The test results were obtained from many shots not just a single shot.
The bullet was assigned 1. Doppler radar measurement results for a Lapua GB Scenar This tested bullet experiences its maximum drag coefficient when entering the transonic flight regime around Mach 1. With the help of Doppler radar measurements projectile specific drag models can be established that are most useful when shooting at extended ranges where the bullet speed slows to the transonic speed region near the speed of sound.
This is where the projectile drag predicted by mathematic modeling can significantly depart from the actual drag experienced by the projectile. Further Doppler radar measurements are used to study subtle in-flight effects of various bullet constructions. Governments, professional ballisticians, defence forces and ammunition manufacturers can supplement Doppler radar measurements with measurements gathered by telemetry probes fitted to larger projectiles.
In general, a pointed projectile will have a better drag coefficient C d or ballistic coefficient BC than a round nosed bullet, and a round nosed bullet will have a better C d or BC than a flat point bullet. Large radius curves, resulting in a shallower point angle, will produce lower drags, particularly at supersonic velocities.
Hollow point bullets behave much like a flat point of the same point diameter. Projectiles designed for supersonic use often have a slightly tapered base at the rear, called a boat tail , which reduces air resistance in flight. Analytical software was developed by the Ballistics Research Laboratory - later called the Army Research Laboratory - which reduced actual test range data to parametric relationships for projectile drag coefficient prediction.
Rocket-assisted projectiles employ a small rocket motor that ignites upon muzzle exit providing additional thrust to overcome aerodynamic drag. Rocket assist is most effective with subsonic artillery projectiles.
For supersonic long range artillery, where base drag dominates, base bleed is employed. Base bleed is a form of a gas generator that does not provide significant thrust, but rather fills the low-pressure area behind the projectile with gas, effectively reducing the base drag and the overall projectile drag coefficient. A projectile fired at supersonic muzzle velocity will at some point slow to approach the speed of sound.
At the transonic region about Mach 1. That CP shift affects the dynamic stability of the projectile. If the projectile is not well stabilized, it cannot remain pointing forward through the transonic region the projectile starts to exhibit an unwanted precession or coning motion called limit cycle yaw that, if not damped out, can eventually end in uncontrollable tumbling along the length axis.
However, even if the projectile has sufficient stability static and dynamic to be able to fly through the transonic region and stays pointing forward, it is still affected.
The erratic and sudden CP shift and temporary decrease of dynamic stability can cause significant dispersion and hence significant accuracy decay , even if the projectile's flight becomes well behaved again when it enters the subsonic region.
This makes accurately predicting the ballistic behavior of projectiles in the transonic region very difficult. Because of this, marksmen normally restrict themselves to engaging targets close enough that the projectile is still supersonic. According to Litz, "Extended Long Range starts whenever the bullet slows to its transonic range. As the bullet slows down to approach Mach 1, it starts to encounter transonic effects, which are more complex and difficult to account for, compared to the supersonic range where the bullet is relatively well-behaved.
The ambient air density has a significant effect on dynamic stability during transonic transition.
Though the ambient air density is a variable environmental factor, adverse transonic transition effects can be negated better by a projectile traveling through less dense air, than when traveling through denser air. Projectile or bullet length also affects limit cycle yaw. Longer projectiles experience more limit cycle yaw than shorter projectiles of the same diameter.
Another feature of projectile design that has been identified as having an effect on the unwanted limit cycle yaw motion is the chamfer at the base of the projectile.
At the very base, or heel of a projectile or bullet, there is a 0. The presence of this radius causes the projectile to fly with greater limit cycle yaw angles. To circumvent the transonic problems encountered by spin-stabilized projectiles, projectiles can theoretically be guided during flight. These projectiles are not spin stabilized and the flight path can be course adjusted with an electromagnetic actuator 30 times per second.
Due to the practical inability to know in advance and compensate for all the variables of flight, no software simulation, however advanced, will yield predictions that will always perfectly match real world trajectories. It is however possible to obtain predictions that are very close to actual flight behavior. For a typical. At those shorter to medium ranges, transonic problems and hence unbehaved bullet flight should not occur, and the BC is less likely to be transient.
Testing the predictive qualities of software at extreme long ranges is expensive because it consumes ammunition; the actual muzzle velocity of all shots fired must be measured to be able to make statistically dependable statements. Sample groups of less than 24 shots may not obtain the desired statistically significant confidence interval.
The normal shooting or aerodynamics enthusiast, however, has no access to such expensive professional measurement devices. Authorities and projectile manufacturers are generally reluctant to share the results of Doppler radar tests and the test derived drag coefficients C d of projectiles with the general public.
Some of the Lapua-provided drag coefficient data shows drastic increases in the measured drag around or below the Mach 1 flight velocity region. This behavior was observed for most of the measured small calibre bullets, and not so much for the larger calibre bullets. This is a limiting factor for extended range shooting use, because the effects of limit cycle yaw are not easily predictable and potentially catastrophic for the best ballistic prediction models and software.
Presented C d data can not be simply used for every gun-ammunition combination, since it was measured for the barrels, rotational spin velocities and ammunition lots the Lapua testers used during their test firings. Changes in such variables and projectile production lot variations can yield different downrange interaction with the air the projectile passes through that can result in minor changes in flight behavior.
This particular field of external ballistics is currently not elaborately studied nor well understood. The method employed to model and predict external ballistic behavior can yield differing results with increasing range and time of flight.
To illustrate this several external ballistic behavior prediction methods for the Lapua Scenar GB The table shows the Doppler radar test derived drag coefficients C d prediction method and the Lapua Ballistics 6 DoF App predictions produce similar results.
The Pejsa drag model closed-form solution prediction method, without slope constant factor fine tuning, yields very similar results in the supersonic flight regime compared to the Doppler radar test derived drag coefficients C d prediction method.
The G7 drag curve model prediction method recommended by some manufacturers for very-low-drag shaped rifle bullets when using a G7 ballistic coefficient BC of 0.
Decent prediction models are expected to yield similar results in the supersonic flight regime. Wind has a range of effects, the first being the effect of making the projectile deviate to the side horizontal deflection. From a scientific perspective, the "wind pushing on the side of the projectile" is not what causes horizontal wind drift.
What causes wind drift is drag.
Drag makes the projectile turn into the wind, much like a weather vane, keeping the centre of air pressure on its nose. This causes the nose to be cocked from your perspective into the wind, the base is cocked from your perspective "downwind.
Wind also causes aerodynamic jump which is the vertical component of cross wind deflection caused by lateral wind impulses activated during free flight of a projectile or at or very near the muzzle leading to dynamic imbalance. Like the wind direction reversing the twist direction will reverse the aerodynamic jump direction. A somewhat less obvious effect is caused by head or tailwinds.
A headwind will slightly increase the relative velocity of the projectile, and increase drag and the corresponding drop. In the real world, pure head or tailwinds are rare, since wind is seldomly constant in force and direction and normally interacts with the terrain it is blowing over.
This often makes ultra long range shooting in head or tailwind conditions difficult. The vertical angle or elevation of a shot will also affect the trajectory of the shot. Ballistic tables for small calibre projectiles fired from pistols or rifles assume a horizontal line of sight between the shooter and target with gravity acting perpendicular to the earth.
Therefore, if the shooter-to-target angle is up or down, the direction of the gravity component does not change with slope direction , then the trajectory curving acceleration due to gravity will actually be less, in proportion to the cosine of the slant angle.
As a result, a projectile fired upward or downward, on a so-called "slant range," will over-shoot the same target distance on flat ground. The effect is of sufficient magnitude that hunters must adjust their target hold off accordingly in mountainous terrain. A well known formula for slant range adjustment to horizontal range hold off is known as the Rifleman's rule.
The Rifleman's rule and the slightly more complex and less well known Improved Rifleman's rule models produce sufficiently accurate predictions for many small arms applications.
Simple prediction models however ignore minor gravity effects when shooting uphill or downhill. The only practical way to compensate for this is to use a ballistic computer program. Besides gravity at very steep angles over long distances, the effect of air density changes the projectile encounters during flight become problematic. The more advanced programs factor in the small effect of gravity on uphill and on downhill shots resulting in slightly differing trajectories at the same vertical angle and range.
No publicly available ballistic computer program currently accounts for the complicated phenomena of differing air densities the projectile encounters during flight. Air pressure , temperature , and humidity variations make up the ambient air density. Humidity has a counter intuitive impact. Since water vapor has a density of 0.
An easy way to remember that water vapor reduces air density is to observe that clouds float. Gyroscopic drift is an interaction of the bullet's mass and aerodynamics with the atmosphere that it is flying in.
Even in completely calm air, with no sideways air movement at all, a spin-stabilized projectile will experience a spin-induced sideways component, due to a gyroscopic phenomenon known as "yaw of repose. For a left hand counterclockwise direction of rotation this component will always be to the left. This is because the projectile's longitudinal axis its axis of rotation and the direction of the velocity vector of the center of gravity CG deviate by a small angle, which is said to be the equilibrium yaw or the yaw of repose.
The magnitude of the yaw of repose angle is typically less than 0. As the result of this small inclination, there is a continuous air stream, which tends to deflect the bullet to the right. Thus the occurrence of the yaw of repose is the reason for the bullet drifting to the right for right-handed spin or to the left for left-handed spin.
This means that the bullet is "skidding" sideways at any given moment, and thus experiencing a sideways component. Doppler radar measurement results for the gyroscopic drift of several US military and other very-low-drag bullets at yards The table shows that the gyroscopic drift cannot be predicted on weight and diameter alone.
In order to make accurate predictions on gyroscopic drift several details about both the external and internal ballistics must be considered. Factors such as the twist rate of the barrel, the velocity of the projectile as it exits the muzzle, barrel harmonics, and atmospheric conditions, all contribute to the path of a projectile.
Spin stabilized projectiles are affected by the Magnus effect , whereby the spin of the bullet creates a force acting either up or down, perpendicular to the sideways vector of the wind. In the simple case of horizontal wind, and a right hand clockwise direction of rotation, the Magnus effect induced pressure differences around the bullet cause a downward wind from the right or upward wind from the left force viewed from the point of firing to act on the projectile, affecting its point of impact.
The Magnus effect has a significant role in bullet stability because the Magnus force does not act upon the bullet's center of gravity, but the center of pressure affecting the yaw of the bullet. The Magnus effect will act as a destabilizing force on any bullet with a center of pressure located ahead of the center of gravity, while conversely acting as a stabilizing force on any bullet with the center of pressure located behind the center of gravity.
The location of the center of pressure depends on the flow field structure, in other words, depending on whether the bullet is in supersonic, transonic or subsonic flight. What this means in practice depends on the shape and other attributes of the bullet, in any case the Magnus force greatly affects stability because it tries to "twist" the bullet along its flight path.
Paradoxically, very-low-drag bullets due to their length have a tendency to exhibit greater Magnus destabilizing errors because they have a greater surface area to present to the oncoming air they are travelling through, thereby reducing their aerodynamic efficiency.
This subtle effect is one of the reasons why a calculated C d or BC based on shape and sectional density is of limited use. Another minor cause of drift, which depends on the nose of the projectile being above the trajectory, is the Poisson Effect.
This, if it occurs at all, acts in the same direction as the gyroscopic drift and is even less important than the Magnus effect. It supposes that the uptilted nose of the projectile causes an air cushion to build up underneath it. It further supposes that there is an increase of friction between this cushion and the projectile so that the latter, with its spin, will tend to roll off the cushion and move sideways. This simple explanation is quite popular. There is, however, no evidence to show that increased pressure means increased friction and unless this is so, there can be no effect.
Even if it does exist it must be quite insignificant compared with the gyroscopic and Coriolis drifts. Both the Poisson and Magnus Effects will reverse their directions of drift if the nose falls below the trajectory.
When the nose is off to one side, as in equilibrium yaw, these effects will make minute alterations in range. The Coriolis effect causes Coriolis drift in a direction perpendicular to the Earth's axis; for most locations on Earth and firing directions, this deflection includes horizontal and vertical components.
The deflection is to the right of the trajectory in the northern hemisphere, to the left in the southern hemisphere, upward for eastward shots, and downward for westward shots. Coriolis drift is not an aerodynamic effect; it is a consequence of the rotation of the Earth. The magnitude of the Coriolis effect is small.
The magnitude of the drift depends on the firing and target location, azimuth of firing, projectile velocity and time of flight.
Viewed from a non-rotating reference frame i. When viewed from a reference frame fixed with respect to the Earth, that straight trajectory appears to curve sideways. The direction of this horizontal curvature is to the right in the northern hemisphere and to the left in the southern hemisphere, and does not depend on the azimuth of the shot.
The horizontal curvature is largest at the poles and decreases to zero at the equator. It causes eastward-traveling projectiles to deflect upward, and westward-traveling projectiles to deflect downward. The effect is less pronounced for trajectories in other directions, and is zero for trajectories aimed due north or south. In the case of large changes of momentum, such as a spacecraft being launched into Earth orbit, the effect becomes significant.
It contributes to the fastest and most fuel-efficient path to orbit: Though not forces acting on projectile trajectories there are some equipment related factors that influence trajectories.
Since these factors can cause otherwise unexplainable external ballistic flight behavior they have to be briefly mentioned. Lateral jump is caused by a slight lateral and rotational movement of a gun barrel at the instant of firing. It has the effect of a small error in bearing.
The effect is ignored, since it is small and varies from round to round. Lateral throw-off is caused by mass imbalance in applied spin stabilized projectiles or pressure imbalances during the transitional flight phase when a projectile leaves a gun barrel off axis leading to static imbalance.
If present it causes dispersion. The maximum practical range [note 4] of all small arms and especially high-powered sniper rifles depends mainly on the aerodynamic or ballistic efficiency of the spin stabilised projectiles used.
Long-range shooters must also collect relevant information to calculate elevation and windage corrections to be able to achieve first shot strikes at point targets. The data to calculate these fire control corrections has a long list of variables including: The ambient air density is at its maximum at Arctic sea level conditions.
Cold gunpowder also produces lower pressures and hence lower muzzle velocities than warm powder.
This means that the maximum practical range of rifles will be at it shortest at Arctic sea level conditions. The ability to hit a point target at great range has a lot to do with the ability to tackle environmental and meteorological factors and a good understanding of exterior ballistics and the limitations of equipment.
Without computer support and highly accurate laser rangefinders and meteorological measuring equipment as aids to determine ballistic solutions, long-range shooting beyond m yd at unknown ranges becomes guesswork for even the most expert long-range marksmen. Interesting further reading: Marksmanship Wikibook. Here is an example of a ballistic table for a. It assumes sights 1. This table demonstrates that, even with a fairly aerodynamic bullet fired at high velocity, the "bullet drop" or change in the point of impact is significant.
This change in point of impact has two important implications. Secondly, the rifle should be zeroed to a distance appropriate to the typical range of targets, because the shooter might have to aim so far above the target to compensate for a large bullet drop that he may lose sight of the target completely for instance being outside the field of view of a telescopic sight.
From Wikipedia, the free encyclopedia. Behavior of projectiles in flight. Play media. It is less common but possible for bullets to display significant lack of dynamic stability at supersonic velocities. Since dynamic stability is mostly governed by transonic aerodynamics, it is very hard to predict when a projectile will have sufficient dynamic stability these are the hardest aerodynamic coefficients to calculate accurately at the most difficult speed regime to predict transonic.
The aerodynamic coefficients that govern dynamic stability: In the end, there is little that modeling and simulation can do to accurately predict the level of dynamic stability that a bullet will have downrange. If a projectile has a very high or low level of dynamic stability, modeling may get the answer right. However, if a situation is borderline dynamic stability near 0 or 2 modeling cannot be relied upon to produce the right answer.
This is one of those things that have to be field tested and carefully documented. Pejsa predictions calculated with Lex Talus Corporation Pejsa based ballistic software with the slope constant factor set at the 0. The range in which a competent and trained individual using the firearm has the ability to hit a target sixty to eighty percent of the time.
In reality, most firearms have a true range much greater than this but the likelihood of hitting a target is poor at greater than effective range. There seems to be no good formula for the effective ranges of the various firearms. Archived from the original on November 7, Retrieved CS1 maint: The test rifle needed Therefore large parts of the book are accessible to amateurs, and many of the results are simple enough for a scientific hand calculator.
In particular there are some beautiful photographs of bullets in flight. The corrections are in a PDF file. If you click on the link below, many browsers will just open the file, as they recognize this standard format. Be patient, it may take a minute or two for the first page to display.
The corrections are copyrighted by the late Donald G. About Us. Ballistic Explorer. Help Page. How To Examples. Free Software.