External ballistics

The external ballistics studies the rail drive-free, unguided ballistic missiles . The statements of the external ballistics apply

for projectiles from barrel weapons ,
for ballistic missile projectiles (short- to medium-range missiles ) after the burnout
and for the trajectory of bombs.

The trajectory of propelled or guided projectiles, rockets and bombs is dealt with in rocket ballistics.

The four main tasks of external ballistics are:

Determination of the projectile trajectory according to the classic trajectory model under normal meteorological and ballistic conditions
Investigation of the rigid body movements of the projectile and their influences
Consideration of the deviations from the normal conditions, i.e. the disturbance calculation
Determination of shooting boards and bombing boards

Basics

The simplest model of external ballistics describes the movement of a mass point in the earth's gravitational field , neglecting the earth's curvature and air resistance on the trajectory parabola . This model is called a parabolic orbit model. His statements are often used as starting values for approximation methods for more difficult models. In addition, this model describes the trajectories of slow projectiles and bombs quite well.

An improvement of the model takes into account the air resistance and the dependence of the gravitational acceleration on the geodetic height, but not the rigid body movements of the projectile. The model obtained in this way is called the Classic Railway Model. It is sufficient for many purposes, such as fire control in general.

The two models are among the models with three degrees of freedom.

the flight time (t ₁ and t ₂ ) for a horizontal shot is equal to the fall time t ₃ from the height of the muzzle s

Fired projectiles are subject to the acceleration of gravity on their path , whereby their movement is composed of the forward movement and the falling movement. The effect of the two components can be clearly demonstrated with a horizontal shot without taking air resistance into account. The movements along the horizontal and the vertical can be observed separately according to the superposition principle .

Shot with terrain angle

Changing the firing range when shooting at a terrain angle

If the target is at a different height than the launch point, the launch angle must be adjusted for the same horizontal distance. For higher-lying targets, the angle of attack or the initial speed must be increased; for lower-lying targets, it must be reduced.
In the illustration, the firing angle is based on the horizontal, the angle to the target. The firing range in a vacuum results from: ${\ displaystyle \ gamma}$ ${\ displaystyle \ theta}$ ${\ displaystyle s_ {1}}$
${\ displaystyle s_ {1} = {\ frac {2 \ cdot {v_ {0}} ^ {2}} {g}} \ cdot {\ frac {\ cos \ theta \ cdot \ sin \ left (\ theta - \ gamma \ right)} {\ cos \ left (\ gamma \ right) ^ {2}}}}$

scattering

Main article: Dispersion (ballistics)

Even if the starting conditions seem to be the same, it is very unlikely that a weapon will hit exactly the same point with successive shots. Due to small, unavoidable variations in the initial parameters of the weapon and ammunition as well as the air , the projectiles move on a slightly different path with each shot. The measure of this deviation is the spread .

Bullet trajectories in the atmosphere

Model atmosphere and standard atmosphere

Knowledge of the physics of the atmosphere is necessary to calculate the orbital elements of a floor. The basis of all aerodynamic calculations in ballistics is a model atmosphere for which the height dependence of its properties is formulated mathematically. The mathematical description of the model atmosphere is based on data from scientific research into the earth's atmosphere, as has long been carried out with aircraft and spacecraft. The aim is to use mathematical formulas to represent the relationships and dependencies of the atmospheric parameters for each altitude. For ballistics, parameters are of interest that help determine the air resistance of a projectile. These are air density and air pressure as well as the speed of sound and the influence of air humidity on these quantities. In the case of high tracks, the decrease in the acceleration due to gravity with increasing height must also be taken into account.

Based on the regularities of the model atmosphere, a standard atmosphere is defined for which these regularities are applied to specific numerical values. For the standard atmosphere, the numerical values of the atmospheric parameters close to the ground are usually determined and their height dependence is determined according to the specifications of the model atmosphere. The parameters include:

Air temperature on the ground
Air pressure on the ground
Air density on the ground (which results from them and the air composition)
Speed of sound on the ground
Acceleration of gravity on the ground
Adiabatic exponent on the ground
Molar mass of air on the ground
dynamic viscosity of the air on the ground
mean free path of the air molecules on the ground
Earth radius

To calculate the air density, the models for the normal atmosphere introduced by the ICAO in 1952 are used, which allow both the current air pressure and the height above mean sea level to be taken into account in the calculations.

Different standard atmospheres are used internationally for ballistic calculations, whereby artillery normal atmospheres adapted to certain fields of application and climatic conditions are used, some of which neglect some of the parameters for practical reasons.

Maximum firing range

For fast, long-range projectiles, there are considerable deviations from the trajectory parabola, primarily due to the air resistance. Without air resistance, a projectile fired at an angle of 45 ° with a muzzle velocity of 700 m / s would travel about 50 km. Light projectiles, such as those made from rifles , however, only achieve a maximum flight range of 3 to 4 km due to the effect of air resistance. Due to the strong braking rifle bullets do not reach their maximum firing range when theoretically optimal firing angle of 45 °, but about 30 to 35 °, and even lighter shot from guns reach their maximum flying distance at launch angles of 20 ° to 30 °.

The projectiles of long-range guns traverse high atmospheric layers on their path , in which the air resistance is significantly lower than near the ground due to the lower air density. In this case, the gun's attachment angle must be more than 45 ° in order to ensure that the projectiles have an angle of 45 ° when reaching the altitude areas of lower air resistance, which, according to the parabolic model, is optimal for achieving a maximum firing range. For range-optimized projectiles with a muzzle velocity of 1000 to 1500 m / s, the launch angle is 50 to 55 °. There is a complicated, non-linear relationship between the initial speed and the optimal launch angle, so if the initial speed were increased further, the angle would decrease again for a maximum range , since the dense air layers are flown through faster.

Aerodynamic influence on projectiles

A fired projectile is surrounded by air in the atmosphere, which leads to the action of various forces that influence the trajectory of the projectile. First and foremost, it is the air resistance that slows down the projectile and causes the path to deviate from the trajectory parabola.

Projectiles such as rockets , arrow projectiles or shotgun barrel projectiles are aerodynamically stabilized. With these bullets, the aerodynamic pressure point lies behind the center of gravity. If these projectiles tilt in the longitudinal axis, they are realigned by the pressure point behind the center of gravity. These bullets have tail units or a slight tail, which shifts the pressure point to the rear.

Most bullets with twist stabilization have the pressure point in front of the center of gravity. This aerodynamic instability creates a tilting moment to the rear so that they would overturn without twisting after leaving the barrel. Due to the twist, they are stabilized by gyroscopic forces similar to a top standing on its tip .

These bullets do not leave the barrel precisely aligned in the direction of flight, but with a small angle of attack . The result of the angle of attack are forces that act at right angles to the drag force. On the one hand, this is a lift force that acts in the plane of the angle of attack; on the other hand, the twist creates a lateral force through the Magnus effect that acts perpendicular to this plane. The lift force always acts in the direction of the angle of attack, so it can also be directed downwards if the angle of attack is aligned accordingly. The transverse forces cause a lateral deflection of the projectile in the direction of twist, i.e. to the right with a right-hand twist. With artillery projectiles, the lateral forces can lead to a lateral acceleration of 0.15 to 0.25 m / s², whereby this value decreases with increasing flight time.

Disproportionate increase in mass m and cross-sectional load Q when increasing the caliber

Although heavy projectiles fired from guns are subject to the same aerodynamic influences and usually have muzzle velocities comparable to handguns, they achieve significantly longer maximum firing ranges of around 20 to 40 km. The reason for this lies in the cross-sectional loading of the floors. If the caliber of a bullet increases, the cross-sectional area increases by the square of the diameter. At the same time, however, the volume of the bullet increases proportionally to the cube of the diameter, and thus its mass also increases proportionally to the cube. With the same bullet shape and density, a tenfold increase in caliber causes a thousandfold increase in mass and a hundredfold increase in the cross-sectional load. Because the weight acting on the cross-sectional area increases more than the air resistance, which increases due to the increase in caliber, the bullet is braked more slowly.

In addition to the cross-sectional load, the drag coefficient ( value) is decisive for the range of a projectile. Projectiles of modern weapons reach speeds from the high subsonic to the high supersonic range. Especially in the range of the speed of sound (around Mach 1) the value rises steeply due to the creation of a bow wave from compressed air (wave resistance). This area of compressed air creates a sonic boom . At higher speeds, the value decreases again, with the characteristic of the wave resistance in the total resistance being up to over 70%. For the calculation of the air resistance of a supersonic projectile, the values of the projectile tip, the projectile body, the tail cone and the tail suction are considered and added separately. ${\ displaystyle c_ {W}}$ ${\ displaystyle c_ {W}}$ ${\ displaystyle c_ {W}}$ ${\ displaystyle c_ {W}}$

The dependence of the value on the speed ( as a function of ) must be taken into account when calculating the path elements of a storey. As a rule, this function is not determined anew for each bullet type, but a standard air resistance law is used, which is then corrected for the respective bullet type using a shape coefficient . ${\ displaystyle c_ {W}}$ ${\ displaystyle c_ {W}}$ ${\ displaystyle v}$

Projectiles designed for the supersonic range such as modern long projectiles or artillery shells often have ogival shaped tips. The ogival tip is the most aerodynamic for supersonic missiles. An ogival tail cannot be implemented structurally in tubular weapons, so that the tail is usually only shaped as a truncated cone (tail cone) in order to reduce the deceleration caused by the tail suction.

To calculate the range , a ballistic coefficient is also determined from the drag coefficient and weight of a projectile . In ballistics there are several definitions of the ballistic coefficient, which must be taken into account in calculations based on this value.

Influence on trajectories

A bullet trajectory in the atmosphere also has a vertical and a horizontal component at launch angles between 0 and 90 °.

horizontal component u and vertical component w of the bullet movement (distance s and height h)

The speed components and are calculated from the firing angle ϑ and the speed v: While the horizontal component of the speed remains constant when shooting in a vacuum ( uniform movement ), it decreases monotonically when shooting under the influence of air resistance , even if the vertical speed component is below the speed minimum increases again. The falling section of the trajectory is therefore generally steeper than the rising section. The horizontal speed component follows the speed of a horizontally flying projectile, the speed of which is constantly decreasing due to air resistance. Analogously, the course of the vertical speed component follows that of a projectile that was shot vertically upwards with the proportional speed value of the vertical component. ${\ displaystyle u}$ ${\ displaystyle w}$
${\ displaystyle u = v \ cdot \ cos \ vartheta}$
${\ displaystyle w = v \ cdot \ sin \ vartheta}$

After crossing the railway summit, projectiles can reach a limit speed, whereby a balance is established between the weight force and the air resistance force.

In the case of trajectories in a vacuum, the minimum projectile velocity occurs in the orbit summit; in the case of trajectories in air, only after the orbit summit, since the deceleration due to air resistance initially outweighs the increase in speed due to the gravitational acceleration. In the case of flat tracks, the bullet can therefore reach the target before this minimum occurs, so that the bullet speed drops permanently to the target without increasing again after the peak of the track.

For more precise calculations, further values are used in ballistics and various disruptive factors are examined more closely.

These disruptive factors are primarily changes in the atmospheric soil values, deviations in the altitude-dependent courses (gradients) of the atmospheric parameters from the normal atmosphere, wind influences and also weather influences such as rain or snowfall. With long-range orbits, the influence of the earth's curvature and rotation must be included in the calculation.

There are no exact mathematical models for calculating the path deviations due to the atmospheric disturbance factors, so that only approximate solutions are possible. Before powerful computer-aided fire control computers were available, the aiming data of the guns were determined and corrected using fire boards . Based on the fixed parameters of initial speed , firing angle and a projectile parameter such as the ballistic coefficient, the guide data are taken from the firing boards depending on the deviations that occur.

The greater the firing range , the greater the influence of the altitude dependence of the atmospheric parameters. Air pressure, air density and air temperature and thus also the speed of sound change with increasing altitude. All these properties are decisive for the air resistance of a projectile, and the knowledge of their height-dependent course is essential for the prediction of the flight path elements. In addition, the air humidity also has an influence on the pressure curve and the speed of sound.

Above about 90 km, the properties of the atmosphere change due to the increasing mean free path in such a way that the atmosphere no longer acts as an ideal gas and fundamental laws of fluid mechanics can no longer be applied. Considerations of projectile trajectories at such heights are primarily the subject of rocket ballistics.

Direct fire and painted area

The area of direct fire, for example from flat fire guns, is within the distance at which the apex of the trajectory lies below the highest point of the target. In this area, external disturbances have the least influence, so that in practice the trajectory of the projectile coincides with the trajectory parabola in a vacuum with sufficient approximation.

If the vertex is higher than the target, the sections of the trajectory that are level with the target are the swept area.

The size of the area of direct fire depends on the speed of the trajectory .

Bombing

Horizontal discharge

When free-falling, unguided bombs are dropped from airplanes, the time at which the bomb is released is crucial for accuracy when it comes to hitting the target. This point in time depends on the altitude and the flight speed of the aircraft. The faster and higher the plane flies, the greater the distance from the target the bomb must be released. When dropping from level flight, the duration of the fall follows from the law of fall, neglecting the air resistance, is the acceleration due to gravity, which is assumed to be constant: ${\ displaystyle h}$ ${\ displaystyle v_ {0}}$ ${\ displaystyle t_ {F}}$ ${\ displaystyle g}$

${\ displaystyle t_ {F} = {\ sqrt {\ frac {2 \ cdot h} {g}}}}$

In this case, the distance that the bomb travels above the ground before it hits the ground results from: ${\ displaystyle s_ {D}}$

${\ displaystyle s_ {D} = v_ {0} \ cdot {\ sqrt {\ frac {2 \ cdot h} {g}}}}$

The bomb must be dropped at this distance from the target.

The calculation of the orbital elements of bombs can be simplified if the ratio of the potential energy and the kinetic energy of the bomb at the moment of its release is determined as an additional parameter . The bomb with the mass is in the aircraft at a height above the ground. The potential energy results from: ${\ displaystyle par}$ ${\ displaystyle W_ {p}}$ ${\ displaystyle W_ {k}}$ ${\ displaystyle m}$ ${\ displaystyle h}$

${\ displaystyle W_ {p} = m \ cdot g \ cdot h}$

Due to the speed of the aircraft, the bomb receives kinetic energy: ${\ displaystyle v}$

${\ displaystyle W_ {k} = {\ frac {m \ cdot v ^ {2}} {2}}}$

The parameter is the quotient of both values: ${\ displaystyle par}$

${\ displaystyle par = {\ frac {W_ {p}} {W_ {k}}}}$

The tangent of the angle of impact of the bomb when dropped from horizontal flight can easily be determined with the help of this parameter: ${\ displaystyle \ vartheta _ {Z}}$

${\ displaystyle \ tan \ vartheta _ {Z} = {\ sqrt {par}}}$

Bomb release at lead angle φ

In the case of a purely optical target acquisition, the bombardier does not determine the distance to the target at which he must release the bomb, but measures the angle between the aircraft and the target until this angle corresponds to the lead angle appropriate for the altitude and speed . The tangent of the lead angle results from: ${\ displaystyle \ phi}$

${\ displaystyle \ tan {\ varphi} = {\ frac {s_ {D}} {h}} = {\ frac {v_ {0}} {h}} \ cdot {\ sqrt {\ frac {2h} {g }}} = {\ frac {2} {\ sqrt {par}}}}$

Release from the climb

With a drop from the climb with the angle , more extensive formulas result. The time from the dropping of the bomb to reaching the railway summit results from: ${\ displaystyle \ vartheta _ {0}}$
${\ displaystyle t_ {G}}$
${\ displaystyle t_ {G} = {\ frac {v_ {0} \ cdot \ sin {\ vartheta _ {0}}} {g}}}$

The flight time results from: ${\ displaystyle t_ {F}}$
${\ displaystyle t_ {F} = t_ {g} \ cdot \ left (1 + {\ sqrt {1 + {\ frac {par} {\ sin ^ {2} \ vartheta _ {0}}}}} \ right )}$

The way to the summit of the railway results from: ${\ displaystyle s_ {G}}$
${\ displaystyle s_ {G} = {\ frac {{v_ {0}} ^ {2}} {2g}} \ cdot \ sin 2 \ vartheta _ {0}}$

The horizontal throwing distance results from: ${\ displaystyle s_ {D}}$
${\ displaystyle s_ {D} = s_ {G} \ left (1 + {\ sqrt {1 + {\ frac {par} {\ sin ^ {2} \ vartheta _ {0}}}}} \ right)}$

literature

Hellmuth Molitz, Reinhold Strobel: External ballistics. Springer 1984, ISBN 3-540-02943-5 .
Felix Poklukar: Models for external ballistics - target ballistics, HTLB Ferlach, 2005 (PDF, 303 kB) ( Memento from March 20, 2018 in the Internet Archive )
Bernd Brinkmann, Burkhard Madea: Handbook of forensic medicine, Volume 1. (Chapter 9.6 External ballistics) ISBN 978-3-540-00259-8 .
Carl Cranz, Ballistics , Chapter 18 (1903) in the Encyclopedia of Mathematical Sciences including its applications at the University of Göttingen, available online

Web links

Commons : Artillery Ballistics - Collection of images, videos and audio files

Commons : External ballistics of various cartridges - collection of images, videos and audio files

Individual evidence

↑ ^a ^b Felix Poklukar: Models for external ballistics - target ballistics, HTLB Ferlach, 2005 (PDF, 303 kB) ( Memento from March 20, 2018 in the Internet Archive ), viewed on December 11, 2009
^ ^A ^b Günter Hauck: Outer ballistics. 1st edition. Military publishing house of the GDR, 1972.
^ Willi Barthold: Jagdwaffenkunde. VEB Verlag Technik, Berlin 1969, edited edition 1979, p. 182.
^ G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 513.
^ G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 133.
^ G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 140.
^ G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 195 ff.
↑ Bernd Brinkmann, Burkhard Madea: Handbook of judicial medicine, volume 1. Chapter 9.6 External ballistics, page 611, (accessed on November 22, 2009)

[Nr.1-1] Felix Poklukar: Models for external ballistics - target ballistics, HTLB Ferlach, 2005 (PDF, 303 kB) ( Memento from March 20, 2018 in the Internet Archive ), viewed on December 11, 2009

[Hauck,_Äußere_Ballistik-2] A ^b Günter Hauck: Outer ballistics. 1st edition. Military publishing house of the GDR, 1972.

[3] Willi Barthold: Jagdwaffenkunde. VEB Verlag Technik, Berlin 1969, edited edition 1979, p. 182.

[4] G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 513.

[5] G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 133.

[6] G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 140.

[7] G. Hauck: Outer ballistics. 1st edition. Military Publishing House of the GDR, 1972, p. 195 ff.

[8] Bernd Brinkmann, Burkhard Madea: Handbook of judicial medicine, volume 1. Chapter 9.6 External ballistics, page 611, (accessed on November 22, 2009)