Ballistic Formulas

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Ballistic Formulas(from http://www.nennstiel-ruprecht.de/bullfly/index.htm )

Abbreviations

A Bullet cross section area; A = pd /4

a Velocity of sound in air, a = a(p,T,h)

B Symbolic variable, indicating bullet geometry

d Bullet diameter

ec Unit vector into the direction of the bullet' s longitudinal axis

et Unit vector into the direction of the tangent to the trajectory

g Acceleration of gravity; g = g( j ,y)

h Relative humidity of air

I x Axial (or polar) moment of inertia of the bullet

I y Transverse (or equatorial) moment of inertia of the bullet

l Bullet length

m Bullet mass

Ma Mach number

p Air pressure

Re Reynolds number

r E Mean radius of the earth; r E = 6 356 766 m

T Absolute air temperature

v w Bullet velocity with respect to wind system

y Altitude of bullet above sea level

a Azimuth angle

d Yaw angle

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Q Angle of inclination of the trajectory

r Air density r = r (p,T,h)

m Absolute viscosity of air; m = m(T)

j Degree of latitude

w Spin rate of bullet (angular velocity)

w E Angular velocity of the earth´s rotation; w E = 7.29.10- rad/s

Azimuth and degree of latitude

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The azimuth a is defined as the angle enclosed between the positive x-axis of a xyz reference frameand the north direction. a is always positive and may take values between 0° and 360°. The xz-planeis parallel to the surface of the earth at the selected location.

j is the degree of latitude and depends on the location on the globe (-90°


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Abbreviations

F Z Centrifugal force

Explanation

The figure above shows a cut through the globe. The formula gives the components of the centrifugalforce in an xyz - reference frame, the y -axis being antiparallel to the force of gravity.

The y - component of the centrifugal force can be regarded as a correction of the force of gravity, theother components are generally neglected in ballistics because of their smallness.


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The Coriolis force

Abbreviations

F c Coriolis force

v Velocity vector with respect to xyz - coordinate system

Vector of the angular velocity of the earth´s rotationwith respect to xyz - coordinate system.

Explanation

The magnitude of the fictitious Coriolis force is so small that it is usually completely neglected and -as a rule of thumb - only has to be considered in ballistics for ranges of 20 km or more (artillery shells

The drag force

Abbreviations

c D Drag coefficient

F D Drag force

Explanation

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The drag force FD is the component of the force FW in the direction opposite to that of the motion of

the centre of gravity (see figure ). The force FW results from pressure differences at the bullet's

surface, caused by the air, streaming against the moving body. In the case of the absence of yaw, the

drag FD is the only component of the force FW .

The drag force is the most important aerodynamic force. Given the atmosphere conditions p,T,h, the

reference area A and the momentary velocity v w , the drag force is completely determined by the the

drag coefficient c D .

The drag coefficient

The drag coefficient cD is the most important aerodynamic coefficient and generallydepends on

- bullet geometry (symbolic variable B ),- Mach number Ma, - Reynolds number Re, - the angle of yaw

The following assumptions and simplifications are usually made in ballistics:

1. Re neglection

It can be shown, that with the exception of very low velocities, theRe

dependency ofc

D can beneglected.

2. dependency

Depending on the physical ballistic model applied, an angle of yaw is either completely neglected( =0) or only small angles of yaw are considered. Large angles of yaw are an indication of instability.For small angles of yaw the following approximation is usually made:

a) c D(B,Ma, ) = c Do(B,Ma) + c D (B,Ma) *2/2

Another theory which accounts for arbitrary angles of yaw is called the "crossflow analogy predictionmethod". A discussion of this method is far beyond the scope of this article, however the general typeof equation for the drag coefficient is as follows:

b) c D(B,Ma, ) = c Do(B,Ma) + F (B,Ma,Re, )

3. Determination of the zero-yaw drag coefficient

The zero-yaw drag coefficient as a function of the Mach number Ma is generally determinedexperimentally either by wind tunnel tests or from Doppler Radar measurements.

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Fig.: Zero-yaw drag coefficient for two military bulletsM80 (cal. 7.62 x 51 Nato)

SS109 (cal. 5.56 x 45)

There is also software available which estimates the zero-yaw drag coefficient as a function of theMach number from bullet geometry. The latter method is mainly applied in the development phase ofa new projectile.

4. Standard drag functions

Generally each bullet geometry has its own zero-yaw drag coefficient as a function of the Machnumber. This means, that specific - time-consuming and expensive - measurements would berequired for each bullet geometry. A widely used simplification makes use of a "standard dragfunction" c Do

standard which depends on the Mach number alone and a form factor i D which depends onthe bullet geometry alone according to:

c Do(B,Ma) = i D(B) * c Do standard (Ma)

If this simplification is applicable, the determination of the drag coefficient of a bullet as a function ofthe Mach number is reduced to the determination of a suitable form factor alone. It will be shown thatthe concept of the ballistic coefficient, widely used in the US for small arms projectiles follows thisidea.

Abbreviations

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c D Drag coefficient; c D(B,Ma,Re, )

c Dos an ar Zero-yaw standard drag function

i D Form factor

The ballistic coefficient (bc )

The 'ballistic coefficient' or bc is a measure for the drag experienced by a bullet moving through theatmosphere, which is widely used by manufacturers of reloading components, mainly in the US.

Although, from a modern point of view, bc s are a remainder of the pioneer times of exterior ballistics,ballistic coefficients have been determined experimentally for so many handgun bullets, that notreatise on exterior ballistics would be allowed to neglect it..

The bc of a test bullet bc test

moving at velocity v is a real number and defined as

the deceleration due to drag of a "standard" bulletdevided by

the deceleration due to drag of the test bullet.

The standard bullet is said to have a mass of 1 lb (0.4536 kg) and a diameter of 1 in (25.4 mm). Thedrag coefficients of the standard bullet can be derived from the G1-function given in literature and willbe named c Do

G1(Ma) .Using

c Dotest

(B,Ma) = i Dtest

(B) * c DoG1

(Ma)

one finds for the bc (assuming "standard" atmosphere conditions)

bc test =1 / i Dtest (B) * mtest / d

2 test

This formula also shows that the bc and the form factor i D of a "test" bullet are two aspects of thesame principal simplification: the substitution of the (unknown) particular drag function of a bullet bythe (given) "standard" drag function of the standard bullet (see also here).

Abbreviations

c Does Zero-yaw drag coefficient of test bullet

c Do1 Zero-yaw G1 standard drag coefficient

i Des Form factor of test bullet

bc es Ballistic coefficient of test bullet

mtest Mass of test bullet in lb

d test Diameter of test bullet in inches

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The ballistic coefficient (bc )

The 'ballistic coefficient' or bc is a measure for the drag experienced by a bullet moving through theatmosphere, which is widely used by manufacturers of reloading components, mainly in the US.

Although, from a modern point of view, bc s are a remainder of the pioneer times of exterior ballistics,ballistic coefficients have been determined experimentally for so many handgun bullets, that notreatise on exterior ballistics would be allowed to neglect it..

The bc of a test bullet bc test moving at velocity v is a real number and defined as

the deceleration due to drag of a "standard" bulletdevided by

the deceleration due to drag of the test bullet.

The standard bullet is said to have a mass of 1 lb (0.4536 kg) and a diameter of 1 in (25.4 mm). Thedrag coefficients of the standard bullet can be derived from the G1-function given in literature and willbe named c Do

G1(Ma) .Using

c Dotest (B,Ma) = i D

test (B) * c DoG1(Ma)

one finds for the bc (assuming "standard" atmosphere conditions)

bc test =1 / i Dtest (B) * mtest / d

2 test

This formula also shows that the bc and the form factor i D of a "test" bullet are two aspects of thesame principal simplification: the substitution of the (unknown) particular drag function of a bullet bythe (given) "standard" drag function of the standard bullet (see also here).

Abbreviations

c Does

Zero-yaw drag coefficient of test bullet

c Do1 Zero-yaw G1 standard drag coefficient

i Des

Form factor of test bullet

bc es Ballistic coefficient of test bullet

mtest Mass of test bullet in lb

d test Diameter of test bullet in inches

The lift force

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Abbreviations

c L Lift coefficient; c L(B,Ma.Re, )

e L Unit vector

F L Lift force

Explanation

The lift force FL (also called cross-wind force) is the component of the wind force FW in the direction

perpendicular to that of the motion of the center of gravity in the plane of the yaw angle . The liftforce vanishes in the absence of yaw and is the reason for the drift of a spinning projectile even in theabsence of wind.

The overturning moment

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The forces F1 and F2 (see previous figure ) form a free couple, which is said to be the

aerodynamic moment of the wind force or simply overturning moment Mw (see ). Thismoment tries to rotate the bullet about an axis through the CG, perpendicular to the axis of symmetry

of the bullet. The overturning moment tends to increase the angle of yaw .

The force FW, which applies at the CG can be split into a force, opposite to the direction of the

movement of the CG (the direction of the velocity vector v), which is called the drag force FD

or simply drag and a force, perpendicular to this direction, which is called the lift force FL orsimply lift.


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Abbreviations

c M Overturning moment coefficient, c M(B, Ma, Re, )

eW Unit vector

MW Overturning moment

Explanation

The point of the longitudinal axis, at which the resulting wind force F1 appears to attack is called thecentre of pressure CPW of the wind force, which, for spin-stabilized bullets is located ahead of theCG. As the flow field varies, the location of the CPW varies as a function of the Mach number. Due tothe non-coincidence of the CG and the CPW, a moment is associated with the wind force. This

moment MW is called overturning moment or yawing moment (see figure ). For spin-stabilized projectiles MW tends to increase the yaw angle and destabilizes the bullet. In the absenceof spin, the moment would cause the bullet to tumble.

The spin damping moment

Abbreviations

c spin Spin damping moment coefficient; c spin(B,Ma.Re)

M S Spin damping moment

Explanation

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Skin friction at the bullet's surface retards its spinning motion. The spin damping moment (also: roll

damping moment) is given by the above formula. The spin damping coefficient depends on bullet

geometry and the flow type (laminar or turbulent).

The Magnus force

Abbreviations

c Mag Magnus force coefficient; c Mag (B,Ma,Re, , )

e M Unit vector

F M Magnus force

Explanation

The Magnus force FM arises from an asymmetry in the flow field, while the air stream against a

rotating and yawing body interacts with its boundary layer and applies at the CPM (see figure ).Depending on the flow field, the CPM may be located ahead or behind the CG. The Magnus forcevanishes in the absence of rotation and in the absence of a yaw angle.

The Magnus force is usually very small and mainly depends on bullet geometry, spin rate, velocityand the angle of yaw. In exterior ballistics, the above expression is used for the Magnus force.

The Magnus force



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For the whole bullet, the Magnus effect (which arises from the boundary layer interaction of the

inclined and rotating body with the flowfield) results in the Magnus force FM

which applies atits centre of pressure CPM. The location of the CPM varies as a function of the flowfield conditionsand can be located either behind or ahead of the CG.

The figure above assumes that the CPM is located behind the CG. Experiments have shown that thiscomes true for a 7.62 x 51 FMJ standard Nato bullet at least close to the muzzle in the highsupersonic velocity regime.


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Abbreviations

c M Overturning moment coefficient, c M(B, Ma, Re, )

eW Unit vector

MW Overturning moment

Explanation

The point of the longitudinal axis, at which the resulting wind force F1 appears to attack is called thecentre of pressure CPW of the wind force, which, for spin-stabilized bullets is located ahead of theCG. As the flow field varies, the location of the CPW varies as a function of the Mach number. Due tothe non-coincidence of the CG and the CPW, a moment is associated with the wind force. This

moment MW is called overturning moment or yawing moment (see figure ). For spin-stabilized projectiles MW tends to increase the yaw angle and destabilizes the bullet. In the absenceof spin, the moment would cause the bullet to tumble.




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The forces F1 and F2 (see previous figure ) form a free couple, which is said to be the

aerodynamic moment of the wind force or simply overturning moment Mw (see ). Thismoment tries to rotate the bullet about an axis through the CG, perpendicular to the axis of symmetry

of the bullet. The overturning moment tends to increase the angle of yaw .

The force FW, which applies at the CG can be split into a force, opposite to the direction of the

movement of the CG (the direction of the velocity vector v), which is called the drag force FD

or simply drag and a force, perpendicular to this direction, which is called the lift force FL or

simply lift.

The Magnus moment

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Abbreviations

c Mp Magnus moment coefficient; c Mp(B,Ma,Re,w,d)

e MM Unit vector

M M Magnus moment

Explanation

As the Magnus force applies at the CPM, which does not necessarily coincide with the CG, a Magnus

moment MM (see figure ) is associated with that force. The location of the centre of pressure ofthe Magnus force depends on the flow field and can be located ahead or behind the CG. The Magnusmoment turns out to be very important for the dynamic stability of spin-stabilized bullets. For the

Magnus moment, the above expression is used in exterior ballistics.

The gyroscopic stability condition

Abbreviations

c Ma Overturning moment coefficient derivative; c Ma(B,Ma)

sg Gyroscopic stability factor



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Explanation

A spin-stabilized projectile is said to be gyroscopically stable, if, in the presence of a yaw angle d, itresponds to an external wind force F1 with the general motion of nutation and precession. In this casethe longitudinal axis of the bullet moves into a direction perpendicular to the direction of the windforce.

It can be shown by a mathematical treatment that this condition is fulfilled, if the gyroscopic stabilityfactor sg exceeds unity. This demand is called the gyroscopic stability condition. A bullet can bemade gyroscopically stable by sufficiently spinning it (by increasing w!).

As the spin rate w decreases more slowly than the velocity v w , the gyroscopic stability factor sg , at

least close to the muzzle, continuously increases. An practical example is shown in a figure .Thus, if a bullet is gyroscopically stable at the muzzle, it will be gyroscopically stable for the rest of itsflight. The quantity sg also depends on the air density r and this is the reason, why special attentionhas to be paid to guarantee gyroscopic stability at extreme cold weather conditions.

Bullet and gun designers usually prefer sg > 1.2...1.5, but it is also possible to introduce too muchstabilization. This is called over-stabilization.

The gyroscopic (also called static) stability factor depends on only one aerodynamic coefficient (theoverturning moment coefficient derivative c Ma) and thus is much easier to determine than the dynamicstability factor. This may be the reason, why some ballistic publications only consider static stability ifit comes to stability considerations.

However, the gyroscopic stability condition only is a necessary condition to guarantee a stableflight, but is by no means sufficient. Two other conditions - the conditions of dynamic stability and thetractability condition must be fulfilled.

Gyroscopic (static) stability factor



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This figure shows the gyroscopic stability factor of the 7.62 x 51 Nato bullet M80, fired at an angleof departure of 32°, a muzzle velocity of 870 m/s and a rifling pitch at the muzzle of 12 inches. TheM80 bullet shows static stability over the whole flight path as the static stability condition sg >1 isfulfilled everywhere. The value of sg adopts a minimum of 1.35 at the muzzle.

Generally it can be stated that if a bullet is statically stable at the muzzle, it will be statically stable forthe rest of its flight. This can be easily understood from the fact, that the static stability factor is

proportional to the ratio of the bullet´s rotational and transversal velocity (see formula ). As thethe rotational velocity is much less damped than the transversal velocity (which is damped due to theaction of the drag), the static stability factor increases, at least for the major part of the trajectory.

Bullet and gun designers usually prefer sg > 1.2 ..1.5 at the muzzle, however it has been observedthat many handgun bullet show excessive static stability.

The dynamic stability condition

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Abbreviations

c D Drag coefficient

c La Lift coefficient derivative

c Mpa Magnus moment coefficient derivative

c mq+c ma Pitch damping moment derivative

sg Gyroscopic (static) stability factorsd Dynamic stability factor

Explanation

A projectile is said to be dynamically stable, if its yawing motion of nutation and precession isdamped out with time, which means that an angle of yaw induced at the muzzle (the initial yaw)decreases.

A dynamic stability factor sd can be defined from the linearized theory of gyroscopes (assuming only a

small angle of yaw) and the above dynamic stability condition can be formulated. An alternate

formulation of this condition leads to the illustrative stability triangle.

sd however depends on five aerodynamic coefficients. Because these coefficients are hard todetermine, it can become very complicated to calculate the dynamic stability factor, which varies as afunction of the momentary bullet velocity.

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The stability triangle

Abbreviations


sd Dynamic stability factor

Explanations

The dynamic stability condition can be expressed in an alternate way. leading to a very

illustrative interpretation of bullet stability.In using a quantity s, according to the above definition, the dynamic stability condition takes a verysimple form (see above formula). This means that for a bullet to be gyroscopically and dynamically

stable, a plot of s vs. sd has to remain completely within the stability triangle (green area in the figurebelow).

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The red areas are regions of gyroscopic stability but dynamic instability: either the slow modeoscillation (left area) or the fast mode oscillation (right area) get umdamped.

The stability triangle

Abbreviations


sd Dynamic stability factor

Explanations

The dynamic stability condition can be expressed in an alternate way. leading to a veryillustrative interpretation of bullet stability.

In using a quantity s, according to the above definition, the dynamic stability condition takes a verysimple form (see above formula). This means that for a bullet to be gyroscopically and dynamically

stable, a plot of s vs. sd has to remain completely within the stability triangle (green area in the figurebelow).

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The red areas are regions of gyroscopic stability but dynamic instability: either the slow mode

oscillation (left area) or the fast mode oscillation (right area) get umdamped.

The tractability condition

Abbreviations

f Tractability factor

f l Low limit tractability factor; fl » 5.7


d p Yaw of repose vector

Explanation

The tractability factor f characterizes the ability of the projectile's longitudinal axis to follow the

bending trajectory (see figure ). The quantity f can simply be defined as the inverse of the yawof repose. It can be shown that the tractability factor f is proportional to the inverse of the gyroscopicstability factor.

Over -stabilized bullet



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This figure schematically shows an over-stabilized bullet on a high-angle trajectory.

An over-stabilized bullet rotates too fast and its axis tends to keep its orientation in space. Thebullet´s longitudional axis becomes uncapable to follow the bending path of the trajectory. Over-stabilization is said to occur, if the angle enclosed between the bullet´s axis of form and the tangent tothe trajectory (the yaw of repose) exceeds a value of approximately 10°.

Over-stabilization of a bullet is most probable, if a bullet has excessive static stability (a high value

of sg and a low value for the tractability factor ) and is fired at a high angle of departure,especially when fired vertically. An over-stabilized bullet on a high-angle trajectory lands base first.

However, when firing bullets from handguns, over-stabilization is of minor importance in normalshooting situation, but must be considered when firing at high angles of elevation.

The yaw of repose

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Abbreviations

c M a Overturning moment coefficient derivative coefficient

d p Yaw of repose vector

Explanation

The repose angle of yaw (or yaw of repose, also called equilibrium yaw) is the angle, by which the

momentary axis of precession deviates from the direction of flight (see figure ). As soon as thetransient yaw induced at the muzzle has been damped out for a stable bullet, the yaw angle d equalsthe yaw of repose.

The magnitude of the yaw of repose angle is typically only fractions of a degree close to the muzzle,but may take considerable values close to the summit especially for high-elevation angles.

The occurrence of the yaw of repose is responsible for the side drift of spin-stabilized projectiles evenin the absence of wind. The spin-dependent side drift is also called derivation.

It can be shown that for right-hand twist, the yaw of repose lies to the right of the trajectory. Thus the

bullet nose rosettes with an average off-set to the right, leading to a side drift to the right.

The above formula for the yaw of repose vector is an approximation for stable bullet flight.

The yaw of repose



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If a bullet flies stable (gyroscopically and dynamically!) and the transient yaw has been damped out,usually after a travelling distance of a few thousands of calibres, the bullet´s axis of symmetry and the

tangent to the trajectory deviate by a small angle, which is said to be the yaw of repose .

For bullets fired with right-handed twist, the longitudinal axis points to the right and a little bit upwardwith respect to the direction of flight, leading to a side drift to the right. The yaw of repose, althoughnormally measuring only fractions of a degree, is the reason for the side deviation of spin-stabilizedbullets.
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Ballistic Formulas