Ballistic Formulas (from http://www.nennstiel-ruprecht.de/bullfly/index.htm )
Abbreviations A
Bullet cross section area; A = pd2/4
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
Ix
Axial (or polar) moment of inertia of the bullet
Iy
Transverse (or equatorial) moment of inertia of the bullet
l
Bullet length
m
Bullet mass
Ma
Mach number
p
Air pressure
Re
Reynolds number
rE
Mean radius of the earth; rE = 6 356 766 m
T
Absolute air temperature
vw
Bullet velocity with respect to wind system
y
Altitude of bullet above sea level
a
Azimuth angle
d
Yaw angle
Q
r m
Angle of inclination of the trajectory Air density r = r (p,T,h) Absolute viscosity of air; m = m(T)
j
Degree of latitude
w
Spin rate of bullet (angular velocity)
wE
Angular velocity of the earth´s rotation; wE = 7.29.10-5 rad/s
Azimuth and degree of latitude
The azimuth a is defined as the angle enclosed between the positive x-axis of a xyz reference frame and the north direction. a is always positive and may take values between 0° and 360°. The xz-plane is 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° <= j <=+90°)
The force of gravity
Abbreviations ej
Unit vector, opposite to the direction of the acceleration of gravity
FG
The force of gravity
Explanation The force of gravity is proportional to the mass of the projectile and the local acceleration of gravity. The force is directed towards the center of the earth and attacks at the CG. The force of gravity is responsible for the bending of the trajectory.
The centrifugal force
Abbreviations FZ
Centrifugal force
Explanation The figure above shows a cut through the globe. The formula gives the components of the centrifugal force 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, the other components are generally neglected in ballistics because of their smallness.
The Coriolis force
Abbreviations Fc
Coriolis force
v
Velocity vector with respect to xyz - coordinate system Vector of the angular velocity of the earth´s rotation with 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 cD
Drag coefficient
FD
Drag force
Explanation
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 vw, the drag force is completely determined by the the drag coefficient cD .
The drag coefficient The drag coefficient cD is the most important aerodynamic coefficient and generally depends 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, the Re dependency of cD can be neglected.
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) cD(B,Ma, ) = cDo(B,Ma) + cD (B,Ma) *
2
/2
Another theory which accounts for arbitrary angles of yaw is called the "crossflow analogy prediction method". A discussion of this method is far beyond the scope of this article, however the general type of equation for the drag coefficient is as follows:
b) cD(B,Ma, ) = cDo(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 determined experimentally either by wind tunnel tests or from Doppler Radar measurements.
Fig.: Zero-yaw drag coefficient for two military bullets M80 (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 the Mach number from bullet geometry. The latter method is mainly applied in the development phase of a new projectile.
4. Standard drag functions Generally each bullet geometry has its own zero-yaw drag coefficient as a function of the Mach number. This means, that specific - time-consuming and expensive - measurements would be required for each bullet geometry. A widely used simplification makes use of a "standard drag function" cDo standard which depends on the Mach number alone and a form factor iD which depends on the bullet geometry alone according to:
cDo(B,Ma) = iD(B) * cDo standard(Ma) If this simplification is applicable, the determination of the drag coefficient of a bullet as a function of the Mach number is reduced to the determination of a suitable form factor alone. It will be shown that the concept of the ballistic coefficient, widely used in the US for small arms projectiles follows this idea.
Abbreviations
cD
Drag coefficient; cD(B,Ma,Re, )
cDostandard
Zero-yaw standard drag function
iD
Form factor
The ballistic coefficient (bc) The 'ballistic coefficient' or bc is a measure for the drag experienced by a bullet moving through the atmosphere, which is widely used by manufacturers of reloading components, mainly in the US. Although, from a modern point of view, bcs are a remainder of the pioneer times of exterior ballistics, ballistic coefficients have been determined experimentally for so many handgun bullets, that no treatise on exterior ballistics would be allowed to neglect it..
The bc of a test bullet bctest moving at velocity v is a real number and defined as the deceleration due to drag of a "standard" bullet devided 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). The drag coefficients of the standard bullet can be derived from the G1-function given in literature and will be named cDoG1(Ma) . Using
cDotest(B,Ma) = iDtest(B) * cDoG1(Ma) one finds for the bc (assuming "standard" atmosphere conditions)
bctest=1 / iDtest(B) * mtest / d2test This formula also shows that the bc and the form factor iD of a "test" bullet are two aspects of the same principal simplification: the substitution of the (unknown) particular drag function of a bullet by the (given) "standard" drag function of the standard bullet (see also here).
Abbreviations cDotest
Zero-yaw drag coefficient of test bullet
cDoG1
Zero-yaw G1 standard drag coefficient
iDtest
Form factor of test bullet
bctest
Ballistic coefficient of test bullet
mtest
Mass of test bullet in lb
dtest
Diameter of test bullet in inches
The ballistic coefficient (bc) The 'ballistic coefficient' or bc is a measure for the drag experienced by a bullet moving through the atmosphere, which is widely used by manufacturers of reloading components, mainly in the US. Although, from a modern point of view, bcs are a remainder of the pioneer times of exterior ballistics, ballistic coefficients have been determined experimentally for so many handgun bullets, that no treatise on exterior ballistics would be allowed to neglect it.. The bc of a test bullet bctest moving at velocity v is a real number and defined as the deceleration due to drag of a "standard" bullet devided 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). The drag coefficients of the standard bullet can be derived from the G1-function given in literature and will be named cDoG1(Ma) . Using
cDotest(B,Ma) = iDtest(B) * cDoG1(Ma) one finds for the bc (assuming "standard" atmosphere conditions)
bctest=1 / iDtest(B) * mtest / d2test This formula also shows that the bc and the form factor iD of a "test" bullet are two aspects of the same principal simplification: the substitution of the (unknown) particular drag function of a bullet by the (given) "standard" drag function of the standard bullet (see also here).
Abbreviations cDotest
Zero-yaw drag coefficient of test bullet
cDoG1
Zero-yaw G1 standard drag coefficient
iDtest
Form factor of test bullet
bctest
Ballistic coefficient of test bullet
mtest
Mass of test bullet in lb
dtest
Diameter of test bullet in inches
The lift force
Abbreviations cL
Lift coefficient; cL(B,Ma.Re, )
eL
Unit vector
FL
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 lift force vanishes in the absence of yaw and is the reason for the drift of a spinning projectile even in the absence of wind.
The overturning moment
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 ). This moment 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 simply lift.
The overturning moment
or
Abbreviations cM
Overturning moment coefficient, cM(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 the centre of pressure CPW of the wind force, which, for spin-stabilized bullets is located ahead of the CG. As the flow field varies, the location of the CPW varies as a function of the Mach number. Due to the 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 spinstabilized projectiles MW tends to increase the yaw angle and destabilizes the bullet. In the absence of spin, the moment would cause the bullet to tumble.
The spin damping moment
Abbreviations cspin
Spin damping moment coefficient; cspin(B,Ma.Re)
MS
Spin damping moment
Explanation
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 cMag
Magnus force coefficient; cMag(B,Ma,Re, , )
eM
Unit vector
FM
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 force vanishes 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, velocity and the angle of yaw. In exterior ballistics, the above expression is used for the Magnus force.
The Magnus force
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 at its centre of pressure CPM. The location of the CPM varies as a function of the flowfield conditions and 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 this comes true for a 7.62 x 51 FMJ standard Nato bullet at least close to the muzzle in the high supersonic velocity regime.
The overturning moment
Abbreviations cM
Overturning moment coefficient, cM(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 the centre of pressure CPW of the wind force, which, for spin-stabilized bullets is located ahead of the CG. As the flow field varies, the location of the CPW varies as a function of the Mach number. Due to the 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 spinstabilized projectiles MW tends to increase the yaw angle and destabilizes the bullet. In the absence of spin, the moment would cause the bullet to tumble.
The overturning moment
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 ). This moment 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 simply lift.
The Magnus moment
or
Abbreviations cMp
Magnus moment coefficient; cMp(B,Ma,Re,w,d)
eMM
Unit vector
MM
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 of the Magnus force depends on the flow field and can be located ahead or behind the CG. The Magnus moment 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 cMa
Overturning moment coefficient derivative; cMa(B,Ma)
sg
Gyroscopic stability factor
Explanation A spin-stabilized projectile is said to be gyroscopically stable, if, in the presence of a yaw angle d, it responds to an external wind force F1 with the general motion of nutation and precession. In this case the longitudinal axis of the bullet moves into a direction perpendicular to the direction of the wind force. It can be shown by a mathematical treatment that this condition is fulfilled, if the gyroscopic stability factor sg exceeds unity. This demand is called the gyroscopic stability condition. A bullet can be made gyroscopically stable by sufficiently spinning it (by increasing w!). As the spin rate w decreases more slowly than the velocity vw, 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 its flight. The quantity sg also depends on the air density r and this is the reason, why special attention has 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 much stabilization. This is called over-stabilization. The gyroscopic (also called static) stability factor depends on only one aerodynamic coefficient (the overturning moment coefficient derivative cMa) and thus is much easier to determine than the dynamic stability factor. This may be the reason, why some ballistic publications only consider static stability if it comes to stability considerations. However, the gyroscopic stability condition only is a necessary condition to guarantee a stable flight, but is by no means sufficient. Two other conditions - the conditions of dynamic stability and the tractability condition must be fulfilled.
Gyroscopic (static) stability factor
This figure shows the gyroscopic stability factor of the 7.62 x 51 Nato bullet M80, fired at an angle of departure of 32°, a muzzle velocity of 870 m/s and a rifling pitch at the muzzle of 12 inches. The M80 bullet shows static stability over the whole flight path as the static stability condition sg>1 is fulfilled 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 for the 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 the the rotational velocity is much less damped than the transversal velocity (which is damped due to the action 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 observed that many handgun bullet show excessive static stability.
The dynamic stability condition
Abbreviations cD
Drag coefficient
cLa
Lift coefficient derivative
cMpa
Magnus moment coefficient derivative
cmq+cma
Pitch damping moment derivative
sg
Gyroscopic (static) stability factor
sd
Dynamic stability factor
Explanation A projectile is said to be dynamically stable, if its yawing motion of nutation and precession is damped 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 to determine, it can become very complicated to calculate the dynamic stability factor, which varies as a function of the momentary bullet velocity.
The stability triangle
Abbreviations sg
Gyroscopic stability factor
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 very simple 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 figure below).
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 stability triangle
Abbreviations sg
Gyroscopic stability factor
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 very simple 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 figure below).
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
fl
Low limit tractability factor; fl » 5.7
sg
Gyroscopic stability factor
dp
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 yaw of repose. It can be shown that the tractability factor f is proportional to the inverse of the gyroscopic stability factor.
Over-stabilized bullet
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. The bullet´s longitudional axis becomes uncapable to follow the bending path of the trajectory. Overstabilization is said to occur, if the angle enclosed between the bullet´s axis of form and the tangent to the 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 normal shooting situation, but must be considered when firing at high angles of elevation.
The yaw of repose
Abbreviations cM a dp
Overturning moment coefficient derivative coefficient 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 the transient yaw induced at the muzzle has been damped out for a stable bullet, the yaw angle d equals the 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 even in 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
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 upward with respect to the direction of flight, leading to a side drift to the right. The yaw of repose, although normally measuring only fractions of a degree, is the reason for the side deviation of spin-stabilized bullets.
