Topic 4 - external ballistics Flashcards
what is the path of a projectile dictated by?
- Firearm projectiles do not follow true ‘ballistic arcs’.
The path of the projectile is dictated by:
* Gravity.
* Angle of launch (elevation).
* Velocity.
* Air density, temperature and humidity.
* Projectile shape (and drag coefficient) plus overall projectile stability.
what is a realistic flight path like?
- A real arc, when air resistance
is taken into consideration. - The second half is more
‘truncated’. = shortened in duration or extent. - Trajectory is uneven either
side of the maximum height.
when can SUVAT equations be used?
- Remember that in the absence of air resistance, calculations can be
conducted using the SUVATS
S = u x t + 1/2 a x t^2
Vx = Sx / Ttotal
Aerodynamic forces and gravity
- Aerodynamic forces and gravity are the two most important factors in calculating the true dynamics of a projectile.
- Aerodynamic drag is typically proportional to the square of the velocity, so drag builds significantly with velocity.
- There are other properties that also contribute to drag, including:
- The profile or shape of the projectile.
- The cross-sectional area of the projectile in the direction of travel.
- Air density.
centre of mass/centre of gravity
- The centre of mass (AKA centre of gravity or CoM/CoG) is the point where the bullet balances its weight (W = mass × gravity). Think of it like the pivot point on a see-saw.
centre of pressure/centre of mass
- Ideally, for the best flight stability, the CoP should be rearward of the CoM and very close to it. Fin stabilisation (more generally known as drag stabilisation) facilitates this by creating additional aerodynamic forces at the back of the projectile.
- Normal ‘spitzer’ bullet shapes actually have the CoP significantly forward of the CoM (as shown in the below figure). This naturally makes the bullet want to
tumble as this creates a major ‘turning moment’. - This is why bullets need gyroscopic stabilisation to overcome the want to tumble and keep the bullet point directed towards the impact point.
Drag Stabilisation
- Drag stabilisation is typically achieved by
adding fins to the projectile, which creates
additional drag forces at the rear of the
projectile (like the rocket in the image, where
the fins make it less classically ‘aerodynamic’
than a bullet shape). - This brings the CoP rearward of the CoM,
leading to a smaller and less influential turning
moment. - This results in a stable flight path without the
need for any gyroscopic stabilisation. - Gyroscopic stability would not be possible with this
design anyway as the fins would not fit in the barrel.
what equation would you use to calculate drag coefficient (Cd)
The total drag force that is experienced by the projectile can be calculated using the following equation:
Fd = 1/2 Cd AV^2 p
Where:
Fd = drag force in N
Cd = drag coefficient (no units)
V = flow velocity (for the air or projectile) in m.s-1
A = cross sectional area in m2
ρ = air density @ sea level, which is about 1.2 kg.m-3
why do we use wind tunnels?
- Wind tunnels can be used to measure the drag
force experienced by a projectile design as air
is blown over a stationary object. - In general, they provide us with the data to
calculate the drag coefficients for a projectile
(or other moving object). - Force gauges can be attached to scale models
of the object (or the real item can be used,
depending on real size). - Using high velocity wind tunnels is an
extremely expensive endeavour – a cheaper
alternative is to use computational fluid
dynamics (CFD).
what equation do we use to calculate drag coefficeint from wind tunnel data?
Cd = 2Fd / AV^2 p
how do we calculate sectional density?
S = m / d^2
what is sectional density? non ballistics definition
Mass of projectile divided by its cross-sectional area.
what is sectional density? ballistics definition
mass of a projectile divided by its maximum diameter squared.
what is a ballistic coefficient (Cb) ?
- This is a measure of the aerodynamic
forces exerted on a particular bullet in flight
and is known as Cb. - The coefficient is specific to an individual
bullet design and size and can be used to
calculate real-time trajectory values during
flight. - It relates the bullet’s sectional density to its
drag coefficient. - Cb has units of kg.m-2
what is modern Cb?
- The most up to date method is derived from the bullet’s cross-sectional
area, drag coefficient, and mass. - In an out-dated system, Cb was quoted as being between zero and one.
- The calculation method we will use yields a value with no theoretical
upper limit. - In reality, most commercially available ammunition falls into the 50 to
500 range (when using SI units in the calculation). - The exception is DU (depleted uranium) ammunition which is much higher due to its very high density (and mass), but this is not commonly encountered and is highly illegal!
how do you calculate Cb?
Cb = Cg / Cd . m / d^2 = Cg / Cd . S
Once you have found Cd, you can then calculate Cb.
- It’s important to note that each Cd value leads to a separate Cb value.
- Never average your Cd and Cb values.
Where:
Cb = Ballistic coefficient
m = Mass of the test bullet
Cd = Drag coefficient
Cg = Drag coefficient of the G1 standard projectile = 0.5191
d = Diameter of the test bullet.
S = Sectional density of the test bullet
As the drag coefficient is determined in a wind tunnel we are usually reliant upon manufacturers for this data.
what is gyroscopic stability?
- One of the major contributors to modern projectile accuracy is gyroscopic (or spin) stabilisation, which we have already discussed in this module.
- With the exception of a brief experiment in the 1983, all modern small arms (aside from shotguns) use the barrel to impart spin stabilisation upon the in-flight projectile.
- The exception here is the “Gyro-jet”, a case-less rocket bullet/hybrid which was spin stabilised after leaving the barrel - a good idea which just couldn’t be made to work with the available propellants.
how is spin rate calculated? (gyroscopic stability)
spin rate = Vm / Tr , in S^-1
- Where: Vm = Muzzle velocity
Tr = Rifling twist rate (length for one 360o rotation, which we called ‘L’ previously) - Note that ‘spin’ relates to the bullet in flight and ‘twist’ relates to the barrel rifling.
- The spin rate is of limited use to a forensic examiner but for the sake of completeness, it’s useful to show how to calculate it here.
- This is the rate of rotation of the bullet about its longitudinal axis after leaving the firearm’s barrel but depends on the twist rate of the barrel, which we have looked at previously as a useful class characteristic of a
firearm.
what is the Greenhill formula?
relationship between the dimensions of a bullet and its optimum twist rate.
T = Cd^2 / L
Where:
T = Optimum twist rate in metres (remember, this is distance for one full rotation).
C = A constant (Use the values of ‘150’ for muzzle velocities under 860 m.s-1 and ‘180’ for over 860 m.s-1
).
d = Bullet diameter in metres.
L = Bullet length in metres.
what is aerodynamic lift?
- This is where a ‘boundary layer effect’ occurs whenever air passes over a curved surface.
- It is how planes fly.
- For example, air moving over the curved
upper wing surface travels further then the
air passing under the flat lower surface. - This creates a pressure drop over the wing which generates lift.
- The amount of lift is proportional to the air speed over the surface and the rate of curvature.
- This can be related to a spinning bullet in a crosswind.
- The bullet will tend to lift or drop at a greater than expected rate in a strong cross wind as the wind may blow over variable shape profiles of the bullet depending on wind angle of interaction.
what is wind deflection?
Two main types of wind deflection are experienced by a projectile:
1. Aerodynamic: Caused by wind flow over the projectile in flight, generating more lift on one side of the bullet due to spin induced pressure difference.
- This is much the same as that experienced by an aircraft wing.
- This effect is very small and only shows up at very long range.
- Windage: Deflection caused by constant wind pressure during projectile flight.
* This has a much more pronounced effect on the overall trajectory.
- A strong breeze can cause several metres of deflection at longer ranges.
how do you calculate wind deflection?
- Bullet wind deflection due to windage can be calculated using the vector addition formula.
R = square root [ (Ax + Bx)^2 + (Ay + By)^2 ]
where:
Ax = Acos 0 with line through
Ay = Asin 0 with line through
Bx = Bcos a that looks like a fish
By = Bsin a that looks like a fish
what are the causes of abnormal flight characteristics?
Rifling-induced instabilities are caused by a low spin rate being
experienced by the projectile and has three main causes:
- Low muzzle velocity: in this case, the bullet engages with the rifling but
never reaches a spin rate that will ensure stability. This could be
caused by a low or defective charge, excessive bore friction, a bullet of
incorrect caliber being used in firearm to name a few reasons. - High muzzle velocity: this is particularly seen with unjacketed bullets.
The bullet velocity may be so high that the soft bullet skids over the
rifling. This is typically caused by a hand-loaded round or an incorrect
bullet choice. This can lead to distinctive rifling marks. - Defective rifling: The bullet fails to engage with the worn or defective
rifling, leading to a low spin rate (compare to Greenhill optimum),
aerodynamic imbalance and potential tumbling.
what is projectile instability?
- Yaw refers to the lateral
movement of the nose of
the bullet away from
the line of flight. - Precession refers to
rotation of the bullet
around the centre of
mass. - Nutation refers to small
circular movement at
the bullet tip due to the tip
not being perfectly round. - Yaw and precession typically decrease as the distance of the bullet from the
barrel increases, since gyroscopic stability will keep them under control.
what causes yaw?
Main causes of yaw include:
- A poorly cast bullet or bad loading,
causing the bullet to be off centre in the
cartridge case neck. - Irregular rifling or non-optimal spin rate. * This is also the main reason why ‘sabot’
rounds (saboted projectiles shown to the
right), though a very good idea, cannot always
be made to work in small caliber applications.
