Math formulas

[MarkSheppard]

Holy @#$%, Am I in the right forum?

Yes you are.

This looks like a Mensa club meeting or rocket science forum. This is too much for my Infantry Grunt brain to handle!

It's actually a quite easy formula to understand, it's basic algebra; unlike a lot of other formulaes out there

[MarkSheppard]

Actually, you got the formula wrong.

It's really

Anderson:
P = 1.044 x V - 0.194 x Ln(L/D) - 0.212 * L

V: velocity (Km/sec)
L: length of the penetrating rod (mm)
D: diameter (mm)
P: Penetration

HOW TO DO IT:

Now to provide the numbers:

V = 0.81 (810 m/s) = Velocity
L = 370.05 = Length of Penetrator (mm)
D = 88 = Diameter of Penetrator (mm)

Roughly that of a German 88mm L56 shell.

So we've got the following equation now:

1.044 x 0.81 - 0.194 x Ln(370.05/88) - 0.212

First thing first, we need to solve the Logarithimic problem; excel is great for those of us who did not pay attention in Math; it has a Logarithm function.

370/88 in Logarithimic is: 1.4363

So now the Formula is:

1.044 x 0.81 - 0.194 x 1.4363 - 0.212 * 370.05

Something that you can solve with basic infantryman math.

That formula results in

131.36mm penetration.

That's actually quite close to the actual performance of the 88mm L56 Panzergranate 39 round; which was 120mm of armor at 30 degrees (139mm total) at 100m.

[Pyros]

MarkSheppard,

Thanks, I already corrected the typo error of the formula.
(btw, you should make a small correction in the formula that you posted because I think that the L is a divider of the penetration, not just a multiplier of the 0.212)


Just to give some depth to the discussion I will add some more parameters to the basic model of Anderson's equation:

Anderson Equation:

P/L = 1.044 x V - 0.194 x Ln(L/d) - 0.212

V: velocity (Km/sec)
L: length of the penetrating rod (mm)
d: diameter (mm)
P: Penetration(mm)


1. The first condition of this formulae is that the velocity should be around 900 m/s and 1800 m/s.

2. Next we should add a modifier (multiplier) for the material of the penetrator (for instance STEEL 0.5, TUNGSTEN 1.0 and DU 1.13)

3. Finally there is also a modifier for the equation related with the scaling factor of the penetrator diameter [Scaling factor = 1+(d/6.5 x 0.05)]

So, the full equation after adding all the above parameters will be as following:

P/L = [1.044 x V(km/sec) - 0.194 x Ln(L/d) - 0.212] x [1+(d/6.5 x 0.05)] x (multiplier for material of penetrator)

Be careful to adjust your Muzzle velocity with the correct V drop (per trajectory flying distance).

You should also add in the above formula the parameter of the "Angle of Obliquity".

At an impact of 60 degree the penetration capability is halved in comparison with an impact at 0 degrees.



cheers,
Pyros

[pdoktar]

Tanknet scientific forum revisited.

[Marek_Tucan]

Where's the mass of the projectile or density or whatever?

[Pyros]

pdoktar said:
Tanknet scientific forum revisited.

This is indeed a good place full of info concerning ballistics

cheers,

[Pyros]

Marek_Tucan said:
Where's the mass of the projectile or density or whatever?

Here...

The Odermatt equation

The Odermatt equation allows the comparison of results from firing tests with differ-ent target inclinations/thicknesses/material properties, penetrator geometries and impact velocities. It also allows the evaluation of influences of penetrator aspect ratio, obliquity and other parameters. Penetrator performance is predictable if penetrator geometrie, density of penetrator and target, ultimate tensile strength of the target, NATO-obliquity and impact velocity are well-known. Reliable extrapolations from a single test result are possible.

In a finite target penetration limit means that the projectile reaches the rear face of the target and spalling opens the penetration channel. The penetrator residue will have a length of approximately one and a half diameter.


Governing parameters

D penetrator diameter [mm]
L overall tungsten penetrator length [mm]
Lw working length of penetrator [mm]
vT impact velocity [m/s]
NATO angle of obliquity [°]
rho_P penetrator density [kg/m3]
rho_T target density [kg/m3]
d target plate thickness [mm]
UTS target material ultimate tensile strength [MPa]
P penetration channel length [mm]


The Odermatt equation is composed of four dimensionless terms with separate representation of the influences of length to diameter ratio A, target oliquity B, density ratio of penetrator to target C as well as material properties and impact velocity D.

Odermatt equation: P/L = A * B * C * D

aspect ratio influence
a good approximation is obtained with the following equation

A = 1+a1*D/Lw*(1-tanh((Lw/D-10)/a2))

with a1 = 3.94 and a2 = 11.2 - valid for Lw/D = 10 up to infinite
Definition of the working length Lw :
The conical tip is replaced by a cylinder of equal mass and diameter D and the remainig length is reduced by 1.5*D.
A good approximation is: Lw = L - 2/3*L_con - 1.5*D

some values of A:

Lw/D A
10 1.394
12 1.270
14 1.185
16 1.126
18 1.085
20 1.057
25 1.020
30 1.007
50 1.000
100 1.000

target obliquity influence
B = (cos °_NATO)^m
the best fit for the exponent is m = -.225

penetrator/target density influence
C = sqrt(rho_P/ rho_T)
this is the well known root density law

Target properties and impact velocity influence
D = exp(-c*UTS/(rho_P*vT^2))

c = 22.1 + 0.01274*UTS - 0.00000947*UTS^2

[JaM]

Thanks Mark. Helped a lot.

[Pyros]

All these formulas are empirical models because it is extremely difficult to design a theoretical model describing the effects of penetration (in relation with the modern warheads and synthetic armor plating).

The principle scientific field which is used for the analysis of the modern ballistic research is the hydrodynamic behaviour (fluid mechanics) of the materials during a hyper-velocity penetration.

cheers,
Pyros