National Physical Laboratory Equation

[eliw00d]

I have heard that Combat Mission made use of this equation in their game, and was wondering if anyone else has used this equation for personal use? It seems to give fairly good results, especially when using multipliers of 0.913 and 0.943 for German production quality ammunition. However, from my understanding, it is only useful for AP and APC/APCBC. Are there any additional equations or multipliers to use when trying to determine APBC, APCR, APDS, APNCR, etc? Any help would be appreciated!

 

Also, when the result becomes negative, is that indicative of a ricochet?

 

I have been using the book WWII Ballistics: Army and Gunnery to apply this equation, among other things.

Edited April 5, 2014 by eliw00d

[Mobius]

I guess you mean a penetration formula. You might look at Nathan Okun's M79APCLC program. It is for the 76mm M79 APC round that may scale up. It would differ for APHE or APCR however.

It looks like he's added a few other programs for different shell shapes.

http://www.navweaps.com/index_nathan/index_nathan.htm

[eliw00d]

Thank you for that resource. I had read about Nathan Okun's spaced plate equation in the book, but this looks like a good start. I will look through it.

 

Any thoughts on the NPL Equation? It seems to fit most German data pretty well, after multipliers.

[Mobius]

Any thoughts on the NPL Equation? It seems to fit most German data pretty well, after multipliers.

I don't use it. I use real data and try to normalize it.

[eliw00d]

Normalize it in what way?

 

Does anyone else have experience with the NPL equation?

[Mobius]

 

Does anyone else have experience with the NPL equation?

Is it the formula found in this report?

http://www.dtic.mil/dtic/tr/fulltext/u2/a955285.pdf

[eliw00d]

I would have to read through that to see, but here is the formula we are looking at: http://www.dtic.mil/dtic/tr/fulltext/u2/301343.pdf

 

On page 12 of the PDF.

 

Basically,

 

(mass * velocity^2) / diameter^3 = 43.4 * sqrt(BHN) * (thickness / diameter) * secant(1.5 * impactAngle) + 929 - (11800 / 65 - impactAngle) - (54000 / (500 - 160 * log10(diameter / 1.565) - BHN))

 

Where thickness and diameter are in inches, and velocity is in feet per second.

 

We are basically looking for some sort of equation (or multiple equations) that gives us the amount of penetration based on projectile diameter, weight, velocity, and angle. It would be nice to also account for different shell types, such as capped and uncapped, as well as against homogeneous and face-hardened armor.

Edited April 6, 2014 by eliw00d

[Mobius]

I suppose that would do it. That report didn't meet my needs as it is summary data.

I found this report better as it had M79 AP test shot data.

http://www.dtic.mil/dtic/tr/fulltext/u2/310022.pdf

 

Now that can be compared to the M62 APHE test shot data.

http://www.dtic.mil/dtic/tr/fulltext/u2/a955289.pdf

[eliw00d]

So,

 

F(e/d,θ) = (sqrt(m) * v * cos(θ)) / (sqrt(e) * d)

 

Would give the penetration coefficient for uncapped, and the same times 41.57 would be for capped? The limit velocity could then be determined with the equation I gave? Then what?

[eliw00d]

Just to show what we have done so far:

local lhs = 3.28084 * shellVelocity * math.sqrt(shellWeight * 2.20462 / math.pow(shellDiameter * 0.0393701, 3)) - 929 + 11800 / ( 65 - impactAngle ) + 54000 / ( 500 - 160 * math.log(shellDiameter * 0.0393701 / 1.565, 10) - plateHardness );

local rhs = ( 43.4 * math.sqrt(plateHardness) ) / ( shellDiameter * 0.0393701 * math.cos(math.rad(1.5 * impactAngle)) );

local penetration = 25.4 * ( lhs / rhs );

The formula has been modified to convert given units (mm, m/s, kg) to formulaic units (inches, fps, lbs), and is written in Lua.

 

Using the example given on p.87 of WWII Ballistics: Armor and Gunnery, which was a 40mm AP with a muzzle velocity of 792 m/s against a plate with a hardness of 260 BHN at 0° gives a result of 86mm (same as the book).

 

We tested the other example and got the same result as the book.

 

When using German shell data, and 275 BHN as the test plate hardness, we used a modifier of 0.914 for 75mm projectiles and 0.943 for 88mm and larger projectiles. The results come pretty close to the data given in the book for 0°, but only for APCBC (strangely). For example, we ended up with ~136mm of penetration for the 7.5cm KwK 40 L/48 at 0° (book gives 135mm), or ~107mm at 30° (tarrif.net gives 99mm, but with a muzzle velocity of 740 m/s instead of 750 m/s).

 

Are we on the right track with this method, using multipliers?

Edited April 7, 2014 by eliw00d

[Mobius]

Does WWII Ballistics: Armor and Gunnery use BHN of 260? I thought they tried to match US data. The US normalized their data to 115,000 tensile strength or 237 BHN.

[eliw00d]

For the NPL Equation, they used 260 BHN as an example.

 

In my calculations for US penetration, I used 240 BHN.

[Mobius]

Hmmm. In my normalized data that normalizes and averages actual penetration data to 270BHN rather than calculates I have the KwK40 at 0° with 750 MV with 126mm.

[eliw00d]

The book gives 133mm @0° for the L/43 and 135mm @0° for the L/48, with muzzle velocities of 740 m/s and 750 m/s, respectively. Of course, one thing I am now taking into account, using another formula in the book, is the velocity of the projectile at a given range (100m). So, 750 m/s becomes ~738 m/s, giving a penetration of ~133mm at 100m @0° for the L/48. Then, 740 m/s becomes ~ 728 m/s, giving a penetration of ~130mm at 100m @0° for the L/43. Given that they used the DeMarre equation for their results, and not the NPL equation, I am not surprised that there is a slight difference.

 

For these results, I used a BHN of 275.

 

Edit: Out of curiosity, how do you normalize them? Do you use slope multipliers?

Edited April 7, 2014 by eliw00d

[Mobius]

 

Edit: Out of curiosity, how do you normalize them? Do you use slope multipliers?

I collect sample data from as many reliable sources as I can find. Then normalized per the slope multiplier, the armor quality and the test criteria. Basically multiply by some factor. (Some is not exact but best guess.) Then average each curve together.

 

You may have visited my Kwk 40 page.

http://www.panzer-war.com/page43.html

 

or my penetration criteria page.

http://www.panzer-war.com/page33.html

 

or my BHN page.

http://www.panzer-war.com/page36.html

[eliw00d]

Most of this was already in the book, so I am not really sure how to apply any of it to test data given on tarrif.net.

 

When you normalize penetration to 0°, do you also account for lateral angle or just the test plate at, for example, 30° from vertical?

 

Maybe it would help to work through an example:

 

7.5cm KwK 40 L/48

 

We only plan on using penetration data for 100m, for which tarrif.net gives 99mm against RHA for APCBC.

 

So, 99mm * 0.914 (production quality ammo) * 1.144 (50% success normalization) * ~1.25 (slope effect for APCBC) = ~129mm at 0°

 

The book gives 135mm at 0°, so it is still off.

 

How different is your approach?

Edited April 10, 2014 by eliw00d

[Mobius]

Have you checked out Bojan's yugo data?

http://www.tank-net.com/forums/index.php?showtopic=18562&hl=%20bojan%20%20yugo

 

See if your results are ballpark actual test results.

 

I simply use the 1/cosine adjustment.

(I guess if you wanted to engineer the value at 100m use data from Spielberger's Panther book you find @ 730m/s the 0°/30° ratio is 1.19 and 1.17 at 665m/s and 1.15 at 605m/s.)

But your .914 * 1.144 is very close to my 1.04 normalization factor.

Edited April 10, 2014 by Mobius

[Mobius]

What am I thinking? The Spielberger's Panther book table has the 0° penetration of the 75mm Pz Gr 39/42.

(For 0° it should be very close to the penetration of PaK 40 Pz Gr 39.)

0° 800m/s=133mm, 730m/s=118, 665m/s=104mm, 605m/s=91mm.

So prorating to 738 m/s gets you to 119.7mm. For 100m multiply by .914 x 1.144 gets =125mm

So you can prorate from these points to find the penetration at various ranges.

Edited April 11, 2014 by Mobius

[MKSheppard]

On 4/7/2014 at 2:08 PM, eliw00d said:

Just to show what we have done so far:

local lhs = 3.28084 * shellVelocity * math.sqrt(shellWeight * 2.20462 / math.pow(shellDiameter * 0.0393701, 3)) - 929 + 11800 / ( 65 - impactAngle ) + 54000 / ( 500 - 160 * math.log(shellDiameter * 0.0393701 / 1.565, 10) - plateHardness );

local rhs = ( 43.4 * math.sqrt(plateHardness) ) / ( shellDiameter * 0.0393701 * math.cos(math.rad(1.5 * impactAngle)) );

local penetration = 25.4 * ( lhs / rhs );

The formula has been modified to convert given units (mm, m/s, kg) to formulaic units (inches, fps, lbs), and is written in Lua.

I'd like to point out the advancement in general technology, if you're still around almost a decade later (eliw00d).

I took your LUA equations and fed them into ChatGPT as:

Create for me a HTML5 Javascript page with user input variables to calculate the following LUA code <snip code>

The code I got was usable -- all I had to do was make a new radians function -- LUA has a built in math.rad function, but Javascript doesn't; and I was off to the races.

Here's the code for the NPL equations in HTML5 Javascript -- just copy and paste this code into a text file on your hard drive that you rename to a HTML file and run it in your browser.

The variables for the Rexford test case of:

Using the example given on p.87 of WWII Ballistics: Armor and Gunnery, which was a 40mm AP with a muzzle velocity of 792 m/s against a plate with a hardness of 260 BHN at 0° gives a result of 86mm (same as the book).

are already pre-loaded into the code.

<!DOCTYPE html>
<html>
<head>
<title>NPL Equation Calculator</title>
</head>
<body>

<h1>NPL Equation Calculator</h1>

<form>
<label for="shellVelocity">Shell Velocity (m/s):</label><br>
<input type="number" id="shellVelocity" name="shellVelocity" value="792"><br>
<label for="shellWeight">Shell Weight (kg):</label><br>
<input type="number" id="shellWeight" name="shellWeight" value="1.08"><br>
<label for="shellDiameter">Shell Diameter (mm):</label><br>
<input type="number" id="shellDiameter" name="shellDiameter" value="40"><br>
<label for="impactAngle">Impact Angle:</label><br>
<input type="number" id="impactAngle" name="impactAngle" value="0"><br>
<label for="plateHardness">Plate Hardness (BHN):</label><br>
<input type="number" id="plateHardness" name="plateHardness" value="260"><br><br>
<input type="button" onclick="calculate()" value="Calculate">
</form> 

<p>Result: <span id="result"></span></p>

<script>

function radians(degrees)
{
return degrees * (Math.PI / 180);
}

function calculate() 
{
  // Get user input values
  var shellVelocity = document.getElementById("shellVelocity").value;
  var shellWeight = document.getElementById("shellWeight").value;
  var shellDiameter = document.getElementById("shellDiameter").value;
  var impactAngle = document.getElementById("impactAngle").value;
  var plateHardness = document.getElementById("plateHardness").value;
  
  // Calculate LHS
  var lhs = 3.28084 * shellVelocity * Math.sqrt(shellWeight * 2.20462 / Math.pow(shellDiameter * 0.0393701, 3)) - 929 + 11800 / ( 65 - impactAngle ) + 54000 / ( 500 - 160 * Math.log(shellDiameter * 0.0393701 / 1.565, 10) - plateHardness );
  
  // Calculate RHS
  var rhs = ( 43.4 * Math.sqrt(plateHardness) ) / ( shellDiameter * 0.0393701 * Math.cos(radians(1.5 * impactAngle)) );
  
  // Calculate penetration
  var penetration = 25.4 * ( lhs / rhs );
  
  // Display result
  document.getElementById("result").innerHTML = penetration.toFixed(2) + " mm penetration using the NPL equation.";
}

</script>

</body>
</html>

 

Edited December 10, 2022 by MKSheppard
fixee

[MKSheppard]

And since we're talking about NPL, here's some scans from the AFV news article in question that introduced it.

Mods; remove this if you think it's breaking copyright, but I don't think that'd be a case, given that the article is what, 30 years old at this point.

[MKSheppard]

I've discovered an error in Rexford/Bird's NPL equation section.

The equation can be used to estimate penetration at various angles and velocities, such as 2 Pounder AP at 2600 fps [792.48 m/s] against 0° armor at 260 BHN. The result is 86mm at 0 yards range. Or U.S. 75mm AP at 2030 fps [618.744 m/s] against 0° armor at 260 BHN, for 111mm at 0 yards range.

I tried replicating these results with the formula he gave:

50% Success Velocity = (d³/w)^0.5 x (43.4 x BHN^0.5 x (t/d) x (cosine (1.5A))^-1 + 916 -(11800/( 65- A )) -54000/(500 - 160 x log (d/1.565) - BHN)

Where:

t = plate thickness in inches
d = projectile diameter in inches
w = projectile weight in pounds
BHN = Brinell Hardness Number
A = impact angle (0° is perpendicular impact)

UK 2 pdr AP (40mm / 1.565in diameter shot; 2.380992 lb / 1.08 kg weight shot; 86mm / 3.38583 inch plate; 260 BHN): I get a V50 of 2,430.428 ft/sec; well below the 2600 fps he gives.

US 75mm AP (76mm / 3in diameter shot; 14.62 lb / 6.634 kg weight shot; 111mm / 4.37008 inch plate; 260 BHN): I get a V50 of 1,722.391 ft/sec; well below the 2030 fps he gives.

However, eliw00d's formula gave correct values (86mm penetration) for 40mm AP shot, but incorrect values (125.99 mm penetration) for 75mm AP shot.

BUT. I figured it out.

It appears that Rexford/Bird entered values for the 75mm M72 APC shot from the 75mm L31 M2 Gun which has a m/v of 563 m/s (1,850 ft/s) -- and printed the results he got (111.33 mm penetration) as the results for 75mm M72 APC shot from the 75mm L40 M3/M6 gun with a m/v of 618 m/s (2,030 ft/s); further confusing people decades down the line. 🙃

EDIT: I have discovered why the 2-pdr results are incorrect when you try using the Rexford/Bird equation. When they were preparing the 2-pdr example, they simply entered "2 pounds" as the weight of the shot, resulting in 2604.911 ft/sec V50.

Whereas checking Wikipedia, we find that the weight of the 2 pounder AP-T Mk IT is 2.375 lbs; and the APCBC/T MK I shot is 2.7 lb.

So when people use actual weights which are now easily available circa 2022; they get confused and think they've messed things up in their maths!

Edited December 12, 2022 by MKSheppard

[Peasant]

Here's an online spreadsheet where you can try out this formula for yourself: https://docs.google.com/spreadsheets/d/10fu5-btgOvtO9sLyGmsqTsURQN5oa--kr76201Kdux8/edit?usp=sharing

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Edited September 5, 2023 by Peasant

[Peasant]

Here is something interesting: if we compare experimental data and the NPL formula, we can see that it's prediction that, on average, ballistic limit for this target does not appreciably change with BHN varying between 240 and 300, matches perfectly.

The amount of kinetic energy absorbed during perforation of the plate is directly related to it's Navy ballistic limit, for example for 780kJ, its 1571 fps.

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The value of R^2 next to 0 indicates that there is very likely no casual connection between the two variables.