Understanding WGs Armor Penetration Curves

[fnord_disc]

This is a work in progress and I would be happy if anybody would be kind enough to contribute. ** THIS POST IS OUTDATED, READ THREAD

 

Introduction

 

It is my personal theory that the developers did not want to make the armor penetration mechanic too complicated compared to real life. I therefore believe that one of the empirical penetration models used historically were taken as-is by the developers and implemented. The simplest formula generally used for such things is the Empirical USN Armor Penetration Formula.

 

Derivation

 

To compare the penetration formula against WoWs, I will draw upon the work in these two threads.

 

http://forum.worldofwarships.eu/index.php?/topic/55486-cleveland-152mm-ap-penetration-table/

http://forum.worldofwarships.eu/index.php?/topic/55502-us-127mm-38-ap-penetration-table/

 

The USN Formula is as follows:

 

P[mm]=0.55616130*v^1.1*w^0.55/(d^0.65)

 

where

 

v is the velocity at impact [m/s]

w is the weight [kg]

d is the diameter of the shell [mm]

 

To calculate the velocity at impact, the differential equation for the drag force m*a=0.5*rho*v^2*c_D*A is solved first for t, then integrated and solved for x. This gives:

 

v(x)=v_0/(2-e^(-k*x))

 

with

 

k=0.5*rho*c_D*A/m

 

Comparison

 

With these two formulas, it is possible to make a comparison against the values determined in the game by arczer25.

 

 

Conclusion

 

Even though several constants had to be guessed, the formula fits the test data extremely well until about 4 kilometers. I suspect that impact angles skew the test data to produce lower armor penetration values. Also, due to how the test data was obtained (limited to certain armor thresholds) it is likely that the values above those 4 kilometers are all significantly too low.

 

Until more test data is available, it is likely that this is the formula used for armor penetration curves.

[herrjott]

Have you tried comparing your formula with the penetration values given in the WG Armada videos?

 

[Tyrendian89]

so, where's our mysteriously beloved Krupp value in this?

[mtm78]

Doesn't Krupp have to do with normalization?

[fnord_disc]

Have you tried comparing your formula with the penetration values given in the WG Armada videos?

 

 

 

The 5km penetration fits okay-ish, the longer ranged ones are lower than my calculation. It's a difference similar to the test data.

 

so, where's our mysteriously beloved Krupp value in this?

 

I don't know.

 

Doesn't Krupp have to do with normalization?

 

Afaik the normalization angle is just a subtraction.

 

There is an additional speed loss due to the length of the ballistic curve. I thought this was negligible, but I'll include this in the formula and make another post.

[Zed_Von_Toza]

Since the game engine is based on the one used in WOT, the armour penetration calculation is probably based on a tank armour penetration model.

 

Maybe something like this http://www.wwiiequipment.com/pencalc/

 

And have you looked at this. http://navweaps.com/index_nathan/index_nathan.php

 

[fnord_disc]

Since the game engine is based on the one used in WOT, the armour penetration calculation is probably based on a tank armour penetration model.

 

Maybe something like this http://www.wwiiequipment.com/pencalc/

 

And have you looked at this. http://navweaps.com/index_nathan/index_nathan.php

 

 

It's definitely not similar to WoT's penetration mechanic in any way. WoWs handles it much more realistically.

[Exustio]

Hey, 

 

I have written the comparison threads on ships, and at some point people did start asking how Krupp/penetration was being calculated. It ended up with a few of us digging into the ballistic calculation formula's that were used at the time. Usually we (just like you) got a formula going (eventhough it was different, and incorperated Krupp) to calculate penetrations. This however, as u have noticed, is only half of the equation, as angle of impact are not taken into account of the penetration formula's. The problem we faced (just like you) is how we calculate ballistic trajectory's ingame (seeing the drag parameter is unknown). As of yet, we have not found a formula that would fit the 2 examples of Budyonny/NO that WG gave us. 

 

I do love how this question (eventhough boring to some, but amongst us mathmaticians, quite intrigueing) keeps popping up, and my hopes is still that someone from the community is able to crack this nut. 

 

Keep on being curious!

 

Exustio

 

P.S. The question of angle of decent because of shell trajectory is a fact, as shell impact inclination is confirmed by WG himself, see following link: 

 

http://wiki.wargaming.net/en/Ship:Gunnery_%26_Armor_Penetration

 

[fnord_disc]

Can someone data mine me the drag coefficients?

 

edit:

 This however, as u have noticed, is only half of the equation, as angle of impact are not taken into account of the penetration formula's. The problem we faced (just like you) is how we calculate ballistic trajectory's ingame (seeing the drag parameter is unknown). As of yet, we have not found a formula that would fit the 2 examples of Budyonny/NO that WG gave us. 

 

I do love how this question (eventhough boring to some, but amongst us mathmaticians, quite intrigueing) keeps popping up, and my hopes is still that someone from the community is able to crack this nut. 

 

Keep on being curious!

 

Exustio

 

P.S. The question of angle of decent because of shell trajectory is a fact, as shell impact inclination is confirmed by WG himself, see following link: 

 

http://wiki.wargamin...mor_Penetration

 

Yes, I know the angle matters. I'm not that stupid.

 

And getting the impact angle is not hard once you nail down the ballistics.

[fnord_disc]

Okay, sorry for the double post, but trust me, it's worth it.

 

I've expanded the calculation further up to include a simple ballistic arc with gravitational force and elevation angle. Since that is definitely too annoying for OpenOffice, the calculation has been shifted to MATLAB. The calculation does not include drag in the y-axis, only in the x-axis. This splits the two counterforces and makes calculating a firing solution easier for me, and since I'm not 100% convinced this is the right direction, I wanted to conserve time. I can expand it later, but for now, it means the speed at impact is somewhat too high and angle too steep, but it shouldn't be off by a lot.

 

Here is an example of Budyonny firing at a 15km target.

 

Budyonny is the only ship for which I have drag coefficients, data mined from Kutuzov (same shells IIRC).

 

W = 55;
D = 0.152;
c_D = 0.321;
V_0 = 950;

 

I played around with the air density and if the air density is set to 0.9, then the following penetration values are obtained using this calculation. It's getting pretty close to the target.

 

target_distance =
        5000

elevation =
    1.9963

p_athit =
  215.6363

impact_speed =
  591.9579

impact_angle =
   -3.1602

effective_ap =
  215.3083

 

 target_distance =
       10000

elevation =
    5.2363

p_athit =
  130.5633

impact_speed =
  375.1452

impact_angle =
  -13.3225

effective_ap =
  127.0497

 

 target_distance =
       15000

elevation =
   10.6229

p_athit =
   96.1052

impact_speed =
  283.9375

impact_angle =
  -38.0717

effective_ap =
   75.6578

[22cm]

Since the game engine is based on the one used in WOT, the armour penetration calculation is probably based on a tank armour penetration model.

 

 

Probably this. Remember that ship walls were already angled by design,  the "V" shape,  that at the normal ranges of the fight the gun arch was quite high,  and the most devastating hits in real life were received through the deck, not the belt.  All the importance of "angling" is coming from WoT,  not from real life,  at 10+ km shells would be coming on a high downward trajectory. 

[fnord_disc]

 

Probably this. Remember that ship walls were already angled by design,  the "V" shape,  that at the normal ranges of the fight the gun arch was quite high,  and the most devastating hits in real life were received through the deck, not the belt.  All the importance of "angling" is coming from WoT,  not from real life,  at 10+ km shells would be coming on a high downward trajectory. 

 

Whether or not the game takes angle into account isn't even the question though. There is no doubt that it does.

 

The question is which formula tells you how many mm of armor the shell penetrates in the first place. Before the angle is calculated. You know, that thing where it just says "202mm" in WoT and where it says nothing in WoWs.

 

WoT uses a linear energy loss model and WoWs definitely does not use the same (hyper-simple) model.

 

edit:

Including a second dimension of drag doesn't materially improve the accuracy of the calculation.

 

For now I just need more drag coefficients.

 

edit2:

 

Comparison generated vs measured.

 

% SHELL CONSTANTS %

 

W = 59;

D = 0.152;

c_D = 0.321;

V_0 = 762;

@ rho 0.9

 

[RichardNixon]

If anyone else is playing around with this stuff, here's an inverse trajectory calculator:

 

http://jaj22.org.uk/trajectory/trajcalc.html

 

Pick a shell, set a target distance and it'll calculate the travel time, impact velocity and angle. Drag is applied in direction of travel, fixed air resistance assumed. Apparently this isn't strictly what WoWS does, although if you simply divide the travel time by 3x then it's very close for the larger shells. The lower calibre shells appear to fly a little further in the same time than predicted, regardless of constant multipliers.

 

There are some drag coefficient values there. Automining them is a bit tricky so they're extracted piecemeal. If you have penetration data for any particular ships then I can get the drag coeffs for them.

 

Edit: Also some close range BB citadel pen data for checking against the armour formula:

SC vs Montana: Pens 409mm @ 15 degrees belt armour at 3.3km

SC vs Iowa: Pens 307mm @ 22 degrees belt armour at 6.1km

[fnord_disc]

target_distance =

        3300

 

elevation =

    1.5692

 

p_athit =

  435.9738

 

impact_speed =

  632.5021

 

impact_angle =

   -2.0022

 

armor_angle =

   -15

 

effective_ap =

  416.9190

 

target_distance =

        6100

 

elevation =

    3.2755

 

p_athit =

  340.8024

 

impact_speed =

  505.6240

 

impact_angle =

   -5.2950

 

armor_angle =

   -22

 

effective_ap =

  302.8565

 

@ rho 0.9 (like with Budyonny and Cleveland, where it fit the data very well) without normalization.

 

Also, I think you have a typo in the muzzle velocity for SC... should be 823, right?

 

I'll look for some more penetration data to compare.

[Zed_Von_Toza]

One last thought.

 

Are you comparing observed vs theoretical?

 

Are you taking into account the material(s) the shell is actually passing through?

 

I mean are you allowing for the different armour layers, I'm pretty sure the game engine factors this in, since each ship has its own armour model, which I believe WG is in the process of improving based on comments in the new patch notes. Actual penetration values would vary from a base theoretical value depending on the properties, density and thickness for example of each individual armour layer. This could account for the slight variations you are seeing. 

[fnord_disc]

One last thought.

 

Are you comparing observed vs theoretical?

 

Are you taking into account the material(s) the shell is actually passing through?

 

I mean are you allowing for the different armour layers, I'm pretty sure the game engine factors this in, since each ship has its own armour model, which I believe WG is in the process of improving based on comments in the new patch notes. Actual penetration values would vary from a base theoretical value depending on the properties, density and thickness for example of each individual armour layer. This could account for the slight variations you are seeing. 

 

I'm assuming a homogenous material since the armor layers mined by GM3D don't have any properties attached to them except a name and a thickness. And an angle of course, but that's a part of the 3D model. What I mean is, there is no values inside the game telling it how good the steel quality is or whether something is cemented or not. At least as far as I know.

 

I doubt the game takes these into account, but it could be the reason, yes.

 

As for why there are differences, well:

 

• WG might include a second axis of drag
• I still don't know what krupp does (if it even does anything related to penetration)
• WG might use some kind of approximation for the ballistic arcs since the explicit solution of alpha(x_target) needs a bunch of messy functions, especially if you have xy-drag and not just x-drag

[Exustio]

Yes, I know the angle matters. I'm not that stupid.

 

And getting the impact angle is not hard once you nail down the ballistics.

 

Never meant it as an insult, and i never said getting the angle was a problem, its the drag coefficient that is the problem. 

 

• I still don't know what krupp does (if it even does anything related to penetration)

 

As i have said before, i did some digging in what krupp is and posted it. See following quote: 

 

 

This question was an expected one, however i can not give u any definitive answer. However, i have come up with the following "idea" of what it would mean. 

 

The name Krupp was determined by the Krupp steel company in the 2nd WW by the family Krupp. This company developed a steel which was stronger then any other, and soon enough the steel was used in every major naval power in the 2nd WW. 

 

What i believe is that Projectile krupp originated from the testing of armour penetration with different caliber guns/armour by the company. This resulted in them defining a constant named "Resistance constant".

 

The Resistance Constant represents the quality of the armour and the capacity of a shell to penetrate it. 

 

Assuming this, we come up with the following formula: 

 

Where: 

B= Armor penetration
V= Velocity
P= Mass Shell 
K= Resistance Constant
D= Caliber Gun

 

This will allow us to calculate average penetrations per gun.

 

For now, my clan seems to think i am correct in these assumptions. Both myself and clan members have tested these value's and they seem to be in the green. 

 

Keep in mind, these are guesses. I have yet to hear from a WG employee if what i assume is correct, so don't go flame when im wrong. 

 

Hope this might clear up some questions. 

 

Exustio

 

As is stated in my original post of 2015, this is all one theory. What i also have stated, is that the only way for us to know what formula WG uses to calculate penetrations is for them to react to a post. Then again, this assumption is a year old. 

 

Also keep in mind, that Krupp will not work on HE, as HE is never meant to penetrate. Otherwise, its all up to speculation. These penetration numbers only apply to situations where drag/angle of impact are optimal, meaning 0 drag, and a 90 degree impact trajectory. 

 

 

[Namolis]

The normalization angles (and auto-ricochet angles, less relevant to this) were fairly recently datamined by Vaexa over at reddit: https://www.reddit.com/r/WorldOfWarships/comments/4mt1pa/0561_live_ap_normalisation_autobounce_values/

 

Notice that the Tachibana has a (probably bugged) normalization value of 68 degrees, so it will essentially always strike dead on. If a penetration corridor can be established for her (ie. points both of max pen distance and min non-pen distance), that would eliminate the effect of falling angle and provide a nice sanity check on your formulas.

[daki]

 

I'm assuming a homogenous material since the armor layers mined by GM3D don't have any properties attached to them except a name and a thickness. And an angle of course, but that's a part of the 3D model. What I mean is, there is no values inside the game telling it how good the steel quality is or whether something is cemented or not. At least as far as I know.

 

I doubt the game takes these into account, but it could be the reason, yes.

 

I am almost certain that in some old patch articles I read that steel quality is also taken into account - though I guess it should be a small impact

[fnord_disc]

 

Never meant it as an insult, and i never said getting the angle was a problem, its the drag coefficient that is the problem. 

 

 

As i have said before, i did some digging in what krupp is and posted it. See following quote: 

 

 

As is stated in my original post of 2015, this is all one theory. What i also have stated, is that the only way for us to know what formula WG uses to calculate penetrations is for them to react to a post. Then again, this assumption is a year old. 

 

Also keep in mind, that Krupp will not work on HE, as HE is never meant to penetrate. Otherwise, its all up to speculation. These penetration numbers only apply to situations where drag/angle of impact are optimal, meaning 0 drag, and a 90 degree impact trajectory. 

 

 

 

Can you explain to me which units the formula takes? Because if I assume metric, this is what happens for South Carolina at 0 meters.

 

823 m/s MV

394.6 kg mass

2408 krupp

0.305 m caliber

 

823*sqr(394.6)/(2408*sqr(0.305)) = 12.3... m? mm? cm? none of these fit

 

I've looked it up and apparently caliber and penetration are in decimeters. This would produce 390mm penetration @ 0m @ 0° for South Carolina, and by that measure she shouldn't be able to penetrate Montana's citadel, but she can. So this formula can't be 100% correct.

 

The normalization angles (and auto-ricochet angles, less relevant to this) was fairly recently datamined by Vaexa over at reddit: https://www.reddit.com/r/WorldOfWarships/comments/4mt1pa/0561_live_ap_normalisation_autobounce_values/

 

Notice that the Tachibana has a (probably bugged) normalization value of 68 degrees, so it will essentially always strike dead on. If a penetration corridor can be established for her (ie. points both of max pen distance and min non-pen distance), that would eliminate the effect of falling angle and provide a nice sanity check on your formulas.

 

Thanks.

 

 

 

Also, if the Krupp formula you gave me is compared to the USN formula, it's easy to see that the exponents are pretty similar: 0.55 and 0.65 vs 0.5 and 1.1 vs 1, so it would be understandable if the Krupp formula is used and not USN and I still get similar results. I just need to get the scaling factor to work.

[kennydbt]

all you need to know it is rigged then a T VI DD can rip apart a T VII BB but a T VI CA get 50 hits on a T VII for no damaged at all it rigged.

[Khul]

all you need to know it is rigged then a T VI DD can rip apart a T VII BB but a T VI CA get 50 hits on a T VII for no damaged at all it rigged.

 

 

[RichardNixon]

Also, I think you have a typo in the muzzle velocity for SC... should be 823, right?

 

Ugh, very old typo. Broke my earlier attempts to match the formula. Physical drag may fit the travel time evidence.

 

I suspect the "Krupp" number is used as a constant multiplier, with ~2400 as a divisor, but there are a lot of unknowns to fit. If you want to demonstrate that it does something then Murmansk vs Omaha should be a good test.

 

[fnord_disc]

 I suspect the "Krupp" number is used as a constant multiplier, with ~2400 as a divisor, but there are a lot of unknowns to fit. If you want to demonstrate that it does something then Murmansk vs Omaha should be a good test.

 

I don't have either of those ships.

 

As a constant multiplier: Hm, yes, perhaps... I'm getting pretty good results by just ignoring it, and since most shells are around 2400...

 

edit: curves edited out and redone further down.

[fnord_disc]

All curves generated with rho=0.9.

 

Budyonny (experimental data is WG video, therefore no impact angle in test data)

 

 

Cleveland (experimental data from arczer25)

 

 

USN 12.7cm / 38 (experimental data from arczer25)