For you folks that like math and statistics here is one way to go from C.U.P or PSI. IF somebody wants this article with all the formatting send me a PM and I will forward it in PDF format.
BB
Correlating PSI and CUP
Denton Bramwell
Having inherited the curiosity gene, I just can’t resist fiddling with things. And one of
the things I can’t resist fiddling with is firearms. I think I am the only kid in town that
asked for, and got, a Fabrique Scientific strain gauge system for Christmas, and promptly
stuck it on his trusty 30-06. So I suppose that it is only natural that I’d be curious about
how CUP and PSI work. That’s what this article is about.
History
The Lyman reloading manual is one of my favorites. It’s clearly written, a pleasure to
read, and it sheds some interesting light on the history of terminology in the measurement
of chamber pressure. Before about the 1960's the only measurement system we had for
chamber pressure was the copper crusher method. Up until that time, what we now call
CUP was commonly known by two different names: CUP and PSI. The two were used
practically interchangeably. Of course, this use of PSI was incorrect. It wasn't much of a
problem until piezoelectric and strain gauge systems became commonly available. These
systems, of course really do measure in PSI. When they arrived on the scene, it caused a
lot of concern and confusion. “For years, 52,000 PSI (crusher method with erroneous
designation) had been pub lished as maximum for the 270 Win. Suddenly, there were new
publications showing 65,000 PSI …as maximum.”1
If you look at any publications before about 1965, and they say that PSI and CUP are not
the same, and that you should not attempt to convert one to the other, they are talking
about the old, incorrect use of the term PSI, not the modern, correct use of PSI from
strain gauges and piezoelectric pressure meters.
What is Correlation?
If you’re on one of the reloading bulletin boards, and say that PSI (modern use) and CUP
are correlated, you’d best be wearing your asbestos underwear. There are a lot of people
that “know” that the two systems aren’t correlated, and will tell you so in no uncertain
terms. Math and physics aren’t on their side, as we shall see. I suspect that their
“knowledge” comes from old information, published to straighten out the problems that
came from incorrectly calling CUP PSI.
If two variables are correlated, you can estimate one from the other. The opposite of this
is “statistically independent”, which means that you can’t estimate one from another.
Actually, it is very hard to come up with numbers that are completely statistically
independent, or uncorrelated. Usually the question is not whether things are correlated,
but how well they are correlated. If you plot my weight vs. my belt size for the past 20
years (please don’t!), you’ll find that one variable reasonably predicts the other. My belt
size and weight, then, are correlated. They won’t be perfectly correlated, and they might
not be linearly correlated, but they will be well correlated.
1 Lyman 47th Reloading Handbook, p92
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A figure of merit for correlation is the R2 value. In the simple case of linear regression,
an R2 of .8 means that 80% of the variation in one variable is “controlled” by the other,
and the remaining 20% of the variation is unaccounted for. Run regression on a pair of
columns of random numbers, and you’ll get R2 values from a fraction of a percent to a
few percent. Run it on a very precise micrometer’s reading vs. the marked values on a
set of gauge blocks spanning a couple of inches, and you’ll get something very close to
100%.
It’s a fact that two variables that are both well correlated with a third variable must be
well correlated to each other. So if the copper crusher system is well correlated with
peak chamber pressure, and the piezoelectric PSI system is well correlated with peak
chamber pressure, then CUP must be well correlated with piezoelectric PSI. It cannot be
otherwise.
All measurement systems lie, at least a little bit. Like all measurement systems, the CUP
method and the PSI method both have a certain amount of random error in them. From
published data, (Lyman manual, p91), it is easy to estimate the random error in both
systems. The bottom line is that the random error associated with the CUP system has a
standard deviation of about 2,000 PSI (correct usage), and the piezoelectric system has a
standard deviation of about 1,300 PSI.
This random variation in the measurement systems accounts for part of the puzzlement in
attempting the conversion. The 7x57 Mauser is rated 46,000 CUP, and 51,000 PSI. The
300 Savage is also rated 46,000 CUP, but 47,000 PSI. Random error in both
measurement systems accounts for this discrepancy. Because there is random error in
both measurement systems, any conversion will be approximate, rather than absolutely
precise.
While it is true that the deformation of the copper pellet in the crusher system is
influenced by all the pressure that happens during the discharge of a bullet, it is also true
that the main thing that the CUP system measures is peak chamber pressure. The
deformation that happens "off-peak" is properly regarded as measurement system error,
and it is minimal, as I will show a bit down the page.
Searching for Correlation Between PSI and CUP
Testing for this correlation is easy. All we need is a set of measurements where the same
event was measured in both systems, and we need that set of measurements to span a
large enough range that we can “see” the correla tion above the random error that is
present.
Measurements taken simultaneously on several examples of a single handload would
about the worst possible choice of data sets. Careful handloaders try very hard to
minimize variation. Ideally, the pressure variation from cartridge to cartridge is zero. In
practice, the range of pressures is so small that a regression on that data would be
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completely swamped by random measurement error, which is significant in both the CUP
and piezoelectric systems.
There is a much better alternative. There are cases where SAAMI has set maximum
pressures for rifles in both CUP and PSI. That data set spans a few tens of thousands of
PSI, and, assuming that SAAMI was careful in how they set the limits, it is much better
for our purpose. I have access to about 30 such data pairs, and that is enough to provide a
reasonable estimate of the conversion factor.
Cartridge ANSI CUP ANSI PSI
222 rem 46000 50000
22-250 rem 53000 65000
243 win 52000 60000
25-06 rem 53000 63000
257 roberts 45000 54000
264 win mag 54000 64000
270 win 52000 65000
280 rem 50000 60000
284 win 54000 56000
30 carbine 40000 40000
300 savage 46000 47000
300 win mag 54000 64000
30-06 springfield 50000 60000
303 british 45000 49000
30-30 win 38000 42000
308 win 52000 60000
32 win special 38000 42000
338 win mag 54000 64000
35 rem 35000 33500
375 h&h mag 53000 62000
444 marlin 44000 42000
45-70 government 28000 28000
6.5 rem mag 53000 65000
6mm rem 52000 65000
7mm express Rem 40000 45000
7mm rem mag 46000 51000
7mm SE vH 52000 61000
7x50 R 52000 61000
8mm rem mag 37000 35000
8x50R 54000 65000
Submitting the SAAMI/ANSI numbers to regression, we get this:
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Analysis of Variance
Source DF SS MS F P
Regression 1 3.302E+09 3.302E+09 357.419 0.000
Error 28 258713297 9239761
Total 29 3.561E+09
An R2 value of .927 puts an end to all discussion about whether PSI and CUP are
correlated. They are. To prove otherwise, you would have to prove that .927 is a lot
closer to zero than it is to one, and you’d have to show that the data pattern in the graph is
much more like a shotgun pattern than it is like a straight line. An F value in the low
teens is usually enough to show statistical significance, and we have an F value of 357.4.
If two variables are well correlated, there is always a formula for converting from one to
the other. The formula for converting from CUP to PSI is shown at the top of the graph.
Since the numbers you are converting do not precisely represent actual chamber pressure,
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the results you get from the conversion will not be precise. About 2/3 of the time, the
formula will land you within 3,000 PSI, so exercise appropriate caution. Also, do not
attempt to use this conversion for handguns or shotguns, or to use it outside the range
shown. We don’t yet know how the conversion works outside the data we have studied.
Let’s go through a couple of examples to show how the formula works. The formula PSI
= -17,902 + 1.516 x CUP is useful if you have data published in CUP, and want to
compare with data published in PSI, Or, if you’re like me, and have instrumented one or
more rifles with strain gauges, you might want to use published CUP data to set an
approximate limit for your loads in PSI. My lovely 6.5x55 Swede is rated at 46,000
CUP, and has no PSI rating. What should I use for a limit in PSI? Multiplying 46,000 by
1.516, and subtracting 17,902 gives me an upper limit of about 51,834 PSI. If I graduate
to a .416 Rigby, which is rated at 42,000 CUP, the same calculation gives us 45,770 PSI.
Reversing the math the 7mm Weatherby Magnum is rated at 65,000 PSI, with no
corresponding CUP number. Converting 65,000 PSI results in a stout 54,685 CUP.
For reasons unknown to me, the 223 Rem doesn’t appear in either of the data sets I have
access to. It is also statistically very different from the rest of the data.
There is also a separate European CIP standard, which uses a different procedure, and
produces different results. Data for 191 cartridges is readily available. Their curve and
10000 20000 30000 40000 50000 60000
70000
60000
50000
40000
30000
20000
10000
CIP CUP
CIP PSI
S = 584.737 R-Sq = 99.7 % R-Sq(adj) = 99.7 %
CIP PSI = -2806.88 + 1.20911 CIP CUP
Regression Plot
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formula look like this:
The European CIP conversion is much more precise than the US SAAMI conversion. If
you eliminate the statistically peculiar 280 FI NE, 310 Cadet Rifle, 38-40 Win, 44-40
Win, 7x50 R, 7x75 R SE vH, 8mm Rem Mag, and the 32 Rem, all other conversions
from CUP to PSI are within about 850 PSI. The precision of the conversion, and the fact
that the same exact values pop up again and again in the residuals indicates that the
Europeans have probably actually just been using one system, and converting by linear
formula to produce the second set of numbers.
Conclusions
1. PSI (correct use) is highly correlated to CUP. Evidence: R^2 = .927 makes it
impossible to successfully argue otherwise.
2. CUP is mainly an indicator of peak chamber pressure: Evidence: The way that
piezoelectric systems are commonly used, they report purely peak chamber pressure.
The CUP system is highly correlated with the piezoelectric system. If the “off-peak”
deformation of the copper pellet were large, the correlation to the piezoelectric
system would be poor.
3. SAAMI did a pretty consistent job of setting maximum pressure limits in both
systems. Evidence: The two are highly correlated. Basically, they got pretty close to
the same answer both ways.
4. You can convert from one system of measurement to the other. Evidence:
Definition of "correlated". Basically, correlated means that you can estimate one
variable from the other. The opposite of this is "statistically independent", which
means that you can't.
5. The formula for the conversion is the one shown above. Evidence: Produces the
"least squares fit" for the two systems, and it produces an R2 of .927. You can test the
formula by plugging in any of the CUP numbers shown above. The formula will give
you back a PSI number that is close to the one shown in the table.
6. Work remains to be done in refining the SAAMI conversion. Evidence: An R2 of
92.7% is produced, leaving 7.3% of the variation to be explained. Measurement
system error probably sets the limit of the R2 that can be obtained at around 98%.
That leaves 5.3% of the variation unexplained. Perhaps someone can discover what
the unaccounted for variable is.
7. The first example of something disproves all claims that it does not exist. The
formula exists, and it works. So all claims that it does not exist cannot be true.
