I reject your incomplete saami plot. I have disagreed with your opinions from the start of this thread. nothing personal, but I always try to adhere to the scientific method.
luck,
murf
Converting CUP to PSI
- Thread starter denton
- Start date
- Status
luck,
murf
mcb
Member
You saw a curve in the data and assumed it was created by a squared term, why not a X^3 or X^4? or a combination of multiple terms, a higher order polynomial. Again you're just guessing at the shape in the data and using a least squares to fit is as well as it can be fit. But your just guessing not basing the equation your fitting on a principal of physics.
If I showed you velocity vs drag force data you would know to guess a quadratic function to fit to the data. If I showed you data for the power of a radio signal vs range from source you would use an inverse squared function to fit a curve to the data. If I showed you alpha particle rate over time from a radioactive sample you would know to fit an exponential curve to the data. What is the physical principal that ties these two very different measurements , Crusher method vs Transducer method, together?
I think you are also over estimating the error and its impact on the SAAMI published numbers. When SAAMI did the work to create these publish MAP specifications it was not base on one test of 10 shots it was establish by taking the average over hundreds maybe thousands of rounds fired, from multiple controlled batches of ammo, fired though many different test barrels. They had enough data to statistically eliminate or identify any bias due to a bad sensor, bad test barrel, bad ammo or bad technician and enough data to get statically meaningful averages and standard deviations. There are controls on how much ammunition can vary, for production, control lots, and proof lots. This can be seen in the other data that is provided with the MAP service pressure the MPLM and MPSM along with other data on the proof pressure charts.
Your paper talks about estimation (which is an accurate way to describe your formula) but somehow that got lost in this thread and you are mostly talking about a "conversion" here. Conversion is a very different thing from estimation, if not always in terms of practical significance, at least in concept.
There are problems with your estimation formula that are common problems with simple fits.
When the spread on the data is roughly the same across the entire range of the data, the magnitude of the variation can approach the size of the actual values on the low end even if it is relatively small compared to the values at the top end of the range.
So the "noise" which doesn't really affect the overall correlation much, and is small (relative to the values) at the top end of the range, is actually quite large (relative to the values) at the low end of the range. In post 9, look at the values near 17KCUP. They range from about 17KPSI to 25KPSI. That's a spread of 8KPSI when the average value is only about 21KPSI. At the top end of the scale, 8KPSI is only about 12% of the values, but at that point on the scale it's getting close to 40% of the values.
In addition, the sparseness of the data in the middle of the plot is somewhat problematic. This makes it easier to fit with a good correlation to a mostly linear fit because the data "bunches" at the ends of the line strengthen the fit but there are fewer errors in the middle of the plot to weaken the correlation numbers.
It's not hard to set up a roughly linear example with a uniform error spread and starting underlying value that are equal but that still provides R^2 values of around 0.96 as long as the data is sparse in the middle of the plot. Obviously very good correlation, but the estimates will give values at the low end of the line that have a potential error spread equal to the underlying values they're supposed to be estimating.
Correlation does provide useful information, but it's not the whole story by a long shot.
SAAMI does not state "there is no relationship between CUP and PSI". They state that there is no direct conversion formula between the two. You have not proven otherwise even if you have come up with simple estimation algorithm that satisfies at least one person.
There's a similar strawman in your paper where you state that people argue that there is no correlation between the two data sets. I don't see many claims that correlation doesn't exist. However, the existence of strong correlation doesn't guarantee that a conversion formula exists, it only indicates that an estimator can be devised that provides values that are much better than random chance.
buck460XVR
Member
- Joined
- Feb 6, 2007
- Messages
- 10,049
CUP and PSI are both measurements of pressure. Yes, we all know that they are different methods of measuring pressure, but as I said before, the pressure created does not change just because you change the method or the measurement unit. CUP, PSI, rally doesn't matter....... the ammo is creating the exact same pressure within the confines of the platform. You can measure temperature by more than half a dozen ways other than the old glass thermometers most of us think of, but the real temperature does not change because of it. This was my point. That and trying to keep it simple. One thing that always amazes me whenever folks start a thread about reloading. Folks always want to make it a very technical and confusing thing. I think most of the time folks do it in an attempt to impress us with their knowledge. Thing is.....for the large majority of folks that reload, it is necessary trivia. Kinda like knowing how our computers work. How many here can build or repair their computer, much less describe how each and every component works....yet here we are, using them with proficiency and communicating on the internet. How many have written the software that we use everyday, yet we have no problem using that software. When I was 12 years old, I bought my first Lee Load-All for 16 ga. shotgun shells. Wasn't a brain surgeon or rocket scientist, but still turned out decent ammo for Grouse, Pheasants and Ducks. All I had to do was follow the directions and the tested and proven recipes given to me by reputable sources. Exactly what most of us do nowadays. Follow the directions, while keeping it simple.
Jim Watson sums it up the way I tried to in my first post.
...as long as we rely on folks with expertise in the field and go by their direction and parameters, we do not really need to know how they get that info. While it may please us to know how, and it may interest us, it is not really needed.
denton
Member
There is more truth than poetry in that statement. If you have well qualified load data, and all you're interested in is getting loads that work, then understanding the connection between the two systems of measure isn't important. If you're trying to relate P O Ackley's measurements to modern measurements, or if you're simply curious, then you may want to understand the connection.
The following statements are equivalent. When you say one, you have said the others:
1. Variable X is correlated with variable Y.
2. The value of Y depends at least partially with the value of X.
3. There is a curve that makes a better than random prediction of Y, given X.
4. There is an equation or formula that at least approximately connects X and Y.
5. You can convert from X to Y. The conversion may not be perfect, but it is a better answer than you can get by taking a simple average.
The hierarchy of the quality of information is:
1. Opinion or appeal to authority is the weakest quality of information.
2. Logic, including mathematics, is the next strongest quality of information.
3. The actual observed data is the strongest quality of information. Logic bows to data. Opinion or authority bows to logic. The data bow only the the quality of the measurement system that collected it.
SAAMI's statement that there is no conversion is equivalent to saying there is no correlation. It is an authority statement (and they did use the term correlation elsewhere). The scatterplot is data. The data and the authority statement cannot be reconciled. Data rules over authority, so the data win.
Now... how do we know we shouldn't use a higher order polynomial? The short answer is that we don't. But it doesn't matter. R^2 = .96 and P = 0 is an extremely strong result, so much so that any improvement you can make on it is going to be tiny and not worth the effort.
There is no requirement that the data be evenly distributed over the domain.
Did SAAMI indeed fire thousands of rounds to make the transition from CUP to PSI? I'd be interested in learning more about what they actually did. If the measurement errors are random, then increasing the number of measurements does indeed improve precision.
mcb
Member
Technically neither CUP nor SAAMI transducer methods are direct measures of gas pressure. In both cases the pressure act on an intermediate mechanism and the pressure is inferred from the resulting deformation or force measurement. Only the CIP transducer method directly measures the gas pressure by direct impingment of gases on a sense element through a perforated cartridge.
I think this is a large part of why there is no mathematical conversion. CUP was assumed to be and accurate pressure measurement until better methods came along and showed that CUP does not capture all the data, or obfuscate that data in the integral nature of the method. The fact that a lower pressure peak that has a longer duration can result in the same CUP measurement as a higher pressure peak that has a shorter duration means data is being lost/obfuscated by the method. Basically the CUP methods has a bandwidth limit (in an analog sense) that is being exceeded thus loosing data.
On a tangent and inspite of my above criticism of the CUP method I wonder if any one has tried a multidimensional regression to improve the attempted conversion?
Every one does a straight one dimensional function. A low order polynomial fit directly from CUP to PSI. What if you extended it to also include other parameters that might be influencing the data. Account for bore diameter, case volume and maybe even case diameter at the point the CUP piston or conformal transducer contacts the case. Do a multi dimensional regression and see if that fits the data better than current one dimensional model. Would be amusing to try.
Last edited:
That's the crux of the matter. CUP is not actually a direct measurement of pressure.
If the quoted statement were true, then the rest of your post would be accurate.
That is just absolutely not true. The presence of correlation does not contradict a statement that there is no conversion. Nor does the statement that there is no conversion imply there is no correlation.
Correlation implies that it is possible to estimate a value with better than random chance, as your estimation formula estimates the value of a PSI number from a CUP number with better accuracy than just random chance. That is not the same thing as a conversion which is a one-to-one mapping of one type of unit to another as we can do with F, C and K in temperature or inches and cm in distance.
1. Conversion (making a units change) is not the same thing as correlation (demonstration of a statistical relationship). It is certainly possible for very strong correlation to exist in data sets when there is absolutely zero possibility of creating a conversion between the two. If you really don't understand the difference, that would be more than adequate proof that anything you say about conversions and correlation should be treated as suspect.
2. SAAMI does not state that there is no correlation between CUP and PSI.
These two facts, taken together, mean that the (obvious) existence of correlation between CUP and PSI does not contradict SAAMI's claims.
I had the opportunity to discuss this topic with Dr. Oehler awhile back and at one point in the past, he did a lot of work related to the differences between CUP and PSI. He did devise a method to calculate a CUP value from the entire pressure curve. In other words, it is possible to look at an entire pressure curve (an adequately sampled function of time) measured in PSI and to generate a single CUP number from that data set. It's not possible to convert from a single PSI peak number to a CUP number nor is it possible to run the calculation backwards and generate an entire pressure curve from a single CUP value. That may be intuitively obvious to some, but for those who don't see it, Oehler demonstrated it. He's written on the topic and, based on my experience, is more than willing to discuss it if you talk to him in person.
Last edited:
Jim Watson
Member
Not the way it went.
For some decades the crusher rig was the only method. The crusher blanks were calibrated by dead load or hydraulics, ammunition readings were given in psi. See Sharpe, Hatcher, and Whelen for examples.
The term CUP was only introduced after the piezoelectric transducer came into use and it was seen that the two devices gave different numbers and therefore needed different units.
The US Army is not a SAAMI member and persisted in showing crusher readings as psi for some time, which convinced the Internet Generation that .308 WCF was loaded 20% "hotter" than 7.62 NATO... with no gain in velocity.
Then consider the pre-metrication British system with an axial gauge, crushers in the bolt head, results in long tons per square inch. Gough Thomas correlated US and British data for shotguns and concluded the British proof ton was about 2800 psi (LUP in current terms.)
denton
Member
Jim, I need some of what you've got. Nobody paid much attention when I pointed out that the term CUP only came into existence when more modern transducers, actually calibrated in PSI, came into use. According to my sources, your explanation is correct. Until we had modern equipment, it was assumed that CUP actually measured PSI. And it does until you get up around 40K PSI or so. Above that, it falls short.
JohnKsa, the dime finally dropped and I think I understand your angst. Maybe I can resolve this.
The general form is Y = f(x1, x2,...xn) + noise. You're requiring the noise term be zero, and that is not a requirement for correlation or conversion. Sometimes it does happen, such as when converting degrees F to C, but it is often not the case.
To illustrate, here's some data from an 8mm large ring Mauser.
For that particular bullet and rifle, I can convert back and forth between PSI and muzzle velocity. Tell me the PSI, and I'll give you the MV, and vice versa.
The conversion contains noise with a standard deviation of 18.8 FPS, so the conversion is less than perfect, but it is a conversion nevertheless.
Another example, possibly better: We convert back and forth between mass and weight without even thinking about it. Balance scales measure mass, and will give you the same answer whether you are on the moon or on Earth. But we typically mark them directly in grains, which is a unit of weight. That works, as long as you're in a place where the acceleration of gravity is normal. Weight = mass * gravity. But the acceleration of gravity is not constant over the face of the Earth. You can find points that are .5% apart. So the conversion of mass to weight has a noise term, and is less than perfect. But it is good enough for most practical purposes.
It's comforting to know that if you need to reload on the moon, you can if you bring a balance scale.
Edited to add: CUP was a fairly successful attempt to measure PSI. Yes, the pellet is deformed during the whole time the chamber is pressurized, but the off-peak deformation does not amount to much. If you run a regression of area under the curve vs. peak pressure, you find a very strong correlation. Most of the area under the curve happens near the peak pressure. Trust me, I've run the calculations. The off-peak deformation is a minor problem.
Last edited:
It most certainly is not a requirement for correlation. In fact, if the noise term is zero, it doesn't make much sense to talk about correlation because there's no need to talk about correlation, you can just talk about converting one to the other using a straight formula. No need for discussing noise or estimation.
But a conversion is simply changing between units. There isn't any noise involved because the actual values are the same, the only thing that's changing is the units. Changing from F to C, for example, involves no noise because the only change is the numbers that represent them.
CUP to PSI is not a direct conversion because the CUP number does not actually represent the same value as the PSI figure. It's not directly measuring pressure, it's measuring the deformation of a copper pellet as a function of the entire pressure vs. time curve.
It's not merely a conversion--a change of units--it's actually measuring something different.
That's nonsense. I can see from the data that it is quite simple to calculate/estimate PSI from muzzle velocity and vice versa, but saying that is a "conversion" makes no sense at all. Muzzle velocity is clearly mostly a function of PSI for a given firearm, but the idea of "converting" from PSI to fps is nonsensical. You might as well talk about converting miles to pounds or mass to lumens.
If you're going to redefine "conversion" to mean any process involving calculating or estimating one value from another (e.g. "Converting" mass and velocity to kinetic energy using the formula for kinetic energy, or "converting" the amount of time a person can stand on one foot with their eyes closed to age.) then you need to state your nonstandard definition up front. And, if you're going to use nonstandard definitions, you need to be very careful to avoid using them to indict/contradict other people/entities who are not using your special definition.
By normal definitions:
1. Conversion (making a units change) is not the same thing as correlation.
2. SAAMI does not state that there is no correlation between CUP and PSI.
higgite
Member
- Joined
- Dec 17, 2009
- Messages
- 2,080
Seems to me if there was a conversion from transducer measured chamber pressure units (PSI) to copper crusher measured chamber pressure units (CUP), SAAMI certainly has the resources, justification and expertise to figure it out, but they haven’t. Instead, they say, "the pressure values determined by one method cannot be mathematically converted to values for another, despite claims to the contrary." And even if such a conversion did exist and SAAMI was so incompetent as to not be aware of it, I certainly wouldn't use their data to derive it. Just MHO.
denton
Member
Well, it's been fun and it's been civil, but I guess we are not going to agree.
I believe this is the first time you've added the qualifier limiting conversion to a units change. That's a conversion, but certainly a limited subset of the universe of conversions. All sorts of conversions are possible, including those used by NIST to establish base units of measure. The meter is defined in terms of time, and the ampere is defined in terms of force. Each of those conversions is from one type of measure to something quite different, and each contains uncertainty in the form of noise.
My favorite conversion is going to the range and converting money to smoke and noise.
I don't have a collection of every pronouncement that SAAMI has made, but they did state that there was no correlation between CUP and PSI, and they did use that word. It's not a true statement. Expecting any organization to be perfectly correct in everything everyone says is not realistic, so finding an error is no great surprise.
As I said, it's been civil, and I appreciate the conversation. Be at peace, my friend.
higgite
Member
- Joined
- Dec 17, 2009
- Messages
- 2,080
Please cite where SAAMI says there is no correlation between CUP and PSI. I looked but the closest I could find is where they said the 2 values can't be converted one to another. Perhaps I didn't search the right document. Thanks.
Mine as well.
And, while I can convert money to smoke and noise as well as the next guy, I can only correlate amount of time spent to amount of fun had. More = more. Jim Watson
Member
Somebody once got a pretty good correlation between women's hemlines - back when women wore skirts - and the status of the stock market.
What was the correlation coefficient for psi vs CUP? All calibers included, no disregarding .357 Magnum because we don't know why it looks funny or .45-70 for which the psi standard is the same as CUP, etc.
What was the correlation coefficient for psi vs CUP? All calibers included, no disregarding .357 Magnum because we don't know why it looks funny or .45-70 for which the psi standard is the same as CUP, etc.
Gentlemen, there is a correlation beween Peak Pressure and CUP.
To repeat, there is a correelation.
To ignore that while looking at the data is the epitome of ". . . there are none so blind as will not see."
Is the correlation perfect ? No, there is an error bar of uncertainty
Is the correlation perfectly causational ? No, one is Peak Pressure, the other is proportional to Pressure over time (work) in crushing a metal cylinder
But to look at the data . . . and say correlation isn't there . . . is akin to shrieking Earth-centered religious dogma in the age of Copernicus.
To repeat, there is a correelation.
To ignore that while looking at the data is the epitome of ". . . there are none so blind as will not see."
Is the correlation perfect ? No, there is an error bar of uncertainty
Is the correlation perfectly causational ? No, one is Peak Pressure, the other is proportional to Pressure over time (work) in crushing a metal cylinder
But to look at the data . . . and say correlation isn't there . . . is akin to shrieking Earth-centered religious dogma in the age of Copernicus.
Last edited:
denton
Member
Putting the 357 back in the mix, R^2 is .945, and the correlation coefficient is the square root of that, .972.
As a mostly-outside observer to this learned discussion, I wonder if most of the participants would agree with the following:
- Pressure in internal ballistics has variability over time. As the whole ignition process progresses, pressure rises and falls (and sometimes rises and falls more than once).
- That pressure variation can be graphed as a curve.
- In terms of projectile velocity, the total area under the curve is the input that drives projectile velocity (with friction and other factors acting as drags).
- In terms of catastrophic firearm failure, however, the primary risk is not excessive area under the whole curve - it is peak pressure exceeding the yield strength (I think that's the right metallurgical term - I could have it wrong, though) of some part of the "pressure vessel" that is the barrel and chamber and breach face. As long as the forces exerted at the peak don't exceed the yield strength of the steel, the elasticity of the steel will cause it to return to shape from whatever stretch occurs during firing and the gun will be undamaged.
- If the peak exceeds the yield strength, however, then the steel will not rebound and will be damaged permanently. If it exceeds it by enough, the steel will stretch enough to rupture, and you get a kaboom.
- CUP does not directly measure peak pressure - it measures some integral of some of the area under the curve. It may measure only portions of the curve above a certain pressure level, but it is measuring more than a transient peak.
- A transducer is measuring many, many points along the pressure curve - basically as fast as whatever the "refresh rate" of the transducer is. As such, it can measure peak pressure - or something very, very close to peak pressure - directly. Each data point is probably its own integral of pressure over time, but the time snapshots are so small that, for most purposes (and certainly compared to CUP), they can be considered instantaneous.
- If one compares the peak pressure measured by a transducer to a CUP measure, the relationship between them will vary by the shape of the curve. A curve with a steeper spike/peak will show a different mathematical relationship to CUP than a curve with a shallower curve of the same peak pressure.
denton
Member
My initial work on this dates back 17 years. That was two hard drives ago! What we have at this point is my memory, the Mahin quote that Reloadron posted, and a similar statement from Lyman's #48, p97. Mahin was a technician at Sierra, and wrote articles for them. Lyman #48 is their statement, not SAAMI's. So it's an imperfect answer, but all I have at the moment.Please cite where SAAMI says there is no correlation between CUP and PSI. I looked but the closest I could find is where they said the 2 values can't be converted one to another. Perhaps I didn't search the right document. Thanks.
Mine as well.
denton
Member
Mainly agreed, but with some exceptions....
CUP does not directly measure peak pressure - it measures some integral of some of the area under the curve.
This may be strictly true, but not very relevant. The CUP measurement was designed as a pressure measurement, and is listed by SAAMI as a pressure measurement. Standards from other organizations also use the copper crusher to estimate peak pressure. The error caused by the off-peak deformation is small, and CUP is at least a decent measure of peak pressure, though not as good as modern methods.
If one compares the peak pressure measured by a transducer to a CUP measure, the relationship between them will vary by the shape of the curve. A curve with a steeper spike/peak will show a different mathematical relationship to CUP than a curve with a shallower curve of the same peak pressure.
This is widely assumed, but as far as I know, unsubstantiated. What is substantiated is that CUP estimates PSI pretty well at lower pressures, and then systematically underestimates it at higher pressures. If you look at SAAMI's data, pistol rounds with fast powders are assumed to give results consistent with rifle rounds with slow powders, so I don't think SAAMI believes it.
It's not "my" qualifier. When talking about a "direct conversion" (SAAMI"s wording), that's what is meant. Calculating values as a function of other values is not typically called a "direct conversion" unless it is converting units.
Good examples that highlight the difference between conversion and calculation.
The meter is defined in terms of distance. The number of wavelengths (distance) of a particular wavelength of light. It would be reasonable to talk about "converting" meters to 'wavelengths of Krypton-86 in a vacuum' because they are both measures of distance.
The ampere is defined in terms of a number of parameters, of which force is one. As such, it is possible to calculate current (amperes) from force (as well as from some other parameters) under controlled conditions but it makes no sense to talk about "converting" from force to current because they are two different quantities.
Most of that is true (as we can readily see from your data there can definitely be a large error in some cases), but it's not especially relevant.
- It is "decent" and was used for many years because nothing better was available.
- What it was designed to do is not relevant because that is not actually what it does.
- SAAMI continues to list it as a legacy value. A couple of notes. 1.) It's ironic that you use SAAMI for support in some cases while suggesting that they're incompetent when it suits your argument better to take that position. 2.) It's worthwhile to point out that the reason that SAAMI has to continue listing the obsolete CUP data for individual cartridges the values is because there isn't a direct conversion. If there were, they could just provide the conversion and there would be no need to continue carrying those obsolete values in their materials.
ABSOLUTELY and INCONTROVERTIBLY correct!
There is most definitely correlation between the values. The issue is that the claim has been made that if there is correlation, it must be possible to come up with a "direct conversion" in contradiction to SAAMI's (and others) claims. That claim is false. The presence of correlation does NOT imply that a "direct conversion" is possible unless one redefines "direct conversion" to mean something much more general than it's standard definition.
denton
Member
I value the comity and civil discourse we enjoy here, and think it's more valuable than resolving disagreements. It really doesn't bother me to have someone respectfully disagree with me, as you have. But I am going to take one more swing at resolving our difference. Maybe it will make a difference somehow.
There was an older, simpler time when the seven standard units of measure were artifacts. The standard kilogram was an object, platinum IIRC, that had a mass of a kilogram. A meter was the distance between two scratches on an iridium bar. That is no longer the case. The last of the old artifacts to go was the kilogram, though I have no idea what replaced it. All NIST basic standard quantities are now defined by conversions.
The meter is no longer defined by the iridium bar or as so many wavelengths of light. It too is now a conversion. The conversion is expressed in terms of c, the speed of light, and time. That is, a constant is multiplied by time to get distance. Time and distance are two very different things, yet there is a conversion.
Consider what happens when we measure peak pressure with a piezoelectric transducer: The surface area of a steel rod converts pressure to force. The mechanical properties of the crystal convert force (or more properly stress) to strain. The piezoelectric properties of the crystal convert strain to surface charge. The dielectric properties of the crystal convert surface charge to voltage. And thus we get volts as an output that we can display on an oscilloscope.
If I'm understanding your claim properly, none of those is actually a conversion because the input variable is not the same type of quantity as the output variable. Pressure is not force, force is not strain, strain is not surface charge, and surface charge is not voltage.
Don't you think that maybe, just possibly, conversion is more general than you take it to be? If not, then what is your word other than conversion for the chain of events in a piezoelectric pressure measurement device?
mcb
Member
Step away for a day and half and look at all that I missed. Since my last post in the thread I went to a Cheer competition for my daughter, got rained out trying to hunt some Coyotes and since the hunting got rained out I did my taxes. None-the-less... and hopefully not to far where the thread has gone...
In a post a bit earlier I talked about having some physical basis for the equation you select when you are trying to fit a function to a data set, Such as in this case were we are trying to create a conversion from one measurement to another that are clearly derived from the same physical input but are not the same thing. I am putting forward that you can't just pick a random equation simple because it gives you a good R^2 value and assume it is the correct equation for the data. Ideally you have some physics based principles to help guide the selection of the equations to be fitted to the data. ie my examples early like knowing that the signal strength fades with the inverse square law and thus trying to fit another equation to the data is going to lead to a bad function, even if you end up with a decent R^2.
In an attempt to illustrate that, I started playing in Excel doing some trendline fits using the internal functions in the graph function. Then I started manually doing the fits and manually calculating the R^2 in the spreadsheet to try to refresh my mind about where that all comes from. I have never had a formal statics class just what I picked up in various engineering classes along the way and a two or three day class on Minitab a few years ago. So after some playing I had a system in Excel that would let me create a custom equation and use Excel's internal Solver to implement a least squares regression to solve for the equation's coefficient and the resulting R^2. Next I created a data set for my example. It is completely made up from a simple functions meant to represent time on the x-axis and the y-axis represents some 1-dimensional displacement. No units required just a simple 1-dimensional displacement over time function. I then added a bit of noise to the data set, to simulate a sensor. The noise is random caped at +/- 2% of full scale, in this case a random number between .46 and -.46 since ful scale was ~23. (units and scale are arbitrary) This made up function is sampled at 10 hz and then each sample has a random amount of noise added to make a noisy data set.
The first thing I did to the data set was throw it into a graph and let Excel add a simple linear fit (Click the Spoiler tag for the graph). Excel fit the equation y=a*x+b using its internal regression to solve for a and b. The equation and R^2 values are on the graph and it looks like a pretty good fit. There is definitely a linear correlation between time and displacement and the equation looks like it describes that pretty well bases on the R^2. But the calculated error between the linear equation and the actual data has some pretty bad local error especially at the low end. The average absolute error is 7.2% with an Min/Max of -26% to +58%. That seem considerably higher than we would expect from our +/-2% sensor/noise.
But now a little bit more information about the system. The position sensor is actually measuring the position of one mass of a mass-spring-mass system that is freely sliding down our 1-dimensional world with no gravity. (we are ignoring friction etc for simplicity's sake). With this addition insight into the physics involved, a better equation can be selected. Clearly there is a linear aspect to the motion (supported by the initial linear regression) but since there is a mass-spring-mass aspect to the system a harmonic component to the motion is likely. This lead to the creation of an equation based on what is know about the physics of the system. Something like this. y = a*x + b + c*sin(d*x+e). I used my least square setu and the internal solver to find the values for a, b, c, d, e. and plotted that. Click the spoiler
Now that looks much better. Having some physical intuition into what was creating the data helped create a function that fit the data much much better even though the R^2 value is only marginally better. The error in this new equation results in an average absolute error of only 2.3% and a min/max of -11% and +10%.
So coming back to the original topic. The above long winded illustration is a very simple case of what I think we have with the CUP to PSI conversion. There is a lot of really complex physics going in both measurement systems to arrive at a single number. They are both clearly correlate as you would expect since they are both derived from the same physical input (ie the the time variant chamber pressure) but there is enough physics/mechanics going on between the gas pressure event and the resulting CUP or transducer measurement (especially the CUP measurement) that writing a meaningful equation that can capture those physics is not trivial, if even possible, especially when you only use the two single number measurements and none of the other data available (bore diameter, case volume etc). The complexity of the measurement systems is why I talked about adding more inputs to the system to try to capture some more information that might help achieve a better fit to the conversion equation we want. As @JohnKSa talks about earlier that Mr Ohler used the entire pressure vs time curve from a transducer measurement and could arrive at the pretty accurate CUP prediction. That makes total sense you could basically simulate a CUP measurement based on the representation of the entire pressure curved captured by the transducer. A finite element model, assuming you have a good finite element model for plastic deformation of copper (not trivial but get easier and easier with advances in FEA software) could take the pressure vs time curve as input and replicate the CUP measurement in a 3D computer model. The converse is not true of the CUP measurement. Too much data is lost in the CUP measurement to take the CUP measurement and use it to create a simulation of what the transducer would see. All the temporal data is lost in the CUP measurement, its all mushed together (pun intended) into one value. It's not that the two measurement are not related or correlated its that the relations between those two measurement is so complex that a simple polynomial fit simple cannot capture the complexity of that relationship especially in light of how much information is lost in the CUP measurement.
-too much rambling, hope that helps more than it confuses. Hopefully there are not too many typos and bad grammar as my brain is fried on rain, taxes, and least square regressions.
**This was a really simple and slightly contrived example and even then I had to start with pretty good guesses at my values for a, b, c, d, e. With an equation this complex there were lots of local minimums in the solution manifold that would get my solver stuck and come up with some strange solutions. Occasionally it would basically turn off the harmonic part of the equation and end up with an equation very similar to the linear solution. And just for completeness the original equation used to create the clean data set was y = 2*x + 3+ 1*sin(2.1 * (2 * pi) + 135*(pi/180)). The (2 * pi) and the (pi/180) is simple converting the frequency and phase angle to radians that Excel's sin function requires.
In a post a bit earlier I talked about having some physical basis for the equation you select when you are trying to fit a function to a data set, Such as in this case were we are trying to create a conversion from one measurement to another that are clearly derived from the same physical input but are not the same thing. I am putting forward that you can't just pick a random equation simple because it gives you a good R^2 value and assume it is the correct equation for the data. Ideally you have some physics based principles to help guide the selection of the equations to be fitted to the data. ie my examples early like knowing that the signal strength fades with the inverse square law and thus trying to fit another equation to the data is going to lead to a bad function, even if you end up with a decent R^2.
In an attempt to illustrate that, I started playing in Excel doing some trendline fits using the internal functions in the graph function. Then I started manually doing the fits and manually calculating the R^2 in the spreadsheet to try to refresh my mind about where that all comes from. I have never had a formal statics class just what I picked up in various engineering classes along the way and a two or three day class on Minitab a few years ago. So after some playing I had a system in Excel that would let me create a custom equation and use Excel's internal Solver to implement a least squares regression to solve for the equation's coefficient and the resulting R^2. Next I created a data set for my example. It is completely made up from a simple functions meant to represent time on the x-axis and the y-axis represents some 1-dimensional displacement. No units required just a simple 1-dimensional displacement over time function. I then added a bit of noise to the data set, to simulate a sensor. The noise is random caped at +/- 2% of full scale, in this case a random number between .46 and -.46 since ful scale was ~23. (units and scale are arbitrary) This made up function is sampled at 10 hz and then each sample has a random amount of noise added to make a noisy data set.
The first thing I did to the data set was throw it into a graph and let Excel add a simple linear fit (Click the Spoiler tag for the graph). Excel fit the equation y=a*x+b using its internal regression to solve for a and b. The equation and R^2 values are on the graph and it looks like a pretty good fit. There is definitely a linear correlation between time and displacement and the equation looks like it describes that pretty well bases on the R^2. But the calculated error between the linear equation and the actual data has some pretty bad local error especially at the low end. The average absolute error is 7.2% with an Min/Max of -26% to +58%. That seem considerably higher than we would expect from our +/-2% sensor/noise.
But now a little bit more information about the system. The position sensor is actually measuring the position of one mass of a mass-spring-mass system that is freely sliding down our 1-dimensional world with no gravity. (we are ignoring friction etc for simplicity's sake). With this addition insight into the physics involved, a better equation can be selected. Clearly there is a linear aspect to the motion (supported by the initial linear regression) but since there is a mass-spring-mass aspect to the system a harmonic component to the motion is likely. This lead to the creation of an equation based on what is know about the physics of the system. Something like this. y = a*x + b + c*sin(d*x+e). I used my least square setu and the internal solver to find the values for a, b, c, d, e. and plotted that. Click the spoiler
Now that looks much better. Having some physical intuition into what was creating the data helped create a function that fit the data much much better even though the R^2 value is only marginally better. The error in this new equation results in an average absolute error of only 2.3% and a min/max of -11% and +10%.
So coming back to the original topic. The above long winded illustration is a very simple case of what I think we have with the CUP to PSI conversion. There is a lot of really complex physics going in both measurement systems to arrive at a single number. They are both clearly correlate as you would expect since they are both derived from the same physical input (ie the the time variant chamber pressure) but there is enough physics/mechanics going on between the gas pressure event and the resulting CUP or transducer measurement (especially the CUP measurement) that writing a meaningful equation that can capture those physics is not trivial, if even possible, especially when you only use the two single number measurements and none of the other data available (bore diameter, case volume etc). The complexity of the measurement systems is why I talked about adding more inputs to the system to try to capture some more information that might help achieve a better fit to the conversion equation we want. As @JohnKSa talks about earlier that Mr Ohler used the entire pressure vs time curve from a transducer measurement and could arrive at the pretty accurate CUP prediction. That makes total sense you could basically simulate a CUP measurement based on the representation of the entire pressure curved captured by the transducer. A finite element model, assuming you have a good finite element model for plastic deformation of copper (not trivial but get easier and easier with advances in FEA software) could take the pressure vs time curve as input and replicate the CUP measurement in a 3D computer model. The converse is not true of the CUP measurement. Too much data is lost in the CUP measurement to take the CUP measurement and use it to create a simulation of what the transducer would see. All the temporal data is lost in the CUP measurement, its all mushed together (pun intended) into one value. It's not that the two measurement are not related or correlated its that the relations between those two measurement is so complex that a simple polynomial fit simple cannot capture the complexity of that relationship especially in light of how much information is lost in the CUP measurement.
-too much rambling, hope that helps more than it confuses. Hopefully there are not too many typos and bad grammar as my brain is fried on rain, taxes, and least square regressions.
**This was a really simple and slightly contrived example and even then I had to start with pretty good guesses at my values for a, b, c, d, e. With an equation this complex there were lots of local minimums in the solution manifold that would get my solver stuck and come up with some strange solutions. Occasionally it would basically turn off the harmonic part of the equation and end up with an equation very similar to the linear solution. And just for completeness the original equation used to create the clean data set was y = 2*x + 3+ 1*sin(2.1 * (2 * pi) + 135*(pi/180)). The (2 * pi) and the (pi/180) is simple converting the frequency and phase angle to radians that Excel's sin function requires.
denton
Member
Your nerd gene is showing!
Great fun, isn't it?The following might possibly be useful to you, as you explore:
A pure statistician doesn't care about causality. His only task is to explain the data. A high R^2, a low P value and small residuals, and he's done for the day. If the model holds over some region of interest, that is all that is required.
The engineer/scientist immediately wants to know what the underlying mechanism is. The power of that is that it allows a model that works (or at least can be tested) outside the region that has been explored.
It's two entirely different mindsets. The good news is that they work very well together, each feeding off the other.
As a matter of conjecture, I suppose that CUP might be underestimating the higher pressures because the deformation of the copper pellet work hardens it. At lower pressures, CUP tends to overestimate pressure a little. At higher pressures, it underestimates significantly. Work hardening would fit that model.
Muzzle velocity is proportional to the area under the pressure curve. You have friction and engraving force to account for, but those are constants for a given rifle. Energy lost to creating angular momentum is a fraction of a percent, and that can be treated as a small constant as well. Muzzle velocity is HIGHLY correlated with peak pressure. This is because most of the acceleration happens near the peak. (Using Bullseye in a 30-06 is not a case that I've tested, though). So the argument that CUP is a measure of area under the curve might be true, but not important. Even if it truly measures area under the curve, it's still close enough to indicating peak pressure.
If you're trying to fit a model, the simplest model that gives adequate correlation is the one to be preferred.
Of course, you can substitute x' for x, and do all manner of logs, exponents, etc. But two things limit your ability to make good models:
1. As you get beyond squared terms, a little error in your estimate of the coefficient can make a big error in the model.
2. You do polynomials by letting x1 be x, x2 be x^2, and x3 be x^3, and so on. Then you do multiple linear regression. The problem there is that as you add more terms to multiple linear regression, the model gets more and more iffy.
You may get a kick out of 2^K factorial experiments. That's a extremely strong framework that allows you to explore multiple input variables with great confidence. It will also easily let you discover interactions, which are often important.
- Status
