Ballistics Coefficients Discussion

[rsterne]

Distance is not necessary, and as you say, that curve would have to be smoothed as well.... Therefore I plotted the LabRadar velocity datapoints vs. time (as you say, the increments are conveniently constant), and used Excel to produce the trendline equation shown on the graph.... I then used that equation to produce the smoothed velocities for each time increment, which I used to compute the rest.... I converted everything to Metric units for simplicity, and the steps I used are as follows:

Convert the temperature, humidity and pressure to air density D in kg/m^3 using this calculator.... http://wahiduddin.net/calc/calc_da_rh.htm ....
Calculate the speed of sound in m/s for the test temperature (in *C) using Vs = 331.3 x SQRT((C+273.15)/273.15)....
ICAO standard atmosphere values (15*C, 0%H, 29.92"Hg), for reference, give D = 1.225 kg/m^3, and Vs = 340.3 m/s....
Convert the smoothed velocities to m/s by multiplying the fps by 0.3048....
Convert the pellet weight in grains to mass M in kg by multiplying by 0.0000648....
Calculate the area A of the pellet and convert to m^2 by using (cal.^2)/4 x PI x 0.000645....

Compute the incremental acceleration a = (dV/dT) for each smoothed velocity by taking one velocity (in m/s), subtracting the previous, and dividing the difference by the time increment (0.002 sec. except for the first increment).... result in m/s^2....

Use F=Ma to compute the drag force F in Newtons for each increment....

Determine the average velocity of each increment by adding the start and end (smoothed) velocities together and dividing the sum by 2, this is used for V in the Cd calculation below....

Calculate the Cd for each increment using the following formula....

Cd = 2 x F / (D x A x V^2)

I checked the units, and they all cancel, leaving the Cd as a dimensionless number the way it should be.... If you see something wrong in my approach, please point it out.... Incidently, I have two more columns, one to compute the local Mach number for each Cd calculation by dividing the average increment velocity V used by Vs for the test conditions, and then another one to convert that back to fps corrected to ICAO for the graph by multiplying the Mach # by 1116.45 fps.... The "X" axis on the graph above is fps in ICAO conditions, not the fps as recorded by the LabRadar....

Bob

[rgb1]

I'm not accustomed to having time as a variable so I did not use it. The "two chrony"
approach, my normal mode of operation, uses (dV/dx)(dx/dt) and therefore an entirely
different equation. What you have outlined is textbook perfect, yet the answers are
incorrect. Obviously, as the saying goes,..........the devil is in the details. I have cross
checked your method with mine and get complete agreement. As an example, using
data posted on June 1 for 34 gr......... at ~ 770 f/s both will give a Cd value of .303
and a drag force of .0645 lb (based on an air density of .0677 lb/cu ft). Does this
help?

                                                                                      Ron

[rsterne]

Well, we are obviously getting completely different results, nice to know mine is wrong.... but better if I knew why....
CORRECTIONS IN RED, I USED THE WRONG PELLET WEIGHT.... DUH!....

From the following thread on the Yellow.... http://www.network54.com/Forum/79537/message/1433166096/A+sample+one+shot+data+ToF%2C+Velocity%2C+range+to+65+yd%2C+JSB+Heavy+Kings+34+gr
For the LabRadar velocity increment from 770.18 fps down to 769.31 fps, ie from 0.035021 to 0.037021 sec. and 9.07 to 9.58 yards, my numbers are as follows,
Specified conditions, 13*C, 58%H, 890hPa.... which gives me an air density of 1.080 kg/m^3 = 0.0674 lb/ft^3.... You used 0.0677, close enough....
Smoothed velocities.... 769.91 to 769.07 fps.... average velocity 234.541 m/s
Time delta = 0.002000 sec.
Acceleration is (769.911-769.071) / 0.002 = 0.84 / 0.002 = 420.0 f/s^2, my spreadsheet uses 0.25618 / 0.002 = 128.09 m/s^2
Mass is 25.4 gr. = 0.001641 kg.... If I did the conversion correctly, you should have 0.0001128 slugs....
Mass is 34.0 gr. = 0.002203 kg.... which is 0.0001510 slugs....
Diameter is 0.250", so the area is (.25)^2 / 4 x PI= 0.04908 in^2, which is 0.0000317 m^2
Drag force F = Ma = 128.09 x 0.001641 = 0.2108 N.... That converts to 0.0474 lbf, yet you have 0.0645 lbf.... so there we have completely different values, the obvious source of our difference....
Drag force F = Ma = 128.09 x 0.002203 = 0.2822 N.... which is 0.0634 lbf.... slight variation from your 0.0645 lbf, likely because of different smoothing equations....
My calculation for Cd is as follows....

Cd = 2 x F / (D x A x V^2)
= 2 (0.2108) (0.2822) / (1.080 x 0.0000317 x (234.541)^2)
= 0.4216 0.5644 / (0.0000342 x 55009)
= 0.4216 0.5644 / 1.881
= 0.2241 0.3001

I would appreciate it if you could do a similar detailed description of your calculations so that I can see where my error is.... It would appear to be in the calculation of the drag force....

Bob

[rgb1]

Perhaps this will explain it.........

[rsterne]

*LOL*, YA THINK ? ? ?

Classic case of GIGO.... Here is the correct graph for the 34 gr. JSB King Heavy....



Thanks for picking that up, Ron.... sometimes we overlook the obvious.... 

I'm curious, did you use a linear trendline, as I notice our Cd value at 770 fps (773 fps ICAO) is different by about 1%.... I have 0.300 and you got 0.303.... I would love to see a plot of your Cd vs Velocity results....

Bob

[rgb1]

Third order polynomial will offer a better fit, have
even used fifth order at times.

[rsterne]

Really interesting our Cd curves are opposite, yours low at ~750 fps, and mine high at ~725 fps.... Yours also has a lot larger range, (4.2%), compared with mine at only 1.5% (if I drop the first reading with the large delta T).... I assume we are both using the same LabRadar data from the thread I linked above, so I have no explanation why we have different smoothing curves.... I tried a 3rd order polynomial, and there was basically no difference.... (note, see below)....

Bob

[rsterne]

Well I found a small problem in that I forgot to revert the datapoints from which I took the trendlines to the original raw data.... That makes hardly any difference in a 2nd order Polynomial, but it changes the 3rd order Polynomial to closely resemble yours, Ron.... Here are all the Cd graphs, using from a linear trendline to a 5th order Polynomial, it makes a HUGE difference in which trendline you choose....







This leaves me with a slight problem about which trendline to choose, of course.... I hate it when it becomes more art than science.... Fortunately, logic tells us that the correct one is the one that is going through a very shallow minimum, which I'm sure is why you chose the 3rd order one, as will I.... It bears in mind that once we have all the data right up to Mach 1.2, the curve we are seeing here will be virtually a straight line compared to the huge increase in drag to come in the transonic region....

I am now confident that I have a spreadsheet that will provide a workable solution to analyzing the LabRadar data.... and I look forward to the day when I have one.... Thank you for your patience and allowing me to work through my spreadsheet until we are getting consistent results....

Bob

[rgb1]

I'm unable to produce graphs such as yours, something still isn't correct(?).
Second, third and fourth order should be nearly identical. Fifth and higher.....
not so much........the very thing you're trying to filter out begins creeping back
in. To make some sort of check, greatly expand the graph.......I blow it up to
at least several feet wide whereby it requires considerable scrolling to see
everything. Granted, it's a pain in the butt, but at least you know where you
stand. And, I guess it probably goes without saying, ........knowing what the
graph should look like ahead of time is indeed a big help.

                                                                                                Ron

[rsterne]

As you are aware, I have no control over how Excel calculates the trendline equations, only the number of terms.... As I understand it, as you increase the number of terms in the trendline equation, it follows the velocity fluctuations more and more closely.... That is why it has more peaks and valleys.... As you said, the raw velocity data wanders around a fair amount, with some values being higher than previous ones (ie the pellet is speeding up in that half yard increment), which we know is impossible, that is the instrument margin of error creeping in.... This is particularly noticeable the further downrange you go.... As I explained earlier, without smoothing the data, the calculated Cd varies over such a huge range to be useless, at times going negative.... Here is the last part of the data (Velocity vs Time) expanded as you suggest.... Blue dots are the raw data, red line the linear trendline, black line the third order trendline....



I think we are stretching the limits of the equipment, and our ability to analyze the data it is producing... The good news is that it won't matter in the long run, because once we look at the big picture, all the little lumps and bumps will disappear.... Once we move up to the transonic range, the pellet will be losing velocity so quickly that the instrument margin of error will be insignificant, or at least greatly reduced.... We will be measuring Cd up in the region of 1.0, if my previous testing is any indication, so when the vertical scale on the graph reflects that, the small swings we are currently seeing will all but disappear.... This is what this data will look like in the big picture, with the scales set to what will be necessary to cover the range of values I expect....



I am confident that we are on the right track, and look forward to being able to process more data (without having to type in and proof read every number)....

Bob

[Pellgunfun]

May I ask a really dumb question?

Is the Drag Coefficient 'inverse' of the Ballistic Coefficient???  So as the Drag Coefficient goes up the Ballistic Coefficient goes down.  Is that right?

I opened the Custom profile in Chairgun and while looking at the different profiles in the edit profile window I noticed the graph and was trying to make sense of it.

Have you guys concluded the the GA pellet profile in Chairgun Pro is wrong?  If so what should it be?  Or is that what this discussion hopes to answer once and for all?

[rsterne]

They move in the opposite direction, yes, but the BC is a comparison of the Cd for your bullet compared to the Cd for the "Standard" bullet.... If the drag profile of your bullet doesn't match the selected profile, then the BC will change at different velocities....

From the measurements I have made so far, I think the GA profile in ChairGun is too low in terms of Cd at virtually all velocities, in other words the pellets I have tested have more drag.... If you compare the GA profile to the widely used G1, you will note that they used the G1 for Mach 1 and above, but reduced the drag slightly below Mach 1.... While this reflects the larger increase in drag seen in the transonic region, it isn't an accurate representation of reality.... Our pellets have more drag than the G1 profile subsonic, and a LOT more drag supersonic....

Using the GA profile is probably better than using the G1 profile, but the BC you use will have to be lower to reflect your pellet behavior.... If you look back at Post 83 on Page 5 of this thread, you will see how steeply the drag increases in the transonic range compared to the G1 profile.... Note that it is slightly higher in the subsonic range, while the GA profile is even lower than the G1 there.... Therefore none of the profiles used are very good, and the purpose of this thread is to improve on that....

Bob

[rkr]

I'm digging up an old thread here but since the question is directly related to this data I'll give it a go. In case of that JSB Heavy King pellet, what would be the ideal speed in order to minimize the wind drift? Would it be 750 fps that's faster end at the bottom of the curve or would the increase in speed negate the increase in drag (and thus decrease in BC) making somewhere around 900 fps optimal (assuming the linear increase lasts to that point)?

[rsterne]

It will depend on the range you are shooting.... If at very close range, then about 750 fps would be optimum, but the further away the target, the faster you need to start, so that the AVERAGE drag is the least across the entire distance.... This is complicated further by the fact that the wind has more effect nearer the muzzle than the target, because once the pellet is deflected, it carries that new (angled) path downrange, with the offset distance from the line of sight increasing with distance.... If you plot muzzle velocity vs. drift, you will get a curve like this....



As you increase the distance, the optimum muzzle velocity increases.... That curve used the G1 Drag Model....

Bob

[rkr]

Thanks, Bob - I knew I had forgotten something - the decrease in velocity in this case. So for 100Y shooting with Heavies it would make sense to start in the 800-850 fps range so that the pellet will spend most of it's flight in the lowest drag area speedwise.

[rsterne]

Possibly even 900 fps or close for 100 yard work.... Only once the entire drag curve is determined will we have the exact answer to that question.... Like anything else that goes through a minimum, some distance either side of that carries little penalty, so even if the optimum is 880 fps, 900 or even 920 may be so close that the flatter trajectory is worth it.... eg. on the above graph, 850 and 900 have the same drift, so may as well choose 900.... and 950 is only 1/8" more.... At 1000 fps, the drift is 3/8" more, however....

Bob

[MicErs]

Possibly even 900 fps or close for 100 yard work.... Only once the entire drag curve is determined will we have the exact answer to that question.... Like anything else that goes through a minimum, some distance either side of that carries little penalty, so even if the optimum is 880 fps, 900 or even 920 may be so close that the flatter trajectory is worth it.... eg. on the above graph, 850 and 900 have the same drift, so may as well choose 900.... and 950 is only 1/8" more.... At 1000 fps, the drift is 3/8" more, however....

Bob

Thank you Bob, this IS valuable engineering you are doing here.  Thank you for your time.

[RONIN001]

Did the Ballistic Coefficient / Modified Drag Model come to end or move to a new topic thread?

[rsterne]

Died for lack of data at the moment....

Bob

[RONIN001]

Are you still looking for 1000fps+ data or are concentrating on normal pellet speeds and slower data?  If its the latter I may be able to supply some data if that is the case (like Harry's data) when I get the time?  Tell me what exactly we need in simple black and white terms.   LOL