I'll let robert answer the programming stuff...

If you look into commercial programs gaussian distribution is the most common for gun accuracy. To apply anything else you would need really extensive testing and fitting.

I have done a shooting diary as well (see Group and velocity statistics thread) and I have now around 500 groups in my database. When it comes to statistics it's not much but it is something. In avarage all the real world group measures I have saved fit very well to gaussian distribution. So I would say it really is pretty good probability distribution for the application.

If you accept that holes distribute according to gaussian distribution you can calculate lot's of things. For example I have done extensive testing with monte carlo comparing different group size measures. There are lots of them. To mention few: radial extreme spread (the most common and worst), ES X and Y directions, radial standard deviation, SD X and Y directions, diagonal, Circular error probability (which is best I have found). There are at least two important things which define the best measure: 1. effectivenes which means that how many rounds you have to shoot for two different loads to define which is more accurate. 2. independence which means how number of shots or group shape affects to measure in question.

The best feature in SD based group measures is that they are not depended on number of shots in the group like extreme spread measures. For example with monte carlo you can calculate that 10-shot group is around 1,6x (if my memory serves me right) larger than 3-shot group. And that's only true if groups are round in average. While CEP is in average approx. the same for both 3- or 10-shot groups.  If group ellipcity factor is not higher than 4.

Quote from: 375CT on June 24, 2014, 07:29:08 AM
Mman, if you don't me asking, why you state your work is not mathematically exact?

Read the article about CEP. Derivation includes few approximations at least on ellipcity and bias. Then again I have assumed that all the dispersion factors are independed. This is pretty good approximation but not exactly true as roberts states on Hit probability thread.

Excellente discussion and insight!

Just a simple question, since I'm not that familiar with Excel programming (just the basics), do you think that using a random number generator (RNG), for using a common programming language, could do the same? I mean, assuming that the RNG is based on a Gaussian distribution?

On the other hand, what other kind of probability distribution do you guys thinks is worth exploring for a realistic scenario?

I find this subject very interesting for developing some sort of BfX simulator, besides the excellent work already done by Robert.

Mman, if you don't me asking, why you state your work is not mathematically exact?

You are digging into fundamental questions about rifle internal ballistics. I been interested in this for years and when I started using finite element method (FEM) for guns I had a dream of 'perfect' gun design. That design would minimize vibrations and ultimately leave only intermediate and external ballistic effects to harm accuracy.

First thing you have to accept is that you can't use analytical methods to accurately estimate recoil nor vibrations. Only sophistic numerical methods like Finite element method are capable of taking into account everything from time depended excitation to flexible materials and supports. Analytical methods like your excel spreadsheet can be pretty good in estimating what are total forces, moments and torques when whole recoil phenomenon is over. However you should be interested in what happens while bullet is still in the barrel. That's far more complicated since material flexibility plays major role in that. Usually only minor part of the recoil is transferred to your shoulder during barrel time. Rest of it comes after and that has only psychological effect (which can be important as well). See picture below. It is calculated recoil force felt on your shoulder for gun and ballistics very similar to your's. This is also very sensitive to hold, recoil pad and even your clothing and body composition. The point still remains; usually most part of the recoil comes after.



From the Internet you can find lot's of general rules what kind of designs can minimize or optimize vibrations. However first thing you learn doing FEM is that gun vibrations are actually really sensitive to many things. For example change of load, stock or rifle scope can significantly alter vibration behavior. This leads to conclusion that general rules that always have a positive effect to accuracy are very hard to come by.

Okay, how did I do in my way to ultimate gun design? As I'm still here writing and not enriched by my inventions so pretty poorly I guess. I have still found many useful trends that work most of the time. Beyond that you have to calculate everything case by case. Usually that requires calibration of calculation model with measurements. However in the past I have got great results even without calibration. I still believe luck has played a some role in those situations. Couple of the most successful cases are combination gun projects where I was able to calculate how two free floating barrels should be assembled in relation to each other so that zero would be the same. According to calculation relative vibration amplitude was about 6 MRAD and same zero was achieved withing 0,5 mrad. The other barrel was shotgun barrel so accuracy was good enough. At the moment I have couple of gun projects going on where I have optimized barrel profile for bending vibration. Load testing will give me feedback how it did work.

About "BARREL VIBRATIONS SIMULATOR" you posted; it's first time I see it. It still obvious that it can't be useful in optimizing barrel profile. Reason is simple; too few input parameters. Only barrel profile is taken into account and excitation can't be altered to mention couple of things that would alone ruin the accuracy. I still agree many points mentioned in the text below calculator.

See Hit probability thread as well. My tool is based on analytical formulas. I have compared the results with litz's tool and the difference is usually less than 5%. I have done some assumptions that are not mathematically exact. However it's still pretty close. In theory benefit is that calculation time is reduced significantly compared to monte carlo method. However I'm not an expert in programming and my code is not as effective as robert's. My program also includes more variables. That's why theoretical benefit in calculation time is not redeemed in practice.

The limitation also is that I can only calculate round targets while with monte carlo target shape is not limited.

Here is as an article about circular probability error which I have used. It is also far better measure for gun accuracy than extreme spread when it comes to probability of hitting something.

Robert, how many iterations you need to do with monte carlo before you get hit probability say in accuracy of 1%? 99 % of the time?

I started shooting 230gr Berger hybrid bullets in my 300WM for use at long range in F-Open competition. While not driving these bullets hard at 2850 fps the 10kg rifle still generates considerable recoil. This set my accuracy back initially until I learned some new skills to cope with the increased recoil.

One of the things I have set out to do is understand the forces involved. I am hoping this might lead to understanding stock design and the best place to add weight to the rifle to maximise the allowed limit of 10kg. So far I have put together a spreadsheet for free recoil calculation, (force and velocity) and also for "torque". The spreadsheet can be downloaded from here: http://fclassdu.com/origin/wp-content/uploads/2014/06/Rifle-Torque-Recoil-Calculator.xls

To state the obvious, recoil comes into consideration for accuracy only for the time the bullet is in the barrel. That usually means for the first few milliseconds and the first few millimeters of rear recoil movement. The rotation of the barrel due to the torque effect for my loads is approx 0.06 degrees as the bullet leaves the muzzle.

I am also very interested in barrel vibration and wonder if anyone using this forum has looked at calculating the effects of barrel profile, weight and load effects on barrel oscillations. I found this link really interesting. http://www.geoffrey-kolbe.com/articles/rimfire_accuracy/barrel_vibrations.htm

Ian

Version 3.0.5 is live. I've added a full metric feature and a tertiary special functions page that gives deltas for -1,-2,-3,+1,+2,+3InHg for wind and drop. 3.0.7 will be out in a few days with more improvements.

Ok,
I understand what you understand - nothing special meant here. Let me make things less abstract.

Basically what Monte Carlo does is to generate "outcomes", e.g. where a  bullet ends, for, varying conditions - the input parameters. The input parameters are varied according to a certain probability distribution. If for example a shooter is able to aim with an accuracy of 1 MoA then people like me model that as a bell-shaped (Gaussian) probability distribution that averages to 1 MoA mean inaccuracy. With this we acknowledge the fact that in some cases the shooter aims very well and in others badly. 

In a Monte Carlo simulation then, the computer draws a random number and converts that to an inital horizontal and vertical aiming angle. This is done in such a way that average inaccuracy is 1 MoA. After the horizontal and vertical launch angles are chosen, the bullets trajectory is calculated.

If, for instance, in addition the effects of a non-constant wind is being studied, one might model that primitively as a wind distribution that is half of the time zero and the other half 1 m/s. In a Monte Carlo calculation then one first draws a random number that determines the launch angle, and an another random number is used to determine if there is wind or not in the trajectory calculation (with the afforementioned spreadsheet I deal on a more sophisticated way with non constant wind along a trajectory). Hence we end up with a set of bullet end coordinates that  were affected by both wind and the shooters abilities.

In practise, the more inputs one generates according to a probability distribution, the less the details of a probability distribution matters. The distribution might as well be flat, meaning that the probability for a certain input value is constant in a certain range and zero elsewhere.

Quote from: admin on June 12, 2014, 01:38:37 PM
Maybe "Monte Carlo" is a  definition issue. In this case, i imagined that input parameters are generated in such a way that they are consistent with a certain probability distribution. Then a trajectory calculated and evaluated. The process is repeated hundreds of times. This is what one might describe as Monte Carlo.

Nice reply, much clear now. Will take a look to your workbook and for sure coming back with more questions, if you don't mind 

Maybe "Monte Carlo" is a  definition issue. In this case, i imagined that input parameters are generated in such a way that they are consistent with a certain probability distribution. Then a trajectory calculated and evaluated. The process is repeated hundreds of times. This is what one might describe as Monte Carlo.

Quote from: admin on June 11, 2014, 01:16:00 AM
WEZ is a Monte Carlo simulation.

I find it hard to opinate about it - it does what it says.

Quite frequently I use a similar tool understand ballistic effects in a target shoot competition setting.

It is available as download on the BfX site. Look for the descriptive text:

"This workbook hosts a simulator with which you can estimate your results in a (multi distance, multi target) match, e.g. the NRA 90 shot full bore regional match course."

Robert,

Thanks for the answer, however how do you know WEZ is based off of Montecarlo? I've read all the available literature by Litz and that word is not present even once.

thanks!