So there is a long running and vigorous argument about the choice between using fast and slow powders for a give load. By load, we mean a shell constructed to launch a desired mass of shot at a specified muzzle velocity. As reloaders already know, there is quite a large variety of recipes available. Even when you narrow it down to one shell type and mass of shot there are numerous recipes that all accomplish pretty much the same velocity. For example if you look at Lyman's shotshell reloading book (4th edition) there is in excess of 80 loads for Winchester Compression Formed hulls and 1 1/8oz of lead shot. The loads cover a wide variety of primers, wads, powder and powder charges. The velocities only vary by ~100fps between the lowest and highest of these recipes but the pressures vary between 6400psi to as high as 11,500psi.

Now if your looking for specific guidance to select a powder I am afraid you will be disappointed with this article. This article is to explore some of the interesting dynamics that happens between the time the primer ignites to the time the wad/shot column leaves the muzzle. There will be no grand life changing conclusion, but hopefully some interesting insight into how pressure affects the wad/shot column while traveling down the barrel.

One of the arguments that always come out of the powder burn-rate argument is that powder X feels softer than powder Y for the same resulting shot mass launched at the same velocity. Since both loads have the same total recoil impulse they are arguing they can feel the difference in the acceleration vs time curve caused by the difference in pressure vs time. Many claim that the slower burning with lower peak pressure of powder X is the reason. Others argue that you cannot sense the difference because if they both have the same mass and the same velocity then they have the same total recoil and feel the same.

Again I don't think I will definitively bust either of these notions as much of the "feel" of recoil is very subjective and particular to each person. What I do believe I can show is that although two loads might truly have the same total recoil impulse that the events that happen over that incredibly short event of setback can actually be different. Whether the average human can feel that differences in an event that takes 2-3 milli-seconds is questionable.

Back in 2002 a man by the name of Neil Winston published two interesting studies on shotguns. The second study looked at pressure and recoil. I am going to borrow some data from and build on his study. The studies can be found at the following web address: http://www.claytargettesting.com/

In the past I have been very critical of his second study on recoil. {http://www.claytargettesting.com/study2/pages/study2.html} As I have revisited the information several times since my initial criticism of the study; my opinion has become less harsh but I still believe the Mr Winston missed a very good opportunity to tell a more complete story.

In the third part of his study, he presents the pressure curve for two loads that have the same total recoil impulse. The two loads have the same mass and velocity well within any reasonable engineering standard. The two loads have velocities that are less than 0.7% different. The interesting thing is the two loads use two dramatically different powders. One is using the fast burning Red Dot (Alliant). The other is using the relatively slow burning PB (IMR). Therefore, the loads have pressure profiles that look noticably different.

In **Figure 1** you can see the pressure verse time profiles of these two different loads. The two loads have the same total recoil. An 1 1/8oz of shot doing ~1225 fps has 2.677 slug-ft/sec recoil impulse. Despite the dramatically different pressure verse time curves the resulting total impulse is the same. This was the conclusion that Mr Winston made in his study. He showed that his free to recoil test system had the same resulting velocity after firing both loads and concluded that the recoil is the same despite the differences in the pressure profile. The conclusion is correct in my opinion. What he missed was the interesting dynamics that happened between primer ignition and the exit of the muzzle. He looked at the total recoil results of the setback event but never looked at what happened during the event. He had the data right in front of him and just never explored it.

I hope to use that data to show what the wad/shot column experienced during the setback event.

So next, I am going to explain how I set up a numerical simulation of the setback event and walk through the results. All the programming was written in MatLab.

First thing I did was to extract the pressure curves from Mr Winston study. I simply digitized the curve shown in **Figure 1** and created two MatLab functions that fit a curve to the digitized data points. When the functions where passed a time index they would return a pressure value.

With those functions created and tested, I set about creating a dynamic simulation of the setback event. Start with Newton Second law, from this we will derive the equation of motion for the shot charge moving down the barrel.**F = m a***F* = force*m* = mass*a* = acceleration

The mass is 1 1/8oz of shot.

The force is a function of time and is the result of the pressure (*P*) generated by the combustion of the powder charge. So for any instant during the simulation the *F* should be the pressure times the bore area. Think of the shogun barrel as the cylinder wall and the wad/shot column as a piston in the cylinder.**F(t) = P(t)*A***A* = area of the bore

So now if we substitute our time dependent force function into the *F = m a* we get an equation of motion that looks like this.**P(t) A = m a**

We now have an equation of motion that describes the acceleration of the wad/shot column as a function of time. Since acceleration (a) is the second derivative of position with time we have a differential equation.**P(t) A = m d ^{2}x/dt**

^{2}

Divide the equation through by mass (

*m*) and we get.

**d**^{2}x/dt^{2}= P(t) A / mFrom this equation we can create a simple simulation. By numerically integrating the equation of motion twice with respect to time we can create, the acceleration vs time, velocity vs time and position vs time profiles of the load as it moves down the barrel.

In the MatLab code I did just that. My integration was a very simple brute force methode. I had the computing horsepower and the pressure curves were smooth with no discontinuities so a moderately fine time step and simple rectangular integration resulted in acceptable good integration.

The original test data was taken using a 30inch test barrel so I integrated until the position reach 30 inches.

The results were promising but off. Both loads resulted in nearly identical velocities but the velocity was high ~1405fps. Given the pressure data coming from digitizing a plot from another paper I was encouraged the velocities were so close to each other.

I reasoned that the cause for the velocity being high was due to the fact that the above model has no friction in it along with a few other secondary forces like crimp opening, the additional friction in the forcing cone and so on. Friction being the highest of all the secondary force action on the wad/shot column as it moves down the barrel.

Therefore, I derived a simple friction model. Sliding friction is normal modeled as a normal force times a coefficient of friction. I did not have a good value for the coefficient for friction between polyethylene wad and a steel barrel. More so, I did not have any idea the magnitude of the normal force between the wad/shot column and barrel would be.

I conjectured that the normal force generating the friction would be proportional to the force the pressure was putting on the base of the wad/shot column. My reasoning here is that the shot under acceleration of this force would try to spread out. Image trying to pile up shot on the top of your reloading bench the shot under just 1g of acceleration would rather spread out than stack. How hard that shot tries to spread out should be proportional to how hard the pellets are being pushed down. Using those assumptions we come up with a friction model that would look like the following starting with the traditional model for sliding friction.

**F**_{f}= u F_{n}

*F*= force of friction_{f}*u*= coefficent of friction*F*= Normal Force_{n}*F*_{n}= C_{n}F(t)*= some constant that relates axle force to the normal force (ie the force spreading the shot out.)*

**C**_{n}We already know that

**F(t) = P(t) A**

So our friction model looks like:

**F**_{f}(t) = u C_{n}P(t) A

The catch here is that we do not know what

*u*or

*Cn*are. They are constants (we hope and assume). But we can back into them since we also already know what what the exit velocity value is from Mr Winston study. Therefore, we combine the two constants into a fudge factor and do just that fudge the number until the resulting velocity is correct. Yes this is a bit clunky but reasonably valid assuming we use just one fudge factor for both pressure profiles representing the idea that if our friction model is reasonable it should work for both loads without changing it. So now our friction model is

**F**_{f}(t) = C_{f}P(t) A

**C**_{f}**= fudge factor, the combination of**

**u C**_{n}And this actually worked reasonable well. With a fudge factor of

*C*0.13 we get a resulting muzzle velocity of both loads that are within 8fps of Mr Winston's results. Not bad given the digitization of his data plot and the simple friction model.

_{f}=Now we get to the interesting stuff. First position data. This is the results of double integrating of the resulting acceleration resulting from the force of the pressure and force of friction.

The **Figure 2** plot looks remarkably like the plot at the end of Mr Winston's part 4 of his recoil study. It should both lines quickly become nearly parallel and by the time we get to ~2.5 msec they pretty much are since both loads have the same velocity (the slope of the position vs time plot is velocity). Just as Mr Wisnton test sled demonstrated by having the same resulting velocity from firing both loads. The total recoil impulse is the same. What is interesting and Mr Winston fail to completely demonstrate is why that slight time shift happened. To see that we need to look at the velocity and accelerations both as functions of time and of their position down the length of the barrel. Next velocity

Here we see more differences in the characteristic of the velocity profile of the two loads. We can see as expected the slower burning PB takes its old sweet time to get rolling but slowly and surely catches up to the Red Dot load. Its harder to see the difference in the Velocity vs Position plot so I made one more graph (**Figure 4**) that is simply the difference in velocity of the two load vs position.

The thing to see take away from **Figure 3** and **Figure 4** is that if you're looking at loading for a short barrel shotgun fast powders will gain you a little velocity, not much but a little compared to slower powders.

Now we move on to some more dramatic differences in the acceleration of the shot column. First verse time and second verse the position in the barrel.

From the acceleration plot in **Figure 5** the big take away is that for a very brief period the Red Dot load is accelerating at a peak of over 64,000g's were the PB load gets just above 47,000g's of acceleration. That means that the poor pellets in the Red Dot load see over a third more force as they are mashed together and against the wad and barrel. Without doing more analysis it difficult to say for sure but its very likely, the increase in peak forces could cause a greater number of pellets to be deformed, especially near the bottom of the shot column where they have all the mass of the other pellets infront of them.

Finally I would like to plot pressure and acceleration (**Figure 6**)on the same plot. I have scaled the pressure and acceleration with non-standard units to make the plots visually clearer. The point of this plot is to show that acceleration and pressure are proportional to each other. The only real take away is that the higher the peak pressure is the higher the peak acceleration the shot load will experience. This may be of interest if your trying to reduce shot deformation but remember that all of these event happen on a time scale that is at or shorter than most humans can resolve temporally.

## Conclusion

Recoil is a nasty little thing to quantify in a complete and in-depth manner. The easiest and probably most objective way it to simply state it as the total recoil impulse (ie change in momentum) which would be Mass times Velocity. It's simple and for comparison sake accurate enough assuming all other factors are the same, such as the same weight firearm same action type, same shooter, similar clothing, and so on. As I stated earlier that although the wad/shot column experience dramatically different forces in the two loads examined above there is probably only a few exceptional individuals that can "feel" the difference between two loads that have the same total recoil impulse but have different pressure vs time curves. Temporally the setback event is so fast that it is at the edge of our ability to resolve, but human tissue has properties that change fairly dramatically with the rate at which they experience loads so I think its possible but rare to find such an individual.

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