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This page (http://www.sigmazone.com/Taguchi_L12_SpudGun.htm) has a great article on an experiment with a pneumatic spud gun.
StatsMan: Welcome to the forum!
Excellent page. I've been waiting for someone to do a multiple variable experiment and then attempt to tease out the individual affects. Good experiment and an excellent write-up of the method.
Its going to take me a while to fully digest your page but a few things;
1. Any statistical analysis? The gun's performance appears remarkably consistent given you are using range as the performance measure. I would have expected that since range is dependent on even small errors in the launch angle, wind conditions, etc. that there would have been more shot-to-shot variability. Looks like you did a heck of a good job to do things consistently.
2. You might consider building your own chronometer to measure the muzzle velocity directly. Muzzle velocity would remove one of your test parameters (launch angle) from the system and still give what you are most interested in (performance). There a several descriptions of homemade chrono's about, like mine, or SoftChrono, or just search this forum for a couple other approaches. You can also buy a shooting chrono (google chrony) for ~$70. (But building your own for a couple of bucks is much more interesting.)
3. It would be interesting to compare your results with the predictions made by the Gas Gun Design Tool, GGDT (the gold standard for pneumatic gun modeling programs).
4. Have you considered looking at any "synthetic variables" in your analysis? For example (chamber volume)/(barrel volume) (aka the CB ratio) or the kinetic energy? (Might be hard to get a KE value from the range data.) KE is interesting because it is direct energy measurement, unfortunately it is non-linear with velocity. CB ratio is a tricky variable since when used to predict performance it is not only non-linear but, in some cases, not a monotonic function. Another interesting variable would be chamber energy, e.g, (pressure)(volume).
Now, since you posted the results of an amazing study, I feel I am required to immediately ask you to do some more experiments.
Mod your valve for faster opening and see how much the performance increases and how the sprinkler mod interacts with the other variables. Blowgun mod of a sprinkler valve is described at link1 and link2 (don't bother permanently plugging the solenoid's port, just leave the solenoid installed).
StatsMan: Are you the author of http://www.sigmazone.com/Taguchi_L12_SpudGun.htm ?
Spent the time and effort (now my brain hurts) to read your page in detail.
A couple random thoughts;
1. It would be nice if the guns dimensions were listed in a table somewhere, including; pipe schedule, nominal ID and lengths.
2. Way cool, you've got two non-parametric variables!
3. There are some assumptions to this approach to reducing the number of measurements. (a) The variables must be independent in their affects. Probably true for the studies you did, but that isn't always the case. (b) The variables must be linearly (or nearly so) related to the independent variable. This isn't true for a couple of your variables.
4. Your results for the barrel length affects are exactly what would be expected given the other characteristics of the guns. The longer barrel gives poorer performance, particularly with the smaller chamber. For your smaller chamber the volume is 198ci. I believe the longer barrel's volume is about 353ci (2.5"ID x 6'). If you pressurize the chamber to 20 PSIG (~35 PSI absolute) then as the T-ball moves down the barrel to the point where it is just about to leave the muzzle the volume has increased to 198+353=551ci. Simple minded application of the ideal gas law says the pressure in the gun will have dropped to (V<sub>2</sub>/V<sub>1</sub>)P<sub>1</sub>=198/551*35=12.3 PSI absolute pressure. Since there is ~14.7 PSI absolute pressure in front of the g-ball trying to push it back into the chamber the shell is actually decelerating before it reaches the muzzle. You need either shorter barrels or higher chamber pressures.
5. Another way to look at the above is that the performance versus barrel length is not only not linear but it isn't even monotonic, there is an optimal barrel length (all other parameters being held constant). Barrels shorter or longer than the optimal length decrease performance. Mathematical approaches like Taguchi's method, cant deal with that kind of a relationship between a parameter and the response. I believe you need to come up with a way to linearize the relationship between the parameter and the response.
6. How did you measure the distances? Are the quoted numbers the distance where the ball hit the ground or the "roll-out" distance where the ball stopped?
7. Do you have the full response equation? Something like;
Distance = (a)(Air Volume) + (b)(Valve) + (c)(Barrel Length) + (d)(Angle) + ...
The comparison of the coefficients (a, b, c, ...) would be really interesting, especially if the parameters are normalized so that the magnitudes are directly comparable. (I suppose this type of analysis is beyond the capability of Excel.)
Overall, the study is an amazing example of both spud gunning and multiparametric analysis. It is right up there with the best spudgun studies, for example Burnt Latke's.
Do some more
I've built the model for the cited data. Only four of the parameters are needed. The param's along with their contribution to the model are;
<table><tr><td>Parameter</td><td>incremental R^2</td><td>total R^2</td></tr><tr><td>Pressure</td><td>0.648</td><td>0.648</td></tr><tr><td>Air Vol.</td><td>0.229</td><td>0.877</td></tr><tr><td>Angle</td><td>0.0674</td><td>0.945</td></tr><tr><td>Wadding</td><td>0.0414</td><td>0.986</td></tr></table>
The R^2 values indicate how much of the spread in the distance the projectile flew is explained by the particular parameter (incremental R^2) and how much is explained by the parameter and the ones before it (total R^2).
So, pressure is the largest affect and accounts for 65% of the spread in the distances. Air volume is the second largest affect and accounts for 23% of the spread. These two parameters together account for 88% (total R^2) of the spread in the data. Launch angle is the next most important, it picks up another ~7% for a total accounting of ~95% of the data. The last parameter, the wadding, picks up a bit more but is probably down at the level of noise. With all four of the parameters 98.6% of the data is explained by the four parameters.
The model's equation that predictes the distance is;
Distance = (10.349)(Pressure) + (0.259)(Air Vol.) + (-4.444)(Angle) + (52.22)(Wadding) - 40.15
Pressure in PSIG
Air Vol. in cubic inches
Angle in degrees (0deg = horizontal ?)
Wadding 1=paper, 2=cloth (non-parametric)
A graph of the predicted distance versus the actual distances is below;
As you can see, the model almost perfectly predicts the behavior of the cannons.
Unfortunately, the model is of no practical use for predicting the performance of guns that differ from the guns used to generate the data. This includes the inability to correctly model the affect of launch angles that are outside the 45 to 60 degree range. The model won't even correctly handle launch angles outside the 45-60 range for the same cannons.
This whole approach depends on a linear, or nearly so, relationship between the parameters and the measured performance (distance in this case). Unfortunately, none of the parameters are actually related to distance in a linear way.
The air volume and pressure parameters are related to the distance as asymptotes. That is, at low volume or pressure ranges the distance changes roughly linearly with these two parameters. However, at high values of volume or pressure the distance will approach a limiting value asymptotically.
The barrel length and launch angle are also not related to the distance in a linear way. Both of these parameters have an optimal value. The gun's performance drops off when these parameters are higher or lower than the optimal value. A plot of distance versus either launch angle or barrel length gives a graph with a hump in it.
The optimal launch angle for a golf ball is probably in the vicinity of 35 degrees. Launch angles greater or less than ~35 degrees will give decreased distances. This type of model can not handle a variable of this type. As far as the model is concerned, shorter barrels and decreasing launch angles always increase the performance of the gun.
It is too bad that so few of the relationships in spudguns are linear. The Tauchi method is amazingly powerful at reducing the number of measurements that need to be done. In this particular study with 2 values for each of 8 gun parameters there are a total of 256 possible configurations. If each configuration is fired four times (to get averages) you would have to do a total of 1024 firings to collect the data. The cited study did almost the same thing with just 48 firings.
In order to use this type of a model a way is needed to convert the non-linear responses into linear ones. In general, that type of conversion means you have to add variables to the system. For example, the optimal launch angle might be expressed in the Taguchi matrix as;
1/|angle<sub>optimal</sub> - angle<sub>actual</sub>|
For the angle column you would then need to calculate both the columns coefficient and the value of angle<sub>optimal</sub>.
Does anyone have any thoughts on how to linearize the variables? In particular, launch angle, pressure, chamber volume and barrel length.
I would think that people have run into this type of problem before. Whether they are using the Taguchi approach or some other method of reducing the number of experiments (measurments) required to determine a model.
Edit: 2^8 is 256 not 1024 (well duh)
Some answers to your questions...
1. I did do regression on the resulting L12 but thought it too complex for the introductory article. Explaining regression in a short time is non-trivial but I was planning on trying in a later article. We did try very hard to reduce extraneous variation.
2. In a later article on Full Factorial DOE on Potato Cannon, I did measure velocity using high speed photography. Your comments about removing a variable are dead on and I was actually considering a follow up on Critical Design Parameters and how to isolate parts of the design. Your chronometer is incredible. I think I will try building one following your plans.
3. So would I. I didn't know it existed but I do now.
4. Very good question. The ultimate goal is to understand the Critical Design Variables, but it isn't always so easy. You pegged the velocity vs. distance, if you combine velocity and launch angle, then you can predict distance. The real Critical Design Parameter (CDP) is velocity which then is an input to distance. Your question about synthetic variables gets to the point of what really drives performance. Is it the volume of the chamber, or is it the ratio of the chamber and the barrel. If it was, then I could remove a variable.
Great tip on the modification of the solenoid valve. I will definitely pursue this, probably this weekend.
Next Post Answers.
Yes, I am the author.
1. I used standard 2" and 3" PVC. The barrel length was the key variable and it is 4 feet and 6 feet (approximately).
3. Again, you are correct. The Taguchi L12 makes the assumption of linearity (or nearly so) and a few of them are probably not linear. However, the L12 is good for finding the most significant effects on the performance. A follow on non-linear DOE like a Central Composite Design or a Box-Behnken would be in order to truly model the gun.
4. I had a feel for the effect of barrel length. It is intuitive that there should be an optimal length of the barrel, given a pressure and amount of air in the compression chamber. If you go too short, there is not time for acceleration. Too long and it is slowing down.
5. If the relationship is non-linear or interactive (which is certainly is) then an L12 would not find the optimal. It could get you into the ball park but not the true optimum. However, a three level design like a Central Composite or a Box Behnken could.
6. Distances were measured with two 300' measuring tapes and were where the ball came to rest. I wanted to do were the ball struck but with my kids around, I didn't want them in the line of fire.
7. I uploaded the full analysis using DOE Proin The worksheet labeled "Regression Table(2)" has the key results. The equation would be -8.633 + .259*Air Volume -10.574*Valve ....
It is important to note that I didn't use the L12 to produce a model but rather to learn which variables were the key ones with respect to performance. This is sometimes called "screening" as you are trying to understand what really drives performance and what is a minor effect. I am new to Spud Guns so I didn't have much to go on. I should really take this experiment and follow up with a design that is intended to produce a model such as a Central Composite Design using the most significant terms from this model.
By the way, I now have a gun that is combustion based that is made of clear PVC. I video taped this gun and posted it on youtube it is a Night Potato Cannon Video . I would like to do a DOE on this gun, but accurately metering the amount of hair spray would be difficult. Any ideas (fuel injector maybe)?
Glad to see a follow up on your first post. I was afraid it was going to be a "drive-by" post.
I've built a technical combustion gun and I probably should look at the higher level (Central Composite Design or a Box-Behnken or ...) analysis protocols. Currently, I'm considering looking at;
1. Fuel mixing; poorly mixed, perfectly mixed
2. Chamber fan +/- during firing
3. Number of sparks (1,2,3)
4. Location of single spark (breech end, center, barrel end of chamber).
5. Fuel ratio (perhaps 5 steps spanning 3% to 8% v/ fuel).
6. CB ratio (I've go a barrel with multiple spud detectors so I can do a CB study with a single shot of the gun.)
7. Single versus double bevel muzzle knife.
8. Projectile mass and/or friction.
It'll take a very long time to work through all the variables using traditional techniques.
The video is cool. What was the frame rate, gun dimensions etc? Would be interesting to calculate the flame speed versus time. Your video is a heck of a lot clearer than mine
Easy thing to do is just skip the hair spray and use propane or butane. The normal way to measure the fuel accurately (and more importantly repoducably) is to use a fuel meter, like this example, or just search these forums for "fuel meter".
A much cheaper (but just as accurate) method is to use a ~$1 syringe like this or this.
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