# Potato Cannons and the Taguchi L12: A Design of Experiments Case Study

by Philip Mayfield

In the week between Christmas and New Year's Eve 2007, I found myself with a nearly empty golf course, beautiful central Florida weather, ample free time with my children, and a potato cannon. A potato cannon is a simple device which has the whimsical purpose of propelling a projectile through the air. While the potato cannon's namesake is the potato, it can also be used with other objects such as tennis balls, golf balls, etc. Click on the picture below to see a short video of the potato cannon in action. You might have to click twice.

While at first glance the potato cannon might seem to lack any serious purpose, people are strangely drawn to and captivated by the device. In the six months since I built my first cannon, I have demonstrated it to approximately 10 people, and no less than three have gone on to build their own.

Safety Note: A potato cannon can cause severe injury. Do not build one without expert guidance. The projectiles (potato, tennis ball, golf ball, etc.) travel at a high velocity which can cause serious injury on impact.

My potato cannon is pneumatic powered, which means that it uses compressed air as the propellant. Here is a short video that explains how the cannon works.

The Experiment

When designing new products, you may be presented with a new technology or a new application and not fully understand all of the variables that might affect the performance of the design. This occurred to me in the development of my potato cannon. I couldn't help but identify the variables of the potato cannon's design that would impact its performance. My very nature dictated that I continue on with an experiment to quantify and better understand these variables.

Critical Design Parameters

What is a Critical Design Parameter (CDP)? A CDP is a design variable that has a significant impact on the performance of a design. Here are some simple examples.

 Product/Design Performance Metric Critical Design Parameters Engine Power Output Cylinder Volume, RPM, Air/Fuel Ratio Portable Music Player Playtime (time until battery dies) Available Battery Power Battery Available Battery Power Acidity of the Electrolyte Golf Driver Drive Distance Coefficient of Restitution Potato cannon Distance of Potato Shot Pressure (psi) in Compression Chamber

A CDP can be a physical/electrical/chemical/software characteristic that the engineer controls (hopefully) in the design. A discussion of how to identify CDPs will be the subject of another article.

I did have one problem with my cannon, and that was the potato itself. The potatoes are not uniformly round, do not all have the same mass, have different aerodynamic characteristics, and finally, they are not golf course friendly. Therefore, to better understand my variables, I substituted a golf ball for a potato.

I selected eight variables for my first experiment. Here is a short video that explains the eight experimental variables.

Experimental Range Selection

For each of the variables, there is a "low" and "high" value. For example, in the picture below you can see the two different compression chambers. The smaller chamber (with a green center and end) is approximately 198 cubic inches of volume while the larger chamber is 672 cubic inches of volume. In DOE parlance we would say the "low" is 198 cubic inches and the "high" is 672 cubic inches. The same goes for barrel length. The "low" is a 4ft barrel length and the "high" is a 6ft barrel length.

It is important to note that the "low" and "high" do not have to correspond to measureable entities. For example, the wad type is either paper or cloth. In the video, I explained that we need a "wad" much like an old time musket to get the golf ball to come out of a 2.5" barrel. This begs the question "what type of wad is better"? I decided to experiment with a cloth wad (small rag) and a paper wad (about 5 pieces of notebook paper in a ball). For the DOE, the paper wad is the "low" and is set to 1, while the cloth wad is the "high" and is set to 2.

Taguchi L12

The experimental design I chose to use is the Taguchi L12. The L12 allows for experimentation with up to 11 variables and does so with only 12 runs. The experimental sequence of the L12 is below.

For the remainder of this article it might be helpful if you download the data file by clicking here. The data is in a single Excel workbook which has two worksheets. If you print these two sheets it is easier to follow the article.

 Row # Air Volume Valve Barrel Angle Pressure Wad type Voltage Ball Type 1 198 1 4 45 20 1 9 1 2 198 1 4 45 20 2 27 2 3 198 1 6 60 40 1 9 1 4 198 2 4 60 40 1 27 2 5 198 2 6 45 40 2 9 2 6 198 2 6 60 20 2 27 1 7 672 1 6 60 20 1 27 2 8 672 1 6 45 40 2 27 1 9 672 1 4 60 40 2 9 2 10 672 2 6 45 20 1 9 2 11 672 2 4 60 20 2 9 1 12 672 2 4 45 40 1 27 1

Variable Key
Air Volume: In Cubic Inches
Valve: Two different valves, same manufacturer and model
Barrel: Length of the barrel in feet
Angle: The angle of the launch in degrees
Pressure: The pressure of the Air Volume in pounds per square inch (psi)
Wad Type: 1 = Paper, 2 = Cloth
Voltage: The voltage applied to open the valve, in volts
Ball Type: 1 = White Ball at \$5 per ball, 2 = Pink Ball at \$1 per ball

Note that I have 8 experimental variables and I will experiment with them all in 12 runs. Take a good look at the experimental runs in the L12. At first glance, the L12 appears to be random; however, this is definitely not the case. The L12 exhibits a mathematical property called orthogonality which means that each of the variables is independent (a good explanation of orthogonality is here). The L12 allows us to get independent estimates of each of the 8 variables, but in only 12 experiments. This is an extremely important property of the L12 and greatly simplifies the analysis.

If you decided to experiment with only one variable at a time, you would need to do at least 16 experiments. You might first shoot the cannon with 20 psi (experiment #1) and then with 40psi (experiment #2) and measure the difference. You might then move on to angle, and shoot the cannon at 45 degrees (experiment #3) and then 60 degrees (experiment #4). Since I have 8 variables and I would need a minimum of two levels each, this would require 16 experiments.

The L12 is more efficient than single factor testing and will turn out to provide much more information. With my experiment test plan in hand, it is now time to collect some data. Put another way, I need to "build" a cannon for each of the rows in the L12, shoot the cannon, and then measure the distance the ball travels. Specifically, for row #8 I need to create a cannon with air volume= 672, use valve #1, a 6 ft barrel, 45 degree launch angle, 40 psi, Wad Type = 2 (cloth), 27 Volt release, and a white ball.

I should note that if you ever decide to do this, 672 cubic inches takes a while to fill to 40psi with a battery powered pump. If you decide to use an air chamber this big, be prepared to wait for it to fill.

After several hours of data collection, and a few good bruises from being struck by the t-ball, we have the data.

 Factor A B C D E F G H Distance Row # Air Volume Valve Barrel Angle Pressure Wad type Voltage Ball Type Y1 Y2 Y3 Y4 1 198 1 4 45 20 1 9 1 89.75 87.167 86.583 78.583 2 198 1 4 45 20 2 27 2 120.25 127.42 129.17 134.67 3 198 1 6 60 40 1 9 1 202.33 206.17 206.92 216.58 4 198 2 4 60 40 1 27 2 175.58 205.75 209.33 224.42 5 198 2 6 45 40 2 9 2 321 332 308.67 293.58 6 198 2 6 60 20 2 27 1 89.333 80.833 66.167 78.333 7 672 1 6 60 20 1 27 2 130.02 133.67 145.5 151 8 672 1 6 45 40 2 27 1 458.33 469.58 473.33 465.83 9 672 1 4 60 40 2 9 2 433.17 381.58 377.25 367.83 10 672 2 6 45 20 1 9 2 188.83 155.08 158.75 169.25 11 672 2 4 60 20 2 9 1 161.58 159.58 164.75 178.17 12 672 2 4 45 40 1 27 1 400.83 415.33 439.42 442.33

I should note that we measured the length of each shot to the nearest inch and then entered the data into Excel using formulas. For example, the first distance is 89.75, which would be 89 feet 9 inches. This was entered into Excel as =89+9/12. This is important to note, as the next observation is 87.167 feet. We did not collect data to the thousands of a foot, this is simply 87 feet 2 inches.

Data Analysis

A visual inspection of the data is quite interesting. I wish my golf game was as consistent as my potato cannon. The four runs I did with row #1 vary from approximately 79 feet to 90 feet. Looking from row to row you can also see a wide range of shots. Rows #1 and #6 are in the 80-90ft range while Rows #8, #9, and #12 are all in the 400ft+ range.

Visually, can you see if a CDP appears to change the distance? For example, look at the first 6 runs where Air Volume = 198 cubic inches compared to the last 6 runs where Air Volume = 672 cubic inches. The data for the first six rows (Air Volume = 198) ranges from around 80ft to a maximum of 320ft, while the data for the last six rows (Air Volume = 672) ranges from 130ft to 470ft. If we calculate the average of the first six rows and compare it to the average of the last six rows, we get the following.

Average of the first six rows when Air Volume is 198 cubic inches = 170ft
Average of the last six rows when Air Volume is 672 cubic inches = 293ft
Impact of changing Air Volume from 198 cubic inches to 672 cubic inches is 293ft - 170ft = 123 ft

At this point most people throw out an objection that we can't compare the first six runs with the last six runs, as the other 7 variables were changing during each six run block. This is why the previously mentioned property orthogonality is so important. Take a close look at the first six runs.

 Row # Air Volume Valve Barrel Angle Pressure Wad type Voltage Ball Type 1 198 1 4 45 20 1 9 1 2 198 1 4 45 20 2 27 2 3 198 1 6 60 40 1 9 1 4 198 2 4 60 40 1 27 2 5 198 2 6 45 40 2 9 2 6 198 2 6 60 20 2 27 1

Note that in the first six runs when Air Volume = 198, half of the runs have valve = 1 and the other half have valve = 2. Similarly, half of the runs have Barrel = 4ft and the other half have Barrel = 6ft. If you look closely, each of the variables will have half of their observations at their "low" values and the other half at their "high" values. There is a bit more to orthogonality than this, but suffice it to say that because the L12 is orthogonal, the variables are independent, which allows us to calculate the change from Air Volume = 198 to 672 and measure the difference.

If we continue to calculate the averages for each variable at their low and high, and plot the results, we end up with something called a Marginal Means plot which will look like this.

A Marginal Means plot is a graphical representation of the averages of the data we just discussed. Remember that the average when Air Volume is 198 cubic inches = 170ft and the average when Air Volume is 672 cubic inches = 293ft. A marginal means "plots" those two points and draws a line between the two. The left most line in the Marginal Means plot for Air Volume is a graphical representation of the impact of changing from 198 cubic inches to 672 cubic inches. The next variable, Valve Type (two different valves from the same manufacturer) went from approximately 235ft to 225ft. Barrel (4ft vs. 6ft length) made an even smaller change, with almost no perceptible difference.

A marginal means plot is very nice as it gives you a graphical representation of all 8 variables that is easy to interpret. The larger the difference between the "low" and the "high" the longer the line. The variables with very long lines, such as Pressure, Air Volume, Angle, and Wad Type have a larger impact on how far the ball goes. The factors with very short and almost flat lines, such as Valve and Barrel have little to no impact on how far the ball will go.

What worked out the way I expected

I expected Angle, Pressure, and Air Volume to be large contributors to distance. However, I didn't expect pressure to be so much larger by comparison. When Air Pressure goes from 20 psi to 40 psi, the golf ball went on average 128 ft and 334 ft, respectively. This is a difference of 206 feet for a 20 psi change. This equates to 10 feet of ball travel for each additional psi. Armed with this information, I will need a more accurate way to measure pressure as the current pressure gauge is difficult to read more than +/- 2psi. A similar analysis of Angle indicates that I need a better way to measure and control angle.

I expected the difference between valves to be small or non-existent. The effect is small which is good news when trying to obtain consistent shots between cannons.

What was different than expected

Voltage and Ball Type are very interesting and it is difficult to say if they truly impact distance or if they are simply noise. The difference between 9V and 27V is around 20 feet on average and is in the expected direction, that is 27V launches the ball farther than 9V. The ball type is also quite interesting. The expensive ball did fly on average about 15 feet farther than the less expensive ball. Again, it is difficult to say if these are truly significant differences or in the experimental noise. In another article I will discuss the smaller effects from a Taguchi L12 and how we might understand them a little better. At this point, I would say that the effect of both voltage and ball type is inconclusive.

I did not expect Wad Type to be significant and certainly not #4 on the list. To be honest, I was expecting the paper to be a better wad, while cloth turned out to be the clear winner. My intuition tells me the cloth wad was better as less air could escape around the cloth during launch. The ideal wad would have no mass, allow no air to escape through or around itself, and would exert no normal force on the walls of the barrel (i.e., it would require no force to insert or remove from the barrel). In other words, the wad would be the perfect energy transference device.

I expected the barrel length to be significant but it turned out have little to no effect. My reasoning was that the longer the barrel, the longer the ball can be accelerated and the farther the ball should go. However, the barrel has the smallest effect of all of the inputs, and the short barrel actually had an average of about 4 ft more distance than the long barrel. Perhaps the ball has accelerated to its maximum speed by the time it gets four feet into the launch. Or perhaps there is an interaction that the Taguchi L12 can not pick up. I suspect there is an interaction between the barrel length and the air volume, but I will have to do additional experimentation to be certain.

Conclusion

After running this simple experiment, I have a much better understanding of the Critical Design Parameters (CDPs) that affect the potato cannon. The Taguchi L12 allows you to evaluate up to 11 variables in 12 experiments, making it highly efficient. The results of the L12 will give you an indication of the relative importance of your CDPs. If you are working on a design and do not understand your variables, the Taguchi L12 is a highly efficient method of generating knowledge.