by Philip Mayfield

Abraham Lincoln's son Tad had a toy cannon which, according to folklore, he fired on the Cabinet after receiving a pretend military commission. Tad's cannon is currently on display at the Lincoln Museum in Springfield, IL. Over the years, cannons have served many non-combat uses including signaling, firing salutes, musical instruments (Tchaikovsky's *1812 Overture*), and a spirited reminder on the infrequent occasion of a touchdown by West Point's football team (go Air Force). We now have a new peaceful use for the cannon... it is quite useful for explaining a statistical tool called Design of Experiments (DOE).

My tennis ball cannon is a simple device with the sole and seemingly fruitless purpose of propelling a tennis ball over very great distances. To see an example of a tennis ball cannon click on the video below (you might have to click twice).

A tennis ball cannon is a natural extension of a potato cannon, with a simple substitution of a tennis ball instead of a potato. For a short explanation of how a tennis ball or potato cannon works, click on the video below (you might have to click twice).

*Safety Note: A potato cannon or tennis ball 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 first DOE with the potato cannon was a Taguchi L12 (full article here). In this article, I am going to extend this work and more fully explore the Air Volume and Barrel Length.

As you know from the video, a pneumatic cannon works by releasing compressed air through a valve to propel a tennis ball down a barrel. I would like to fully understand how the size of the compression chamber and the barrel length impact the speed of the tennis ball. On the left side of the picture below are two compression chambers of different sizes. The smaller of the two (with green ends and middle) has an air volume of approximately 198 cubic inches. Just to the left of it, the longer chamber has a volume of approximately 672 cubic inches. The size of the compression chamber or the "Air Volume" is one of our experimental variables in our DOE. The second experimental variable is the length of the barrel. On the right side of the picture are two barrels. One is 4ft long and the other is 6ft long. The design of the potato cannon allows me to pair any barrel with either the small or large compression chambers.

The output of my experiment is the speed in which the potato cannon propels the tennis ball. Note that I am always providing 40 psi of pressure in the compression chamber and that my projectile is always a tennis ball. Click below to see a video of the tennis ball cannon in action.

To maximize the amount of information gained, I selected a design called a 2^{2} Full Factorial. Admittedly, this is an odd name; however, the name describes the structure of the experiment. The first 2 represents the number of levels and the second 2 indicates there are two factors. A complete explanation of the design alternatives is beyond the scope of this article, but you can see the actual design below.

Row # | Barrel | Air Volume |

1 | 4 | 198 |

2 | 4 | 672 |

3 | 6 | 198 |

4 | 6 | 672 |

If you take a good look at this design, you will see that all combinations of Barrel Length and Air Volume are in the experiment, thus the name "Full Factorial". This experiment will allow us to quantify the effect of these two variables in a very few runs. Below are four pictures of the actual cannon used that correspond to the four rows in design.

Row #1 Barrel = 4 feet Air Volume = 198 Cubic Inches | Row #2 Barrel = 4 feet Air Volume = 672 Cubic Inches |

Row #3 Barrel = 6 feet Air Volume = 198 Cubic Inches | Row #4 Barrel = 6 feet Air Volume = 672 Cubic Inches |

For each shot, I videotaped the tennis ball using a Sony video camera that has a feature called "Smooth Slow Record". This increases the frame rate to 240 frames per second instead of the normal 60 frames per second. This makes the video a little darker and grainier, but measuring the time required for the ball to travel 21 feet is much easier with the slow motion video. Click on the videos below to see the experiments for rows 3 and 4 (you may have to click twice).

Video of Row #3 (slow motion)

Video of Row #4 (slow motion)

The data for the DOE is below.

Factor | A | B | Velocity | ||||

Row # | Barrel | Air Volume | Y1 | Y2 | Y bar | ||

1 | 4 | 198 | 85.91 | 81.82 | 83.865 | ||

2 | 4 | 672 | 90.43 | 95.45 | 92.94 | ||

3 | 6 | 198 | 44.06 | 55.43 | 49.745 | ||

4 | 6 | 672 | 107.39 | 101.07 | 104.23 |

Note that Y1 and Y2 are replicates or two different observations for each row. When Barrel Length = 4ft and Air Volume = 198 cubic inches, the velocity for the first shot was 85.91 mph and the velocity for the second shot was 81.82 mph. The average of each row is in the column "Y bar".

As always, I start with a visual inspection of the data. The velocity of the tennis ball is very consistent within row but changes a good deal from row to row. For example, the average velocity for Row #4 is more than double the average velocity for Row #3. Two very simple graphical analyses from this experiment would be the marginal means plot and the interaction plot. Let's start with the Marginal Means plot, which is below.

The marginal means above shows the effect of changing Barrel Length from 4ft to 6ft and changing Air Volume from 198 cubic inches to 672 cubic inches. The black line on the left shows the average effect of changing barrel length (note the 4 and 6 on the horizontal axis) and the blue line on the right shows the average affect of changing Air Volume (note the 198 and 672 on the horizontal axis). How was this graph constructed? Let's start with the first plotted point, which is for Barrel Length = 4ft. This point is identified on the plot as "(83.9+92.9)/2=88.4". The point is plotted at 88.4 and thus tells us that the cannon shot the ball an average of 88.4 feet when Barrel Length = 4ft. Below is the data from the experiment; note that I have colored the rows red where Barrel = 4ft .

Factor | A | B | Velocity | ||||

Row # | Barrel | Air Volume | Y1 | Y2 | Y bar | ||

1 | 4 | 198 | 85.91 | 81.82 | 83.865 | ||

2 | 4 | 672 | 90.43 | 95.45 | 92.94 | ||

3 | 6 | 198 | 44.06 | 55.43 | 49.745 | ||

4 | 6 | 672 | 107.39 | 101.07 | 104.23 |

In rows #1 and #2, where Barrel = 4ft, the average speed of the ball was 83.865 mph and 92.94 mph (look in the Y bar column). The average of these two numbers is 88.4 which is where the point is plotted on the marginal means plot. The dot for Barrel = 6ft is plotted at the average of 49.745 and 104.23 which happens to be 76.95 feet, and is the second plotted point on the marginal means plot. If we connect the two dots, we get the black line and a graphical representation of the effect of changing the barrel from 4ft to 6ft. Put in very simple terms, on average, changing the barrel length from 4ft to 6ft resulted in the ball traveling approximately 10 mph slower. The affect of changing the air volume from 198 cubic inches to 672 cubic inches resulted in the ball traveling approximately 30 mph faster.

Let's assume for the moment the goal is to maximize the speed of the tennis ball. If we were to look at our marginal means plot above, the shorter 4ft barrel has the higher average than the longer 6ft barrel and the larger 672 cubic inch air volume has a larger average than the smaller 198 cubic inch air volume. So, if we want to maximize the speed of the tennis ball, we should build a cannon with Barrel Length = 4ft and Air Volume = 672 cubic inches. **However, this disagrees with the raw data.** Here is the raw data again.

Factor | A | B | Velocity | ||||

Row # | Barrel | Air Volume | Y1 | Y2 | Y bar | ||

1 | 4 | 198 | 85.91 | 81.82 | 83.865 | ||

2 | 4 | 672 | 90.43 | 95.45 | 92.94 | ||

3 | 6 | 198 | 44.06 | 55.43 | 49.745 | ||

4 | 6 | 672 | 107.39 | 101.07 | 104.23 |

From the marginal means plot we would choose Barrel Length = 4ft and Air Volume = 672; however, that corresponds with row #2. This is somewhat disturbing since Row #4 is clearly a faster shot. If you are skimming over this you might want to go back and read the previous paragraph and this one again. They contain a very key point. What went wrong? Why did the marginal means plot point to building a different cannon than the raw data? The issue is that we have yet to discuss the concept of an interaction; a Marginal Means plot does not tell the entire story. Below is the interaction plot of Barrel and Air Volume.

The blue line represents the velocity of the tennis ball when Air Volume = 672 cubic inches, while the black line is when Air Volume = 198 cubic inches. Note that the Barrel Length is on the X Axis. The data for this plot comes directly from the four rows in our DOE data. In other words, when Barrel Length = 4ft and Air Volume = 198 cubic inches, the velocity was 83.9 in both our raw data and in the plot above.

The interesting part of this graph is that the larger 672 cubic inch air volume is always faster than the smaller 198 cubic inch chamber. **However, the 6ft barrel produces both the slowest and fastest velocity shots. **When the 6ft barrel is paired with a 672 cubic inch air volume the maximum speed is obtained. When the 6ft barrel is paired with the 198 cubic inch air volume, the slowest shot is obtained.

I chose this example as this physical interaction is relatively easy to understand. There is some friction (and normal force if you happen to be an engineer) between the ball and the barrel. With increased air volume in the compression chamber, the tennis ball will be "pushed" harder over the length of the launch. With the small 198 cubic inch air volume, there isn't sufficient air to continuously accelerate the ball. When using the 6ft barrel, the ball is actually slowing at the end of the barrel. However, with the larger air volume, there is sufficient air to accelerate the ball past the 4ft mark, thus producing a longer shot. If our goal is to maximize the distance of the shot, we would choose a 6ft barrel and a 672 cubic inch compression chamber.

Conclusions

Design of Experiments as a highly efficient method to generate the maximum amount of information in a minimum number of experimental resources. With only eight total shots from this pneumatic air cannon, we now have far more information than had we tinkered with the variables instead.

Interactions are not limited to tennis ball cannons. They occur commonly in physical, chemical, and electrical applications. Some of the basic laws of physics, such as force = mass * acceleration and voltage = current * resistance are interactions. Full factorial designs like this one are capable of modeling these interactions and providing a detailed view of our system.

Quantum XL is our flagship product that performs Design of Experiments.

Related Articles

Taguchi L12 on the Pneumatic Air Cannon