btrettel's Nerf blog

random thoughts about the ballistics of foam darts

Archive for May, 2013

Drag coefficient of Nerf darts

Posted by btrettel on May 31, 2013

The drag coefficient tells you how much drag affects an object moving in a fluid. A higher drag coefficient means that the drag force is larger, which’ll slow down an object more. Lower drag darts will have more range and travel faster.

What’s the drag coefficient of a Nerf dart? Many tests have been done on cylinders. I’ve plotted the results of these tests below. The data comes from this paper and this book.

This is a plot as a function of fineness ratio (L/d), which is the ratio of the dart’s length to its diameter. This’ll allow the results to scale up for any size dart.

Nerf darts seem to be much closer to the flat nose curve above than the smooth nose one. Tests done by Daniel Beaver have shown that Nerf darts with a length of 1.25 inches have a drag coefficient of about 0.67.

This plot shows that darts have the least drag in the range of L/d = 1.75 to 3. The increase in drag for darts shorter than that is much steeper than the increase for longer darts. The variability also increases for shorter darts, which could reduce accuracy. Based on this information, we should make our darts about 2.5 times as long as their diameter to have low drag and good accuracy.

The curve fits I developed above use this equation:

C_\text{d} = \frac{\displaystyle C_{\text{d},0} \left(1 + a_1 (L/d)\right) + a_2 (L/d)^2}{\displaystyle 1 + a_1 (L/d) + a_3 (L/d)^2}

For the flat nose, C_{\text{d},0} = 1.104, a_1 = -0.988, a_2 = 0.402, and a_3 = 0.413. For the smooth nose, C_{\text{d},0} = 0.427, a_1 = -0.579, a_2 = 0.060, and a_3 = 0.248. C_{\text{d},0} is the drag coefficient of a cylinder with effectively zero length.

Typical darts were longer at one time. Perhaps Nerfers have slowly have figured out that shorter darts perform better. In 2004, the average Nerf dart was 2.1 inches long (L/d = 4.0). In 2008, the average dart was 1.7 inches long (L/d = 3.2). Now I’d say darts are even shorter. Daniel Beaver’s tests used darts with a fineness ratio of 2.4, which is about what I suggest based on this data.

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Nerf test gun

Posted by btrettel on May 23, 2013

I am developing a Nerf gun purely for test purposes. My main goal is to get consistent performance. I am using a few methods to do this: regulated pressure, reducing the effect of the speed of opening the trigger valve by piloting my QEV with another smaller QEV (that is in turn piloted by a smaller trigger valve), and using stiff aluminum barrel to reduce the effect of barrel vibrations.

The first major set of tests I want to do will examine fishtailing and dart stability. I want to determine when darts fishtail. I have some some simple theoretical analysis, but the results have been trivial: darts possibly are unstable when the center of gravity is behind the center of pressure. That doesn’t say that much. The center of pressure moves as a function of the dart velocity. It’s commonly reported that faster darts fishtail more easily. Is the movement of the center of pressure the main effect causing the instability here? Or does the speed cause other effects?

Towards this, I have done some dimensional analysis. Hopefully this analysis will help me decide how to conduct my experiment. I have determined that the important parameters are fineness ratio, Reynolds number, a dimensionless distance difference between the dart center of pressure and center of mass, a dimensionless mass, a dimensionless moment of inertia, and the fractional location of the dart center of gravity (or pressure)

Unfortunately, there is one effect which is hard to characterize: muzzle blast. If there’s some extra pressure when the dart leaves the barrel, then this pressure could knock the dart off balance. I can’t measure muzzle blast at the moment, but I know that muzzle blast should be approximately minimized when the optimal barrel length is used. Thus, the first part of my tests is to find the optimal barrel lengths for a variety of different pressures. I figure this is a good restriction as we’d all want to use the optimal barrel length anyway.

I intend to finish the test gun this weekend and do some preliminary tests to help figure out what gas chamber volume, barrel lengths, etc., are appropriate for future tests.

This likely will be the first test gun I build. I plan to use what I learn from this test gun to develop a second test gun. The second test gun will use higher precision regulators and pressure gauges.

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