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I recall watching a show about an aeronautical engineering contest to build model-sized aircraft able to lift a set weight to a set altitude with the lightest possible aircraft. The program prominently mentioned that model aircraft will always be less efficient than "real" ones due to two major effects, one is the length of the wing which is an important measure of lift efficiency.

The other was something like "the minimum gauge problem", which stated that you couldn't linearly scale down the gauge of the materials with overall size because the materials will no longer support themselves at some point, so the material weight in a small aircraft will often be relatively higher than in a full-sized one. They used the term in a way that suggested it was a widely understood rule-of-thumb.

Today I wished to use the "minimum gauge problem" as a metaphor, but when I googled it... nothing. Well, lots of stuff actually, just nothing like the concept I'm looking for. I've seen a cogent hit on the fact that the smallest subway you can build is still too large for most suburbs, and this is the basic concept I'm thinking of, but in this case, the "gauge" in question is rail gauge.

Am I misremembering this, or is my google-fu failing?

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  • $\begingroup$ It's far from widely understood. Wrinkling has been getting some serious study in the past 50 years, and is frequently counterintuitve. If you take a thin annulus and tension the perimeter uniformly, the interior will wrinkle ??! researchgate.net/figure/… $\endgroup$
    – Phil Sweet
    May 29, 2023 at 0:48

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The issue on the lower end is that we don't know how to make and handle materials that are micro-meters thick. Manufacturing methods do not continuously scale down, either, how will you rivet, weld, or machine tiny components?

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  • $\begingroup$ I'm pretty sure this was well above micrometers. The C150 has wing skins .025", so if we're talking about a 1/10th scale model (for instance), that's .0025" and I believe that is within our capabilities? $\endgroup$ Nov 21, 2022 at 16:35
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This should be the other way around. If you scale up, self weight goes up with 3rd power, but load bearing cross-section areas only with 2nd power, so you need better materials or thicker members. Galileo knew this and postulated that this effectively limited size of animals and plants, you can check out Galileo's square-cube law.

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  • $\begingroup$ But that is for solids, in this case we're talking about skinning. The example they used was the aluminum skin of an airplane, where the skin itself is a structural member but as you scale down it eventually turns into aluminum foil and is no longer useful for structure. $\endgroup$ Nov 21, 2022 at 16:33
  • $\begingroup$ I am not sure why the aluminum skin would not be useful for structure after scale down. Only reason could be manufacturing problems at lower scale as Tiger Guy hinted. $\endgroup$ Nov 21, 2022 at 17:34
  • $\begingroup$ Because of buckling, The cross section second moment of area decreases with the 3rd power of thickness, but the compressive strength decreases linearly with thickness. so panel size decreases faster than scaling ratio. $\endgroup$
    – Phil Sweet
    May 29, 2023 at 0:33
  • $\begingroup$ Lame' wrinkling - researchgate.net/figure/… $\endgroup$
    – Phil Sweet
    May 29, 2023 at 0:44
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I think what you're really after is called the square-cube law. It is a slight formalization of the rule that "Stuff doesn't scale.".

Imagine a simple cylindrical beam in compression with a large solid mass providing a load. If we scale the whole thing up by a factor k, the cylinder is now k times taller, and k times the diameter. The linear dimensions of the load are all k times larger.

The cross sectional area of the cylinder is now k2 larger -- this is related to the load carrying capability of the cylinder.

However, the volume of the load is now k3 larger -- if it is a solid mass, the load scales with k3.

So this is a situation where the load carrying capability scales with k2 and the load scales with k3. In the end, this means that things that work at one scale fail to work at another scale.

Many things can limit min gauge for a structure. It is tied to application and manufacturing process. In general, it is a shortcut for capturing things we don't otherwise want to analyze.

The min gauge for skins on light aircraft is typically .030". This set by the fact that we usually want to use countersunk flush-set rivets. Rivets come in different sizes, but anything smaller than a #3 is very difficult to work with. A #3 rivet is 3/32" in diameter. When you cut a countersink (not dimple) into sheet, if the countersink is too big, you end up with a knife-edge at the bottom of the hole. This creates a stress concentration and a fatigue problem. If you do the trig on #3 flush head rivets in a lap joint with countersunk holes, the min gauge is .030".

You can see numerous things that could change in the above situation to relieve the .030" min gauge. However, if you don't want to change those things, then it presents a limit.

If you high-speed machine your skin panels out of billet, your min gauge may be set by your machining tolerances and thermal effects during manufacture. Perhaps you are only willing to take the skin to .100" using this kind of process.

A (space industry) colleague of mine argued that the idea of min gauge did not exist -- he said you could always go thinner and cited the use of chem milling to get ultra-thin skins where needed (now outlawed in Europe). Satellites also generally work in an environment limited to highly trained technicians in bunny suits -- once they make it to orbit, they don't have to worry about loads ever again.

Composites can have a min gauge too -- you can't have less than one layer in any location. You can go to a lighter fabric, but that drives costs and only gets you so far.

Sometimes a min gauge is set by a load condition that isn't normally considered. For example, a mechanic walking on the top surface of a wing. Or when that mechanic drops a screwdriver. Or a passenger in high heels walking down the aisle of an aircraft.

None of these are the loads we think about as engineers. However, they might set a minimum material thickness condition.

Sometimes new loads appear as you scale a solution down. Small UAV's and R/C aircraft have to deal with handling loads that would be crazy to think about for large scale aircraft. For example, the flight loads on a UAV aileron servo and linkage are probably much smaller than the load when a person comes along, grabs the aileron, and wiggles it around. Likewise, any UAV, drone, or R/C aircraft small enough to be picked up and carried in one hand likely will be. Even if that means picking it up by the wingtip and swinging it around -- imagine that load on a 747!

Min gauge of .030" for a small aircraft isn't a problem -- that is pretty close to what the skin needs to be anyway (whether for handling or flight loads). However, if you were to scale that structure down to a small UAV or R/C aircraft, it would be crazy. That .030" skin would be way over built.

Instead, we make the R/C aircraft out of balsa and foam, with skins of mylar or other super-thin plastic. These are materials that work great for small aircraft, but would never scale up to a 747.

In the end, scaling is a fascinating problem. None of it is linear, and none of it will get you very far.

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