Factors of Scale

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Scaling Laws by Dave Harding offers a good explanation and derivation of scaling factors.

Dynamically Similar Values is a simple table from Scale Flyers of Minnesota that I find handy, which lists scaling factors and values for different scales, although it doesn't provide the analysis and derivations as the article above. The table only lists values for 1/2 to 1/10, but gives the factors so you can easily calculate any scale. Note that there is an error in the explanation of scale speed. The example given is for 1/5 scale, but the numbers are for 1/6.

Builder 2010 said:
The B-17 cruised at 187 mph. In scale RC model meets some of the judges are wanting the planes to fly at scale speeds too. So for the 1:16 B-17 I built the scale speed would be a bit under 12mph. Instead the model flew at about 80mph.
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From this table, for the 1/16 B-17, the scale factor would be sqrt(1/16) = 1/4, converting 187 mph to 46.75 mph scale speed. While perhaps still a bit slow, this is much more reasonable and can be more closely approximated. The Harding article uses a power/drag calculation and produces a factor of approx. 0.3, or 55.4 mph, a very similar result.
 
Shortround6 said:
Or if you stick the wing of a 210 on a 172 how much of your payload do you give up to stay within gross weight?
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What you have is almost exactly an RG Cardinal with a 172 engine, its empty weight increased approx 200 lbs and its drag increased by 10-15%, probably for performance reasons mandating a reduced gross weight. Can you spell "lead sled"?
Cheers,
Wes
 
pbehn said:
Thou art a cad.
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Gentlemen, we have us a callsign! "Please join with us in inaugurating this noble and honored warrior, heretofore known as 'GregP' into the ancient and honorable order of CAD, with all the privileges and duties thus pertaining, and avow that we hereby dub thee 'AutoDesk', to be so addressed in all communication henceforth. Please kneel, Sir, for the ceremony of the sword. Thou shalt soon discover that thy helmet, flight suit, locker, ready room seat, and aircraft have been appropriately labeled."
Cheers,
Wes
 
Okay... so if I read F Fascinated 's post correctly
  • Fuselage length is to the cube law: x^3
  • Wingspan scales to the power of 3.2238
    • Does that include the fuselage in the span, or does that just cover each wing?
  • Spar stress for equal g-load increases by 2.25 for every doubling in size (basically the mean of x^3.2 and x^2)
  • Wing Loading increases by 2.25 for the doubling of A/C weight; stall speed increases by 1.5?
I feel like something's wrong with my interpretation
 
XBe02Drvr said:
Like trying to reduce a multi-variable, multi-step design process down to a simple rule of thumb? You might be right about that. "Ain't no free (or even cheap) lunch, lad."
Cheers,
Wes
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You don't mean "Its complicated" by any chance?
 
pbehn said:
You don't mean "Its complicated" by any chance?
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Naw, "complicated" is too simplistic and imprecise a term. "Multi-variable, multi-step" sounds more impressive, don't you think? Maybe I'll add "muti-faceted" as well, just for the effect.
Cheers,
Wes
 
XBe02Drvr said:
Naw, "complicated" is too simplistic and imprecise a term. "Multi-variable, multi-step" sounds more impressive, don't you think? Maybe I'll add "muti-faceted" as well, just for the effect.
Cheers,
Wes
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I spent my working life on the various ramifications of the Iron-Carbon equilibrium diagram. Anyone can produce this diagram and every time I tested products for some reason there were exceptions to it, simply because you never get just iron and carbon you get a whole lot of other "stuff" too, that's when it gets complicated. As a rule of thumb any ton of Iron/Carbon will break your toes if you don't wear safety boots,, hope that helps.
 
There are no rules of thumb, there are rules and laws of physics, aerodynamics, thermodynamics, thermoconductivity, metallurgy and dozens of other fields all of these impinge on each other. In every discussion you look for a "killer" number or rule of thumb. Life isn't like that.
 
Scaling up or scaling down will at some point involve application of the law of diminishing returns. like everything else, changing the variable of size will at some point reach a point of nil or even negative return.

An interesting spin off from this is the question of 'what would happen if a man were shrunk to the size of an ant?" or the reverse "what would happen if an ant were the size of a human?" In both cases the subject would die, though the ants demise would be rather quicker. in both cases the subject has vastly exceeded the optimal size of that system…..it has broken the parameters of diminishing returns for that system.

How Strong Would a Man-Sized Ant be? | Curious Meerkat
 
Galileo (one of those people so famous, we don't use his surname ) first looked at this.

When you scale up an object -- multiplying all its dimensions by the same value -- the volume and mass multiply by the third power of the scaling factor and surface area by the square.

A simple example: cube of 6061 aluminum 1 meter on a side will have a volume of 1 cubic meter, a mass 2,720 kg (density from www.engineeringtoolbox.com), and a surface area of 6 square meters, so area divided by volume is 6. Scale it up by a factor of two, it will be 2 meters on a side, have a volume of 8 cubic meters, a mass of 21,760 kg, and a surface area of 24 square meters: its area divided by volume will be 3.

In reality, aircraft are not scaled perfectly linearly: light aircraft will not have things like chemically milled skins, but will tend to have skins that are proportionately thicker than larger aircraft, instruments don't scale -- an airspeed indicator or an altimeter in 747 isn't larger than the one in a Bonanza -- etc.
 
And, as I've asked before, and will ask again since clearly, nobody has seen the posts: I read the PDF that F Fascinated , put up. I read it, and attempted to interpret the data on the sheet, to the best I understand it, and it feels like something's not right in my interpretation of the math.

My skill at math as we know is not good, and I'm curious if I screwed up in any way
Jeez! Who do I gotta blow to get a straight answer ?
 
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I don't understand how you arrived at some of these values. If 'K' is your scaling factor, all lengths scale the same, K^1. If you are scaling x 2, then fuselage length, wingspan, are all x 2. (2^1 = 2)

I also don't understand what you are asking by "wing loading". Do you mean you are doubling the A/C weight while keeping all else the same? Then you are also doubling the wing loading. This is not a scaling issue.

If you are reading this from the PDF, then note that the table at the end of the article lists the scaling for SIZE doubling. He argues that weight increases faster than volume, by K^3.22. This results in the wing loading increasing by K^1.22 (The weight increases by K^3.22, the area increases by K^2. Weight / Area = K^3.22/K^2 = K^1.22) But again, this is for a doubling of SIZE, not weight. The weight goes up disproportionately. I think you are reading the stall speed correctly, except again, you are not doubling the weight. This is where you are getting confused.

Hmmm, I don't exactly get his "multipliers" when I do the calculations. In his chart, Multiplier = 2^y (he is using "2" because he is doubling the size in this example). I suspect he is rounding either the scaling factor or the multiplier (or both!) in the chart, but not in his calculation. Read the final three paragraphs of the article, beginning with "Table 2 is a summary...." He explains it correctly, even if his calculations aren't exactly exact.
 
Fascinated said:
I don't understand how you arrived at some of these values.
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That was a possibility that I had been concerned with. I derived the figures based on what I interpreted out of the pdf file: Scaling Laws by Dave Harding.
If 'K' is your scaling factor, all lengths scale the same, K^1. If you are scaling x 2, then fuselage length, wingspan, are all x 2. (2^1 = 2)
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Of course, that I get fine. And if you're scaling for volume, then K^3, and if everything doubles 2^3 = 8.

I was just curious about weights and amount of lift required: According to the text: The scaling for span seemed to increase by K^3.2238. Figure 3 on the second page listed (0.0001*span^3.2238).

When it came to stresses imposed on the aircraft, it seemed to be the quotient of the weight and area so if a design was scaled up by 2, and the weight increased to the power of 3.2238 and the weight to the square you'd have (2^3.2238)/(2^2) = about 2.2356 (The author used 9/4 for simplicity and got 2.25) and stated that lift would increase to the square of the weight so about 1.5283 (he used 1.5)
I also don't understand what you are asking by "wing loading". Do you mean you are doubling the A/C weight while keeping all else the same?
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Well, as I understood it, wing-loading is the weight of A/C divided by the wing-area. That kind of left me scratching my head.
If you are reading this from the PDF, then note that the table at the end of the article lists the scaling for SIZE doubling. He argues that weight increases faster than volume, by K^3.22.
Click to expand...
That I grasped
This results in the wing loading increasing by K^1.22 (The weight increases by K^3.22, the area increases by K^2. Weight / Area = K^3.22/K^2 = K^1.22) But again, this is for a doubling of SIZE, not weight.
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So, basically all proportions being equal the wing loading keeps increasing by K^1.22 instead of K^1, and flying speed increasing to the square-root of this.

If I do the math right, I get for a plane that weighs 6500 pounds with a wing-area of 260 ft^2, and an aspect ratio of 6 (typical for propeller driven fighters), you'd end up wingspan of around 39.5 feet and a wing-loading of about 25.

So if I was to increase the wingspan by 1.5, which is around 59'3", weight would increase (all things being equal) to around 23984 (A/C weight * 1.5^3.22), wing-area to a hair short of (wing area * 1.5^2), and a wing-loading of 40.93. These numbers don't look right... if K^1.22 was right I'd have a wing-loading of 88.7 right?
I think you are reading the stall speed correctly, except again, you are not doubling the weight.
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If I recall doubling the weight would cause the stall speed to increase to the square root of 2. So a 10,000 pound plane with a stall speed of 100 knots, would stall at 141.4 knots if it's weight was increased to 20,000 pounds (same wing area).
 
Zipper730 said:
I was just curious about weights and amount of lift required: According to the text: The scaling for span seemed to increase by K^3.2238. Figure 3 on the second page listed (0.0001*span^3.2238).
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I'm not sure where his (0.0001*span^3.2238) came from. He has plotted weight (oz.) vs. span (in.) on a log-log plot. So I think he has calculated the slope to derive that relationship. On his line, 100 oz. crosses at about 64 in. So, (0.0001*64^3.2238)~=66.5. Ok, but I think the coefficient will only be correct for this set of units (ounces vs. inches).

K=1.5, right? So K^1.22 = 1.5^1.22 = 1.64 approx. Your initial wing loading was 25 * 1.64 = 41, pretty much what you got the first way. Unless I'm missing something. I really have to write this all down carefully to keep it straight.

I think you're right on the stall speed.
 
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