Wednesday, December 18, 2013

Working with scale effects

What are scale effects?

If you've been in modelling very long, you've seen scale effects at work. If you're a fan of bad science fiction or horror movies, you've probably seen scale effects being ignored.
A scale effect is a physical effect that changes with the scale of the model - specifically those that change other than in proportion to the scale. The best known scale effect is probably the one described by the square-cube law: the weight a structure can hold goes up with the square of its size, but its weight goes up with the cube. So if you double the size of a model (or an ant), it will weigh eight times as much as the standard version, but its legs will only be able to hold up four times as much weight. Which fact is conveniently ignored in movies about giant ants, robots and similar things.

Computing scale effect numbers

The scale factor is the change in size: a 1/48th scale model has a scale factor of 1/48. The scale effect number is the exponent to raise the scale factor to to figure out how much the given measurement will change with that scale factor.
All physical quantities can be expressed as the product of measurements of length, mass and time. That's why scientists talk about the cgs (centimeter/gram/second), mks (meter/kilogram/second) and fps (foot/pound/second) systems. If we can express some measurement of interest in those terms - it doesn't really matter which units we use - we can calculate a scale effect number for it. So we'll use the l*ength-*m*ass-t*ime system.
Given the expression of a measurement as a product of lmt terms, computing it's scale effect number is straightforward. Multiply the exponent of each of l, m and t by their scale effect number, then sum the products. A result of 0 means this quantity doesn't scale. A result of 1 means it changes proportionally to the scale. A result of -1 means it changes in inverse proportion to the scale. Results of 2, 3, ... mean it changes proportionally to the square, cube, etc. of the scale.

Length/Mass/Time scaling

The scale effect number for length is just the scale factor itself, since length changes in direct proportion to the scale.
Time's scale effect number requires a little physics knowledge to find. A pendulum's period is proportional to the square root of it's length. So if you make a pendulum at ¼th scale, its period is ½ that of the full scale model. So a scale second for a ¼th-scale model should be ½ a second. The power you have to raise ¼th to to get ½ is .5, meaning the scale effect number is .5, or the square root.
Mass is a bit harder. However, density provides us a clue. It doesn't change with scale. Density is mass divided by the volume. Volume is the cube of length, but in the denominator, so it's exponent is -3. The exponent for mass is 1. So for density to correctly scale, mass must have a scale effect number of 3.

A table of scale effect numbers

With those numbers and the scale effect formula in hand, we can calculate the scale effect number for a variety of measurements. We'll include the length, mass and time scale effect numbers for convenience.
Measurement Dimensions Scale effect number
Length l 1
Mass m 3
Time t .5
Area 2
Volume 3
Density l⁻³m 0
Speed lt⁻¹ .5
Acceleration lt⁻² 0
Force mlt⁻² 3

Automating the calculation

If you don't want to do these calculations, and keeping the table handy is a bit much for you, the author of the tool/programming language/app Frink has added a feature so it can do them for you. If you're doing physical calculations with a tool that doesn't keep track of units for you, you should check out Frink. It's amazing, and so is the author.

If you have the latest version of Frink, you can write a function to calculate the scale effect number:

getScaleEffect[x] := getExponent[x, "length"] + 3 * getExponent[x, "mass"] + getExponent[x, "time"] /2

This will compute the scale effect number for the quantity x. You could now write a two-argument function that takes a quantity and a scale, and computes the scale value by multiplying it by the scale raised to the power of the above sum:


getScaleValue[x, scale] := x * scale ^ getScaleEffect[x]

Scale effect numbers examples

Let's walk through one or two calculations for some measurements to see how this is done. If the measurement you want to know the scale effect for isn't in the table above, you can calculate it yourself. You'll need the appropriate lmt formula. This can probably be found in a good engineering or physics text, or on wikipedia or Wolfram Alpha. That will probably be in cgs or mks, but that doesn't really matter.

Supported weight

The weight[^1] that a vertical object can support is proportional to the area of its cross section. The area of the cross section is proportional to the square of the length, so the scale factor for weight support is 2 × 1+ 0 × 3 + 0 × .5, or simply 2, which agrees with the table. So the amount those ants legs can support will go up by the square of their size.

Scale speed

People talk about scale speed for model cars. We can now see what that means. Speed has a scale effect number of .5. For a 1/16th scale model car, we take the square root of 1/16th to get ¼th. To translate from real speed to scale speed, we multiply by 4, since we're scaling up. So if the car is moving at a real speed of 20 mph, it's scale speed would be 80 mph. To work it out the long way, each mile represents 16 miles at scale, and each hour represents 4 hours at scale (because the scale effect number for time is .5). That's 20 × 16 or 320 miles in 4 hours, or 80 mph.

A practical example

So, let's see if we can figure out how a model's scale will affect how much wind we can fly it in.

Force of the wind

So, what's the scale effect number for the force of the wind on your model aircraft? Checking wikipedia, we find that the force of the wind is the product of the density of the air, the surface area of the craft, and the wind speed squared. Working that out the long way, that's (l-³m¹)l²(lt⁻¹)², or l⁷m⁻¹t⁻², making the scale effect number -3 × 1 + 1 × 3  + 1 * 2 + 1 * 2 + -2 × .5 = 3. Using the table entries for density, area and speed , it's 0 + 2 + 2 × .5 = 3. Or, we could just use the Force entry in the table, which is 3.
While that was an illuminating exercise, this is scale force, meaning it uses the wind's scale speed. It tells you how the force will change with scale, assuming the wind speed will do the same.

Force of the wind, take two

What we really want to know is how much more difficult the model will be to fly as we scale it down. This is a much more difficult question to answer, because difficulty is a subjective measure.
We can deal with that by picking a measurable point and declaring that more difficult than that as difficult, and less difficult as not difficult. For instance, we can pick the wind speed at which a model can no longer fly against the wind, because the wind force is greater than the force the copter can provide to fly. This point isn't fixed by scale, as the model's capabilities can vary quite a bit at the same scale. Where a 100-size 3-channel coax helicopter you picked up in the mall can't fly in anything but a dead calm, a 100 size collective pitch model tuned for fast forward flight can quite easily handle a stiff breeze.
But setting that aside, this one is easy to figure out. We're comparing a non-scale speed to a scale speed, so the scale effect number is .5.
This seems about right. The Blade Nano QX flies nicely in 5 mph winds, and can handle 10 mph winds, but not well - at least not in my hands. The Blade 350QX - at roughly 4 times the size - should behave similarly at 10 * √4 ≅ 20 mph. And sure enough, in 20 mph winds - with a GoPro 3 in the waterproof case attached - and my hands on the controls is about as manageable as the Nano QX in 10 mph winds.

Summary

Ok, we've covered how you calculate scale effect numbers, explained the ever-popular scale speed concept, shown some examples of calculating scale effect numbers, and how to use this to get some practical information. If you run into something that you can't handle with that base, drop me a note about it.