Original subtitles

Next I want to explore with you the effects of sizing the platform.

What does it mean to have a larger platform?

Well clearly it becomes bigger and becomes heavier, and

therefore the truss to weight ratio changes.

What does it mean to have a smaller platform?

Well, the truss to weight ratio might get better.

Or does it?

So there are a few things you might consider.

First the mass, the inertia of the platform.

The maximum amount of thrust it can exert and the maximum moment it can generate.

Lets look at the characteristic length l.

Which is roughly half the diameter of the vehicle.

If you look at the mass of the vehicle,

it scales as the cube of the characteristic length.

And the moments of inertia go as the fifth power of the characteristic length.

Very simply, mass scales as volume and volume goes as l cubed.

Moments of inertia go as mass times length squared, and

therefore it's scaled as l to the fifth.

If you look at the total thrust applied by the rotors,

this scales as the area spanned by a single rotor.

It also scales as the square of the blade hit speed.

So if omega is the rotor speed, and

r is the rotor radius, then the product of omega and

r gives you the blade hit speed and the thrust scales as velocity squared.

In short, the thrust scales as r squared times v squared,

where v now is the blade tip speed.

Let's now look at the moment that can be generated by a vehicle like this.

While clearly if you apply a thrust f on each rotor,

the moment that you can apply scales as force times length.

So if f is the thrust and

l is the characteristic length then the moment scales as f times l.

Now let's assume that the rotor size scales as a characteristic length.

And this is reasonable to do because this is a geometric constraint.

In this setting the thrust goes as l squared times v squared.

And the moment goes as l cubed times v squared.

Now let's look at the max acceleration and the max angular acceleration,

which we can calculate by taking the total thrust, dividing it by the mass and

the total moment and dividing it by the moment of inertia.

If you substitute the appropriate scaling rules, the mass going as l cubed,

the inertia going as l to the fifth,

you quickly realize that the maximum accelerations, linear and

angular, go as v squared over l and v squared over l squared.

How does the blade fit speed, v, scale as the character stick length?

Well, there are a couple of ways of looking at it.

If you look at the scaling experiments we've done in our lab, we've found that

the blade tip speed scales as the square root of the characteristic length.

So this is generally true at the scales that

we play around with in our laboratory.

These are smaller vehicles, and this might not hold for

much larger platforms like commercial helicopters.

This paradigm is often called Froude scaling.

In contrast to that,

in aerodynamics there's a different paradigm called Mach scaling.

So Froude scaling suggests that the blade tip speed goes as

the square root of length.

Mach scaling suggests that the blade tip speed is roughly constant,

independent of length.

With these two assumptions, you can calculate the maximum thrust,

in one case it goes as l cubed and the other case it goes as l squared.

And you can calculate the maximum acceleration and

the maximum angular acceleration.

In both cases you will see that the angular acceleration

increases as you scale down the size of the platform.

And this in fact results in greater agility.

So, that's really the key idea.

The smaller you make the vehicle,

the larger the acceleration you can produce in the angular direction.

And this allows greater agility.