All language subtitles for 07_effects-of-size.en

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.