>> Which one was that?

That was this one.

All right, you do [INAUDIBLE].

Right, your momentum internal forces and fuel slosh,

as complicated as equations are.

Partial differential equations and surface interactions and every stoke equations and

who knows what else.

It only internal forces and we've shown up angular momentum

of a system cannot change unless you apply external forces.

So, that's why when you're loosing energy, you will asymptotically get to

this kind of the case which you can never bring the craft to rest.

All right, so those are all quick answer questions that we've had.

So good, these little pole hole plots, they are quite common,

you see these sometimes with different analysis.

We'll see after, it's going to do a spin or similar configuration.

We're not spending too much time on one but [INAUDIBLE] classic literature,

you'll often find these kind of illustrations.

And I wanted to make sure you're familiar with that.

Here the thing we did last time was Torque-Free Motion.

These are the equations where we assume to have a body of frame that's principal,

therefore the inertia tensor is diagonal, right.

We get this nice form.

Now, if we have an axis symmetric body,

you can see from here what happens on the right hand side.

You always have differences of inertia, and if it's axis symmetric that basically

means two inertias have to be equal about like this pen,

about orthogonal axis are equal.

Therefore one of the terms is going to go to zero and

immediately we know that omega dot is zero and that's going to be a constant.

Very, and the trick how to solve this last time was,

we want to go from these equations.

We know Omega 3 is constant, and

to solve this they're now to couple first order differential equation

Omega 1 that depends on Omega 2, and Omega 2 that depends on Omega 1.

Anybody remember the trick how we got to solve these two

coupled first order differential equations?

Yeah? >> One of the equations?

>> We differentiate them again, which can seem counterintuitive.

We're trying to get rid of dots,

in fact we add dots first because it makes it easier.

By differentiating them we get basically omega 1 dot,

omega 3's a constant, you get omega 2 dot.

Now we get to plug in the coupled one that depends on omega 1, and you end up with

a differential equation that's only omega 1s, but we traded two coupled.

First order differential equations for

two uncoupled second order differential equations right.

So that was a trade off but this one is a classic spring mass system so

you can solve this very easily.

Anybody who has differential equations should know how to solve this all right,

which is what we did.

So this is a classic answer if you have an axisymmetric body And

when we do a numerical simulation, just looking at the tumble rates,

you can see one of them flat lines.

That's expected, so

the spin about the axis of symmetry is always going to be constant.

And the other two give you this nice sinusoidal response, but

if you look at the attitude, it's not easy to predict.