6 Example: Phase Space Plot of Duffing Equation

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Del curso dictado por University of Colorado Boulder

Kinetics: Studying Spacecraft Motion

14 calificaciones

University of Colorado Boulder

Kinetics: Studying Spacecraft Motion

14 calificaciones

As they tumble through space, objects like spacecraft move in dynamical ways. Understanding and predicting the equations that represent that motion is critical to the safety and efficacy of spacecraft mission development. Kinetics: Modeling the Motions of Spacecraft trains your skills in topics like rigid body angular momentum and kinetic energy expression shown in a coordinate frame agnostic manner, single and dual rigid body systems tumbling without the forces of external torque, how differential gravity across a rigid body is approximated to the first order to study disturbances in both the attitude and orbital motion, and how these systems change when general momentum exchange devices are introduced. After this course, you will be able to... *Derive from basic angular momentum formulation the rotational equations of motion and predict and determine torque-free motion equilibria and associated stabilities * Develop equations of motion for a rigid body with multiple spinning components and derive and apply the gravity gradient torque * Apply the static stability conditions of a dual-spinner configuration and predict changes as momentum exchange devices are introduced * Derive equations of motion for systems in which various momentum exchange devices are present Please note: this is an advanced course, best suited for working engineers or students with college-level knowledge in mathematics and physics.

De la lección

Torque Free Motion

The motion of a single or dual rigid body system is explored when no external torques are acting on it. Large scale tumbling motions are studied through polhode plots, while analytical rate solutions are explored for axi-symmetric and general spacecraft shapes. Finally, the dual-spinner dynamical system illustrates how the associated gyroscopics can be exploited to stabilize any principal axis spin.

Module 2 Introduction1:22

1: Torque Free Motion Polhode Plots33:27

1.1 Example: Special Polhode Plots3:00

2: Torque Free Motion Axisymmetric Solution5:51

3: Torque Free Motion General Inertia Case14:25

4: Torque Free Motion Integrals of Motion6:16

5: Torque Free Motion Phase Space Plots9:07

5 Example: Phase Space Plots for Varying Energy Levels4:58

6: Torque Free Motion Attitude Precession11:43

6 Example: Phase Space Plot of Duffing Equation4:35

Optional Review: Torque Free Motion10:11

Conoce a los instructores

Hanspeter Schaub
Glenn L. Murphy Chair of Engineering, Professor
Department of Aerospace Engineering Sciences

The Duffing equation gets a little bit more interesting.

So, I'm just showing here v is equal to

a unit a and a unit b, so I just have,

basically, x dot is equal to x double dot plus x plus x cubed.

That's what I'm simulating in this one.

But, this gives me now that phase space plot

we're talking about with the Duffing equation,

and there's a reversal of signs as well.

This part would be stabilizing;

this part is actually destabilizing.

That's kind of what we had with the omega two motion.

That was that case,

and this is the phase space.

If these have opposite signs,

your x double dot can go to zero if all the x's are zero,

which is kind of makes sense.

That's right here, and around

it we get something that looks a lot like

an oscillator equations, nearly circular stuff.

But, you can also get an equilibrium if x is, you know,

if x is equal to one,

then you have minus one plus one cubed

just still one gives you equal to zero.

So, plus one is one and minus one.

So, those are two equilibriums that we can read out

from this phase space plot,

that we can do.

Then, we can have some fun.

We can play with this and make this a little bit bigger.

So here, I'm plotting the same phase space plot

of all the possible, you know, lots of initial conditions.

If you started up here,

this is what would happen to your curves

and I'm plotting for a fixed amount of time.

This is a way then I can show you,

numerically, the separatrix motions.

We saw it in the video, those,

that spinning handle, right?

It hung out there for a certain amount of time,

and then it went back to the other side, and came back.

And, we talked about how you hang out near

the separatrix motion, as you get there.

So, here, what I've got something is I can put

an initial condition and it quickly solves

the differential equations,

and shows me the resulting phase space motion.

If I put my initial condition up here,

you know, starts very big.

I had a cube, an unstable cubic term,

so that one would go off to infinity, as expected.

Right? If I'm really,

if I have small departures,

stable linear term keeps me stable but if I get too far,

at some point the unstable cubic will dominate.

And that's what's happening.

If I reduce my integration time,

you will see that curve gets smaller.

So, you can see now what happens.

Here, I mean, the linear part,

I'm just doing a fraction of an orbit.

But, as I start to get close to the separatrix motion,

you can see I don't get very far.

I'd have to get really precise.

So, as I increase,

I'm not exactly on it.

But you can see as I'm increasing,

that curve gets close and really hangs out

there for a while, and eventually it moves back.

That's precisely what we saw happening

with that handle that the astronauts spun, right?

It hung out there for a while.

If I could get more precisely on it,

I mean it is very very sensitive.

If I'm off, it hangs out there even longer.

And if I were perfectly on it,

it would hang out there forever,

and never quite get there.

If you're off infinitesimally,

and you go in a whole different place, right?

And that's the definition of chaos,

infinite sensitivity to initial conditions.

You're smidgen to left, and you go off there.

You smidgen right, you go somewhere completely different.

That's why this is just a, hopefully,

visual illustration of why we're talking

about separatrix motions and the conditions.

So, it's kind of fun to play with.

But as we come in and have enough time to actually

complete my phase space plots.

I'm also showing speed in here somewhat,

so as I'm coming in.

I don't need that.

Anyway.

What was I talking about?

Speeds. Yes, this is showing something about the speeds.

Here you got green is faster,

blues and purples is very fast,

but once you come in,

past the greens, yellows,

so the reds that gets much slower again.

If you're close to the equilibrium,

you just have a nice slow,

just a little bit of a wobble.

And if you get close here you really slow down.

But then around it, we speed up to flip back again.

So this is just kind of a nice way to close out

the torque-free motion part for a single rigid body.

We've seen the video that illustrated it.

We've seen up whole hold plots that do it graphically.

We've seen mathematical formulations where we did

the axisymmetric and general cases,

and the different predictions that you can do

about these different motions and how things behave.

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