0:05

What other insight do we have,

Â if we assume, as we did with the pole hold plots?

Â So without loss in generality, remember,

Â we can always flip the principal axis such that

Â the largest inertia is axis one,

Â intermediate inertia is axis two and the least inertia axis

Â is axis number three, right?

Â So, I assume this ordering.

Â Then, we found earlier the B's

Â for one and three had to be positive,

Â meaning they have very strong stability for large departures.

Â Here, that's negative, something we have to consider.

Â But intermediate axis, we know,

Â this one is always unstable, right?

Â That was the separatrix stuff, so not too surprising.

Â It's actually linearly stable, nonlinearly unstable

Â and the other two's you don't have it defined.

Â It depends on the situation,

Â on how you're doing that and how you tumbling it for

Â that, you know, the energy levels to get the right sign.

Â So, it's different ways we can do this.

Â So, when you're looking at these systems for small departures,

Â we said, well, you know,

Â k times x is always going to be bigger close to x's around zero,

Â than another a times x is going to be bigger than b times x cubed

Â if x gets small enough.

Â As x gets very very small, x cubed is even smaller, right,

Â whereas x is going to be bigger in that sense.

Â At some point, this magnitudes always

Â crisscross and as x gets large,

Â at some point, x cubed will dominate.

Â So, you can use these coefficients to argue

Â if you have a particular sign.

Â If A1 is positive, that means I've got a positive spring stiffness.

Â That omega one motion should stay nicely bounded and stable,

Â nice oscillatory part.

Â If you have big departures and B is negative,

Â that means once you go too far from the origin,

Â it doesn't just oscillate at some point,

Â it's going to just take off to infinity.

Â That's what you find out in this analysis.

Â Or vice versa, if a is negative, that means, well,

Â it won't be stable.

Â Small departures drive you away from the origin,

Â but if B happens to be positive, in that case,

Â remember you can't have your x's then

Â go to infinity because at some point

Â x cubed is going to dominate

Â and it will try to bring it back.

Â And, in nonlinear system you get things called limit cycles,

Â and it doesn't converge to zero,

Â but these two terms fight each other

Â and you get these cyclic motions that happen in the end.

Â That's what it converges to.

Â So, the take off, this whole class is on nonlinear systems

Â that talk about these things.

Â So yeah, B1 can always be restoring if it's positive,

Â and we showed that.

Â Phase plane plots.

Â Let's talk about that.

Â Many of you have seen them, but not everybody,

Â not everybody's had quite an engineering background.

Â So, if I have a simple mx double dot plus kx system, all right,

Â what is a phase space plot?

Â What's on the horizontal axis?

Â Gamma x.

Â x.

Â And what's on the vertical axis?

Â x-dot, right?

Â So you solve this.

Â So, after some math or numerically,

Â you find what

Â x is going to be, and you have

Â also a solution what x-dot is going to be.

Â If it's numerical, you could,

Â you know, you're always getting

Â your positions and your rates and you're integrating,

Â so you could keep track of all of that.

Â The phase space plot

Â is nothing but a visualization of my position and rates.

Â So, I'm not plotting it versus time, but I'm plotting

Â added positions and velocities compared themselves directly.

Â So, if you have an oscillator equation

Â I give it an initial deflection here,

Â that's where I'm starting,

Â and I have a sinusoidal response, right?

Â Initially, I take the spring.

Â I deflect it and then I just gently let go.

Â The system would just oscillate, right?

Â What happens to the phase space plot?

Â What does that curve look like, for spring mass system?

Â Andrei.

Â Circle.

Â It's a circle, for this case, right?

Â So, at some point, this

Â potential energy goes into kinetic energy

Â right and then it slows down again.

Â And then, the net positions become negative

Â and then it becomes positive again and

Â you're just going to circle around, right?

Â If I give it less deflection,

Â you have a curve that does this.

Â If you give it more deflection,

Â you have bigger curves, right?

Â Do these curves ever intersect?

Â If you start here,

Â could this curve do this and go through here?

Â Then it's a different system.

Â If it's the same dynamical system,

Â can you have phase space lines intersect?

Â And the answer is really no.

Â The challenge with that, if they intercept,

Â if you had a different phase space plots, right?

Â It means if some curve does this and some other curve does this,

Â when you hit this intersection points which way do you go?

Â It's a second order dynamical system.

Â If I'm giving it a position velocity, the accelerations are

Â dictated by equations of motion.

Â You can't be going in two different directions.

Â Maybe unless you go on quantum stuff or something, you know.

Â But for classic mechanics systems, if given a position velocity,

Â there is a very distinct acceleration

Â and that velocity will evolve in a certain direction.

Â So, you don't typically see these lines intersect,

Â but you get these flow diagrams that you can draw,

Â and they will show you how the stuff evolves.

Â OK?

Â So, That's just a phase space, there should be a dot here.

Â That's what a phase space plot is.

Â With a phase space plot, let's still look at x double dot + kx,

Â just x double dot + the natural frequency squared x equal to zero.

Â What is an equilibria, no, actually that's not the one I'm looking for.

Â Sin(x) equal to zero.

Â This is basically your planar

Â and normalized version of a planar pendulum.

Â That was this problem we're talking about earlier.

Â With this system, what are my equilibriums going to be?

Â When is x doubled dot and x dot gonna go to zero?

Â If what?

Â When is x double dot gonna be zero?

Â That's the easy one.

Â Zero and 180.

Â Thank you.

Â So, if, x is equal to zero or pi.

Â So, on a phase space plot, plus or minus pi, right,

Â because you can go upside down this way or this way,

Â and if it hangs straight down, let's call that zero.

Â Right?

Â We know if we linearize this,

Â it's just x double dot plus x,

Â and you guys were arguing earlier, in that case,

Â around the equilibrium,

Â that's at the zero equilibrium

Â I would have something where my phase space plots look like circles.

Â Alright?

Â And they have a certain direction.

Â But if you go here,

Â and you'd linearize around that.

Â Here, you'd end up with x double dot equal to minus x,

Â x double dot minus x is equal to zero,

Â which gives you negative stiffness,

Â and it's actually hyperbolically unstable.

Â So there's these asymptotes you can draw,

Â and that's good for both of them.

Â So as we have larger departures,

Â these phase space plots can all look like this.

Â And, if you're on these lines,

Â you would do weird stuff,

Â where you could tumble from one with enough speed

Â and hit the, you know,

Â you can go from here and give it enough initial speed

Â to where you would hit over here again,

Â and do different kinds of motions, right?

Â But, that's what these phase space look like,

Â so you can quickly identify equilibriums.

Â If you do this and there's programs that will give you this flow,

Â the phase space flow, and you will see the dead spots.

Â Those are the equilibriums and around it,

Â the flows visually very quickly tell you this looks stable

Â neighboring stuff or no neighboring.

Â Some of it comes in, other stuff goes crazy, right?

Â So, that's my quick review, if you haven't seen phase space plots.

Â So, these kind of things with linear and cubic part can be used for that, as well.

Â So this is one of the phase space plots we could talk about,

Â where they flow.

Â Omega's nearby, those omega two's look good,

Â but then there's a limiting case, as well.

Â And it turns out, with this omega two this was the separatrix case,

Â but that was the intermediate axis case.

Â There's a limit.

Â If I'm doing a pure spin about omega two,

Â that's one of the conditions, and then anything inside is

Â a non-pure spin about the intermediate axis,

Â and you can, it turns out, you can show,

Â this paper showed, I'm not doing that in class.

Â But there's different limiting conditions

Â that they can come up with,

Â physical motions can never be outside here.

Â You have to be within this arc.

Â You can never actually exceed initial conditions.

Â It will give you nonphysical initial states that you'd have to have to get there.

Â